Special Relativity and Quantum Theory

Fundamental Theories of Physics

An International Book Series on The Fundamental Theories 0/ Physics: Their

Clarification, Development and Application

Editor: ALWYN VAN DER MERWE

University 0/ Denver, U.SA.

Editorial Advisory Board:

ASIM BARUT, University o/Colorado, U.SA.

HERMANN BONDI, Natural Environment Research Council, UK.
BRIAN D. JOSEPHSON, University 0/ Cambridge, UK.
CLIVE KILMIS1ER, University 0/London, UK.
GONTER LUDWIG, Philipps-Universitat, Marburg, F.R.G.
NATHAN ROSEN, Israel Institute o/Technology, Israel
MENDEL SACHS, State University 0/New York at Buffalo, U.sA.
ABDUS SALAM, International Centre/or Theoretical Physics, Trieste, Italy
HANS-JORGEN TREDER, Zentralinstitut/ur Astrophysik der Akademie der
Wissenschajten, GD.R.
Special Relativity
and Quantum Theory
A Collection ofPapers on the Poincare Group

dedicated to Professor Eugene Paul Wigner on the 50th

Anniversary of His Paper on Unitary Representations of
the Inhomogeneous Lorentz Group (completed in 1937
and published in 1939)

edited by

M.E.Noz
Department ofRadiology,
New York University, U.S.A.

and
y. S. Kim
Department of Physics and Astronomy,
University of Maryland, U.S.A.

KLUWER ACADEMIC PUBLISHERS

DORDRECHT / BOSTON / LONDON
Library of Congress Cataloging in Publication Data
Library
Specl~l relativity and
Specul qUintu s theory: e• collection of papers on the
.nd Qu~ntu~
Poincare
POlne ar i group I edited
edlnd by M.E. Noz. V.S. KII.
p. U.
CD. --- (Fundanental theories
- CFunduenul theorIes of physics)
physICS)
"Dedlc~ted to Professor
'Oedlcned Profenor Eugene Wj~ner
Wl gn er on the 50th anniversary of
his paper on unlUry
hl$ unitary representations
represent.ttons of the lnho~ogeneous
tnho.ogeneous Lorentz
(co.pleted In 1937 end
group (co.plettd and published
publ1shed in In 19391."
1939)."
Inc I udes b1bllograph
InclUdes bib 110graph 11es.
es.
ISBN -13: 978-94-010-7885-8
ISSN-13: 978-94-010-7865-8
Ouantun flelel
1. Ou.ntu. field theory--Congresses.
theory- - Congresns. 2. Special
Spee 1al relatiVity
rela t lvlty
(Physlcsl--Congresses. 3.
(Physlcsl--Congresse5. 3 . POincare
POlnc.re serles--Congresses.
s er lu--Con gr esses. 4. Lorentz
Lor en n
groups--Congresses.
groups- -Con gresses. 5. 5 . Wigner.
WIgnlr. Eugene Paul. 1902-
1902- I. Noz,
1.
Marilyn E. II.
Mar1lyn 11. Kl~,
Kl l , Y.
V. S. 111.
Ill. Wigner.
Wlgner, Eugene Paul,
P.ul , 1902-
Series.
IV. SerIes.
OCt74.46.S64 1988
OCI74.46.S64
530. I'' 2--dcI9
530.1 2--dc 19 88-13278

ISBN-13: 978-94-010-7872-6
978-94-010-7865-8 e-ISBN-13 978-90-277-2782-4
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Table of Contents

Preface ix

Introduction xi

Chapter I: Perspective View of Quantum Space-Time Symmetries

1. E. P. Wigner, Relativistic Invariance and Quantum Phenomena. Rev.
Mod. Phys, 29,255 (1957). 3
2. P. A. M. Dirac, The Early Years of Relativity. in Albert Einstein:
Historical and Cultural Perspectives: The Centennial Symposium in
Jerusalem. edited by G. Holton and Y. Elkana (Princeton Univ. Press,
Princeton, New Jersey, 1979). 17

Chapter II: Representations of the Poincare Group

1. E. Wigner, On Unitary Representations of the Inhomogeneous
Lorentz Group. Ann. Math. 40.149 (1939). 31
2. V. Bargmann and E. P. Wigner, Group Theoretical Discussion of
Relativistic Wave Equations. Proc. Nat. Acad. Sci. (U.S.A.) 34, 211
(1948). 103
3. P. A. M. Dirac, Unitary Representations of the Lorentz Group. Proc.
Roy. Soc. (London)A183, 284 (1945). 118
4. S. Weinberg, Feynman Rules for Any Spin, Phys. Rev. 133. B1318
(1964). 130
5. Y. S. Kim, M. E. Noz, and S. H. Oh, Representations of the Poincare
Group for Relativistic Extended Hadrons, J. Math. Phys. 20, 1341
(1979). 145
6. Y. S. Kim, M. E. Noz, and S. H. Oh, A Simple Methodfor Illustrating
the Difference between the Homogeneous and Inhomogeneous
Lorentz Groups, Am. J. Phys. 47,892 (1979). 149
vi Table of Contents

Chapter III: The Time-Energy Uncertainty Relation

1. P. A. M. Dirac, The Quantum Theory of the Emission and Absorption
of Radiation. Proc. Roy. Soc. (London) A114. 243 (1927). 157
2. P. A. M. Dirac, The Quantum Theory of Dispersion. Proc. Roy. Soc.
(London) A114, 710 (1927). 180
3. E. P. Wigner, On the Time-Energy Uncertainty Relation. in Aspects
of Quantum Theory. in Honour of P. A. M. Dirac's 70th Birthday,
edited by A. Salam and E. P. Wigner (Cambridge Univ. Press,
London, 1972). 199
4. P. E. Hussar, Y. S. Kim, and M. E. Noz, Time-Energy Uncertainty
Relation and Lorentz Covariance. Am. J. Phys. 53. 142 (1985). 210

Chapter IV: Covariant Picture of Quantum Bound States

1. P. A. M. Dirac, Forms of Relativistic Dynamics. Rev. Mod. Phys. 21,
392 (1949). 219
2. H. Yukawa, Quantum Theory of Non-Local Fields. Part I. Free
Fields. Phys. Rev. 77. 219 (1950). 227
3. H. Yukawa, Quantum Theory of Non-Local Fields. Part II.
Irreducible Fields and Their Interaction. Phys. Rev. 80. 1047 (1950). 235
4. H. Yukawa, Structure and Mass Spectrum of Elementary Particles. I.
General Considerations. Phys. Rev. 91. 415 (1953). 241
5. H. Yukawa, Structure and Mass Spectrum of Elementary Particles. II.
Oscillator Model. Phys. Rev. 91. 416 (1953). 244
6. G. C. Wick, Properties of the Bethe-Salpeter Wave Functions. Phys.
Rev. 96. 1124 (1954). 247
7. Y. S. Kim and M. E. Noz, Covariant Harmonic Oscillators and the
Quark Model. Phys. Rev. D 8.3521 (1973). 258
8. M. J. Ruiz, Orthogonality Relation for Covariant Harmonic-
Oscillator Wave Functions. Phys. Rev. D 10. 4306 (1974). 265
9. F. C. Rotbart, Complete Orthogonality Relations for the Covariant
Harmonic Oscillator. Phys. Rev. D 23.3078 (1981). 268
10. D. Han and Y. S. Kim, Dirac's Form of Relativistic Quantum
Mechanics. Am. J. Phys. 49, 1157 (1981). 272
Table of Contents vii

Chapter V: Lorentz-Dirac Deformation in High Energy Physics

1. R. Hofstadter and R. W. McAllister, Electron Scattering from the
Proton. Phys. Rev. 98, 183 (1955). 279
2. K. Fujimura, T. Kobayashi, and M. Namiki, Nucleon Electromagnetic
Form Factors at High Momentum Transfers in an Extended Particle
Model based on the Quark Model. Prog. Theor. Phys. 43, 73 (1970). 282
3. R. P. Feynman, The Behavior of Hadron Collisions at Extreme
Energies in High Energy Collisions. Proceedings of the Third
International Conference, Stony Brook, New York, edited by C. N.
Yang et al. (Gordon and Breach, New York, 1969). 289
4. J. D. Bjorken and E. A. Paschos, Inelastic Electron-Proton and 'Y-
Proton Scattering and the Structure of the Nucleon. Phys. Rev. 185.
1975 (1969). 305
5. Y. S. Kim and M. E. Noz, Covariant Harmonic Oscillators and the
Parton Picture. Phys. Rev. D 15. 335 (1977). 313
6. P. E. Hussar, Valons and Harmonic Oscillators. Phys. Rev. D 23.
2781 (1981). 317

Chapter VI: Massless Particles and Gauge Transformations

1. S. Weinberg, Feynman Rules for Any Spin II. Massless Particles.
Phys. Rev. 134, B882 (1964). 323
2. S. Weinberg, Photons and Gravitons in S-Matrix Theory: Derivation
of Charge Conservation and Equality of Gravitational and Inertial
Mass. Phys. Rev. 135, BI049 (1964). 338
3. D. Han, Y. S. Kim, and D. Son, E(2)-Like Little Group for Massless
Particles and Neutrino Polarization as a Consequence of Gauge
Invariance. Phys. Rev. D 26, 3717 (1982). 346

Chapter VII: Group Contractions

1. E. Inonu and E. P. Wigner, On the Contraction of Groups and Their
Representations, Proc. Nat. Acad. Sci. (U.S.A.) 39, 510 (1953). 357
2. D. Han, Y. S. Kim, M. E. Noz, and D. Son, Internal Space-Time
Symmetries of Massive and Massless Particles. Am. J. Phys. 52, 1037
(1984). 372
3. D. Han, Y. S. Kim, and D. Son, Eulerian Parametrization of
Wigner's Little Groups and Gauge Transformations in Terms of
Rotations in Two-Component Spinors, J. Math. Phys. 27. 2228
(1986). 379
viii Table of Contents

4. Y. S. Kim and E. P. Wigner, Cylindrical Group and Massless

Particles. J. Math. Phys. 28,1175 (1987). 387

Chapter VIII: Localization Problems

1. T. D. Newton and E. P. Wigner, Localized States for Elementary
Systems. Rev. Mod. Phys. 21,400 (1949). 395
2. A. S. Wightman, On the Localizability of Quantum Mechanical
Systems. Rev. Mod. Phys. 34, 845 (1962). 402
3. D. Han, Y. S. Kim, and M. E. Noz, Uncertainty Relations for Light
Waves and the Concept of Photons. Phys. Rev. A 35. 1682 (1987). 430

Chapter IX: Lorentz Transformations

1. V. Bargmann, L. Michel, and V. L. Telegdi, Precession of the
Polarization of Particles Moving in a Homogeneous Electromagnetic
Field. Phys. Rev. Lett. 2, 435 (1959). 443
2. J. Kupersztych, Is There a Link Between Gauge Invariance. Relativis-
tic Invariance, and Electron Spin? Nuovo Cimento 3IB. 1 (1976). 447
3. L. Parker and G. M. Schmieg, Special Relativity and Diagonal
Transformations, Am. J. Phys. 38, 218 (1970). 458
4. L. Parker and G. M. Schmieg, A Useful Form of the Minkowski
Diagram, Am. J. Phys. 38, 1298 (1970). 463
5. B. Yorke, S. L. McCall, and J. R. Klauder, SU(2) and SU(l,/)
Interferometers, Phys. Rev. A 33,4033 (1986). 468
6. D. Han, Y. S. Kim, and D. Son, Thomas Precession, Wigner Rota-
tions and Gauge Transformations, Class. Quantum Grav. 4, 1777
(1987). 490
7. D. Han, Y. S. Kim, and M. E. Noz, Linear Canonical Transforma-
tions of Coherent and Squeezed States in the Wigner Phase Space.
Phys. Rev. A 37, 807 (1988). 497
Preface

Special relativity and quantum mechanics are likely to remain the two most
important languages in physics for many years to come. The underlying language
for both disciplines is group theory. Eugene P. Wigner's 1939 paper on the Unitary
Representations of the Inhomogeneous Lorentz Group laid the foundation for
unifying the concepts and algorithms of quantum mechanics and special relativity.
In view of the strong current interest in the space-time symmetries of elementary
particles, it is safe to say that Wigner's 1939 paper was fifty years ahead of its time.
This edited volume consists of Wigner's 1939 paper and the major papers on the
Lorentz group published since 1939. .
This volume is intended for graduate and advanced undergraduate students in
physics and mathematics, as well as mature physicists wishing to understand the
more fundamental aspects of physics than are available from the fashion-oriented
theoretical models which come and go. The original papers contained in this
volume are useful as supplementary reading material for students in courses on
group theory, relativistic quantum mechanics and quantum field theory, relativistic
electrodynamics, general relativity, and elementary particle physics.
This reprint collection is an extension of the textbook by the present editors entitled
"Theory and Applications of the Poincare Group." Since this book is largely
based on the articles contained herein, the present volume should be viewed as a
continuation of and supplementary reading for the previous work.
We would like to thank Professors J. Bjorken, R. Feynman, R. Hofstadter, J.
Kuperzstych, L. Michel, M. Namiki, L.Parker, S. Weinberg, E.P. Wigner, A.S.
Wightman, and Drs. P. Hussar, M. Ruiz, F. Rotbart, and B. Yurke for allowing us to
reprint their papers. We are grateful to Mrs. M. Dirac and Mrs. S. Yukawa for
giving us permission to reprint the articles of Professors P.A.M. Dirac and H.
Yukawa respectively.
We wish to thank the Annals of Mathematics for permission to reprint Professor
Wigner's historic paper. We thank the American Physical Society, the American
Association of Physics Teachers, The Royal Society of London, il Nuovo Cirnento
and Progress in Theorectical Physics for permission to reprint the articles which
appeared in their journals and for which they hold the copyright. The excerpt from
Albert Einstein: Historical and Cultural Perspective: The Centennial Symposium in
Jerusalem is reprinted with permission of Princeton University Press; that from

ix
x Preface

High Energy Collisions is reprinted with pennission of Gordon and Breach Science
publisher, Inc. and that from Aspects of Quantum Theory with pennission of
Cambridge University Press.
Introduction

One of the most fruitful and still promising approaches to unifying quantum
mechanics and special relativity has been and still is the covariant formulation of
quantum field theory. The role of Wigner' s work: on the Poincare group in quantum
field theory is nicely summarized in the fourth paragraph of an article by V.
Bargmann et al. in the commemorative issue of the Reviews of Modem Physics in
honor of Wigner's 60th birthday [Rev. Mod. Phys. 34, 587 (1962)], which
concludes with the sentences:
"Those who had carefully read the preface of Wigner's great
1939 paper on relativistic invariance and had understood the
physical ideas in his 1931 book on group theory and atomic
spectra were not surprised by the tum of events in quantum field
theory in the 1950' s. A fair part of what happened was merely a
matter of whipping quantum field theory into line with the
insights achieved by Wignerin 1939".

It is important to realize that quantum field theory has not been and is not at present
the only theoretical machine with which physicists attempt to unify quantum
mechanics and special relativity. Indeed, Dirac devoted much of his professional
life to this important task, but, throughout the 1950's and 1960's, his form of
relativistic quantum mechanics was overshadowed by the success of quantum field
theory. However, in the 1970's, when it was necessary to deal with quarks confined
permanently inside hadrons, the limitations of the present form of quantum field
theory become apparent. Currently, there are two different opinions on the
difficulty of using field theory in dealing with bound-state problems or systems of
confined quarks. One of these regards the present difficulty merely as a
complication in calculation. According to this view, we should continue developing
mathematical techniques which will someday enable us to formulate a bound-state
problem with satisfactory solutions within the framework of the existing form of
quantum field theory. The opposing opinion is that quantum field theory is a model
that can handle only scattering problems in which all particles can be brought to
free-particle asymptotic states. According to this view we have to make a fresh start
for relativistic bound-state problems.
These two opposing views are not mutually exclusive. Bound-state models
developed in these two different approaches should have the same space-time
symmetry. It is quite possible that independent bound-state models, if successful in

Xl
xii Introduction

explaining what we see in the real world, will eventually complement field theory.
One of the purposes of this book is to present the fundamental papers upon which a
relativistic bound-state model that can explain basic hadronic features observed in
high-energy laboratories could be build in accordance with the principles laid out by
Wigner in 1939.
Wigner observed in 1939 that Dirac's electron has an SU(2)-like internal space-time
symmetry. However, quarks and hadrons were unknown at that time. Dirac's form
of relativistic bound-state quantum mechanics, which starts from the representations
of the Poincare group, makes it possible to study the O(3)-like little group for
massive particles and leads to hadronic wave functions which can describe fairly
accurately the distribution of quarks inside hadrons. Thus a substantial portion of
hadronic physics can be incorporated into the O(3)-like little group for massive
particles.
Another important development in modern physics is the extensive use of gauge
transformations in connection with massless particles and their interactions.
Wigner's 1939 paper has the original discussion of space-time symmetries of
massless particles. However, it was only recently recognized that gauge-dependent
electromagnetic four-potentials form the basis for a finite-dimensional non-unitary
representation of the little group of the Poincare group. This enables us to associate
gauge degrees of freedom with the degrees of freedom left unexplained in Wigner's
work. Hence it is possible to impose a gauge condition on the electromagnetic
four-potential to construct a unitary representation of the photon polarization
vectors.
Wigner showed that the internal space-time symmetry group of massless particles is
locally isomorphic to the Euclidian group in two-dimensional space. However,
Wigner did not explore the content of this isomorphism, because the physics of the
translation-like transformations of this little group was unknown in 1939. Neutrinos
were known only as "Dirac electrons without mass", although photons were known
to have spins either parallel or antiparallel to their respective momenta. We now
know the physics of the degrees of freedom left unexplained in Wigner's paper.
Much more is also known about neutrinos today that in 1939. For instance, it is
firmly established that neutrinos and anti-neutrinos are left and right handed
respectively. Therefore, it is possible to discuss internal space-time symmetries of
massless particles starting from Wigner's E(2)-like little group. Recently, it was
observed that the O(3)-like little group becomes the E(2)-like group in the limit of
small mass and/or large momentum.
Indeed, group theory has become the standard language in physics. Until the
1960's, the only group known to the average physicist had been the three-
dimensional rotation group. Gell-Mann's work on the quark model encouraged
physicists to study the unitary groups, which are compact groups. The Weinberg-
Salam model enhanced this trend. The emergence of supersymmetry in the 1970's
has brought the space-time group closer to physicists. These groups are non-
compact, and it is difficult to prove or appreciate mathematical theorems for them.
Introduction xiii

The Poincare group is a non-compact group. Fortunately, the representations of this

group useful in physics are not complicated from the mathematical point of view.
The application of the Lorentz group is not restricted to the symmetries of
elementary particles. The (2 + 1)-dimensional Lorentz group is isomorphic to the
two-dimensional symplectic group, which is the symmetry group of homogeneous
linear canonical transformations in classical mechanics. It is also useful for
studying coherent and squeezed states in optics. It is likely that the Lorentz group
will serve useful purposes in many other branches of modem physics.
This reprint volume contains the fundamental paper by Wigner, and the papers on
applications of his paper to physical problems. This book starts with Wigner's
review paper on relativistic invariance and quantum phenomena. The reprinted
papers are grouped into nine chapters. Each chapter starts with a brief introduction.
Chapter I

Perspective View of Quantum Space-Time Symmetries

When Einstein formulated his special theory of relativity in 1905, quantum

mechanics was not known. Einstein's original version of special relativity deals
with point particles without space-time structures and extension. These days, we
know that elementary particles can have intrinsic space-time structure manifested by
spins. In addition, many of the particles which had been thought to be point
particles now have space-time extensions.
The hydrogen atom was known to be a composite particle in which the electron
maintains a distance from the proton. Therefore, the hydrogen atom is not a point
particle. The proton had been regarded as a point particle until, in 1955, the
experiment of Hofstadter and McAllister proved otherwise. These days, the proton
is a bound state of more fundamental particles called the quarks. We still do not
know whether the quarks have non-zero size, but assume that they are point
particles. We assume also that electrons are point particles. However, it is clear
that these particles have intrinsic spins. The situation is the same for massless
particles. For intrinsic spins, the Wigner's representation of the Poincare group is
the natural scientific language.
As for nonrelativistic extended particles, such as the hydrogen atom, the present
form of quantum mechanics with the probability interpretation is quite adequate. If
the proton is a bound state of quarks within the framework of quantum mechanics,
the description of a rapidly moving proton requires a Lorentz transformation of
localized probability distribution. In addition, this description should find its place in
Wigner's representation theory of the Poincare group.
This Chapter consists of one article by Wigner on relativistic invariance of quantum
phenomena, and one article by Dirac. As he said in his 1979 paper, Dirac was
concerned with the problem of fitting quantum mechanics in with relativity, right
from the beginning of quantum mechanics. Dirac suggests that the ideal mechanics
should be both relativistic and deterministic. It would be too ambitious to work with
both the relativistic and deterministic problem at the same time. Perhaps the easier
way is to deal with one aspect at a time. Then there are two routes to the ideal
mechanics, as are illustrated in Figure 1. The current literature indicates that it
would be easier to make quantum mechanics relativistic than deterministic. In this
book, we propose to study the easier problem first.
2 CHAPTER I

Relativistic Ideal
Quantum Mech. Mechanics

u
+=
.>
II)
Poincare'
+= I Group )
.!2
CI)
0::

Quantum Mech. Quantum Mech.

Present Form Deterministic Deterministic

FIG. I. Two different routes to the ideal mechanics. Covariance and detenninism are
the two main problems. In approaching these problems, there are two different
routes. In either case, the Poincare group is likely to be the main scientific language.
PERSPECTIVE VIEW OF QUANTUM SPACE-TIME SYMMETRIES 3

Reprinted from REVIEWS OF MODER" PHYSICS, Vol. 29, No. J, 255-268, July, 1957
Pri.ted in U. S. A.

Relativistic Invariance and Quantum Phenomena·

EUGENE P. WIGNER

Palmer Physical Laborawry, Princeton University, Princeton, NffUJ Jersey

INTRODUCTION is perhaps irritating. It does not alter the fact that the

T HE principal theme of this discourse is the great

difference between the relation of special relativ-
question of the consistency of the two theories can at
least be formulated, that the question of the special
relativistic invariance of quantum mechanics by now
ity and quantum theory on the one hand, and general
relativity and quantum theory on the other. Most of has more nearly the aspect of a puzzle than that of a
the conclusions which will be reported on in connection problem.
with the general theory have been arrived at in col- This is not so with the general theory of relativity.
laboration with Dr. H. Salecker,I who has spent a The basic premise of this theory is that coordinates
year in Princeton to investigate this question. are only auxiliary quantities which can be given
The,difference between the two relations is, briefly, arbitrary values for every event. Hence, the measure-
that while there are no conceptual problems to separate ment of position, that is, of the space coordinates, is
the ~eory of special relativity from quantum theory, certainly not a significant measurement if the postulates
there IS hardly any common ground between the general of the general theory are adopted: the coordinates can
theory of relativity and quantum mechanics. The be given any value one wants. The same holds for
statement, that there are no conceptual conflicts mome~ta. Most of us have struggled with the problem
between quantum mechanics and the special theory, of how, under these premises, the general theory of
should not mean that the mathematical formulations relativity can make meaningful statements and predic-
of the two theories naturally mesh. This is not the case tions at all. Evidently, the usual statements about
and i~ required the very ingenious work of Tomonaga: future positions of particles, as specified by their
Schwmger, Feynman, and Dyson' to adjust quantum coordinates, are not meaningful statements in general
mechanics to the postulates of the special theory and relativity. This is a point which cannot be emphasized
this was so far successful only on the working level. strongly enough and is the basis of a much deeper
What is meant is, rather, that the concepts which are dilemma than the more technical question of the
used in quantum mechanics, measurements of positions, Lorentz invariance of the quantum field equations.
mom~nta, and the like, are the same concepts in terms It pervades all the general theory, and to some degree
of whIch the special relativistic postulate is formulated. we mislead both our students and ourselves when we
Hence, it is at least possible to formulate the require- calculate, for instance, the mercury perihelion motion
ment of special relativistic invariance for quantum without explaining how our coordinate system is fixed
theories and to ascertain whether these requirements in space, what defines it in such a way that it cannot
are met. The fact that the answer is mOre nearly no be rotated, by a few seconds a year, to follow the
than yes, that quantum mechanics has not yet been perihelion's apparent motion. Surely the x axis of our
fully adjusted to the postulates of the special theory, coordinate system could be defined in such a way that
it pass through all successive perihelions. There must
• Address of retiring president of the American Physical
Society, January 31, 1957.
be some assumption on the nature of the coordinate
I This will he reported joiutly with H. Salecker iu more detail system which keeps it from following the perihelion.
in another journal. This is not difficult to exhibit in the case of the motion
• See, e.g., J. M. Jauch and F. Rohrlich, T~ T'-ry of Pro"""
aM EJect""" (Addison-Wesley Press Cambridge Massachusetts of the perihelion, and it would be useful to exhibit it.
1955). ' " Neither is this, in general, an academic point, even
255

Reprinted from Rev. Mod. Phys, 29, 255 (1957),

4 CHAPTER I

256 EUGENE P. WIGNER

though it may be academic in the case of the mercury ahout the number of polarizations of a particle and the
perihelion. A difference in the tacit assumptions which principal purpose of the following paragraphs is to
fix the coordinate system is increasingly recognized to illuminate it from a different point of view.' Instead of
be at the bottom of many conflicting results arrived at the question: "Why do particles with zero rest-mass
in calculations based on the general theory of relativity. have only two directions of polarization?" the slightly
Expressing our results in terms of the values of co- different question, "Why do particles with a finite
ordinates became a habit with us to such a degree that rest-mass have more than two directions of polariza-
we adhere to this habit also in general relativity where tion?" is proposed.
values of coordinates are not per se meaningful. In The intrinsic angular momentum of a particle with
order to make them meaningful, the mollusk-like zero rest-mass is parallel to its direction of motion,
coordinate system must be somehow anchored to that is, parallel to its velocity. Thus, if we connect
space-time events and this anchoring is often done with any internal motion with the spin, this is perpendicular
little explicitness. If we wllnt to put general relativity to the velocity. In case of light, we speak of transverse
on speaking terms with quantum mechanics, our first polarization. Furthermore, and this is the salient point,
task has to be to bring the statements of the general the statement that the spin is parallel to the velocity
theory of relativity into such form that they conform is a relativistically invariant statement: it holds as
with the basic principles of the general relativity theory well if the particle is viewed from a moving coordinate
itself. It will be shown below how this may be attempted. system. If the problem of polarization is regarded from
this point of view, it results in the question, "Why
RELATIVISTIC QUANTUM THEORY OF ELEMENTARY can't the angular momentum of a particle with finite
SYSTEMS
rest-mass be parallel to its velocity?" or "Why can't
The relation between special theory and quantum a plane wave represent transverse polarization unless
mechanics is most simple for single particles. The it propagates with light velocity?" The answer is that
equations and properties of these, in the absence of the angular momentum can very well be parallel to
interactions, can be deduced already from relativistic the direction of motion and the wave can have trans-
invariance. Two cases have to be distinguished: the verse polarization, but these are not Lorentz invariant
partiCle either can, or cannot, be transformed to rest. statements. In other words, even if velocity and spin
If it can, it will behave, in that coordinate system, are parallel in one coordinate system, they do not
as any other particle, such as an atom. It will have an appear to be parallel in other coordinate systems.
intrinsic angular momentum called J in the case of This is most evident if, in this other coordinate system,
atoms and spin S in the case of elementary particles. the particle is at rest: in this coordinate system the
This leads to the various possibilities with which we
are familiar from spectroscopy, that is spins 0, !, 1,
!, 2, ... each corresponding to a type of particle.
If the particle cannot be transformed to rest, its
velocity must always be equal to the velocity of light.
Every other velocity can be transformed to rest. The
rest-mass of these particles is zero because a nonzero
rest-mass would entail an infinite energy if moving
t

v
with light velocity.
Particles with zero rest-mass have only two directions
of polarization, no matter how large their spin is. This
contrasts with the 2S+ 1 directions of polarization for
particles with nonzero rest-mass and spin S. Electro-
magnetic radiation, that is, light, is the most familiar
example for this phenomenon. The "spin" of light is 1,
but it has only two directions of polarization, instead
of 2S+ 1 = 3. The number of polarizations seems to
i
jump discontinuously to two when the rest-mass
decreases and reaches the value O. Bass and Schrodinger" FIG. 1. The short simple arrows illustrate the spin, the double
followed this out in detail for electromagnetic radiation, arrows the velocity of the particle. One obtains the same state,
no matter whether one first lmp:arts to it a velocity in the direction
that is, for S= 1. It is good to realize, however, that of the spin, then rotates it (R(")A (0, ..)), or whether one first
this decrease in the number of possible polarizations is rotates It, then gives a velocity in the direction of the spin
purely a property of the Lorentz transformation and (A (", ..)R(")). See Eq. (1.3).
holds for any value of the spin. 4 The essential point of the argument which follows is contained
There is nothing fundamentally new that can be said in the present writer's paper, Ann. Math. 40, 149 (1939) and more
explicitly in his address at the Jubilee of Relativity Theory,
• L. Bass and E. Schriidinger, Proc. Roy. Soc. (London) A232, 1 Bern, 1955 (Birkhauser Verlag, Basel, 1956), A. Mercier and
(1955). M. Kervaire, editors, p. 210.
PERSPECTIVE VIEW OF QUANTUM SPACE-TIME SYMMETRIES 5

it E L A T I V 1ST I C I N V A R I A NeE ... N [) Q!J ANT 11 MPH E NOM E N A 257

I )
direction of its motion any more. In the nonrelativistic
case, that is, if all velocities are small as compared with
i f the velocity of light, the spin will still be parallel to z
and it will, therefore, enclose an angle with the particle's
direction of motion. This shows that the statement that

1//
the spin is parallel to the direction of motion is not
invariant in the nonrelativisitic region. However, if
the original velocity of the particle is close to the light
velocity, the Lorentz contraction works out in such a
way that the angle between spin and velocity is given by

i tan (angle between spin and velocity)

= (1- ../,,)1 sin", (1)
FIG. 2. The particle is fint given a small velocity in the direction
of ita spin, then increasing velocities In a prependicular direction
(upper part of the figure). The direction of the spin remains where " is the angle between the velocity v in the
essentially unchanged; it includes an increasingly large angle moving coordinate system and the velocity in the
with the velocity as the velocity In the perpendicular direction coordinate system at rest. This last situation is illus-
increases. U the velocity imparted to the particle is large (lower
part of the figure), the direction of the spin seems to follow the trated at the bottom of the figure. If the velocity of
direction of the velocity. See Eqs. (1.8) and (1.7). the particle is small as compared with the velocity
of light, the direction of the spin remains fixed and is
angular momentum should be parallel to nothing. the same in the moving coordinate system as in the
However, every particle, unless it moves with light coordinate system at rest. On the other hand, if the
velocity, can be viewed from a coordinate system in particle's velocity is close to light velocity, the velocity
which it is at rest. In this coordinate system its angular carries the spin with itself and the angle between
momentum is surely not parallel to its velocity. direction of motion and spin direction becomes very
Hence, the statement that spin and velocity are small in the moving coordinate system. Finally, if the
parallel cannot be universally valid for the particle particle has light velocity, the statement "spin and
with finite rest-mass and such a particle must have velocity are parallel" remains true in every coordinate
other states of polarization also. system. Again, this is not a consequence of any physical
It rp.ay be worthwhile to illustrate this point some- property of the spin, but is a consequence of the
what more in detail. Let us consider a particle at rest properties of Lorentz transformations: it is a kind of
with a given direction of polarization, say the direction Lorentz contraction. It is the reason for the different
of the z axis. Let us consider this particle now from a behavior of particles with finite, and particles with
coordinate system which is moving in the - z direction. zero, rest-mass, as far as the number of states of
The particle will then appear to have a velocity in the polarization is concerned. (Details of the calculation
z direction and its polarization will be parallel to its are in Appendix I.) .
velocity (Fig. 1). It will now be shown that this last The preceding conSideration proves more than was
statement is nearly invariant if the velocity is high. intended: it shows that the statement "spin and
It is evident that the statement is entirely invariant velocity are parallel for zero mass particles" is invariant
with respect to rotations and with respect to a further and that, for relativistic reasons, one needs only one
increase of the velocity in the z direction. This is state of polarization, rather than lwo. This is true as
illustrated at the bottom of the figure. The coordinate far as proper Lorentz transformations are concerned.
system is first turned to the left and then given a The second state of polarization, in which spin and
velocity in the direction opposite to the old z axis. velocity are antiparallel, is a result of the reflection
The state of the system appears to be exactly the same symmetry. Again, this can,be illustrated on the example
as if the coordinate system bad been first given a of light: right circularly polarized ligbt appears as
velocity in the - s direction and then turned, which is right circularly polarized light in all Lorentz frames of
the operation illustrated at the top of the figure. The reference which can be continuously transformed into
state of the system appears to be the same not for any each other. Only if one looks at the right circularly
physical reason but because the two coordinate systems polarized light in a mirror does it appear as left
are identical and they view the same particle (see circularly polarized light. The postulate of reflection
Appendix I). symmetry allows us to infer the existence of left
Let us now take our particle with a high velocity in circularly polarized light from the existence of right
the z direction and view it from a coordinate system circularly polarized light-if there were no such
which moves in the - y direction. The particle now will reflection symmetry in the real world, the existence
appear to have a momentum also in the y direction, its of two modes of polarization of light, with virtually
velocity will have a direction between the y and z identical properties, would appear to be a miracle.
axes (Fig. 2). Its spin, however, will not be in the The situation is entirely different for particles with
6 CHAPTER I

258 EUGENE P. WIGNER

nonzero mass. For these, the 2S +! directions of

polarization follow from the in variance of the theory
with respect to proper Lorentz transformations. In
particular, if the particle is at rest, the spin will have
different orientations with respect to coordinate
systems which have different orientations in space.
Thus, the existence of all the states of polarization follow
from the existence of one, if only the theory is invariant
with respect to proper Lorentz transformations. For
particles with zero rest-mass, there are only two
states of polarization, and even the existence of the
second one can be inferred only on the basis of reflection
symmetry.
REFLECTION SYMMETRY
The problem and existence of reRection symmetry A"",ii Co

have been furthered in a brilliant way by recent

theoretical and experimental research. There is nothing
essential that can be added at present to the remarks
and conjectures of Lee, Yang, and Oehme, and all
that follows has been said, or at least implied, by F,IG. 3. The right side is the mirror image of the left side,
Salam, Lee, Yang, and Oehme.' The sharpness of the according to the interpretation of the parity experiments" which
break with past concepts is perhaps best illustrated by maintains the reflection as a symmetry element of aU physical
laws. It must be assumed that the reflection transforms matter
the cobalt experiment of Wu, Ambler, Hayward, into antimatter: the electronic ring current becomes a positronic
Hoppes, and Hudson. ring current, the radioactive cobalt is replaced by radioactive
The ring current-this may be a permanent current anticobalt.
in a:' superconductor--neates a magnetic field. The Co
source is in the plane of the current and emits {J particles If it is true that a symmetry plane always remains a
(Fig. 3). The whole experimental arrangement, as symmetry plane, the initial state of the Co experiment
shown in Fig. 3, has a symmetry plane and, if the could not have contained a symmetry plane. This would
principle of sufficient cause is valid, the symmetry not be the case if the magnetic vector were polar-in
plane should remain valid throughout the further fate which case the electric vector would be axial. The charge
of the system. In other words, since the right and left density, the divergence of the electric vector, would then
sides of the plane had originally identical properties, become a pseudoscalar rather than a simple scalar as in
there is no sufficient reason for any difference in their current theory. The mirror image of a negative charge
properties at a later time. Nevertheless, the intensity would be positive, the mirror image of an electron a
of the {J radiation is larger on one side of the plane than positron, and conversely. The mirror image of matter
the other side. The situation is paradoxical no matter would be antimatter. The Co experiment, viewed
what the mechanism of the effect is-in fact, it is through a mirror, would not present a picture contrary
most paradoxical if one disregards its mechanism and to established fact: it would present an experiment
theory entirely. If the experimental circumstances can carried out with antimatter. The right side of Fig. 3
be idealized as indicated, even the principle of sufficien t shows the mirror image of the left side. Thus, the
cause seems to be violated. principle of sufficient cause, and the validity of sym-
It is natural to look for an interpretation of the metry planes, need not be abandoned if one is willing
experiment which avoids this very far-reaching conclu- to admit that the mirror image of matter is antimatter.
sion and, indeed, there is such an interpretation." It The possibility just envisaged would be technically
is good to reiterate, however, that no matter what described as the elimination of the operations of
interpretation is adopted, we have to admit that the reflection and charge conjugation, as presently defined,
symmetry of the real world is smaller than we had as true symmetry operations. Their product would
thought. However, the symmetry may still include still be assumed to be a symmetry operation and
reflections. proposed to be named, simply, reflection. A few
• Lee, Yang, and Oehme, Phys. Rev. 106,340 (1957). further technical remarks are contained in Appendix
Is The interpretation referred to has been proposed indepen- n. The proposition just made has two aspects: a very
dently by numerous authors, including A. Salam, Nuovo cimento appealing one, and a very alarming one.
5, 229 (1957); L. Landau, Nuclear Phys. 3, 127 (1957); H. D.
Smyth and L. Biedenharn (personal communication). Dr. S. Let us look first at the appealing aspect. Dirac has
Deser has pointed out that the "perturbing possibility" was raised said that the number of elementary particles shows an
already by Wick, Wightman, and Wigner [phys. Rev. 88, 101
(1952)] but was held "remote at that time." Naturally, the ap· alarming tendency of increasing. One is tempted to
parent unanimity of opinion does not prove its correctness. add to this that the number of invariance properties
PERSPECTIVE VIEW OF QUANTUM SPACE-TIME SYMMETRIES 7

R E L A T I V 1ST I C I N V ART A NeE AND QUA N TUM P HEN 0 MEN A 259

also showed a similar tendency. It is not equally appeared to be a perfectly valid concept in spectroscopy
alarming because, while the increase in the number of and in nuclear physics. This concept could be explained
elementary particles complicates our picture of nature, very naturally as a result of the reflection symmetry
that of the symmetry properties on the whole simplifies of space-time, the mirror image of electrons being
it. Nevertheless the clear correspondence between electrons and not positrons. We are now forced to believe
the invariance properties of the laws of nature, and the that this symmetry is only approximate and the
symmetry properties of space-time, was most clearly concept of parity, as used in spectroscopy and nuclear
breached by the operation of charge conjugation. physics, is also only approximate. Even more funda-
This postulated that the laws of nature remain the mentally, there is a vast body of experimental informa-
same if all positive charges are replaced by negative tion in the chemistry of optically active substances
charges and vice versa, or more generally, if all particles which are mirror images of each other and which have
are replaced by antiparticles. Reasonable as this optical activities of opposite direction but exactly
postulate appears to us, it corresponds to no symmetry equal strength. There is the fact that molecules which
of the space-tiine continuum. If the preceding inter- have symmetry planes are optically inactive; there is
pretation of the Co experiments should be sustained, the fact of symmetry planes in crystals. T All these
the correspondence between the natural symmetry facts relate properties of right-handed matter to
elements of space-time, and the invariance properties left-handed maller, not of right-handed matter to
of the laws of nature, would be restored. It is true that left-handed anlimaller. The new experiments leave no
the role of the planes of reflection would not be that to doubt that the symmetry plane in this sense is not
which we are accustomed-the mirror image of an elec- valid for all phenomena, in particular not valid for
tron would become a positron-but the mirror image of (j decay, that if the concept of symmetry plane is at all
a sequence of events would still be a possible sequence valid for all phenomena, it can be valid only in the
of events. This possible sequence of events would be sense of converting matter into antimatter.
more difficult to realize in the actual physical world Furthermore, the old-fashioned type of symmetry
than what we had thought, but it would still be possible. plane is not the only symmetry concept that is only
The restoration of the correspondence between the approximately valid. Charge conjugation was mentioned
natural symmetry properties of space-time on one before, and we are remainded also of isotopic spin,
hand, and the laws of nature on the other hand, is the of the exchange character, that is multiplet system,
appealing feature of the proposition. It has, actually, for electrons and also of nuclei which latter holds so
two alarming features. The first of these is that a accurately that, in practice, parahydrogen molecules
symmetry operation is, physically, so complicated. can be converted into orthohydrogen molecules only
If it should tum out that the operation of time inversion, by first destroying them. 8 This approximate validity
as we now conceive it, is not a valid symmetry operation of laws of symmetry is, therefore, a very general
(e.g., if one of the experiments proposed by Treiman phenomenon-it may be the general phenomenon. We
and Wyld gave a positive result) we could still maintain are reminded of Mach's axiom that the laws of nature
the validity of this symmetry operation by reinterpret- depend on the physical content of the universe, and
ing it. We could postulate, for instance, that time the physical content of the universe certainly shows
inversion transforms matter into meta-matter which no symmetry. This suggests-and this may also be
will be discovered later when higher energy accelerators the spirit of the ideas of Yang and Lee-that all
will become available. Thus, maintaining the validity symmetry properties are only approximate. The
of symmetry planes forces us to a more artificial view weakest interaction, the gravitational force, is the basis
of the concept of symmetry and of the invariance of of the distinction between inertial and accelerated
the laws of physics. coordinate systems, the second weakest known inter-
The other alarming feature of our new knowledge action, that leading to (j d~y, leads to the distinction
is that we have been misled for such a long time to between matter and antimatter. Let me conclude this
believe in more symmetry elements than actually exist. subject by expressing the conviction that the discoveries
There was ample reason for this and there was ample of Wu, Ambler, Hayward, Hoppes, and Hudson,'
experimental evidence to believe that the mirror image and of Garwin, Lederman, and Weinreich '• will not
of a possible event is again a possible event with remain isolated discoveries. More likely, they herald a
electrons being the mirror images of electrons and not revision of our concept of invariance and possibly
of positrons. Let us recall in this connection first how T For the role of the space and time inversion operators in

the concept of parity, resulting from the beautiful classical theory, see H. Zocher and C. Torok, Proc. Nat!. Acad.
Sci. U.S. 39, 681 (1953) and literature quoted there.
though almost forgotten experiments of Laporte,' I See A. Farkas, Orthol,ydrog ... , PIJI'ahydrogen and Heavy
Hydrogen (Cambridge University Press, New York, 1935).
• O. Laporte, Z. Physik 23, 135 (1924). For the interpretation • Wu, Ambler, Hayward, Hoppes, and Hudson, Phys. Rev. lOS,
of Laporte's rule in terms of the quantum-mechanic;a.1 operation 1413(L) (1957).
of inversion, see the writer's GrupPenJ/zeoru urut ihri A nwendungen 10 Garwin, Lederman, and Weinreich, Phys. Rev. lOS, 1415(1.)
auf die Quantm_lumik tier Almosp<kwen (Friedrich Vieweg und (1957); also, J. l.. l'';p'oman and V. L. Tc1egdi, ibid. lOS, 1681 (I.)
Sohn, Braunschw"ig, 1931), Chap. XVIII. (19571.
8 CHAPTER!

EUGENE P. WIGNER

of other concepts which are even more taken for Appendix III.) This shows that the establishment of a
granted. close network of points in space-time requires a
reasonable energy density, a dense forest of world
QUANTUM LIMITATIONS OF THE CONCEPTS OF lines wherever the network is to be established. How-
GENERAL RELATMTY ever, it is not necessary to discuss this in detail because
The last remarks naturally bring us to a discussion the measurement of the distances between the points of
of the general theory of relativity. Tbe main premise the network gives more stringent requirements than
of this theory is that coordinates are only labels to the establishment of the network.
specify spare-time points. Their values have no partic- It is often said that the distances between events
ular significance unless the coordinate system is must be measured by yardsticks and rods. We found
somehow anchored to events in space-time. that measurements with a yardstick are rather difficult
Let us look at the question of how the equations of to describe and that their use would involve a great
the general theory of relativity could be verified. deal of unnecessary complications. The yardstick gives
The purpose of these equations, as of all equations of the distance between events correctly only if its marks
physics, is to calculate, from the knowledge of the coincide with the two events simultaneously from the
present, the state of affairs that will prevail in the point of view of the rest-system of the yardstick.
future. The quantities describing the present state are Furthermore, it is hard to image yardsticks as anything
called initial conditions; the ways these quantities but macroscopic objects. It is desirable, therefore,
change are called the equations of motion. In relativity to reduce all measurements in space-time to measure-
theory, the state is described by the metric which ments by clocks. Naturally, one can measure by
consists of a network of points in splce-time, that is clocks directly only the distances of points which are
a network of events, and the distances between these in time-like relation to each other. The distances of
events. If we wish to translate these general statements events which are in space-like relation, and which
into something concrete, we must decide what events would be measured more naturally by yardsticks,
are, and how we measure distances between" evenls. will have to be measured, therefore, indirectly.
The metric in the general theory of relativity is a It appears, thus, that the simplest framework in
metric in space-time, its elements are distances between space-time, and the one which is most nearly micro-
space-time points, not between points in ordinary space. scopic, is a set of clocks, which are only slowly moving
The events of the general theory of relativity are with respect to each other, that is, with world lines
coincidences, that is, collisions between particles. which are approximately parallel. These clocks tick
The founder of the theory, when he created this concept, off periods and these ticks form the network of events
had evidently macroscopic bodies in mind. Coincidences, which we wanted to establish. This, at the same time,
tbat is, collisions between such bodies, are immediately establishes the distance of those adjacent points which
observable. This is not the case for elementary particles; are on the same world line.
a collision hetween these is something much more Figure 4 shows two world lines and also shows an
evanescent. In fact, the point of a collision between event, that is, a tick of the clock, on each. The figure
two elementary particles can be closely localized in shows an artifice which enables one to measure the
space-time only in case of high-energy collisions. (See distance of space-like events: a light signal is sent out
from the. first clock which strikes the second clock
at event 2. This clock, in tum, sends out a light signal
which strikes the first clock at time I' after the event 1.
If the first light signal had to be sent out at time j
before the first event, the calculation given in Appendix
IV shows that the space-like distance of events 1 and 2
is the geometric average of the two measured time-like
distances j and 1'. This is then a way to measure
distances between space-like events by clocks instead
of yardsticks.
It is interesting to consider the quantum limitations
on the accuracy of the conversion of time-like measure-
ments into space-like measurements, which is illustrated
in Fig. 4. Naturally, the times / and /' will be well
defined only if the light signal is a short pulse. This
FIG. 4. Measurement of space-like distances by means of a implies that it is composed of many frequencies and,
clock. It is assumed that the metric tensor is essentially constant hence, that its energy spectrum has a corresponding
within the space-time region contained in the figure. The space-like width. As a result, it will give an indeterminate recoil
distance between events 1 and 2 is measured by means of the light
signals which pass through event 2 and a geodesic which goes to the second clock, thus further increasing the un-
through event I. Explanation in Appendi.lV. certainty of its momentum. All this is closely related
PERSPECfIVE VIEW OF QUANTUM SPACE-TIME SYMMETRIES 9

R E L A T i V 1ST I C I N V A R I A NeE AND QUA N TUM PH E NOM EN... 261

to Heisenberg's uncertainty principle. A more detailed For example, a clock, with a running time of a day and
calculation' shows that the added uncertainty is of an accuracy of 10-a second, must weigh almost a
the same order of magnitude as the uncertainty inherent gram-for reasons stemming solely from uncertainty
in the nature of the best clock that we could think of, principles and similar considerations.
so that the conversion of time-like measurements So far, we have paid attention only to the physical
into space-like measurements is essentially free. dimension of the clock and the requirement that it
We finally come to the discussion of one of the be able to distinguish between events which are only
principal problems-the limitations on the accuracy a distance I apart on the time scale. In order to make
of the clock. It led us to the conclusion that the inherent it usable as part of the framework which was described
limitatioI1l! on the accuracy of a clock of given weight before, it is necessary to read the clock and to start it.
and size, which should run for a period of a certain As part of the framework to map out the metric of
length, are quite severe. In fact, the result in summary space-time, it must either register the readings at
is that a clock is an essentially nonmicroscopic object. which it receives impulses, or transmit these readings
In particular, what we vaguely call an atomic clock, to a part of space outside the region to be mapped out.
a single atom which ticks off its periods, is surely an This point was already noted by Schriidinger." How-
idealization which is in conflict with fundamental ever, we found it reassuring that, in the most interesting
concepts of measurability. This part of our conclusions case in which 1= ct, that is, if space and time inaccuracies
can be considered to be well established. On the other are about equal, the reading requirement introduces
hand, the actual formula which will be given for the only an insignificant numerical factor but does not
limitation of the accuracy of time measurement, a sort change the form of the expression for the minimum
of uncertainty principle, should be considered as the mass of the clock.
best present estimate. The arrangement to map the metric might consist,
Let us state the requirements as follows. The watch therefore, of a lattice of clocks, all more or less at rest
shall run T seconds, shall measure time with an accuracy with respect to each other. All these clocks can emit
of Tjn=l, its linear extension shall not exceed I, its light signals and receive them. They can also transmit
mass shall be below m. Since the pointer of the watch their reading at the time of the receipt of the light
must be able to assume n different positions, the system signal to the outside. The clocks may resemble oscil-
will have to run, in the course of the time T, over at lators, well in the nonrelativistic region. In fact, the
least n orthogonal states. Its state must, therefore, be velocity of the oscillating particle is about n times
the superposition of at least n stationary states. It is smaller than the velocity of light where n is the .ratio
clear, furthermore, that unless its total energy is at of the error in the time measurement, to the dui-ation
least h/I, it cannot measure a time interval which is of the whole interval to be measured. This last quantity
smaller than I. This is equivalent to the usual un- is the spacing of the events on the time axis, it is also
certainty principle. These two requirements follow the distance of the clocks from each other, divided by
directly from the basic principles of quantum theory; the light velocity. The world lines of the clocks from
they are also the requirements which could well have the dense forest which was mentioned before. Its
been anticipated. A clock which conforms with these branches suffuse the region of space-time in which the
postulates is, for instance, an oscillator, with a period metric is to be mapped out.
which is equal to the running time of the clock, if it We are not absolutely convinced that our clocks
is with equal probabilty in any of the first n quantum are the best possible. Our principal concern is that we
states. Its energy is about n times the energy of the have considered only one space-like dimension. One
first excited state. This corresponds to the uncertainty consequence of this was that the oscillator had to be a
principle with the accuracy t as time uncertainty. one-dimensional oscillator. It is possible that the size
Broadly speaking, the clock is a very soft oscillator, the limitation does not increase the necessary mass of the
oscillating particle moving very slowly and with a clock to the same extent if use is made of all three
rather large amplitude. The pointer of the clock is spatial dimensions.
the position of the oscillating particle. The curvature tensor can be obtained from the
The clock of the preceding paragraph is still very metric in the conventional way, if the metric is measured
light. Let us consider, however, the requirement that with sufficient accuracy. It may be of interest, never-
the linear dimensions of the clock be limited. Since theless, the describe a more direct method for measuring
there is little point in dealing with the question in the curvature of space. It involves an arrangement,
great generality, it may as well be assumed here that illustrated in Fig. S, which is similar to that used for
the linear dimension shall correspond to the accuracy obtaining the metric. There is a clock, and a mirror,
in time. The requirement I=cl increases the mass of the at such a distance from each other that the curvature
clock by nl which may be a very large factor indeed: of space can be assumed to be constant in the interven-
11 E. Schrlklinger, Ber. Preuss. Akad. Wiss. phys.-math. Kl.
m> n'ht/l'=n1h/c't. (2) 1931,238.
10 CHAPTER I

262 EUGENE P. WIGNER

FIG. 5. Direct meas-

urement of the curva-
ture by means of a
dock and mirror. Only
one space-like dimension
is considered and the
curvature assumed to
be constant within the
space-time region con-
tained in the figure. The
explanation is given in
Appendix V.

World l' World line 2 World line I World line 2

ing region. The two clocks need not be at rest with for the accuracy with which the curvature can be
respect to each other, in fact, such a requirement would measured. The result is, as could be anticipated, that
involve additional measurements to verify it. If the the curvature at a poinl in space-time cannot be
space is flat, the world lines of the clocks can be drawn measured a t all j only the average curvature over a
straight. In order to measure the curvature, a light finite region of space-time can be obtained. The error of
signal is emitted by the clock., and this is reflected by the measurement1 is inversely proportional to the
the mirror. The time of return is read on the clock-it two-thirds power of the area available in space-time,
is Ir-and the light signal returned to the mirror. that is, the area around which a vector is carried,
The time which the light signal takes on its second trip always parallel to itself, in the customary definition of
to return to the clock is denoted by I,. The process is the curvature. The error is also proportional to the cube
repeated a third time, the duration of the last roundtrip root of the Compton wavelength of the clock. Our
denoted by I,. As shown in Appendix V, the radius of principal hesitation in considering this result as defini-
curvature, a, and the relevant componenl ROID1 of tive is again its being based on the consideration of
the Ri~mann tensor are given by only one space-like dimension. The possibilities of
measuring devices, as well as the problems, may be
1,- 21,+1, 11 substantially different in three-dimensional space.
----=-=llnR OlOl)l. (2)
122 a Whether or not this is the case, the essentially
nonmicroscopic nature of the general relativistic
If classical theory would be valid also in the micro- concepts seems to us inescapable. If we look at this
scopic domain, there would be no limit on the accuracy first from a practical point of view, the situation is
of the measurement indicated in Fig. 5. If h is infinitely rather reassuring. We can note first, that the measure-
small, the time intervals 11, I" I, can all be measured ment of electric and magnetic fields, as discussed by
with arbitrary accuracy with an infinitely light clock. Bohr and Rosenfeld,12 also requires macroscopic, in
Similarly, the light signals between clock and mirror, fact very macroscopic, equipment and that this does
however short, need carry only an infinitesimal amount not render the electromagnetic field concepts useless
of momentum and thus deflect clock and mirror for the purposes of quantum electrodynamics. It is
arbitrarily little from their geodesic paths. The quantum true that the measurement of space-time curvature
phenomena considered before force us, 'however, to requires a finite region of space and there is a minimum
use a clock with a minimum mass if the measurement for the mass, and even the mass uncertainty, of the
of the time intervals is to have a given accuracy. In measuring equipment. However, numerically, the
the present case, this accuracy must be relatively situation is by no means alarming. Even in interstellar
high unless the time intervals I" I" I, are of the same space, it should be possible to measure the curvature
order of magnitude as the curvature of space. Similarly,
the deflection of clock and mirror from their geodesic "N. Bohr and L. Rosenfeld, Kg!. Danske Videnskab. Selskah
Mat.-fys. Medd. 12, No. 8 (1933). See also further literature
paths must be very small if the result of the measure- quoted in L. Rosenfeld's article in Niels Eo"" and lJu Development
ment is to be meaningful. This gives an effective limit oj Physics (Pergamon Press, London. 1955).
PERSPECTIVE VIEW OF QUANTUM SPACE-TIME SYMMETRIES 11

R E L A T I V 1ST I C I N V A R I A NeE AND QUA N TUM PH E NOM E N A 263

in a volume of a light second or so. Furthermore, the axis, we look at the particle in the standard state from
mass of the clocks which one will wish to employ for a coordinate system moving with the velocity v in
such a measurement is of the order of several micro- the -. direction. If we wish to have a particle at rest
grams SO that the finite mass of elementary particles but with its spin in the yz plane, including an angle a
does not cause any difficulty. The clocks will contain with the • axis, we look at the standard state from a
many particles and there is no need, and there is not coordinate system the y and z axes of which include an
even an incentive, to employ clocks which are lighter angle a with the y and. axes of the coordinate system
than the elementary particles. This is hardly surprising in which the standard state was defined. In order to
since the mass which can be derived from the gravita- obtain a state in which both velocity and spin have the
tional constant, light velocity, and Planck's constant, aforementioned direction (Le., a direction in the y.
is about 20 micrograms. plane, including the angles a and ir-a with the y
It is well to repeat, however, that the situation is and z axes), we look at the standard state from the
less satisfaQ:ory from a more fundamental point of point of view of a coordinate system in which the
view. It remains true that we consider, in ordinary spin of the standard state is described as this direction
quantum theory, position operators as observables and which is moving in the opposite direction.
without specifying what the coordinates mean. The Two states of the system will be identical only if the
concepts of quantum field theories are even more Lorentz frames of reference which define them are
weird from the point of view of the basic observation identical. Under this definition, the relations which
that only coincidences are meaningful. This again is will be obtained will be valid independently of the
hardly surprising because even a 20-microgram clock properties of the particle, such as spin or mass (as
is too large for the measurement of atomic times or long as the mass in nonzero so that the standard state
distances. If we analyze the way in which we "get exists). Two states will be approximately the same if the
away" with the use of an absolute space concept, we two Lorentz frames of reference which define them
simply find that we do not. In our experiments we can be obtained from each other by a very small
surround the microscopic objects with a very macro- Lorentz transformation, that is, one which is near
scopic framework and observe coincUknces between the identity. Naturally, all states of a particle which
the particles emanating from the microscopic system, can be compared in this way are related to each other
and parts of the framework. This gives the collision inasmuch as they represent the same standard state
matrix, which is observable, and observable in terms of viewed from various coordinate systems. However, we
macroscopic coincidences. However, the so-called shall have to compare only these states.
observables of the microscopic system are not only not Let us denote by A (O,I") the matrix of the trans-
observed, they do not even appear to be meaningful. formation in which the transformed coordinate system
There is, therefore, a boundary in our experiments moves with the velocity -. in the • direction where
between the region in which we use the quantum .=c tanhl"
concepts without worrying about their meaning in
face of the fundamental observation of the general
theory of relativity, and the surrounding region in
A(O'¥'}=II~o smh¥,
c?~¥' coshl"
s~hl"ll· (1.1)
which we use concepts which are meaningful also in
the face of the basic observation of the general theory Since the x axis will play no role in the following
of relativity but which cannot be described by means of consideration, it is suppressed in (1.1) and the three
quantum theory. This appears most unsatisfactory rows and the three columns of this matrix refer to the
from a strictly logical standpoint. y', .', d and to the y, " cl axes, respectively. The
matrix (1.1) characterizes the state in which the
APPENDIX I particle moves with a velocity v in the direction of the
It will be necessary, in this appendix, to compare z axis and its spin is parallel to this axis.
various states of the same physical system. These Let us further denote the matrix of the rotation by
states will be generated by looking at the same state- an angle I" in the yz plane by
the standard state-from various coordinate systems.
Hence every Lorentz frame of reference \ will define a
state of the system-the state as which the standard (1.2)
state appears from the point of view of this coordinate
system. In order to define the standard state, we We refer to the direction in the yz plane which lies
choose an arbitrary but fixed Lorentz frame of reference between the y and. axes and includes an angle {J with
and stipulate that, in this frame of reference, the the z axis as the direction {J. The coordinate system
particle in the standard state be at rest and its spin which moves with the velocity -. in the {J direction is
(if any) have the direction of the z axis. Thus, if we obtained by the transformation
wish to have a particle moving with a velocity v in
the • direction and with a spin also directed along this A ({J,¥'}=R({J}A (O,¥,}R( -{J}. (1.3)
12 CHAPTER I

264 EUGENE P. WIGNER

In order to obtain a particle wbich moveb in the direc- This transformation does not have the form (1.4). In
tion {} and is polarized in this direction, we first rotate order to bring it into that form, it has to be multiplied
the coordinate system counterclockwise by {} (to have on the right by R(o), i.e., one bas to rotate the spin
the particle polarized in the proper direction) and ahead of time. The angle 0 is given by the equation
impart it then a velocity - v in the {} direction. Hence,
it is the transformation tanh '1" v'
tan.=--=-(l-.'N)1 (1.7)
sinh I" v
sin{} cosh I" sin{} sinh I"
cos{} cosh I" cos{} sinh I"
II (1.4)
and is called the angle between spin and velocity.
For V«c, it becomes equal to the angle which the
sinh I" coshl" ordinary resultant of two perpendicular velocities, v
which characterizes the aforementioned state of the and v', includes with the first of these. However, •
particle. It follows from (1.3) that becomes very small if v is close to c; in this case it is
hardly necessary to rotate the spin away from the z
T({},I")=R({})A(O'I")=R(~)T(O,I") (1.5) axis before giving it a velocity in the z direction.
so that the same state can be obtained also by viewing These statements express the identity
tbe state characterized by (1.1) from a coordinate
A (!r,I"/)A (O,I")R(.) = T({},I"") (1.8)
system that is rotated by{}. It follows that the statement
"velocity and spin are parallel" is invariant under which can be verified by direct calculation. The right
rotations. This had to be expected. side represents a particle with parallel spin and velocity,
If the state generated by A (O,\O)=T(O,I") is viewed the magnitude and direction of the latter being given
from a coordinate system which is moving with the by the well-known equations
velocity u in the direction of the z axis, the particle
will still appear to move in the z direction and its spin ."=c tanh 1"" = (.'+v"-.'v"/c')1 (1.Sa)
will remain parallel to its direction of motion, unless and
u>v in which case the two directions will become sinh 1"/ v/
antiparallel, or unless u= v in which case the statement tan{}=--=---- (1.8b)
becomes meaningless, the particle appearing to be tanhl" v(l-v"/c')1
at rest. Similarly, the otber states in which spin and Equation (I) given in the text follows from (1.7) and
velocity are parallel, i.e., the states generated by the (1.8b) for fY'Vc.
transformations T({},I") , remain such states if viewed The fact that the states T(t'J,I")>{;o (where >{;o is the
from a coordinate system moving in the direction of the standard state and 1"» 1) are approximately invariant
particle's velocity, as long as the coordinate system is under all Lorentz transformations is expressed mathe-
not ·moving faster than the particle. This also had to matically by the equations,
be expected. However, if the state generated by T(O,<P)
is viewed from a coordinate system moving with velocity R{{})· T(O,I")>{;o= T({},I")>{;o, (1.Sa)
v' = c tanhl'" in the - y direction, spin and velocity will
nol appear parallel any more, provided Ihe velocily v A (0,'1")· T(O,I")>{;o= T(O,I"'+<P)>{;o, (1.9a)
of Ihe parlicle is nol close 10 lighl velocily. This last and
proviso is the essential one; it means tbat the bigh (1.9b)
velocity states of a particle for which spin and velocity
are parallel (i.e., the states generated by (1.4) with a which give the wave function of the state T(O,I")>{;o,
large 1") are states of this same nature if viewed from a as viewed from other Lorentz frames of reference.
coordinate system which is not moving too fast in the Naturally, similar equations apply to all T(a,I")>{;o.
direction of motion of the particle itself. In the limiting In particular, (1.Sa) shows that the states in question
case of the particle moving with light velocity, the are invariant under rotations of the coordinate system,
aforementioned states become invariant under all (1.9a) that they are invariant with respect to Lorentz
Lorentz transformations. transformations with a velocity not too' high in the
Let us first convince ourselves that' if the state direction of motion (so that 1"'+ 1">>0, i.e., 1'" not too
(1.1) is viewed from a coordinate system moving in large a negative number). Finally, in order to prove
the - y direction, its spin and velocity no longer appear (1.9b), we calculate the transition probability between
parallel. The state in question is generated from the tbe states A (!r,I"')· T(O,I")>{;o and T({}"P")>{;o where
normal state by the transformation t'J and 1"" are given by (l.8a) and (l.8b). For this,
(1.8) gives
A <tr,<P')A (0,'1')
(A (!r, 1"') . T(O,I")>{;o, T({},I"")>{;o)
Il °
cOSb<P' sinh<p sinhl'"
= cosh I" (1.6) = (T({},I"")R(.)-l,fo, T(IJ,I"")l/to)
Sinhl'" sinh<p coshl'" = (R(o)-l,fo,./to)-+(l/to,y,o).
PERSPECfIVE VIEW OF QUANTUM SPACE-TIME SYMMETRIES 13

R E L A T I V 1ST I C I N V A R I A N C E AND QUA N TUM PH E NOM EN A 265

The second line follows because T(rJ,rp") represents a The three operations I, T, C, together with their
coordinate transformation and is, therefore, unitary. products TC (Liiders' time inversion of the second
The last member follows because .-.0 as <p->oo as kind), IC, IT, ITC and the unit operation form a
can be seen from (1.7) and R(O) = 1. group and the products of the elements of this group
The preceding consideration is not fundamentally with those of the proper Poincare group were considered
new. It is an elaboration of the facts (a) that the to be the symmetry operations of all laws of physics.
subgroup of the Lorentz group which leaves a null-vector The suggestion given in the text amounts to eliminating
invariant is different from the subgroup which leaves the operations I and C separately while continuing to
a time like vector invariant' and (b) that the representa- postulate their product IC as symmetry operation.
tions of the latter subgroup decompose into one The discrete symmetry group then reduces to the unit
dimensional representations if this subgroup is "con- operation plus
tracted" into the subgroup which leaves a null-vector IC, T, and ICT, (2.1)
invariant."
and the total symmetry group of the laws of physics
APPENDIXD becomes the proper Poincare group plus its products
Before the hypothesis of Lee and Yang" was put with the elements (2.1). This group is isomorphic
forward, it was commonly assumed that there are, in (essentially identical) with the unrestricted Poincare
addition to the symmetry operations of the proper group, i.e., the product of all Lorentz transformations
Poincare group, three further independent symmetry with all the displacements in space and time. The
operations. The proper Poincare group consists of all quantum mechanical expressions for the operations of
Lorentz transformations which can be continuously the proper Lorentz gl'oup and its product with IC are
obtained from unity and all translations in space-like unitary, those for T and ICT (as well as for their
and time-like directions, as well as the products of all products with the elements of the proper Poincare
these transformations. It is a continuous group j the group) antiunitary. Liiders" has pointed out that,
Lorentz transformations contained in it do not change under certain very natural conditions, ICT belongs to
the direction of the time axis and their determinant is the symmetry group of every local field theory.
1. The three independent further operations which were
considered to be rigorously valid, were APPENDIX m
Space inversion I, that is, the transformation Let us consider, first, the collision of two particles
x, y, z->- x, - y, - 3, without changing particles into of equal mass m in the coordinate system in which the
antiparticles. average of the sum of their momenta is zero. Let us
Time inversion T, more appropriately described assume that, at a given time, the wave function of
by Liiders" as Umkehr der Bewegungsrichtung, which both particles is confined to a distance I in the direction
replaces every velocity by the opposite velocity of their average velocity with respect to each other. If
so that the position of the particles at +1 becomes we consider only this space-like direction, and the time
the same as it was, without time inversion, at -/. axis, the area in space-time in which the two wave
The time inversion T (also called time inversion of functions will substantially overlap is [see Fig. 6(a)]
the first kind by Liiders' ·) does not convert particles
into antiparticles either. (3.1)
Charge conjugation C, that is, the replacement of
positive charges by negative charges and more where Vmi. is the lowest velocity which occurs with
generally of particles by antiparticles, without chang- substantial probability in the wave packets of the
ing either the position or the velocity of these par- colliding particles. Denoting the average momentum
ticles'" The quantum-mechanical expressions for the by ji (this has the same value for both particles) the
symmetry operations I and C are unitary, that for T half-width of the momentum distribution by a, then
is antiunitary. Vml.= (ji-a)(m'+(ji-a)'/c2)-I. Since I cannot be below
k/a, the area (3.1) is at least
u E. Inonu and E. P. Wigner, Proc. Nat!. Acad. Sci. U.S·
39,510 (1953). . h' (m'+(ji-a)'/c')1
.. T. D. Lee and C. N. Yang, Phys. Rev. 104, 254 (1956). See (3.1a)
also E. M. Purcell and N. F. Ramsey, Phys. Rev. 78, 807 (1950).
.. G. Lilders, Z. Physik 133,325 (1952). U' ji-a
II G. Lilders, Kg!. Danske Videnskab. Selskab MatAys. Medd.
28, No.5 (1954). (Note that the area becomes infinite if 6>ji.) The
"All three symmetry operations were first discussed in detail
by J. Schwinger, Phys. Rev. 74, 1439 (I94K). See also H. A. the present writer, Z. Physik 43, 624 (1927) and Nachr. Akad.
Kramers, Proc. Acad. Sci. Amsterdam 40, 814 (1937) and W. Pauli'. Wi... Gallingen, Math.-physik. 1932,546. See also T. D. Newton
article in NiJs Bolw aoul 'M v...Iop"..,., of Physia (Pergamon and E. P. Wigner, Revs. Modem Phy •. 21, 400 (1949); S.
Press, London, 1955). The significance of the tint two symmetry Watanabe, Revs. Modern Phys. 27, 26 (1945). The concept of
operations (and their connection with the concepts of parity and charge conjugation is based on the observation of W. Furry, Phys.
the Kramers degeneracy respectively), were first pointed out by Rev. 51, 125 (1937).
14 CHAPTER I

266 EUGENE P. WIGNER

,,
L x

\
,,
,, I

"
I

(b)

FIG. 6. (al Localization of a collision of two particles of equal mass. The full lines indicate the effective boundaries
of the wave packet of the particle traveling to the right, the broken lines the effective boundaries of the wave packet
of the particle traveling to the left. The collision can take place in the shaded area of space-time. (bl Localization
of a collision between a particle with finite mass and a particle with zero rest-mass. The full lines. at a distance
A apart in the % direction, indicate the boundary of the particle with uro rest-mass, the broken lines apply to the
wave packet of the particle with nonzero rest-mass. The collision can take place in the shaded area.

minimum of (3.1a) is, apart from a numerical factor ties. Hence p'" hiX. The kinetic energy of the particle
with finite restmass will be of the order of magnitude
t' h'c
amln"'-(m'+P'Ic')I", ,(3.2) !{m'C<+ (p+hll)'c')I+Hm'C<+ (p-hll)'c')I_ mc', (3.4)
p' EI(E+mc')1
since hll is the momentum uncertainty. Since I ~ >.,
where E is the kinetic energy (total energy minus one can neglect p in (3.4) if one is interested only in the
rest-energy) of the particles. order of magnitude. This gives for the total kinetic
The kinetic energy E permits the contraction of the energy,
wave functions of the colliding particles also in direc- E",hc/;"+ (m'C<+h'c'/l')I_mc', (3.5)
tions perpendicular to the average relative velocity, while the area in Fig. 6(b) is of the order of magnitude
to an area h'c'IE(E+2mc'). Hence, again apart from a
numerical factor, the volume to which the collision a= (Xlc) (l+<lvXlc), (3.6)
can be confined in four dimensional space-time becomes where <lv is the uncertainty in the velocity of the second
particle
v mio = - - - - - (3.3)
pHil ji-h/I
EI(E+mc')! ------. (3.6a)
(m'+(p+hll)'lc')! (m'+cp-hll)'Ic')1
E is the average kinetic energy of the particles in the
coordinate system i.n which their center of mass is, This can again be replaced by (hll)(m'+h'/l'c')-I.
on the average, at rest. Equation (3.3) is valid apart For given E, the minimum value of a is assumed if the
from a numerical constant of unit order of magnitude kinelic energies of the two particles are of the same
but this constant depends on Elmc'. order of magnitude. The two terms of (3.6) then become
Let us consider now the opposite limiting case, about equal and 1/>.", (EI(m+E»)I. The -minimum
the collision of a particle with finite rest-mass m with a value of a, as far as order of magnitude is concerned,
particle with zero rest-mass. The collision is viewed is again given by (3.2). Similarly, (3.3) also remains
again in the coordinate system in which the average valid if one of the two particles has zero rest-mass.
linear momentum is zero. In this case, one will wish to The two-dimensional case becomes simplest if both
confine the wave function of the particle with finite particles have zero rest-mass. In this case the wave
rest-mass to a narrower region I than that of the particle packets do not spread at all and (3.2) can be im-
with zero rest-mass. If the latter is confined to a region mediately seen to be valid. In the four-dimensional case,
of thickness >., [see Fig. 6(b)], its momentum and (3.3) again holds. However, its proof by means of
energy uncertainties will be at least hlX and helX explicitly constructed wave packets (rather than
and these expressions will also give, apart from a reference to the uncertainty relations) is by no means
numerical factor, the average values of these quanti- simple. It requires wave packets which are confined in
PERSPECflVE VIEW OF QUANTUM SPACE-TIME SYMMETRIES 15

RELA T IV I STI C I NVAR lANCE AN D QUANTU M PHENOMENA 267

every direction, do not spread too fast and progress

essentially only into one half space (one particle going
toward the right, the other toward the left). The World II.,.
construction of such wave packets will not be given in of mirror
detail. They are necessary to prove (3.2) and (3.3)
more rigorously also in the case of finite masses; the 4
preceding proofs, based on the uncertainty relations World liN
I 1of clock
show only that a and ~ cannot be smaller than the right I I
sides of the corresponding equations. It is clear, in fact, I I
that the limits given by (3.2) and (3.3) would be very : I
difficult to realize, except in the two-dimensional case __ "~ I 1
and for the collision of two particles with zero restmass. I
t--re------l I
In all other cases, the relatively low values of ami. t---,.----J
and V mi. are predicated on the assumption that the I '4----......J1
wave packets of the colliding particles are so constituted FIG. 7. Analysis of the experiment of Fig. S. The figure repreoeots
that they assume a minimum size at the time of the a view of the hyperboloid of deSitter space, viewed along its
axis. Every point of the rlane which is outside the circle COm>-
collision. At any rate, (3.2) and (3.3) show that only sponds to two points 0 the deSitter world with one spatial
collisions with a relatively high collision energy, and dimension, those with oppositely equal times. The first light
high energy uncertainty, can be closely localized in signal is emitted at I, reaches the mirror at I', and returns to
the clock at 2. The paths of the second and third light sigoals
space-time. are 22'3 and 33'4.
APPENDIX IV
polar angles" "'. The metric form in terms of these is
Let us denote the components of the vector from
event 1 to event 2 by Xi, the components of the unit a'
vector along the world line of the first clock at event 1 ds' = ---I1r- rtW. (5.2)
by ei. The components of the first light signal are r-a'
xi+lei, that of the second light signal Xi-I' ei. Hence
Two points of deSitter space correspond to every pair"
gik(xi+lei)(x.+te.) =0 (4.1) '" (except ,=a): those with positive and negative
T= (r-a') I. This will not lead to any confusion as all
gi·(xi-l'ei)(x.-I'e.)=O. (4.2)
events take place at positive T. The null lines (paths of
Elimination of the linear terms in 1 and I' by multiplica- light signals) are the tangents to the r= a circle.
tion of (4.1) with I' and (4.2) with 1 and addition gives The experiment described in the text can be analyzed
by means of Fig. 7. For the sake of simplicity, the
2gi·XiX.+ 211' gikeie.= O. (4.3) clock and mirror are assumed to be "at rest," i.e.,
their world lines have constant polar angles which
Since e is a unit vector g'·eie.= 1 and (4.3) shows that will be assumed as 0 and 6, respectively. The first light
the space-like distance between points 1 and 2 is (11')1. signal travels from 1 to l' and back to 2, the second
from 2 to 2' and back to 3, the third from 3 to 3' and
APPENDIX V back to 4. The polar angle of the radius vector which
is perpendicular to the first part 22' of the world line
Since the measurement of the curvature, described
of the second light signal is denoted by"'•. The con-
in the text, presupposes cons/ani cuna/we over the
struction of Fig. 7 shows that angle q,.' which the world
space-time domain in which the measurement takes line of the mirror includes with the radius vector
place, we use a space with constant curvature, or, rather, perpendicular to the second part 2'3 of the second
part of a space with constant curvature, to carry out light signal's world line is
the calculation. We consider only one spatial dimension,
i.e., a two-dimensional deSitter space. This will be (5.3)
embedded, in the usual way, in a thr~-dimensional
space'8 with coordinates x, y, T. The points of the
The angles "'I, "'t'. "'" ",.'
have similar meanings; they
are not indicated in the figure in order to avoid over-
deSitter space then form the hyperboloid crowding. For reasons similar to those leading to
(5.3), we have
r+y'-r'=a', (5.1)

where a is the "radius of the universe." As coordinates ""=""+6=",.+2a (5.3a)

of a point we use x and y, or rather the corresponding ""=,,,.-26 (5.3b)
"See, e.g., H. P. Robertson, Revs. Modem Phys. 5,62 (1933). ""=",.+26=,,,.+40. (5.3c)
16 CHAPTER!

268 EUGENE P. WIGNER

The radial coordinates of the points I, 2, 3, 4 are Similar expressions apply for the traveling times of the
denoted by rl, r" r., r. first and third light signals; all </> can be expressed by
r,=a/co&/l,. (5.4) means of (5.3a), (5.3b), (5.3c) in terms of </>. and 8.
This allows the calculation of the expression (3). For
Tbe proper time I, registered by the clock, can be small a, one obtains
obtained by integrating tbe metric form (5.2) along
the world line </>=0 of the clock
I=a In[r+(r-a,)tJ. (5.5) ---"'-, (5.7)
I.' a
Hence, the traveling time I, of tbe second light signal
becomes and Riemann's invariant R=2/a' is proportional to
r.+ (r,'-a') I co&/l.(l +sin</>.) the square of (5.7). In particular, it vanishes if the
I,=a In a In . (5.6)
r.+(r.·-a')t coSr/>.(l+sin</>.) expression (3) is zero.
PERSPECTIVE VIEW OF QUANTUM SPACE-TIME SYMMETRIES 17

P. A. M. Dirac

THE EARLY YEARS OF

RELATIVITY

I AM VERY HAPPY to have this opportunity of talking at the Ein-

stein Symposium because I am a very great admirer of Einstein.
In this Symposium we have heard many historical talks. His-
torians collect all the documents they can find referring to their
subject, assess those documents, collate them, and then give us
a detailed account of what happened. I am not going to give you
a talk of that nature, because I am not a historian. What I would
like to talk to you about is the arrival of relativity as it appeared
to someone who lived through it.
I was in England at the time, and for the most part I shall be
talking about the coming of relativity to England, although I think
it happened very much the same in other Western countries. What
I want to emphasize is the tremendous impact that relativity had
on the general public. I think that the historians have not em-
phasized this sufficiently in their talks.
First, I must describe the background. The time I am speaking
of is the end of the First World War. That war had been long and
terrible. It had been a war in which not very much had happened,
from the military point of view. We had battle lines that remained
almost static year after year, but we also had tremendous cas-
ualties, ever increasing casualties, and nearly everyone had lost
close relatives or friends. Then the end of this war came, rather
suddenly and unexpectedly, in November 1918. There was im-
mediately an intense feeling of relaxation. It was something dread-
ful that was now finished. People wanted to get away from think-
ing about the awful war that had passed. They wanted something
new. And that is when relativity burst upon us.

79

Reprinted from Albert Einstein: Historical and Cultural Perspectives: The Centennial
Symposium in Jerusalem, (1979).
18 CHAPTER!

P. A. M. DIRAC

I can't describe it by other words than by saying that it just

burst upon us. It was a new idea, a new kind of philosophy, and
it aroused interest and excitement in everyone. The newspapers,
as well as the magazines, both popular and technical, were con-
tinually carrying articles about it. These articles were mainly
written from the "philosophical" point of view. Everything had
to be considered relatively to something else. Absolutism was
just a bad idea that one had to get away from. There was no public
radio in those days, so we did not have that kind of propaganda;
but all the written material that we had was devoted to bringing
out this new idea of relativity. In most of what people wrote there
was no real physics involved. The people writing the articles did
not understand the physics.
At that time I was sixteen years old and a student of engineering
at Bristol University. You might think it was rather unusualfor
someone only sixteen years old to be a student, but it was not
unusual when you consider the times. All the young men had
been taken away from the universities to serve in the army. There
were some professors left, those who were too old to serve in the
army and those who were not physically fit; but they had empty
classrooms. So the younger boys were pushed on, as far as they
were able to absorb the knowledge, to fill up these empty class-
rooms. That is how I came to be an engineering student at the
time.
I was caught up in the excitement of relativity along with my
fellow students. We were studying engineering, and all our work
was based on Newton. We had absolute faith in Newton, and now
we learned that Newton was wrong in some mysterious way. This
was a very puzzling situation. Our professors were not able to
help us, because no one really had the precise information needed
to explain things properly, except for one man, Arthur Eddington.
Eddington was an astronomer. He had kept in touch with his
friend Willem de Sitter, an astronomer in Holland. Holland was
neutral at that time, so they could write to each other. De Sitter
had kept in contact with Einstein, and in that roundabout way
Eddington was in contact with Einstein and had heard all about
the development of relativity theory.
At the end of 1918, the. end of the war, relativity was not really
a new idea. It had dated from 1905, a good many years before the
war started. But no one had heard about relativity or about Ein-
stein previously, except for a few specialists at the universities.
No one in the engineering faculty at Bristol had heard anything

80
PERSPECTIVE VIEW OF QUANTUM SPACE-TIME SYMME1RIES 19

EARLY YEARS OF RELATIVITY

about these things, and it was all completely new to us. Then we
had Eddington to explain things.
Eddington was very good at popular exposition. He had a great
talent for it, and he applied his talent to explaining the founda-
tions of relativity theory to the general public. He told us about
the need to consider that we cannot communicate instanta-
neously with people at a distance. We could communicate only
with the help of light signals and make a note of the time the
light signals were sent out and the time we received an answer,
and we had to work on that. Eddington also told us about the
Michelson-Morley experiment. It was quite an old experiment,
but one whose importance had not been previously appreciated.
Alben A. Michelson and Edward W. Morley had attempted to
determine the velocity of the earth through the ether, but, sur-
prisingly, their experiment failed to give any definite answer. The
only way to account for this failure was to suppose a very peculiar
behavior of measuring rods and clocks. Moving measuring rods
had to be subject to a kind of contraction, called the Lorentz-
Fitzgerald contraction. Clocks had to have their rates slowed up
when they were moving.
All this was very hard to explain to the general public, but still
Eddington and other people wrote innumerable articles about it,
doing the best they could. The engineering students were not very
much better off. We were told that in some way the absolute
scheme of things that we had been using in our engineering stud-
ies had to be modified, but there was no very definite way to make
the modification. We were not given any definite equations.
Now, Eddington was an astronomer and was very interested in
the general theory of relativity, and especially in testing its as-
tronomical consequences. The theory predicts a motion of the
perihelion of Mercury. It was easy to work this out and to check
that the observations agreed with the Einstein theory. This was
the first big triumph of the Einstein theory.
Then there was the question of observing whether light would
be deflected when it passed close by the sun, another requirement
of the Einstein theory. This is something that can be tested ob-
servationally only at a time of total eclipse. Eddington had heard
about these things already in 1916, and he immediately set to
work to find out if there was a favorable total eclipse coming up
soon. He found that there would be a very favorable one occurring
in May 1919-very favorable because the time of totality was long
and also because the sun was then in a very rich field of stars, so

81
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P. A. M. DIRAC

that there would be many stars to observe. Eddington immediately

set to work to plan expeditions to make the needed observations.
Of course, he knew very well that there would be no hope of
making such observations while the war was on, but he was
hoping that the war would be over in time. The war was over in
November 1918, which was in time, and the eclipse expeditions
were sent out. They brought back their photographic plates and
measured them up!; and they got results in agreement with the
Einstein theory. Then the public really went wild. The new theory
was proved, and everyone was so excited about it.
I shall mention just one example to illustrate the sort of en-
thusiasm that prevailed. In a detective story called The Bishop
Murder Case, an important clue was provided by a piece of paper
on which some of the Einstein equations were written down. The
theory of relativity was woven into the plot of the story, with the
result that the book had a big sale. All the young people were
reading it. That shows you the tremendous excitement that per-
vaded all fields of thought. It has never happened before or since
in the history of science that a scientific idea has been so much
caught up by the public and has produced so much enthusiasm
and excitement.
Yet with all that was written about relativity, we still had very
little accurate information. Professor Broad at Bristol gave a series
of lectures about relativity, which I attended, but they dealt mostly
with philosophical aspects. He did, however, give some infor-
mation about the geometry of special relativity, in particular about
Minkowsky's space, and I began to get some definite information
about the theory from him.
We really had no chance to understand relativity properly until
1923, w4en Eddington published his book, The Mathematical
Theory of Relativity, which contained all the information needed
for a proper understanding of the basis of the theory. This math-
ematical information was interspersed with a lot of philosophy.
Eddington had his own philosophical views, which, I believe, were
somewhat different from Einstein's, but developed from them.
But there it was, and it was possible for people who had a
knowledge of the calculus, people such as engineering students,
to check the work and study it in detail. The going was pretty
tough. It was a harder kind of mathematics than we had been
used to in our engineering training, but still it was possible to
master the theory. That was how I got to know about relativity
in an accurate way.

82
PERSPECTIVE VIEW OF QUANTUM SPACE-TIME SYMMETRIES 21

EARLY YEARS OF RELATIVITY

Eddington's book did not give any information about the strug-
gles that Einstein had gone through in order to set up his theory.
It just gave us the completed result. I have been very interested
in the lectures given by historians at the various Einstein sym-
posia, which enable me to understand better both Einstein's strug-
gles and also his appreciation of the need for beauty in the math-
ematical foundation. Einstein seemed to feel that beauty in the
mathematical foundation was more important, in a very funda-
mental way, than getting agreement with observation.
This was brought out very clearly in the early work about the
theory of the electron. Hendrik Lorentz had set up a theory for
the motion of electrons that was in agreement with Einstein's
principles, and experiments were made by Walter Kaufmann to
see whether this theory was in agreement with observation. The
resulting observations did not support the theory of Lorentz and
Einstein. Instead, they supported an older theory of the electron,
given by Max Abraham. Lorentz was completely knocked out by
this result. He bewailed that all his work had gone for nothing.
Einstein seems not to have reacted very much to it. I do not know
just what he said-that's a question for the historians to decide.
I imagine that he said, "Well, I have this beautiful theory, and
I'm not going to give it up, whatever the experimenters find; let
us just wait and see."
Well, Einstein proved to be right. Three years later the exper-
iments were done again by someone else, and the new experiment
supported the Lorentz-Einstein view of the electron. And some
years after that, a fault was found in the apparatus of Kaufmann.
So it seems that one is very well justified in attaching more im-
portance to the beauty of a theory and not allowing oneself to be
too much disturbed by experimenters, who might very well be
using faulty apparatus.
Let us return to the general theory of relativity. The observa-
tions of the eclipse expeditions supported the theory. Right from
the beginning there was agreement. We then had a satisfactory
basis for the development of relativity. At that time I was a re-
search student and very much enjoyed the new field of work that
was opened up by relativity. One could take some previous piece
of work that had been expressed in nonrelativ,istic language and
tum it into the relativistic formalism, get a better understanding
of it, and perhaps find enough material to publish a paper.
It was about that time, I think in 1924, that A. H. Compton
visited Cambridge and spoke at the Cavendish Laboratory about

83
22 CHAPTER!

P. A. M. DIRAC

his experiments on the Compton effect. These experiments in-

volved both the light quantum hypothesis and some of the fun-
damental relations of special relativity. I can remember very well
this colloquium. I found Compton's results very convincing, and
I think most of the members of the audience- were also convinced
by Compton, although there were a few people who held out
against the light quantum hypothesis.
The situation was completely changed in 1925, when Werner
Heisenberg introduced his new quantum mechanics. This was a
theory in which it was soon found out that the basic idea was to
have dynamic variables that do not satisfy the commutative law
of multiplication. That is to say, a x b is not equal to b x a.
Quite a revolutionary idea, but an idea that rapidly proved to have
great success.
Now, in spite of that success, Einstein was always rather hostile
to quantum mechanics. How can one understand this? I think it
is very easy to understand, because Einstein had been proceeding
on different lines, lines of pure geometry. He had been developing
geometrical theories and had achieved enormous success. It is
only natural that he should think that further problems of physics
should be solved by further developments of geometrical ideas.
Now, to have a x b not equal to b x a is something that does
not fit in very well with geometrical ideas; hence his hostility to
it.
I first met Einstein at the 1927 Solvay Conference. This was
the beginning of the big discussion between Niels Bohr and Ein-
stein, which centered on the interpretation of quantum mechan-
ics. Bohr, backed up by a good many other physicists, insisted
that one can use only a statistical interpretation, getting proba-
bilities from the theory and then comparing these probabilities
with observation. Einstein insisted that nature does not work in
this way, that there should be some underlying determinism. Who
was right?
At the present time, one must say that, according to Heisen-
berg's quantum mechanics, we must accept the Bohr interpre-
tation. Any student who is working for an exam must adopt this
interpretation if he is to be successful in his exams. Once he has
passed his exams, he may think more freely about it, and then
he may be inclined to feel the force of Einstein's argument.
In this discussion at the Solvay Conference between Einstein
and Bohr, I did not take much part. I listened to their arguments,
but I did not join in them, essentially because I was not very

84
PERSPECTIVE VIEW OF QUANTUM SPACE-TIME SYMMETRIES 23

EARLY YEARS OF RELATIVITY

much interested. I was more interested in getting the correct

equations. It seemed to me that the foundation of the work of a
mathematical physicist is to get the correct equations, that the
interpretation of those equations was only of secondary impor-
tance.
Right from the beginning of quantum mechanics, I was very
much concerned with the problem of fitting it in with relativity.
This turned out to be very difficult, except in the case of a single
particle, where it was possible to make some progress. One could
find equations for describing a single particle in accordance with
quantum mechanics, in agreement with the principle of special
relativity. It turned out that this theory provided an explanation
of the spin of the electron.
Also, one could develop the theory a litde further and get to
the idea of antimatter. The idea of antimatter really follows di-
recdy from Einstein's special theory of relativity when it is com-
bined with the quantum mechanics of Heisenberg. There is no
escape from it. With just a single, uniform line of argument one
goes right up from special relativity to antimatter.
This was all very satisfactory so long as one considered only a
single particle. There remained, of course, the problem of two or
more particles interacting with each other. Then one soon found
that there were serious difficulties. Applying the standard rules,
all one could say was that the theory did not work. The theory
allowed one to set up definite equations. When one tried to in-
terpret those equations, one found that certain quantities were
infinite according to the theory, when according to common sense
they should be finite. That was a very serious difficulty in the
theory, a difficulty that still has not been completely resolved.
Physicists have been very clever in finding ways of turning a
blind eye to terms they prefer not to see in an equation. They
may ·go on to get useful results, but this procedure is gf course
very far from the way in which Einstein thought that nature
should work.
It seems clear that the present quantum mechanics is not in
its final form. Some further changes will be needed, just about as
drastic as the changes made in passing from Bohr's orbit theory
to quantum mechanics. Some day a new quantlJlll mechanics, a
relativistic one, will be discovered, in which we will not have
these infinities occurring at all. It might very well be that the
new quantum mechanics will have determinism in the way that
Einstein wanted. This determinism will be introduced only at the

85
24 CHAPTER!

P. A. M. DIRAC

expense of abandoning some other preconceptions that physicists

now hold. So, under these conditions I think it is very likely, or
at any rate quite possible, that in the long run Einstein will turn
out to be correct, even though for the time being physicists have
to accept the Bo1;u probability interpretation, especially if they
have examinations in front of them.
There are two other subjects that I would like to talk about.
The first concerns an incompleteness in the Einstein theory of
gravitation. This Einstein theory gives us field equations, but one
cannot solve them and obtain precise results without having some
boundary conditions, that is, some conditions that one can use
to refer to what space is like at,a veri great distance. To establish
the boundary conditions, we. need to have a theory of cosmology,
a theory that tells us what the universe as a whole looks like
when we smooth out the irregularities caused by stars and gal-
axies, and so on. Einstein himself realized from the beginning the
need for such boundary conditions, and he proposed a model of
the universe, a model in which space is limited, although un-
bounded. That was a new idea for people to get used to, one on
which Eddington brought to bear his remarkable powers of ex-
plaining to try to get it across to the general public. The idea is
not that difficult to comprehend; you can think of the surface of
the earth, a region of space in two dimensions that is finite but
still unbounded.
Einstein set up a cylindrical model of the universe, but it was
soon found that it had to be abandoned because of its static char-
acter. Observations showed us that things that are very far away
are receding from u& with velocities that increase as the distance
increases. Einstein's model did not account for that phenomenon.
Therefore, it had to be abandoned.
Soon afterward, another model was proposed by de Sitter. De
Sitter's model did give correctly the distant matter receding from
us, but it gave zero for the average density of matter, which of
course is in disagreement with observation. Therefore, this model
also had to be abandoned.
Many other models were worked out by Alexander Friedman,
Georges Lemaitre, and others, all consistent with Einstein's basic
field equations. Among all these other models, I want to call your
at~ntion to one that was brought out jointly by Einstein and de
Sitter. This model was satisfactory in that it gives distant matter
receding from us and also gives a non-zero average density of

86
PERSPECTIVE VIEW OF QUANTUM SPACE-TIME SYMMETRIES 25

EARLY YEARS OF RELATIVITY

matter about in agreement with observation. In all its basic fea-

tures it appears to be satisfactory.
This Einstein-<ie Sitter joint model I would like to propose as
the one that should be generally accepted, because it is the sim-
plest of all the models that are not in some elementary way in
disagreement with the facts. We ought to keep to the simplest
model until something turns up that causes us to depart from it.
The Einstein-<ie Sitter model proposes that the universe started
with a big bang and will go on expanding forever. Many models
claim that the universe will expand to a certain point and then
contract again. This is an unneeded complication, and there is no
observational evidence supporting it at the present. I feel that
such models should not be taken as seriously as the simple Ein-
stein-<ie Sitter model. The latter has the L:haracter that three-
dimensional space is flat. It also has the character that the pr~ssure
in the smoothed-out universe is very small.
I should mention that all these different models will have their
influence on the solutions of the Einstein equations we need in
order to account for phenomena occurring in the solar system.
But the differences are exceedingly small. They would not affect
the wonderful agreement that we have between observation and
Einstein's theory in the description of the solar system.
There is one other topic that I would like to talk about: the
problem of the unification of the gravitational field and the elec-
tromagnetic field. Einstein was very much concerned with this
unification. Those were the only two fields that he had to con-
sider. Nowadays, physicists have other fields, but these other
fields all involve short-range forces, forces that are significant only
for particles extremely close together, lying inside an atomic nu-
cleus. The gravitational force and the electromagnetic force are
long-range forces. These forces falloff inversely proportional to
the square of the distance. In some ways, they are more important,
I think, than the other forces. One feels that there should be a
close connection between them.
Einstein had achieved his very great success in accounting for
the gravitational force in terms of geometry, and he thought that
some generalization of the geometry would bring in the electro-
magnetic force also. A method of doing this was very soon dis-
covered by Hermann Weyl. Weyl made a rather simple general-
ization in the geometry of Riemann, which Einstein was using.
He supposed that the distance between two neighboring points
does not have an absolute value. There is no natural absolute unit

87
26 CHAPTER!

P. A. M. DIRAC

to which one could refer it. But one can transport it hom one
location in space to another, and the equations that govern this
transport are such that, if you go around a closed loop and get
back to your starting point, the final distance does not agree with
the starting value. That led to a generalization of the geometry,
which was soon found to provide just what was needed in order
to bring in the electromagnetic field.
This seemed to;be a very wonderful solution of the problem,
but then there w~ a difficulty. Atomic events do provide a natural
scale for measuring distances. You could refer all yow distances
to this atomic scale, and then there would be no poipt at all in
having the uncertainty in distances introduced by Weyl's geom-
etry. So this very beautiful theory of Weyl was reluctantly aban-
doned.
Einstein worked for the rest of his life on trying to solve ~s
problem of unifying the gravitational and the electromagnetic
fields. He tried one scheme after another. All were unsatisfactory.
Other people have joined in this work without achieving any
greater success than Einstein did. I have been wondering whether
Einstein was limiting his ideas too much in these attempts to
unify the gravitational and the electromagnetic fields. It seems
to me that it is quite possible that one will have to bring in
cosmological effects to arrive at a satisfactory solution of this
problem. The way cosmological effects would show up would be
this.
Let us accept the Einstein theory as it stands for all problems
involving just classical theory, and only when we go over to atomic
problems, let us require that some modification is needed. This
modification can be expressed by saying that Planck's constant
is not really a constant in the cosmological sense but must be
considered as varying with the epoch, that is, with the time since
the origin of the universe, the "Big Bang." If we have Planck's
constant h varying, then we must also have the charge on the
electron e varying, because e21hc is a dimensionless constant that
plays an important role in physics and is observed to have the
value 11137, and it seems to be really constant. Thus, if h is
varying, e must vary according to its square root. On this basis
one could set up a new theory.
Before introducing such a drastic revision in basic ideas, it is
desirable to have some confirmation of it by observation. If you
have these atomic constants varying when referred to the Einstein

88
PERSPECTIVE VIEW OF QUANTUM SPACE-TIME SYMMETRIES 27

EARLY YEARS OF RELATIVITY

picture, it will mean that atomic clocks will not keep the same
time as the time of the Einstein theory. The time of the Einstem
theory is the time that governs the motion of the planets around
the sun, what astronomers call ephemeris time. So one could look
to see whether there is any difference between atomic clocks and
clocks based on ephemeris time.
Astronomers have been studying this question. In particular,
T. C. Van Flandern, working at the Naval Research Observatory
in Washington, has spent many years studying lunar motion, re-
ferred to both ephemeris time and atomic time. Lunar motion
has been observed with atomic clocks since 1955. However, Van
Flandern's results up to the present are still not conclusive. One
must wait a little longer to see whether a difference really exists
between the two ways of measuring time.
There is another possibility of checking these new ideas about
variation of constants that has been followed up by I. I. Shapiro.
His method consists in sending radar signals to one of the planets
and observing the reflected radar signals, and then timing the to
and fro journey with an atomic clock. In effect, he is observing
distances in the solar system with atomic clocks, and there should
be some discrepancy showing up if the atomic clocks do not keep
the same time as ephemeris time.
Lwas'talking to Shapiro just before I came to this Symposium
and asked him about the latest information concerning his work.
There is a very good chance of observing radar waves reflected
from Mars because of the Viking Lander that landed on Mars in
1976, which can be used to send back reflected radar waves. Sha-
piro said that the time base that he has for the observations with
the Viking expedition (just about two years) is not long enough
for him to give a definite answer to this question. There were also
observations of Mars made previously with a Mariner expedition,
which give him a considerably longer time base, eight years in-
stead of two. But he told me that another year or two of work
would be needed to evaluate his results before he could answer
the question of variation of the constants.
That is the situation at the present time. I am sorry that I cannot
offer anything more definite than that. There ate hopes that Sha-
piro will come out with a definite result, and if it is a positive
result, we shall have a new basis for looking at the question of
the unification of the gravitational and the electromagnetic fields.
It might be possible to revive Weyl's geometry, which was aban-

89
28 CHAPTER I

P. A. M. DIRAC

doned only because it clashed with the distances provided by

atomic events. So it seems that if this approach turns out to be
correct, the whole theory of unifying these distances with grav-
itational theory would have to oe gone into again. There may very
well be greater chances of success.

90
Chapter II

Representations of the Poincare Group

This Chapter consists of the fundamental paper of Wigner on the Poincare group
and some of the resulting papers. In his original paper, Wigner constructed
subgroups of the Lorentz group whose transformations leave the four-momentum of
a given particle invariant. These subgroups are called the little groups. In their
1948 paper, Bargmann and Wigner formulated Wigner's little groups in terms of the
infinitesimal generators. The little groups for massive, massless, and imaginary-
mass particles are locally isomorphic to the three-dimensional rotation group, the
two-dimensional Euclidean group, and the (2 + I)-dimensional Lorentz group.
In his 1945 paper, Dirac suggested the four-dimensional harmonic oscillator to
construct representations of the Lorentz group. In their 1979 papers, Kim, Noz, and
Oh constructed a representation of the O(3)-like little group for massive hadrons in
the quark model.
Quantum field theory has its place in the development of relativistic quantum theory,
and its strength and weakness are well known. As is well known, the Poincare
group is the basic language for quantum field theory, and there are many papers
which will indicate this point. In one of his 1964 papers, Weinberg gives a lucid
treatment of the role of the representations of the Poincare group in Feynman
diagrams.
REPRESENTATIONS OF THE POINCARE GROUP 31

ON UNITARY REPRESENTATIONS OF THE

INHOMOGENEOUS LORENTZ GROUP

By E. Wigner

(Received December 22, 1937)

Parts of the present paper were presented at the Pittsburgh Symposium on

Group Theory and Quantum Mechanics. Cf. Bull. Amer. Math. Soc., 41,
p.306, 1935.

Reprinted from Ann. Math. 40, 149 (1939).

32 CHAPTER II

1. Origin and Characterization of the Problem

It is perhaps the most fundamental principle of Quantum Mechanics that

the system of states forms a linear manifold, 1 in which a unitary scalar
product is defined. 2 The states are generally represented by wave
functions 3 in such a way that <1> and constant multiples of <1> represent the
same physical state. It is possible, therefore, to normalize the wave
function, i.e., to multiply it by a constant factor such that its scalar product
with itself becomes 1. Then, only a constant factor of modulus 1, the
so-called phase, will be left undetermined in the wave function. The linear
character of the wave function is called the superposition principle. The
square of the modulus of the unitary scalar product ('1',<1» of two
normalized wave functions 'I' and <1> is called the transition probability from
the state 'I' into <1>, or conversely. This is supposed to give the probability
that an experiment performed on a system in the state <1>, to see whether or
not the state is '1', gives the result that it is '1'. If there are two or more
different experiments to decide this (e.g., essentially the same experiment,
performed at different times) they are all supposed to give the same result,

IThe possibility of a future non linear character of the quantum mechanics must be
admitted, of course. An indication in this direction is given by the theory of the positron, as
developed by PAM. Dirac (proc. Camb. Phil Soc. 30, 150, 1934, cf. also W. Heisenberg,
Zeits. f. Phys. 90, 209, 1934; 92, 623, 1934; W. Heisenberg and H. Euler, ibid. 98, 714,
1936 and R. Serber, Phys. Rev. 48, 49, 1935; 49, 545, 1936) which does not use wave
functions and is a non linear theory.

2Cf. PAM. Dirac, The Principles of Quantum Mechanics, Oxford 1935, Chapters I and
II; J. v. Neumann, Mathematische Grundlagen der Quantenmechanik, Berlin 1932, pages
19-24.

3The wave functions represent throughout this paper states in the sense of the
"Heisenberg picture," i.e. a single wave function represents the state for all past and
future. On the other hand, the operator which refers to a measurement at a certain time t
contains this t as a parameter. (Cf. e.g. Dirac, 1.c. ref. 2, pages 115-123). One obtains the
wave function <ps(t) of the Schroedinger picture from the wave function <PH of the
Heisenberg picture by <ps(t) = exp(-iHUij<pH • The operator of the Heisenberg picture is Q(t) =
exp(iHUijQexp(-iHUij, where Q is the operator in the Schroedinger picture which does not
depend on time. Cf also E. Schroedinger, Sitz. d. Koen. Preuss. Akad. p. 418, 1930.

The wave functions are complex quantities and the undetermined factors in them are
complex also. Recently attempts have been made toward a theory with real wave functions.
Cf. E. Majorana, Nuovo Cim. 14, 171, 1937 and P. A. M. Dirac, in print
REPRESENTATIONS OF THE POINCARE GROUP 33

i.e., the transition probability has an invariant physical sense.

The wave functions form a description of the physical state, not an

invariant however, since the same state will be described in different
coordinate systems by different wave functions. In order to put this into
evidence, we shall affix an index to our wave functions, denoting the
Lorentz frame of reference for which the wave function is given. Thus cI>[
and cl>l<' represent the same state, but they are different functions. The first
is the wave function of the state in the coordinate system I, the second in
the coordinate system 1'. If cI>[ = 'IIl'the state cI> behaves in the coordinate
system I exactly as 'II behaves in the coordinate system I'. If cI>[ is given, all
cl>1' are determined up to a constant factor. Because of the invariance of the
transition probability we have
(1)
and it can be shown4 that the aforementioned constants in the cI>[, can be
chosen in such a way that the cl>1' are obtained from the cI>[ by a linear
unitary operation, depending, of course, on I and I'.
cl>l' = D(l', l)cI>[. (2)

The unitary operators D are determined by the physical content of the

theory up to a constant factor again, which can depend on 1 and 1'. Apart
from this constant however, the operati6ns D(1', I) and D(I'I' II) must be
r
identical if arises from I by the same Lorentz transformation, by which
1'1 arises from II' If this were not true, there would be a real difference
between the frames of reference I and II' Thus the unitary operator D(I', l)
= D(L) is in every Lorentz invariant quantum mechanical theory (apart
from the constant factor which has no physical significance) completely
determined by the Lorentz transformation L which carries 1 into l' = Li.
One can write, instead of (2)
<l>LI = D(L)cI>/' (2a)
By going over from a first system of reference 1 to a second l' = Ll1 and
then to a third 1" = L2Lli or directly to the third l" = (L2L 1)1, one must
obtain-apart from the above mentioned constant-the same set of wave
functions. Hence from

4E. Wigner, Gruppentheorie und ihre Anwendungen auf die Quantenmechanik det
Atoms-pektren. Braunschweig 1931, pages 251-254.
34 CHAPTER II

D(l", l')D(l', 1)<\>[

<\>(, = D(I", 1)<\>1

it follows
D(l", 1')D(l', I) = roD(l", 1) (3)
or
(3a)
where co is a number of modulus 1 and can depend on L2 and L 1. Thus the
D(L) form, up to a factor, a representation of the inhomogeneous Lorentz
group by linear, unitary operators.

We see thusS that there corresponds to every invariant quantum

mechanical system of equations such a representation of the
inhomogeneous Lorentz group. This representation, on the other hand,
though not sufficient to replace the quantum mechanical equations
entirely, can replace them to a large extent. If we knew, e.g., the operator
K corresponding to the measurement of a physical quantity at the time t =
0, we could follow up the change of this quantity throughout time. In
order to obtain its value for the time t = t 1, we could transform the original
wave function <\>1 by D(1', 1) to a coordinate system l' the time scale of
which begins a time tl later. The measurement of the quantity in question
in this coordinate system for the time 0 is given-as in the original one-by
the operator K. This measurement is identical, however, with the
measurement of the quantity at time tl in the original system. One can say
that the representation can replace the equation of motion, it cannot
replace, however, connections holding between operators at one instant of
time.

It may be mentioned, finally, that these developments apply not only in

quantum mechanics, but also to all linear theories, e.g., the Maxwell
equations in empty space. The only difference is that there is no arbitrary
factor in the description and the co can be omitted in (3a) and one is led to
real representations instead of representations up to a factor. On the other
hand, the unitary character of the representation is not a consequence of
the basic assumptions.

SE. Wigner, l.c. Chapter XX.

REPRESENTATIONS OF TIlE POINCARE GROUP 35

The increase in generality, obtained by the present calculus, as compared

with the usual tensor theory, consists in that no assumptions regarding the
field nature of the underlying equations are necessary. Thus more general
equations, as far as they exist (e.g., in which the coordinate is quantized,
etc.) are also included in the present treatment. It must be realized,
however, that some assumptions concerning the continuity of space have
beer! made by assuming Lorentz frames of reference in the classical sense.
We should like to mention, on the other hand, that the previous remarks
concerning the time-parameter in the observables, have only an
explanatory character and we do not make assumptions of the kind that
measurements can be performed instantaneously.

We shall endeavor, in the ensuing sections, to determine all the continuous

unitary representations up to a factor of the inhomogeneous Lorentz group,
i.e., all continuous systems of linear, unitary operators satisfying (3a).6

2. Comparison With Previous Treatments

and Some Immediate Simplifications

A. Previous treatments

The representations of the Lorentz group have been investigated

repeatedly. The first investigation is due to Majorana,7 who in fact found
all representations of the class to be dealt with in the ~resent work
excepting two sets of representations. DiracS and Proca gave more
elegant derivations of Majorana's results and brought them into a form
which can be handled more easily. Klein's work9 does not endeavor to

tYrhe exact definition of the continuous character of a representation up to a factor will be

given in Section 5A. The definition of the inhomogeneous Lorentz group is contained in
Section 4A.

7E. Majorana, Nuovo Cim. 9, 335,1932.

sp. A. M. Dirac, Proc. Roy. Soc. A. 155, 447, 1936; Ai. Proca, 1. de Phys. Rad. 7, 347,
1936.

9K1ein, Arkiv f. Matern. Astr. och Fysik, 25A, No. 15, 1936. I am indebted to Mr.
Darling for an interesting conversation on this paper.
36 CHAPTER II

derive irreducible representations and seems to be in a less close

connection with the present work.

The difference between the present paper and that of Majorana and Dirac
lies-apart from the finding of new representations-mainly in its greater
mathematical rigor. Majorana and Dirac freely use the notion of
infinitesimal operators and a set of functions to all members of which
every infinitesimal operator can be applied. This procedure cannot be
mathematically justified at present, and no such assumption will be used in
the present paper. Also the conditions of reducibility and irreducibility
could be, in general, somewhat more complicated than assumed by
Majorana and Dirac. Finally, the previous treatments assume from the
outset that the space and time coordinates will be continuous variables of
the wave function in the usual way. This will not be done, of course, in the
present work.

B. Some immediate simplifications

Two representations are physically equivalent if there is a one to one

correspondence between the states of both which is 1. invariant under
Lorentz transformations and 2. of such a character that the transition
probabilities between corresponding states are the same.

It follows from the second conditionS that there either exists a unitary
operator S by which the wave functions <1>(2) of the second representation
can be obtained from the corresponding wave functions <1>(1) of the first
representation
<1>(2) = S<1>(1) (4)
or that this is true for the conjugate imaginary of <1>(2). Although, in the
latter case, the two representations are still equivalent physically, we shall,
in keeping with the mathematical convention, not call them equivalent.

The first condition now means that if the states <1>(1), <1>(2) = S<1>(I)
corres]:>Ond to each other in one coordinate system, the states n(1)(L)<1>(I)
and D(2)(L)<1>(2) correspond to each other also. We have then
D(2)(L)<1>(2) = SD(l)(L)<1>(I) = SD(1)(L)S-I<1>(2). (4a)

As this shall hold for every <1>(2), the existence of a unitary S which
REPRESENTATIONS OF TIlE POINCARE GROUP 37

transforms 0(1) into 0(2) is the condition for the equivalence of these two
representations. Equivalent representations are not considered to be really
different and it will be sufficient to find one sample from every infinite
class of equivalent representations.

If there is a closed linear manifold of states which is invariant under all

Lorentz transformations, i.e., which contains O(L)", if it contains "', the
linear manifold perpendicular to this one will be invariant also. In fact, if
<I> belongs to the second manifold, O(L)<I> will be, on account of the unitary
character of O(L) , perpendicular to O(L)'I" if 'I" belongs to the first
manifold. However, O(L-l)", belongs to the first manifold if", does and
thus O(L)<I> will be orthogonal to O(L)O(L-l)", = co", i.e. to all members of
the first manifold and belong itself to the second manifold also. The
original representation then "decomposes" into two representations,
corresponding to the two linear manifolds. It is clear that, conversely, one
can form a representation, by simply" adding" several other representations
together, i.e. by considering as states linear combinations of the states of
several representations and assume that the states which originate from
different representations are perpendicular to each other.

Representations which are equivalent to sums of already known

representations are not really new and, in order to master all
representations, it will be sufficient to determine those, out of which all
others can be obtained by "adding" a finite or infinite number of them
together.

Two simple theorems shall be mentioned here which will be proved later
(Sections 7A and 8C respectively). The first one refers to unitary
representations of any closed group, the second to irreducible unitary
representations of any (closed or open) group.

The representations of a closed group by unitary operators can be

transformed into the sum of unitary representations with matrices of finite
dimensions.

Given two non equivalent irreducible unitary representations of an

arbitrary group. If the scalar product between the wave functions is
invariant under the operations of the group, the wave functions
belonging23 to the first representation are orthogonal to all wave functions
belonging to the second representation.
38 CHAPTER II

C. Classification of unitary representations

according to von Neumann and MurraylO

Given the operators D(L) of a unitary representation, or a representation up

to a factor, one can consider the algebra of these operators, i.e. all linear
combinations
a1D (L 1) + a2D(L 2) + a3D(~) + '"
of the D(L) and all limits of such linear combinations which are bounded
operators. According to the properties of this representation algebra, three
classes of unitary representations can be distinguished.

The first class of irreducible representations has a representation algebra

which contains all bounded operators, i.e. if 'V and <\> are two arbitrary
states, there is an operator A of the representation algebra for which A'll =
<\> and A'If = 0 if 'If is orthogonal to'll. It is clear that the center of the
algebra contains only the unit operator and multiply thereof. In fact, if C is
in the center one can decompose C'I' = a'll + 'If so that 'If shall be
orthogonal to'll. However, 'If must vanish since otherwise C would not
commute with the operator which leaves 'v invariant and transforms every
function orthogonal to it into O. For similar reasons, a must be the same
for all'll. For irreducible representations there is no closed linear manifold
of states, (excepting the manifold of all states) which is invariant under all
Lorentz transformations. In fact, according to the above definition, a <\>'
arbitrarily close to any <\> can be represented by a finite linear combination
a1D (L 1)'I' + a2D(~)'I' + ... + a,p(Ln)'I'·
Hence, a closed linear invariant manifold contains every state if it contains
one. This is, in fact, the more customary definition for irreducible
representations and the one which will be used subsequently. It is well
known that all finite dimensional representations are sums of irreducible
representations. This is not true,lO in general, in an infinite number of
dimensions.

The second class of representations will be called factorial. For these, the
center of the representation algebra still contains only multiples of the unit
operator. Clearly, the irreducible representations are all factorial, but not
conversely. For finite dimensions, the factorial representations may

lOp. 1. Murray and 1. v. Neumann, Ann. of Math. 37, 116, 1936; J. v. Neumann, to be
published soon.
REPRESENTATIONS OF THE POINCARE GROUP 39

contain one irreducible representation several times. This is also possible

in an infinite number of dimensions, but in addition to this, there are the
"continuous" representations of Murray and von Neumann.1° These are
not irreducible as there are invariant linear manifolds of states. On the
other hand, it is impossible to carry the decomposition so far as to obtain
as parts only irreducible representations. In all the examples known so far,
the representations into which these continuous representations can be
decomposed, are equivalent to the original representation.

The third class contains all possible unitary representations. In a finite

number of dimensions, these can be decomposed first into factorial
representations, and these, in tum, in irreducible ones. Von Neumann lO
has shown that the first step still is possible in infinite dimensions. We can
assume, therefore, from the outset that we are dealing with factorial
representations.

In the theory of representations of finite dimensions, it is sufficient to

determine only the irreducible ones, all others are equivalent to sums of
these. Here, it will be necessary to determine all factorial representations.
Raving done that, we shall know from the above theorem of von
Neumann, that all representations are equivalent to finite or infinite sums
of factorial representations.

It will be one of the results of the detailed investigation that the

inhomogeneous Lorentz group has no "continuous" representations, all
representations can be decomposed into irreducible ones. Thus the work
of Majorana and Dirac appears to be justified from this point of view a
posteriori.

D. Classification of unitary representations

from the point of view of infinitesimal operators

The existence of an infinitesimal operator of a continuous one parametric

(cyclic, abelian) unitary group has been shown by Stone. 11 Re proved that
the operators of such a group can be written as exp(iRt) where R is a
(bounded or unbounded) hermitian operator and t is the group parameter.

11M. H. Stone, Proc. Nat. Acad. 16, 173, 1930, Ann of Math. 33,643, 1932, also J. v.
Neumann, ibid, 33, 567,1932.
40 CHAPTERll

However, the Lorentz group has many one parametric subgroups, and the
corresponding infinitesimal operators HI' H2, ... are all unbounded. For
every H an everywhere dense set of functions <\> can be found such that Hi<\>
can be defined. It is not clear, however, that an everywhere dense set can
be found to all members of which every H can be applied. In fact, it is not
clear that one such <\> can be found.

Indeed, it may be interesting to remark that for an irreducible

representation the existence of one function <\> to which all infinitesimal
operators can be applied, entails the existence of an everywhere dense set
of such functions. This again has the consequence that one can operate
with infinitesimal operators to a large extent in the usual way.

Proof: Let Q(t) be a one parametric subgroup such that Q(t)Q(t') =

Q(t+t'). If the infinitesimal operator of all subgroups can be applied to <\>,
the
(5)

exists. It follows, then, that the infinitesimal operators can be applied to

R<\> also where R is an arbitrary operator of the representation: Since
R-IQ(t) R is also a one parametric subgroup
limrl(R-IQ(t)R - I)<\> = lim}{1.r1(Q(t) - l)R<\>
t=O t=O

also exists an hence also (R is unitary)

limrl(Q(t) - l)R<\>.
t=O

Every infinitesimal operator can be applied to R<\> if they all can be applied
to <\>, and the same holds for sums of the kind
a1R 1<\> + a2R2<\> + ... + a"Rn<\>' (6)
These form, however, an everywhere dense set of functions if the
representation is irreducible.

If the representation is not irreducible, one can consider the set No of such
wave functions to which every infinitesimal operator can be applied. This
set is clearly linear and, according to the previous paragraph, invariant
under the operations of the group (i.e. contains every R<\> if it contains <\».
The same holds for the closed set N generated by No and also of the set P
REPRESENTATIONS OF THE POINCARE GROUP 41

of functions which are perpendicular to all functions of N. In fact, if <1>1' is

perpendicular to all <l>n of N, it is perpendicular also to all R-l<1>n and, for
the unitary character of R, the R<I>p is perpendicular to all <l>n' i.e. is also
contained in the set P.

We can decompose thus, by a unitary transformation, every unitary

representation into a "normal" and a "pathological" part. For the former,
there is an everywhere dense set of functions, to which all infinitesimal
operators can be applied. There is no single wave functions to which all
infinitesimal operators of a "pathological" representation could be
applied.

According to Murray and von Neumann, if the original representation was

factorial, all representations into which it can be decomposed will be
factorial also. Thus every representation is equivalent to a sum of factorial
representations, part of which is "normal," the other part "pathologicaL"

It will turn out again that the inhomogeneous Lorentz group has no
pathological representations. Thus this assumption of Majorana and Dirac
also will be justified a posteriori. Every unitary representation of the
inhomogeneous Lorentz group can be decomposed into normal irreducible
representations. It should be stated, however, that the representations in
which the unit operator corresponds to every translation have not been
determined to date (cf. also section 3, end). Hence, the above statements
are not proved for these representations, which are, however, more truly
representations of the homogeneous Lorentz group, than of the
inhomogeneous group.

While all these points may be of interest to the mathematician only, the
new representation of the Lorentz group which will be described in section
7 may interest the physicist also. It describes a particle with a continuous
spin.

Acknowledgment. The subject of this paper was suggested to me as early

as 1928 by P. M. Dirac who realized even at that date the connection of
representations with quantum mechanical equations. I am greatly indebted
to him also for many fruitful conversations about this subject, especially
during the years 1934/35, the outgrowth of which the present paper is.

I am indebted also to 1. v. Neumann for his help and friendly advice.

42 CHAPTER II

3. Summary of Ensuing Sections

Section 4 will be devoted to the definition of the inhomogeneous Lorentz

group and the theory of characteristic values and characteristic vectors of a
homogeneous (ordinary) Lorentz transformation. The discussion will
follow very closely the corresponding, well-known theory of the group of
motions in ordinary space and the theory of characteristic values of
orthogonal transformations. 12 It will contain only a straightforward
generalization of the methods usually applied in those discussions.

In section 5, it will be proved that one can determine the physically

meaningless constants in the D(L) in such a way that instead of (3a) the
more special equation
(7)
will be valid. This means that instead of a representation up to a factor, we
can consider representations up to the sign. For the case that either Ll or
~ is a pure translation, Dirac 13 has given a proof of (7) using infinitesimal
operators. A consideration very similar to his can be carried out, however,
also using only finite transformations.

For representations with a finite number of dimensions (corresponding to

an only finite number of linearly independent states), (7) could be proved
also if both Ll and ~ are homogeneous Lorentz transformations, by a
straightforward application of the method of Weyl and Schreier.14
However, the Lorentz group has no finite dimensional representation
(apart from the trivial one in which the unit operation corresponds to every
L). Thus the method of Weyl and Schreier cannot be applied. Its first step
is to normalize the indeterminate constants in every matrix D(L) in such a
way that the determinant of D(L) becomes 1. No determinant can be
defined for general unitary operators.

lZCf. e.g. E. Wigner, l.c. Chapter III. O. Veblen and J.W. Young, Projective Geometry,
Boston 1917. Vol 2, especially Chapter VII.

13p. A. M. Dirac, mimeographed notes of lectures delivered at Princeton University,

1934/35, page Sa.

14H. Weyl, Mathem. Zeits. 23, 271; 24, 328, 377, 789, 1925; O. Schreier, Abhandl.
Mathem. Seminar Hamburg, 4,15, 1926; 5,233, 1927.
REPRESENTATIONS OF TIlE POINCill GROUP 43

The method to be employed here will be to decompose every L into a

product of two involutions L = MN with M2 = N2 = 1. Then D(M) and
D(N) will be normalized so that their squares become unity and D(L) =
D(M)D(N) set. It will be possible, then, to prove (7) without going back to
the topology of the group.

Sections 6, 7, and 8 will contain the determination of the representations.

The pure translations form an invariant subgroup of the whole
inhomogeneous Lorentz group and Frobenius' method 15 will be applied in
Section 6 to build up the representations of the whole group out of
representations of the subgroup, by means of a "little group." In Section
6, it will be shown on the basis of an as yet unpublished work24 of J.v.
Neumann that there is a characteristic (invariant) set of "momentum
vectors" for every irreducible representation. The irreducible
representations of the Lorentz group will be divided into four classes. The
momentum vectors of the
1st class are time-like,
2nd class are null-vectors, but not all their components will be zero,
3rd class vanish (i.e., all their components will be zero),
4th class are space-like.
Only the first two cases will be considered in Section 7, although the last
case may be the most interesting from the mathematical point of view. I
hope to return to it in another paper. I did not succeed so far in giving a
complete discussion of the 3rd class. (All these restrictions appear in the
previous treatments also.)

In Section 7, we shall find again all known representations of the

inhomogeneous Lorentz group (i.e., all known Lorentz invariant equations)
and two new sets.

Sections 5, 6, 7 will deal with the "restricted Lorentz group" only, i.e.
Lorentz transformations with determinant 1 which do not reverse the
direction of the time axis. In section 8, the representations of the extended
Lorentz group will be considered, the transformations of which are not
subject to these conditions.

15G. Frobenius, Sitz. d. Koen. Preuss. Akad. p. 501, 1898, I. Schur, ibid, p. 164, 1906;
F. Seitz, Ann. of Math. 37, 17, 1936.
44 CHAPTER II

4. Description of the Inhomogeneous Lorentz Group

A.

An inhomogeneous Lorentz transformation L = (a, A) is the product of a

translation by a real vector a
x'i = xi + ai U=I,2,3,4) (8)
and a homogeneous Lorentz transformation A with real coefficients
4
x'i = I A i,0k· (9)
k=l
The translation shall be performed after the homogeneous transformation.
The coefficients of the homogeneous transformation satisfy three
conditions: (1) They are real and A leaves the indefinite quadratic form
2 2 2 2. .
-xCXz-x3+x 4 mvanant:
AF'A' = F (10)
where the prime denotes the interchange of rows and columns and F is the
diagonal matrix with the diagonal elements -1, -1, -1, +1. -(2) The
determinant I~I = 1 and -(3) A44 > o.

We shall denote the Lorentz-hermitian product of two vectors x and y by

{x, Y} = - x1Yl* - xiY2 * - x:Y'3 * - x4Y4·* (11)

(The star denotes the conjugate imaginary.) If {x, x} < 0 the vector x is
called space-like, if {x, x} = 0, it is a null vector, if {x, x} > 0, it is called
time-like. A real time-like vector lies in the positive light cone if x4 > 0; it
lies in the negative light cone if x4 < O. Two vectors x and y are called
orthogonal if {x, y} = O.

On account of its linear character a homogeneous Lorentz transformation

is completelY defined if Av is given for four linearly independent vectors
v(l), v(2), v(3), v(4).

From (11) and (10) it follows that {v, w} = {Av, Aw} for every pair of
vectors v, w. This will be satisfied for every pair if it is satisfied for all
pairs vO), v(k) of four linearly independent vectors. The reality condition is
satisfied if (Av(i))* = A(v(i)*) holds for four such vectors.
REPRESENTAnONS OF TIlE POINCARE GROUP 45

The scalar product of two vectors x and y is positive if both lie in the
positive light cone or both in the negative light cone. It is negative if one
lies in the positive, the other in the ne~ative light cone. Since both x and y
are time-like IX412 > IXI12 + Ixi + IX31 ; IY4 12 > IY I 12 + IY2 12 + IY31 2. Hence,
by Schwarz's inequality Ix4*Y41 > IXtYI
* + x2*Y2 + x3*Y31 and the sign of the
scalar product of two real time-like vectors is determined by the product of
their time components.

A time-like vector is transformed by a Lorentz transformation into a time-

like vector. Furthermore, on account of the condition A44 > 0, the vector
v(O) with the components 0, 0, 0, 1 remains in the positive light cone, since
the fourth component of Av(O) is A44 . If v(1) is another vector I6 in the
°
positive light cone {v(1), v(O)} > and hence also {Av(1), Av(O)} > and
Av(l) is in the positive light cone also. The third condition for a Lorentz
°
transformation can be formulated also as the requirement that every vector
in (or on) the positive light cone shall remain in (or, respectively, on) the
positive light cone.

This formulation of the third condition shows that the third condition holds
for the product of two homogeneous Lorentz transformations if it holds for
both factors. The same is evident for the first two conditions.

From AFA' = F one obtains by multiplying with A-I from the left and A'-I
= (A-I)' from the right F = A-IF(A-l), so that the reciprocal of a
homogeneous Lorentz transformation is again such a transformation. The
homogeneous Lorentz transformations form a group, therefore.

One easily calculates that the product of two inhomogeneous Lorentz

transformations (b, M) and (c, N) is again an inhomogeneous Lorentz
transformation (a, A)
(b, M)(c, N) = (a, A) (12)
where
a·I (l2a)

l~herever a confusion between vectors and vector components appears to be possible,

upper indices will be used for distinguishing different vectors and lower indices for
denoting the components of a vector.
46 CHAPTER II

or, somewhat shorter

A = MN; a b + Me. (12b)

B. Theory of characteristic values and characteristic vectors

of a homogeneous Lorentz transformation

Linear homogeneous transformations are most simply described by their

characteristic values and vectors. Before doing this for the homogeneous
Lorentz group, however, we shall need two rules about orthogonal vectors.

°
[1] If {v, w} = and {v, v} > 0, then {w, w} < 0; if {v, w} = 0, {v, v} = 0,
then w is either space-like, or parallel to v (either {w, w} < 0, or w = cv).

Proof:
*
v 4w4 v *lWl + v 2*w 2 + v 3w3'
* (13)
By Schwarz's inequality, then
1v 4121w412::;; (I v i + 1vi + 1v 312)(lwlI2 + IW212 + IW312). (14)

For IV412 > IVll2 + IV212 + IV312 it follows that IW412 < IWll2 + IW212 + IW312. If
IV412 = IVll2 + IV212 + IV312 the second inequality still follows if the
inequality sign holds in (14). The equality sign can hold only, however, if
the first three components of the vectors v and w are proportional. Then,
on account of (13) and both being null vectors, the fourth components are
in the same ratio also.

[2] If four vectors v(l), v(2). v(3), v(4) are mutually orthogonal and linearly
independent, one of them is time-like, three are space-like.

Proof: It follows from the previous paragraph that only one of four
mutually orthogonal, linearly independent vectors can be time-like or a
null vector. It remains to be shown therefore only that one of them is
time-like. Since they are linearly independent, it is possible to express by
them any time-like vector
4
v(t) =I,akv(k).
k=l
The scalar product of the left side of this equation with itself is positive
REPRESENTAnONS OF THE POINCARE GROUP 47

and therefore

or
~~)Xi{V(k), vUp«k»} > 0 (15)
k

and one {v(k), v(k)} must be positive. Four mutually orthogonal vectors
are not necessarily linearly independent, because a null vector is
perpendicular to itself. The linear independence follows, however, if none
of the four is a null vector.

We go over now to the characteristic values 'A of A. These make the

determinant IA - 'All of the matrix A - 'AI vanish.

[3] If 'A is a characteristic value, 'A*, 'A-I and 'A*-1 are characteristic values
also.

Proof: For 'A* this follows from the fact that A is real. Furthermore, from
IA - 'All = 0 also IA' - 'All = 0 follows, and this multiplied by the
determinants of AF and F-I gives
IAFI·IA'-'All·lFr l = IAFA'r-l-'AAI = Il-'AAI = 0,
so that 'A-I is a characteristic value also.

[4] The characteristic vectors VI and v2 belonging to two characteristic

values 'AI and Ivz are orthogonal if'A* Ilvz * 1.
Proof:

[5] If the modulus of a characteristic value A is IAI :f:. 1, the corresponding

characteristic vector v is a null vector and Aitself real and positive.

From {v, v} = {Av, Av} = 1'A12 {v, v} the {v, v} = 0 follows immediately
*
for I 'A I 1. If 'A were complex, 'A * would be a characteristic value also.
The characteristic vectors of 'A and A* would be two different null vectors
and, because of [4], orthogonal to each other. This is impossible on
48 CHAPTER II

account of [1]. ThUS'). is real and va real null vector. Then, on account of
the third condition for a homogeneous Lorentz transformation, '). must be
positive.

[6] The characteristic value'). of a characteristic vector v of length null is

real and positive.

If '). were not real, ').* would be a characteristic value also. The
corresponding characteristic vector v* would be different from v, a null
vector also, and perpendicular to v on account of [4]. This is impossible
because of [1].

[7] The characteristic vector v of a complex characteristic value '). (the

modulus of which is 1 on account of [5]) is space-like: {v, v} < O.

Proof: ').* is a characteristic value also, the corresponding characteristic

*"
vector is v *. Since ('). *) *'). = ').2 1, {v*, v} = O. Since they are different,
at least one is space-like. On account of {v, v} = {v*, v*} both are space-
like. If all four characteristic values were complex and the corresponding
characteristic vectors linearly independent (which is true except if A has
elementary divisors) we should have four space-like, mutually orthogonal
vectors. This is impossible, on account of [2]. Hence

[8] There is not more than one pair of conjugate complex characteristic
values, if A has no elementary divisors. Similarly, under the same
condition, there is not more than one pair A, A-I of characteristic values
whose modulus is different from 1. Otherwise their characteristic vectors
would be orthogonal, which they cannot be, being null vectors.

For homogeneous Lorentz transformations which do not have elementary

divisors, the following possibilities remain:

(a) There is a pair of complex characteristic values, their modulus is 1, on

account of [5]

(16)
and also a pair of characteristic values "-:" ').4' the modulus of which is not
1. These must be real and positive:

"-:, =
*
~>O. (16a)
The characteristic vectors of the conjugate complex characteristic values
REPRESENTATIONS OF THE POINCARE GROUP 49

are conjugate complex, perpendicular to each other and space-like so that

they can be normalized to -1

VI = V 2*; {Vl,V2} = {VI' *

Vi} 0
(17)
{ VI' V I} = { V 2' V 2} = -1
those of the real characteristic values are real null vectors, their scalar
product can be normalized to 1

V3 = V3
* V4 =
*
{ V3' V4} = 1
V4 (17a)
{ V 3' V 3} =v 4' v 4} = o.
{

Finally, the former pair of characteristic vectors is perpendicular to the

latter kind
(17b)
It will turn out that all the other cases in which A has no elementary
divisor are special cases of (a).

Figure 1: Position of the characteristic values for the general case a) in

the complex plane. In case b), "-3 and A4 coincide and are
equal 1; in case c), Al and ~ coincide and are either + 1 or -1.
In case d) both pairs "-3 = A4 = 1 and Al = 2 = 'f:. 1 coincide.
50 CHAPTER II

(b) There is a pair of complex characteristic values AI' "-2 = All = A~, Al :F-
A;, 111.11 = 1"-21 = 1. No pair with 1"-:,1 :F- 1, however. Then on account of [8],
still "-3 = ~* which gives with 1"-31 = 1, "-3 = ± 1. Since the product
11.1"-2"-311.4 = 1, on account of the second condition for homogeneous
Lorentz transformations, also 11.4 = "-3 = ± 1. The double characteristic
value ± 1 has two linearly independent characteristic vectors v3 and v4
which can be assumed to be perpendicular to each other, {v3' v4} = o.
According to [2], one of the four characteristic vectors must be time-like
and since those of Al and "-2 are space-like, the time-like one must belong
to ± 1. This must be positive, therefore "-3 = 11.4 = 1. Out of the time-like
and space-like vectors {v3' v3} = -1 and {v4' v4} = 1,0ne can build two
null vectors v4 + v3 and v4 - v3. Doing this, case (b) becomes the special
case of (a) in which the real positive characteristic values become equal "-3
,\-1
= 1\.4 = 1.

(c) All characteristic values are real; there is however one pair "-3 *
11.4
= ~,

= ~1, the modulus of which is not unity. Then {v3' v3} = {v4' v4} = 0 and
11.3 > 0 and one can conclude for Al and "-2, as before for "-3 and 11.4 that Al
= 11.2 = ± 1. This again is a special case of (a); here the two characteristic
values of modulus 1 become equal.

(d) All characteristic values are real and of modulus 1. If all of them are
+1, we have the unit matrix which clearly can be considered as a special
case of (a). The other case is Al = "-2 = -1, "-3 = 11.4 = +1. The
characteristic vectors of Al and "-2 must be space-like, on account of the
third condition for a homogeneous Lorentz transformation; they can be
assumed to be orthogonal and normalized to -1. This is then a special case
of (b) and hence of (a) also. The cases (a), (b), (c), (d), are illustrated in
Fig. 1.

The cases remain to be considered in which A has an elementary divisor.

We set therefore
Aeve = Aeve; Aewe = Aewe+ve. (18)
It follows from [5] that either IAel = 1, or {ve, ve} = O. We have {ve, we} =
{Aeve' Aewe} = IAi{ve, we} + {ve, ve}. From this equation
{v e, v e} = 0 (19)
follows for 1Ael = 1, so that (19) holds in any case. It follows then from [6]
REPRESENTATIONS OF THE POINCARE GROUP 51

that Ae is real, positive and ve' we can be assumed to be real also. The last
equation now becomes {ve' we} = A2e{ve' we} so that either Ae = 1 or {ve'
we} = O. Finally, we have
{we' we} = {Aewe,Aewe}
A2e {we' we} + 2Ae{we' Ve} + { Ve' Ve}'
=

This equation now shows that

{we' v e } = 0 (19a)
even if Ae = 1. From (19), (19a) it follows that we is space-like and can be
normalized to
{we' we} -1. = (19b)
Inserting (19a) into the preceding equation we finally obtain
Ae = 1. (19c)

[9] If Ae has an elementary divisor, all its characteristic roots are 1.

From (19c) we see that the root of the elementary divisor is 1 and this is at
'*
least a double root. If A had a pair of characteristic values Al 1, A2 =
A~l, the corresponding characteristic vectors VI and v2 would be
orthogonal to ve and therefore space-like. On account of [5], then IAll =
1"-21 = 1 and {VI' v2} = O. Furthermore. from {we' VI} = {Aewe' Aevl} =
AI{We• VI} + AI{Ve• VI} and from {ve, VI} = 0 also {we' VI} = 0 follows.
Thus all the four vectors VI' v2' ve' we would be mutually orthogonal.
This is excluded by [2] and (19).

Two cases are conceivable now. Either the fourfold characteristic root has
only one characteristic vector, or there is in addition to ve (at least) another
characteristic vector v l' In the former case four linearly independent
vectors ve' we' ze' xe could be found such that
Ae v e = ve

Atfe = ze + we Aexe = xe + ze
However {ve, xe} = {Aeve' Aexe} = {ve, xe} + {ve, ze} from which {ve,
ze} = 0 follows. On the other hand
{we,ze} = {Aewe,Aeze}
= {We,Ze} + {we,we}+{v e,ze}+{v e'we },
This gives with (19a) and (19b) {ve' ze} = 1 so that this case must be
52 CHAPTER II

excluded.

(e) There is thus a vector VI so that in addition to (18)

Aevl = VI (18a)
holds. From {we' VI} = {~we' Aevl} = {We' VI} + {Ve' VI} follows
{v e, VI} = O. (19d)
The equations (18), (18a) will remain unchanged if we add to we and VI a
multiple of ve. We can achieve in this way that the fourth components of
both we and VI vanish. Furthermore, VI can be normalized to -1 and added
to we also with an arbitrary coefficient, to make it orthogonal to VI.
Hence, we can assume that
vI4=we4=0; {vl,vl}=-I; {we,vl}=O. (1ge)
We can finally define the null vector ze to be orthogonal to we and VIand
have a scalar product 1 with ve
{ze' ze} = {ze' we} = {ze' v I} = 0; {ze' v e} = 1. (19.1)
Then the null vectors ve and ze represent the momenta of two light beams
in opposite directions. If we set Aeze = aVe + bWe + cZe + dv 1 the
conditions {ze' v} = {Aeze' Aev} give, if we set for v the vectors ve' we' ze'
VI' the conditions c = 1; b = c; 2ac - b2 - d2 = 0; d = O. Hence
Aeve = ve Aewe = we + ve
Ae v l = VI A~e = ze + we + 1I2ve· (20)
A Lorentz transformation with an elementary divisor can be best
characterized by the null vector ve which is invariant under it and the space
part of which forms with the two other vectors we and VI three mutually
orthogonal vectors in ordinary space. The two vectors we and VI are
normalized, VIis invariant under Ae while the vector ve is added to we
upon application of Ae. The result of the application of Ae to a vector
which is linearly independent of ve' we and VI is, as we saw, already
determined by the expressions for Aeve' Aewe and AeVI.

The Ae(Y) which have the invariant null vector ve and also we (and hence
also VI) in common and differ only by adding to we different multiples YVe
of ve' form a cyclic group with Y= 0, the unit transformation as unity:
Ae(y)AlY) = Ae(Y + i)·
The Lorentz transformation M( a) which leaves vIand we invariant but
replaces ve by aVe (and ze by a-1ze) has the property of transforming Ae(Y)
REPRESENTATIONS OF TIlE POINCARE GROUP 53

into
M( a)Ae(y)M( ar1 = Ae( ay). (+)
An example of Ae(Y) and M(a) is
1 0 0 0
o1 Y Y
Ae(Y) = 0 -y 1-1ft( -1ft(
oy 1fzr 1+1fzr

10 o o
o1 o o
M (ex) = 0 0 1fz(ex+a.-1) 1fz(a.-a1)
o 0 1fz(a.-a.-1) 1fz(a.+a.-1)

These Lorentz transformations play an important role in the

representations with space like momentum vectors.

A behavior like (+) is impossible for finite unitary matrices because the
characteristic values of M(ar 1Ae(y)M(a) and ~(y) are the same-those of
Ae(ya) = Ae(Y)CX the a th powers of those of Ae(Y). This shows very simply
that the Lorentz group has no true unitary representation in a finite number
of dimensions.

c. Decomposition of a homogeneous Lorentz transformation

into rotations and an acceleration in a given direction

The homogeneous Lorentz transformation is, from the point of view of the
physicist, a transformation to a uniformly moving coordinate system, the
origin of which coincided at t = 0 with the origin of the first coordinate
system. One can, therefore, first perform a rotation which brings the
direction of motion of the second system into a given direction-say the
direction of the third axis-and impdii. it a velocity in this direction, which
will bring it to rest. After this, the two coordinate systems can differ only
in a rotation. This means that every homogeneous Lorentz transformation
can be decomposed in the following way17

17ef. e.g. L. Silberstein, The Theory of Relativity, London 1924, p. 142.

54 CHAPTERll

A = RZS (21)
where Rand S are pure rotations, (i.e. ~4 = R4i = Si4 = S4i = 0 for i 4 *
and R44 = S44 = 1, also R' = R-1, S' = S-1) and Z is an acceleration in the
direction of the third axis, i.e.

1 000
o 100
Z= OOab
o0 b a
with a2 - b2 = 1, a> b > O. The decomposition (21) is clearly not unique.
It will be shown, however, the Z is uniquely determined, i.e. the same in
every decomposition of the form (21).

In order to prove this mathematically, we chose R so that in R -1 A = I the

first two components in the fourth column 114 = 124 = 0 become zero: R-1
shall bring the vector with the components A 14 , A24 , A34 into the third
1

axis. Then we take 134 = (Ai4 + A~ + A~4)i and 144 = A44 for b and a to
form Z; they satisfy the equation I~ - 1~4 = 1. Hence, the first three
components of the fourth column of J = Z-1 I = Z-1 R-l A will become zero
2 2 2 2
and J44 = 1, because of J44 - J14 - J24 - J34 = 1. Furthermore, the first three
components of the fourth row of J will vanish also, on account of J~4 - J~l -
J~2 - J~3 = 1, i.e. J = S = Z-1 R -1 A is a pure rotation. This proves the
possibility of the decomposition (21).

The trace of AA' = RZ2R-l is equal to the trace of Z2, i.e. equal to 2a2 +
2b2 + 2 = 4a2 = 4b2 + 4 which shows that the a and b of Z are uniquely
determined. In particular a = 1, b = 0 and Z the unit matrix if AA' = 1, i.e.
A a pure rotation.

It is easy to show now that the group space of the homogeneous Lorentz
transformations is only doubly connected. If a continuous series A(t) of
homogeneous Lorentz transformations is given, which is unity both for t =
o and t = 1, we can decompose it according to (21)
A(t) = R(t)Z(t)S(t). (21a)
It is also clear from the foregoing, that R(t) can be assumed to be
continuous in t, except for values of t, for which A14 = A24 = A34 = 0, i.e.
for which A is a pure rotation. Similarly, Z(t) will be continuous in t and
REPRESENTATIONS OF TIlE POINCARE GROUP 55

this will hold even where A(t} is a pure rotation. Finally, S = Z-IR-IA will
be continuous also, except where A(t} is a pure rotation.

Let us consider now the series of Lorentz transformations

(2Ib)
where the b of Z(t}S is s times the b of Z(t}. By decreasing s from I to 0
we continuously deform the set Al (t) = A(t} of Lorentz transformations
into a set of rotations Ao(t} = R(t}S(t}. Both the beginning Ao(O} = I and
the end As( I} = I of the set remain the unit matrix and the sets As(t} remain
continuous in t for all values of s. This last fact is evident for such t for
which A(t) is not a rotation: for such t all factors of (2Ib) are continuous.
But it is true also for to for which A(to) is a rotation, and for which, hence
Z(to} = I and As(to} = A 1(to} = A(to}. As Z(t)is everywhere continuous,
there will be a neighborhood of to in which Z(t} and hence also Z(t}S is
arbitrarily close to the unit matrix. In this neighborhood As(t) = A(t).
S(tt 1Z(tt 1Z(t)SS(t) is arbitrarily close to A(t}; and, if the neighborhood is
small enough, this is arbitrarily close to A(to} = As(to}.

Thus (2Ib) replaces the continuous set A(t} of Lorentz transformations by

a continuous set of rotations. Since these form an only doubly connected
manifold, the manifold of Lorentz transformations can not be more than
doubly connected. The existence of a two valued representation 18 shows
that it is actually doubly and not simply connected.

We can form a new group14 from the Lorentz group, the elements of which
are the elements of the Lorentz group, together with a way A(t),
connecting A(l) = A with the unity A(O) = E. However, two ways which
can be continuously deformed into each other are not considered different.
The product of the element "A with the way A(t}" with the element "I
with the way I(t}" is the element AI with the way which goes from E along
A(t) to A and hence along AI(t} to AI. Clearly, the Lorentz group is
isomorphic with this group and two elements (corresponding to the two
essentially different ways to A) of this group correspond to one element of
the Lorentz group. It is well known,I8 that this group is holomorphic with

18Cf. H. Weyl, Gruppentheorie und Quantenmechanik, 1st. ed. Leipzig 1928, pages
110-114, 2nd ed. Leipzig 1931, pages 130-133. It may be interesting to remark that
essentially the same isomorphism has been recognized already by L. Silberstein, l.c. pages
148-157.
56 CHAPTER II

the group of unimodular complex two dimensional transformations.

Every continuous representation of the Lorentz group' 'up to the sign" is a

singlevalued, continuous representation of this group. The transformation
which corresponds to "A with the way A(t)" is that d(A) which is
obtained by going over from d(E) = d(A(O» = 1 continuously along d(A(t»
to d(A(I» = d(A).

D. The homogeneous Lorentz group is simple

It will be shown, first, that an invariant subgroup of the homogeneous

Lorentz group contains a rotation (i.e. a transformation which leaves x4
invariant).- We can write an arbitrary element of the invariant subgroup in
the form RZS of (21). From its presence in the invariant subgroup follows
that of S·RZS·S-l = SRZ = TZ. If ~ is the rotation by 1t about the first
axis, X;Z~ = Z-l and ~TZ~-l = ~T~X;Z~ = ~TX;Z-l is
contained in the invariant subgroup also and thus the transform of this with
Z, i.e. Z-l~TX1t also. The product of this with TZ is T~T~ which
leaves x4 invariant. If T~T~ = 1 we can take TY1tTY1t. If this is the
unity also, TXxTXx = TY1tTY1t and T commutes with XxY1t, i.e. is a
rotation about the third axis. In this case the space like (complex)
characteristic vectors of TZ lie in the plane of the first two coordinate axes.
Transforming TZ by an acceleration in the direction of the first coordinate
axis we obtain a new element of the invariant subgroup for which the
space like characteristic vector will have a not vanishing fourth
component. Taking this for RZS we can transform it with S again to
obtain a new SRZ = TZ. However, since S leaves x4 invariant, the fourth
component of the space like characteristic vectors of this TZ will not
vanish and we can obtain from it by the procedure just described a rotation
which must be contained in the invariant subgroup.

It remains to be shown that an invariant subgroup which contains a

rotation, contains the whole homogeneous Lorentz group. Since the three-
dimensional rotation group is simple, all rotations must be contained in the
invariant subgroup. Thus the rotation by 1t around the first axis X 1t and
also its transform with Z and also
ZX;z:-l.X1t = Z.X;z:-lX1t = Z2
is contained in the invariant subgroup. However, the general acceleration
in the direction of the third axis can be written in this form. As all
REPRESENTATIONS OF TIlE POINCARE GROUP 57

rotations are contained in the invariant subgroup also, (21) shows that this
holds for all elements of the homogeneous Lorentz group.

It follows from this that the homogeneous Lorentz group has apart from
the representation with unit matrices only true representations. It follows
then from the remark at the end of part B, that these have all infinite
dimensions. This holds even for the two-valued representations to which
we shall be led in Section 5 equ. (520), as the group elements to which the
positive or negative unit matrix corresponds must form an invariant
subgroup also, and because the argument at the end of part B holds for
two-valued representations also. One easily sees furthermore from the
equations (52B), (52C) that it holds for the inhomogeneous Lorentz group
equally well.

S. Reduction of Representations Up to a Factor to

Two-Valued Representations

The reduction will be effected by giving each unitary transformation,

which is defined by the physical content of the theory and the
consideration of reference only up to a factor of modulus unity, a
, 'phase," which will leave only the sign of the representation operators
undetermined. The unitary operator corresponding to the translation a will
be denoted by T(a), that to the homogeneous Lorentz transformation A by
d(A). To the general inhomogeneous Lorentz transformation then D(a, A)
= T(a)d(A) will correspond. Instead of the relations (12), we shall use the
following ones.
T(a)T(b) ro(a, b)T(a + b) (22B)

d(A)T(a) meA, a)T(Aa)d(A) (22C)

d(A)d(l) = meA, /)d(Al). (22D)

The m are numbers of modulus 1. They enter because the multiplication
rules (12) hold for the representative only up to a factor. Otherwise, the
relations (22) are consequences of (12) and can in their return replace (12).
We shall replace the T(a), d(A) by Q(a)T(a) and Q(A)d(A) respectively,
for which equations similar to (22) hold, however with
mea, b) = 1; meA, a) = 1; meA, l) = ;tl. (22')
58 CHAPTER II

A.

It is necessary, first, to show that the undetermined factors in the

representation D(L) can be assumed in such a way that the m(a,b), m(A,a),
m(A,I) become -apart from regions of lower dimensionality-continuous
functions of their arguments. This is a consequence of the continuous
character of the representation and shall be discussed first.

(a) From the point of view of the physicist, the natural definition of the
continuity of a representation up to a factor is as follows. The
neighborhood 0 of a Lorentz transformation Lo = (b, I) shall contain all the
transformations L = (a, A) for which lak - bkl < 0 and IAik - Iikl < The o.
representation up to a factor D(L) is continuous if there is to every positive
number e, every normalized wave function <\> and every Lorentz
transformation Lo such a neighborhood 0 of Lo that for every L of this
neighborhood one can find an n of modulus> 1 (the n depending on L
and <\» such that (u<l>' u<l» < e where
u<l> = (D(Lo) - QD(L)<\». (23)
Let us now take a point Lo in the group space and find a normalized wave
function <\> for which I(<\>, DLO<\»I > 1/6. There always exists a <\> with this
property, if I(<\>, D(~o)<\»1 < 1/6 then 'V = a<\> + ~D(Lo)<\> with suitably
chosen a and ~ will be normalized and I('V, D(Lo)'V)1 > 1/6. We consider
then such a neighborhood N ofLo for all L of which IC<\>, DCL)<\»I > 1112. It
is well known 19 that the whole group space can be covered with such
neighborhoods. We want to show now that the D(L)<\> can be multiplied
with such phase factors (depending on L) of modulus unity that it becomes
strongly continuous in the region N.

We shall chose that phase factor so that (<\>, D(L)<\» becomes real and
positive. Denoting then
(D(L 1) - D(L»<\> = U<l>'
the (V<l>' V<l» can be made arbitrarily small by letting L approach
sufficiently near to L 1, if Ll is in N. Indeed, on account of the continuity,
as defined above, there is an n = eik such that (u, u) < e if L is sufficiently

1"rhis condition is the "separability" of the group. Cf. e.g. A. Haar, Ann. of Math., 34,
147,1933.
REPRESENTATIONS OF THE POINCARE GROUP 59

near to LI where
u = (D(L I ) - eikD(L))$. (23')
Taking the absolute value of the scalar product of u with $ one obtains

I($,D(L I )$) - cos k($, D(L)$) - isink($,D(L)$)1 = 1($,u)1 ~..fE,

because of Schwartz's inequality. If only ..fE < 1112, the k must be smaller
than rc/2 because the absolute value is certainly greater than the real part
and both ($, D(L I )$) and ($, D(L)$) are real and greater than 1112.

As the absolute value is also greater than the imaginary part, we

sink < 12..fE
On the other hand,
U~ = u + (e ik - 1)D(L)$,
and thus

(b) It shall be shown next that if D(L)$ is strongly continuous in a region

and D(L) is continuous in the sense defined at the beginning of this
section, then D(L}1jI with an arbitrary", is (strongly) continuous in that
region also. We shall see, hence, that the D(L), with any normalization
which makes a D(L)$ strongly continuous, is continuous in the ordinary
sense: There is to every L l , e and every 'V a 8 so that (V"" V",) < e where
U1jI=(D(L l )-D(L))'V
if L is in the neighborhood 8 of L I .

It is sufficient to show the continuity of D(L)'V where 'V is orthogonal to $.

Indeed, every 'JI' can be decomposed into two terms, "" = a$ + B'V the one
of which is parallel, the other perpendicular to $. Since D(L)$ is
continuous, according to supposition, D(L)'JI' = aD(L)$ + BD(L)", will be
continuous also if D(L)", is continuous.

The continuity of the representation up to a factor requires that it is

possible to achiev, that (u"" u",) < e and (u", +~, ~ +~) < e where
60 CHAPTER II

U\jI = (23a)

(23b)
with suitaly chosen n's. According to the foregoing, it also is possible to
choose Land Ll so close that (Vql' V<)l) < E.

Subtracting (23') and (23a) from (23b) and applying D(L)-l on both sides
gives
(n\jl- Q\jI+<)l)'!' + (1- ~+<)l)<\> = D(Lrl(u\jI+<)l- u\jI- U<)l)
The scalar product of the right side with itself is less than 9E. Hence both
1 1 1

I~ - n\jl + <)ll < 3E2 and 11 - ~ + <)ll < 3£2 or 11 - ~I < 6£2. Because of V\jI
1 1

= ~ - (1 - ~)D(L),!" the (V\jI' V\jI)2 < (u\jl' U\jl)2 + 11 - ~I and thus (V\jI'
V\jI) < 49£.

This completes the proof of the theorem stated under (b). It also shows
that not only the continuity of D(L),!, has been achieved in the
neighborhood of Lo by the normalization used in (a) but also that of D(L),!,
with every '!', i.e., the continuity ofD(L).

It is clear also that every finite part of the group space can be covered by a
finite number of neighborhoods in which D(L) can be made continuous. It
is easy to see that the ro of (22) will be also continuous in these
neighborhoods so that it is possible to make them continuous, apart from
regions of lower dimensionality than their variables have. In the following
only the fact will be used that they can be made continuous in the
neighborhood of any a, b, and A.

B.

(a) We want to show next that all T(a) commute. From (22B) we have
T(a)T(b)T(a)-l = e(a, b)T(b) (24)
where c(a, b) = ro(a, b)/ro(b, a) and hence
e(a, b) = e(b, arlo (24a)

Transforming (24) with T(a') one obtains

REPRESENTATIONS OF TIlE POINCill GROUP 61

T(a')T(a)T(b)T(a)-IT(a')-1 = c(a, b)T(a')T(b)T(a')-1

or
ro(a',a)T(a' + a)T(b)ro(a',ar 1T(a' +ar1 = c(a,b)c(a',b)T(b)
or
c(a,b)c(a',b) ':' c(a+a', b). (25)

It follows 20 from (25) and the partial continuity of c(a, b) that

4
c(a, b) = exp(21tiI alk(b» (26)
k=l
and, since this is equal to c(b, at 1 = exp(-21t(Lbkfk(a»
4
I (alk(b) + blk(a» = n(a, b) (27)
k=1
where n(a, b) is an integer. Setting in (27) for b the vector e(A.) the A.
component of which is 1, all the others zero and for fk(e(A.» = -fkA.
!;.,(a) = n(a, e()..» + IalkA,
k
and putting this back into (27) we obtain
4 4
IfkA(a)..bk + b)..ak) + I a~(b, e(k» + b~(a, e(k» = n(a, b). (28)
k,)..=l k=l
Assuming for the components of a and b such values which are
transcendental both with respect to each other and the fU (which are fixed
numbers), one sees that (28) cannot hold except if the coefficient of every
one vanishes
fa + J,.,.k = 0; n(b, e(k» = 0, (29)
so that (26) becomes
4
c(a, b) = exp(21ti} fkA,a,.bJ. (30)
k,!:l
It is necessary now to consider the existence of an operator d(A) satisfying
(22C). Transforming this equation with the similar equation containing b
instead ofa

20G. Hamel, Math. Ann. 60, 460, 1905, quoted from H. Hahn, Theorie der reellen
Funktionen. Berlin 1921, pages 581-583.
62 CHAPTERll

d(A)T(b)d(A)-ld(A)T(a)d(A)-ld(A)T(b)-ld(A)-l
= ro(A, b)T(Ab)ro(A, a)T(Aa)ro(A, b)-lT(Abr l
= ro(A, a)c(Ab, Aa)T(Aa),

while the first line is clearly d(A)c(b, a)T(a)d(At l = ro(A, a)c(b, a)T(Aa)
whence
c(b, a) = c(Ab, Aa) (31)
holds for every Lorentz transformation A. Combined with (30) this gives
~(fleA.a~A. - 2.JvIlAv';"JUI.a~A.) = n'(a, b),
where n'(a, b) is agliin an integer. As this equation holds for every a, b

fleA. = 2.Jvll~';"JlA.; f = A'fA

VIl
must hold also, for every Lorentz transformation. However, the only form
invariant under all Lorentz transformations are multiples of the F of (10).
Actually, because of (29), f must vanish and c(a, b) = 1, all the operators
corresponding to translations commute
T(a)T(b) = T(b)T(a). (32)

It is well to remember that it was necessary for obtaining this result to use
the existence of d(A) satisfying (22C).

(b) Equation (32) is clearly independent of the normalization of the T(a).

If we could fix the translation operators in four linearly independent
directions e(1), e(2), e(3), e(4) so that for each of these directions
T(ae(k»T(be(k» = T«a + b)e(k» (33)
be valid for every pair of numbers a, b, then the normalization
T(al il) + a2e(2) + a3e(3) + a4e(4»
= T(al e(l»T(a2e(2»T(a3e(3»T(a4e(4» (33a)
and (32) would ensure the general validity of
T(a)T(b) = T(a + b). (34)

As the four linearly independent directions e(1), ... ,e(4) we shall take four
null vectors. If e is a null vector, there is, according to section 3, a
homogeneous Lorentz transformation21 Ae such that Aee = 2e.

21The index e denotes here the vector e for which Aee = 2e; this Ae has no elementary
divisor.
REPRESENTATIONS OF TIlE POINCARE GROUP 63

We normalize T (e) so that

d(Ae)T(e)d(Ae)-l = T(el (35)

This is clearly independent of the normalization of d(Ae)' We further

normalize for all (positive and negative) integers n
d(Ae)nT(e)d(Aern = T(2 ne). (35a)
It follows from this equation also that
T(2 ne)2 = d(Ae)nT(e)2d(Ae)-n = d(Ae)nd(Ae)T(e)d(Ae)-ld(Aern
= T(2n+I e). (36)

This allows us to normalize for every positive integer k

T(k·2-ne) = T(2- ne)k (35a)
in such a way that the normalizaton remains the same if we replace k by
2mk and n by n + m. This ensures, together with (36), the validity of
T(ve)T(lle) = T«v + Il)e)
(36a)
d(Ae)T(Ve)d(Ae)-1 = T(2ve)
for all dyadic fractions v and Il.

It must be shown that if VI' v2' v3' ... is a sequence of dyadic fractions.
converging to 0, lim T(vie) = 1. From T(a)·T(O) = coCa, O)T(a) it follows
that T(O) is a constant. According to the theorem of part (A)(b), the T(ve),
if multiplied by proper constants Oy will converge to 1, i.e., by choosing
an arbitrary <\>, it is possible to make both (l-nvT(ve»<\> = u and
(I-11vT(ve»'d(Aet1<j> = u' arbitrarily small, by making v small. Applying
d(Ae) to the second expression, one obtains, for (36a), that (1 - OyT(2ve»<j>
= d(A£)u' is also small. On the other hand, applying T(ve) to the first
expression one sees that (T(ve) - OyT(2ve»<\> = T(ve)u approaches zero
also. Hence. the difference of these two quantities (1 - T(ve»<\> goes to
zero, i.e. T(vie)<j> converges to <\> if vI' v2' v3' ... is a sequence of dyadic
fractions approaching O.

Now VI' v2' v3' ... be a sequence of dyadic fractions coverging to an

arbitary number a. It will be shown then that T(vie) converges to a
multiple of T(ae) and this multiple of T(ae) will be the normalized T(ae).
Again, it follows from the continuity that there are such Q i that QiT(vie)<\>
converges to T(ae)<\>. The QflT(VjetIQiT(Vie)<\> will converge to <\>,
64 CHAPTER II

therefore, as both i and j tend to infinity. However, according to the

previous paragraph, T«vi - vj)e)C\> tends to C\> and thus Q{lQi tends to 1. It
follows that Qi- 1 converges to a definite number Q. Hence Q(l·QiT(Vie)C\>
converges to QT(ae)C\> which will be denoted, henceforth, by T(ae). For
the T(ae), normalized in this way, (33) will hold, since if Ill' 1l2' Il3 ... are
dyadic fractions converging to b, we obtain, with the help (36a)
T(ae)T(be)C\> = lim T«vi + lli)e)C\> = T«a + b)e)C\>.
ij=co
This argument not only shows that it is possible to normalize the T(ae(k»
and hence by (33a) the T(a) so that (34) holds for them but, in addition to
this, that these T(a) will be continuous in the ordinary sense.
c.

It is clear that (34) will remain valid if one replaces T(a) by

exp(21ti{a,c})T(a) where c is an arbitrary vector. This remaining freedom
in the normalization of T(a) will be used to eliminate the coCA, a) from
(22C).

Transforming (22C) d(A)T(a)d(Ar l = coCA, a)T(Aa) with d(M) one obtains

on the left side co(M, A)d(MA)T(a)co(M, Atfd(MArl = co(MA, a)T(MAa)
while the right side becomes coCA, a)co(M, Aa)T(MAa). Hence
co(MA, a) = co(M, Aa)co(A, a). (37)
On the other hand, the product of two equations (22C) with the same A but
with a and b respectively, instead of a yields with the help of (34)
coCA, a)co(A, b) = co(A(a + b».
Hence
coCA, a) = exp(21ti{a, AA)}),
where f(A) is a vector which can depend on A. Inserting this back into
(37) one obtains
{a, f(MA)} = {Aa, f(M)} + {a, f(A)} + n,

{a, f(MA) - A-lAM) - f(A)} = n,

where n is an integer which must vanish since it is a linear function of a.
Hence
f(MA) = A-lAM) + AA). (38)
If we can show that the most general solution of the equation is
REPRESENTATIONS OF THE POINCARE GROUP 65

(39)
where Vo is a vector independent of A, the meA, a) wil~ become meA, a) =
exp(2ni{(A - l)a, vo}). Then meA, a) in (22C) will disappear if we replace
T(a) by exp(2ni{a, vo} )T(a).

The proof that (39) is a consequence of (38) is somewhat laborious. One

can first consider the following homogeneous Lorentz transformations.

-S3 C3 0 0
o 0 C 3 S3
o 0 S3 C 3

where ci = cos ~; si = sin ai; Ci = ChYi; Si = ShYi' All the X(a, y)

commute. Let us choose, therefore, two angles aI' Y1 for which 1 - X(al'
Y1t 1 has a reciprocal. It follows then from (38)
X(a, yrY(X(a l , YI»+f(X(a, y) = X(a 1, ylrY(X(a, y)) +j{X(a l , Y1))
or
JCXCa, y» = [l-XCal' y1r 1rl[l-XCa, y)-llttXca1, Y1» (41)
f(X(a,y)) = (l-X(a,y)-l)v x'
where Vx is independent of a, y. Similar equations hold for the f(Y(a,y»
and f(Z(a,Y». Let us denote now X(n, 0) = X; Yen, 0) = Y; zen, 0) =
Z. These anticommute in the following sense with the transformations
(40):
YX(a, y)Y = ZX(a, y)Z = X(a, yrl. (42)

From (38) one easily calculates

f(YX(a, y)Y) = (YX(a, yr 1 + 1)f(Y) + Yj{X(a, y»,
or, because of (41) and (42), after some trivial transformations
66 CHAPTERll

(1 - X(a, y»(1 - y)(vX - vy) = O. (43)

As a, y can be taken arbitrarily, the first factor can be dropped. This

leaves (1 - Y)(vx - Vy) = 0, or that the first and third components of Vx
and Vy are equal. One similarly concludes, however, that (1 - X)(Vy - vx)
= 0 and thus that the first three components of vx, Vy and also of Vz are
equal.

For YI = Y2 = Y3 = 0 the transformations (40) are the generators of all

rotations, i.e. all Lorentz transformations R not affecting the fourth
coordinate. As the 4-4 matrix element of these transformations is 1, the
exyression (1 - R -I)v is independent of the fourth component ?f v and (1 -
R- )vx = (1 - RI)Vy = (1 - R-I )vZ. It follows from (38) that If f(R) = (1 -
R-I)vX and f(S) = (1 - S-I)vX' then f(SR) = (1 - R-I S-I)vX. Thus f(R) = (1
- R-I)vX is valid with the same Vx for all rotations.

Now
f(X(a, y)R) = R-I(1-X(a, y)-I)vx+(I-J(I)vX
= (1-(X(a, y)R)-I)vX.
One easily concludes from (38) that the f(E) corresponding to the unit
operation vanishes and f(A-I) = - Af(A). Hence f(R-IX(a, yrI) = (1 -
X(a,yR)vx; and one concludes further that for all Lorentz transformations
A = RX( a,y)S, (39) holds with Vo = -vx if Rand S are rotations. However,
every homogeneous Lorentz transformation can be brought into this form
(Section 4C). This completes the proof of (39) and thus of ro(A, a) = 1.

D.

The quantities ro(a, b) and ro(A, a) for which it has just been shown that
they can be assumed to be 1, are independent from the normalization of
d(A). We can affix therefore an arbitrary factor of modulus 1 to all the
d(A), without interfering with the normalizations so far accomplished. In
consequence hereof, the ensuing discussion will be simply a discussion of
the normalization of the operators for the homogeneous Lorentz group and
the result to be obtained will be valid for the group also.

Partly because the representations up to a factor of the three dimensional

rotation group may be interesting in themselves, but more particularly
REPRESENTATIONS OF THE POINCARE GROUP 67

Figure 2:
because the procedure to be followed for the Lorentz group can be
especially simply demonstrated for this group, the three dimensional
rotation group shall be taken up first.

It is well known that the nomalization cannot be carried so far that meA, I)
= 1 in (22D) and there are well known representations for which meA, I) =
± 1. We shall allow this ambiguity therefore from the outset.

One can observe, first, that the operator corresponding to the unity of the
group is a constant. This follows simply from d(A)d(E) = meA, E)d(A).
The square of an operator corresponding to an involution is a constant,
therefore.

The operator corresponding to the rotation about the axis e by the angle 1t;
normalized so that its square be actually 1, will be denoted bye; e 2 = 1.
The e are-apart from the sign-uniquely defined.

A rotation R about v by the angle a. is the product of two rotations by 1t

about el and e2 where el and e2 are perpendicular to v and e2 arises from
e 1 by rotation about v and with al2. Choosing for every v an arbitrary el
perpendicular to v, we can normalize, therefore
68 CHAPTERn

(44)
Now d(R) commutes with every deS) if S is also a rotation about v. This is
proved in equations (24)-(30). The fu in (30) must vanish on account of
(29).

Figure 3:
(Also, both Rand S can be arbitrarily accurately represented as powers of
a very small rotation about v). Hence, transforming (44) by deS) one
obtains
(44a)

Now d(S)e1d(Sr l corresponds to a rotation by 1t about an axis,

perpendicular to v and enclosing an angle ~ with el' where ~ is the angle
of rotation of S. Since the square of d(S)e Id(Sr l is also 1, (44a) is simply
another way of writing d(R) = e3e4 as a product of two e and we see that
the normalization (44) is independent of the choice of the axis e 1 (Cf. Fig.
2).

For computing d(R)d(T) we can draw the planes perpendicular to the axes
of rotation of R and T and use for d(R)= eRe C such a development that the
axis ec of the second involution coincide with the intersection line of the
REPRESENTATIONS OF TIlE POINCARE GROUP 69

above-mentioned planes, while for d(T) = e~T we choose the first

involution to be a rotation about this intersection line (Fig. 3). Then, the
product

d(R)d(T) = ±eiceir = ±eir (45)

will automatically have the normalization corresponding to (44). This
shows that the operators normalized in (44) give a representation up to the
sign.

For the Lorentz group, the proof can be performed along the same line,
only the underlying geometrical facts are less obvious. Let A be a Lorentz
transformation without elementary divisors with the characteristic values
e 2iy, e-2iy, e2x, e-2X and the characteristic vectors VI' v2 = v~, v3' v4' as
described in section 4B.

We want to make A = MN with M2 = N2 = 1. For AN = M, we have

ANAN = 1 and thus ANA = N. Setting NVi = Lk(lik:vk' we obtain ANAvi =
L Ak~"-ivi = L ~vk· Because of the linear independence of the vk this
amounts to Ai~~ =~: all ~ are zero, except those for which Ai~ = 1.
As in none of the cases (a), (b), (c), (d) of section 4B is Al or A.z reciprocal
to one of the last two A, the vectors VI and v2 will be transformed by N
into a linear combination of vI and v2 again, and the same holds for v3 and
v4. This means the N can be considered as the product of two
transformations N = NsNt, the first in the vlv2 plane, the second in the
v3v4 plane. (Instead of v I v2 plane one really should say VI + v2' iVI - iV2
plane, as v I and v2 are complex themselves. This will be meant always by
vlv2 plane, etc.). The same holds for M also.

Both Ns and Nt must satisfy the first and third condition for Lorentz
transformations (cf. 4A) and both determinants must be either 1, or -1.
Furthermore, the square of both of them must be unity.

If both determinants were +1, the Nt had to be unity itself, while Ns could
be the unity or a rotation by 1t in the vlv2 plane. Thus VI' v2' v3' v4 would
be characteristic vectors of N itself.

If both determinants are -1 (this will tum out to be the case), Ns is a

reflection on a line in the vlv2 plane and Nt a reflection in the v3v4 plane,
interchanging v3 and v4. In this case VI' v2' v3' v4 would not all be
characteristic vectors of N.
70 CHAPTER II

If VI' v2' v3' v4 are characteristic vectors of N, they are characteristic

vectors of M = AN also. Then both M and N would be either unity, or a
rotation by 1t in the vIv2 plane. If both of them were rotations in the vIv2
plane, their product A would be the unity which we want to exclude for the
present. We can exclude the remaining cases in which the determinants of
Ns and Nt are +1 by further stipulating that neither M nor N shall be the
unity in the decomposition A = MN.

Hence N is the product of a reflection in the vI v2 plane

AT I
iVS V =
I.
liT
S V'
LYSv = Sv (46a)
where sv and s/v are two perpendicular real vectors in the vIv2 plane
S'v = eiv vI + e-iv v 2; Sv = i(eivv 1 - e-ivv 2)' (46b)
and of a reflection in the v3v4 plane
N( J! = (J!; NtJ! = -til' (46c)
where again til' t' J! are real vectors in the v3 v4 plane, perpendicular to each
other, \t being space-like, t' J! time-like:
(J! = eJ! v 3 + e-J! v 4; tJ! = eJ! v 3 - e-J! v 4· (46d)
Thus N becomes a rotation by 1t in the purely space like SytJ! plane. The M
can be calculated from M = AN

(46e)

M(J! =~(J! =A(J! =~J!+2Xv3+e-ll-2Xv4

«(
= ze2X«(J! + tJ!) +ie2X J! -tJ!) = Ch2X·(J! + Sh2X·tJ!

MtJ! = -Sh2X·(J!-Ch2X·tw
Thus M also becomes a product of two reflections, one in the v I v2 = s'vSv
the other in the v3v4 = t/J!tJ! plane. This completes the decomposition of A
into two involutions. One of the involutions can be taken to be a rotation
by 1t in an arbitrary space like plane, intersecting both the vIv2 and the
v3v4 planes, as the freedom in choosing v and Il allows us to fix the lines
sV' and \t arbitrarily in those planes. The involution characterized by (46)
will be called NVJ! henceforth. The other involution M is then a similar
REPRESENTATIONS OF TIlE POINCARE GROUP 71

rotation, in a plane, however, which is completely determined once the

svt/! plane is fixed. It will be denoted by My (it is, in fact My/! =
Nv+'»l+X). One sees the complete analogy to the furee dimensional case if
one remembers that y and X are the half angles of rotation.

The d(M) and deN) so normalized that their squares be 1 shall be denoted
by d l (My/!) and d l (Nv/!). We must show that the normalization for
d(A) = ±d1(Mv/!)d1(Nv/!) (47)
is independent of v and Jl. For this purpose, we transform
(47 a)
with deAl) where Al has the same characteristic vectors as A but different
characteristic values, namely eiv , e-iv , e/! and e-/!. Since A1MooAl- 1 = Mv/!
and AINooAI-I = Nv/! we have d(AI)dl(Moo)d(Altl = rodiMy/!) where ro
= ± 1, as the squares of both sides are 1. Hence, (47a) becomes if
transformed with deAl) just
d(A1)d(A)d(A1)-1 ±d1(Mv/!)d1(Nv/!).
= (47b)
The normalization (47) would be clearly independent of v and Jl if deAl)
commuted with d(A).

Again, the argument contained in equations (24)-(30) can be applied and

shows that
(48)
holds for every y, X, v, Jl. However, the exponential in (48) must be 1 ify
1
= 0; V = 27t/n; X = iIlJl since in this case A = A7. Thus exp (-41t2ifJl) = 1 for
every Jl and f = 0 and the left side of (47b) can be replaced by d(A); the
normalization in (47) is independent ofv and Jl.

In order to have the analogue of (45), we must show that, having. two
Lorentz transformations A = My/!Nv/! and I = Pa.~~~ we can choose v, Jl
and ex, ~ so that Nv/! = Pa.~ i.e. that the plane of rotation Sytf.L of NvlJ.
coincide with the plane of rotation ofPa.~. As the latter plane can be made
to an arbitrary space like plane intersecting both the wIw2 and the w3w4
planes (where WI' w2' w3' w4 are the characteristic vectors of I), we must
show the existence of a space like plane, intersecting all four planes vlv2'
v3v4' wlw2' w3w4. Both the first and the second pair of planes are
orthogonal.
72 CHAPTER II

One can show22 that if A and I have no common null vector as

characteristic vector, there are always two planes, perpendicular to each
other which intersect four such planes. One of these is always space like.
It is possible to assume, therefore, that both NYJl and Pa~ are the rotation
by 1t in this plane. Thus
d(A)d(1) = ±dl(MYJl)dl(NYJl)dl(Pa~)dl(Qa~) (49)
= ±d1(MyJl)d1(Qa~)'
and d(A)d(I) has the normalization corresponding to the product of two
involutions, neither of which is unity. This is, however, also the
normalization adopted for d(AI). Hence
d(A)d(1) = ±d(AI) (49a)
holds if A, I and AI are Lorentz transformations corresponding to one of
the cases (a), (b), (c) or (d) of section 4B and if A and I have no common
characteristic null vector. In addition to this (49a) holds also, assuming
deE) = ± 1, if any of the transformations A, I, AI is unity, or if both
characteristic null vectors of A and I are equal, as in this case the planes
v3v4 and w3w4 and also vlv2 and wlw2 conincide and there are many
space like planes intersecting all.

22We first suppose the existence of a real plane p intersecting all four planes v I v2' V3V4'
WI wZ' w3w4· If p intersects VI Vz the plane q perpendicular to p will intersect the plane v3v4
perpendicular to v I V2' Indeed, the line which is perpendicular to both p and v I v2 (there is
such a line as p and VI Vz intersect) is contained in both q and v3v4 . This shows that if there

.
is a plane intersecting all four planes, the plane perpendicular to this will have this property
also .

If the plane p-the existence of which we suppose for the time being--contains a time-like
vector, q will be space-like (Section 4B, [I)). Both in this case and if p contains only
space-like vectors, the theorem in the text is valid. There is a last possibility, that p is
tangent to the light cone, i.e. contains only space like vectors and a null vector Y. The
space-like vectors of p are all orthogonal to v, otherwise p would contain time-like vectors
also. In this case the plane q, perpendicular to p will contain v also. The line in which v I vz
intersects p is space-like and orthogonal to the vector in which v3 v4 intersects p. The latter
intersection must coincide with v, therefore, as no other vector p is orthogonal to any
space-like vector in it. Hence, v is the intersection of p and v3v4 and is either v3 or v4' One
can conclude in the same way that v coincides with either w3' or w4 also and we see that if
p is tangent to the light cone the two transformations A and I have a common null vector as
characteristic vector. Thus the theorem in the text is correct if we can show the existence of
an arbitrary real plane p intersecting all four planes VI vz• v3v4 • WI wZ' w3w4 ·
REPRESENTATIONS OF THE POINCARE GROUP 73

Figure 4: Fig. 4 gives a projection of all lines into the xl x2 plane.

One sees that there are, in general, two intersecting planes,
only in exceptional cases is there only one.

Let us draw a coordinate system in our four dimensional space, the Xl Xz plane of which is
the v 1v2 plane, the x3 and x4 axes having the directions of the vectors v3 - V4 and v3 + v4'
respectively. The three dimensional manifold M characterized by x4 = 1 intersects all
planes in a line, the v1v2 plane in the line at infinity of the Xl x2 plane, the v3v4 plane in the
x3 axis. The intersection of M with the w 1w2 and w3w4 planes will be lines in M with
directions perpendicular to each other. They will have a common normal through the
origin of M, intersecting it at reciprocal distances. This follows from their orthogonality in
the four dimensional space.

A plane intersecting v I v2 and v3 v4 will be a line parallel to x l x2 through the x3 axis. If we

draw such lines through all points of the line corresponding to WI w2 ' the direction of this
line will turn by 1t if we go from one end of this line to the other. Similarly, the lines going
through the line corresponding to w3w4 will tum by 1t in the opposite direction. Thus the
first set of lines will have at least one line in common with the second set and this line will
correspond to a real plane intersecting all four planes v l v2' v3v4• w l w2• w3w4 . This
completes the proof of the theorem referred to in the text.
74 CHAPTERll

If A and I have one common characteristic null vector, v3 = w3' the others,
v4 and w4 respectively, being different, one can use an artifice to prove
(49a) which will be used in later parts of this section extensively. One can
find a Lorentz transformation J so that none of the pairs I - J; A - U; AU -
J-l has a common characteristic null vector. This will be true, e.g. if the
characteristic null vectors of J are v4 and another null vector, different
from v3' w4 and the characteristic vectors of AI. Then (49a) will hold for
all the above pairs and
d(A)d(I) = ±d(A)d(I)d(J)d(r1) = ±d(A)d(IJ)d(r 1)
= ±d(A/J)d(r1) = ±deAl).

This completes the proof of (49a) for all cases in which A, I and AI have
no elementary divisors. It is evident also that we can replace in the
normalization (47) the d 1 by d. One also concludes easily that d(M)2 is in
the same representation either + 1 for all involutions M, or -1 for every
involution. The former ones will give real representations, the latter ones
representations up to the sign.

If A has an elementary divisor, it can be expressed in the ve' we' ze' VI

scheme as the matrix (Cf. equ. (20»

oIIlhOi
II0
Ae = 0 0 1 0
o0 0 1

and can be written, in the same scheme, as the product· of two Lorentz
transformations with the square 1

1 -1 1f2 0 1 000
0-1 1 0 0-1 0 0
Ae = Mello = 0 0 1 0 o0 1 0
o0 0-1 o 0 0-1

We can normalize therefore d(A) = ± d(Mo)d(No). If A can be written as

REPRESENTATIONS OF TIlE POINCill GROUP 75

the product of two other involutions also Ae = M1N1 the corresponding

normalization will be identical with the original one. In order to prove
this, let us consider a Lorentz transformation J such that neither of the
Lorentz transformations J, Nol, N 1J, AeJ = MoNol = M 1N 1J have an
elementary divisor. Since the number of free parameters is only 4 in case
(e), while 6 for case (a), this is always possible. Then, for (45a)
d(Mo)d(No)d(J) = ±d(Mo)d(Nr/) = ±d(Mrflr/)
= ±d(M1N 1J) = ±d(M1)d(N1J) = ±d(M1)d(N1)d(J)

and thus d(Mo)d(N o) = ± d(M 1)d(N 1). This shows also that even if AI is in
case (e), meA, I) = ± 1, since (49) leads to the correct normalization.

If A = MN has an elementary divisor, I not, d(A)d(I) still will have the

normalization corresponding to the product of two involutions. One can
find again a J such that neither of the transformations J, J- 1, U, NU, MNIJ,
have an elementary divisor. Then
d(M)d(N)d(/) = ±d(M)d(N)d(/)d(J)d(J)-1

±d(M)d(N)d(IJ)d(J)-1 = ±d(M)d(NIJ)d(J)-1

d(AlJ)d(r 1).
The last product has, however, the normalization corresl'onding to two
involutions, as was shown in (49a), since neither AU, nor J- 1 is in case (e).

Lastly, we must consider the case when both A and I may have an
elementary divisor. In this case, we need a J such that neither J, r1, U
have one. Then, because of the generalization of (49a) just proved, in
which the first factor is in case (e)
d(A)d(I) = ±d(A)d(I)d(J)d(r 1) = ±d(A)d(IJ)d(r1)
= ±d(A/J)d(r1)

which has the right normalization.

This completes the proof of

m(A,/) = ±1 (50)
for all possible cases, and the normalization of all D(L) of a representation
of the inhomogeneous Lorentz group up to a factor, is carried out in such a
way that the normalized operators give a representation up to the sign. It is
even carried so far that in the first two of equations (22) m = 1 can be set.
We shall consider henceforth systems of operators satisfying (7), or more
76 CHAPTER II

specifically, (22B) and (22C) with (\l(a, b) = (\l(A, a) = 1 and (22D) with
(\l(A, I) = ± 1.
E.

Lastly, it shall be shown that the renormalization not only did not spoil the
partly continuous character of the representation, attained at the first
normalization in part (A) of this section, but that the same holds now
everywhere, in the ordinary sense for T(a) and, apart from the ambiguity of
sign, also for d(A). For T(a) this was proved in part (B)(b) of this section,
for d(A) it means that to every AI' E and <\> there is such a 0 that one of the
two quantities

(51)
if A is in the neighborhood 0 of AI' The inequality (51) is equivalent to

«1=Fd(Ao»<\>,(FFd(Ao»<\» < E, (51a)

where Ao = A~lA now can be assumed to be in the neighborhood of the

unity. Thus, the continuity of d(A) at A = E entails the continuity
everywhere. 23 In fact, it would be sufficient to show that the d(X), dey)
and d(Z) corresponding to the transformations (40) converge to ± 1, as a, Y
approach 0, since one can write every transformation in the neighborhood
of the unit element as a product A = Z(O, Y3)Y(0, Y2)X(0, YI)X(al' O)Y(az,
O)Z(~, 0) and the parameters al"", <\>3 will converge to 0 as A
converges to 1. However, we shall carry out the proof for an arbitrary A
without an elementary divisor.

For d(A), equations (46) show that as A approaches E (Le., as <\> and X
approach zero) both Moo and Noo approach the same involution, which we
shall call K. Let us now consider a wave function l.!, = <\> + d l (K)<\> or, if this
vanishes", = <\> - dl(K)<\>. We have dl(K)", = ± ",. If A is sufficiently near
to unity, d l (Noo)'" will be sufficiently near to Od l (K)", = ± 0", and all we
have to show is that 0 approaches ± 1. The same thing will hold for
d l (Moo)· Indeed from d 1(Noo)'" - 0", = u it follows by applying d 1(Noo)
on both sides", - 0 2", = (dl(Noo) + O)u. As (u, u) goes to zero, 0 must
go to ± 1, and consequently, also dl(Noo)'" goes to '" or to -",. Applying

231. von Neumann, Sitz. d. Kon. Preuss. Akad. p. 76,1927.

REPRESENTATIONS OF TIlE POINCARE GROUP 77

d l (MOO) to this, one sees that d l (MoO)d l (NOO)'I' = d(A)'I' goes to ± 'I' as A
goes to unity. The argument given in (A) (b) shows that this holds not only
for 'I' but for every other function also, i.e. dA converges to ± I = deE) as
A approaches E. Thus d(A) is continuous in the neighborhood of E and
hence everywhere.

According to the last remark in part 4, the operators ± d(A) form a single
valued representation of the group of complex unimodular two
dimensional matrices C. Let us denote the homogeneous Lorentz
transformation which corresponds in the isomorphism to C by C Our task
of solving the equs. (22) has been reduced to finding all single valued
unitary representation[s] of the group with the elements [a,C] = [a,l] [O,C],
the multiplication rule of which is [a,C I ] [b,C 2] = [a + Cl b,C I C 2]. For the
representations of this group D[a, C] = T(a)d[C] we had
T(a)T(b) = T(a+b)
d[C]T(a) = T(Oz)d[C] (52a)
d[C 1]d[C 2] = d[C 1C2]
It would be more natural, perhaps, from the mathematical point of view, to
use henceforth this new notation for the representations and let the d
depend on the C rather than on the C or A. However, in order to be
reminded on the geometrical significance of the group elements, it
appeared to me to be better to keep the old notion. Instead of the equations
(22B), (22C), (22D) we have, then
T(a)T(b) = T(a+b) (52B)
d(A) T(a) = T(Aa) d(A) (52C)
d(A) d(l) = ±d(Al). (52D)

6. Reduction of the Representations of the

Inhomogeneous Lorentz Group to Representations
of a "Little Group"

This section, unlike the other ones, will often make use of methods, which
though commonly accepted in physics, must be further justified from a
rigorous mathematical point of view. This has been done, in the
meanwhile, by 1. von Neumann in an as yet unpublished article and I am
much indebted to him for his cooperation in this respect and for his
78 CHAPTER II

readiness in communicating his results to me. A reference to this paper24

will be made whenever his work is necessary for making inexact
considerations of this section rigorous.

A.

Since the translation operators all commute, it is possible24 to introduce

such a coordinate system in Hilbert space that the wave functions q,(p, l;)
contain momentum variables PI' P2' P3' P4 and a discrete variable l; so that
T(a)q,(p, s) = ei{p,a}q,(p, S). (53)
P will stand for the four variables PI' P2' P3' P4'

Of course, the fact that the Lorentzian scalar product enters in the
exponent, rather than the ordinary, is entirely arbitrary and could be
changed by changing the signs of PI' P2' P3'

The unitary scalar product of two wave functions is not yet completely
defined by the requirements so far made on the coordinate system. It can
be a summation over S and an arbitrary Stieltjes integral over the
components of p:
(\jf, q,) = tJ \jf(p, S)*q,(p, S)df(p, /;). (54)

The importance of introducing a weight factor, depending on p, for the

scalar product lies not so much in the possibility of giving finite but
different weights to different regions in p space. Such a weight
distribution g(p, /;) always could be absorbed into the wave functions,
replacing all q,(p, S) by -Yg(p,S)·q,(p,S). The necessity of introducing the
f(p, /;) lies rather in the possibility of some regions of p having zero weight
while, on the other hand, at other places points may have finite weights.
f
On account of the definite metric in Hilbert space, the integral df(p, S)
over any region r, for any S, is either positive, or zero, since it is the scalar
product of that function with itself, which is 1 in the region r of p and the
value l; of the discrete variable, zero otherwise.

24J. von Neumann, Ann. of Math. to appear shortly.

REPRESENTATIONS OF THE POINCARE GROUP 79

Let us now define the operators

P(A)$(P, /;;) = $(A-lp, /;;). (55)

This equation defines the function P(A)$, which is, at the point p, S, as
great as the function <\> at the point A-Ip, /;;. The operator peA) is not
necessarily unitary, on account of the weight factor in (54). We can easily
calculate
P(A)T(a)$(p, s) = T(a)<\>(A-lp, S) = ei{A-lp, a}$(A-lp, S),

T(Aa)P(A)<\>(P,/;;) = ei{P,Aa}p(A)$(p, /;;) = ei{P,Aa}$(A-Ip, /;;),

so that, for {A-Ip, a} = {p, Aa}, we have

P(A)T(a) = T(Aa)P(A). (56)
This, together with (52C), shows that d(A)P(Ayl = Q(A) commutes with
all T(a) and, therefore, with the multiplication with every function of p,
since the exponentials form a complete set of functions of PI' P2' P3' P4.
Thus
d(A) = Q(A)P(A), (57)
where Q(A) is an operator in the space of the Salone24 which can depend,
however, on the particular value of p in the underlying space:
Q(A)$(P, /;;) = ~Q(P,A)l;ll<\>(P, Tl)· (57a)
Tl
Here, Q(p, A)l;ll are the components of an ordinary (finite or infinite)
matrix, dependlllg on p and A. From (57), we obtain
d(A)<\>(P,s) = LQ(P, A)1;llP (A)<\>(p, 11) (57 b)

= "£..,;Q(P, Tl
A)1;l1<\>(A -Ip, n).
Tl
As the exponentials form a complete set of functions, we can approximate
the operation of multiplication with any function of PI' P2' P3' P4 by a
linear combination
f(P)<\> = LCnT(an)$· (58)
n

If we choose f(p) to be such a function that

f(P) = f(Ap) (58a)
the operation of multiplication with f(p) will commute with all operations
80 CHAPTER II

of the group. It commutes evidently with the T(a) and the Q(p, A), and on
account of (56) and (58), (58a) also with peA). Thus the operation of (58)
belongs to the centrum of the algebra of our representation. Since,
however, we assume that the representation is factorial (cf. 2), the centrum
contains only multiples of the unity and
!(P)$(P, ~) = c$(P, ~). (58b)
This can be true only if $ is different from zero only for such momenta p
which can be obtained from each other by homogeneous Lorentz
transformations, because f(p) needs to be equal to f(p') only if there is a A
which brings them into each other.

It will be sufficient, henceforth, to consider only such representations, the

wave functions of which vanish except for such momenta which can be
obtained from one by homogeneous Lorentz transformations. One can
restrict, then, the definition domain of the $ to these momenta.

These representations can now naturally be divided into the four classes
enumerated in section 3, and two classes contain two subclasses. There
will be representations, the wave functions of which are defined for p such
that
(l ){p, p} = P<0 (3) p = 0

(2) {p,p}=P=O; p:;eO (4) {p, p} = < O.

The classes 1 and 2 contain two sub-classes each. In the pOSltIve
subclasses P+ and 0+ the time components of all momenta are P4 > 0, in
the negative subclasses P_ and 0_ the fourth components of the momenta
are negative. Class 3 will be denoted by 0Q' If P is negative, it has no
index.

From the condition that d(A) shall be a unitary operator, it is possible to

infer24 that one can introduce a coordinate system in Hilbert space in such
a way that

Jr
df(p, ~) = J Ar
dj(p, 11) (59)

if Q(p, A)~n :;e 0 for the p of the domain r. Otherwise, r is an arbitrary

domain in llie space of PI' P2' P3' P4' and Ar is the domain which contains
Ap if r contains p. Equation (59) holds for all ~, 11, except for such pairs
for which Q(p, A)~T} = O. It is possible, hence, to decompose the original
representation in such a way that (59) holds within every reduced part.
REPRESENTATIONS OF THE POINCARE GROUP 81

Neither T(a) nor d(A) can have matrix elements between such 11 and /; for
which (59) does not hold.

In the third class of representations, the variable p can be dropped entirely,

and T(a)$(/;) = $(/;), i.e., all wave functions are invariant under the
operations of the invariant subgroup, formed by the translations. The
equation T(a)$(/;) = $(/;) is an invariant characterization of the
representations of the third class, i.e., a characterization which is not
affected by a similarity transformation. Hence, the reduced parts of a
representation of class 3 also belong to this class.

Since no wave function of the other classes can remain invariant under all
translations, no representation of the third class can be contained in any
representation of one of the other classes. In the other classes, the
variability domain of p remains three dimensional. It is possible,
therefore, to introduce instead of PI' P2' P3' P4 three independent variables.
In the cases 1 and 2 with which we shall be concerned most, PI' P2' P3 can
be kept for these three variables. On account of (59), the Stieltjes integral
can be replaced by an ordinary integral24 over these variables, the weight
1

factor being IP4 1- 1 = (P + pi + p; + p;)-2

{'JI, $} = t fff: 'JI(P, /;)*$(P, /;)lp41-1dPldP2dP3· (59a)

In fact, with the weight factor Ip4 1- 1 the weight of the domain r i.e., Wr =
f f fr 1P41-ldPldP2dP3 is equal to the weight of the domain W AT as
required 25 by (59). Having the scalar product fixed in this way, P(A)
becomes a unitary operator and, hence, Q(A) will be unitary also.

We want to give next a characterization of the representations with a given

P, which is independent of the coordinate system in Hilbert space. It
follows from (53), that in a representation with a given P the wave
functions 'JII' 'JI2' ... which are different from zero only in a finite domain

25The invariance of integrals of the character of (59a) is frequently made use of in

relativity theory. One can prove it by calculating the Jacobian of the transformation
2 2 21
P'i = AilPI + Ai7P2 + A i'3fJ3 +(P + PI + P2 + P3)'2 (i=1,2,3)
1

which comes out to be (P + pi + P~ + p;)2(P + p'i + p'; + p';)~. Equ. (59a) will not be
used in later parts of this paper.
82 CHAPTER II

of p, form an everywhere dense set, to all elements of which the

infinitesimal operators of translation can be applied arbitrarily often
limh-n(T(he)-I)n'l' = limh-n(eih{p, e} - l)n'l' = in{p, e}n'l', (60)
h=O h=O
where e will be a unit vector in the direction of a coordinate axis or
oppositely directed to it. Hence for all members 'I' of this everywhere
dense set

lim
h=O k
I±h-2(T(2he k) - 2T(he k) + 1)'1' = (pi + p~ + pi - p~)'I'
= -P'l', (61)
where t1c is a unit vector in (or opposite) the kth coordinate axis and the ±
is + for k = 4, and - for k = 1,2,3.

On the other hand, there is no <\> for which

lim
h=O k
I±h- (T(2he
2 k) - 2T(hek + I)<\> (6Ia)

if it exists, would be different from -P<\>o Suppose the limit in (6Ia) exists
and is -P<\> + <\>'. Let us choose then a normalized '1', from the above set,
such that ('1',<\>') = 0 with 0 > 0 and an h so that the expression after the lim
sign in (6Ia) assumes the value -P<\> + <\>' + u with (u, u) < 0/3 and also the
expression after the lim sign in (61), with oppositely directed ek becomes
-P'I' + u' with (u', u') < 0/3. Then, on account of the unitary character of
T(a) and because ofT(-a) = T(at!
('I', L±h- (T(2he 2T(he
2 k) - k) + 1)<1»,

(L±h- (T(-2he
k
2 k) - 2T(-hek ) + 1)'1',<1»,
k
or
-P('I',<\» + ('1',<\>') + ('I',u) -P('I',<\» + (u',<\»,
which is clearly impossible.

Thus if the lim in (6Ia) exists, it is -P<\> and this constitutes a

characterization of the representation which is independent of similarity
transformations. Since, according to the foregoing, it is always possible to
find wave functions for a representation, to which (6Ia) can be applied,
every reduced part of a representation with a given P must have this same
P and no representation with one P can be contained in a representation
with an other P. The same argument can be applied evidently to the
positive and negative sub-classes of class 1 and 2.
REPRESENTATIONS OF THE POINCARE GROUP 83

B.

Every automorphism L -7 L° of the group allows us to construct from one

representation D(L) another representation
(62)
This principle will allow us to restrict ourselves, for representations with
finite, positive or negative P, to one value of P which can be taken
respectively, to be +1+ and -1. It will also allow in cases 1 and 2 to
construct the representations of the negative sub-classes out of
representations of the positive sub-classes.

The first automorphism is aO = aa, AO = A. Evidently Equs. (12) are

invariant under this transformation. If we set, however,
TO(a)<\> = T(aa)<\>; dO(A)<\> = d(A)<\> ,
then the occurring p
TO(a)<\> = T(aa)<\> = ei{p,aa}<\> = ei{ap,a}<\>,
will be the p occurring for the unprimed representation, multiplied by a.
This allows, with a real positive a, to construct all representations with all
possible numerical values of P, from all representation with one numerical
value of P. If we take a negative, the representations of the negative sub-
classes are obtained from the representations of the positive sub-class.

In case P = 00 evidently all representations go over into themselves by the

transformation (62). In case P = 0+ and P = 0_ it will turn out that for
positive a, (62) carries every representation into an equivalent one.

c.

On account of (53) and (56), (57), the equs. (52B) and (52C) are
au~om~tical1y satisfied and the Q(p, A)~T\ must be determined by (52D).
ThIS gIves
L Q(P,A)r.,TlQ(A-Ip,l)Tle<\>(I-1A-Ip,S)
TIe
= ±IQ(p,AI)r.,e<\>(I-1A-1p,S). (63)
e
Since this must hold for every <\>, one would conclude
84 CHAPTERll

LQ(P,A)~11Q(A-IP))11e = ±Q(P,AI)~e· (63a)

11
Actually, this conclusion is not justified, since two wave functions must be
considered to be equal even if they are different on a set of measure zero.
Thus one cannot conclude, without further consideration, that the two sides
of (63a) are equal at every point p. On the other hand,24 the value of
Q(p,A)~'I1 can be changed on a set of measure zero and one can make it
continuous in the neighborhood of every point, if the representation is
continuous. This allows then, to justify (63a). It follow from (63a) that
Q(p, 1)~'I1 = <\'11.

Let us choose 15 now a basic Po arbitrarily. We can consider then the

subgroup of all homogeneous Lorentz transformations which leave this Po
unchanged. For all elements 'A" t of this "little group," we have
L Q(PO,'A,)~11Q(Po,t)11e = ±Q(Po,'A,t)~e
11
q('A,)q(t) = ±q('A,t) , (64)
where g('A,) is the matrix q('A,)~'I1 = Q(po, 'A,)~'I1. Because of the unitary
character of Q(A) , the Q(po' A)~'I1 is unitary matrix and q('A,) is unitary also.

If we consider, according to the last paragraph of Section 5, the group

formed out of the translations and unimodular two-dimensional matrices,
rather than Lorentz transformations, the ± sign in (64) can be replaced by a
+ sign. In this case, 'A, and t are unimodular two-dimensional matrices and
the little group is formed by those matrices, the corresponding Lorentz
transformations ~, t to which leave Po unchanged ~Po = t Po = Po.

Adopting this interpretation of (64), one can also see, conversely, that the
representation q('A,) of the little group, together with the class and P of the
representation of the whole group, determines the latter representation,
apart from a similarity transformation. In order to prove this, let us define
for every p a two-dimensional unimodular matrix a(p) in such a way that
the corresponding Lorentz transformation

(65)
brings Po into p. The a(p) can be quite arbitrary except of being an almost
everywhere continuous function of p, especially continuous for p = Po and
a(po) = 1. Then, we can set
REPRESENTATIONS OF TIlE POINCARE GROUP 85

d( a(p )-1 )~(pO' s) = ~(p, S),

d(a(p))~(p, S) = ~(pO' S)· (66)
This is equivalent to setting in (58)
Q(p,a(p)) = 1 (66a)
and can be achieved by a similarity transformation which replaces <1>(p, S)
by ~ Q(po, a(ptl)t;l1~(P, 11)· As the matrix Q(po, a(ptl) is unitary, this
is a unitary transformation. It does not affect, furthermore, (53) since it
contains p only as a parameter.

Assuming this transformation to be carried out, (66) will be valid and will
define, together with d(A), all the remaining Q(p, A) uniquely. In fact,
calculating d(A)~(p, S), we can decompose A into three factors

(67)

The second factor ~ = a(pt1Aa(k 1p) belongs into the little group:
&.(prlfi(klp)po = &.(pr11klp = O(prlp = po. We can write, therefore
(kIp = p')
d(A)<1>(P, S) = d(a(p))d(~)d(a(p'))-l~(p, S)
= d(~)d(a(p'))-l<1>(po, S) (67a)
= Lq(~)~Ttd(a(p'rl)~(po' 11) = Lq(~)~Tt<1>(P', 11)·
Tt Tt

This shows that all representations of the whole inhomogeneous Lorentz

group are equivalent which have the same P and the same representation of
the little group. Further than this, the same holds even if the
representations of the little group are not the same for the two
representations but only equivalent to each other. Let us assume ql (A) =
sq2(A)s-1. Then by replacing <1>(p,~) by ~s(~, 11)<1>(P, 11) we obtain a new
form of the representation for which (53) still holds but q2(~) for the little
group is replaced by ql (~). Then, by the transformation just described (Eq.
(66)), we can bring d(A) for both into the form (67a). The equivalence of
two representations of the little group must be defined as the existence of a
unitary transformation which transforms them into each other. (Only
unitary transformations are used for the whole group, also).

On the other hand, if the representations of the whole group are equivalent,
the representations of the little group are equivalent also: the
representation of the whole group determines the representation of the
86 CHAPTER II

little group up to a similarity transformation uniquely.

The representation of the little group was defined as the set of matrices
Q(po' A)1;11 if the representation is so transformed that (53) and (66a) hold.
Having two equivalent representations D and SDS- I = DO for both of
which (53) and (66a) holds, the unitary transformation S bringing the first
into the second must leave all displacement operators invariant. Hence, it
must have the form (57a), i.e., operate on the Sonly and depend on p only
as on a parameter.
S<j>(p,s) = LS(P)~l1<j>(P,ll). (68)
'11

Denoting the matrix Q for the two representations by Q and QO, the
condition SD(A) = DO(A)S gives that
LS(P)~'I1Q(P,A)119 = LQ°(P,A)~l1S(A-lp)119
11 11
holds, for every A, for almost every p. Setting A = a(PI) we can let p
approach PI in such a way that (68a) remains valid. Since Q is a
continuous function of p both Q(p, A) and QO(p, A) will approach their
limiting value 1. It follows that there is no domain in which
S(PI) = S(a(Plrlpl) = SCpo) (69)
would not hold, i.e., that (69) holds for almost every Pl. Since all our
equations must hold only for almost every p, the S(P)1;1l can be assumed to
be independent of p and (68a) then to hold for every p also. It then follows
that the representations of the little group D and DO are transformed into
each other by S1;1'r

The definition of the little group involved an arbitrarily chosen momentum

vector po. It is clear, however, that the little groups corresponding to two
different momentum vectors Po and p are holomorphic. In fact they can be
transformed into each other by a(p): If A is an element of the little group
leaving p invariant then a(pt l Aa(p) = p is an element of the little group
which leaves Po invariant. We can see furthermore from (67a) that if A is
in the little group corresponding to p, i.e. Ap = P then the representation
matrix q(P) of the little group of Po' corresponding to p, is identical with
the representation matrix of the little group of p, corresponding to A =
a(p )pa(pt 1. Thus when characterizing a representation of the whole
inhomogeneous Lorentz group by P and the representation of the little
group, it is not necessary to say which Po is left invariant by the little
REPRESENTATIONS OF TIlE POINCARE GROUP 87

group.
D.
Lastly we shall determine the constitution of the little group in the
different cases.

1+. In case 1+ we can take for Po the vector with the components 0, 0, 0, 1.
The little group which leaves this invariant obviously contains all rotations
in the space of the fIrst three coordinates. This holds for the little group of
all representations of the first class.

00. In case 00' little group is the whole homogeneous Lorentz group.

-1. In case P = -1 the Po can be assumed to have the components 1,0,0,

O. The little group then contains all transformations which leave the form
- x22 - x32 + x24 tnvanant,
. . .
I.e., . the 2 + 1 d·ImenSlona
IS . I homogeneous
Lorentz group. The same holds for all representations with P < O.

0+. The determination of the little group for P = 0+ is somewhat more

complicated. It can be done, however, rather simply, for the group of
unimodular two dimensional matrices. The Lorentz transformation
corresponding to the matrix (~ ~) with ad - bc = 1 brings the vector with
, ,
the components xl' x2, x3' x4' into the vector with the components xl' x2'
I I 18
x3' x4 ' where

(ab)
cd
(X4+~3X1+ix2)
xClx2 x4-x3
(a:<)
b d
= (~4+~/3Xll+iX».
x Cix'2 x'4-x 3
(70)

The condition that a null-vector Po' say with the components 0, 0, 1, 1 be

invariant is easily found to be lal 2 = 1, c = O. Hence the most general
element of the little group can be written

°
( e-i~12 (x +el~/2
iy')ei~/2)
'
(71)

with real x, y, ~ and

as t(x, y) 8(~) where
° ~ ~ < 41t. The general element (71) can be written

(71a)

The multiplication rules for these are

88 CHAPTER II

= t(X + X', Y + y'),

t(x, y )t(x', y') (71b)
8(~)t(x, y) = t(XCOS ~ + ysin~, -xsin ~ + ycos ~)8(~), (71c)
8(~)8(W) = 8(~ + W). (71d)
One could restrict the variability domain of ~ in 8(~) from 0 to 21t. As
8(21t) commutes with all elements of the little group, it will be a constant
and from 8(21t)2 = 8(41t) = 1 it can be 8(21t) = ± 1. Hence 8(~ + 21t) =
±8(~) and inserting a ± into equation (71d) one could restrict ~ to 0 :5; ~ <
21t

These equations are analogous to the equations (52)-(52D) and show that
the little group is, in this case, isomorphic with the inhomogeneous
rotation group of two dimensions, i.e. the two dimensional Euclidean
group.

It may be mentioned that the Lorentz transformations corresponding to

t(x,y) have elementary divisors, and constitute all transformations of class
e) in 4B, for which ve = po' The transformations 8(~) can be considered to
be rotations in the ordinary three dimensional space, about the direction of
the space part of the vector po. It is possible, then, to prove equations (71)
also directly.

7. The Representations of the Little Groups

A. Representations of the three dimensional rotation group

by unitary transformations.

The representations of the three dimensional rotation group in a space with

a finite member of dimensions are well known. There is one irreducible
representation with the dimensions 1,2,3,4, ... each, the representations
with an odd number of dimensions are single valued, those with an even
number of dimensions are two-valued. These representations will be
denoted by DU) (R) where the dimension is 2j + 1. Thus for single valued
representations j is an integer, for double valued representations a half
integer. Every finite dimensional representation can be decomposed into
these irreducible representations. Consequently those representations of
the Lorentz group with positive P in which the representation of the little
group-as defined by (64)-has a finite number of dimensions, can be
decomposed into such representations in which the representation of the
REPRESENTAnONS OF TIlE POINCARE GROUP 89

little group is one of the well known irreducible representations of the

rotation group. This result will hold for all representations of the
inhomogeneous Lorentz group with positive P, since we shall show that
even the infinite dimensional representations of the rotation group can be
decomposed into the same, finite, irreducible representations.

In the following, it is more appropriate to consider the subgroup of the two

dimensional unimodular group which corresponds to rotations, than the
rotation group itself, as we can restrict ourselves to single valued
representations in this case (cf. equations (52». From (70), one easily
sees 18 that the condition for (~ ~) to leave the vector with the components
0, 0, 0, 1 invariant is that it shall be unitary. It is, therefore, the two
dimensional unimodular unitary group the representations of which we
shall consider, instead of the representations of the rotation group.

Let us introduce a discrete coordinate system in the representation space

and denote the coefficients of the unitary representation by q(R)kA. where R
is a two dimensional unitary transformation. The condition for the unitary
character of the representation q(R) gives

(72)

(72a)

This show also that Iq(R)u1 ~ 1 and the q(R~A. are therefore, as functions
of R, square integrable:
J Iq(R)k1.J2dR

exists if f ... dR is the well known invariant integral in group space.

Since this is finite for the rotation group (or the unimodular unitary group),
it can be normalized to 1. We then have
L JIq(R)kifdR
k 1
= >J
1:'
Iq(R)k1.fdR = 1. (73)

The (2j + 1)2 D(j)(R)kl form,26 a complete set of normalized orthogonal

functions for R. We set

26H. Weyl and F. Peter, Math. Annal. 97, 737, 1927.

90 CHAPTER II

(74)

We shall calculate now the integral over group space of the product of
D(j)(R)*kl and
q(RS)/q.L = ~ q(R)ki..q(S)~. (75)

The sum on the right converges uniformly, as for (72a)

00 00 00 1

~ Iq(R)ki..q(s)~1 ~ (~lq(R)ki..12~ Iq(S)~12)2

00 1

~ (~lq(S)~12)2

can be made arbitrarily small by choosing an N, independent of R, making

the last expression small. Hence, (75) can be integrated term by term and
gives

JDV)(R)~(RS)/qJ.dR = L JDV)(R)~(R)ki..q(S)~dR. (76)

Substituting LmD(j)(RS)kmD~S-l)ml for D(j)(R)kl one obtains

L DW(S-l):UJDV)(RS):nq(RS)/q.LdR (77)
m

.t.-q(S);.~J D(})(R)~(R)ki..dR.

In the invariant integral on the left of (77), R can be substituted for RS and
we obtain, for (74) and the unitary character

~DV)(S)lmct = ~q(S)~CJ:;. (78)

Multiplying (78) by D(h)(S):, the integration on the right side can be

carried out term by term again, since the sum over A. converges uniformly
00 00 00 1 00 1

~ IC~(S)~I ~ (~ICJ:;12~ Iq(S)~12)2 ~ (~1C;:;12)2.

This can be made arbitrarily small, as even L;.L
kl(2j+l)-1ICJ:;1 2
o

converges, for (74) and (72a). The integration of (78) yiel;!s thus
REPRESENTAnONS OF TIlE POINCARE GROUP 91

t: cj~ln = OjhOIPJt· (79)

From q(R)q(E) = q(R) follows q(E) = 1 and then q(RI) = q(Rt l = q(R)t.

This, with the similar equation for D(j)(R) gives

(80)

or

C~ = cj(;. (81)
On the other hand q(E)kA. = Ou yields
I,CJ:k = Ok).. (82)
jk

These formulas suffice for the reduction of q(R). Let us choose for every
finite irreducible representation D(j) an index k, say k = O. We define then,
in the original space of the representation q(R) vectors v(kjl) with the
components
kl fa,..k3
Cjkl' Cjkl' L.jkl' ... •

The vectors v(kjl) for different j or 1 are orthogonal, the scalar product of
those with the same j and 1 is independent of 1. This follows from (79) and
(81)

(v<Jl/l'), v (kjl» = ~cj;;,c~ (83)

= ~ C~c7-~k = OiJ~,OlrC~.

The v(kjl) for all k, j, 1, form a complete set of vectors. In order to show
this, it is sufficient to form, for every v, linear combination from them, the
v component of which is 1, all other components O. This linear
combination is
~CVk (Jcjl) (84)
£.J jlk V •
kjl

In fact, the A component of (84) is, on account of (79) and (82)

92 CHAPTERll

~ vk Jc').. ~CvA s::

£.J CilkCild = £.J ill = U VA•
(85)
kil il

However, two v with the same j and t but different first indices k are not
orthogonal. We can choose for every j and t, say t = 0 and go through the
vectors v(ljO), v(2jO), . .. and, following Schmidt's method, orthogonalize
and normalize them. The vectors obtained in this way shall be denoted by

w(niO) = } _J v (f..}O) (86)

"'( (,rnA .

Then, since according to (83) the scalar products (V(kjl), V(Aj/) do not
depend on t, the vectors

(86a)

will be mutually orthogonal and normalized also and the vectors w(njl) for
all n, j, t will form a complete set of orthonormal vectors. The same holds
for the set of the conjugate complex vectors w(njl)*. Using these vectors as
coordinate axes for the original representation q(R), we shall find that q(R)
is completely reduced. The v component of the vector q(R)v(kjl)* obtained
by applying q(R) on v(kj/)* is

(87)

The right side is uniformly convergent. Hence, its product with (2h+l)
n(h)(R);n can be integrated term by term giving

L f (2h + 1)D(h)(R);nq(R)vIlC~dR (88)

11

= °hj.(),nCfi!·
Thus we have for almost all R

.4, D(j)(R) i v (kii)*)v, (88a)

1
or
q(R) V (kil)* L D(;)(R) il v (kjl)*. (88b)
i
Since both sides are supposed to be strongly continuous functions of R,
(88b) holds for every R. In (86a), for every n, the summation must be
REPRESENTATIONS OF THE POlNCill GROUP 93

carried out only over a finite number of A... We can write therefore
immediately
q(R)w(njl)* = LDW(R)ilw(njl)*. (89)
i

This proves that the original representation decomposes in the coordinate

system of the w into well known finite irreducible representations O(j)(R).
Since the w form a complete orthonormal set of vectors, the transition
corresponds to a unitary transformation.

This completes the proof of the complete reducibility of all (finite and
infinite dimensional) representations of the rotation group or unimodular
unitary group. It is clear also that the same consideration applies for all
f
closed groups, i.e., whenever the invariant integral dR converges.

The result for the inhomogeneous Lorentz group is: For every positive
numerical value of P, the representations of the little group can be, in an
1
irreducible representation, only the 0(0), 0(2), 0(1), ... , both for P + and
for P_. All these representations have been found already by Majorana and
by Dirac and for positive P there are none in addition to these.

B. Representations of the two dimensional Euclidean group

This group, as pointed out in section 6, has a great similarity with the
inhomogeneous Lorentz group. It is possible, again24 , to introduce
, 'momenta" , i.e. variables l;, 11 and v instead of 1; in such a way that
t(x, y)cp(Po, l;, 11, v) = ei(x~+JTl)cp(Po' l;, 11, v). (90)
Similarly, one can define again operators R(~)
R(~)cp(Po' ~, 11, v) = CP(Po, ~', 11', v), (91)
where
~' =~cos~ - 11sin~, (91a)
11' = ~sin ~ + 11cos~.
Then O(~)R(~tl = S(~) will commute, on account of (71c), with t(x, y)
and again contain ~, 11 as parameter only. The equation corresponding to
(57a) is
O(~)cp(Po' ~, 11, v) = LS(~) v CJ)CP(Po' ~', 11', 00). (92)
94 CHAPTER II

One can infer from (90) and (92) again that the variability domain of S, 11
can be restricted in such a way that all pairs S, 11 arise from one pair So, 110
by a rotation, according (91a). We have, therefore two essentially
different cases:
S2 + 112 ~ *0 a.)

S2 + 112 ~ 0, i.e. S = 11 = O. b.)

The positive definite metric in the S. 11 space excludes the other
possibilities of section 6 which were made possible by the Lorentzian
metric for the momenta, necessitated by (55).

Case b) can be settled very easily. The "little group" is, in this case, the
group of rotations in a plane and we are interested in one and two valued
irreducible representations. These are all one dimensional (eis~)
S(~) = eis~ (93)
where s is integer or half integer. These representations were also all
found by Majorana and by Dirac. For s = 0 we have simply the equation
1
1]<1> = 0, for s = ± '2 Dirac's electron equation without mass, for s = ± 1
Maxwell's electromagnetic equations, etc.

In case a) the little group consists only of the unit matrix and the matrix
(01_?) of the two dimensional unimodular group. This group has two
irreducible representations, as (1) and (-1) can correspond to the above
two dimensional matrix of the little group. This gives two new
representations of the whole inhomogeneous Lorentz group, corresponding
to every numerical value of 8. Both these sets belong to class 0+ and two
similar new sets belong to class 0_.

The final result is thus as follows: The representations P +j of the first

subclass P + can be characterized by the two numbers P and j. From these
P is positive, otherwise arbitrary, while j is an integer or a half integer,
positive, or zero. The same holds for the subclass P_. There are three
kinds of representations of the subclass 0+. Those of the first kind 0 +s can
be characterized by a number s, which can be either an integer or a half
integer, positive, negative or zero. Those of the second kind 0+(8) are
single valued and can be characterized by an arbitrary positive number 8,
those of the third kind 0'/8) are double-valued and also can be
characterized by a positive 8. The same holds for the subclass 0_. The
representations of the other classes (00 and P with P < 0) have not been
REPRESENTATIONS OF THE POINCARE GROUP 95

determined.

8. Representations of the Extended Lorentz Group

A.

As most wave equations are invariant under a wider group than the one
investigated in the previous sections, and as it is very probable that the
laws of physics are all invariant under this wider group, it seems
appropriate to investigate now how the results of the previous sections will
be modified if we go over from the "restricted Lorentz group" defined in
section 4A, to the extended Lorentz group. This extended Lorentz group
contains in addition to the translations all the homogeneous
transformations X satisfying (10)
XFX' = F (10')
while the homogeneous transformations of section 4A were restricted by
two more conditions. From (10') it follows that the determinant of X can
be + 1 or -1 only. If its -1, the determinant of Xl = XF is +1. If the
four-four element of Xl is negative, that of X2 = -Xl is positive. It is
clear, therefore, that if X is a matrix of the extended Lorentz group, one of
the matrices X, XF, -X, -XF is in the restricted Lorentz group. For F2 =
1, conversely, all homogeneous transformations of the extended Lorentz
group can be obtained from the homogeneous transformations of the
restricted group by multiplication with one of the matrices
1, F, -1, -F. (94)
The group elements corresponding to these transformations will be
denoted by E, F, I, IF. The restricted group contains those elements of the
extended group which can be reached continuously from the unity. It
follows that the transformation of an element L of the restricted group by
F, I, or IF gives again an element of the restricted group. This is,
therefore, an invariant subgroup of the extended Lorentz group. In order
to find the representations of the extended Lorentz group, we shall use
again Frobenius' method. lS

We shall denote the operators corresponding in a representation to the

homogeneous transformations (94) by d(E) = 1, d(F) , d(I), d(IF). For
deriving the equations (52) it was necessary only to assume the existence
96 CHAPTER II

of the transformations of the restricted group, it was not necessary to

assume that these are the only transformations. These equations will hold,
therefore, for elements of the restricted group, in representations of the
extended group also. We normalize the indeterminate factors in d(F) and
del) so that their squares become unity. Then we have d(F)d(l) =
ood(I)d(F) or del) = rod(F)d(I)d(F). Squaring this, one obtains 002 = ± 1.
We can set, therefore
d(IF) = d(l)d(F) = ±d(F)dCl) (95)
d(F)2 = d(l)2 = 1; d(IF)2 = ±1.
Finally, from
d(F)D(L 1)d(F) = oo(L 1)D(FL 1F) (96)
we obtain, multiplying this with the similar equation for L2
oo(L1)oo(Lz) = oo(L1L 2)
which, gives oo(L) = 1 as the inhomogeneous Lorentz group (or the group
used in (52B)-(52D)) has the only one dimensional representation by the
unity (1). In this way, we obtain
d(F)D(L)d(F) = D(FLF), (96a)
d(I)D(L)d(l) = D(IU) , (96b)
d(IF)D(L)d(IF)-1 = D(IFLFl). (96c)

B.

Given a representation of the extended Lorentz group, one can perform the
transformations described in section 6A, by considering the elements of
the restricted group only. We shall consider here only such representations
of the extended group, for which, after having introduced the momenta, all
representations of the restricted group are either in class 1 or 2, i.e. P ;::: 0
but not 00, Following then the procedure of section 6, one can find a set of
wave functions for which the operators D(L) of the restricted group have
one of the forms, given in section 6 as irreducible representations. We
shall proceed, next to find the operator d(F). For the wave functions
belonging to an irreducible D(L) of the restricted group, we can introduce
a complete set of orthonormal functions 'l'1(P, ~), 'l'2(P, ~), .. '. We then
have
D(L)wip, ~) = :LD(L)J.Ik'l'J.l(P, ~). (97)
J.l
REPRESENTATIONS OF TIlE POINCARE GROUP 97

The infinite matrices D(L)/lk defined in (97) are unitary and form a
representation which is eqUIvalent to the representation by the operators
D(L). The D(L), d(F) are, of course, operators, but the D(L)Jlk are
components of a matrix, i.e. numbers. We can now form the wave
functions d(F)'VI' d(F)2' d(F)'V3' ... and apply D(L) to these. For (96a)
and (97) we have
D(L)d(F)'Vk = d(F)D(FLF)'Vk
d(F)ID(FLF)J.llc'VJ.l (97a)
J.l
= ID(FLF)J.llcd(F)'V w
J.l

The matrices DO(L)Jlk = D(FLF)llk give a representation of the restricted

group (FLF is an element of the restricted group, we have a new
representation by an automorphism, as discussed in section 6B). We shall
find out whether DO(L) is equivalent D(L) or not. The translation
operation D° is
TO(a) = d(F)T(a)d(F) = T(Fa) (98)
which, together with (53) shows that D° has the same P as D(L) itself. In
fact, writing
UI<l>(P, s) = <l>(Fp, s) (99)
one has V~l = VI and one easily calculates VITO(a)V I = T(a). Similarly
for VIdO(A)V I one has
U1dO(A)U1<l>(P, s) = U1d(FAF)U1<l>(P, s)
=d(FAF)Ul<l>(Fp.~) = LQ(Fp. FAF)~TPl<l>(FA-lp.11) (99a)
~ Q(Fp. F AF)~T]<l>(A-fP.11)·
= L.J
This means thai the similarity transformation with VI brings TO(a) into
T(a) and dO (A) into Q(Fp, FAF)P(A). Thus the representation of the "little
group" in VIdO(A)V I is
qO('A,) = Q(FPo,FAF).
For this latter matrix. one obtains from (67a)
qO('A,) = Q(FPo.F'AF) = q(a(Fpo)-IFAFa(Fpo» = q('A 0 ) (100)

where 'A° is obtained from 'A by transforming it with Fa(FpO)'

98 CHAPTERll

The representations DO(L) and D(L) are equivalent if the representation of

q(A) is equivalent to the representation which coordinates q(AO) to A. The
a(Fpo) is a transformation of the restricted group which brings PO into
a(FpO)po = FPO' (Cf. (65).) This transformation is, of course, not uniquely
determined but if a(FpO) is one, the most general can be written as
a(Fpo)t, where tpo = Po is in the little group. For q(t-1a(Fpot1Aa(FPo)t)
= q(t-1)q(a(Fpot1Aa(FPo»q(t), the freedom in the choice of a(FPO) only
amounts to a similarity transformation of qO(A) and naturally does not
change the equivalence or non equivalence of qO(A) with q(A).

For the case P+' we can choose Po in the direction of the fourth axis, with
components 0, 0, 0, 1. Then Fpo = Po and a(Fpo) = 1. The little group is
the group of rotations in ordinary space and FAF = A. Hence qO(A) = q(A)
and DO(A) is equivalent to D(A) in this case. The same holds for the
representations of class P_.

For 0+ we can assume that Po has the components 0, 0, 1, 1. Then the

components of Fpo are 0, 0, -1, 1. For a(FPO) we can take a rotation by 1t
about the second axis and Fa(Fpo) will be a diagonal matrix with diagonal
elements 1, -1, 1, 1, i.e., a reflection of the second axis. Thus if A is the
transformation in (70), AO = a(FpotlFAFa(Fpo) is the transformation for
which

(101)

This is, however, clearly AO = A*. Thus the operators of qO(A) are obtained
from the operators q(A) by (cf. (71a»
to(x, y) = t(x, -y) (lOla)
80(~) = 8(-~).
For the representations O+s with discrete s, the qO(A) and q(A) are clearly
inequivalent as 8o(~) = (e-is~) and 8(~) = (eis~), except for s = 0, when
they are equivalent. For the representations 0/3), 0'/3), the qO(A) and
q(A) are equivalent, both in the single valued and the double valued case,
as the substitution 11 ~ -11 transforms them into each other. The same
holds for representations of the class 0_. If DO(L) and D(L) are equivalent
D-IDO(L)U = D(L), (102)
the square of U commutes with all D(L). As a consequence of this, U2
REPRESENTATIONS OF TIlE POINCARE GROUP 99

must be a constant matrix. Otherwise, one could form, in well known

manner,27 an idempotent which is a function of U2 and thus commutes
with D(L) also. Such an idempotent would lead to a reduction of the
representation D(L) of the restricted group. As a constant is free in U, we
can set
U2 = 1 (102a)

c.

Returning now to equation (97a), if DO(L) = D(FLF) and D(L) are

equivalent (P > 0 or 0+, 0_ with continuous B or s = 0) there is a unitary
matrix UJ.l.V' corresponding to U, such that
LD(FLF)IqJ.Ullv = L UIqJ.D(L)llv
Il Il
(102b)

Let us now consider the functions

<\>v = 'l'v + L UIlvd(F)'l'1l' (103)
Il

Applying D(L) to these

D(L)<\>v = D(L)'l'v + L Ul1yD(L)d('F)<\>1l
L Ullvd(F)D(FLF)'l'1l
Il
D(L)'l'v + (103a)
Il
I.D(L) llv'l'l1 + I.Ullvd(F)D(FLF)IqJ.'l'k

ID(L)llv('l'1l + {UIqJ.d(F)'l'k) = L D(L)llv<\>1l'

Il k Il

Similarly

27J. von Neumann, Ann. of Math. 32,191,1931; ref. 2, p. 89.

100 CHAPTERTI

d(F)<\>v = d(F)'ljIv + 2, U ~v'I'~

~
(l03b)
= 2, U~v('I'~ + 2, Uk)!d(F)'I'k) = I U~v<\>w
~ k ~

Thus the wave functions <\> transform according to the representation in

which D(L~llv corresponds to Land U IlV to d(F). The same holds for the
wave functIons
<\>'v = 'l'v - I U~vd(F)'I'w (104)
Il

except that in this case (-UIlV ) corresponds to d(F). The 'l'v and d(F)'I'v
can be expressed by the <\> and <\>'. If the 'I' and d(F)'I' were linearly
independent, the <\> and <\>' will be linearly independent also. If the d(F)'I'
were linear combinations of the '1', either the <\> or the <\>' will vanish.

If we imagine a unitary representation of the group formed by the Land

FL in the form in which it is completely reduced out as a representation of
the group of restricted transformations L, the above procedure will lead to
a reduction of that part of the representation of the group of Land FL, for
which D(L) and D(FLF) are equivalent.

If D(L)llv and DO(L)IlV are inequivalent, the 'l'k and d(F)'I'v = 'I"v are
orthogonal. This is again a generalization of the similar rule for finite
unitary representation~.28 One can see this in the following way:
Denoting Mkv = ('I'k' 'l'v) one has
Mkv = ('I'k' "'v) = (D(L)'I'/r.' D(L)'I"v)

= ~D(L);PO(L)A.vM,.v.;

M D(L)tMD (L). °
Hence
(105)
From these, one easily infers that MMt commutes with D(L), and MtM
commutes with DO(L). Hence both are constant matrices, and if neither of

28Cf. e.g. E. Wigner, ref. 4, Chapter XII.

REPRESENTATIONS OF THE POINCARE GROUP 101

them is zero, M and Mt are, apart from a constant, unitary. Thus D(L)
would be equivalent DO(L) which is contrary to supposition. Hence MMt
= 0, M = 0 and the 'I' are orthogonal to the d(F)'I' = ",. Together, they give
a representation of the group formed by the restricted Lorentz group and
F. If they do not form a complete set, the reduction can be continued as
before.

One sees, thus, that introducing the operation F "doubles" the number of
dimensions of the irreducible representations in which the little group was
the two dimensional rotation group, while it does not increase the
underlying linear manifold in the other cases. This is analogous to what
happens, if one adjoins the reflection operation to the rotation groups
themselves. 29

D.

The operations d(l) can be determined in the same manner as the d(F) were
found. A complete set of orthonormal functions corresponding to an
irreducible representation of the group formed by the Land FL shall be
denoted by '1'1' '1'2' . .. . For this, we shall assume (97) again, although
the D(L) contained therein is now not necessarily irreducible for the
restricted group alone but contains, in case of 0 +s or O_s and finite s, both s
and -so We shall set, furthermore
d(F)'I'k = L d (F)I1k'VW (106)
fI. .

We can form then the functions d(I)'Vl' d(I)'V2' .. '. The consideration,
contained in (97a) shows that these transform according to D(ILI)Jlk for the
transformation L of the restricted group:
D(L)d(I)'Vk = LD(ILl)l1kd(l)'Vw (106a)
fI.
Choosing for L a pure translation, a consideration analogous to that
performed in (98) shows that the set of momenta in the representation L ~
D(ILI) has the opposite sign to the set of momenta in the representation
D(L). If the latter belongs to a positive subclass, the former belongs to the
corresponding negative subclass and conversely. Thus the adjunction of

291. Schur, Sitz. d. kon. Preuss. Akad. pages 189,297, 1924.

102 CHAPTER IT

the transformation I always leads to a "doubling" of the number of states,

the states of "negative energy" are attached to the system of possible
states. One can describe all states '1'1' '1'2' ... , d(I)'I'l' d(I)'I'2' . .. by
introducing momenta PI' P2' P3' P4 and restricting the variability domain of
p by the condition {p, p} = P alone without stipulating a definite sign for
P4'

As we saw before, the d(I)'I'l' d(I)'I'2' are orthogonal to the original set of
wave functions '1'1' '1'2' .. '. The result of the application of the operations
D(L) and d(F) to the '1'1' '1'2" .. (i.e., the representation of the group
formed by the L, FL) was given in part C. The D(L)d(I)'I'k are given in
(106a). On account of the normalization of d(l) we can set
d(/)d(I)'I'k = 'l'k' (106b)
For d(F)d(I)'I'k we have two possibilities, according to the two possibilities
in (95). We can either set
d(F)d(I)'I'k = d(/)d(F)'I'k = I.d(F)~(I)'I'w (107)
J.L
or
d(F)·d(I)'I'k = -d(I)d(F)'I'k = - I.d(F)~(I)'I'w (107a)
J.L

Strictly speaking, we thus obtain two different representations. The

system of states satisfying (107) could be distinguished from the system of
states for which (107a) is valid, however, only if we could really perform
the transition to a new coordinate system by the transformation I. As this
is, in reality, impossible, the representations distinguished by (107) and
(107a) are not different in the same sense as the previously described
representations are different.

I am much indebted to the Wisconsin Alumni Research Foundation for

their aid enabling me to complete this research.

Madison, Wis.
REPRESENTATIONS OF THE POINCARE GROUP 103

Group Theoretical Discussion of

Relativistic Wave Equations

by V. Bargmann and E.P. Wigner

Princeton University

Read before the Academy, November 18, 1947

Reprinted from Proc. Nat. Acad. Sci. (U.S.A.) 34, 211 (1948).
104 CHAPTERll

I ntroduction. 1- The wave functions, 'V, describing the possible states of a

quantum mechanical system form a linear vector space V which, in
general, is infinite dimensional and on which a positive definite inner
product (<\>,'V) is defined for any two wave functions <\> and 'V (Le., they
form a Hilbert space). The inner product usually involves an integration
over the whole configuration or momentum space and, for particles of
higher spin, a summation over the spin indices.

If the wave functions in question refer to a free particle and satisfy

relativistic wave equations, there exists a correspondence between the
wave functions describing the same state in different Lorentz frames. The
transformations considered here form the group of all inhomogeneous
Lorentz transformations (including translations of the origin in space and
time). Let 'VZ' and 'Vz be the wave functions of the same state in two
Lorentz frames I' and I, respectively. Then 'VI' = U(L)'Vz, where U(L) is a
linear unitary operator which depends on the Lorentz transformation L
leading from I to I'. By a proper normalization, U is determined by L up to
a factor ± 1. (For all details the reader is referred to the paper of reference
2, hereafter quoted as (L).) Moreover, the operators U form a single- or
double-valued representation of the inhomogeneous Lorentz group, i.e., for
a succession of two Lorentz transformations L 1, L2, we have
(1)

Since all Lorentz frames are equivalent for the description of our system, it
follows that, together with'll, U(L)'II is also a possible state viewed from
the original Lorentz frame I. Thus, the vector space V contains, with every
'V, all transforms U(L)'V, where L is any Lorentz transformation.

The operators U may also replace the wave equation of the system. In our
discussion, we use the wave functions in the "Heisenberg" representation,
so that a given 'V represents the system for all times, and may be chosen as
the "Schroedinger" wave function at time 0 in a given Lorentz frame I. To
find 'lit ' the Schroedinger function at time to' one must therefore transform
o
to a frame I' for which t' = t - to' while all other coordinates remain
unchanged. Then'Vt = U(L)'V, where L is the transformation leading from
o
I to 1'.

A classification of all unitary representations of the Lorentz group, i.e., of

all solutions of (1), amounts, therefore, to a classification of all possible
relativistic wave equations. Such a classification has been carried out in
REPRESENTAnONS OF TIlE POINCARE GROUP 105

(L). Two representations U(l) and qL) = VU(L)V- 1, where V is a fixed

unitary operator, are equivalent. If the system is described by wave
functions V, the description by

'ij = VVI (2)

is isomorphic with respect to linear superposition, to forming the inner

-
product of two wave functions, and also to the transition from one Lorentz
--
frame to another. In fact, if VI' = U(L)VI' then ~, = VVr = Q:L)'ij. Thus,
one obtains classes of equivalent wave equations. Finally, it is sufficient
to determine the irreducible representations since any other may be built
up from them.

Two descriptions which are equivalent according to (2) may be quite

different in appearance. The best known example is the description of the
electromagnetic field by the field strength and the four vector potential,
respectively. It cannot be claimed either that equivalence in the sense of
(2) implies equivalence in every physical aspect. Thus, two equivalent
descriptions may lead to quite different expressions for the charge density
or the energy density in configuration space (cf. Fierz,3) because (2) only
implies global, but not local, equiValence of the wave functions. It should
be emphasized, however, that any selection of one among the equivalent
systems or the superposition of non-equivalent systems in any particular
way involves an explicit or implicit assumption as to possible interactions,
the positive character of densities, etc. Our analysis is necessarily
restricted to free particles and does not lead to any assertions about
possible interactions.

The present discussion is not based on any hypothesis about the structure
of the wave equations provided that they be Lorentz invariant. In
particular, it is not necessary to assume differential equations in
configuration space. But it is a result of the analysis in (L) that every
irreducible wave equation is equivalent (in the sense of (2)) to a system of
differential equations. For the relation of the present point of view to other
treatments of the subjects see reference 11.

In the present note, we shall give, for every representation of (L), a

differential equation the solutions of which transform according to that
representation. We also will discuss in some detail the infinitesimal
operators which generate the irreducible representations determined in (L),
and we shall characterize these representations, and hence the covariant
differential equations, by certain invariants constructed from the
106 CHAPTERll

infinitesimal operators. This is of some interest, because the infinitesimal

operators are closely related to dynamical variables of the system.
L. Garding4 has recently shown that even in the infinite dimensional case
one can rather freely operate with infinitesimal transformations. In
particular, it immediately follows from his discussion (although it is not
explicitly stated in his note) that the familiar commutation rules remain
valid.

1. The Infinitesimal Operators of the Lorentz Group.-The metric tensor is

assumed in form ~4 = 1, gIl = g22 = g33 = -1, gld = 0 (/c:tl) and gkI = gld'
The scalar product of two four vectors a, b will be denoted by {a,b} =
akbk. Both c, the velocity of light, and f\ Planck's constant divided by 21t,
are set equal to 1.

The Infinitesimal Operators Pk and Mid' A translation in the xt-direction is

generated by Pk' a rotation in the (Jc - xl) plane by Mid = -Mlk (k, I =
1, ... ,4). These operators are Hermitian, and the unitary operators U
which represent the finite Lorentz transformation are obtained by
exponentiation; thus U = exp (-iapk) corresponds to a translation by the
amount a. in the direction x/t;: Clearly, Pk are the four momenta of the
system, and M23 , M3I , MI2 the three components of the total angular
momentum. The following commutation rules hold (where [A, B] = AB -
BA)
[Mkl.Mmnl = i(gl,,!Jkn - gm!Jln + gJalrflm - glJtf/cm), (3a)
IPk,P11= 0 [MId,Pml = i(glrrPk- gknl'l)' (3b)
We now define four operators wk by
(wI, w2,~, w 4 ) = (v234' v3I4' vI24' v32I)' (4a)
vldm = PJ!vflm + pzMmk + P,,!J1d
= MlrrPk + MmJI'l + Mk/Pm' (4b)
Note that wk is a "pseudo-vector," i.e., it is a vector only with respect to
Lorentz transformations of determinant 1. By (3),
[MId,wm] = i(glmwk- g/cmwI) IPk,wl]' (5)
It follows from (3) and (5) that the two operators,
P=pkPk; mvklm = -wkwk>
W = (1I6)v ld (6)
commute with all the infinitesimal operators Mkl and Pk' Therefore, they
have constant values (i.e., they are multiples of the unit operator) for every
irreducible representation of the Lorentz group. (The familiar arguments
REPRESENTAnONS OF THE POINCARE GROUP 107

which establish this for finite dimensional representations can be carried

over to the infinite dimensional case. (Cf. V. Bargmann, reference 5, p.
602.»

W may also be written in the form

W = 1/2Mk#ldp"Rm-M~lmp"PI' (7)
(This quantity was first introduced by W. Pauli, cf. Lub~mski.6) The scalar
product ...JcPk vanishes.

2. Summary of the Results of (L).-( a) For every irreducible representation

the states", may be expressed as functions", (p, ~) of the momentum
vector P and an auxiliary variable ~ which may assume a finite or an
infinite number of values. The momenta P are either all zero, or they vary

°
over the manifold pkpk = P, with a constant value P. We confine ourselves
*'
to the cases in which P 0, and either P > or P = 0, because the
remaining cases are unlikely to have direct physical significance.5.

(b) To every inhomogeneous Lorentz transformation I= 'A.k~l + ak (in

vector form: y = Ax + a) corresponds a unitary operator U(L) defined by
U(L)",(P,~) = e-i{a,p}Q(p, A)",(A-lp,~), (8)
where Q(P,A) is a unitary operator which may depend on P but affects only
the variable~. The inner product (<\>,"') is obtained by an integration over
the manifold pkpk = P and by a summation or integration over the variable
S·
(c) The subgroup of the homogeneous Lorentz transformations which keep
a fixed momentum vector Po unchanged is cal1ed "little group." (The little
groups defined by different vectors Po are isomorphic.) The unitary
operators Q(Po,A) (where Apo = Po) form an irreducible representation of
the little group and determine the irreducible representation U(L) of the
inhomogeneous Lorentz group.

In all cases the operators Mid have the form M k/ + Sk/' where the
o

Mk/ = i(Pk".E--- prh) = i(Pk8r-Plgk.)1.... (9)

o apl apk Ij g apj

act on the variables p and correspond to the orbital angular momenta,

while the Sid act on the variables S and correspond, to the spin angular
108 CHAPTER II

momenta. Both Mid and Sid satisfy the commutation rules (3a). Since the
o

Mid do not contribute to vldm (cf. (4b», we have

o

(10)
or, introducing the three-dimensional vector operators,

S= (S23,S31,SI2); S' = (SI4,S24,S34); P= (pl,p2,p3); (lOa)

;=(wl,w2,w3); w 4=p.S; ;=p4S-<p x S').
Clearly, Mid may also be replaced by Sid in the expression (7) for W.

For a fixed momentum vector Po the operators wk are the infinitesimal

generators of the little group. Since wkpk = 0, only three of them are
linearly independent.

3. Classification of the Irreducible Representations.- We now turn to a

brief summary of the main results, including the characterization of the
representations in terms of the operators p and w. A more detailed
discussion will follow in the succeeding sections.

The classes found in (L) (§§ 7, 8) are these:

I. Ps ' Particles of finite mass and spin s.-Here P = m 2 > O. In the rest
system of the particle, the momentum vector has only the one non-
vanishing component p4 = ± m, hence, by (lOa), W = m2S2. The operator
p-l W represents the square of the spin angular momentum, and has the
value s(s + 1) (s = 0, 112, 1, ... ) for an irreducible representation. For a
given momentum vector there are 2s + 1 independent states. The
representation U(L) is single €>r double valued according to whether s is
integral or half integral. The lowest cases (s = 0, 112, 1) correspond to the
Klein-Gordon, Dirac and Proca equations, respectively.

II. Os' Particles of zero rest mass and discrete spin.-These

representations may be considered limiting cases of the representations Ps
for m ~ O. Then both P and W are equal to zero, and do not suffice to
characterize these representations. For a given momentum vector, there
*
exist 2 independent states if s 0 (corresponding to two different states of
polarization), and there is only one state if s = O. Right and left circularly
polarized states are described by the operator equations wk = sPk' and wk =
REPRESENTATIONS OF THE POINCARE GROUP 109

-sPk respectively, so that the representation Os is characterized by P0, =

wkwi = s2pkPl' the lowest cases (s = 0, 112, 1) correspond to the scalar

wave equation, the neutrino equation, and Maxwell's equations,
respectively.

III. 0(8) and 0'(8). Particles of zero rest mass and continuous
spin.-Here, P = 0, W = 8 2, where 8 is a real positive number. For a given
momentum vector there exist infinitely many different states of
polarization, which may be described by a continuous variable. The
representation 0(8) is single valued, while 0'(8) is double valued.

To construct these representations explicitly, we shall select, in each case,

one among the equivalent sets of wave equations, define a Lorentz
invariant inner product (<\l,W), and prove the operator relations stated
above. We shall operate in momentum space; this is particularly simple,
because the momenta (but not the coordinates) are defined by the Lorentz
group, as infinitesimal translations.

4. The Class Ps.-(a) s = 0. Here, the variable ~ assumes only one value
and may therefore be omitted. Consequently, Q(P, A) = 1 (cf. reference 8),
and for the little group the trivial one-dimensional representation is
obtained. Hence, Ski = 0, and wk = 0. The wave equation reduces to pkpk
= m2 ; the inner product (<\l,W) is determined by the norm (W,W)of a wave
function,
(11)

the integral being extended over both sheets of the hyperboloid

pkpk = P = m 2. The expression (11) is Lorentz invariant, because dO. is an
invariant volume element in momentum space. For the wave function in
configuration space, one finds

W(x) =
-3

(21t)""f J
e-i(p.x)W(p)dn, (12)
where x stands for xl, x2 , x3, x4. It is well known that (W,W) cannot be
simply expressed in configuration space, because for the Klein-Gordon

°
equation the density is indefinite, and the integral over the density in
configuration space coincides with (11) only if W(P) = whenever p4 < 0.
1
(b) s 'iN with N = 1, 2, 3, .. '. For particles of higher spin we use the
=
equations first derived by Dirac7 in the form essentially given in reference
8. We use for ~ the N four-valued variables Sl' .... , SN in which the wave
110 CHAPTERll

function 'Jf{p; ~1' ... , ~N) is symmetric. We define for every ~v four-
dimensional matrices Yvk of the same nature as are used in Dirac's electron
theory:
y/yJ + y}y/ = 2gkIi (k,l = 1,2,3,4). (13)

The ywith different lower indices V commute. The r, T, -t are skew

Hermitian, t, is Hermitian. The wave equations then are
Y}Pk'V = m'V (v = 1,2, , . , ,N). (14)

It follows from any of these equations in well known fashion that

gkIpJI'I'V = pkPi'¥ =m2'V. (14a)
The infinitesimal operators of displacement are the P, those of four-
dimensional rotation the MkI = MkI + Ski with MkI of (9) and
o o

(15)
v
where the

Yvk = gklYvl (ISa)

satisfy the same relations (13) as do the Y/

The invariant scalar product is

JL
('V,'V) = I 'V*Yl4.y24 , .. YN4'VldQ (16)
In fact, (16) is in~ant both with respect to the operators M and also with
o

respect to the S. The latter condition means that

«1 + ieSkI)'V, (1 + ieSkI)'V) = ('V, 'V),
up to terms with e2 , This formula can be verified by observing that, if both
k and I are space like Ski is a Hermitian matrix and commutes with the
product of the t. If either k or I is 4, Ski is skew Hermitian, but
anticommutes with product of the t.
It follows that (16) is invariant with
respect to the proper Lorentz transformations. Its invariance with respect
to reflections, etc" can also be shown.

The absolute sign in (16) is necessary to make it positive definite. We now

shall give (16) a new form which is based on the set of identities
REPRESENTATIONS OF TIlE POINCARE GROUP 111

(17)
where Av is a skew Hermitian matrix involving only the fIrst v of the t
(and the p). We can prove (17) best by induction: applying P4YV+14 to
(17) gives, by means of(14),
(P4)V+IYV+14yv4 ... Y24Y1 4'V = mVp4Yv+14'V+P4YV+14Av'V (17a)
=mV+l'V+ (-mvp{YV+lk+P4YV+14Av)'V (k= 1,2,3).
The last bracket is Av+l: it is skew Hermitian and involves only the fIrst
v + 1 of the Y so that (17) is established by induction. Setting v = N in
(17), multiplying with yand summing over the 1; yields
P4~'V*Y14Y24. ··YN4'V = ~1'V12+ L,'V*AN'V. (17b)
~ ~ ~

Because of the skew Hermitian nature of AN' the last term is imaginary.
Since the two other terms of (17b) are real, they must be equal. As a
result, we can write for (16) also
('V,'V) = JIm/p4~ 1'V12dn. (18)

At the same time, (18) permits us to give another form to the scalar
product,
('V,'V) = J1P4rN-ltl'V12dPldP2dP3' (18a)
which differs from (18) or (16) by the positive constant m-N . It may be
worth noting here that the absolute signs in (16), and in the defInition (11)
of dn (or in (18a», can be omitted in case of an odd N. This makes it
possible to defIne a simple positive defInite scalar product in coordinate
space by means of (12). In particular, for N = I, (16) (or (ISa» equals the
integral of 1'V21over ordinary space. In case of even N (integer spin s) no
simple positive defInite scalar product can be defined in coordinate space.

It is now established that the solutions of (14) form a Lorentz invariant set
in which a positive definite scalar product (16) or (18a) can be defined.
We shall now determine the representation of 2 to which the solutions
belong and will also calculate the invariants P and W.

In order to define a little group, we choose as momentum Po with the

components 0, 0, 0, m. The little group then becomes the group of
rotations in ordinary space. If we assume that the t are diagonal, with
diagonal elements 1, 1, -1, -1, equation (14) shows that only those
components of 'V can be different from zero which correspond to the fIrst
112 CHAPTERll

two rows of 'Yv' There are 2N such components, the rest of the 4N
components of 'II must vanish. Even these components will not be
independent: as a result of the symmetry of the 'I' in the ~, all components
of 'II will be equal in which the same number 1C of the N indices ~
correspond to the first row of the 'Y, the N - K other indices to the second
row. Since K can assume any of the values between 0 and N, there are N +
1 such components. If P4 = -m, the same considerations will hold, except
that the last two rows of'Y will play the role which the first two rows play
in case ofP4 = m.

In order to determine the transformation properties of these N + 1 = 2s + 1

independent components under the elements of the little group, we note
that the space like M give zero if applied to 'II with a purely time like P =
o
Po' We need only to calculate, therefore, the effect of the Sid on '1'. Since,
t,
in particular, 1I2i'Yt'Y2 commutes with but is not identical with it, we can
assume that it is diagonal and has the diagonal elements 112, -112, 112, -112.
If the sum of such 1I2i'Yl'Y2 is applied to the component of 'II in which 1C of
the ~v correspond to the first row, N - K to the second row, this component
will be multiplied by 1I2K - 1I2(N - K) = K - s. Since 1C runs from 0 to N =
2s, the M12'1' = S12'1' will run from -s'l'to s'l'. Hence the representation of
the little group in question is D(s), as was postulated.

Because of (lOa), W becomes m 2(S23 2 + S31 2 + S122) or, since the S23'
S31' S12 are the infinitesimal operators ofD(s), we have W = m 2s(s + I) as
given9 in §3. The value ofP is m2 because of (14a).

5. The Class 0s-(a) s = O. The corresponding discussion in the preceding

section may be literally applied to this case, with the exception that m = 0
and that the integral (11) is to be extended over the light cone.

(b) The wave equations can be obtained by setting m = 0 in (14). The

infinitesimal operators continue to be given by (9) and (15). The scalar
product must be defined by (18a) because (16) vanishes for all",. The
invariance of this scalar product follows from the invariance of (18a) for
finite mass because, except at PI = P2 = P3 = 0, the wave function is
continuous in m.

The essential difference between finite and zero mass is that, in the latter
case, not only the infinitesimal operators but also the wave equation are
REPRESENTATIONS OF THE POlNCARE GROUP 113

invariant under anyone of the operators <I>y = iyylYy2y}yy4. As a result,

for m = 0, the linear manifold defined by (14) can be decomposed into
invariant manifolds by giving definite values to the <I>y. In particular,
we shall be concerned henceforth with the manifold defined by (14) and
ry'" = '" (v = 1,2" .. ,N), (19a)
and with the other one for which
(v=12"'N)
" , (19b)
holds. Both manifolds are invariant under proper Lorentz transformations
but go over into each other by reflections: they correspond physically to
right and left circular polarization. lO

Let us now again choose a particular momentum vector Po in order to

define the little group. The covariant components of Po shall be 0, 0, 1, 1.
The wave equations (14) then can be written, after multiplication with y},
in the form
(v = 1,2, ... ,N). (20)

It is now advantageous to assume that the --(f are diagonal, their diagonal
elements being 1, 1, -1, -1. Equation (20) then expresses the fact that", for
the Po in question is different from zero only if all 1; have values
corresponding to the first two rows of the y. Since the r commute with the
ff but are not identical with them, they may be also assumed to be
diagonal, with diagonal elements 1, -1, 1, -1. Hence, in the manifold
defined by (20) and (19a) all components of", vanish (for P = Po) unless
all 1; have values corresponding to the first rows of the r.
the manifold
(20), (19a) is one dimensional for given p. The same holds for the
manifold defined by (20), (19b) except that in this case "'(Po; SI' ... , sn)
differs from zero only if all Shave values corresponding to the second row
of the y. For given momentum, '" has only two independent components.

The infinitesimal operators of the little group are M 12, M13 - M 14 , M 23 -

M24 which leave Po invariant. The corresponding M give again zero if
o

applied to '" at p = Po' The S corresponding to the second of the above

operators (cf. (15), (15a)) is a sum of matrices 1I2i(y}y} + y}y}). It
vanishes if applied to our", as can be seen by applying yy ly} to (20). The
same holds for M23 - M 24 . On the other hand, 1I2iy}y} gives 112", if
applied to the '" of (20), (19a), and gives - 112", if applied to the '" of (20),
114 CHAPTERlI

(19b). One sees this most easily by applying 1I2iyv ly/ to (20) and
making use of (19). As a result, M12'1' = ± 1I2N'I' = ±S'l' for the two
manifolds in question: these indeed belong to the representation Os' of the
inhomogeneous Lorentz group.

The value of the invariant P is zero. The above also involves a calculation
of the w for the'll at P = Po: we have w3'1' = M12'1' = ± s'l', wI'll = (M42 +
M23 )'I' = 0, w2'1' = (M31 + M I4 )'I' = 0, - w4'1' = M12'1' = ± s'l'. It follows
that the value of the second invariant W = - (w4)2 + (w 1)2 + (w2)2 + (w3)2
is also zero for all the manifolds Os; these cannot be characterized by P
and W. However, these manifolds can be characterized by the equation P =
owith the additional set
wk = sPk and wk = -sPk' (21)
the + applying to (19a), the - to (19b). Both these equations are invariant
with respect to proper Lorentz transformations. If reflections are to be
included, one can combine them into wkwl = s2pII'I'

6. The Class O(S). - Here, the auxiliary is a space like four vector ~ of
length I, orthogonal to p. The scalar function 'I'(p,~) is determined by the
equations 11
gklpII'I'l' = 0; gklpk~l'l' = 0; gkl~k~l'l' = -'1', (22)
Pka'l'ra~k = -is'll, (22a)
with a real positive constant S. By (22a), for every real number p,
'I'(p,~ + pp) =e-ip:::'I'(p,~). (23)

The infinitesimal operators of displacement are the Pk' those for rotations
are the M of (9) plus the
o

(24)

In order to find the invariant scalar product, we introduce, for every vector
P on the light cone, two real space like vectors u(l)(p) and u(2)(p) of length
one, orthogonal to p and to each other, so that
{u(r)(p),p} = 0, {u(r)(p),u(s)(p)} = 8rs (r,s = 1,2). (25)
Then ~ is a linear combination of p, u(l)(p), u(2)(p),
REPRESENTATIONS OF THE POINCARE GROUP 115

(26)
where a and the ~ are real. {~,~} = -1 implies ~12 + ~l = 1, hence ~I +
i~2 = ei't with a suitable real angle 'to ,!,(p,~) is therefore a function of p, a,
't,
(27)
The choice of the t/(p) is, of course, not unique. Let v(r)(p) be another
system of vectors which satisfy (25). They may be expressed in the form
(26), i.e.,
v(r)(p) = K'p +LAsrU(s)(p) (r,s = 1,2).
s

By (25), the matrix Asr is orthogonal. In terms of the v(r) , ~ = a'p +

LrWrv(r)(p), where Wr = LsAs,~s' In particular
WI + iW 2 = /t'; 't' = ±('t + A) (28)
A depending on the ASf' By (23), I<I>(P, a, 't)1 = I<I>(P, 0, 't)I, and we define
the norm of,!, by
('!','!') = f 1<I>(P,0,'t)1 2dOd't. (29)
This expression is independent of the choice of the u(r). In fact, let <I>(P, a,
't) = <I>/(P, a', 't') where the primed variables refer to another set v(r). Then
1<1>(P , 0, 't)1 = 1<1>'(P, a', 't/)1 = 1<1>(P, 0, 't')I, and Idi/d'tl = 1. To prove the
Lorentz invariance of (29) we proceed as follows: If a homogeneous
Lorentz transformation maps p on A-Ip, and ~ on A-l~, we may, in
particular, choose the u(r)(p) in the new system to be the transforms of the
original ones; then the coefficient a, ~1' ~2 in (26), and hence 't remain
unchanged, and the integral (29) is invariant.

If we choose as the basic vector again Po with the components 0, 0, 1, 1,

the infinitesimal operators of the little group are again M 12, M13 - M 14 ,
M 23 - M24 . The M parts of these give zero for p = Po' the S parts of the
o

latter two are

S13 - S14

= -i~k·~- + .l...) + i(~3 - ~4).l..., (30a)

d~3 d~4 d~l
116 CHAPTER II

S23-S24
= -i~2(l.... + l....) + i(~3 - ~~. (30b)
a~3 a~ a~2
Because of (22a), the first term gives, if applied to 'I' at P = Po just 5~1'Y
and 5~2'1', respectively. The second terms vanish because of the second
equation of (22). Hence", is not invariant under the "displacements" M13
- M14 and M23 - M24 in ~ space, and the sum of the squares of the
"momenta" is (~12 + ~l) 52 = 52 because of the last equation of (22).
This is also the value ofW, while P = O.

7. The Class O'(5).-Since the discussion of this last case follows the
pattern of the preceding section we confine ourselves to stating the main
results. We introduce, in addition to the vector ~, a discrete spin variable 1;
which can assume four values. The wave equations become
Y'Pk'l' = 0;
gldp61'l' = 0; gld~k~''l' = -'I'. (31)

(31a)
The parameters (l and 't are introduced as before. The norm is given by
J
('I', 'II) = P4-2I, IcI>(P,0,'t)1 2dP 1dP2 dP3d't. (32)
(Cf. (18a) and (29).) Again W", = s2v, P'If = o.
It may be remarked that the scalar product has a simple positive definite
form in coordinate space for these equations. 11

1 All the essential results of the present paper were obtained by the two authors
independently, but they decided to publish them jointly.
2 Wigner, E.P., Ann. Math., 40, 149-204 (1939).
3 Fierz, M., Helv. Phys. Acta, XII, 3-37 (1939).
4 Garding, L., Proc. Nat. Acad. Sci., 33,331-332 (1947).
5 Gelfand L., and Neumark, M., J. Phys. (USSR), X, 93-94 (1946); Harish-Chandra, Proc.
Roy Soc. (London), A, 189, 372-401 (1947); and Bargmann, Y., Ann. Math., 48, 568-640
(1947), have determined the representations of the homogeneous Lorentz group. These are
representations also of the inhomogeneous Lorentz group. In the quantum mechanical
interpretation, however, all the states of the corresponding particles are invariant under
translations and, in particular, independent of time. It is very unlikely that these
REPRESENTATIONS OF THE POINCARE GROUP 117

representations have immediate physical significance. In addition, the third paper contains
a determination of those representations for which the momentum vectors are space like.
These are not considered in the present article as they also are unlikely to have a simple
physical interpretation.
6 Lublmski, J. K. Physica, IX, 310-324 (1942).
7 Dirac, P. A., M., Proc. Roy. Soc., A, 155,447-459 (1936).
8 The literature on relativistic wave equations is very extensive. Beside the papers quoted
in reference 11, we only mention the book by the Broglie, L., Theorie generaie des
particuies b. spin (Paris, 1943), and the following articles which give a systematic account
of the subject: Pauli, W., Rev. Mod. Phys., 13, 203-232 (1941); Bhabha, H.J., Rev. Mod.
Phys., 17, 203-209 (1945); Kramers, H. A., Belinfante, F. 1., and Lubanski, J. K., Physica,
Vill, 597-627 (1941). In this paper, the sum of (14) over all v was postulated; (14a) then
has to be added as an independent equation (except for N = 1). Reference 11 uses these
equations in the form given by Kramers, Belinfante and Lublmski.
9 One may derive this result in a more elegant way, without specializing the coordinate
system. For the sake of brevity, we omit this derivation.
10 de Wet, 1.S., Phys. Rev., 58, 236-242 (1940), in particular, p. 242.
11 Wigner, E. P., ZPhysik, (1947).
118 CHAPTER II

Unitary representations of the Lorentz group

By P. A. M. DIRAc, F.R.S., Bt John'8 College, Cambridge

(Received 31 May 1944)

Certain quantities are introduced which are like tensors in space·time with an infinite
Nlumerable number of eemponents and with an invariant positive definite quadratic form
for their equ8red length. Some of the main properties of these quantities are dealt' with.
and some applications to quantum mechanics are pointed out.

1. INTRODUCTION
Given any group, an important mathematical problem is to get a matrix representa-
tion of it, which means to make each element of the group correspond to a matrix
in such a way that the matrix corresponding to the product of two elements is the,
product of the matrices corresponding to the factors. The matrices may be looked
upon as linear transformations of the co-ordinates of a vector and then each element
of the group corresponds to a linear transformation of a field of vectors. Of specia.l
interest are the unitary representations, in whlch the linear transforma.tions leAve
invariant a positive definite quadratic form in the co-ordinates of a vector.
The Lorentz group is the group of linear transformations of four real varia:bles
;0' ;1' ;1' ;" such that ;~ -;~ - ;=.- ;~ is invariant. The finite representations of this
group, i.e. those whose matrices have a finite number of rows and columns, are all
well known, and are dealt with by the usual tensor analysis and its extension spinor
analysis. None of them is unitary. The group has also some infinite representations
which are unitary. These do not seem to ha.ve been studied much, in spite of their
possible importance for physical applications.
The present paper gives a new method of attack on these representations, which
was suggeste9 by Fock's quantum theory of the harmonic oscillator. It leads to a
new kind of tensor quantity in space-time, with an infinite number of components
and a positive definite expression for its squared length.

2. THREE-t>IMENSIONAL THEORY
This section will be devoted to some preliminary work applying to the rotation
group of three-dimensional Euclidean space. Take an ascending power series
(1)
in a real variable 61 with real coefficients ar' Consider these coefficients to be the
co-ordinates of a vector in a certain space of an infinite number of dimensions, and
define the. squared length of the vector to be
l:ci r!a~, (2)
The series (2) must converge for the vector to be a finite one.
[ 284 ]

Reprinted from Proc. Roy. Soc. (London), A183, 284 (1945).

REPRESENTATIONS OF THE POINCARE GROUP 119

Unitary repre&entationa of the Lorentz group

Take two more simila.r power series l;' b.U and l;' c,U in the real variables E.
s.
and and consider their coefficients to be the co-o:n!nates of vectors in two more
vector spaces, with squared lengths defined by the corresponding formula to (2).
Now multiply the three vector spaoestogether. A general vector in the product
space will be a sum of products of vectors of the three original vector SP~. and
its co-ordinates A,.., can be represented as the coefficients in a power series
p = ~o A,..,ErE:g (3)

in the three variables EI' Ell' Ea. The s~uared length of such a vector is
~ci r! s!t! A~., (4)

and two vectors with co-ordinates A,.., and B,.., have a scalar product
~or!s!t!A,..,B,..,. (5)

If the variables El' E., Sa are subjected to a linear transformation, going over into
say, the power series (3) will go over into a power series in E~, E~, E~,
E~, E~, E~,

p = ~A:",sirs~·E~.
in which the coefficients A' are linear functions of the previous coefficients A. Thus
each linear transformation of the fs generates a. linear transformation of the
coefficients A.
The theorem will now be proved: A li~r trans/ormation 0/ EI' EI' E. which leo.1Je8
EI + EI +EI invariant generate8 a linear trans/ormation 0/ the eoeJficientB A which
leo.fJe8 the sqwred length (4) invariant. Consider first the infinitesimal transformation

(6)
which leaves ~~ + ~I + ~~ invariant, e being a small quantity whose square is negligible.
Substituting into (3), one gets
p = ~A""(sir~~·+reEir-IE~HI-seEir+lE~·-I)Et
Hence A;'" = A,.., + (r + I) eAr+1,a-1,I- (s + I) eA r - I ..+1,,'
in which A ret with a negative suffix is counted as zero. Thus
~r!s! t! A;':' = ~r! s! t! [A:. + 2(r+ l)eA,..,A r+1,a_1,I- 2(s+ l)eAr_~1.tA,..,].
The last two terms in the [ ] here cancel, as may be seen by substituting r - I for r
in the former and s - I for 8 in the latter, and hence the squared length (4) is invariant
for the tra.ns:formation (6). Any linear transformation of E1, Ea, Ea whioh leaves
EJ+Q+EI invariant can be built up from the infinitesima.l transformation (6) and
Bimila.r infinitesimal transformations with E1, EI and Ea permuted, together with
possibly a reflexion El = - Ei, E. = Ei. E. = E~, which obviously leaves the squared
length (4) invariant, and hen,ce the theorem is proved.
The group of transfon:n:ations of the E's which leave Ef + EI + Elinva.ria.nt is the
rotation group in three-dimensional Euclidean &p&oe, 80 the tra.ns:formations of the
120 CHAPTER II

286 P. A .. M. Dirac
coefficients A provide a representation of this rotation group. One may restrict the
function P to be homogeneous, of degree u say, and then the representation is a
finite one. The coefficients A then form the components of a symmetrical tensor
of rank u, the connexion with the usual tensor notation being effected by taking
Ara to be U!fr!8!t! times the usual tensor component with the suffix I occurring
r times, 2 occurring 8 times and 3 occurring t times, as may be seen from the
invariance of expreasion;(3) with (61,6,,63) transforming like a vector.
One can make a straightforward generalization of the foregoing theory by illtro-
ducing other triplets of variables, 1/1' 1/" 1/3 and '1' 'I' '8 sa.y, which transform to-
gether with 61' 6a, 6a, and setting up a power series in all the variables. The trans-
formations of the coefficients of these more general power series will provide further
representations of the three-dimensional rotation group. If such a more general
power series is restricted to be homogeneOus, its coefficients will form the components
of an unsymmetrical tensor.

3. FOUR-DIMENSIONAL THEORY
Take a descending power series
ko/6o+ kl/fJ+ka/sg+ ks/fJ+ ... (7)

in a real variable 60 with real coefficients k n . Consider these coefficients to be the

co-ordinates of a vector in a certain space of an infinite number of dimensions, and
defuie the squared length of the vector to be
~ k!/nL (8)

Multiply this vector space into the vector space of the preceding section. A general
vector in the product space will have co-ordinates Anrst which can be represented
'a,s the coefficients in a power series

Q= ~o Anrst6on-1S~6;s~ (9)
in the four variables SO' SI' 6" Sa. The squared length of such a vector is
~cin!-lr!8!t!A:"'., (10)

and two vectors with co-ordinates Anrae and Bnrae have a scalar product
~"'n'-lr'8't'
*'0 • • • • .an..
A B nrae· (11)

The series (7) may be extended baokwards to inoludesome terms with non-
negative powers of 60,80 that coefficients k" occur with negativen-values, lea.djng
to coeffici,nts Anrae with negative n-values. S~ce n I is infinite for n negative, these
new coefficients
I'
do not contribute to the squared length of a veotor or the scalar
product of two vectors. Thus the terms with non-negative powers of €o should be
counted as corresponding J;o the vector zero, and whether they are present in an
expansion or not does not matter.
REPRESENT ATIONS OF THE POINCARE GROUP 121

U mtary representations of the Lorentz group 287

Now apply a Lorentz transformation to the fa,
(12)
the ~'a satisfying certain conditions so that €~ - sf - €I- €~ is invariant. This makes
sgg~ go over into a finite polynomial in s~, s;, €~, si, and £0,,-1 go over into-'

(13)

which may be expanded in ascending powers of s~, s;,

S~ and descending powers
of~. (The question of the convergence of this expansion is.left to the next ~ction,
so as not to break the main argument here.) The power series (9) then goes over
into a series
(14)
in ascending powers of 61' S2' Sa and descending powers of So' with coefficients A'
which are linear functions of the previous coefficients A. There may be terms in
(14) with non-negative powers of £0, but on account of what was said above these
can be dtscarded. if such terms are present in the original series (9), they will not
affect any of the coefficients with non-negative n-values in (14).
The Lorentz transformation thus generates a linear transformation of the coeffi-
cients An... with non-negative n-values. The theorem will now be proved: The
transformation of the coeJficients A leaves the sqWlred length (10) invariant.
Consider first the infinitesimal Lorentz transformation
60 = S~ + €S;" 61 = s; + €S~, 62 = 6~, Sa = S~. (15)
Substituting into (9), one gets
Q = 1:Anret[s~ -n-ls~r - (n + I) es~ -n-ls~r+1 + res~ -ns~r-lH~·s~'.
Hence
in which Anr" with a negative value for r, 8 or t is counted as zero. Thus
1:n!-lr! s! t! A~r.
= 1:n!-lr! 8! t! [A!r"- 2neA n... A n_l,r_W+ 2(r+ 1)€An+l,r+l...,A ....].

The last two terms in the [ ] here cancel, as may be seen by substituting n + 1 for
in the former and r-l for r in the latter; and hence the squared length (10) is
1/,

invariant for the transformation (15). Any Lorentz transformation can be built up
from infinitesimal transformations like (15) and three-dimensional rotations like
those considered in the preceding section, together with possibly a reflexion, which
obviously leaves the squared length (10) invariant, and hence the theorem is proved.
The transformations of the coefficients A thus provide a unitary representation
of the Lorentz group. The coefficients themselves form the components of a new kind
of tensor quantity in space-time. I propose for it the name expansor, because of its
connexion with binomial expansions. One may restrict the function (9) to be
homogeneous and one then gets a simpler kind of expansor, which may be called a
 122 CHAPfERII

288 P. A. M. Dirac
lwmogeneoU8 exptJnBOr. The analogy with the three-dimensional case suggests that
one should look upon a homogeneous expansor as a symmetrical tensor in space-
time with the suffix 0 occurring in its components a negative number of times.
The foregoing theory can be generalized, like the three-dimensional theory, by
the introduction of otl1er quadruplets of /Variables, 1/0' 1/1' 1/", 1/8 and '0' '1' '2' '8 say,
which transform tog~ther with 60' 61' 62' 68' One can then set up a power series in
all the variables, ascending in those variables with suffixes 1, 2 and 3, and descending
in those variables with suffix O. The transformations of the coefficients of these more
general power series will provide further unitary representations of the Lorentz
group, and the coefficients themselves will form the components of more general
expansors.
There is another generalization which may readily be made in the theory, namely,
to take the values of the index n in (9) to be not integers, but any set of real numbers
no, no+l, no+ 2, ... extending to infinity. In the formula (10) for the squared length
n! is then to be interpreted as r(n+ 1). The terms with negative n-values can no
longer be discarded. The expressi?n for the 'squared length is still positive definite
if the minimum value of n is greater than -1, which is the case if the function (9)
IS homogeneous and its degree is negative. The resulting representation is then still
unitary. If, however, the function (9) is homogeneous and its degree is positive,
there will be a finite number of negative terms in the expression for the squared
length. The resulting kind of representation may be called nearly 'Unitary.

4. SOME THEOREMS ON CONVERGENCE

If the series (2) is convergent, then (1) is convergent for all values of 61' Similarly,
if (4) is c~nvergent, then (3) is convergent for all values of 61' 62' 68' On the other
hand, if (8) is convergent, (7) need not be convergent for any value of 60' in which
case, of course, it does not define a function of 60. Similarly, if (10) is convergent,
(9) need not be convergent for any values for the 6'S. Thus, corresponding to a general
expansor A ..... , there need not exist any function Q of the fs. However, it will now
be proved that if the 8eries (9) i8 Iwmogenwus and (10) i8 convergent, then (9) i8
ab80lutely convergent for all values of the 6'8 sati8fying
;~-;j- 61-;~ > O. (16)

If" the series (9) is homogeneous of degree u -1, it may be written

(17)

w4ere ~8 means a sum over all values of r, 8 and t satisfying

r+8+t = n+u. (18)

With this notation the series (10) may be written

(19)
REPRESENTATIONS OF THE POINCARE GROUP 123

Unitary representf.!,tions oj the lhre:ntz group 289

Now apply Cauchy's inequality
'(X1YI +X,Ys +xaYa + ... )1 ~ (xl + xl +xl+ ... ) (Yi+y~ +yi+ ... ) (20)
in the following way. Take (r+8+t)!/,.!8!t! of the x's in (20) to be each equal to
gr;;g~ and the corresponding y's to be each equal to A".. r!8!t!/("+8+t}!, and do
this for all r, 8, t subject to (18) for a fixed value of n. Then (20) becomes
(~8An .. gg:g~)I~ (;1 + gl + ;)"+u (n+u) !-I ~8r! 8! t! A~r.. (21)
If the sum with respect to n in (19) converges, there mU8t be 80me number K such that
n!-l ~8r! 8! t! A!,.. < K
for all n. Then from (21)
"Ic-n-I~ A
~o S
crcactl<Kllclu-l{
nr.~1!;>IS8 !:>O
n! }l{~+~+gi}l("+U)
(n+u)! g~ J

which shows that (17) is convergent when (16) is satisfied. One may take go, gl' g" g.
and the A's to be all positive without disturbing the argument, 80 (17), considered
as a quadruple series in n, r, 8, t, is ab80lutely convergent.
ThU8 for any homogeneous expansor of finite length, there exists a function Q
given by (9) defined within the light-cone (16). In transforming this function with
a Lorentz transformation (12), it is legitimate to use the expansion of (13) in
ascending powers of g;, g~, g~ and descending powers of g~, for the following reason.
Suppose g" is within the light-cone and take for definiteness go> O. Then

~~+~~+~~+~~>~ (n)
One can change the sign of any of the co-ordinates gi, g~, gi, leaving g~ unchanged
and ~p will still lie within the light-cone with ~o > 0, 80 (22) must still be satisfied.
Hence
~g~-I~~; I...L I~g; I-I~gi 1 >0,
which shows that the expansion of (13) in the required manner is absolutely con-
vergent. The use of this expansion in the preceding section is thus justified for the
case when (9) is .homogeneous, the new coefficients A being determined by the
transformed function Q within the light-cone. The justification for (9) not homo-
geneous then follows since terms of different degree do not interfere. It should be
noted that all the foregoing arguments are valid also when the values of n are not
integers.
There are some expansors which are invariant under all Lorentz transformations,
namely those whose components are the coefficients of (~- gt- -
~ gl)-l expanded
in ascending powers of gl' gl' ga and descending powers of go' For such an expansion
to be possible, l must not be a negative integer or zero, but it can be any other real
number. Again, there are expansors which transform like ordinary tensors under
Lorentz transformations, namely those whose components are the coefficients in
the expansion of
(23)
 124 CHAPTER II

290 P. A. M. Dirac
where 1 is any homogeneous integral polynomial in Eo, €l' €,' €., and I is restricted.
as before. (23) transforms like the polynomial I, and thus like a tensor of order
equaJ to the degree of I. One would expect a.ll these expansors to be of infinite
length, a.s otherwise one could set up a positive definite form for the squared length
of a tensor in space-time. A forma.l proof that they are is a.s follows.
Suppose the 1 in (23) is of degree u and express it a.s
1 = gu+€i,gu-J +(€I- ~')gu_I+€o(€I-~2)gu_a+ (€~- ~1)lgu-4 + ... ,
where ~I = €f + €I + €land each of the g's is a polynomial in €1' fl' fa only, of degree
indicated by the suffix. The successive terms here contribute to (23) the amounts

_ ~""
- --"'-0
l(l+ 1)(1+ 2) ... (1+ m-l) gu;8m
m!

f (Ell _ ;2)-l+1 = 1:"" (l- 1) l(l+ 1) ... (I + m - 1) gu_a;l(m+1~

guo' 0 0 m--l (m+ I)! Ei<m+l)-1 '

and so on. These expa.nsions show that, for large values of m, the terms arising from
gu-I' 9u-4., ... are of smaller order than the corresponding terms arising from gu,
and the terms arising from gu-a, flU-5' ... are of smaller order than the corresponding
terms arising from g_I' 80 that in testing for the convergence of the series which
gives the squared length, only the gu and g..-1 terms need. be taken into account.
(It will be· found that the convergence conditions are not sufficiently critical to be
affected by this neglect.) Now express gu and g..-1 a.s
g. = 8.. +~·8u_I+;'8.. _, + ... , 9u-l = 8.._1+;·8U-3+~'8u-5+ ... ,
where the 8's are solid harmonic functions of El' fl' €., of degrees indicated by the
suffixes. Each of them gives a contribution to (23) of the form
(m+l-l)I ~I(m~)
8u_Ir;Ir(f~-;')-l = 8"_11'1::_0 ~!(l-I)!· ffm-tll (24)

(m+l-l)I ;I(m~)
or 8u_Ir_1 Eo;Ir(€I-;I)-l =8 U - .......1 1::_0 m!(I-I)!· €i<m-tll- 1 ' (25)

Using the result (41) of the appendix, one finds for the squared lengths of the
expa.nsors whose components are the coefficients of (24) and (25), series of the form
(m+I-·l)114m-tr(m+r)! (m+u-r+l)!
c1:m m!I(I-I)!~ (2m+2l-1)!

(m+l-l) !1I4m-tr(m+r)! (m+ u-r-l)!

and c1:m mlll (l-I)!1 (2m+2l-2)! '
REPRESENTATIONS OF THE POINCARE GROUP 125

Unitary representations oj the Lorentz group 291

respectively. In both these series the ratio of the (m+ l)th term tothemthis 1 +u/m
for large m, and since u is neoessa.rily positive or zero, the series diverge and the
expansors are of infinite length. From the result (40) of the appendix, expansors
&88Ociated with solid harmonics of different degrees are orthogonal to one another,
and hence the total expansor with the coefficients of (23) as components mUjt also
be of infinite length.

5. A TRANSFORMATION OJ!' V.A.BIABLES

Up to the present, expansors have always been considered in connexion with
certain ;-functions, the components of the expansor being the coefficients of the
;-function. In the case of integral n-values, one can make a transformatiQn of
variables of the kind which is familiar in quantum mechanics, and get the expansors
connected with some other functions, which serves to clarify some of their properties.
Introduce the four operators x"' defined by
21xo = ;0-0/0;0' 21x, = E,+0/0E, (r = 1,2,3). (26)
Under Lorentz transformations they transform like the components of a 4-vector.
The operators o/OX,. may be taken to be
(27)
as this gives the correct commutation relations between all the operators. One can
now represent ;-flillctions with integral n-values by functions of the x's according
to the following scheme.
Take as starting point the E-function Eo l . It vanishes when operated on by
0/oE1 , 0/0f,1' or %Es, and also effectively vanishes when multiplied by ;0' since the
result of the multiplication can be discarded.. It must therefore be represented by
a function ofthe x's which vanishes when operated on by x, +%x, with r = 1, 2 or 3,
or by Xo + %xo, and thus by a multiple of e-l<"hlll), where Xl = xl+x:+;4. Take
(28)
where the sign == means 'represented by'. Multiply the left-hand side of (28) by
the operator (- %Eo)";r f,g, and the right-hand side by the operator equal to it
according to (26) and (27). The result is, after dividing through by n!,

Eon-1 f,~ f,: ~ == 7rIn !-12-i<n+<"+HIl(Xo - ~)"

OXo
(Xl - _~x (X - ~\' (X8 - ~)' e-l<zl+lI1)
ax;} 1 OX"J OX 8

(29)
say. This gives the function F of the x's which represents Eon-1gf,g~. A general
f,-function is now represented thus,
~AnratEon-lg f,g~ == ~ .... F,.,..(xo, Xl' X" x 8)· (30)

In this way a general expansor .An,. gets connected with a function of the x's.

Vol. 183. A. 20
126 CHAPTER II

292 P. A. M. Dirac
The chief interest of this connexion is that the law for the scalar product of two
expansors becomes very simple when expressed in terms of the functions of the x's.
The scalar product is the inte{1ral oj the product oj the two Junctions oj the x' 8 over aU
Xo. Xl, XI and X 3• To prove this, first evaluate the integral

f
_aoao (d)'"
Z- dz e-1zl .(d)""
z - dz e-izl dz, (31)

where the dot has the meaning that operators to the left of it do not operate on
functions to the right of it. For m > O. (31) goes over by partial integration into

f ao
,-ao
(d)"-l
z--
dz
e-lzl. (d)( d)"" e-h1dz
z+~ z--
dz dz

= fao
-ao (d)"'-l
z- dz e-hl .{(z- dzd)""(z+ dzd) +2m' (d)""-l)
z- dz e-h1dz

= 2m'f'" (z_~)m-l e-izi. (z_~)""-l e-h1dz.

-ao dz dz
Applying this procedurem times, one gets zero if m' <m and 2"'m!11'1 if m' = m.
Since there is symmetry between m and m', the result is

J~", (z-:z)'" e-izl.(z- !)"" e-1.1dz = 2"'m!1T i <t",,,,,.

Now substitute for z each of the variables xo, Xl' XI' X. in turn, with m equal to n, r, 8, t
and m' equal to n'. r'. 8'. t' respectively. and multiply the four equations 80 obtained.
The result, after dividing through by 7T1n !12''''''''H+I, is

ffffFftnlFn'. ..-rdzodztdzadza = n!-lr! 8! t! 6nn,6,." 6... 6,.,.

This proves the theorem for the case when the expansors each have only one non-
vanishing ~mponent, which is sufficient to prove it generally.
One can obviously g~t a unitary representation of the Lorentz group by con-
sidering the transformations of a set of vectors in a space of an infinite number of
dimensions, where each vector corresponds to a function of four Lorentz variables
xo, Xl' X., X•• and where the squared length of a vector is defined &8 the integral of the
square of the function over all xo. Xl' XI' x 3 • The above work shows how this obvious
unitary representation is connected with the expansor theory.

6. APPLIOATIONS TO QUANTUM MEOHANICS

The four x's of the preceding section may be looked upon &8 the co-ordinates
of a four-dimensional harmonic oscillator, the four operators ioj'iJz,. being the con-
jugate.momenta. y. and the energy of the oscillator may be taken to be the Lorentz
invJriant (32)
The 1, 2. 3 components of the oscillator thus have positive energies and the 0 com-
ponent negative energy.
REPRESENTATIONS OF THE POINCARE GROUP 127

U 11/f,Wry repre8emtUlOn8 01 t/l,e Lorentz grO'Up 293

The state of the oscillator for which the 0, 1, 2, 3 components are in the nth, rth,
"th, tth quantum states respectively is then represented by the function FlU'll defined
by (29), with a suitable normalizing factor. This representative may be transformed
to the ,-representation and becomes So"-l€r,:g. Thus a state of the oscillator for
which each of its components is in a quantum state corresponds to an expa.n8Ol' with
one non-vanishing component. A general state of the Q8cilla.tor therefore corresponds
to a general e~r with integral n-values. A stationary state of the oscillator
corresponds 1i9 a homogeneous expansor, the degree of the expansor giving the
energy of the state with neglect of zero-point energy.
Four-dimensional harmonic 08cilla.tors of the above type occur in the theory of
the electromagnetic field. Each Fourier coQlppnent of the field, specified by a par-
ticu1a.r frequency and a particula.r direction of motion of the waves, provides one
such oscillator, its four components coming from the four eleotroma.gI1etio potentials.
Thus a state of the electromagnetio field in quantum mechanics is desoribed by a
nUmber of expansors, one for each Fourier component. By using the electromagnetio
equation which gives the value of the divergence of the potentials, one can eliminate
in a non-relativistic way the 0 component and one othe.component of, each of the
four-dimensional oscillators, 80 that only two-dimensional osoillators are left. This
circumstance has made it possible for people to develop quantum electrodynamics
without using expansors.
Another possible application of expansors is to the spins of particles. The wave
function describing a particle may be a function of the four co-ordinatea x,. of the
particle in space-time and also of the four variables €,. whose coefficients" are the
components of an expan80r. As a simple example of relativistic wave equations for
such a particle in the absence of external forces, one may consider

(33)

The first of these is the usual equation for the motion of the particle as a whole. The
second shows that at eaoh point in space-time the wave function", is homogeneous
in the fs of degree -1. The third shows that the state for which the momentum-
energy four-vector of the particle has the value PI' is represented by the wave
function

in three-dimensional vector notation. This may be expanded as

(34)

For the state for which the particle is at rest, PI = PI = Pa = 0, Po = m, and '"
reduces to
128 CHAPTER II

294 P. A. M. Dirac
This Vr is spherically symmetrical, showing that when the particle is at rest it has
no spin. But when the particle is moving, it is represented by the general Vr (34)
and has a finite probability of a non-zero spin. In fact, taking for simplicity
PI = Pa = 0, the particle has a probability (mpflp'G+l)lof being in a state of spin
corresponding to the transformations of ~f under three-dimensional rotations.
This example shows there is a possibility of a particle having no spin when at rest
but acquiring a spin when moving, a state ofaffairs which was not allowed by previous
theory. It is desirable fihat the new spin possibilities opened up by the present theory
should be investigated to see whether they could in some ca.ses give an improved
description of Nature. The present theory of expansors applies, of course, only to
integral spins, but probably it will be possible to set up a corresponding theory of
two-valued representations of the Lorentz group, which will apply to half odd
integral spins.

APPENDIX

The rules (5), (II) for forming scalar products are not always convenient for direct
use. There are various ways of transforming them and making them more suitable
for practical application. One such way has been given (Dirac 1942, equation 3·22)
for the ca.se of a single ~ with ascending power series. Another way, applicable to
the ca.se of, homogeneous functions of Sl' Sa, Sa, is provided by the following.
By partial integration with respect to S', one gets, for m> 0,

If the integrals are made precise in the sense of Cesaro, which means neglecting
oscillating terms like f"ei £{' for S' infinite, this gives

Taking m ~ 110 and applying the partial integration process 110 times, one gets

II: ~"'fn e.
II)
iEf' df elS' = i"n!II:",s"'-f& elfE' df elS'

= 21Ti"n! I:..,sm-f& ct(s) ds

= 21Ti"n! 8m... (35)

It follows that if A and B are homogeneous functions of ~l' S,,' Sa of degree 'ft, their
scalar Ilroduct according to (5) is

(AB) = (21T)-3i-VIJ.. .... A(SlStSa) B(sis;s~)ei(M~+f.E.+Ea~dSldSi dSadS~dSadS~. (36)

REPRESENTATIONS OF THE POINCARE GROUP 129

Unitary representations of the Inrentz group 295

As an application of this rule, take
A = (~+a+~)"8"-Ir' B = (~+a+€I>-8,,-..,
where the 8's are solid harmonic functions. Then, using three-dimensional vector
notation, (36) gives
(A B) = (217)-3 ,-uJf. . . .~~'.. 8,,_Ir(E) 8,,_ ..(E') ei(U') dEl ... dE;. (3~)
From Green's theorem

fff[ei(U')(a~ + a~ + :~) (~1r8"_Ir(E)} - ~1r8v_Ir(E) (a~ + :~ + :~) eiCU')] dE1dE.dE.

equals a surface integral of an oscillating kind which is to be counted &8 vanishing

at infinity. This result reduOOB to

4r(u-r+ i)fff:oo·~i(r-1)8"_Ir(E) eiCU') dEl dE. dEa + ~'Ifff:"" ~1r8"_Ir(E) eiCU')dEldEl~Ea

= o.
(37) becomes, for r, 0,
ff. . . .~l(r-I)~'I(.-1)8,,_Ir(E)8v_
80 8>

(AB) = (217)-3i-U+l4r(u-r+i) .. (f)ei(U')dEl ... dE••

(38)

Now suppose 8 ~ r and apply the procedure by which (37) W&8 changed to (38) f'
times. The result is
(AB) = (217)-3i-U+llr 4rr!(u-r+ i)! (u- 2r+ l) !-1

x ff.. . . ~'I(a-r) 8,,_tr(E) 8,,_se(E') ei(U') dEl··· dE;. (39)

where n! means r(n+ 1) for n not an integer. H 8>r, the procedure can be applied
once more, and then shows that
(AB) = 0 for r=+=8. (40)
H 8 = r, (39) shows that (AB) = c4rr!(u-r+i)!, (41)
where c depends only on u - 2r and on the two 8 functions.

REFERENOE

Dirac, P. A. M. 1942 Proc. Roy. Soc. A, 180, 1-40.

130 CHAPTER II

PHYSICAL REYIEW VOLUME 133. NUMBER .cia 9 MARCH 1904

Feynman Rules for Any Spin*

STEVEN WE'NBERGt
Department oj Physics, University of California, Berkeley, California
(Received 21 October 1963)

The explicit Feynrnan rules are given for massive particles of any spin j, in both a 2j+l-component and
a 2(2j+l)-component formalism. The propagators involve matrices which transform like symmetric trace-
less tensors of rank 2j; they are the natural generalizations of the 2X2 four-vector qjJ and 4X4 four-vector
'YJJ. for j=i. Our calculation uses field theory, but only as a convenient instrument for the construction of a
Lorentz-invariant S matrix. This approach is also used to prove the spin-statistics theorem, crossing sym-
metry, and to discuss T, C, and P.

I. INTRODUCTION (b) For (x- y) spacelike,

T HIS article will develop the relativistic theory of

higher spin, from a point of view midway he tween [JC (x),JC (y)]=O . (1.5)
that of the classic Lagrangian field theories and the
The necessity of (a) is rather obvious if we use (1.3) to
more recent S-matrix approach. Our chief aim is to
rewrite (1.1) as
present the explicit Feynman rules for perturbation
calculations, in a formalism that varies as little as
possible from one spin to another. Such a formalism (-i)"1
S= L -,-
00
d'x!·· ·d'x"T{JC(x,)·· ·JC(x,,)}. (1.6)
should be useful if we are to treat particles like the 3-3 11-0 11.
resonance as if they were elementary, and is perhaps in-
indispensable if we are ever to construct a relativistic But (a) is certainly not sufficient, because the 0 func-
perturbation theory of Regge poles. tions O(x,-x;) implicit in the definition of the time-
Our treatment! is ba~ed on three chief assumptions. ordered product are not scalars unless their argument is
(1) Perturbation theory. We assume that the S matrix timelike or lightlike. Condition (b) guarantees that no
can be calculated from Dyson's formula: oever appears with a spacelike argument.
(3) Particle interpretation. We require that JC(x) be
(-i)"J oo
S=L - - constructed out of the creation and annihilation opera-
00
dt,·· ·dt"T{H'(t,)·· ·H'(t,,)}. (Ll)
n=() ttl -:>0 tors for the free particles described by H o. The only
known way of making sure that such an JC(x) will
Here we have split the Hamiltonian H into a free- satisfy the restrictions 2(a) and 2(b), is to form it as a
particle part Ho and an interaction H', and define H'(t) function of one or more fields ",,,(x), which are linear
as the interaction in the interaction representation: combinations of the creation and annihilation operators,
and which have the properties:
H'(t)=exp(iHr/.)H' exp( -iHr/.). (1.2)
(a) The fields transform according to
(2) Lorentz int'ariance of the S matrix. We require
that S be invariant under proper orthochronous Lorentz (1.7)
transformations. This certainly imposes a much stronger
restriction on H 0 and H' than that they just transform
like energies. A sufficient and probably necessary con- where D"m[A] is some representation of A.
dition for the invariance of S is: (b) For (x-y) spacelike

H'(t)= 1 d'xJC(x,t) , (1.3)

where [ ]± may be either a commutator or anticom-
(1.8)

where: mutator. Condition 3 (a) enables us to satisfy 2 (a) by

(a) JC(x) is a scalar. That is, to every inhomogeneous coupling the ",,,(x) in various invariant combinations,
Lorentz transformation x" -> A",x'+a" there corre- while 3(b) guarantees the validity of 2(b), provided
sponds a unitary operator U[A,a] such that that JC(x) contains an even number of fermion field
U[A,a]JC(x)U-![A,a]=JC(Ax+a) . (1.4) factors. (There are some fi,ne points about the case
X= y which will be discussed in Sec. V.)
Equations (1.7) and (1.8) will dictate how the fields
* Research supported in part by the U. S. Air Force Office of
Scientific Research, Grant No. AF-AFOSR-232-63. are to be constructed. We have not pretended to derive
t Alfred P. Sloan Foundation Fellow. these equations as inescapable consequences of assump-
1 I have recently learned that a similar approach is used by
E. H. Wichmann in the manuscript of his forthcoming book in tions (1)-(3), but our discussion suggests strongly that
quantum field theory. they can be understood as necessary to the Lorentz
B131S

Reprinted from Phys. Rev. 133, B1318 (1964).

REPRESENTATIONS OF THE POINCARE GROUP 131

FEYNMAN RULES FOR ANY SPIN B1319

invariance of the S matrix, withont any recourse to TABLE I. The scalar matrix n(q) = (_)2iII'I1'2·· 'Ql'lQl/2 " •• for
spins j~3. In each case J is the usual 2j+l-dimensional matrix
separate postulates of causality or analyticity.' representation of the angular momentum. The propagator for a
Nowhere have we mentioned field equations or La- particle of spin j is Seq) ~ -i( -im)-,jTI(q)!q'+m'-i,.
grangians, for they will not be needed. In fact, our
refusal to get enmeshed in the canonical formalism has lI(O)(q) ~ 1
a number of important physical (and pedagogical) TIO!2) (q) ~'1'-2 (q. J)
advantages: II''' (q) ~ -q'+2(q· J) (q. J _gO)
(1) We are able to use a 2j+l-component field for a TIIs;" (q) = -q2(g"-2q· J)H[(2q- J)'-q'][3'1'-2q' J]
massive particle of spin j. This is often thought to be n(2) (g) ~ (_q2)'- 2g'(q· J) (q- J -'1')
impossible, because such fields do not satisfy any free- +j(q. J)[(q- J)'_q2J[q. J-2qoJ
field equations (besides the Klein-Gordon equation). TIl ,2) (q) ~ (_q2)2(q"_ 2q' J) - ,q'[(2q- J)'- q'][3'1'-2q· J]
The absence of field equations is irrelevant in our ap- 1
proach, because the fields do satisfy (1.7) and (1.8); a +--{ (2q· J)'- q'][(2q· J)2-9 q2J[5'1'- 2q· JJ
free-field equation is nothing but an invariant record of 120
which components are superfluous. II (3) (g) = (-q')'+2( -q')(q. J) (q. J-gO)
The 2j+ I-component fields are ideally suited to weak - ~g'(q' J)[ (q. J)'-q'J[q' J - 2qoJ
interaction theory, because they transform simply 4
+-(q. J)[ (q- J)2_ q'J[(q- J)'-4q'J[q· J -3'1'J
under T and CP but not under CorP. In order to 4.1
discuss theories with parity conservation it is con-
venient to use 2 (2j+ I)-component fields, like the Dirac
field. These do obey field equations, which can be de-
rived as incidental consequences of (1.7) and (1.8). symmetry. Section V is devoted to a statement of the
(2) Schwinger" has noticed a serious difficulty in the Feynman rules. The inversions T, C, and P are studied
quantization of theories of spin j~ ~ by the canonical in Sec. VI. They suggest the use of a 2 (2j+ I)-com-
method. This can be taken to imply either that particles ponent field whose propagator is calculated in Sec. VII.
with .i~ ~ cannot be elementary, or it might be inter- More general fields are considered briefly in Sec. VIII.
preted as a shortcoming of the Lagrangian approach. The propagator for 2j+l- and 2(2j+I)-component
(3) Pauli's proof' of the connection between spin and fields involves a set of matrices which transform like
statistics is straightforward for integer j, but rather symmetric traceless tensors of rank 2j, and which form
indirect for half-integer j. We take the particle inter- the natural generalizations of the 2X2 vector {a,l} and
pretation of "'n(X) as an assumption, and are able to the 4X4 vector 'I" respectively. These matrices are
show almost trivially that (1.8) makes sense only with discussed in two appendices, where we also derive the
the usual choice between commutation and anticom- general formulas for a spin .i propagator. The 2.i+ I
mutation relations. Crossing symmetry arises in the X2j+ I propagators for spin j~3 are listed in Table I,
same way. and the 2(2j+1)X2(2j+1) propagators for j~2 are
(4) By avoiding the principle of least action, we are listed in Table II.
able to remain somewhat closer throughout our de- This article treats a quantum field as a mere artifice
velopment of field theory to ideas of obvious physical to be used in the construction of an invariant S matrix.
significance. It is therefore not ur.likely that most of the work pre-
At any rate the ambiguity in choosing JC(x) is no sented here could be translated into the language of
worse than for ",(x). The one place where the La- pure S-matrix theory, with unitarity replacing our
grangian approach does suggest a specific interaction assumptions (I) and (3).
is in the theory of massless particles like the photon and
graviton. Our work in this paper will be restricted to TABLE II. The scalar matrix <9(q) = _i i qIlIQ"'2' .. qjJzi
2i-yjJJjJ2 ··jJ2
massive particles, but we shall come back to this point for spins j ~ 2. In each case
in a later article.
The transformation properties of states, creation and ~~[tj) ~(iJJ.
annihilation operators, and fields are reviewed in Sec. The propagator for a particle of spin j is
II. The 2j+ I-component field is constructed in Sec. III Seq) ~ -im-21[ rJ>(q)+m,jJI<f+m'-i,.
sO that it satisfies the transformation rule (1.7). The
"causality" requirement (1.8) is invoked in Sec. IV, rJ>(O)(q)~1
yielding the spin-statistics connection and crossing
rJ>(l/"(q) ~'1'i3-2(q '~h5i3
rJ>(l)(q) ~ -g'i3+2(q·~) (q'~i3-q"~5m
2 In this connection, it is very interesting that a Hamiltonian
without particle creation and annihilation can yield a Lorentz- rJ>(3!2l (q) ~ -q'('1'i3-2q· ~Y5i3)+![(2q. ~)'- q'][3'1'i3-2q ·~.h5i3]
invariant S matrix, but not if we use perturbation theory. See
rJ>(2)(g) ~ (-q')'i3-2g'(q' ~)[w gji3-q"y..s]
R. Fong and J. Sucher, University of Maryland (to be pubhshed).
3 J. Schwinger, Phys. Rev. 130, 800 (1963). +Hw,m[(q-gj)'-q'][q' ~i3-2q"y..s]
4 W. Pauli, Phys. Rev. 58, 716 (1940).
132 CHAPfERII

B1320 STEVE," WEINBERG

II. LORENTZ TRANSFORMATIONS The coefficients D.,.W are

In our noncanonical approach it is essential to begin D.,.W[R]=(u'IU[RJlu). (2.9)
with a description of the Lorentz transformation prop-
erties of free-particle states, or equivalently, of creation In (2.8), R is the pure rotation L-I(Ap)AL(p) (the
and annihilation operators. The transformation rules so-called "Wigner rotation") so that DW[R] here is
are simple and unambiguous, and have been well nothing but the familiar 2j+ 1-dimensional unitary
understood for many years, 5 but it will be useful to matrix representation' of the rotation group.
review them once again here. A general state containing several free particles will
The proper homogeneous orthochronous Lorentz transform like (2.8), with a factor [w'/w]"'D for each
transformations are defined by particle. These states can be built up by acting on the
bare vacuum with creation operators a*(p,u) which
satisfy either the usual Bose or Fermi rules':
g,.A'AA'p=gAp, (2.1)
[a(p,u),a*(p',u')]±= 6..,a'(p-p'), (2.10)
detA= 1; AOo>O.
so the general transformation law can be summarized
These will be referred to simply as "Lorentz transforma- by replacing (2.8) with
tions" from now on. OU( metric is
U[A]a*(p,u)U-I[A]
g;;=8 i;; goo=-1; giO=gOi=O. (2.2)
= [w(Ap)/w(p)J'/2 L;., D.,.W[L-I(Ap)AL(p)]a*(Ap,u').
To each A there corresponds a unitary operator U[A], (2.11)
which acts on the Hilbert space of physical states, and
has the group property Taking the adjoint and using the unitarily of DW[R]
gives
U[A.]U[A,]= U[AoA,]. (2.3)
U[A]a(p,u)U-I[A]
Of particular importance for us is the "boost" L(p),
= [w(Ap)/w(p) ]'/'L;.,D ..,W[L-I(p)A-'L(Ap)]a(Ap,u').
which takes a particle of mass m from rest to mo-
men.um p: (2.12)
L';(p) = 6,;+p,p;[coshO-1], It will be convenient to rewrite (2.11) in a form
L'o(p)=LO,(p)=p,sinh9, (2.4) closer to that of (2.12). Note that the ordinary complex
LOo(p)= coshO. conjugate of the rotation-representation D is given by
a unitary transformationS
Here p is the unit vector p/ I pi, and
DW[R]*=CDW[R]c-', (2.13)
sinh8= I pl/m, cosh8=w/m=[Y+m'J'/'/m. (2.5)
where C is a 2j+1X2j+1 matrix with
Strictly speaking, this should be called L(p/m).
We can use L(p) to define the one-particle state of C*C=(-)';j ctC=1. (2.14)
momentum p, mass m, spin j, and z-component of spin u
(u= j, j-1, ... , -j) by [With the usual phase conventions, C can be taken as
the matrix
I p,u) = [mlw(p)]'/'U[L(p)]lu), (2.6)
C ..,= (- );+'8.,,_.,
where lu) is the state of the particle at rest with J.=u.
Our normalization is conventional, but we won't need this here.] Since DW[R] is unitary,
(2.13) can be written
(p,ulp',u')=8'(p-p')6..,. (2.7)
D.,.Ul[R]={CDW[R-I]C-I} .., (2.15)
The effect of an arbitrary Lorentz transformation A'.
on these one-particle states is so (2.11) becomes

U[A] I p,u)= [m/w(p)J'/2U[A]U[L(p)Jlu) U[AJa* (p,u )U-I[A]

= [w(Ap)IW(P)]'/2L;.,{CDW[L-I(p)A-IL(Ap)]C-I} ..,
= [m/w(p)]"'U[L(Ap)]U[L-'(Ap)AL(p)] Iu)
Xa*(Ap,u'). (2.16)
= [m/w(p)J'2 Eo, U[L(Ap)]lu')
X (u'l U[L-' (Ap)AL(p)JI u), I See, for example, M. E. Rose, FJemenlary Theory of Angular
and finally Momen/um (John Wiley and Sons, Inc., New York, 1957), p. 48 II.
1 We use an asterisk to denote the adjoint of an operator on

U[A] I p,u)= [w(Ap)/w(p)]'" L:"IAp,u') the physical Hilbert space, or the ordinary complex conjugate of
a (; number or a c-number matrix. A dagger is used to indicate the
XD.,.W[L-I(Ap)AL(p)]. (2.8) adjoint of a ,-number matrix. Other possible statistics than
allowed by (2.10) will not be considered here.
• E. P. Wigner, Ann. Math. 40, 149 (1939). I Reference 6, Eq. (4.22) .
REPRESENTATIONS OF THE POINCARE GROUP 133

FEY,\M."'\ RULES FOR ANY srI"" B1321

We speak of one particle as being the antiparticle It follows from (2.3) that
of another if their masses and spins are equal, and all
their charges, baryon numbers, etc., are opposite. We [J i,J;] = i'ij,J k , (2.24)
won't assume that every particle has an antiparticle, [Ji,K;] = i'ij,K., (2.25)
since this is a well-known consequence of field theory,
which will be proved from our standpoint in Sec. IV. [Ki,Kj] = -i'i;.I.. (2.26)
But if an antiparticle exists then its states will trans- The J generate rotations and the K generate boosts.
form like those of the corresponding particle. In par- In particular, the unitary operator for the finite boost
ticular, the operator b*(p,<T) which creates the anti- (2.4) is
particle of the particle destroyed by a(p,<T) transforms U[L(p)J=exp(-ip·KO). (2.27)
by the same rule (2.16) as a*(p,<T):
The co.nmutation rules (2.24)-(2.26) can be de-
U[Ajb*(p,<T)U-'[A] coupled by defining a new pair of non-Hermitian
= [w(Ap)/w(p) Jl/2 Lu' (CDUl[L-1( p)A-IL(Ap) ]C-l} uu' generators:
Xb*(Ap,<T'). (2.17) A=~[HiK], (2.28)

To some extent this is a convention, but it has the ad- B=![J-iKJ, (2.29)
vantage of not forcing us to use different notation for with commutation rules
purely neutral particles and for particles with distinct
antiparticles. AxA=iA, (2.30)
It cannot he stressed too strongly that the trans- BxB=iB, (2.31)
formation rules (2.12) and (2.17) have nothing to do
with representations of the homogeneous Lorentz group, [Ai,BjJ=O. (2.32)
but only involve the familiar representations of the
ordinary rotation group. If a stranger asks how the spin +
The (2A 1) (2B+ I)-dimensional irreducible repre-
states of a moving particle with j = 1 transform under sentation (A ,B) is defined for any integer values of
some Lorentz transformation, it is not necessary to ask 2A and 2B by
him whether he is thinking of a four-vector, a skew (a,bl AI a',b')=Dbb'Jaa,(A), (2.33)
symmetric tensor, a self-dual skew symmetric tensor,
or something else. One need only refer him to (2.16) or (a,bIBla',b')=Daa,Jbb'(B), (2.34)
(2.8), and hopc that he knows the j = 1 rotation
matrices.
where a and b run by unit steps from - A to A and +
from - B to +B, respectively, and JW is the usual
The complexities of higher spin enter only when we 2j+ I-dimensional representation of the rotation group:
try to use a(p,<T) and b*Cp,O') to construct a field which
transforms simply under the homogeneous Lorentz (.T/iJ±iJ /I)u'u= Du' .u±l[ (j=F<T) (j±<T+ I)Jl/2, (2.35)
group. We will need to usc only a little of the classic (J,(j)u'u= Du'u<T.
theory of the representations of this group, but it will
be convenient to recall its vocabulary. Any representa- The representations (A,B) exhaust all flnite dimen-
tion is specified by a representation of the infinitesimal sional irreducible representations of the homogeneous
Lorentz transformations. These arc of the form Lorentz group. None of them are unitary, except for
(0,0).
(2.18) We will be particularly concerned with the simplest
irreducible representations (j,O) and (O,j). These are
where the w's form an infinitesimal "six-vector"
respectively characterized by
(2.19) J -> JU), K -> -iJU), for (j,0) (2.36)
The corresponding unitary operators are of the form and
J -> JW, K -> +iJW, for (O,j), (2.37)
U[Hw J= H (i/2)J"w"', (2.20)
where J<il is given as always by (2.35). We denote the
2j+ I-dimensional matrix representing a finite Lorentz
transformation A by DW[AJ and DW[AJ in the (j,O)
It is very convenient to group the six operators J" into and (O,j) representations, respectively. The two repre-
two Hermitian three-vectors sentations are related by
(2.22) (2.38)

Ki=J,o=-Jo,. (2.23) In particular the boost L(p) is represented according to

134 CHAPTER II

B1322 STEVEK WEINBERG

(2.27) and (2.36) or (2.37) by section. It is clear that this is the most general linear
combination of the a's and the b*'s which has the simple
DCil[L(p)J=exp( _po JCilO) , (2.39)
Lorentz transformation property
DCil[L(p)J=exp(+p. JCilO) , (2.40)
U[A,a]I'.(x)U-l[A,a]
with sinhB= I pi/m. For pure rotations both D(j)[R] =L' D••,(i)[A-l]I'.,(Ax+a). (3.7)
and DCil[RJ reduce to the usual rotation matrices.
[We choose to combine a and b*, so that I'.(x) also
behaves simply under gauge transformations.]
Ill. 2j+l-COMPONENT FIELDS
In terms of the original creation and annihilation
We want to form the free field by taking linear com- operators, the field is

f
binations of creation and annihilation operators. The
transformation property under translations required by d'p
1'. (x) = (2 ..)-3/' - - -
(1. 7) forces us to do this by setting the field equal to [2W(p)Jl/2
some sort of Fourier transform of these operators. But
(2.12) and (2.17) show that each a(p,.-) and b*(p,.-) XL [W •• ,(il[L(p)]a(p,..')e'P"
behaves under Lorentz transformations in a way that "
depends on the individual momentum p, so that the +I1{DW[L(p)]C-1) ..,b*(p,..')e-'P"] , (3,8)
ordinary Fourier transform would not have a covariant
character. In order to construct fields with simple We have already derived a formula [Eq. (2.39)J for
transformation properties, it is necessary to extend the wave function appearing in (3.8):
D(j)[RJ to a representation of the homogeneous Lorentz
group, so that the p-dependent factors in (2.12) and D..,W[L(p)]= {exp( _po J(/)O) ..,.
(2.17) can be grouped' with the a(p,,-) and b*(p,.-).
There are as many ways of doing this as there are The field obeys the Klein-Gordon equation
representations of the Lorentz group, but for the present (O'-m')I'.(x)=O, (3.9)
we shall use the (j,O) representation defined by (2.36)
and (2.35). [The (O,j) representation will be considered but it does not obey any other field equations, As dis-
in Sec. VI, the (j,0)® (O,j) in Sec. VII, and the general cussed in the introduction, we consider this to be a
case in Sec. VIII. J distinct advantage of the (j,O) representation, because
Having extended the definition of the 2j+1X2j+l any field equation [except (3,9)J is nothing but a con-
matrix DCil in this way, we can split the rotation matrix fession that the fi,eld contains superfluous components,
appearlUg in (2.12) and (2.17) into three factors If a particle has no antiparticle (including itself)
then we have to set '1=0 in (3.6) and (3,8), In the
D(j)[L-l(p)A-1L(Ap)]
other extreme, a theory with full crossing symmetry
=DCil-1[L(p)]D(i)[A-']DW[L(Ap)]. (3.1) would have IIII = I~ I· We will now show that the choice
This allows us to write (2.12) and (2.17) as' of ~ and 11 is dictated by requirement (1.8), and hence
essentially by the Lorentz invariance of the S matrix,
U[A]a(p,.-)U-l[A]= :E., D•• ,W[A-l]a(Ap,.-') , (3.2)
U[A]Ii(p,.-)U-l[A] = L, D•• , (/>[A-l]1i (Ap,.-') , (3.3) IV. CROSSING AND STATISTICS
with We are assuming, on the basis of their particle in-
terpretation, that the a's and b's satisfy either the usual
a (p,.-) = [2w(p)J'/' :E., D••,W[L(p)]a(p,u') , (3.4)
Bose commutation or Fermi anticommutation rules:
Ii(p,.-)= [2w(p)J'/':E.' {DW[L(p) ]C-'} •• ,b'(p,.-'). (3.5)
[a(p, ..),a* (p, ..')J±= a(p- p')a.."
The operators a and Ii transform simply, so the field (4.1)
[b(p, .. ),b*(p, ..')J±= a(p- p')a.., ,
can be constructed now by a Lorentz invariant Fourier

f
transform with all others vanishing. It is then easy to work out
d'p the commutation or anticommutation rule for the field
1'. (x) = (211')-'/' - - defined by (3.8):
2w(p)
[q>.(x),q>.,t(y)J±

".'If
X[~(p, .. )eip·'+'1Ii(p,..)e-ip"J, (3.6)

with constants ~ and 11 to be determined in the next =-- -d'p

- I I .., Ci)(p,w(p)
(211')' 2w(p)
9 This step corresponds to Stapp's replacement of the S matrix
by the "M-functions." See H. Stapp, Phys. Rev. 125, 2139 (1962) X{ I ~12exp[ip· (x-y)J± 1I1I'exp[ -ip· (x-y)J) ,
for j=i; and A. O. Barut, I. MU2inich, D. N. Williams, Phys.
Rev. 130, 442 (1963) for general j. (4.2)
REPRESENTATIONS OF THE POINCARE GROUP 135

FEYNMAN RULES FOR ANY SPIN B1323

where the matrix II (p) is given by The field is now in its final form:
m-2iII(p,w)=DW[L(p)]DW[L(p)]t (4.3)

=exp( -2p· J8) (4.4)

lOa (x) = (2"')-'I'f~
[2W(p)JI12
[coshO= pO/m=w(p)/m].
xL: [Daa,<il[L(p)]a(p,O"')e ipox
This matrix is evaluated explicitly in the Appendix a'
and given for j ~ 3 in Table 1. For our present purposes,
the important point is that

II ..,(p) = (- )'jt ..,m· .. ·..jP •• P.,.·· P.,j, (4.5) The commutator or anticommutator is

where t is a constant symmetric traceless tensor. It [lOa(X),'!'a,t(y)]±

follows then from (4.2) that =i( -im)-2it ..,.·.2 .. ·.2ia ••a.,.· ·a"jA(x-y) , (4.12)
[lOa (x), lOa' t (y)]±
where A is the usual causal function

X f d'p
--(\~I'exp[ip·(x-y)]
2w(p)
±(-),jh:'exp[-ip,(x-y)]l. (4.6)
V. THE FEYNMAN RULES
It is well known' that such an integral will vanish Suppose now that the interaction Hamiltonian is
outside the light-cone if, and only if, the coefficients of
given as some invariant polynomial in the lOa(X) and
exp[ip· (x-y)] and exp[ -ip· (x-y)] are equal and
opposite, i.e., their adjoints. For example, the only possible non-
derivative interaction among three particles of spin it,
(4.7) j" and js would be
Thus the requirement of causality leads immediately
to the two most important consequences of field theory:
(a) Statistics: Eq. (4.7) makes sense only if

(4.8)

so a particle with integer spin must be a boson, with a the "vertex function" being given here by the usual
(-) sign in (4.1), while a particle with half-integer spin 3j symbol.
must be a fermion, with a (+) sign in (4.1).10 The S matrix can be calculated from (1.1) by using
(b) Crossing: Eq. (4.7) also requires that Wick's theorem as usual to derive the Feynman rules:

I~I = I'll· (4.9) (a) For each vertex include a factor (-i) times
whatever coefficients appear with the fields in X(x).
Thus every particle must have an antiparticle (perhaps For example, each vertex arising from (5.1) will con-
itself) which enters into interactions with equal coupling tribute a factor
strength. There is no reason why we cannot redefine
the phase of a(p,O") and b*(p,O") and the phase and
normalization of lOa(X) as we like, so Eq. (4.9) allows
j,) . (5.2)
us to take
0",

i;='1/=1 (4.10) (b) For each internal line running from a vertex at
x to a vertex at y include a propagator
without any loss of generality.
(r{ 10. (x) lOa' tty)} )o=8(x-Y)(lOa(X)lOa' t(y»o
10 As a demonstration that the causality requirement cannot be
satisfied with the wrong statistics, this is certainly inferior to the +(- )2i8(y-X)(IO.,t(y)IO.(X»0 (5.3)
more modern proof of P. N. Burgoyne, Nuovo Cimento 8, 007
(\958). Our purpose in this section i. to show that causality can (c) For an external line corresponding to a particle
be satisfied, but only with the right statistics and with crossing
symmetry. of spin j, J,=p., and momentum p, include a wave
136 CHAPTER II

B1324 STEVEN WEINBERG

function
1
-----.D./j)[L(p)] exp(ip·x) [particle destroyed],
[2w(p)]112(211")'/2
1
- - - - - D•• w*[L(p)] exp( -ip·x) [particle created],
[2W(p)]'/2(211-)'12
(5.4)
1
-----I[DW[L(p)JG-IJ,. exp(-ip·x) [antiparticle created],
[2w(p) J'12(211")'/'
1
-----[DUl[L(p)]C-I].: exp(ip·x) [antiparticle destroyed].
[2w(p)J1I2(211")'I'

These wave functions can be calculated from Eq. where -iAC(x-y) is the usual spin-zero propagator:
(2.39). In conjunction with (4.4), this tells us that
-iAC(x)=i8(x)<4(x)+i8( -x)<4( -x)
D(j)[L(p)]=m-'IJIW{p') , (5.5) =![~,(x)+iE(X)~(X)J (5.9)
and, as usual,
where the 4-vector p' is defined to have 8' =8/2, i.e.,
E(x)=8(x)-8( -x),
p'= {ji[im(w-m)JI/2Hm(w+m)JI'}. (5.6) ~,(x)=i[~+(x)+<4( -x)], (5.10)
The matrix II(j) is calculated in the Appendix; see also ~(x)=<4(x)-<4( -x).
Table I.
(d) Integrate over all vertex positions x, y, etc. and It is well known that ~C(x) is scalar, because E(X) is
sum over all dummy indices II, II', etc. scalar unless x is spacelike, in which case ~(x)=O.
(e) Supply a (-) sign for each fermion loop. Using the tensor transformation rule (A.5) for the
The problem still remaining is to calculate the pro- t·,· .. we find that
pagator (5.3). An elementary calculation using (4.11) DW[A]S(x)D(j)[AJt=S(Ax). (5.11)
and (4.3) gives
This is just the right behavior to guarantee a Lorentz-
(""(x)",,. try»~. invariant S matrix.

= f
d3p
(2".)-'m-'; --IT ••• (p) exp[ip· (x-y)]
2w{p)
But unfortunately the propagator (5.3) arising from
Wick's theorem is not equal to the covariant propagator
Sex) defined by (5.8), except for j=O and j=i. The
(", •• t(y) ",.{x». trouble is that the derivatives in (5.8) act on the E

f
function in ~C(x) as well as on the functions A and A,.
d3p This gives rise to extra terms proportional to equal-
= (2".)-'m-2; - - I I ••• {p) exp[ -ip· (x-y)J.
2w(p) time 6 functions and their derivatives. These extra
terms are not covariant by themselves, but are needed
Formula (4.5) for IT(p) lets us write this as to make Sex) covariant; we must conclude then that
(5.3) is not covariant.
(",.(x) ", •• t(y»o For example, for spin 1 Eq. (5.3) gives
=i( -im)-';t••. •IPI "·· li8Pl8., . ·8"'i~+(X-Y), (5.7)
(T{ ",.(x) ", •. t(y)} )o=iim-'t.... ,
(- )';(1'., '(y)",,(x»o X [8.8,~, (x- y)+iE(X- y)8.8,~(x-y)],
= i( -im)-2it/lrl,,IIl,u2" ·fJ2i0Jl.ldp,,· • . dIJ2iLl-t (y- x),
while (5.8) gives

J
where
S .., (x-y) = !im-'t••••'8.8,
1 d'p
i<4(x)=- --exp{ip·x). X[~,(X-y)+iE(X-y)A(x-y)].
(211")' 2w(p)
The difference can be readily calculated by using the
At this point we encounter an infamous difficulty. familiar properties of ~(x). We find that
If the 8 function in (5.3) could be commuted past the
derivatives in (5.7), then the propagator (5.3) would be (T{ ",.(x)"", t (y)}).
=S•• , (x-y)-2m-'t ••,oo8'(x-y) , (5.12)
S ••• (x-y) = -it -im)-'it...• ..'··· ..i
X8Pl8"," ·8pzi~C{X-y), (5.8) and the second term is definitely not covariant in the
REPRESENTATIONS OF THE POINCARE GROUP 137

FEYNMA:>1 RULES FOR ANY SPIN B1325

sense of Eq. (5.11). [This problem does not arise for spin the covariant propagator
0, where there are no derivatives, nor for spin !, where -it -im)-2it••,·1••...•• j
S••, (:1:-Y) =
there is just one derivative and the extra tenn is pro-
portional to Xii.lii.,.· ·aP2j~C(x-y). (5.8)

t"~(x-y)ii•• (x-y) = 2t"~(x-y)~(t'-)f1) =0. Similar modifications are required when X(x) includes
derivative interactions.
But it does occur for any j:?,1.] The Feynman rules could also be stated in momentum
This problem has nothing to do with our noncanonical space. The propagator (5.8) would then become
approach or our use of 2j+ 1-component fields. For ex-
ample, in the conventional theory of spin 1 (using the
four-component (!,!) representation) the propagator is
S •• , (q)= f d·xe-'··xS .., (x)
= -i( -m)-2 i Il ••, (q)/q'+m'-i.. (5.13)
(T{A"(x)A,(y)})o
= - (i/2)[(g".-m-2ii"ii')~I(X-Y) The monomials II(q) are calculated in the Appendix,
+i.(x-y)(g",-m-2iiiJ,)~(x-y)] and presented explicitly for j~3 in Table 1.
= -i(g",-m-2ii"ii,)~C(x-y)- 2m-2~"O~,O~'(x-y);
VI. T, C, AND P
so here also there appears a noncovariant tenn like The effect of time-reversal (T), charge-conjunction
that in (5.12). The general reason why the S matrix (C), and space-inversion (P) on the free-particle states
turns out to be noncovariant is that condition (1.5) is is well known. It can be summarized by specifying the
not really satisfied by an interaction like (5.1) if any of transformation properties of the annihilation operators:
the spins are higher than !, because the commutators
(4.12) of such fields are too singular at the apex of the Ta(p,O')T-l=1JT L., C•• ,a( -p, 0"), (6.1)
light cone. Tb(p,O')T-l=~T L., C •• ,b( - p, 0"), (6.2)
The cure is well known. We must add noncovariant
"contact" tenns to x (x) in such a way as to cancel out Ca(p,O')C-l=1J cb(p,O'), (6.3)
the noncovariant terms in the propagator. If we used a Cb(p,O')C-l=~ca(p,O'), (6.4)
Lagrangian fonnalism, then such noncovariant contact
tenns would be generated automatically in the transi- Pa(p,O')p-l=1Jpa( - p, 0'), (6.5)
tion from £(x) to X(x), although the proofll of this
Pb(p,O')P-l=~pb( -p, 0'). (6.6)
general Matthews theorem is very complicated. For
our purposes it is only necessary to remark that we take The ~'s and ~'s are phase factors 12 representing a degree
the invariance of the S matrix as a postulate and not a of freedom in the definition of these inversions. The
theorem, so that we have no choice but to add contact operator T is antiunitary, while C and P are unitary.
terms to X(x) which will just cancel the noncovariant The matrix C••' was defined in Sec. II, and has the
parts of the propagator, such as the second term in properties
(5.12). CJWC-l=-JW*, (6.7)
In summary, we are to construct the S matrix ac-
C*C=(-)2i; CtC=1 (6.8)
cording to the Feynman rules (a)-(e), but with the
slight modifications: where JW are the usual 2j+ 1- dimensional angular-
(a') Pay no attention to the noncovariant contact momentum matrices.
interactions; compute the vertex factors using only the In order to describe the effect that C and P have on
original covariant part of X(x). the field '1'. (x), it will be necessary to introduce a
(b') Do not use (5.3) for internal lines; instead use second 2j+1-component field:

x.(x)= (2".)-3/2f~ L [D ••'(il[L( -p)]a(p,O")e'P'x+( - )2iL{DW[L( -P)]C-'I ••,b*(p'O")e-'''']' (6.9)

[2W(p)]'/2 .' .'

This is the field that we would have constructed in- the (O,j) representation of the Lorentz group:
stead of 'I'.(x) had we chosen to represent the "boost"
generators by
U[A]X.(X)U-l[A] = L., D••,W[A-l]X., (Ax) , (6.10)

(2.37) (6,11)

instead of Eq. (2.36). The field x.(x) transfonns under 12 For a general discussion of these phases, see G. Feinberg and
S. Weinberg, Nuovo Cimento 14, 571 (1959). The discussion there
II See, for example, H. Umezawa, Quantum Field Theory (North- was limited to (0,0), (!,!), and (!,O)ffi(OJ) fields, but can be
Holland Publishing Company, Amsterdam, 1956), Chap. X. easily adapted to the general case.
138 CHAPTER II

B1326 STEVEN WEINBERG

the matrix jj appearing instead of D because we use spinless field:

(2.40) instead of (2.39). Like I".(x), the field x.(x)
CPTI".(x)T-IP-IC-I=~C~P~TI".t( -x), (6.25)
obeys the Klein-Gordon equation (and no other field
equation) and commutes with its adjoint outside the CPTx.(x)T-IP-IC-I=~C~P~T( - )'iX. t ( -x), (6.26)
light-cone. It also has causal commutation relations
with 1". (x), but only because of our choice of the sign permitting a great simplification in the proof of the
(- )'i in Eq. (6.9). CPT theorem.] The use of 2j+1-component fields
The effect of T, C, and P on I".(x) and x.(x) can be (either 1". or x.) for massive particles as well as for
readily calculated by use of the formula: neutrinos would seem very appropriate in theories of
the weak interactions, where CP and T are conserved
DW[L(p)]*=CDW[L( - p)]C-I. (6.12) but C and P are not.
We find that:
VII. 2(2j+l)-COMPONENT FIELDS
TI".(x)T-I=~T L:., C ••' 1".' (x, -tJ), (6.13) Any parity-conserving interaction must involve both
TX.(x)T-I=~T L:., C..,X., (x, -tJ), (6.14) the (j,0) field I".(x) and the (O,j) field x.(x). It is
therefore convenient to unite these two (2j+1)-
CI".(X)C-I=~CL:., C••,-IX.,t(X) , (6.15) component fields into a single 2(2j+1)-component
CX.(X)C-I=~C( - )'iL:., C••,-II".' t(x) , (6.16) field:

PI".(X)P-I=~pX.( - X, tJ), (6.17) y,(X)=[I"(X)] . (7.1)

X(x)
pX.(X)P-I=~pl".( - x, tJ), (6.18)
This field transforms according to the (j,0)® (O,j)
provided that the antiparticle inversion phases are representation, i.e.,
chosen as
U[A]y,.(x)U-'[AJ= L:~ !I)apW[A-I]y,p(Ax) , (7.2)
~T=~T*j ~c=~c*j ;;p=~p*(_)'i. (6.19)

°]
where
Any other choice of the fj would result in the creation
DW[A]
and annihilation parts of 1". and X. transforming with
different phases, destroying the possibility of simple
transformation laws." .
!I)W[AJ= [
.° _,
DW[A]
(7.3)

If a particle is its own antiparticle then we call it the representations DW and jjw being defined by
"purely neutral," and set (2.36) and (2.37) respectively. The representation !I)w
can be defined also by specifying that the generators
a(p,u)=b(p,u). (6.20) of rotations are to be represented by
In this special case the (j 0) and (0 j) fields are related
by
X.t (x) = L:., C.., 1".' (x) , (6.21)
3 w= [ °°JW]
JW'
(7.4)

I".t(x) = (- )'iL:., C••' X.' (x) . (6.22) and that the generators of boosts are represented by
The fields are not Hermitian, except of course for j=O. ~W=-haw, ~~
Nevertheless, Eq. (6.20) requires the phases ~I to be where 'Y. is the 2(2j+1)-dimensional matrix:
equal to the corresponding'1I, and (6.19) then implies
that these phases can only take the real values ± 1,

°-1
except that ~ p must be ±i for purely neutral fermions. 'Y5=[1 0J. (7.6)
We see that the fields I"'(x) and x.(x) transform
separately under T, and also under the combined This satisfies (2.24)-(2.26) because 'Y.'= 1.
operation CP: The (j,0)® (OJ) representation (7.3) differs from the
CPI".(X)P-IC-I=~C~p L:., C..,-II".,t(-x, tJ), (6.23) (j,O) and (O,j) representations in the important re-
spect that !I)t is equivalent to !I)-I:
CPX. (x) p-IC-I
!I)(;)[A]!=/1!I)(;)[A-']B, (7.7)
=~~p(- )'iL:., C ••,-IX.,t(-x, tJ). (6.24)
where
[Under CPT the transformation law is just that of a
(7.8)
13 An important consequence is that a particle-antiparticle pair
has intrinsic parity
qpijp=(_)'i,
[See Eq. (6.11).J This has the consequence that
a well-known result that would be inexplicable on the basis of
nonrelativistic quantum mechanics. U[AJ~.(X)U-l[AJ=L:p ~p(Ax):l)p.(j)[AJ, (7.9)
REPRESENTAnONS OF THE POINCARE GROUP 139

r E Y " ~j A" R C L E S FOR A" Y S P 1:'1 B1327

where if, is the covariant adjoint where II (g) and fi(g) are defined by (A.10) and (A.41).
In the 2(2j+ I)-dimensional matrix notation this reads
if,(x)=·P(x)fI. (7.10)
The T, C, and P transformation properties of ",.(x) [-y.,.2···.,'8,,8. 2 · ·8. 2 ,+m'J)f(x) =0, (7.19)
can be read off immediately from (6.13)-(6.18):
where the generalized 'I matrices, '1.1• 2"', are defined by
T"'(x)T-'=~Te",(X, -x") (7.11)
~ce-I{3"'*(x) (bosons), (7.20)
C",(X)C-I = (7.12)
~ce-l'15f!l/t*(X) (fermions),
P"'(X)P-I=~p{3"'( - x, x"), (7.13) and are discussed and evaluated in Appendix B.
with The field'" obviously obeys causal commutation re-
C 0 lations, since 'P and X commute with both \Ot and xt
e= [ ] (7.14) at spacelike separations. Its homogeneous Green's
o C functions are
A purely neutral particle will have a field which satisfies
the reality condition ("'a(x)if,~(y)o= (271')-'m- 2]j.!!....M a~(p)
e-'fI"'*(x) (bosons) 2w(p)
"'(x) = (7.15)
e- I'I5{3"'*(X) (fermions). Xexp{ip· (x-y)), (7.21)
Its inversion phases ~T, ~C, ~ P must be real, except that
~ p = ±i for purely neutral fermions.
(if,~(y)"'.(x)o= (271')-'1II- 2 ]j d'p .Y a6 (p)
2w(p)
The field "'(x) of course satisfies the Klein-Gordon
equation Xexp{ip· (y-x)), (7.22)
(OLm')",.(X) =0. (7.16)
where
But "'(x) has twice as many components as the opera-
tors a(p,er) and b* (p,er), so it has a chance of also satis-
m'] U(P)]
M(p)= [ , (7.23)
fying some other homogeneous field equation. In fact, fi(p) m']
it does. LTsing (A.12) and (:\.40), we can easily show
that the (j,O) and (O,j) fields are related by (-m)2 j U(p) ]
X(p)= [ _ = (-)2]M(_p). (7.2-1)
n(-i8)\O(x)=m2jx(x), (7.17) U(p) (-m)2]

(7.18) The "raw" propagator is then

- - - - - - _.. _ - - - - - - - - - - - -

. J3 p
= (271')-3m-,jj --[O(x-ylM a6( -i8) exp{ip· (x-y)l+O(y-x)Ma~( -i8) exp{ip· (y-x))]. (7.25)
2w(p)

As discussed in Sec. V, this is not the covariant propagator to be used in conjunction with the Feynman rules. We
must add certain noncovariant contact terms to (7.25) which allow us to move the derivatives in M (-i8) to the
left of the 0 functions. The true propagator is

Sa~(x-)') = (271')-"",-2 J M a~( -i8) j d'p

--[O(x-y) exp{ip· (x-y))+O(y-x) exp{ip· (y-x)}]
2w(p)
=-im-'jMa~(-ia)t:,.C(x-y), (7.26)
------------

where t:,.C(x) is the invariant j=O propagator (5.9). In momentum space we replace a. by iq., so that
This can be written in a more familiar form by using
(B.13) ; we find that
Seq) = -im-'J[ CP(q)+m 2j]N+m L i.,
where
Sex) =im-2i['1","2 ". 2j8.,a.,.· ·8"j-m,j]t:,.C(x). (7.33)
It is easy to see from (B.4) that this has the correct
General formulas for CP(g) are given in Appendix B;
transformation property:
the results for j ~ 2 are in Table 2. The wave functions
;I)Ul[A]S(x) ;I)(jH[A] = S(Ax) . for creation and annihilation of particles and anti-
140 CHAPTER II

B1328 STEVEN WEINBERG

particles can be read off from (7.1), (4.11), and (6.9), or going back to a and b*

f
or alternatively found from the solutions of (7.19).
This whole formalism reduces to the Dirac theory for
j=i·
1/t.(x) = d'p ~ [u.(p,u)a(p,u)e'P"
+vn(p,u)b*(p,u)e-iP"J, (8.8)
VIII. GENERAL FIELDS
We started in Sec. III by introducing a field I"(x) where the "wave functions" in (8.8) are
which transforms according to the (j,0) representation. u.(p,u)= (2 ... )-'/2[2w(p)j'/2 L D.m[L(p)]it..(u) , (8.9)
Then, in order to discuss parity conserving theories,
we introduced the (O,j) field x(x) in Sec. VI and used
it in Sec. VII to construct a field 1/t(x) which transforms V.(p,u) = (2...)-·/2[2w(p)J-l/2 L Dnm[L(p)J
m,tT'
under the (reducible) representation (j,0)~ (O,j). These
particular fields have the advantage of depending very XV.. (U')C.,.-l. (8.10)
simply and explicitly on the particular value of j, but
1", x, and 1/t are certainly not otherwise unique. In fact, This field transforms correctly
the usual tensor representation of a field with integer j U[A,aJ1/t.(x)U-l[A,aJ= L mD.,.[A-IJ1/tm(Ax+a). (8.11)
is (j12,j/2), while the Rarita-Schwinger representation
for half-integer j is based on the (2j+ 1)2-dimensional It obeys the Klein-Gordon equation, and mayor may
reducible representation: not obey other field equations as well. The causality
condition (1.8) can be satisfied if we choose
2j -l 2j-l)
m,O)~(O,t)J® ( -4-'-4- . L u.(u)Um*(u) = L v.(u)V,.*(u) , (8.12)

Our simpler fields agree with these conventional repre- and if we use the usual connection between spin and
sentations only for the case j = t. statistics. We will not pursue these matters further here.
We now consider the general case. Let D.m[AJ be The chief point to be learned from this general con-
any representation (perhaps reducible) of the Lorentz struction is that the wave functions (8.9), (8.10) which
group. Assume that when A is restricted to be a rota- enter into the Feynman rules are always determined by
tion R, the representation D[RJ contains a particular the matrices Dnm[L(p)J representing a boost.
component DW[R]. By this we mean that there must
be a rotation basis of vectors u.(u), such that ACKNOWLEDGMENTS

Lm D.m[RJUm(u) = L, un(u')D.,.W[R]. (8.1) I am very grateful to P. N. Burgoyne, K. M. Watson,

and E. H. Wichmann, for conversations on this subject.
We can form an operator a.(p) analogous to (3.4):
APPENDIX A: SPINOR CALCULUS FOR ANY SPIN
an(P) = [2W(P)J1I2 Lm D.m[L(p)JUm(u)a(p,u) (8.2)
Everyone knows that the three Pauli matrices to-
which transforms simply:
gether with the 2X2 unit matrix make up a four
U[AJa.(p) U-I[AJ= L .. D.m[A-lJam(Ap). (8.3) vector t":
(AI)
[Use (8.1) and (2.12).J For the antiparticles we can use in the sense that
another basis vm(u), which in general mayor may not
be the same as the Um(u), but which must also satisfy D(1/2)[AJt"D(1/2)[AJt =A,"I'. (A2)
LmD.m[RJvm(u)=L, v.(u')D.,.W[R]. (8.4) Here A is a general proper homogeneous Lorentz trans-
formation, and D(1I2)[AJ is the corresponding 2X2
The operator fl.(p) analogous to (3.5) is now formed as matrix in the (t,0) representation, defined by repre-
fl.(p) = [2w(p)J'/2 L D.m[L(p)J senting the generators of infinitesimal transformations
(I,tI',m as
XVm(u')C.,.-lb*(p,u). (8.5) 1 i
J=--a, K=---a. (A3)
Using (8.4) and (2.17), we see that it transforms just 2 2
like a.(p):
This famous construction of the vector t" is the basis
U[AJfJ.(p)U-I[AJ= Lm D.m[A-l]fJ".(Ap). (8.6) of the familiar spinor calculus, which can also be em-
The field is constructed as the invariant Fourier ployed in a rather cumbrous fashion to discuss spins
transform higher than t.

f d3p
1/t.(x) = (2...)-'/2 --[a.(p)e'P·'+fJ.(p)e-ip"J, (8.7)
2w(p)
We shall instead show here that this construction
of a vector out of two-dimensional matrices can be
directly generalized to the construction of a tensor of
REPRESENTATIONS OF THE POINCARE GROUP 141

FEYNMAN RULES FOR ANY SPIN BB29

rank 2j out of 2j+l-dimensional matrices." We shall we define a scalar matrix

also show that the commutator and propagator of a
II._.U) (q) = (- )2it......' .. ··'iq..q., . . ·qm' (AlO)
(2j+ l)-component field of spin j are proportional to
this tensor. We will prove that if q is in the forward light-cone,
We first prove that for any integral or half-integral j
there exists a set of quantities q'=-m2; q">O, (All)
then
fI' fI'= j, j-l, "', - j ) IIW(q)=m2iDW[L(q)]'=m2i exp( -28q· JU» (Al2)
{(ftl,PIIl!" 'P'~i (
).11, ).12, •• " ).I2j= 0, 1, 2,3 where
with the properties: q=qjlql,
(A13)
sinh8= Iql/m,
(a) t is symmetric in all ).I's.
(b) t is traceless in all ).I'S, i.e., and JW is the usual 2j+ l-dimensional representation
of the angular momentum. [The constant factor in
(M) (Al2) is of course arbitrary, but is chosen here so that
(c) t is a tensor, in the sense that the normalization of t will be as simple as possible.]
DW[A]tm'···.'iDW[A]f Proof of (Al2) :
=A"t1A,l''J·· ·A'2/,2itJllIl2"''''#, (A5) The transformation law (AS) implies that
where DW[A] is the 2j+l-dimensional matrix corre- DU)[A]IIW(q)DU)f[A]=IIW(Aq). (Al4)
sponding to A in the (j,0) representation. [These
Let us fix q to have the rest-value q=q(m), where
DW[A] are the same as used in the text, and are
defined by Eqs. (2.36) and (2.35). Ordinary matrix q"(m)=m; q(m)=O. (AlS)
multiplication is understood on the left-hand side of
(AS). Eq. (AS) reduces to (A2) for j = t.] (a) If A is a rotation then DCil[A] is the unitary
matrix
Exislence Proof: DW[A]=exp(ie' JU» , (Al6)
Let u. be a 2j + l-dimensional basis transforming where JW is the usual 2~+l-dimensional representa-
according to the (j,0) representation of the Lorentz
tion of the angular momdntum vector J. The vector
group, i.e., (AlS) is rotation-invariant, so (Al4) gives
u.-> L.- D._.W[A]u._. (A6)
[JW,IIW(q(m»]=O. (Al7)
The quantities u.u.* evidently furnish a (2j+l)2-
dimensional representation, the direct product of the But the three matrices JW are irreducible, so Schur's
(j 0) representation with its complex conjugate. But Lemma tells us that IIW(q(m» must be proportional
this is to the unit matrix. We will fix its normalization so that
(j,O)® (O,j) = (j,j) (A7)
(Al8)
so the quantities u.u.* transform under the (j,j) and therefore
representation. The (j,j) representation consists of all (Al9)
symmetric traceless tensors of rank 2j, so it must be
Equation (Al4) therefore gives
possible to form such a tensor basis by taking linear
combinations of the u.u.*, i.e., IIW(Aq(m»=m2iDW[A]D<ilf[A]. (A20)
TJ.llPI·· 'P 2i(U) = Ln tarJlIP'" " P2iU"U/ (A8) (b) If A is the "boost" L(p) defined by Eq. (2.4),
then DW[A] is the Hermitian matrix
in such a way that the transformation (A6) gives
DW[L(p)]=exp( -Up· JW) (A2l)
and
But this requires that the t coefficients must satisfy L(p)q(m)=p. (A22)
Eq. (AS). They must also be symmetric and traceless
Formula (Al2) now follows immediately.
with respect to the ).Ii, because T(u) is symmetric and
The exponential in (Al2) may be calculated as a
traceless for all u. Q.E.D.
polynomial of order 2j in the matrix
Having proved the existence of the I's, we must now
establish a formula which will allow us to calculate z=2(q· JW). (A23)
them, and which will also provide a connection with the Recall that z is an Hermitian matrix with integer eigen-
Green's functions of field theory. For any four-vector q, values 2j, 2j-2, "', -2j, and that therefore
14 These are a special case of the matrices constructed by Barut,
Muzinich, and Williams, Ref. 4, by induction from the j =! case. (z-2j) (z-2j+2)· .. (z+2j)=O. (A24)
142 CHAPTER II

BI330 STEVEN WElNBERG

This can be rewritten to give z2fH as a polynomial of Setting this equal to - I'q, then gives (At).
order 2j in z. It follows then that IT(j)(q) must itself
be such a polynomial, since all powers of z beyond the To go through this sort of calculation for general j
2jth in the Taylor series for the exponential can be would be tedious and difficult. We shall approach the
reduced to polynomials in z of order 2j. problem of representing exp( - 20z) as a polynomial in z
For example, in the case of spin j=!, Eq. (A24) more directly. First split it into even and odd parts,
gives, z'= I, so that exp( -Oz) =coshOz-sinhOz. (A26)
exp( -zO) = l-zO+!O'-tzO'+' .. =coshO-z sinhO. We consider separately the cases of j integer and
half-integer.
Then (AI2) gives
1. Integer Spin
ITO/2) (q) =m[coshO- 2(g· JO/2» sinhOJ The eigenvalues 2j, 2j- 2, etc., of the Hermitian
=q'-2(q.J(lf2». (A2S) matrix z= 2(g· J) are even integers. If follows thatl '

i-I Z2 (Z2- 2') (Z2- 4')' .. (Z2- (2n )')

coshzO= 1 +L: sinh'n+20, (A27)
,,-0 (211+2) 1
i-I (Z2- 22) (z2-42) . .. (Z2- (211)2)
sinhzO= z coshe L: sinh2n+IO. (AZ8)
n-O (Zn+1) 1

Using (A26) and (AI2) gives for all q:

i-I (_ q2)j--l-n
Wi) (q) = (_q2)i+ L: (Zq. J)[(2q. J)2- (Zq)2J[(Zq. J)2- (4q)2}
",", (211+2) 1
X [(2q· J)2- (2nq)2J[2q· J - (2n+ 2)f] (A29)
or
( _ q2)J~1 (_ q2)H
IT(j)(q) = (-q2)i+ _ _ (2q· J)[2q·J-2fJ+--(2q· J)[(2q' J)'- (2q)2J[2q· J-4fJ
21 41
(_q2)f-3
+ - - ( 2 q · J)[(2q· J)2- (2q)2J[(2q· J)2- (4q)2J[2q· 1-6fJ+···. (A30)
61
The series (A30) cuts itself off automatically after j+ I terms. The terms we have listed are sufficient to calculate
IT for j = 0, I, 2, 3; the results are in Table I.

2. Half-Integer Spin
The eigenvalues 2j, 2j- 2, etc., of z= 2(q· J) are now odd integers. It follows that"
i-l/' (z2-1 2)(z'-32). "(Z'-(211-1)2) ]
coshzO= coshe[ 1+ L: sinh 2"O , (A31)
n-l (Zn)!
;-1/2 (z2-1')(z'-3 2) ... (Z2_ (2n-l)2) ]
sinhzO= z sinhB[ 1+ L: sinh2"O . (A32)
n-l (ZII+I)1

Using (AZ6) and (AI2) now gives:

;-1/2 (_q2);-n-I/2
IT(j)(q) = (_q2)i-1I'[f- 2q· 1]+ L:
,,_1 (211+1) 1
X[ (Zq' J)'- q2J[(Zqo 1)'- (3q)2}. -[(2q' J)2_ ([211-1Jq)2J[(2n+ l)f- 2q· JJ, (A33)

"For (A27) and (A31) see, for example, H. B. Dwight, Table oj Integrals and Other Mathematical Data (The Macmillan Company,
New York, 1961), fourth edition, formulas 403.11 and 403.13, respectively. Equations (A28) and (A32) can be checked by
differentiating with respect to 8; we get (A27) and (A31). I would like to thank C. Zemach for suggesting the existence of such ex-
pressions and a method of deriving them.
REPRESENTATIONS OF THE POINCARE GROUP 143

FEY 1\ ~l .\ r\ R lJ L E 5 FOR .\:\ Y 5 r I" B1331

or
1
IIUI(q) = (_q2)H12[q'-2q. J]+-( _q2)H/2[(2q· J)'-q'][3q"-2q· J]
31
1
+-( _q2)i- 5/ 2[(2q· J)2-q'][(2q· J)'- (3q)'][5q"-2q' J]+" '. (A34)
51

The series (A34) cuts itself off after j+~ terms. The It follows immediately from (A12) and (A40) that
terms we have listed suffice to caleula te II for j = ~, t ! ; lI Ul (q)fi(j)(q) = fi(j) (q)IIU)(q) = (_q2)';, (A46)
the results are in Table I.
Having calculated II(g), the coefficients lem'" may Substitution of (A1O) and (A41) into (A46) gives
be determined by inspection, For example, in the case
j = 1, Eq. (A30) gives tIl- I1l2 " 'Mit v1 1''/,-' 'P2 j qPIQ/J-2' .• qP2iq~lq~2' •• q~2i
= lPIP2" 'P2it"1~2" '~2iq/JlqP2' •• Q/J2jQ"lQ"2' •• Q"2i
JICl)(q) = -q'+2(q. J)(q. J _gil), (:\35)
= (-q')';. (A47)
Setting this equal to t.'q.q, gives
Since this holds true for any q, we can use it to derive
100=1 formulas for any symmetrized product of t and t. For
tU'=lill=+J, (A36) j=!:
lij= {J,,1;} -O'j.
(A48)

Observe that this is traceless, because APPENDIX B: DIRAC MATRICES FOR ANY SPIN
V=[2J'-3]-1=2(J'-2)=O, (A3i) We will use the 2j+ l-dimensional matrices I·'" ,
\I'e won't bother extracting the Ie'''' for j> 1, because t"'''' discussed in Appendix A to construct a set of
it is II(q) that we really need to know. 2(2j+ 1)-dimensional matrices:
We could have gone through this whole analysis using
the (O,j) instead of the (j,O) representation in (A5), (Ill)
In that case we should have defined a symmetric trace-
less object 1·'·'" '." which is a tensor in the sense that
DUI[A]lm'"'.'IDCi)[A]t
=.A~/l.\~l2 ... AII2jIJ.2itvIV2' "1I2j, (A38)
~,=G -~l (B2)

~=C ~l
where D(j)[A] is the matrix corresponding to A in the
(B3)
(O,j) representation:
D(jI[A] = D())["~-IJt, (:\39)
Their properties follow immediately from the work of
The fundamental formula (:\12) would then read Appendix A,
fi(iI (q) =m2jD(jl[L(q)]2= m2iD(jJ[L( - q)]2
1. Lorentz Transformations
= m'i exp(20q· JUI) , (A40)
where It follows from (AS), (A38), and (A39) that the ~'s
are tensors, in the sense that
Hence !DU)[A]~.,.',' '.';!D(j)-l[A]
tIl1 fJ'2 ... /J.2i=(±)tJl.1iJ.2 ......2
j ,
(A42) =A~tlA~l2 .. ·A"2/2j'Y~1~2 "~2j, (B4)
+
the sign being 1 or -1 according to whether the /-I'S
where !D Ci) is the (j,O)!JJ (O,j) representation
contain altogether an even or an odd number of space-
like indices. There is another relation between barred
and unbarred matrices which follows from (6.i): (B5)
fiCiI(q)*= ClI(j)(q)C-l , (A43)
and so Obviously ~5 is a scalar
:DCii[A }y,:DCi)-l[A]=~" (B6)
Equation (:\44) in conjunction with (0\42) yields the
reality condition but ~ is not, because
(A45) ~= _i-2;,),00",o (1l7)
144 CHAPTER II

BI332 STEVEN WEINBERG

2. Symmetry and Tracelessness and for half-integer j:

The t and t are symmetric and traceless in the I" in- (p(j) (q) = (_q')i-I/2[q"/l- 2q '~'Yr.,Il]
dices, so 'Y is also :
i-l/2 ( - l/)j-o-I/2
'Y .... ··· ..i ='Y.1' . .'··· ..i' (BB)
(any permutation),
+L [(2q·~)2_q2]
g...,'Y.... ···.'i=O. (B9) n91 (2n+I)!

3. Algebra X[(2q.~)2_(3q)2}. -[(2q.~)2-([2n-l]q)2

I have not studied the algebra generated by these 'Y X[(2n+I)q"/l-2q·~'Yr.,Il]. (BI7)
matrices in detail, but there is one simple relation that
can be derived very easily. It follows from (A47) that The results for j ~2 are presented in Table II.
for any q:
-yJlIJl'J.·' 'Jl1i'Y"1Jl'A" ·,I1iq"'lQI'2· .. Q",2;Q'lQ'2' . 'q""i= (q2)2( (Bi0) 5. Spin ! and 1
Cancellation of the q's gives the symmetrized product Table II gives
of two 'Y's as a symmetrized product of g". For example,
it follows from (B lO) that (pCI/2) (q)= -i'Y'q.=qo/l- 2 (q·3hr.,ll ,
so that
j=!: {-Y','Y'}=2g", (Bll)
j= 1 : {-y'P,'Y'~} + {-y",'YP~}+{-y''''YP'}
= 2[g"gp~+g'Pg'~+g"~g'p], (BI2)
and so on. (BIB)

4. Evaluation
Comparison with (AlO) and (A4I) shows that
(p(q) = -t~j'Y""" ·"iq..q.. ' .. q",/
Tbis is just the standard representation of the Dirac
matrices with 'Y5 diagonal.
= [_ 0 n(q)]. (BI3) For spin I, Table II gives
II (g) 0 (pCl)(q)='Y"q.q.= -1//l+2(q'3) (q. 3/l-q°'Yr.,Il) ,
The matrix n(q) was evaluated in Appendix A, and so that
fi (q) is just 'YOO=/l ,
fi(q) =n( - q, qO). (BI4)
'Y'''='Y''=3;'Yr.,Il, (B19)
It follows that we can calculate (P(q) from the formulae
(A29) and (A33) for n(q), by making the substitution 'Yii= {3i,3i}/l-~'j-/l.

JW-.3Cj)'Y5
,
where 3Cil=[JCj)
0
OJ
JCi)
(ElS)
Notes added in proof. (I) The external-line wave
functions are much simpler in the Jacob-Wick helicity
formalism. They are given for both massive and massless
and then multiplying the whole resulting formula on particles in a second article on the Feynman rules for
the right by fl. We find that for integer j: any spin (Phys. Rev., to be published). (We also give
(pCj) (q) = ( - q')i/l general rules for constructing Lorentz-invariant inter-
actions involving derivatives, field adjoints, etc.) (2)
i-l ( - 1/)'"-1-0
It is not strictly necessary to introduce 2 (2j+ I)-com-
+L (2q·~)[(2q·~)2_(2q)2]
ponent fields in order to satisfy P and C conservation,
0-<) (2n+2)!
because the x. fields in (6.15) and (6.17) may be ex-
X[(2q·~)2- (4q)'} .. [(2q·~)2- (2nq)2] pressed in terms of «3. by using (7.17). I would like to
X[2q'~/l- (2n+2)q"'Yr.,Il], (BI6) thank H. Stapp for a discussion on this point.
REPRESENT ATIONS OF THE POINCARE GROUP 145

Representations of the Poincare group for relativistic

extended hadrons
Y. S. Kim
Center for Theoretical PhysIcs. Depurtment of PhysIcs and Astronomy. Unwersiry of Maryland. College
Pa'k. Maryland 20742

Marilyn E. Noz
Department of Radiology. New York University. New York, New York 10016

S.H.Oh
L'aboratory oj Nuclear Science and Department of Physics. Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139
(Received 7 June 1978; revised manuscript received 23 October 1978)

Representations of the Poincare group are comtructed from the relativistic harmonic oscillator wave
functions which have been effective in descnhing the physics of internal quark motions in the relativistiC
quark model. These wave functions are solutions of the Lorentz-invariant harmonic oscillator differential
equation m the "cylindrical" coordmate system moving with the hadronic velocity in which the time-
separation variable is treated separately. This result enables us to assert that the hadronic mass spectrum
is generated by the internal quark level excltatlon, and that the hadronic spin is due to the internal
orbital angular momentum. An addendum relegated to PAPS contains diSCUSSions of detailed calculational
aspects of the Lorentz transformation, and of solutions of the oscillator equation which are diagonal in the
Casimir operators of the homogeneous Lorent7 group. It is shown there that the representation of the
homogeneous Lorentz group consists of solutions of the oscillator partial differential equation in a
"spherical" coordinate sy~tem in which the Lorentz-mvariant Minkowskian distance between the
constituent quarks is the radial variable.

I. INTRODUCTION harmonic oscillator potential of unit strength, and then dis-

cuss the generators of the Poincare group applicable to the
In building models of relativistic extended hadrons, we
entire system. In Sec. III, we present the oscillator wave
have to keep in mind the fundamental fact that the overall
functions which are diagonal in the invariant Casimir opera-
space-time symmetry structure is that of the Poincare
tors of the Poincare group.
group.' In our previous papers on physical applications of
the relativistic harmonic oscillator,' our primary purpose
was to devise a calculational scheme for explaining experi- II. FORMULATION OF THE PROBLEM
mental observations. As was pointed out by Biedenharn et In our previous papers on physical applications of the
al.,' the question of the Poincare symmetry has not been relativistic harmonic oscillators, we started with the follow-
systematically discussed. ing Lorentz-invariant differential equation;
The purpose of the present paper is to address this sym- 12[D, +D,l ~~x, ~x,)'+mo'I¢(x"x,)=O, (I)
metry problem. We are considering a model hadron consist-
ing of two spin less quarks bound together by a harmonic where x, and x, are the space-time coordinates for the two
oscillator potential. In this case, we are led to consider the spinless quarks bound together by a harmonic oscillator po-
center-of-mass coordinate which specifies the space-time lo- tential with unit spring constant. In order to simplify the
cation of the hadron, and the relative coordinate which above equation, let us define new coordinate variables
specifies the internal space-time separation between the
quarks.
x = !(x, + x,), x = (I!2V2)(x, ~ x,). (2)
The X coordinate represents the space-time specification of
Both the hadronic and internal coordinates are subject
the hadron as a whole, while the x variable measures the
to Poincare transformations consisting of translations and
relative space-time separation between the quarks. In terms
Lorentz transformations. The hadronic coordinate under-
of these variables, Eq. (I) can be written as
goes Poincare transformation in the usual manner. Howev-
er, the internal coordinate is invariant under translations. a' + m o' + -I
[ -~ ( a'
-~ ~x~' )] ¢(X,x) = o. (3)
This coordinate should, nonetheless, satisfy the Poincare ax/ 2 ax/
symmetry as a whole. We discuss in this paper the role of this
The above equation is separable in the X and x variables.
internal coordinate, and show that internal excitations gen-
Thus we write
erate the hadronic mass spectrum, and that the internal an-
gular momentum corresponds to the spin of the hadron. ¢(X,x) =f(X)I/J(x), (4)
In Sec. II, we formulate the problem using a model ha- wheref(X) and t/!{x) satisfy the following differential equa-
dron consisting of two spinless quarks bound together by a tions respectively;

1341 J. Math. Phys. 20(7), July 1979 0022·2488179/071341·04$01.00 © 1979 American Institute of PhySK;s 1341

Reprinted from 1. Math. rhys. 20, 1341 (1979).

146 CHAPTER II

III. PHYSICAL WAVE FUNCTIONS AND

ta~.' +mo' +(..1 + I)]/(X) =0, (5)
REPRESENTATIONS OF THE POINCARE
GROUP
.l (~ - x/)I/J(x) = (..1 + 1)rP(x). (6)
2 ax/ In constructing wave functions diagonal in the Casimir
The differential equation ofEq. (5) is a Klein-:Gordon equa- operators of the Poincare group, we note first that the opera-
tion, and its solutions are well known.f(X) takes the form tor which acts on the wave function in the subsidiary condi-
tion of Eq. (9) commutes with these invariant operators:
I(X) = exp( ± ip.X), (7)
[p',ppa!l = 0, (13)
with
(14)
p' = m o' + (..1 + I), (8)
Therefore, the wave functions satisfying the condition ofEq.
where p is the 4-momentum of the hadron. p' is, of course,
(9) can be diagonal in the Casimir operators.
the mass of the hadron and is numerically constrained to
take the values allowed by Eq. (8). The separation constant..1 In order to obtain the solutions explicitly, let us assume
is determined from the solutions of the harmoni'c oscillator without loss of ~enerality that the hadron moves along the z
differential equation ofEq. (6). The physical solutions of the direction with the velocity parameter [3. Then we are led to
oscillator equation satisfy the subsidiary condition consider the Lorentz frame where the hadron is at rest, and
the coordinate variables are given by
pI'a!I/Jp(x) = 0, (9)
x'=x, y'=y,
where
z' =(Z-/3I)/(I-/3')'''. (15)
at =X + ..!.... I' = (I - /3z)/(1 - [3')'''.
• • axP The Lorentz-invariant oscillator equation of Eq. (6) is sep-
The physics of this subsidiary condition has been ext~nsive\y
arable in the above variables. In terms of these primed varia-
discussed in the literature.'"
bles, we can construct a complete set of wave functions
The space-time transformation of the total wave func-
I/Jp(X) = Ib(X')/, (y')/,(z')lk(1 '). (16)
tion ofEq. (4) is generated by the following ten generators of
the Poincare group. The operators where

P =i~ (10)
I,(z') = (V-;2'n!) '12H ,(Z') exp( - z"/2),
• ax· 1,(1')= (V-;2 k k!) -II2Hk (I')eXp( -1"/2).
generate space-time translations. Lorentz transformations,
If the excitation numbers. b.... ,k are allowed to take all possi-
which include boosts and rotations, are generated by
ble nonnegative integer values, the solutions in Eq. (16) form
M." = L :,. + L,n' (11) a complete set. However, the eigenvalues A takes the form
where A=b+s+n-k. (17)
Because the coefficient of k is negative in the above expres-
L:.,=i(Xp~ -x,~),
r ax" ax" sion, A has no lower bound, and there is an infinite degener-
acy for a given value of A.
L." = i(X• ..!.... - X, ...!....).
ax" ax" In terms of the primed coordinates, the subsidiary con-
The translation operators p. act only on the hadronic dition of Eq. (9) takes the simple form
coordinate, and do not affect the internal coordinate. The
;n.
operators L and L,n' Lorentz-transform the hadronic and (:r, + l}p(X) =0. (18)
internal coordinates respectively. The above ten generators
This limits/, (I ') to/.(1 '), and the eigenvalue A becomes
satisfy the commutation relations for the Poincare group.
A=b+s+n, (19)
In order to consider irreducible representations of the
Poincare group, we have to construct wave functions which The physical wave functions satisfying the subsidiary condi-
are diagonal in the invariant Casimir operators of the group, tion ofEq. (9) or (18) have nonnegative values of A.
which commute with all the generators ofEqs. (10) and (II). As far as the x'. y', z' coordinates are concerned, they
The Casimir operators in this case are form an orthogonal Euclidean space. and/.(x'),f,(y'),/,,(z')
PllP'l and Wf'~j' (12) form a complete set in this three-dimensional space. The
Hermite polynomials in these Cartesian wave functions can
where then be combined to form the eigenfunctions of W' which, ill
terms of the primed coordinate variables. takes the form
W'=M'(L')'. (20)
The eigenvalues of the aboveP' and W' represent respective-
ly the mass and spin of the hadron. where

1342 J. Malh. Phys.• Vol. 20. No.7, July 1979 Kim, Noz, and Oh 1342
REPRESENTATIONS OF THE POINCARE GROUP 147

The functional forms ofEqs. (23) and (24) are relatively

simple, and they suggest that this representation of the Poin-
care group corresponds to the solution of the lorentz-invar-
iant oscillator differential equation in a "cylindrical" coordi-
nate system moving with the hadronic velocity where the t'
FIG. I Elliptic and hyper-
variable is treated separately. We are then led to the question
bolic localizations in space- of why this fact was not known.
time. The wave functions in
the present paper are ellipti·
Even though the above representations take simple
cally localized. and undergo forms, the wave functions contain the following nonconven-
Lorentz deformation as the tional features. The first point to note is that they are written
hadron moves. The Lorentz as functions ofthex' ,y', z', t ' variables. The transverse varia-
invariant form XllX I ,. to
HYPERBOLIC blesx',y' are simply x andy respectively. However,z' and t'
which we are accustomed, is
hyperbolically localized. and are linear combinations of z and t. Because the physical
is basically different from the meaning of the time-separation variable was not clearly un-
form used in the present derstood, the 1 dependence discouraged us in the past from
paper
using it explicitly in representation theory. The explicit use
of this variable in the present paper is based on the progress
that has been made in our physical understanding of this
time-separation variable in terms of measurable quantities,
and in terms of the relativistic wave functions carrying a
covariant probability interpretation.'
Another factor which used to discourage the use ofthe t
L; = . ,a
- lCi;".xj aX
variable was that we are accustomed to its appearance
through the form
k
and M is the hadronic mass. xJ.lx/-l=t 2 -r, (25)

The physical wave functions now take the form where

",,"'m(x) = (lhT)'''[exp( - t ")]R,,(r')Y'm(O ',,p '), (21) r' = x' + y' + z'.
where r', () " ,p , are the radial and spherical variables in the In terms of this form, it is very inconvenient, if not impossi-
three-dimensional space spanned by x', y', z'. R.,(r') is the ble, to describe functions which are localized in a finite
normalized radial wave function for the three-dimensional space-time region.
isotropic harmonic oscillator, and its form is well known. In contrast to the above hyperbolic case, the wave func-
The above wave function is diagonal in W' for which the tions which we constructed in this paper are well localized
eigenvalue is / (/ + I)M', and / represents the total spin of the within the region
hadron in the present case. The quantum number m corre-
sponds to the helicity.
(z" + t ") < 2, (26)

Since the eigenvalue p' of the Casimir operator P' is due to the Gaussian factor appearing in the wave functions.
constrained to take the numerical values allowed by Eq. (8), This elliptic form was obtained from the covariant
the hadronic mass is given by expression

M' = m o' + (), + I). (22) _ x"x" + 2(x·p/M)' = x" + y" + z', + 1 ". (27)

Ifwe relax the subsidiary condition ofEq. (18), we in- The x' and y' variables have been omitted in Eq. (26) because
deed obtain a complete set. In this case, A of Eq. (17) can they are trivial. In terms ofz and t, the above inequality takes
become negative for sufficiently large values of k. For A> 0, the form

)'j < 2.
the solutions become
l/fii'''''(x) = [V-;':-2'k,j ,/2H,(t')[exp(-t"/2)] [ 1 - /3 (z + t)' + 1 + /3 (z - t (28)
1+/3 1-/3
We are therefore dealing with the function localized within
)( RA , <.I(r') Y'm(fJ',,p '). (23) an elliptic region defined by this inequality, and can control
For..l < 0, the solutions take the form the 1 variable in the same manner as we do in the case of the
spatial variables appearing in nonrelativistic quantum me-
ifI''''' (x) = [V--;2" "(k - A)! j 'I'H, _"(I ') chanics. This localization property together with the hyper-
bolic case is illustrated in Fig. 1. .
X [exp( - t ")]Ru(r')Y'm«()',,p '). (24)
The eigenvalues of P' and W' are again me' + (A + I) and
IV. CONCLUDING REMARKS
/ (l + I)M' respectively. In both of the above cases, k is al- We have shown in this paper that the wave functions
lowed to take all possible integer values. used in our previous papers are diagonal in the Casimir oper-

1343 J. Malh. Phys .. Vol. 20. NO.7. JuLy 1979 Kim, Noz, and Oh 1343
148 CHAPTER II

ators of the Poincare group, which specify covariantly the Kim and M.E. Noz, Prog. Theor. Phys. 57, IJ73 (1977); 60, 801 (1978);
mass and total spin of the hadron. These wave functions are Y.S. Kim and M.E. Noz, Found. Phys. 9, 375 (1979); Y.S. Kim, M.E. Noz,
and S.H. Oh, "Lorentz Deformation and the Jet Phenomenon," Found.
well localized in a space-time region, and undergoes elliptic Phys. (to be published). For review articles written for teaching purposes,
Lorentz deformation. see Y.S. Kim and M.E. Noz, Am. J. Phys. 46, 480, 486 (1978). For a review
written for the purpose of form,ulating a field theory of extended hadrons,
An addendum to this paper containing a discussion of see T.J. Karr, Ph.D. thesis (University of Maryland, 1916).
Lorentz transformation of the physical wave function and a 'L.e. Biedenham and H. van Dam, Phys. Rev. D 9, 471 (1974).
construction of the representation of the homogeneous Lo- 'T. Takabayashi, Phys. Rev. 139, BI381 (1965); S.lshidaandl.Otokozawa,
rentz group is relegated to PAPS.' It is shown there that Prog. Theor. Phys. 47, 2117 (1972).
'See AlP document no. PAPS lMAPA-20-I336-12 for twelve pages of
solutions of the oscillator equation diagonal in the Casimir discussions of the Lorentz transformation of the physical wave functions,
operators of the homogeneous Lorentz group are localized and of the representations of the homogeneous Lorentz group. Order by
within the Lorentz-invariant hyperbolic region illustrated in PAPS number and journal reference from American Institute of Physics,
Fig. I. Physics Auxiliary Publication Service, 335 East 45th Street, New York,
N.Y. 10017. The price is $1.50 for each microfiche (98 pages), or $5 for
photocopies of up to 30 pages with SO. 15 for each additiona1 page over 30
pages. Airmail additional. Make checks payable to the American Institute
'E.P. Wigner, Ann. Math. 40,149 (1939). of Physics. This material also appears in Current Physics Microfilm, the
'Y.S. Kim and M.E. Noz, Phys. Rev. D 8,3521 (1973), 12, 129 (1975); IS, monthly microfilm edition of the complete set of journals published by
335(1977); Y.S. Kim, 1. Korean Phys. Soc. 9, 54(1976); 11, I (1978); Y.S. AlP, on the frames immediately following this journal article.

1344 J. Math. Phys., Vol. 20, No.7, July 1979 Kim, Noz, and Oh 1344
REPRESENT ATIONS OF THE POINCARE GROUP 149

A simple method for illustrating the difference between the homogeneous and
inhomogeneous Lorentz groups
Y. S. Kim
Center for Theoretical Physics, Department of Physics and Astronomy, University of Maryland. College
Park. Maryland 20742

Marilyn E. Noz
Department of Radiology, New York University, New York. New York 10016

S. H. Oh
Laboratory for Nuclear Science and Department of Physics. Massachusetts Institute of Technology.
Cambridge. Massachusetts 02139
(Received 5 January 1979; accepted 12 June 1979)

A method is proposed for illustrating the difference between the Poincare

(inhomogeneous Lorentz) and homogeneous Lorentz groups. Representations of
the Poincare group are constructed from solutions of the relativistic harmonic
oscillator equation whose physical wave functions have been effective in
describing basic high-energy hadronic features in the relativistic quark model. It
is shown that the Poincare group can be represented by solutions of the
relativistic oscillator equation in a "moving cylindrical" coordinate system in
which the time-separation variable in the hadronic rest frame is treated
separately. Representations which are diagonal in the Casimir operators of the
homogeneous Lorentz group are also constructed from solutions of the same
oscillator differential equation in a hyperbolic coordinate system. It is pointed out
that the difference between the Poincare and homogeneous Lorentz groups
mainfests itself in the coordinate systems in which the oscillator differential
equation in separable.

I. INTRODUCTION tively by two different sets of solutions of the same Lo-

rentz-invariant partial differential equation. It is shown that
In our previous papers in this journal, we discussed the the (homogeneous) Lorentz group can be represented by
possibility of using the relativistic harmonic oscillator for- the solutions of the harmonic oscillator equation in a hy-
malism in teaching group theory, I and also in discussing perbolic coordinate system in which the Minkowskian
high-energy physics in the first-year course in quantum space-time distance between the quarks is a "radial" vari-
mechanics. 2 It was emphasized in Ref. I that the oscillator able, while the Poincare group is represented by the solu-
formalism with its mathematical simplicity can serve as an tions of the same differential equation in a "cylindrical"
illustrative example in teaching one of the difficult theorems coordinate system in which the time-separation variable in
in group theory. As in the case of Ref. I, we are interested the hadronic rest frame is treated separately.
here in a group theory course for graduate students who In Sec. II, we formulate the problem in terms of the group
have some background in quantum mechanics and special generators applicable to solutions of the Lorentz-invariant
relativity_ In such a group theory course, the inhomogeneous differential equation for the hadron consisting of two quarks
Lorentz group, which is often called the Poincare group, bound together by a harmonic oscillator force. It is pointed
occupies an important place because its representations play out that the Poincare group is not a direct product but is
the fundamental role of specifying covariantly the mass and instead a semidirect product of a translation and a Lorentz
spin of a relativistic particle. 3 transformation, and therefore that the Casimir operators
The purpose of the present paper is to give a concrete of the Poincare group are quite different from those of the
example for illustrating how the Poincare group is different Lorentz group.
from a direct product of a translation and the homogeneous In Sec. 111, we construct the wave functions which are
Lorentz group which we call here simply the Lorentz group. diagonal in the Casimir operators of the Poincare group.
There are at present two different ways to tell students how It is shown that these wave functions are solutions in a
the Poincare group is different from the Lorentz group. One "moving cylindrical" coordinate system in which the
way is to point out that the Casimir operators for the former time-separation variable is treated separately. It is shown
are different from those of the latter 4 Another approach also that the physical solutions form a subset of a complete
is to show that the homogeneous Lorentz group is only a basis for the Poincare group. Transformation proper{ies of
little group of the Poincare group and tell students that this the physical solutions are also discussed.
little group has no physical significance 5 However, it would Section IV deals with solutions of the oscillator equation
be better if we could use concrete examples to illustrate this which are diagonal in the Casimir operators of the homo-
difference. geneous Lorentz group. Unlike the Poincare case, the so-
In this paper, we use the covariant harmonic oscillator lutions forming a representation of the homogeneous Lo-
formalism as an illustrative example. We show that the rentz group are obtained in the hyperbolic coordinate
Poincare and Lorentz groups can be represented res pec- system.

892 Am. J. Phys. 47( 10). Oct. 1979 0002-9505/79 J100892-06$00.50 © 1979 American A~sociation of Physics Teachers 892

Reprinted from Am. 1. Phys. 47,892 (1979).

150 CHAPTER II

II. STATEMENT OF THE PROBLEM a Lorentz transformation A. We call this a Poincare

transformation and write it as
Because of its mathematical simplicity, the harmonic
oscillator has served as a concrete solution to many physical A= (T,A). (8)
theories. It played an important role also in the development For two successive Poincare transformations A, and A2,
of relativistic theories of particle physics where hadrons are
regarded as bound states of quarks. 6,7 Let us consider in this A 2A, = (T2(A2T,),A2A,),
(9)
paper a hadron consisting of two quarks held together by '" (T2T"A 2A,).
a harmonic oscillator potential of unit strength, and start
with the differential equation 7 For this reason, the Poincare group is not a direct product
but is only a semidirect product of T and A.
12[0, + 021 - (I/16)(x, - X2)2 + mellq,(x"x2) = o. In terms of the operators applicable to the wave function
(I) of Eq. (4), space-time translations are generated by

This partial differential equation has many different solu- p . () (10)

tions depending on the coordinate system in which the "=, ()X"'
equation is separated, and on the boundary conditions. The
and the Lorentz transformation generators take the form
purpose of this paper is to discuss two different sets of so-
lutions which are useful for teaching group theory. (II)
In order to simplify the above equation, let us define new
coordinate variables where
X = (x, +
x2)/2,
x = (x, - x2)/2V2 .
(2) L:, = i (X. ()~, -X, ()~.), (12)

The X coordinate represents the space-time specification

of the hadron as a whole, while the x variable measures the L.'=+'()~'-x,()~"). (13)
relative space-time separation between the quarks. In terms
of these variables, Eq. (I) can be written The above four translation and six Lorentz-transformation
operators form the ten generators of the Poincare group.

!()~; + mel + ~ (::; - x;)] q,(X,x) = O.

The invariant Casimir operators which commute with all
(3) of the Poincare group generators are
p2 = p.p.,
This equation is separable in the X and x variables.
Thus and
q,(X,x) = f(X)if;(x), (4) (14)
andf(X) and "'(x) satisfy the following differential equa- where W. = f.,a~P'MaP.
tions, respectively: The solutions of the oscillator equation which are diag-
onal in the above Casimir operators have been discussed in
(~
()X"
+ mel + (A + l»)f(X) = 0, (5) the literature.',l,s In Sec. III of this paper, we discuss ex-
plicitly the transformation properties of the wave functions
representing the Poincare group.
~ (()~; - x;) if;(x) = (A + I )if;(x). (6) Since the relative coordinate x defined in Eq. (2) is not
affected by translation operations, it is difficult to resist the
Equation (5) is a Klein-Gordon equation, and its solution temptation to construct as well solutions which are diagonal
takes the form in the operators which commute only with the six L" op-
erators of Eq. (13). The Casimir operators for this homo-
f(X) = exp(± ip"X"), (7) geneous Lorentz group are
with
c, = (I/2)U'L." (IS)
p2 = M2 = mel + (A + I),
C2 = (1/4)f.,apU'La P. (16)
where M and p are, respectively, the mass and four mo-
mentum of the hadron. The eigenvalue A is determined from These operators are basically different from those of the
the solutions of Eq. (6). Poincare group.' We shall see this difference more explicitly
The wave function if;(x) of Eq. (6) is a solution of the by constructing representations of the homogeneous Lo-
harmonic oscillator differential equation whose physical rentz group in Sec. IV.
solutions have been discussed extensively in the literature.
Because it is separable in many different coordinate sys-
tems, the oscillator differential equation given in Eq. (6) III_ REPRESENTATIONS OF THE POINCARE
generates many solutions of mathematical interest, in ad- GROUP
dition to those which carry a physical interpretation.
We shall use these properties of the oscillator equation In this section we discuss wave functions which are di-
to illustrate the following important point to be covered in agonal in the Casimir operators of Eq. (14). Since the ha-
group theory courses for physicists. The most general dron in this case has a definite four momentum p specified
space-time transformation consists of a translation T and in Eq. (7), we are naturally interested in the Lorentz frame

893 Am. J. Phys., Vol. 47, No. 10, October 1979 Kim, Noz. and Oh X93
REPRESENTATIONS OF THE POINCARE GROUP 151

LORENTZ where
r' = (X'2 + y'2 + Z'2)1/2,
Rp,(r') is the normalized radial wave function for the os-
cillator, and its form is well known. The total wave func-
tion now takes the form
>/;~'mk(X) = R p ,(r')Y'!'((J',1>')fdl'), (23)

Fig. I. Concentration regions for the wave functions representing the

with A = Jl - k. This function generates the eigenvalues M2
Poincare and homogeneous Lorentz groups. The concentration region in and 1(1 + I )M2 for the Casimir operators p2 and WI, re-
both cases is dictated by the Gaussian factor in the wave function. In the spectively. The physical wave functions discussed in the
case the Poincare group, the wave function is concentrated within an ellipse literature are those of the above functions with k = O. Due
specified in the figure. If {1 = O. the ellipse becomes a circle. As f'increases, to the Gaussian factors, the above wave functions decrease
the ellipse becomes more eccentric. Unlike the Poincare case, the wave rapidly outside the region:
functions representing the homogeneous lorentz group are concentrated
within" the hyperbolic region around the light cones. This concentration (24)
region does not depend on {3, and is therefore Lorentz invariant.
If we ignore the trivial transverse coordinates, the above
inequality becomes

in which the hadron is at rest 9

for this frame are
The coordinate variables I(I
z'2 + ('2 = - --(3
- (z + 1)2 + -+(3 1
- (z - 1)2) <2
21+(3 1-{3 .
X' = x, y' = y. (25)
0' = (z - (3t)/( I - (32)1/2, (17) This {3-dependent elliptic region is illustrated in Fig. I.
t' = (t - (3z)/(1 - (32)1/2. The wave functions given in Eq, (23) can serve a useful
purpose in illustrating the group theoretical identities given
where Ii = p,/Po. We assume here that the hadron moves in the classic paper of Naimark which is standard reference
along the z axis. material in group theory courses for physicists,IO.11 Ref-
In terms of these rest-frame variables. we can write the erence 10 gives a detailed discussion of the recurrence
Casimir operator Wl of Eq. (14) simply as relations for the generators of the Lorentz transformation
W 2 = M2(L')2, (18) as well as those for the well-known case of the rotation
group. The L p , of Eq. (13) applicable to the x coordinate
where represents three rotation and three boost generators. The
, , () rotation operators are L'j, where i,j = 1,2,3. The boost
L/ = if.ijkXj--;-' generators consist of K, = L,o' The rotation generators take
()Xk
the form
The Casimir operator p2 is independent of the variable x,
but is constrained to take the eigenvalue of the oscillator (26)
equation given in Eq. (6). In terms of the rest-frame coor-
dinate variables, we can write the p2 operator applicable and the boost generators take the form
to the wave function >/;(x) as

p-,_
- nlo
, + -I (()2
--'-' - ")
XI-' . (19) K = -i
I
(x'()t~ + t~)(}Xi'
(27)
2 ()x,t
With the above forms of the Casimir operators, it is not where i = 1,2,3.
difficult to construct normalizable wave functions which The rotation around the n axis through an angle ~ is
are diagonal in Wl and p'. First, we observe that the os- represented by
cillator equation of Eq. (6) is Lorentz invariant and is sep-
arable also in the x'. y'. z'. t' variables. We can therefore R(n.O = exp[ -i~o· L), (28)
choose the cylindrical coordinate system in which the ('
and its mathematics is well known. The Lorentz boost along
variable is treated separately, and write the wave function the direction n by '7 is
as
>/;(x) = f(t')g(x'), (20) T(o,1)) = exp[ -i'7o' K), (29)

wheref(t') is a solution of the one-dimensional harmonic where

oscillator equation and takes the form
fdt'l = (v7r2 k k,)-1/2H k (t')exp(-t'2/2). (21)
As in the case of rotation, let us discuss first the effeGt of
This portion of the wave function is not affected by the the generators K, on the wave functions. It is more conve-
Casimir operator Wl. g(x') satisfies the three-dimensional nient to work with K J and K ±.IO where
isotropic oscillator equation. and the solutions diagonal in
Wl can be written (30)

g(x') = R",(r') Yf"(Ii',1>'). (22) If we apply these operators to the wave functions of Eq (23)

S94 Am. J. Ph}:>. .. Vol. 47. ~o 10. October 1979 Kim. Noz, and Oh 894
152 CHAPIERII

with {3 =0, we obtain It is shown in Ref. I that this expression can be simplified
iK .I,A' -
3';'lm - 3
A((I +(21m ++ .1)(/- m + 1))'/2
1)(21 + 3)
(J
YI+ I( .</»
m
to

(I + 1)(1- m) )'/2 yr.-,«(J.</». (31) ,

if;iiO(z.t) = fAZ({3)if;t"'(z.t). (38)
+B3 ((2/+ 1)(2/-1)
The remaining problem is to determine the coefficient
iK if;At = A ((I ±(21m ++ 1)(2/
1)(1 ± m + 2»)'/2 Ym ±I«(J ) t n ({3), Using the orthogonality relation, we can write
± /m '" + 3) I+I.</> AZ ((3) = f dzdtY;iiO(z,t)if;3+k"(z,t)
=;: Ht Lkn!(~ +
(l'Fm)(l'Fm-I»)'/2 "
+ B", ( (21 + 1)(2/- I) yr.-, «(J.</».
k)!Y (39)
The notation for the wave function in the above expression X f dzdIHn+k(z)H,(t)Hn(z')
is slightly different from that used in Eq. (23), but this
difference should not cause any confusion. The coefficients
of the spherical harmonics given above have been calculated
X exp (- ~ (z2 + z'2 + t 2 + 1'2»).
before by Naimark.1O What is new in the present work is In the above integral, the Hermite polynomials and the
that the coefficients A and B can be calculated in terms Gaussian form are mixed with the kinematics of Lorentz
explicit functions. These coefficients take the form transformation. However, if we use the generating function
A3 = Q-IFt'(r.O. B3 = QI+,Ft'(r,t). for the Hermite polynomial as Ruiz did in his paper,' 2 this
A", = 'F Q-IFt'(r.t). B± = ± QI+IFt'(r.t). (32) integral can be calculated easily, and

AZC(3) = (I - (32)(n+I)/2{3' (nn~k~)!r2

where
(40)
QI=t..£.+r..£.+J~. Thus Eq. (37) can be rewritten
i)r i)t r
FAk(r t) = if;t!(x)
/. Y7'(O.</»·
W'(z,t) = (;:t 2
mn/2(1 - (32)(n+I)/2

In order to understand the nature of the Lorentz trans-

formation T(n,'l) more clearly, let us concentrate our efforts X (Jo~:! Hn+k(Z)H,(t))
on the case where the boost is along the z direction. The
form of K3 applied to if;~'7 in Eq. (31) indicates that the X exp (- ~ (z2 + 12»), (41)
helicity quantum number m is conserved under this boost
transformation. However, the transformed wave function Let us now examine the implications of the above result.
contains all Possible values of I. This is a reflection of the Since the expression in Eq. (41) requires a sum over the
following nonvanishing commutator: longitudinal excitations equal to or higher than n, the Lo-
rentz transformed wave function with a given I value in the
[(L)2,(L')2] ~ O. (33) moving frame is a sum Over all corresponding I values of the
The forms of A3 and B3 in Eq. (32) indicate that the wave function at rest. This result is not inconsistent with
transformed wave function will contain radial and timelike K3if; of Eq. (31) and the nonvanishing commutator of Eq.
wave functions which are different from those given ini- (33).
tially. In order to see this and other points more clearly, let In the hadronic rest frame. the wave function with k >
us consider a finite boost along the z axis. For this purpose, ois not a physical solution, Therefore the wave function of
we go back to Eq. (23), and consider a Lorentz transfor- Eq, (41) is a sum of nonphysical solutions in the rest frame.
mation of a wave function without timelike oscillations for However, after the summation, these wave functions in the
which the calculation is relatively simple. rest frame form a physical wave function corresponding to
Because only the z and t components are affected by the a hadron moving with velocity parameter (3. The wave
boost along the z direction, we have to rewrite the wave function of Eq. (41) satisfies the subsidiary condition
function in terms of the Cartesian variables and their
Hermite polynomials. The portion of the wave function
which is affected by this transformation is
()~, + t') I/;p(x) = 0, (42)

whose physical interpretation has been extensively discussed

I ) 1/2 Hn(z)exp ( - 2"I (z2
if;3'O(z,t) = (1r2nn! + t 2)) . (34) in the literature. 1.2
For the Lorentz transformation of more general wave
The superscript 0 indicates that there are no timelike ex- functions with k > O. the calculation becomes more com-
citations: k = O. We now consider the transformation plicated. However, the mathematics is essentially the
if;'ii°(z.t) = T({3)if;3°(z.t). same.
(35)
Y;'ii°(z.t) = if;3°(z'.t'). (36)
IV. REPRESENTATIONS OF THE
and ask what T({3) does on 1/;8'O(z,t). In order to answer this HOMOGENEOUS LORENTZ GROUP
question, we write Eq. (35) as
The purpn,c of this paper is to show that the represen-
Y;~O(Z./) = L A~'~d(3)if;Z'·"(z.t). (37) tations of me Poincare group discussed in Sec. III are dif-
n'.k' ferent from those of the Lorentz group, The best way to

895 Am. 1. Phys .• Vol. 47. No. 10, October 1979

Kim. Noz, and Oh 895
REPRESENTATlONS OF THE POINCARE GROUP 153

teach this point is to construct solutions of the same dif- The radial wave function in this case takes the form
ferential equation, Eq. (6), which are diagonal in the
Casimir operators of the Lorentz group given in Eqs. (I5) R~..(p) = pnLtn+il(p2)exp(-p2/2), (55)
and (16). with f = 2(2/l + n), /l = 0,1,2,.... L~n+I)(p2) is the gen-
I n terms of the rotation and boost generators, the Casimir eralized Laguerre function,13
operators take the form With this preparation, we now write the "angular"
C I =L2_K2,C2 =L.K. (43) function B as

If we evaluate C 2 using the explicit expression for Land K, B~(a,O,</» = A~(a)Y7'(O,cP). (56)
this operator vanishes for the present spinless case. In order For the timelike region where t II>" we use the nota-
to construct solutions diagonal in C I, we use a hyperbolic tion
coordinate with the Lorentz invariant distance
A~(a) = T~(a), (57)
(44)
and for the spacelike region,
where
A~(a) = S~(a). (58)
, = (x 2 + y2 + z2)1/2, t = ± pcosha, , = psinha, Then T~(a) and S~(a) satisfy the following differential
(45) equations, respectively:
for It I >', and
~ (sinh2aT~) - [n(n + 2) + 1(1 + l)lT~ = 0,
t = p sinha, , = pcosha, (46) (ja

for It!
<,. For both cases, we use the usual three-dimen- (59)
sional spherical coordinate for x,y,z:
~ (cosh2aS~) - [n(n + 2) - 1(1 + I )lS~ = o.
x = , sinO cos</>, (ja
y = ,sinO sin</>, (47) (60)
Z = 'cosO. If 1 = 0, the solutions to the above equations take the
form
In terms of p, a, 0, cP, the differential equation of Eq. (6)
takes the form T~(a) = sinh(n + I )n/sinha,
(61)
S~(a) = cosh(n + I )a/cosha .

p3 (jp (p3 (jl/;)

..!..~ (jp
+ {..!.. (K2 -
\P2 Ll) - p2) I/; = fl/;, (48) For nonvanishing values of I,

where f = ±2().. + I) for the timelike and spacelike cases I d)l T~(a),
T~(a) = (sinha)l (-'-h--d
respectively. The form of L is well known. The operator (Ll SIn Q:' a
(62)
- K2) takes the form
d)'
I - - S~(a).
S~(a) = (coshn)l (' -
cosha dn
(U - K') = -.-1- ~ (sinh 2a ~) - -.-'- L2, (49)
slnh 2 a ()a ()lY smh 2 a
The solutions given in Eqs. (61) and (62) become infinite
for It I >'. and when a ~ 00. This means that the Lorentz harmonics are
singular along the light cones. At this point, we are tempted
, ') I ( () h' () ) to make n imaginary in order to make T~(n) and Sn(a)
(L- - K- = cosh'" ()" cos -" ()" + coshI 2" I '
,-,
normalizable. In fact, this and other interesting possibilities
(50) have been extensively discussed in the literature. 14 However,
if n takes noninteger values, the radial wave function be-
for It I <,. We arc interested in representations which arc comes singular along the light cones. In either case, the
diagonal in the above operators. light-cone singularity is unavoidable.
In order to construct the desired representation, we solve The wave functions which arc diagonal in the Casimir
the partial differential equation given in Eq. (4g) by sepa- operators ('I arc now
rating the variables
I/;'!;,(x) = R~(p)A;,(,,)Y7'(O.cP), (63)
I/;(x) = R(p)B(a./I.</», (51 )
where R~, A~ arc given in Eqs. (55), and (57), and (58),
In terms of R(p) and B(n,(I,</», Eg. (48) is separated
respectively.
into
The localization property of the above solution is dictated
-- p3_(») --,+p--l
[ I (j ( ry , 1R(p) =0, (52) by the Gaussian factor in the radial function R~(p), and is
illustrated in Fig. I. Unlike the case of wave functions
,,' (jp
(jp p'
representing the Poincare group, the hyperbolic localization
and
region is independent of the hadronic velocity and is thus
(L2 - K2)B(n,lI,</>1 = ryB(",1I,</». (53) Lorentz invariant. We can of course carry out the mathe-
matics of the operators L, and K, applied to the wave
In order that the radial equation have regular solu-
functions given in Eq. (63). However, it is not yet clear
tions,
whether this wave function carries any physical interpre-
ry = n(n + I). n = 0,' ,2,. (54) tation.)

896 Am. J. Phys., Vol. 47. No. 10, OClober 1979 Kim. NOI, and Oh 896
154 CHAPTER II

V. CONCLUDING REMARKS 'E. P. Wigner. Ann. Math. 40,149 (1939).

4L. Michel, in Group Theoretical Concepts and Methods in Elementary
We have discussed in this paper the difference between Particle Physics; edited by F. Gursey (Gordon and Breach, New York,
the representations of the Poincare group and those of the 1963).
Lorentz group starting from the same differential equation. Sf. R. Halpern, Special Relativity and Quantum Mechanics (Prentice-
The Poincare group is represented by solutions in a moving Hall. Englewood Cliff. NJ. 1968).
'H. Yukawa. Phys Rev. 91,416 (1953); K. Fujimura. T. Kobayashi. and
"cylindrical" coordinate system, while the representations
M. Namiki. Prog. Thcor. Phys. 43, 73 (1979); Y. S. Kim and M. E. NOl.
of the Lorentz group are formed from solutions in a hy- Phys. Rev. 0 15,335 (1977).
perbolic coordinate system. This result may serve a useful 1R. P. Feynman. M. Kislinger, and F. Ravndal, Phys. Rev. D 3, 2706
purpose in a group theory course for physicists. (1971).
As was clearly stated by Michel,4 confusion concerning 'Y. S. Kim. M. E. NOl. and S. H. Oh. J. Math. Phys. 20, 1341
this difference is not uncommon in the literature. This fact (1979).
was not recognized in Refs. I and 2. The present paper 9For more detailed discussions of this coordinate system. see Y. S. Kim
clarifies this important point. and M. E. NOl. Phys. Rev. 08,3521 (1973); Found. Phys. 9, 375
(1979). See atso. Y. S. Kim. Phys. Rev. 0 14, 273 (1976).
10M. A. Naimark. Uspehi Mat. Nauk 9, 19 (1954); Am. Math. Soc. Trans.
ACKNOWLEDGMENT 6, 379 (1957).
11 W. Miller, Symmetry Groups and Their Applications (Academic. New
In carrying out the calculations for Sec. IV, we have York and London, 1972).
been guided by M. Rubin's unpublished note. We thank 12M. J. Ruil, Phys. Rev. 0 10,4306 (1974).
Professor Rubin for making his note available to us. J3W. Magnus and F. Oberhettinger. Formulas and Theorems for the
Functions in Mathematical Physics (Chelsea. New York. 1949).
Iy. S. Kim and M. E. NOl. Am. J. Phys. 46, 480 (1978). 14E. G. Kalnins and W. Miller. J. Math. Phys. 18. 1 (1977). and the rer-
'Y. S. Kim and M. E. No•• Am. J. Phys. 46, 484 (1978). erences contained in this paper.

897 Am . .I. Phys .• Vol. 47. No. 10. October 1979 Kim, Noz. and Oh R97
Chapter III

The Time-Energy Uncertainty Relation

According to Heisenberg's uncertainty principle, the position and momentum are

noncommuting q-numbers. However, the time variable is a c-number. This causes
a problem when we make a Lorentz transformation. If the time variable is purely a
c-number in one Lorentz frame, it is no longer a c-number in different Lorentz
frames. This was first pointed out by Dirac in 1927. Indeed, the question of where
the time-energy uncertainty relation stands in quantum mechanics is constantly
debated in the literature.
As Wigner pointed out in 1972, the reason why the time energy uncertainty relation
is so peculiar is that there are not many physical phenomena which can be regarded
as direct consequences of the time energy uncertainty relation. In 1985, Hussar,
Kim, and Noz showed how the time-energy uncertainty relation can be combined
covariantly with the position-momentum uncertainty relation in the relativistic quark
model, and pointed out there are direct consequences of this uncertainty relation in
high-energy physics.

155
THE TIME-ENERGY UNCERTAINTY RELA TION 157

Emission and Absorption of Radiation. 243

The Quantum Theory of the Emission and Abs011Jtion of

Radiation.
By P. A. M. DIRAC,St. John's College, Cambridge, and Institute for
Theoretical Physics, Copenhagen.
(Communicated by i'l". Bohr, For. Mem. KS.-Received February 2, 1927.)

§ 1. Introduction and SlUnm.ary.

The new quantum theory, based on the assumption that the dynamical
variables do not obey the commutative law of multiplication, has by now been
developed sufficiently to form a fairly complete theory of dynamics. One can
treat mathematically the problem of any dynamical system composed of a
number of particles with instantaneous forces acting between them, provided it
is describable by a Hamiltonian function, and one can interpret the mathematics
physically by a quite definite general method. On the other hand, hardly
anything has been done up to the present on quantum electrodynamics. The
questions of the correct treatment of a system in which the forces are propa-
gated with the velocity of light instead of instantaneously, of the production of
an electromagnetic field by a moving electron, and of the reaction of this field
on the electron have not yet been touched. In addition, there is a serious
difficulty in making the theory satisfy all the requirements of the restricted

Reprinted from Proc. Roy. Soc. (London) A114, 243 (1927).

158 CHAPTER III

244 P. A. M. Dirac.

principle of relativity, since a Hamiltonian function can no longer be used.

This relativity question is, of course, connected with the previous ones, and it
will be impossible to answer anyone question completely without at the same
time' answering them all. However, it appears to be possible to build up a
fairly aatisfactory theory of the emission of radiation and of the reaction of
the radiation field on the emitting system on the basis of a kinematics and
dynamics which are not strictly relativistic. This is the main object of the
present paper. The theory is non-relativistic only on account of the time
being counted throughout as a c-number, instead of being treated symmetrically
with the space co-ordinates. The relativity variation of mass with velocity
is taken into account without difficulty.
The underlying ideas of the theory are very simple. Consider an atom inter-
acting with a field of radiation, which we may suppose for definiteness to be
confined in an enclosure so as to have only a discrete set of degrees of freedom.
Resolving the radiation into its Fourier components, we can consider the energy
and phase of each of the components to be dynamical variables describing the
radiation field. Thus if Er is the energy of a component labelled rand er
is the corresponding phase (defined as the time since the wave was in a standard
phase), we can suppose each Er and er to form a pair of canonically conjugate
variables. In the absence of any interaction between the field and the atom,
the whole system of field plus atom will be describaple by the Hamiltonian
H=~rEr+Ho (1)
equal to the total energy, Ho being the Hamiltonian for the atom alone, since
the variables E" eT obviously satisfy their canonical equations of motion
.
Er =
aH
-as . oH
=0, e = oE =
T 1.
r r

When there is interaction between the field and the atom, it could be taken into
account on the classical theory by the addition of an interaction term to the
Hamiltonian (1), which would be a function of the variables of the atom and of
the variables En eT that describe the field. This interaction term would give
the effect of the radiation on the atom, and also the reaction of the atom on the
radiation field.
In order that an analogous method may be used on the quantum theory,
it is necessary to assume that the variables E r, er are q-numbers satisfying
the standard quantum conditions erE, - Erer = ih, etc., where h is (21t')-1
times the usual Planck's constant, like the other dynamical variables of the
problem. This assumption immediately gives light-quantum properties to
THE TIME-ENERGY UNCERTAINTY RELATION 159

Emission and Absorption oj Radiation. 245

the radiation.· For if v, is the frequency of the component r, 21tY,6, is an

angle variable, so that its canonical conjugate E,/27tv, can only assume a
discrete set of values differing by multiples of h, which means that E, cal!-
change only by integral multiples of the quantum (27th) V,. If we now add a~
interaction term (taken over from the clasical theory) to the Hamiltonian (1),
the problem can be solved according to the rules of quantum mechanics, and
we would expect to obtain the correct results for the action of the radiation
and the atom on one another. It will be shown that we actually get the correct
laws for the emission and absorption of radiation, and the correct values for
Einstein's A's and B's. In the author's previous theory,t where the energies
and phases of the components of radiation were c-numbers, only the B's could
be obtained, and the reaction of the atom on the radiation could not be taken
into account.
It will also be shown that the Hamiltonian which describes the interaction
of the atom and the electromagnetic waves can be made identical with the
Hamiltonian for the problem of the interaction of the atom with an assembly
of particles moving with the velocity of light and satisfying the Einstein-Bose
statistics, by a suitable choice of the interaction energy for the particles. The
number of particles having any specified direction of motion and energy, which
can be used as a dynamical variable in the Hamiltonian for the particles, is
equal to the number of quanta of energy in the corresponding wave in the
Hamiltonian for the waves. There is thus a complete harmony between the
wave and light-quantum descriptions of the interaction. We shall actually
build up the theory from the light-quantum point of view, and show that the
Hamiltonian transforms naturally into a form which resembles that for the
waves.
The mathematical development of the theory has been made possible by the
author's general transformation theory of the quantum matrices_t Owing
to the fact that we count the time as a c-number, we are allowed to use the notion
of the value of any dynamical variable at any instant of time. This value is

• Similar assumptions have been used by Bom a.nd Jordan [' Z. f. Physik,' vol. 34,
p. 886 (1925» for the purpose of taking over the classical formula. for the emission of radiation
by a dipole into the quantum theory, and by Bom, Heisenberg and Jordan [' Z. f. Physik,'
voL 35, p. 606 (1925» for calculating the energy fluctuations in a field of bla.ck-body
radiation.
t ' Roy. Soc. Proc.,' A, voL 112, p. 661, § 5 (1926). This is quoted later by, loco cit., I.
t ' Roy. Soc. Proc.,' A, voL 113, p. 621 (1927). This is quoted later by loco cit., II. An
6118entially equivalent theory has been obtained independently by Jordan [' Z. f. Physik.'
vol. 40, p. 809 (1927)]. See also. F. London •. Z. f. Physik,' vol. 40. p. 193 (1926).
160 CHAPTERlll

246 P. A. M. Dirac.
a q-number, capable of being represented by a generalised" matrix" according
to many different matrix schemes, some of which may have continuous ranges
of rows and columns, and may require the matrix elements to involve certain
kinds of infinities (of the type given by the () functions·). A matrix scheme can
be found in which any desired set of constants of integration of the dynamical
system that commute are represented by diagonal matrices, or in which a set of
variables that commute are represented by matrices that are diagonal at a
specified time. t The values of the diagonal elements of a diagonal matrix
representing any q-number are the characteristic values of that q-number. A
Cartesian co-ordinate or momentum will in general have all characteristic values
from - 00 to + 00 , while an action variable has only a discrete set of character-
istic values. (We shall make it a rule to use unprimed letters to denote the
dynamical variables or q-numbers, and the same letters primed or multiply
primed, to denote their characteristic values. Transformation functions or eigen-
functions are functions of the characteristic values and not of the q-numbers
themselves, so they should always be written in terms of primed variables.)
Iff(~, '1) is any function of the canonical variables ~b ''lk> the matrix repre-
sentingf at any time t in the matrix scheme in which the ~I: at time t are diagonal
matrices may be written down without any trouble, since the matrices repre-
senting the ~I: and '1)1: themselves at time t are known, namely,
I
f.(2)
~k (r:'~") = ~k' () (~'r:"),

''l1(~'~'') = -ih 0(~1' -~l") ... O(~k-l' -~k-l") 0' (~k' -~I:") 0 (~+l' -~l:+l") .. '
Thus if the Hamiltonian H is given as a function of the ~k and 'Y)b we can at
once write down the matrix H(~' ~"). We can then obtain the transformation
function, (~' /a.') say, which transforms to a matrix scheme (a.) in which the
Hamiltonian is a diagonal matrix, as (~' /«') must satisfy the integral equation

JH (f~") d~" (~" /(1.') = W(<<') . (~' /«'), (3)

of which the characteristic values W(<<') are the energy levels. This equation
is just SchrOdinger's wave equation for the eigenfunctions (~' /«'), which becomes
an ordinary differential equation when H is a simple algebraic function of the

• Loc. cit. II, § 2.

t One can have a matrix scheme in which a set of variables that commute are at all times
represented by diagonal matrices if one will sacrifice the condition that the matrices must
satisfy the equations of motion. The transformation function from such a scheme to one
in which the equations of motion are satisfied Will involve the time explicitly. See p. 628
in loco eil., II.
THE TIME-ENERGY UNCERTAINTY RELATION 161

Emission and Abs01ption of Radiation. 247

~"and lJk on accoWlt of the special equations (2) for the matrices representing
~I: and "'II:' Equation (3) may be written in the more general form

JH (~'~") d~" (~"fa') = ik 0(;' fa') fat, (3')

in which it can be applied to systems for which the Hamiltonian involves the
time explicitly.
One may have a dynamical system specified by a Hamiltonian H which
cannot be expressed as an algebraic function of any set of canonical variables,
but which can all the same be represented by a matrix H(~'~"). Such a problem
can still be solved by the present method, since one can still use equation (3)
to obtain the energy levels and eigenfunctions. We shall find that the Hamilto-
nian which describes the interaction of a light-quantum and an atomic system is
of this more general type, so that the interaction can be treated mathematically,
although one cannot talk about an interaction potential energy in the usual
sense.
It should be observed that there is a difference between a light-wave and the
de Broglie or SchrOdinger wave associated with the light-quanta. Firstly, the
light-wave is always real, while the de Broglie wave associated with a light-
quantum moving in a definite direction must be taken to involve an imaginary
exponential. A more important difference is that their intensities are to be
interpreted in different ways. The number of light-quanta per Wlit volume
associated with a monochromatic light-wave equals the energy per unit volume
of the wave divided by the energy (j7th)v of a single light-quantum. On the
other hand a monochromatic de Broglie wave of amplitude a (multiplied into
the imaginary exponential factor) must be interpreted as' representing a 2 light-
quanta per unit volume for all frequencies. This is a special case of the general
rule for interpreting the matrix analysis, * according to which, if (~' fa') or
~.' (~,,') is the eigenfunction in the variables ~k of the state a' of an atomic
system (or simple particle), I ~A' (~k')11 is the probability of each ~I: having the
value ~k" [or I ~... (~') II d~l' d;2' .. , is the probability of each ~" lying between
the values ~k' and ~k' + d;,,', when the ~k have continuous ranges of character-
istic values] on the assumption that all phases of the system are equally probable.
The wave whose intensity is to be interpreted in the first of these two ways
appears in the theory only when one is dealing with an assembly pf the associated
particles satisfying the Einstein-Bose statistics. There is thtis no such wave
associated with electrons.
• Loc. cit•• II, §§ 6, 7.
162 CHAPTERll

248 P. A. M. Dirac.

§ 2. The Perturbation of an Assembly of Independent Systems.

We shall now consider the transitions produced in an atomic system by an
arbitrary perturbation. The method we shall adopt will be that previously
given by the author,t which leads in a simple way to equations which determine
the probability of th~ system being in any stationary state of the unperturbed
system at any time.t This, of course, gives immediately the probable number
of systems in that state at that time for an assembly of the systems
that are independent of one another and are all perturbed in the same way.
The object of the present section is to show that the equations for the rates
of change of these probable numbers can be put in the Hamiltonian form in a
simple manner, which will enable furt,her developments in the theory to be
made.
Let Ho be the Hamiltonian for the unperturbed system and V the perturbing
energy, which can be an arbitrary function ot the dynamical variables and may
or may not involve the time explicitly, so that the Hamiltonian for the perturbed
system is H = Ho + V; The eigenfunctions for the perturbed system must
satisfy the wave equation
ih oljijat = (Ho +
V) q"
+
where (Ho V) is an operator. If Iji = ~,arljir is the solution of this equation
that satisfies the proper initial conditions, where the q,,'s are the eigenfunctions
for the unperturbed system, each associated with one stationary state labelled
by the suffix r, and the a,'s are functions of the time only, then Iar 12 is the prob-
ability of the system being in the state r at any time. The a/s must be nor-
malised initially, and will then always remain normalised. The theory will
apply directly to an assembly of N similar independent systems if we multiply
each of these a,'s by Nl so 88 to make ~r Ia r 12 = N. We shall how have that
Ia r 12 is the probable number of systems in the state r.
The equation that determines the rate of change of the a,'s is§
ihdr = ~.Vr,a" (4)
where the Vr.'s are the elements of the matrix representing V. The conjugate
imaginary equation is
- ihar* = ~.Vr.·a.· = ~,a.*V". (4')
t Loc. cit. I.
l The theory has recently been extended by Born [' Z. f. Physik,' vol. 40, p. 167 (1926)]
so 88 to take into account the adia.batic changes in the stationary states that may be
produced by the perturbation as well 88 the transitions. This extension is not used in
the present paper.
§ Loc. cil., I, equa.tion (25).
THE TIME-ENERGY UNCERTAINTY RELATION 163

Emi3sion and Absorption oj Radiation. 249

If we regard a,. and i"

a,.* as canonical conjugates, equations (4) and (4') take
the Hamiltonian form with the Hamiltonian function Fl = "E.na,*V,,p,,
namely,

We can transform to the canonical variables N" +, by the contaet trans-

formation

This transformation makes the new variables N, and +, real, N, being equal
to apr· = I a,. the probable number of systems in the state r, and
12, +,1"
being the phase of the eigenfunction that represents them. The Hamiltonian
F 1 now becomes
F1_- ....
6J" "
V.N IN le,(+,-,,)/A
r, ,

and the equations that determine the rate at which transitions occur hav~ the
canonical form
N = _ aFl .~
, a!fo,' 4>, = aN'
,
A more convenient way of putting the transition equations in the Hamiltonian
form may be obtained with the help 'of the quantities

W, being the energy of the state r. W~ have Ib, III equal to Ia, 12, the probable
number of systems in the state r. For b, we find
ihb,= W,b, + iha,e-iW.l/ A

= W,b, +~. V,P,ei(w.-W,)t/A

with the help of (4). If we put Yr. = v"ei(W,-w.) tilt, so that v" is a constant
when V does not involve the time explicitly, this reduces to
ih b, = W,b, + r..Vrob.
= ~.H,.,b" (6)
where H,. = W, 8,. + v", which is a matrix element of the total Hamiltonian
H = Ho + V with the time factor ei(w.-w.)t/It removed, so that Hr. is a constant
when H does not involve the time explicitly. Equation (5) is of the same form
as equation (4), and may be put in the Hamiltonian form in the same way.
It should be noticed that equation (5) is obtained directly if one writes down
the SchrOdinger equation in a set of variables that specify the stationary states
of the unperturbed system. If these variables are ~, and if H(~'~")denptes
VOL. CXIV.-A. 8
164 CHAPTERm

250 P. A. M. Dirac.
a matrix element of the total Hamiltonian H in the (~) scheme, this
8chrOdinger equation would be
in OtV (~')fat = ~r' H (~'~.) tV (~"), (6)

like equation (3'). This difiers from the previous equation (5) only in the
notation, a single 'suffix r being there used to denote a stationary state instead
of a set of numerical values ~k' for the variables ~I:> and b, being used instead
of tV (~'). Equation (6), and therefore also equation (5), can still be used when
the Hamiltonian is of the more general type which cannot be expressed as an
algebraic function of a set of canonial variables, but can still be represented
by a matrix H (~'~") or H".
We now take b, and in b,* to be canonically conjugate variables instead of
Or and ina,*. The equation (5) and its conjugate imaginary equation will
now take the Hamiltonian form with the Hamiltonian function
F= ~,/J,. H,}J,. (7)

Proceeding as before, we make the contact transformation

(8)

to the new canonical variables N" 6" where N, is, as before, the probable
number of systems in the state r, and 6, is a new phase. The Hamiltonian F
will now become
F = 1:" H" N,l N,l ei(B,-',)/",
and the equations for the rates of change of N, and 6, will take the canonical
form
. of . aF
N,= - 06,' 0, = aN'
,
The Hamiltonian may be written
F= ~,W,N, + ~"v.. N,lN,le'(·'-'·)/1a. (9)
The first term 'f"W,N, is the total proper energy of the assembly, and the
second may be regarded as the additional energy due to the perturbation. If
the perturbation is zero, the phases 6, would increase linearly with the time,
while the previous phases "" would in this case be constants.

§3. The Perturbation of an Assembly satisfying the Einstein-Bose Statistics.

According to the preceding section we can describe the effect of a perturba-
tion on an assembly of independent systems by means of canonical variables
and Hamiltonian equations of motion. The development of the theory which
THE TIME-ENERGY UNCERTAINTY RELATION 165

Emission and Absorption of Radiation. 251

na~urally suggests itself is to make these canonical variables q-numbers satisfy-

ing the usual quantum conditions instead of c-numbers, so that their Hamilto-
nian equations of motion become true quantum equations. The Hamiltoruan
function will now provide a Schrodinger wave equation, which must be solved
and interpreted in the usual manner. The interpretation will give not merely
the probable number of systems in any state, but the probability of any given
distribution of the systems among the various states, this probability being,
in fact, equal to the square of the modulus of the normalised solution of the
wave equation that satisfies the appropriate initial conditions. We could,of
course, calculate directly from elementary considerations the probability of
any given distribution when the systems are independent, as we know the
probability of each system being in any particular state. We shall find that the
probability calculated directly in this way does not agree with that ob,tained
from the wave equation except in the special case when there is only one system
in the assembly. In the general case it will be shown that the wave equation
leads to the correct value for the probability of any given distribution when
the systems obey the Einstein-Bose statistics instead of being independent.
We assume the variables b" ihb,* of §2 to be canonical q-numbers satisfying
the quantum conditions
b, . ih b,* - ih b,* . b, = ih
or b,b,* - b,*br = 1,
and b,b, - b,b, = 0, b,*b,* - b,*b, *= 0,
b,b.* - b.*br = 0 (8 ~ r).

The transformation equations (8) must now be written in the quantum form

b, = (N, + 1)i e-;I,/A = e-ill,/I'NrJ }

(10)
b,* = N,*e"'/h = ei',/" (Nr + 1)1,
in order that the N" 6, may also be canonical variables. These equations
show that the Nr can have only integral characteristic values not less than
zero,t which provides us with a. justification for the assumption that the
variables are q-numbers in the way we ha.ve chosen. The numbers of systems
in the different states are now ordinary quantum numbers.

t See § 8 of the author'8 paper , Roy. 800. Proo.,' A, vol. 111, p. 281 (1926). What are
there called the c-number values that a q-number can take are here given the more precise
D&IlUI of the characteristic values of that q-number.

8 2
166 CHAPTERll

252 P. A. M. Dirac.
The Hamiltonian (7) now becomes
F = "J:.,p,*H,p, = "J:.,,N,tei8,/AH,,(N. + l)ie-;S./h
= "J:.,.II,.,N,i (N, +1- 8..). ei (S, - 9,)/h (11)

in which the H" are still c-numbers. We may write this F in the form corre-
8ponding to (9)
(11')

in which it is again composed of a proper energy term "J:.,W,N, and an inter-

action energy term.
The wave equation written in terms of the variables Nr ist

ih ~ tjI (Nt', N 2', N,' ... ) = FtjI (N1', N 2', Na' ... ), (12)

where F is an operator, each a, occurring in F being interpreted to mean ih alaN,' .

If we apply the operator e±iI/,/h to any function f(N 1', N 2', ' " N,', ... ) of the
variables N1', N 2', ... the result is
e±iB./h f(N 1', N 2', ... N,', ... ) = e"b/bN: f(N l ', N 2', ... N,' ... )
= f(N I ', N 2', ... N,' =t= 1, ... ).
If we use this rule in equation (12) and use the expression (ll) for F we obtaint

ih :e tjI(N I ', N 2', Na' ... )

= l:,. H,.N,'i (N,' + 1-:- 8,,)1 tjI (N1', N 2' ... N,' -1, ... N,' + 1, ... ). (13)
We see from the right-hand side of this equation that in the matrix repre-
senting F, the term in F involving eil90 - ',)fA will contribute only to
those matrix elements that refer to transitions in which N, decreases
by unity and N, increases by unity, i.e., to matrix elements of the type
F (N I ', N 2' ... N,' ... N,'; N l ', Nt' ... N,' - 1 ... N,' + 1 ... ). If we find a
solution ~(Nt', N 2' ... ) of equation (13) that is normalised [i.e., one for which
"J:.N ." N.· ... I tjI (N1', N z' ... ) 12 = 1] and that satisfies the proper initial con-
ditions, then 1tjI (N1', N 2' ... ) 12 will be the probability of that distribution in
which Nt' systems are in state 1, N z' in state 2, ... at any time.
Consider first the case when there is only one system in the assembly. The
probability of its being in the state q is determined by the eigenfunction
t We are supposing for definiteness that the label r of the stationary states takes the
values I, 2, 3, ....
+
t Whens = r, '" (N1', N 2 ' ••• N.' - 1. .. N: 1) is to be taken to mean'" (N1'N2 ' ... N: ... ).
TIlE TIME-ENERGY UNCERTAINTY RELATION 167

Emission and Absorption oj Radiation. 253

q,(Nt ', N 2', ••• ) in which all the N"s are put equal to zero except N,,', which is
put equal to unity. .This eigenfunction we shall denote by 4{q}. When it ,.
substituted in the left-hand side of (13), all the terms in the summation on
the right-hand side vanish except those for which, = q, and we are left with

ih ; 4{q} = ~.4{8},
which is the same equation as (5) with 4{q} playing the part of btl. This estab-
lishes the fact that the present theory is equivalent to that of the preceding
section when there is only one system in the assembly.
Now take the general case of an arbitrary number of systems in the assembly,
and assume that they obey the Einstein-Bose statistical mechanics. This
requires that, in the ordinary treatment of the problem, only those e~gen­
functions that are symmetrical between all the systems must be taken into
account, these eigenfunctions being by themselves sufficient to give a complete
quantum solution of the problem.t We shall now obtain the equation for the
rate of change of one of these symmetrical eigenfunctions, and show that it is
identical with equation (13).
If we label each system with a number n, then the Hamiltonian for the
assembly will be HA. = L"H (n), where H (n) is the H of §2 (equal to Ho + V)
expressed in terms of the variables of the nth system. A stationary state of
the assembly is defined by the numbers,1> '2.... '" ...
which are the labels of the
stationary states in which the separate systems lie. The Schr5dinger equation
for the assembly in a set of variables that specify the stationary states will be
of the form (6) [with HA. instead of H), a.nd we can write it in the notation of
equation (5) thus :-

ihb('1'2 ... ) = L,., ••... HA.('1'2 .. ·; 8 18 2 ",) b(8182 ... ), (14:)

where HA.('1'2 ... ; 8 18 2 ... } is the genera.l matrix element of HA. [with the time
factor removed]. This matrix element vanishes when more than one 8" difters
from the corresponding 1',,; equals H,••• when 8.. difters from ' .. and every
other 8" equals 'n; and equals LnH,.,. when every 8.. equals '... Substituting
these values in (14), we obtain
ihb('I'a ... ) = L",L•• #.H,....b('lr 2 ... '",_1 8",r",+1 ... ) + L"H,,,,.b('I'."·)· (15)
We must now restrict b ('1'\1 ... ) to be a symmetrical function of the variables
'1' '2'" in order to obtain the Einstein-Bose statistics. This is permissible
since if b (r1'2 ... ) is symmetrical at any time, then equation (15) shows that
t Loc. cit., I, § 3.
168 CHAPTER ill

254 P. A. M. Dirac.
b('I'1 ... ) is also symmetrical at that time, 80 that b (r1'1 ... ) will remain
symmetrical.
Let Nr denote the number of systems in the state ,. Then a stationary state
of the assembly describable by a symmetrical eigenfunction may be specified
by the numbers NI, Na ... N, ... just as well as by the numbers 'I' '2 ... '" ... ,
and we shall be able to transform equation (15) to the variables NI, Nz ... .
We cannot actually take the new eigenfunction b (NI' Na ... ) equal to the pre-
vious one b ('I'a ... ), but must take one to be a numerical multiple of the
other in order that each may be correctly normalised with respect to its
respective variables. We must have, in fact,

= 1 = l::N,. N•... I b (NI, N2 ... ) 12,

l::". ,.... 1b ('1 'z ... ) 12
and hence we must take Ib (Nl' Nz .... ) 12 equal to the sum of Ib ('l'z ... ) 12 for
all values of the numbers '1"2'" such that there are NI of them equal to 1, N2
equal to 2, etc. There are N !jNI! Na ! ... terms in this sum, where N = S,N,
is the total number of systems, and they are all equal, since b('l'Z ... ) is a.
symmetrical function of its variables '1' r 2 .... Hence we must have

If we make this substitution in equation (15), the left-hand side will become
ih (NI ! N2! ... (N !)l b (NI' Nz ... ). The term Hrm.mb (rlr2 ... rm-I Sm'm+l ... )
in the first summation on the right-hand side will become

where we have written r for rm and S for Sm. This term must be summed for
all values of S except" and must then be summed for, taking each of the values
'1' '2.... Thus each term (16) gets repeated by the summation process until
it occurs a total of Nr times, so that it contributes

N,[N!! Nz ! .. , (N, - I)! ... (N.:t-1)! ... jN !]l H,,b (Nl' Nz ... N,-I ... N, + 1...)
= Nri(N.+ l)i(NI! Nz ! ... jN !)lH,,b (N!, Nz ... N,-1 .. , N. + 1 ... )

to the right-hand side of (15). Finally, the term l::"H,,,,,,b('I' r 2 ... ) becomes

l::,N,lI".b(rl,z ... ) =l::,N.Hr,. (Nl! N2! ... (N !)i beN}> N z' ... J.

Hence equation (15) becomes, with the removal of the factor (Nl! N z L.. (N !)l.

ihb (N!, Nz ... ) = l::,l::.",N,i (N.+l)lH,p (N!, Nt ... Nr-l .. , N. + 1 ... )

+ l::,N,H"b (N!. Nz ... ), (17)
THE TIME-ENERGY UNCERTAINTY RELATION 169

Emission and Absorption of Radiation. 255

which is identical with (13) [except for the fact that in (17) the primes have
been omitted from the N's, which is permissible when we do not require to refer
to the N's as q-numbers]. We have thus established that the Hamiltonian
(11) describes the effect of a perturbation on an assembly satisfying the Einstein-
Bose statistics.

§ 4. The &action of the Assembly on the Perturbing System.

Up to the present we have considered only perturbations that can be repre-
sented by a perturbing energy V added to the Hamiltonian of the perturbed
system, V being a function only of the dynamical variables of that system and
perhaps of the time. The theory may readily be extended to the case when
the perturbation consists of interaction with a perturbing dynamical system,
the reaction of the perturbed system on the perturbing system being taken
into account. (The distinction between the perturbing system and the per-
turbed system is, of course, not real, but it will be kept up for convenience.)
We now consider a perturbing system, described, say, by the canonical
variables J b Wk> the J's being its first integrals when it is alone, interacting
with an assembly of perturbed systems with no mutual interaction, that satisfy
the Einstein-Bose statistics. The total Hamiltonian will be of the form

Hr = Hp (J) + L"H (n),

where Hp is the Hamiltonian of the perturbing system (a function of the J's
only) and H (n) is equal to the proper energy Ho (n) plus the perturbation energy
V(n) of the nth system of the assembly. H (n) is a function only of the variables
of the nth system of the assembly and of the J's and w's, and does not involve
the time explicitly.
The Schrodinger equation corresponding to equation (14) is now

ihb (J', '1'2 .•• ) = LJ" L ... .

, ... HT (J', '1'2 ..• ; J", 818 2 ",) b(J", 818 a ••• ),
in which the eigenfunction b involves the additional variables J,.'. The matrix
element Hor (J', '1'2 .•. ; J", 8182 •.• ) is now always a constant. As before, it
vanishes when more than one 8" differs from the corresponding , ft. When
S", differs from ' . and every other 8" equals '", it reduces to H (J',.; J"8,.).

which is the (J',"'; J"8.) matrix element (with the time factor removed) of
+
H = Ho V, the proper energy plus the perturbation energy of a single
system of the assembly; while when every 8" equals f'", it has the value
+
Hp (J') 8J 'J" L" H (J',,,; J",..). If, as before, we restrict the eigenfunctions
170 CHAPTERll

256 P. A. M. Dirac.

to he symmetrical in the variables rl , r 2 ••• , we can again transform to the

variables Nl , N z ... , which will lead, as before, to the result
lAb (J', NI ', Nt' ... ) = Hp (J') b (J', N'l' Ns' ... )
+l:,J.l:,.,N,'1 (N,' + 1_~,,}1 H (J'r; J's) b(J',N1',Nz' ... N,' -L.N,' + 1...) (18)
This is the Schrodinger equation corresponding to the Hamiltonian function
F -- H P (J) + L r,' U
.J.~.
Nr , + 1-~ ,. )1 e\(8,-8.)/h ,
I (N (19)
in which H" is now a function of the J's and w's, being such that when repre-
sented by a matrix in the (J) scheme its (J' J') element is H (J'r; J"s). (It
should be noticed that H" still commutes with the N's and 6's.)
Thus the interaction of a perturbing system and an assembly satisfying the
Einstein-Bose statistics can be described by a Hamiltonian of the form (19).
We can put it in the form corresponding to (11') by observing that the matrix
element H (J'r; J"8) is composed of the sum of two parts, a part that comes
from the proper energy Ho, which equals W, when J k' = J k ' and ~ = rand
vanishes otherwise, and a part that comes from the interaction energy V,
which may be denoted by v (J'r; J"s). Thus we shall have
Hra = W, a" + tI",
where Vr , is that function of the J's and w's which is represented' by the matrix
whose (J' J') element is tI(J'r; J's), and so (19) becomes
F = Hp (J) + L,W,N, + L,.• v,,N,1 (N, + 1 - ~,,)l e,(8,-8./h. (20)
The Hamiltonian is ,thus the sum of the proper energy of the perturbing system
Hp(J), the'proper energy of the perturbed systems LrW,Nr and the perturba-
tion ene~av
l"'tJ
l;r,. tI,N
r r , + 1 _ ~.., )1 e'(8,-8.)/h .
i (N

§ 5. ,Thwry oj Transitions in a System Jrom One State to Others oj the Same Energy.
Before applying the results of the preceding sections to light-quanta, we
shall consider the solution of the problem presented by a Hamiltonian of the
type (19). The essential feature of the problem is that it refers to a dynamical
system which can, under the influence of a perturbation energy which does
not involve the time explicitly, make transitions from one state to others of
the same energy. The problem of collisions between an atomic system and an
electron, which baa been treated by Born,· is a special case of this type. Born's
methOd is to find a periodic solution of the wave equation which consists, in
so far as it involves the co-ordinates of the colliding electron, of plane waves,
• Born, • Z. f. Phyaik,' vol. 38, p. 803 (1926).
THE TIME-ENERGY UNCERTAINTY RELA TION 171

Emission and Absorption of Radiation. 257

representing the incident electron, approaching the atomic system, which are
scattered or diffracted in all directions. The square of the amplitude of the
waves scattered in any direction with any frequency is then assumed by Born
to be the probability of the electron being scattered in that direction with
the corresponding energy.
This method does not appear to be capable of extension in any simple manner
to the general problem of systems that make transitions from one state to others
of the same energy. Also there is at present no very direct and certain way
of interpreting a periodic solution of a wave equation to apply to a non-periodic
physical phenomenon such as a collision. (The more definite method that
will now be given shows that Born's assumption is not quite right, it being
necessary to multiply the square of the amplitude by a certain factor.)
An alternative method of solving a collision problem is to find a rwn-periodic
solution of the wave equation which consists initially simply of plane waves
moving over the whole of space in the necessary direction with the necessary
frequency to represent the incident electron. In course of time waves moving
in other directions must appear in order that the wave equation may remain
satisfied_ The probability of the electron being scattered in any direction with
any energy will then be determine4l by the rate of growth of the corresponding
harmonic component of these waves. The way the mathematics is to be
interpreted is by this method quite definite, being the same as that of the
beginning of § 2_
We shall apply this method to the general problem of a system which makes
transitions from one state to others of the same energy under the action of a
perturbation. Let Ho be the Hamiltonian of the unperturbed system and
V the perturbing energy, which must not involve the time explicitly. If we
take the case of a continuous range of stationary states, specified by the first
integrals, IX" say, of the unperturbed motion, then, following the method of
§2, we obtain
(21)

corresponding to equation (4). The probability of the system being in a state

for which each IX" lies between rJ.,,' and rJ.,,' + dn.,/ at any time is Ia (<1') 11ldn.1' • k ll' •••
when a (rJ.') is properly normalised and satisfies the proper initial conditions.
If initially the system is in the state <1°, we must take the initial value of a (<1')
to be of the form aO. 8(ot' -ot°). We shall keep aO arbitrary, as it would be
inconvenient to normalise a (ot') in the present case. For a first approximation
172 CHAPTER III

258 P. A. M. Dirac.

we may substitute for a (<<") in the right-hand side of (21) its initial value. This
gIves
ih a(at') = aOY (at'«O) = at0V («'<<0) ei(W (a')-W (a")] 1/,\
where v(«'<<O) is a constant and W(<<') is the energy of the state IX'. Hence
. ei(W(.')-W(.O)U/A - 1
+
ih a (<<') =;: aO 8 (<<' - «0) aOv (at'«O) i [W (<<') _ W (<<O)]/h· (22)

For values of the «k' such that W (at') differs appreciably from W (<<0), a (IX')
is a periodic function of the time whose amplitude is small when the perturbing
energy V is small, so that the eigenfunctions corresponding to these stationary
states are not excited to any appreciable extent. On the other hand, for values
of the «/c' such that W (<<') = W (<<0) and atk' ~ «k° for some le, a (IX') increases
uniformly with respect to the time, so that the probability of the system being
in the state IX' at any time increases proportionally with the square of the time.
Physically, the probability of the system being in a state with exactly the same
proper energy as the initial proper energy W (at°) is of no importance, being
infinitesimal. We are interested only in the integral of the probability
through a small range of proper energy values about the initial proper energy,
which, as we shall find, increases linearly with the time, in agreement with the
ordinary probability laws.
We transform from the variables «I' at2 ... 0Cu to a set of variables that are
arbitrary independent functions of the «'s such that one of them is the proper
energy W, say, the variables W, YI' Y2' ... Y,-I. The probability at any time
of the system lying in a stationary state for which each Yk lies between Yt' and
Yk' + dyJr.' is now (apart from the normalising factor) equal to
dYl' . dy,,' ... dYv-l'
.
r a
I a (at') 12 ~;,I" «,2' ... «v'~) dW'.
, YI ... Yv-I
(23)

For a time that i~ large compared with the periods of the system we shall find
that practically the whole of the integral in (23) is contributed by values of
W' very close to WO = W (<<0). Put
a (<<') = a (W', Y') and a
(<<I', at2' ... «v')/o (W', YI' ... Yv-I') = J (W', y').
Then for the integral in (23) we find, with the help of (22) (provided Yt' ~ Ykf)
for some k)
JI a (W', Y') 12 J (W', Y') dW'

f
="1 aO 12 IV (W', y'; WO, yO) 12 J (W', Y')
[e
i(W'-WO)t/A 1] [-i(W'-WO)t/A 1]
~,:.wo)" - aw'
= J
21ao 12 I v (W',y'; WO,yO)\IJ(W',y')[I-cos(W'- W~tlh]/(W'- WO)".dW'

= 21 aO I" t/h. fIv (WO + hx/t, y'; WO, yO) IIJ (WO +hx/t, Y') {I-cos x)/zl. ax.
THE TIME-ENERGY UNCERTAINTY RELA TION 173

Emiss'ion and Absorption of Radiation. 259

if one makes the substitution (W' - WO)tlh = x. For large values of t this. ....
reduces to
21ao l'l.tlh.1 v (WO, y'; WO, l) \2 J (WO, y') [ .. (I-COS x)/x2. dx

= 21' Ia011tlh.1 v (WO, y'; WO, yO) \1 J (WO, y').

The probability per unit time of a transition to a state for which each y" lies
+
between YJ.:' and Ylc' dylc' is thus (apart frQID. the normalising factor)
21'Iaolzlh.lv(WO,y'; WO,yO)\zJ(WO,y')dYl'.dY2' ... dyu-t', (24)
which is proportional to the square of the matrix element associated with that
transition of the perturbing energy.
To apply this result to a simple collision problem, we take the ex's to be the
components of momentum Pez' PV' PI of the colliding electron and the' y's to
e
be and rp, the angles which determine its direction of motion. If, taking the
relativity change of mass with velocity into account, we let P denote the
+ +
resultant momentum, equal to (pez2 Pv2 PI2)', and E the energy, equal to
(m2c'+p2c2)I, of the electron, m being its rest-mass, we find for the Jacobian

J = aa(p""(E, Pr' PI) =

e, rp)
EP ,
c2 S1ll
e •

Thus the J (WO, y') of the expression (24) has the value
J (WO, y') = E'P' sin 6' 1c2, (25)
where E' and P' refer to that value for the energy of the scattered electron which
makes the total energy equal the initial energy WO (i.e" to that value required
by the conservation of energy).
We must now interpret the initial value of a (ex'), namely, aO a(oc' - exO),
which we did not normalise. According to § 2 the wave function in terms of the
variables exk is b (oc')= a (ex') e-iW'/Ih, so that its initial value is
aO a(oc' - exO) e-iW'ljh = aO 8(p",' - PzO) a(Pv' - PliO) a(po' _ p.O)e-iW'/,h.
If we use the transformation function·
(x'ip') = (2rch)-3/2 eiI.,.P.':r:'J",
and the transformation rule

1\1 (x') = J(X'IP') Ij;(p') dpz' dp./ dp.',

we obtain for the initial wave function in the co-ordinates x, .y, z the value
aO (2rch)-3/2 eiI."ph'/h e-iW'I/h.
• The symbol x is used for brevity to denote x, y, z.
174 CHAPTERll

260 P. A. M. Dirac.
This corresponds to an initial distribution of lao \II (27th)-3 electrons per unit
volume. Since their velocity is J>Oc2/Eo, the number per unit time striking a
unit surface at right-angles to their direction of motion is lao \2 J>Oc2 j(27th)3 EO.
Dividing this into the expression (24) we obtain, with the help of (25),
E'Eo P'
4~ (27th)2 7 1v (p'; pO) \II po sin 6' d6' dep'. (26)

This is the effective area that must be hit by an electron in order that it shall
be scattered in the solid angle sin 0' dO' dep' with the energy E'. This result
differs by the factor (21th)2j2mE' . P' fPo from Born's. * The necessity for the
factor P' fPo in (26) could have been predicted from the principle of detailed
balancing, as the factor 1v (p'; pO) 12 is symmetrical between the direct and
reverse processes.t
§ 6. Application to Light-Quanta.
We shall now apply the theory of §4 to the case when the systems of the
assembly are light-quanta, the theory being applicable to this case since light-
quanta obey the Einstein-Bose statistics and have no mutual interaction. A
light-quantum is in a stationary state when it is moving with constant momen-
tum in-a straight line. Thus a stationary state r is fixed by the three com-
ponents of momentum of the light-quantum and a variable that specifies its
state of polarisation. We shall work on the assumption that there are a finite
number of these stationary states, lying very close to one another, as it would
be inconvenient to use continuous ranges. The interaction of the light-quanta
with an, atomic system will be described by a Hamiltonian of the form (20),
in which Hp (J) is the Hamiltonian for the atomic system alone, and the
coefficients v,. are for the present unknown. We shall show that this form
for the Hamiltonian, with the V r• arbitrary, leads to Einstein's laws for the
emission and absorption of radiation.
The light-quantum has the peculiarity that it apparently ceases to exist
when it is in one of its stationary states, namely, the zero state, in which its
momentum, and therefore also its energy, are zero. When a light-quantum
is absorbed it can be considered to jump into this zero state, and when one is
emitted it can be considered to jump from the zero state to one in which it is
*In a more recent paper (' Na.ohr. Gesell. d. Wisa.,' Gottingen, p. 146 (1926» Bom has
obtained a result in agreement with that of the present paper for non.relativity mechanics,
by using an interpretation of the analysis based on the oon.eervation theorems. I am
indebted to Prof. N. Bohr for seeing an advance copy of this work.
t See Klein and Roeseland, ' Z. f. Physik,' vol. 4, p. (6, equation (4) (1921).
THE TIME-ENERGY UNCERTAINTY RELATION 175

Emission and Absorption of Radiation. 261

physically in evidence, ~ that it appe8.rs to have been created. Since there is
no limit to the number of light-quanta that may be created in this way. we must
suppose that there are an infinite number of light-quanta in the zero state, so
that the No of the Hamiltonian (20) is infinite. We must now have 60 , the
variable canonically conjugate to No, a constant, since

eo = aFjaNo = Wo + terms involving No-lor (No + 1)-1

and W0 is zero. In order that the Hamiltonian (20) may remain finite it is
necessary for the coefficients tirO. VOr to be infinitely small. We shall suppose
that they are infinitely small in such a way as to make vrONot and tlo,No'
finite, in order that the transition probability coefficients may be finite. Thus
we put
vrO(N o+ 1)le-"o/A = v" vo,NoleiBoIA = v,.,
where v, and v,· are finite and conjugate imaginaries. We may consider the
Vr and vr• to be functions only of the J's and w's of the atomic system, since
their factors (No + l)l e-·Io/A and Noie"o/A are practically constants, the rate
of change of No being very small compared with Nu' The Hamiltonian (20)
now becomes
F = Hp(J) + I:,W,N, + I:, .. o[v,N;e,,·JA + v,·(N, + 1)le- il./AJ
+ I:,~oI:... ov,.N,i(N, + 1- 3,s)le'(,·-··IJA. (27)
The probability of a transition in which a light-quantum in the state r is
absorbed is proportional to the square of the modulus of that matrix element of
the Hamiltonian which refers to this transition. This matrix element. must
come from the term v,N,le"'/" in the Hamiltonian, and must therefore be
proportional to N,'l where N,' is the number of light-quanta in state r before
the process. The probability of the absorption process is thus proportional
to N,'. In the same way the probability of a light-quantum in state -r being
emitted is proportional to (N,' + 1), and the probability of a light-quantum in
state r being scattered into state 8 is proportional to N,' (N,' + 1). Radiative
processes of the more general type considered by Einstein and Ehrenfest,t in
which more than one light-quantum take part simultaneously, are not allowed
on the present theory_
To establish a connection between the number of light-quanta per stationary
state and the intensity of the radiation, we consider an enclosure of finite
volume, A say, containing the radiation. The number of stationary states
for light-quanta of a given type of polarisation whose frequency lies in the
t ' Z. f. Physik: vol. 19. p. 301 (1923).
176 CHAPTER ill

262 P. A. M. Dirac.
range v, to v, + dv, and whose direction of motion lies in the solid angle dw,
about the direction of motion for state r will now be Av,2dv,dw,/c'. The energy
of the light-quanta in these stationary states is thus N,' . 21th'll,. Av,'-dv,dw,/c'.
This must equal Ac-1I,dv,dw" where I, is the intensity per unit frequency
range of the radiation about the state 'T. Hence
I, = N,' (21th) '11,3 jc2, (28)
+ +
so that .N,' is proportional to I, and (N,' 1) is proportional to I, (21th)v.3 /cS.
We thus obtain that the probability of an absorption process is proportional to
I" the incident intensity per unit frequency range, and that of an emission
process is proportional to I, + (21th)v,3/c2, which are just Einstein's laws. *
In the same ~ay the probability of a process in which a light-quantum is scattered
+
from a state r to a state s is proportional to I, [I, (21th)v,3/cS], which is Pauli's
law for the scattering of radiation by an electron.t

§ 7. The Probability Coefficients for Emission and Absorption.

We shall now consider the interaction of an atom and radiation from the wave
point of view. We resolve the radiation into its Fourier components, and
suppose that their number is very large but finite. Let each component be
labelled by a suffix 'T, and suppose there are cr, components associated with the
radiation of a definite type of polarisation per unit solid angle per unit fre-
quency range about the component r. Each component r can be described by
a vector potential Ie, chosen so as to make the scalar potential zero. The
perturbation term to be added to the Hamiltonian will now be, according to
the classical theory with neglect of relativity mechanics, a-I ~, It, X" where X,
is the component of the total polarisation of the atom in the direction of K"
which is the direction of the electric vector of the component r.
We can, as explained.in § 1, suppose the field to be described by the canonical
variables N" 6" of which N, is the number of quanta of energy of the com-
ponent r, and 6, is its canonically conjugate phase, equal to 21th'll, times the
6, of § 1. We shall now have It, = a, cos 6,/h, where a, is the amplitude of
K" which can be connected with N, as follows :-The flow of energy per unit

area per unit time for the component r is t1tC-1 a,"v,'-. Hence the intensity

• The ratio of stimulated to spontaneous emission in the present theory is just twice its
value in Einstein's. This is because in the present theory either polarised oomponent of
the incident radiation can stimulate only radiation poIa.rised in the Bame way, while in
Einstein's the two polarised oomponents are treated together. This remark applies al80
to the scattering Pl'OC6ll8.
t Pauli, , Z. f. Physik: vol. 18, p. 272 (1923).
THE TIME-ENERGY UNCERTAINTY RELATION 177

Emission and Absorption of Radiation. 263

per unit frequency range of the radiation in the neighbourhood of the com·
ponent l' is I, = 17tc-1 a,"v,"a,. Comparing this with equation (28), we obtain
4, = 2 (hv,/ca,)IN,i, and hence

K, = 2 (hv,/ca,)l N,i cos 8,/h.

The Hamiltonian for the whole system of atom plus radiation would now be,
according to the classical theory,
F = Hp (J) +~, (27th v,) N, + 2c-1 t, (hv,/ca,)l X,N",l cos 8,/h, (29)
where Hp (J) is the Hamiltonian for the atom alone. On the quantum theory
we must make the variables N, and 8, canonical q-numbers like the variables
J", 'Wt that describe the atom. We must now replace the N,l cos 8,/h in (29)
by the real q-number
1{N,l eilJr/h + e- iBr/1a N,l} = I {N,l eilJr/h + (N, + 1)1 e- ilJr/h}
so that the Hamiltonian (29) becomes
F = Hp (J) + :E, (27th v,) N,+ hi c-;:E, (v,/a,)~ :X, {N,I eilJr /h + (N, + 1)1 e- ilJr/Ia }.
(30)
This is of the form (27), with
tJ, = tJ,* = hi c-; (v,/a,)1 X, (31)
and tJ" = 0 (1', S ~ 0).
The wave point of view is thus consistent with the light-quantum point of view
and gives values for the unknown interaction coefficient tJ" in the light-
quantum theory. These values are not such as would enable one to express
the interaction energy as an algebraic function of canonical variables. Since
the wave theory gives v" = 0 for 1', 8 ¢ 0, it would seem to show that there are
no direct scattering processes, but this may be due to an incompleteness in
the present wave theory.
We shall now show that the Hamiltonian (30) leads to the correct expressions
for Einstein's A's and B's. We must first modify slightly the analysis of §5
ao as to apply to the case when the system has a large number of discrete station-
ary states instead of a continuous range. Instead of equation (21) we shall
now have
ih Ii (<<') = ~." V («'<</F) a (<</F).
If the system is initially in the state «0, we must take the initial value of a (<<')
to be 3....0, which is now correctly normalised. This gives for a first approxi.
mation

which leads to
178 CHAPTERll

264 P. A. M. Dirac.
corresponding to (22). If, as before, we transform to the variables W, Y1'
Y2 ... Y_-1> we obtain (when Y' ~ y~
a (W'y') = v (W', y'; WO, yO) [l_e,<w'-wO)I/h]/(W' - WO).

The probability of the system being in a state for which each YIt. equals Yt'
is ~w' Ia (W' y')\I, If the stationary states lie close together and if the time t

is not too great, we can replace this sum by the integral (~W)-lJ la (W'y/) \2 dW',

where ~W is the separation between the energy levels. Evaluating this integral
as before, we obtain for the probability per unit time of a transition to a state
for which each Ylt = Yk'
27t/h~W . I v·(WO, y'; Wo, yO) 12. (32)
In applying this result we can take the Y's to be any set of variables that are
independent of the total proper energy Wand that together with W define
a stationary state.
We now return to the problem defined by the Hamiltonian (30) and consider
an absorption process in which the atom jumps from the state JO to the state
J' with the absorption of a light-quantum from state r. We take the variable8
y' to be the variables J' of the atom together with variables that define the
direction of motion and state of polarisation of the absorbed quantum, but
not its energy. The matrix element v (WO, y'; WO,..,o) is now
h1!2c- 3/2 (v,/a,)l 12 X, (JoJ')N/,

where:X,. (JoJ') is the ordinary (JoJ') matrix element of i,. Hence from (32) the
probability per unit .time of the absorption process is

~ hv, I X· (JoJ') I IN 0
W . ff'a,'
h~ ,.
To obtain the probability for the process when the light-quantum comes from
any direction in a solid angle dw, we must multiply this expression by the number
of possible directions for the light-quantum in the solid angle dw, which is
dw a,~W/21th. This gives

dw I.v~ I X. (Jol') 12 N,O =

nc-
dw 2 h! 21
1t cv,
X, (JOJ') 12 I,
w.ith the help of (28). Hence the probability coefficient for the absorption
process is 1/21th2cv,2.1X,(JoJ')\2, in agreement with the usual value for Ein-
stein's absorption coefficient in the matrix mechanics. The agreement for
the emission coefficients may be verified in the same manner.
THE TIME-ENERGY UNeERT AINTY RELATION 179

Emission and Absorption of Radiation. 265

The present theory, since it gives a proper account of spontaneous emission,

must presumably give the effect of radiation reaction on the emitting system,
and enable one to calculate the natural breadths of spectral lines, if one can
overcome the mathematical difficulties involved in the general solution of the
wave problem corresponding to the Hamiltonian (30). Also the theory enables
one to understand how it comes about that there is no violation of the law of the
conservation of energy when, say, a photo-electron is emitted from an atom
under the action of extremely weak incident radiation. The energy of inter-
action of the atom and the radiation is a q-number that does not commute with
the first integrals of the motion of the atom alone or with the intensity of the
radiation. Thus one ~annot specify this energy by a c-number at the same
time that one specifies the stationary state of the atom and the intensity of the
radiation by c-numbers. In particular, one cannot say that the interaction
energy tends to zero as the intensity of the incident radiation tends to zero.
There is thus always an unspecifiable amount of interaction energy ,which
can supply the energy for the photo-electron.

I would like to express my thanks to Prof. Niels Bohr for his interest in this
work and for much friendly discussion about it.

Summary.
The problem is treated of an assembly of similar systems satisfying the
Einstein-Bose statistical mechanics, which interact with another different
system, a Hamiltonian function being obtained to describe the motion. The
theory is applied to the interaction of an assembly of light-quanta with an
ordinary atom, and it is shown that it gives Einstein's laws for the emission
and absorption of radiation.
The interaction of an atom with electromagnetic waves is then considered,
and it is shown that if one takes the energies and phases of the waves to be
q-numbers satisfying the proper quantum conditions instead of c-numbers,
the Hamiltonian function takes the same form as in the light-quantum treat-
ment. The theory leads to the correct expressions for Einstein's A's and B's.

VOL. CXIV.-A. T
180 CHAPTER III

710 P. A. M. Dirac.

The Quantum Theory of Dispe·rsion.

By P. A. M. DIRAC, St. John's College, Cambridge; Institute for Theoretical
Physics, Gottingen.

(Communioated by R. H. Fowler, F.R.S.-Received April 4, 1927.)

§ 1. Introduction and Summa·ry.

The new quantum mechanics could at first be used to answer questions con-
cerning radiation only through analogies with the classical theory. In Heisen-
berg's original matrix theory, for instance, it is assumed that the matrix elements
of the polarisation of an atom determine the emission and absorption of radiation
analogo~ly to the Fourier components in the classical theory. In more recent
theories* a certain expression for the electric density obtained from the quantum
mechanics is used to determine the emitted radiation by the same formullll
as in the classical theory. These methods give satisfactory results in many
cases, but cannot even be applied to problems where the classical analogies
are obscure or non-existent, such as resonance radiation and the breadths
of spectral lines.
A theory of radiation has been given by the author which rests on a more
definite basis.t It appears that one can treat a field of radiation as a dynamical
system, whose interaction with an ordinary atomic system may be described
by a Hamiltonian function. The dynamical variables specifying the field are
the energies and phases of ita various harmonic components, each of which
• E. SchrOdinger, ' Ann. d. Physik,' vol. 81, p. 109 (1926) ; W. Gordon, ' Z. f. Physik,'
vol. 40, p. 117 (1926); O. Klein, 'Z. f. Physik,' vol. 41, p. 407 (1927).
t 'Roy. Soc. Proc.,' A, vol. 114, p. 243 (1927). This is referred to later by loco cit.

Reprinted from Proc. Roy. Soc. (London) A114, 710 (1927).

THE TIME-ENERGY UNCERTAINTY RELA TION 181

Quantum Theory of Dispersion. 711

is effectively a simple harmonic oscillator. One must, of course, in the quantuni
theory take these variables to be q-numbers satisfying the proper quantum
conditions. One finds then that the Hamiltonian for the interaction of the
field with an atom is of the same form as that for the interaction of an assembly
of light-quanta with the atom. There is thus a complete formal reconciliation
between the wave and light-quantum points of view.
In applying the theory to the practical working out of radiation problems
one must use a perturbation method, as one cannot solve the Schrodinger
equation directly. One can assume that the term (V say) in the Hamiltonian
due to the interaction of the radiation and the atom is small compared with that
representing their proper energy, and then use V as the perturbing energy.
Physically the assumption is that the mean life time of the atom in any state
is large compared with its periods of vibration. In the present paper we shall
apply the theory to determine the radiation scattered by the atom, considering
also the case when the frequency of the incident radiation coincides with that
of a spectral line of the atom_ The method used will be that in which one finds
a solution of the SchrOdinger equation that satisfies certain initial conditions,
corresponding to a given initial state for the atom and field. In general terms
it may be described as follows :-
If Vm.. are the matrix: elements of the perturbing energy V, where each
suffix m or n refers to a stationary state of the whole system of atom plus field
the stationary state of the atom being specified by its action variables, J say,
and that of the field by a given distribution of energy among its harmonic
components, or by a given distribution of light-quanta}, then each V"... gives
rise to_ transitions from state n to state m*; more accurately, it causes the
eigenfunction representing state m to grow if that representing state n is already
excited, the general formula for the rate of change of the amplitude am of an
eigenfunction beingt
ihj27t • a'm =:E ""'''
V an =:E"vmfta" ';'ri(W.-W.)I/1I , (1)
where V m• is the constant amplitude of the matrix: element Vm,,, and W'" is the

• In loco cit., § 6. it was in error assumed that V..... caused transitions from state m to
state., and consequently the information there obtained about an absorption (or emission)
process in terms of the number of light-quanta existing before the process should really
apply to an emission (or absorption) process in terms of the number of light-quanta in exist.
ence after the process. This change, of course, does not affect the results (namely the
proof of Einstein's laws) which can depend on 1v,"" I t = IV ..... I".
ot Loc. cit., equation (4). In the present paper h is taken to mean just Planck's constant
[instead of (2".)-1 times this qua.ntity as in loco cit.] which is preferable when one has to deal
much with quanta. hv of radiation.
182 CHAPTERm

712 P. A. M. Dirac.
total proper energy of. the state m. To solve these equations one obtains a
first approximation by substituting for the a's on the right-hand side their
initial values, a second approximation by substituting for these a's their values
given by the first approximation, and so on. One or two such approximations
will usually be sufficient to give a solution that is fairly accurate for times
that are small compared with the life time, but may all the same be large
compared with thelperiods of the atom. From the first approximation, namely,
a", = am. + l:"v",,,a,,,, (1 - e2..i (W. - W.II/")/(W", - W,,), (2)

where afll) denotes the initial value of a", one sees readily that when two states
m and n have appreciably different proper energies, the amplitude a", gets
changed only by a small extent, varying periodically with the time, on account
of transitions from state n. Only when twostates, m and m' say, have the same
energy does the amplitude a", of one of them grow continually at the expense
of that of the 9ther, as is necessary for physically recognisable transitions to
occur, and the rate of growth is then proportional to V mm"
The interaction term of the Hamiltonian function obtained in Zoe. cit. [equation
(30)] does not give rise to any direct scattering processes, in which a light-
quantum jumps from one state to another of the same frequency but different
direction of motion (i.e., the corresponding matrix element V m ",' = 0). All
the same, radiation that has apparently been scattered can appear by a double
process in which a third state, n say, with difierent proper energy from m and
m', plays a part. If initially all the a's vanish except am', then a.. gets excited
on accoUnt of transitions from state m' by an amount proportional to V".',
and although it must itself always remain small, a calculation shows that it
will cause am to grow continually with the time at a rate proportional to
V",..V""," The scattered radiation thus appears as the result of the two processes
m' -+ nand n -+ m,. one of which must be an absorption and the other an
emission, in neither of which is the total proper energy even approximately
conserved.
The more accurate expression for the interaction energy obtained in § 3
of the present paper does give rise to direct scattering processes, whose eftect
is of the same order of magnitude as that of the double processes, and must
be added to it. The sum of the two will· be found to give just Kramers' and
Heisenberg's dispersion formula* when the incident frequency does not coincide
with that of an absorption or emission line of the atom. If, however, the
incident frequency coincides with that of, say, an absorption line, one of the
• Kramera and Heisenberg, C Z. f. Phyaik,' vol. 31, p. 681 (1925).
THE TIME-ENERGY UNCERTAINTY RELATION 183

Quantum Theory of Dispersion. 713

terms in the Kramers-Heisenberg formula becomes infinite_ The present
theory shows that in this case the scattered radiation consists of two parts,
of which the amount of one increases proportionally to the time since the inter-
action commenced, and that of th~ other proportionally to the square of this
time. The first part arises from those term!! in the Kramers-Heisenberg formula
that remain finite, with perhaps a contribution from the infinite term, while
the second, which is much larger, is just what one would get from transitions
of the atom to the upper state and down again governed by Einstein's laws of
absorption and emission.
A difficulty that appears in the present treatment of radiation problems
should be here pointed out. If one tries to calculate, for instance, the total
probability of a light-quantum having been emitted by a given time, one
obtains as result a sum or integral with respect to the frequency of the emitted
light-quantum that does not converge in the high frequencies. This difficulty
is not due to any fundamental mistake in the theory, but comes from the fact
that the atom has, for the purpose of its interaction with the field, been counted
simply as a varying electric dipole, and the field produced by a dipole, when
resolved into its Fourier components, has an infinite amount of energy in the
short wave-lengths, owing to the infinite field in its immediate neighbourhood.
If one does not make the approximation of regarding the atom as a dipole,
but uses the exact expression for the interaction energy, then the fact that the
singularity in the field is of a lower order of magnitude and remains constant
is sufficient to make the series or integral converge. The exact interaction
energy is too complicated to be used as a basis for radiation theory at present,
and we shall here use only the dipole energy, which will mean that divergent
series are always liable to appear in the calculation. The best method to adopt
under such circumstances is first to work out the general theory of any effect
using arbitrary coefficients Vm,,, and then to substitute for these coefficients
in the final result their values given by the dipole interaction energy. If one
then finds that the series all converge, one can assume that the result is a correct
first approximation; if, however, any of them do not converge, one must
conclude that a dipole theory is inadequate for the treatment of that particular
e:fIect. We shall find that for the phenomena of dispersion and resonance
radiation dealt with in the present paper, there are no divergent series in the
first approximation, so that the dipole theory is sufficient. If, however, one
tries to calculate the breadth of a spectral line, one meets with a divergent
series, 80 that a dipole theory of the atom is presumably inadequate for the
correct treatment of this question.
184 CHAPTER III

714 P. A. M. Dirac.

§ 2. Prelimirw.ry Formula!.
We consider tJie electromagnetic field to be resolved into its components.
of plane, plane-polarised, progressive waves, each component r having a definite
frequency, direction of motion and state of polarisation, and being associated.
with a certain type of light-quanta. (To save writing we shall in future suppose
the words " directipn of motion" applied to a light-quantum or a component
I

of the field to imply also its state of polarisation, and a sum or integral taken
over all directions of motion to imply also the summation over both states of
polarisation for each direction of motion. This is convenient because the two
variables, direction of motion and state of polarisation, are alw!l-Ys treated
mathematically in the same way.) For an electromagnetic field of infinite
extent there will be a continuous three-dimensional range of these components.
As this would be inconvenient to deal with mathematically, we suppose it to be
replaced by a large number of discrete components. If there are a, components
per unit solid angle of direction of motion per unit frequency range, we can
keep a, an arbitrary function of the frequency and direction of motion of the
component r, provided it is large and reasonably continuous, and shall find
that it always cancels from the final results of a calculation, which fact appears
to justify our replacement of the continuous range by the discrete set.
We can express a, in the form a, = (Av, AN,}-l, where Av, can be regarded
as the frequency interval between successive components in the neighbourhood
of the component r, and AN, is in the same way the solid angle of direction
of motion to be associated with this component. The quantities Av" AN, enable
one to pass directly from sums to integrals. Thus if f, is any function of the
frequency and direction of motion of the component r that varies only slightly
frOnl on~ component to a. neighbouring one, the sum of f, Av, ior all components.
having a specified direction of motion is

(3)

and the sum of f, AN, for all components having a specified frequency is

(3')·

Also the sum off, (a,}-l for all components is

};f,{a,}-l = };f, Av, AN,. J

f,dv,dN,. (3"}
THE TIME-ENERGY UNCERTAINTY RELATION 185

Quantum Theory of Dispe'rsion. 715

If the number· N. of quanta of energy of the component 8 varies only slightJ,y

from one component to a neighbouring one, one can give a meaning 11> the
intensity of the radiation per unit frequency range. By supposing the dis-
creteness in the number of components to arise from the radiation being confined
in an enclosure (which would imply stationary waves and a special function 0",)
one obtainst for the rate of flow of energy per unit area per unit solid angle
per unit frequency range
r"V 83/2
I ...., -- N'" c, (4)
a result which may be taken to hold generally for arbitrary 0", and progressive
waves.t If only those components with a specified direction of motion are
excited, we have instead that the rate of flow of energy per unit area per unit
frequency range is
(5)
while if only a single component s is excited, we have that the rate of flow of
energy per unit area is
I = N,hv,3 je? . Aw, Av, = N/lv,3/c2a,. (6)
In this last case the amplitude of the electric force has the value E given by
E2 = 81tl/c = 87tN/lv.3 /c1a" (7)
and the amplitude a of the magnetic vector potential, when chosen so that the
electric potential is zero, is
a = cE/27tv, = 2 (hv./27t(:tJ.)1 N,t. (8)

§ 3. The Ha.miltonian Function.

We shall now determine the Hamiltonian function that describes the inter-
action of the field with an atom more accurately than in loco cit. We consider
the atom to consist of a single electron moving in an electrostatic field of
potential cpo According to the classical theory its relativity Hamiltonian
equation when undisturbed is
p,,2 + p/' + p.2 _ (W + ecp)2fc2 + m2c2 = 0,
80 that its Hamiltonian function is
H = W = c {m 2c2 + p,,2 + p/ + p.2}! - ecp. (9)
• The rule given in loco cit. that symbols representing c-number values for q -number
variables should be primed need not always be observed if no confusion thus arises, as in
the present case.
t Loc. cit., § 6, equation (28).
t This is justified by the fact that one can obtain the result by e.n alternative method
tha.t does not rquire a finite enclosure, namely by using a quantum-mechanical argument
1Iimilar to that of loco cit. (lower part of p. 259). applied to the case of discrete momentum
wa.lues.
VOL. CXIV.-A. 3 D
186 CHAPTER ill

716 P. A. M. Dirac.
If now there is a perturbing field of radiation, given by the magnetic vector
potential ICc, "., K. chosen so that the electric scalar potential is zero, the
Hamiltonian equation for the perturbed system will be

e)! + (..p'+ce)2 +(P'+ce

(PZ+c"c KV
I \2
KI ) -
(W + ep)2
c2
2
+mc2=o,

r (p, + ~ ",r + ~ rr-

which gives for the Hamiltonian function

H = W = c {m2c2 + (pz + ICz ~+ (.PI "I e,p

= c{[m2c2 + pz2 + p.,,2 + P12] + [2e/c. (Pz"z + .Pv"." + P.K,)

+ e2Jc2 • ("212 + IC,,2 + ICI2)]}l- e,p.
By expanding the square root, counting the second term in square brackets [ ]
as small, and then neglecting relativity corrections for this term, one finds
approxima.tely

H = c [m2c2 + pz2 + p,2 + p,2]l -e,p + e/c . (i"z + fpc, + ZIC,)

+ e2/2mc2. (lCz2 + IC,,2 + ".2)
= 110 + e/c. (.tICZ + il", + z".) + e?j2mc2 • (ICz2+ "."z + ".2), (lO)
where Ho is the Hamiltonian for the unperturbed system given by (9). When
one counts the radiation field as a dynamical system, one must add on its proper
energy ~N,Av, to the Hamiltonian (10).
According to the classical theory, .the magnetic vector potential for any
component r of the radiation is
", = a,. cos 2rtO,/h = 2 {hv,j2rtca,)i N,l cos 2rtOlh (11)
from (8), where 0, increases uniformly with the time such that 6, = hv" and
is the variable that must be taken to be the canonical conjugate of N, when the
radiation field is treated as a dynamical system. The direction of this vector
potential is that of the electric vector of the component of radiation. Hence
the total value of the component of the vector potential in any direction, say
that of the z-axis, is
"21 = ~r", cos otzr = 2 {hj21'tc)I~,cOSotzr (v,/a,)l N,l cos 2rtO,/h, (12)

where CXzr is the angle between the electric vector. of the component r and the
z-axis. In the quantum theory, where the variables N" 0, are q-numbers,
the expression 2N,I cos 2rtO,Ih must be replaced by the real q-number
+ +
N,le2ri1,/A (N, 1)1 e- 2friI./A• With this change one can take over the
TIlE TIME-ENERGY UNCERTAINTY RELA TION 187

Quantu,m, Theory of Dispersion. 117

Hamiltonian (10) into the quantum theory, which gives, when one includes
the term };N,hv"
H= Ho + };N,hv, + eh1/(2rt)} c* . };,:i:, (v,/a,)l [N,le2....'/II + (N, + 1)le- 2..iB,/li]
+ e2 hJ4rtmc3 • };,•• cos~. (v,v./a,a,)~ [N,l e2..i8,/h + (N, + 1)le- 2.-i1,/h]
X [N.!e2ri8./h + (N. + l)te- 2ri8Jh ] (13)

where x, denotes the component of the vector (x, y, z) in the direction of the
electric vector of the component r, i.e.,

x, = x cos IX.". + y cos IX~, + z cos 0(."

and IX,. denotes the angle between the electric vectors of the components r
and s, i.e.

The terms in the first line of (13) are just those obtained in loco cit., equation
(30), and give rise only to emission and absorption processes. The remaining
terms (i.e., those in the double summation) were neglected in loco cit. These
terms may be divided into three sets :-
(i) Those terms that are independent of the 6's, which can be added to the
proper energy Ho + };N,hv,. The sum of all such terms, which can arise
only when r = s, is
e2h/4rtmc3 • };, v,/a, . [N,le2lfiB,/h (N, + 1)1 e-Zlfi/l,/h
+ (N, + 1)te- 2 ui/"/1I N;e2ri8,fI\)
= e2 },/4rtmc3 • };,v,/a,. (2N, + 1).
The terms e2h/4rtmc. "'i.v,/a, . 2N, are negligible compared with };N,hv"
owing to the very large quantity a, in the denominator, while the terms
e2h/4~. };v,/~, may be ignored since they do not involve any of the
dynamical variables, in spite of the fact that the smn 'J:..v,!a" equal to
Jv, dv, dw, from (3"), does not converge for the high frequencies.
(ii) The terms containing a factor of the form cZri (8,- •• ).'li (r ~ s), whose sum. is
e2h/4rtmc3 };,};."" cos IX,. (v, v./ap,)l [N,l (N, + 1)1 e2..i (B,-B.)/h
+ (N,+ 1)1 N,l e- 2..i (B,-B.)/h]
= e2hl2rtm~ "'i.,};.~, cos 0(" (v,v./a,rJ,)l N,l (N,+ 1)1 e2lfi(6,-B.)/h. (14)

These terms, which are the only important ones in the three sets, give rise to
transitions in which a light-quantum jumps directly from a state s to a state r_
3 D 2
188 CHAPTERID

718 P. A. M. Dirac.
Such transitions may be called true seattering processes, to distinguish them
from the double scattering processes described in § l.
(iii) The remaining temls, each of which involves a factor of one or other of
the forms e±4>ri8,''', e±2..i (8. +8.)'1,. These terms correspond to processes in
which two light-quanta are emitted or absorbed simultaneously, and cannot
arise in a light-qua~tum theory in which there are no forces between the light
quanta. The effects of these terms will be found to be negligible, 80 that the
disagreement with the light-quantum theory is not serious.

§ 4. Discussion of the Emission and True Scattering Pmcesses.

We shall consider now the simple emission processes, in order to discuss the
divergent integral that arises in this question. Suppose a light-quantum to be
emitted in state T, with a simultaneous jump of the atom from the state J = J'
to the state J = J". If we label the final state of the whole system of atom
plus field 1n and the initial state k, the value at time t of the amplitude am of
the eigenfunction of the final state will be in the first approximation
(15)
obtained by putting ak. = 1, a... = 0 (1/ ~ k) in equation (2). The only term in
the Hamiltonian (13) that can contribute anything to the matrix element t'mk
+
is the one involving e2..i8,1h, whose (J", Nt', N 2 ' ••• N: 1 ... ; J', Nt', N 2 ' ••• N: ... )
matrix element is eh l J(2rt)i ci . X, (J" J') (v,(cr,)l (N: +
1)~, where i, (J" J') is the
ordinary (J"J') matrix element of i,. If there is no incident radiation we
must take all the N"s zero, which gi\'es
Vmk = ehi ( (2r.) I c1 . :l', (J"J') (v,fcr,)I,
and also

Thus
WIll - Wk = Ho (J") + hv, - Ho(J') = h [v, - v (J' J")]
where v (J' J") = [Ho (J') - Ho (J")]/h is the transition frequency between
states J' and J", if one assumes J' to be the higher one. Hence from (15)

1am \2 = e2 !J'r (J/J") \2 v, 1 - cos 21t [v, - v (J?")] t .

1thcS 'cr, [v, - v (J/J~)]~
To obtain the total probability of any light-quantum being emitted within
the solid angle 8w about the direction of motion of a given light-quantum f'
with this jump of the atom, we mUBt multiply \ am 12 by 8wJ tlw, and sum for
all frequencies. This gives, with the help of (3)

8 L 1am 12 = 8w ~ 1.1: (J/J") 12

w • tlw, 1thc3 '
J'" v dv 1 -
0"
cos 2x [v, -
[v, - V
V (J'JII)]t.
(J/J")]2
(16)
THE TIME-ENERGY UNCERTAINTY RELA TION 189

Quantum Them'!! of Dispersion, 719

The int.egral does not converge for the high frequencies. This is due, as
mentioned in § 1, to the non-legitimacy of taking only the dipole action of the
atom into account, which is what one does when one substitutes for the magnetic
potential in (10) its value given by (12), which is its value at some fixed point
such as the nucleus instead of its value where the electron is momentarily
situated. To obtain the interaction energy exactly, one should put cOs 27t
[6,/h - vr;r/c] instead of cos 2rt6r/h in (11), where ;r is the component of the
vector (x, y, z) in the direction of motion of the component r of radiation.
This will make no appreciable change for low frequencies Vr , but will cause a
llew factor cos 2rtvr;r/c or sin 2rtvr~r/c, whose matrix elements tend to zero
as Vr tends to infinity, to appear in the coefficients of (13). This will presumably
cause the integral in (16) to ~onverge when corrected, as its divergence when
uncorrected is only logarithmic.
Assuming that the integrand in (16) has been suitably modified in the high
frequencies, one sees that for values of t large compared with the periods of the
atom (but small compared with the life time in order that the approximations
may be valid) practically the whole of the integral is contributed by values of v,
close to v (J' J"), which means physically that only radiation close to a transitio?
frequency can be spontaneously emitted. One fmds readily for the total
probability of .the emission, by performing t.he integration,

which leads to the correct value for Einstein's A coefficient per unit solid angle,
namely,

We shall now determine the rate at which true scattering processes occur,
caused by the terms (14) in the Hamiltonian. We see at once that the frequency
of occurrence of these processes is independent of the nature of the atom,
and is thus the same for a bound as for a free electron. The true scattering
is the only kind of scattering that can occur for a free electron, so that we should
expect the terms (14) to lead to the correct formula for the scattering of radiation
by a free electron, with neglect of relativity mechanics and thus of the Compton
effect.
Suppose that initially the atom is in the state J' and all the N's vanish except
one of them, N, say, which has the value Y.,'_ We label this state for the whole
system by k, and the state for which J = J' and N. = N,' - 1, Nr = 1 with
190 CHAPTERm

720 P. A. M. Dirac.

all the other N's zero by m. In the first approximation a", is again
. given by.
(15), where we now have
11",1: = e2hJ2rcmc3 • cos 0:" (vrv.!O',a.)~ N.'~, (17)

WI: = Ho(J') + N,'hv" W.. = Ho (J') + (N,' - 1) hv. + hv,. (18)

Thus
(19)
and hence
2 e4 "vrv'N,I-cos21t(v,-v.)t
\ am \ = 2 2 6 COS- OC,.• - ,( )2 •
21t m c (1,(1.< V, - v,

To obtain the total probability of a scattered light-quantum being in the solid

angle 8w W3 must, as before, multiply 1am \2 by 8w/ /J.w, and sum for all
frequencies v" which gives·

~ ~ \a", 12 ~ ~ e4 2 ~ 1\."- ,
- - - ow,~ cos IX" n,
.,w~v
Jd
V, V,
1 - cos 21t (v; - v,) t
2. (20)
/J.w, 21t m c cr, (V, - V,)

We again obtain a divergent integral, of the same form as before, which we may
assume becomes convergent in the more exact theory. We now have that
practically the whole of the integral is contributed by values of Vr close to v,
and the total probability for the scattering process is

from (6); where I is the' rate of flow of incident energy per unit area. The rate
of emission of scattered energy per unit solid angle is thus

where 0:" is the angle between the electric vectors of the incident and scattered
radiation, which is the correct classical formula.

• The reason why tHere is 0. small probability for the scattered frequency II, ditJeriDg
by a finite amount from the incident frequency ". is because we are considering the scattered
radiation, after the scattering process has been acting for only a finite time I, resolved into
its Fourier components. One sees from the formula (20) that 88 the time' gets greater.
the scattered radiation gets more and more nearly monochromatic with the frequency •••
n one obtained a periodic solution of the SchrOdinger equation corresponding to permaneDt
physical conditiona, one would then find that the scattered frequency was exactly equal
to the incident frequency.
THE TIME-ENERGY UNCERTAINTY RELATION 191

Quantum Theory of IMpersion. 72l

§ 5. Throry of Dispersion.
We shall now work out the second approximation to the solution of equations
(1), taking the case when the system is initially in the state k, so that the fiplt
approximation, given by (2) with a"" = 0",,, reduces to
am = Omk + vmd1 - e2•• (w* - w.)I/")/(W", - W,J.
When one substitutes these values for the aA's in the right-hand side of (1).
one obtains
ih/2rc . tim = e
tlmt 2.' (W. - w.) I,'h
+ L"Vm"tlnk (1 - e2wi (W.- W.)ll 1a ) e2 "';(w.- W.)II" I (W" - Wk )
_ (. _ ~ V m.. Vn'· ') 2".i(W. -W,) 1:"+· ~ tiM" tiM 2"'; (W.-\\".)II"
- Vmk .<J" W" _ Wb e .<J .. W" _ Wk e ,
and hence when m ¢- k
_ ( V m" v.. ,. ) 1 - e2wi (W. - w.)I.'Ia
am - Vmk - ~"W _ W . W - W
\ "k, m k

V V 1 - e2,..;{w.-·W.)I/1I
+ L" W" - mA "k
Wk W'" - WA
• (21)

We may suppose the diagonal elements V"" of the perturbing energy to be zero.
since if they were not zero they could be included with the proper energy W".
There will then be no terms in (21) with vanishing denominators, provided all
the energy levels are different.
Suppose now that the proper energy of the state m is equal to that of the
initial state k. Then the first term on the right-hand side of (21) ceases to be
periodic in the time, and becomes

{Vmk - L" Vm" v..d (W n - Wk)} 2rct/ih,

which increases linearly with the time. The rate of increase consists of a part,
proportional to Vml:> that is due to direct transitions from state k, together with
a sum of parts, each of which is proportional to a V",,,Vnk, and is due to transitions
first from k to n and then from n to m, although the amplitude a" of the eigen-
function of the intermediate state always remains small.
When one applies the theory to the scattering of radiation one must consider
not a single final state with exactly the same proper energy as the initial state,
but a set of final states with proper energies lying close together in a range that
contains the initial proper energy, corresponding to. all the possible scattered
light-quanta with different frequencies but the same direction of motion that
192 CHAPTER ill

722 P. A. M. Dirac.

may appear. One must now determine the total probability of the system
lying in anyone of these final states, which is

l: lam 12 = J(6.W..)-11 am 12 dW n"

where 6.Wm iS'the interval between the energy levels. The second term in the
expression (21) for am may be neglected siuce it always remains small (except
in the case of resonance which will be considered later) and hence

If one assumes that the integral converges, so that for large values of t practically
the whole of it is contributed by values of Wm close to Wk> one obtains

l: Ia 12 =
m kW
6. m mk
I
47t2t v _ ~ V.."Vnl.: \2
"W-W'
"k
(22)

where the quantities on the right refer to that final state that has exactly the
initial proper energy.
We take the states k and m to be the sa.me as for the true sca.ttering process
considered in the preceding section, so that equations (17), (18) and (19) still
hold, and IlWIII = Allv, = AJa, 6.6>,. We can now take the state 11. to be either
the state J = J", N. = N,' - 1, N. = 0 (t =1= s) for any J", which would make
the process k-+ 11. an absorption of an s-quantum and 11.-+ m an emission of an
r-quantum, or the state J = J", N, = N,/, N, = 1, N, = 0 (t':;f! 8, f), which
would make k-+ 11. the emission and 11.-+ m the absorption. In the first case
we should'have
_ e ( hv. \)1,. (J"J/) N 'I
-c . -2-
= ~(. hv, )1 '. (J/J") ,
V"k -
7tca. x. • V .." c 27tca, x,
and
W" = Ho(J") + (N,' -1) hv. W" - W" = h[V(J"J/) - v,]·,
and in the second
_ e (-nv,
,V"k-- -
)1., (J"J/)
iI" VIR..
= ~ ('2kv. ')1,.X. (J/J") N, '1,
C 27tca, C , 7tca,
and
W" = Ho (J") + N.'hv, + hv, w" - WI.: = k[ v(J"J') + v,l.
We shall neglect the other possible states 11., namely those for which the matrix
elements v"'", VIII: come from terms in the double summation in the Hamiltonian
• The frequency" (J"J') is not necess&Jily positive.
THE TIME-ENERGY UNCERTAINTY RELA TION 193

Quantum Theory of Dispersion. 723

(13), as we are working only to the first order in these terms_ (We are working
to the second order only in the emission and absorption terms, which, as we
shall find, is the same as the first order in the terms of the double summation_}
We now obtain for the right-hand side of (22) in which we must take V, = V"

N't e~v.21!
, /lW r h2c6a, 1n cos OCr,
_~ ,,{X,(J'J").;;,(J"J')
J(J"J')
+ X, (J'J") x, (J"J')} 12 (23}
(J"J') + '
'I - 'Is V V,

The most convenient way of expressing this result is to find the amplitude
P (a vector) of the electric moment of that vibrating dipole of frequency V,
that would, according to the classical theory, emit the same distribution of
radiation as that actually scattered by the atom. The number of light-quanta
of the type r (with Vr = v,) emitted by the dipole P in time t per unit solid
angle is

where P, is the component of P in the direction of the electric vector of the light-
quanta 'T. Comparing this with (23) (which must first be divided by /lwr
to change it to the probability of a light quantum being scattered per unit
solid angle) one finds for Pr

_ (81tN,')l I
e2 h {i:, (J'(J"J')-
J") (J" J') ,v, (J' J") (J" J')} I J:,
1t
,j',
P,- ~
(I(;-v,a, 4
2 -CQs(X'.-~J"
m V
+ (J"J')+
V,. V V,

= E~2-I_h_ _~ ,[ (J"J,)]:!Jxr(J'J")x,(J"J')
h ~2 4~m
2 cos X,. J' V l '(J"J')- ~

J')} I
I

X. (J' J") x, (J" (24)

+ V(J"J') + V, ,

using (7), where E is the amplitude of the electric vector of the incident radiation.
We can put this result in a different form by using the following relations,
which follow from the quantum conditions,

~J" [x, (J' JII) x, (J" J') - x, (J' JII) X, (J" J')] = [:t, x, - x. x,] (J' J') =0
(25)
and
~JII [x, (J' JII) X. (J" J') - X, (J' 1") I, (1" J')] = [XT X. - x. x,) (J' J')
= ih,27t1n . cos (Xm (26)
which gives
~J" [x, (J' JII) X. (J" J') V (J" J') + x. (J' JII) X T (J" J') '1(1" J')]
= hj41t2m . cos 'Y-". (27)
194 CHAPTER ill

724 P. A. M. Dirac.

Multiplying (25) by V, and adding to (27), we obtain

~J" [xr (J' J") x, (J" J') {v (J" J') + v,} + x, (J' J") Xr (J" J') {v (J" J') - v,}]
= h/4rt2m . cos OCr,. (28)
With the help of this equation, (24) reduces to
_ e2\ {X, (J'J") x., (J"J') x, (J'J")xr(J"J'}l \
P, -E Ii ~J" v (J"J') - v, + V (J"J') + v. J '
.80 that the vector P is equal to
_ e2 \
P - E Ii ~J"
{x (J'J") x. (J"J')
V (J"J') - V, +
x, (J'J") x (J"J'»)
V (J"J') + v. J '
I (29)

where x without a suffix means the vector (x, y, z). This is identical with
Kramers' and Heisenberg's result.*
In applying the formula (22), instead of taking the final state m of the system
to be one for which the atom is again in its initial state J = J', we can take a new
final state for the atom, J = J'" say. The frequency Vr for the scattered
radiation that gives no change of total proper energy is now
V, = v. - v (J"'J') = v. + v (J"J''') - v (J"J'), (30)
which differs from the incident frequency V" so that we ebtain in this way
the non-coherent scattered radiation. (We assume that this V, is positive
as otherwise there would be no non-coherent scattered radiation associated
with the final state J = J'" of the atom.) In the present case we have Vml: = 0,
corresponding to the fact that the true scattering process does not contribute
to the non-coherent radiation. We now obtain for P" after a similar and almost
identical calculation to that leading to equation (24),

P, = E~_l /1:J "V(rJ') v (J"J"')

h '1,'1,
f x, (J'" J") x, (J" J')
l v (J" J') - v. +
x, (J'" JII) x, (J" J'n
V (J"J') + v, J
I (31)

This result can be put in the form corresponding to (29) with the help of
equations analogous to (25) and (26) referring to the non-diagonal (J"'J')
matrix elements of [x,x. - x,x,] and [xri. - :t,x,]. These equations give,
corresponding to (28),
~J" [x, (J"'J") x, (J"J') {v (J"J') + vr} +x. (J"'J") x, (J"J') {v (J" Jill) - v,}] = o.
• Kramers and Heisenberg, loco cit., equation (18). For previous quantum.theoref;ioal
deductions of the dispersion formula see Born, Heisenberg and Jordan, 'z. f. Physik,'
vol. 35, p. 557, Kap. 1, equation (40) (1926); SchrMinger, loco cit., § ·2, equation (23); and
Klein, loco cit., § 5, equation (82).
THE TIME-ENERGY UNCERTAINTY RELA TION 195

Quantum Theo'ry of Dispe1-sion. 725

When the left-hand side of this equation is subtracted from the summation in
{31) one obtains, on account of the relations

v (J" J') v (J"J"') = v (J" J') [v (JIIJ') + v, - v,]

= [v (JIIJ') - v,] [v (J"J') + v,] + v,v.,
.and
v (JIIJ') v (J"J"') = [v (J"J"') - v,] [v (J"J') + v,] + v,v.
which follow from (30), the result

_ e2 \ {x, (J'"J") x, (J"J') x. (J'"JII) x, (J"J'») I

P r - Eh ~J" v (J"J') - v. + v (J".T') + v, f'

again in agreement with Kramers and Heisenberg.

§ 6. The Case of Resonance.

The dispersion formulre obtained in the preceding section can no longer
hold when the frequency of the incident radiation coincides with that of an
absorpt.ion or emission line of the atom, on account of a vanishing denominator.
One easily sees where a modification must be made in the deduction of the
formulre. Since one of the intermediate states n now has the same energy
.as the initial state k, the term in the second summation in (21) referring to this
n becomes large and can no longer be neglected.
In investigating this case of resonance one must, for generality, suppose
the incident radiation to consist of a distribution of light-quanta over a range
.of frequencies including the resona.nce frequency, instead of entirely of light-
.quanta of a single frequency, as the results will depend very considerably
.on how nearly monochromatic the incident radiation is. Thus one must take
the initial state k of the system to be given by J = J' and N. = N,', where
N: is zero except for light-quanta of a specified direction, and is for these
light-quanta (roughly speaking) a continuous function of the frequency, so
that the rate of flow of incident energy per unit area per unit frequency range
is given by (5). The final state m for a process of coherent scattering is one
for which J = J' again, and a light-quantum s has been ~bsorbed and one r
of approximately the same frequency emitted. Thus we have
(32)
As before, the intermediate states n will be those for which J = J" (arbitrary)
and either the s-quantum has already been absorbed or the r-quantum has
already been emitted. If we take for definiteness the case when the range of
incident frequencies includes only one resonance frequency, and this is an
196 CHAPTER III

726 P. A. M. Dirac.

absorption frequency to the state of the atom J = J', say, then that intermediate
state of the system for which J = J' and for which the s-quantum has already
been absorbed will have very nearly the same proper energy as the initial state.
Calling this intermediate state l we have

W,-W,,=h('1o-v,) (33)

where Vo is the resQ'nance frequency, equal to [H (J') - H (J')]jh.

In equation (21) we can now neglect only those terms of the second summation
for which n ~ l. This gives

which, with the help of (32) and (33), may be written

_ (I :E tJmlltJ..k ') 1 - e2lTi (.,-v.)'

am - ,
V"'k - II -II ,11
If II
_ UT
n k
h ( Vr _ v, )

Vm(Vlk {I - tf.. i(.,,-.,u) I 1- e2.n(v,-.,.)I'1

+h " (V.
~ '10 -
) V. - '10 - Vr - V, j"

We must now determine the total probability of a specified light-quantum r

being emitted with the absorption of anyone of the incident light-quanta s,
which is given by :E •• I am 12, equal to J( ~v,)-ll am 12 dv,. To evaluate this
we require the following integrals

_ 4 2.. (v r - t - sin 2.. ('Ir -

'10) '10) l
- 7t ( )3
Vr - VQ

'"
fo
1- e2.n(v,-v.)1 {I - e-:!lfi(v,-vu)1 _ 1- e-2 ..i<v,-v.)I} dv
('1,-'1,)('10-'1,) '1,-'10 '1,-'1, ,

_ ') f 2.. (v, - '10) t - sin 2.. (v, - '10) t + .1 - cos 2.. (v, - '10) t)
- .o" l. (v, - '10)2 ~ ('I, - '10)2 I'
THE TIME-ENERGY UNCERTAINTY RELATION 197

Quantum Theory of Dispersion. 727

for large t, and with their help obtain,

~.,I an. I
2
=
I Vm/.: - Luil
Vm ..Vnl: 12 4,rt
W.. _ WI: h2~v.
+I Vml VI/': :! ! 41t 21t (v, - vo) t - sin 27: (V, - vo)t
h' !lv, (V,-Vo)S
-.L ')R ( _ L V m .. 11..1.: '.) Vkl'V'm ') J21t (V, - Volt - sin 21t( v, - Volt
I ~
.
17m!.: nil W ft _ ur
n k'
h3 lJ.V,
A ... 1t l (Vr _ Vo )2
+ .1 - cos 21t (V, - volt) (34)
~ (v, - vo )2 f
where the quantities on the right now refer to that incident light-quaBtum s
for which v. = v" and R means the real part of all that occurs in the term after it.
The first of these three t.erms is just the contribution of those terms of the
.dispersion formula (22) that remain finite, the second is that which replaces the
contribution of the infinite term,· and the third gives the interference between
the first two, and replaces the cross terms obtained when one squares the dis-
persion electric moment. One can see the meaning of the second term more
clearly if one sums it for all frequences v, of the scattered radiation in a small
frequency range Vo -(:I.' to Vo + (:I." about the resonance frequency Vo (which
frequency range must be large compared with the theoretical breadth of the
spectral line in order that the approximations may be valid). This is equivalent
to multiplying the term by (~v,)-l and integrating through the frequency range.
If, for brevity, one denotes the quantity 41t I VmIV,1.: \2/h4 ~v, h.v, .by f (v,), the
result is, neglecting terms that do not increase indefinitely with t or that tend
to zero as the (:I.'S tend to zero,
j..
"0 -
+ ""f (v,) 21t (v, - vo) t - sin ~1t (v, - vo) t dVr
.' (v, - vo)
-f( ) JVo+ ·"21t (v, - vo) t - sin 21t (v r - vo) td
- Vo v,
('V r - vol
3
"0 - Q'

+f' (vo) f"o + .." 21t (vr - vol t - sid ~1t (v, - vol t dV r
• "0 -.' (v, - Vol

= f(vo} (21tt)2 [~1t - ~

27ttr:J.
- -12,
1tt(:l..
J+ f' (vo) 21tt log (:1."/(:1.'.
* It should be noticed that this second term does not reduce to the square of the l term
in the summation (22) when "r is not a resonance frequency, but to double this amount.
This difference is due to the fact that processes involving a change of proper energy are not
entirely negligible for the initial conditions used in the present paper, and one such scattering
process, which was neglected in § 5, becomes in the resonance case a. process with no change
of proper energy and is included in the calculation.
198 CHAPTERm

728 Quantum Theory of Dispersion.

Thus the contribution of the second term in (34) to the small frequency range
'10 - a.' to '10 + a." consists of two parts, one of which increases proportionally
to tll and the other proportionally to t. The part that increases proportionally
to tll, na.mely,

is just that which would arise from actual transitions to the higher state of
the atom and down again governed by Einstein's laws, since the probability
that the atom has been raised to the higher state by the time 't' is*
(2 'It)1I1 'Ilk III/h2 flv, ..., and when it is in the higher state the probability per
unit time of its jumping down again with emission of a light-quantum in the
required direction is (27t)21 vm/12 /h2 Av" so that the total probability of the two-
transitions taking place within a time t is

(2'1t): IVIk 12 . (27t): I vm l1 2Ji"t .. d-: = (27t~'1 Vm/VIk 12 it2.

Ii ~v, 1, flv, o· h flv,flv,

The part that increases linearly with the time may be added to the contributions
of the first and third terms, which also increase according to this law. For
values of t large compared with the periods of the atom, the terms proportional
to t will be negligible compared with those proportional to t2, and hence the
resona.nce scattered radiation is due practically entirely to absorptions and
emissions according to Einstein's laws.

• This result and the one for the emission follow at once from formula (32) of loco cit.
TIlE TIME-ENERGY UNCERTAINTY RELATION 199

On the Time-Energy Uncertainty

Relation
Eugene P. Wigner
1. INTRODUCTION AND SUMMARY
There is only one well-known application for the time-energy un-
certainty relation: the connection between the life-time and the energy-
width of resonance states. The relation in question was commonly
known even before quantum mechanics was established; its first
quantum mechanical derivation was based on Dirac's original theory
of the interaction between matter and radiation. l The point which
should be noted is that the uncertainty relation does not apply to time
and energy in abstracto but to the life-time of a definite state of a system.
In the example referred to, this is the state in which an atom is in an
excited state but there is no radiation present.
The preceding formulation of the time-energy uncertainty relation
appears to be different from Heisenberg's well-known position-momen-
tum uncertainty relation. 2 This postulates that the state of any quantum
mechanical system is, at every instant of time, such that the product of
the position and momentum spreads s., and sp

s.,sp ~ tli. (1)

The spreads in question are defined as the positive square roots of

S.,2 = <1foI(x- x o)211fo)/<1foI1fo)

St> 2= <1foI(p-po)2\1fo)/<1fo\1fo) (la)
and holds for every 1fo and all x o, Po. In (la) 1fo is an arbitrary state vector,
x and p position and momentum operators, respectively. The clause
'at every instant of time' in the preceding formulation of the uncertainty
principlt! is, evidently, important since the state vector 1fo in (1) repre-
sents the state of the system under consideration only for a definite
instant of time and this must be the same for both expressions in (la).
237

Reprinted from Aspects in Quantum Theory (1972).

200 CHAPTERllI

238 E. P. WIGNER

It follows that, if one wishes to formulate the time-energy analogue

of the usual position-momentum uncertainty relation, one will have to
restrict one's attention to the situation along a single value of a space
coordinate (which will be chosen as the x coordinate) just as a single
time coordinate entered (1). The simplest form of the relation then
becomes the statement that the product of the energy spread and the
spread in the probability that the definite value of the coordinate x be
assumed at time t, is at least /i12. For a single non-relativistic particle
these spreads 7 and E are defined as the positive square roots of
72_ m/y,(x, y, z, t)12 (t - t o)2 dy dz dt (2)
- m/",(x, y, z, t)12 dy dz dt
E2= <"'I(H _Eo)21"'>I<"'I"'>
or
E2- fHl4>(x, y, z, E)12 (E-Eo)2 dy dz dE
- • JffI4>(x, y, z, E)12 dy dz dE '
(2a)
where
4>(x, y, z, E) = f"'(x, y, z, t)e iEII" dt (2b)
is the Fourier transform of y, from time into energy. It will be seen (as
is pretty evident) that the uncertainty relation

(3)
holds again for all values of to and Eo. However, whereas there are, for
any Xo and Po in (1), state vectors for which the equality sign is valid in
(1), namely those for which the x dependence of", is a Gaussian of
X-X o multiplied with exp ipoX, the inequality sign always holds in (3).
This is a consequence of the fact that the energy is a positive definite
operator (or has, in the non-relativistic case, a lower bound). The lower
limit of TE is, naturally, independent of to but does depend on Eo and
increases substantially as Eo approaches the lower bound of H from
above. Naturally, it increases even further as Eo crosses that bound and
decreases further. All this, as well as equation (2a), will be further
discussed below.

2. A GENERAL OBSERVATION ON THE

INTERCHANGE OF THE SPACE AND TIME
COORDINATES
Actually, the point just mentioned does not constitute the most interest-
ing diff.erence between the position-momentum and the time-energy
uncertainty relations. The difference which appears most interesting to
THE TIME-ENERGY UNCERTAINTY RELA TION 201

TIME-ENERGY UNCERTAINTY RELATIOlIl 239

this writer will be first formulated in a universe with only one spatial
dimension_ In such a universe, it is natural to ask3 for the probal5ility
that, at a definite time, the spatial coordinate of the particle have the
value x_ It is less natural to ask for the probability that the particle be,
at a definite landmark in space, just at the time t. It would be more
natural to ask, instead, for the probability that the particle crosses the
aforementioned landmark at the time t from the left, and also that it
crosses that landmark, at a given time, from the right. The sum of these
probabilities, when integrated over all times t, is 1 for a free particle.
The difference between the two cases arises from the fact that a particle's
world line can cross the t= constant line only in one direction (in the
direction of increasing t); it can cross the x = constant line in both
directions. If we replace 'line' in the last sentence by 'plane', we have
the generalization of the distinction to the actual four-dimensional
universe. In Dirac's theory of the electron,' the probability that the
particle be, at the definite time t, at the point x, y, z is given by ",tariP,
the space-time variables of,p being x, y, z, and t. The probabilities for
crossing the x= constant plane, at y, z and at time t, in the two different
directions, are given, essentially, by lfst ( a o+ a.,),p and lfst( «0 - a",),p.
This point of this paragraph, interesting though it may be, will not be
elaborated further.
Even though the idea of space-time, and hence the similarity between
space and time coordinates, appears natural from the point of view of
relativity theory, initial conditions characterizing the state of the system
at a definite instant of time appear more natural to us than initial con-
ditions giving the state of the system for all times but only for a single
value of one of the spatial coordinates. There are, indeed, valid reasons
for this preference. The transition from the position-momentum
uncertainty relation to the time-energy uncertainty relation is, however,
based on the second type of description of the state of the system. It
would be interesting to develop equations of motion giving the spatial
derivative of the second type of characterization of a state and to
explore the properties of the resulting equations.

3. A GENERALIZATION OF HEISENBERG'S
UNCERTAINTY RELATION
There are uncertainty relations for practically any two non-commuting
operators, but the generalization of Heisenberg's relation to be pointed
out here is a very special one and bears a close resemblance to the
original form of the relation. We denote, first, the variables of ,p by
202 CHAPTER ill

240 E. P. WIGNER

x and T, the latter one standing for all variables different from x. The
relation (1) then remains valid if one replaces'" in (la) by
(4)
cfobeing any function of T and the integration is over all values of the
continuous coordinates implied by T and summation over the discrete
ones. This is, of cour!>e, a well-known fact, most commonly used with
a cfo which is a delta function of all coordinates different from x. The
derivation of the relation (1), given by Robertson,2 remains equally
valid, however, if T is assumed to involve also the time, with ",(x, r)
satisfying Schrodinger's equation and cfo depending on time in an
arbitrary fashion. The right side of (4) is then a generalized transition
amplitude which can be thought of as corresponding to a measurement
lasting a finite length of time, hut not affecting x.
Accepting the generalization of Heisenberg's relation just outlined:
one sees that ~he time-energy uncertainty relation referring to the
life-time of resonance states, which was mentioned in the first paragraph
of this article, is not as different from (3) as it first appeared. It is in fact
included in the generalization of (3) which is the analogue of the
generalization of (1) just pointed to. The generalization replaces in this
case ",(x, y. z, t) by (ul"'(t» = X(t) where u is any state vector. The
time spread T is then the positive square root of
2_ II (ul"'(t»12 (t -
t o)2 dt _ IIx(t)12 (t- t o)2 dt (S)
T - JI(ul",(t»12 dt - Ilx(t)\2 dt .
In order to define the energy spread, we calculate, in analogy to (Zh)
cfo(E)=(271li)-i I ",(t)iEtl"dt. (Sa)
This is a stationary state of energy E. Hence, E2 will be
lL II (ul1>(E»1 2 (E-Eo)2 dE _ II7}(E)12(E-Eo)2dE
(Sb)
E - JI(u\cfo(E»\2 dE - JI7J{E)\2 dE
where
7J{E) = (ulcfo(E» = (27T1i)-i f(ul",(t»eiEt'''dt
= (27T1i)-i fx(t)iE11"dt (Sc)
is the Fourier transform of x. The factor (2711i)-i renders the denomi-
nators of (Sb) and (S) equal. With these definitions, and identifying u
with the state vector of the resonance, (3) will represent the uncertainty
relation giving the minimum energy spread of a resonance with given
life-time. This is the time-energy uncertainty relation mentioned in
the tfirst paragraph of the present article. On the other hand, if one
THE TIME-ENERGY UNCERTAINTY RELATION 203

TIME-ENERGY UNCERTAINTY RELATION 241

inserts a delta function of x for u, the close analogue of the original
Heisenberg uncertainty relations given by (2), (2a), and (3) resuits.
We will go over next to the proof of (3) with the T and € given by
(5) and (5b) and will give some estimates for the lowest possible value
of TE as function of Eo. The lower bound of T€ will turn out, naturally,
to be independent of to and, perhaps somewhat less obviously, depends
only on the ratio Eo/€, not on Eo and € separately.

4. MINIMAL TIME-ENERGY UNCERTAINTY

PRELIMINARY REMARKS
As was mentioned already in the first section, the product of the spread
in the time of presence at a definite plane and the spread in energy, T€,
has to exceed Ji12. (The time of presence at a definite plane was, actually,
generalized in the preceding section to the time of being in any definite
quantum mechanical state.) The rest of this article will be concerned
with the lower bound of n, for a given €, by characterizing the.p which
renders T€ to a minimum. This.p, and the corresponding T€, will in this
case depend on the energy Eo around which the spread of the energy of
.p is defined as E. The minimum of the quantity T€ will be independent
of to around which the spread of the time of presence, that is T, will be
calculated, .p will depend on to only in a trivial way. The last two
statements follow, of course, from time displacement invariance. The
minimum of TE (and the corresponding .p) will, on the other hand,
depend on the choice of E: clearly, if € can be chosen to be very small
as compared with the excess of Eo over the lower bound of the energy,
the existence of the lower bound will have very little significance. The
lower bound /i/2 of szSp Of the usual Heisenberg uncertainty relation
was independent of sp because the momentum p has no lower bound.
In our case, the lower bound of T€ can be expected to increase with
increasing € (or, rather, with the increase of the ratio E/(Eo-Eb) where
Eb is the lower bound of the energy).
Actually, what the subsequent calculations are aimed at is the
determination of
x(t) = (ul.p(t» (6)
or of its Fourier transform

7J(E) = (ul<P<E» = (2111i)-1 JX(t)iE1/ldt (6a)

which render
2 L Ilx(t)12 (t-t o)2dt II7J(E)12 (E-Eo)2dE (7)
T € - UI7J(E)lll dE)ll
s
204 CHAPTER III

242 E. P. WIGNER

to a minimum. In the calculation which follows 7](E) will be the depen-

dent variable because the essential limitation of the problem, the
existence of a lower bound for the energy, is most easily expressed in
terms of 7](E). This must vanish for all E below the lower bound. It
will be convenient to choose an energy scale in which the lower bound
is 0; the transformation to any other lower bound is trivial. As a result,
the integrations over E will extend from 0 to CX) and will not be specified
explicitly.
We wish to establish, next, that for O~E the function 7](E) can be
chosen arbitrarily, subject only to the condition that it approach 0 fast
enough as E increases toward infinity. This requires three assumptions.
First, the Hamiltonian will be assumed to have a continuous spectrum,
extending from its lower bound to infinity. This is surely true for any
isolated system. Second, it will be assumed that, if lu) is expanded in
terms of the characteristic functions IE) of the Hamiltonian
lu)= Jb(E)IE)dE (8)
beE) does not vanish for any positive E. This condition is surely fulfilled
for all states lu) which restrict the system to a spatial plane, i.e., for
the original form (3) of the time-energy uncertainty relation. It is
fulfilled also for the resonance states discussed later but is, of course,
not true for all u. The second assumption, therefore, restricts our
considerations to a certain degree. The third assumption is that the
beE) of (8) does not go to 0 at very large E as fast as exp (- aE2) with
any a. This is again fulfilled in the two aforementioned cases but
restricts u somewhat further.
If the preceding conditions are satisfied, we expand r/J(t) also in
terms of the characteristic functions of the Hamiltonian
r/J(t) = Ja(E)e-iEtIAIE)dE. (9)
If the spectrum of the Hamiltonian is degenerate, the IE) in (9) shall
be the same which appear in the expansion of lu) in (8). We then have,
by (6)
x(t) == JJb(E)·<EIE')a(E')e-iEtIAdEdE'
= Jb(E)·a(E)e-iEt/AdE (10)
and hence
(lOa)
Hence, any 7](E) can be obtained by a proper choice of aCE) as long as
beE) remains different from zero for all E. The square integrability of
aCE) does, though, impose a condition on the way 7] must go to 0 as
THE TIME-ENERGY UNCERTAINTY RELATION 205

TIME-ENERGY UNCERTAINTY RELATION 243

E--+oo. lt will tum out, however, that the 'YJ which will be obtained
under the sole condition that it be square integrable goes to 0 as.,
exp ( - leW). Hence, if u satisfies the last condition imposed above, the
aCE) needed to furnish this 'YJ will be automatically square integrable.
We can, therefore, proceed to the determination of the 'YJ(E) which
gives the smallest TE, permitting 'YJ to be, for positive E, an arbitrary
function of E.

5. MINIMAL TIME-ENERGY UNCERTAINTY.

EQUATION FOR 71 WHICH MINIMIZES
UNCERTAINTY
If 'YJ(E) were finite at the lower bound of the energy (at E=O), or if it
had a discontinuity somewhere, its Fourier transform
X( t) = (2111i)-1 J1J(E)e -iEt/A dE
would go to 0 at t= 00 only as lIt. The integral in (5) then would
become infinite. This surely would not give the minimum value of TE.
lt follows that 71(0) = O. Hence
00
I dt(t- t o)2 ITJ(E)e- iEt/1 dE ITJ(E')·eiE't,A dE'
~=_-~oo ________ ~~~~~~ __________ (11)
211 Ii JI'YJ(E)12 dE
Introducing g=(t-lo)/Ii as variable instead of t, and writing
7Jo(E) = 7J(E)e- iEt.,A (12)
one obtains

J dg JJTJo(E) 7Jo(E')·(82/8E8E')e i
00

1i2 {E'- Elf dEdE' (13)

oo

211 II7Jo(E)12 dE
Since 7Jo(E) and TJo(E')* both vanish at both ends of the integration, at
o and 00, partial integration with respect to E and E' gives
co
1i2 J d~ JJ(Ur]o(E)loE)(Ur]o(E')·loE')ei{E'-ElfdEdE'
2 ___ -00
T - 211 ~-----:;JI-1Jo7.(E)=12-;'dE:::---·---

= 1i2 II Ur]o(E)loEI2 dEl JI1Jo(E)12 dE. (14)

The last line follows from Fourier's theorem and is the expression
which had to be expected. In fact, the calculation was carried out in
206 CHAPTER ill

244 E. P. WIGNER

such detail principally to show the necessity to assume 7J(0) = 0 10

order to obtain (14).
We now have
T2=1i2 II07J0(E)/oEI 2dE/ II7Jo(E)1 2dE (15)
and
(2= II7Jo(E)12(E-Eo)2dE/ Iho(E)1 2dE. (lsa)
If one writes
(16)
with both ex and f3 real, f3 drops out from the expression for (2 and the
denominator of (15). The integral in the numerator becomes

and this will be decreased if one sets of3/oE = O. It follows that the
minimum of T( will be assumed for a real 7Jo and such an 7Jo will be
assumed henceforth. The real nature of 7Jo could have been inferred
also from time inversion invariance.
We are ready to obtain the minimum of T2(2 for given (2 (and Eo). To
obtain it, we set the variation of

>.' 2 -1i2 2 J( O7J0/ OE)2 dE >.' J7J0 2(E - EO)2 dE

2 2 (18)
T (+ (- ( J7J02 dE + I7J0 2 dE
equal to 0; the >..' is a Lagrange multiplier. This gives us the equation

I7J0 2dE [ - 2/i2(2 ~]; + 2>.'(E - Eo)~o]

= [1i2(2 J(07J0/oE) 2dE+>.' J7J02(E-Eo)2dE] 27Jo. (19)
Elimination of the integrals by means of (15) and (lsa) and division by
2T2(2 gives then

(20)
where
(20a)
has been introduced to make (20) somewhat more symmetric; >.. must
be so determined that (lsa) become valid.
Since 7Jo must vanish for both E = 0 and E = 00, (20) is essentially a
characteristic value - characteristic function equation. It is well known,
aQd can be easily verified, that its solution which approaches 0 for very
large E approaches 0 as exp (- lcW) with c= >..IT/Ii(. This verifies the
statement made about 7Jo at the end of the last section.
THE TIME-ENERGY UNCERTAINTY RELA nON 2fJ7

TIME-ENERGY UNCERTAINTY RELA.TION 245

The solution of (20) is easily obtained for very large EoIE and for
Eo=O. In the former case, one can set ,\= 1 and obtains T=Ii/ZE (that
is TE = 1i/2 as expected) and

(21)

This does not quite satisfy the boundary condition at E = 0 but for
large Eo it satisfies it quite closely. Similarly, (15a) is satisfied closely
enough. In the present case, actually, the minimum of TE is independent
of E as long as this remains very much smaller than Eo.
The otlier case in which (20) can be easily solved is Eo=O. In this
case the only solutions of (20) which satisfy the boundary conditions
have the form
'YJo=E exp( - lcW). (22)

One can determine c by (15a) to be

c=3/2E2• (22a)
This then solves the differential equation (20) if one sets again ,\ = 1
and TE becomes
TE=31i/2. (22b)
Because of the low value of Eo, the uncertainty is much larger than in
Heisenberg's relation or for the large Eo of (21). In this case again, the
minimum of TE is independent of E - it is, as we shall see next, indepen-
dent in no other case but the two just considered. Let us then proceed
to the determination of ,\ and the discussion of 710 in the general case.

6. MINIMAL TIME-ENERGY UNCERTAINTY.

DISCUSSION
Let us observe, first, that the lower limit of TE depends on E and Eo
only in terms of Eo/E. The two examples considered in the last section
correspond to the values of 00 and 0, respectively, of this quantity. It
is for this reason that we found the minimum of TE to be independent
of E and Eo separately.
In order to see that the minimum of TE depends .only on Eo/E, let us
denote the solution of (20) for E= 1 and the value eo of Eo by g(E, eo).
This solves (20) with E= 1, Eo=e o and a '\(e o) such that the solutiong
satisfies the condition (15)

(23)
208 CHAPTER III

246 E. P. WIGNER

and the differential equation

the primes denoting differentiation with respect to the first variable and
T the value for which g(O, eo) = 0, and g tending to 0 for largt! E.

We can then set

(24)
i.e., choose eo=Eo/€ and expand the first variable of g by the factor €.
If one then substitutes (24) into (23a), one finds that 7Jo satisfies (20).
Similarly,
JTJO(E)2(E - EO)2dE = Jg(E/€, E oM2(E - Eo)2dE
€3 Jg(E', E oM2(E' - Eo/£)2dE' =€2 Jg(E/€, E oM2dE = €2 J7Jo(E) 2dE.
The before-last member is a consequence of (23). This establishes the
theorem which was plausible anyway. It follows that the wave functions
7Jo of the minimal T€, that is time-energy uncertainty, can all be obtained
by means of (24) by solving (23a) with the subsidiary condition (23).
In order to solve (23a) with this subsidiary condition, one will choose
a .h2 , multiply (23a) with T2, whereupon this will be a simple charac-
teristic value equation for T2. One obtains its lowest characteristic value
and the corresponding characteristic function g by Ritz' or some other
method and compare then the two sides of (23). If the left side is larger
than the right, one will choose a larger .h2 ; if the left side is smaller,
one will try a smaller ,.\-r2 • It should be possible to satisfy (23) after not
too many trials.
It may be of some interest to deduce a few identities between A, T
and the function 7Jo which renders T€ to a minimum. For this purpose,
one multiplies (20) with certain factors and integrates the resulting
equation with respect to E. Multiplication with 7Jo and partial integra-
tion of the first term gives
(1;2/T2) J(07]0/OE)2dE + (.\j€2) J(E - EO)27J02dE = (1 + ,,) J7J02dE. (25)
The integrated terms are 0 in this case because 7Jo(O)=O. Because of
(15) and (15a), this is an identity, " dropping out.
Multiplication of (20) with 07]%E gives in a similar way
1;2 A
2T7J '0(0)2= 2J7JO(E)2(E-Eo)
2 €
dE (2Sa)

showing that the mean value of the energy is always larger than Eo.
THE TIME-ENERGY UNCERTAINTY RELA nON 209

TIME-ENERGY UNCERTAINTY RELATION 247

Finally, multiplication of (20) with E8TJoIaE gives with (15) and (15a)

1 _ ~ A- .\Eo ITJo(E)2(E-Eo) dE __ 1 _ 1 A (25b)

"f 2 E2 171o(E)' dE -"f"f-

This last equation does not contain T, the former one gives an expres~ion
for Ar in terms of 710' Naturally, (25a) and (25b) can be combined in
various ways to eliminate A or the integral in (25a). These equations
play the role of the virial theorem which applies for the wave function
giving minimal position-energy spread and Heisenberg's uncertainty
relation. The last term on the left of (25b) vanishes in the simple cases
considered in the last section: Eo=O in the second case and the integral
in the numerator vanishes for Eo=O, giving A= 1 in both cases. In all
other cases, A< 1.

REFERENCES
1. P. A. M. Dirac, Proc. Roy. Soc. (London) A1l4, 243, 710 (19~7). The
calculation was carried out by V. Weisskopf and E. Wigner, Z. Physik 63,
54 (1930). See also the article of the same authors, ibid. 65, 18 (1930) and
many subsequent discussions of the same subject and of resonance states
decaying by the emission of particles rather than radiation.
2. W. Heisenberg, Z. Physik 43, 172 (1927). For a rigorous derivation, see
H. P. Robertson, Phys. Rev. 34, 163 (1929). The derivation of section 5 is
patterned on that of this article.
3. G. R. Allcock, Ann. Phys. (N.Y.) 53, 253, 286, 311 (1969). Section II of
the first of these articles gives a very illuminating discussion of the ideas
which underlie also the present section. It also contains a review of the
literature of the time-energy uncertainty principle, making it unnecessary
to give such a review here. The review also gives a criticism of some of the
unprecise interpretations of the time-energy uncertainty relation which are
widely spread in the literature. The later parts of the aforementioned articles
arrive at a pessimistic view on th~ possibility of incorporating into the
present framework of quantum mechanics time measurements as described
by Allcock in section II or in the present section. This pessimism, which is
not shared by the present writer, is expressed, however, quite cautiously
and is mitigated by the various assumptions on which it is based.
4. The notation of chapter Xl of P. A. M. Dirac's The Principles of Quantum
Mechanics (Oxford University Press, various editions) is used.
210 CHAPTER III

Time-energy uncertainty relation and Lorentz covariance

P. E. Hussar) and Y. S. Kim
Center Jar Theoretical Physics. Department ojPhysics and Astronomy. University oj Maryland. Col/ege Park.
Maryland 20742
Marilyn E. Noz
DeportmentoJRadiology. New York University. New York. New York 10016
(Received 2 December 1983; accepted for publication 14 March 1984)
The uncertainty relations applicable to space and time separations between the quarks in a hadron
are discussed. It is pointed out that the time-energy uncertainty relation between the time and
energy separations is the same as the relationship between the widths and lifetimes of unstable
states. It is then shown that this relation can be combined with Heisenberg's position-momentum
uncertainty relation to give the uncertainty principle in a covariant form. It is pointed out that this
effect manifests itself in Feynman's parton picture.

I. INTRODUCTION answers to these questions from the existing literature is

that, while the uncertainty relation is to be formulated
The time-energy uncertainty relation in the form from experimental observations: there are not many ex-
(.<it II.<iE )~Ii was known to exist even before the present perimental phenomena which can be regarded as direct
form of quantum mechanics was formulated. 1.2 However, manifestations of the time-energy uncertainty relation.
the treatment of this subject in existing quantum-mechan- The uncertainty relation between the lifetime and the
ics textbooks is not adequate. Students and teachers alike energy width of unstable states was known to exist before
are frequently confused on the following three issues: 1927. 1.2 It is now believed that this phenomenon is indepen-
dent of the Schrodinger equation. t. 7
(a) Is the time-energy uncertainty relation a consequence The Schrooinger equation leads to a form of the time-
of the time-dependent Schrooinger equation, or is this rela- energy uncertainty relation when we calculate transition
tion expected to hold even in systems which cannot be de- rates in the first-order time-dependent perturbation the-
scribed by the Schrooinger equation?'" ory. However, the second-order time-dependent perturba-
(b) While there exists the time-energy uncertainty rela- tion theory requires a separate time-energy uncertainty re-
tion in the real world, possibly with the form [to HI = iii.' lation in the initial condition.
this commutator is zero in the case of Schrooinger quan- It is now widely believed that the time-energy uncertain-
tum mechanics. As was noted by Dirac in 1927,2 the time ty relation is responsible for off-mass-shell intermediate
variable is a c number. Then. is the c number time-energy particles in quantum field theory. In this case, however, we
uncertainty relation universal. or true only in nonrelativis- are talking about particles which are not observable. In-
tic quantum mechanics? deed. the role of the time-energy uncertainty relation in the
(c) If the time variable is a c number and the position present form of quantum field theory requires further in-
variables are q numbers. then the coordinate variables in a vestigation'
different Lorentz frame are mixtures of c and q numbers. The purpose of the present paper is to study the relativis-
This cannot be consistent with special relativity. as was also tic quark model as a physical example in which the time-
pointed out by Dirac. 2 energy uncertainty relation leads to a directly observable
One of the reasons why we are not able to get satisfactory effect. For a hadron consisting of two quarks whose space-

Reprinted from Am. J. Phys. 53, 142 (1985).

 THE TIME-ENERGY UNCERTAINTY RELATION 211

T lator formalism is used for giving a wave-function interpre-

tation to the commutator form of the uncertainty relations.
In Sec. IV, Dirac's light-cone coordinate system is shown
to playa decisive role in defining the uncertainty relations
in a Lorentz-invariant manner. In Sec. V, we discuss briefly
measurable consequences of the time-energy uncertainty
relation combined covariantly with the position-momen-
tum uncertainty relation.

II_ TIME-ENERGY UNCERTAINTY RELATION

APPLICABLE TO INTERNAL SPACE-TIME
SEPARATION COORDINATES IN THE QUARK
MODEL
In order to discuss the uncertainty relations, we need
momentum-energy variables in addition to the space-time
Fig. I. Hadronic and internal coordinate systems. Each of the two quarks coordinates. Let us define the four-momenta:
in the hadron has its own space-time coordinate. These two coordinates
can be translated into the hadronic coordinate IZ, T I and the separation p=p, + p"
coordinate Iz,t I (4)
q = v'l(p, - P2),
time coordinates are X, andx 2 , we can define new variables: where p, andp, are the four-momenta of the first and sec-
ond quarks, respectively. P is the sum of the momenta of
X = (x, + x,)/2, the two quarks and is therefore the four-momentum of the
(I) hadron. q measures the difference between the quark four-
x = lx, - x 2 )!2,'2· momenta.
Then X and x correspond, respectively, to the overall Without loss of generality, we assume that the hadron
hadronic coordinate and space-time separation between has a definite four-momentum and moves along the z or Z
the quarks. As is specified in Fig. I, the coordinate variable direction with velocity parameter /3. Then it is possible to
X specifies the space-time position of the hadron, and the x find the Lorentz frame in which the hadron is at rest. We
coordinate is for the space-time separation between the shall use x" ,y", z", and r "to denote the space-time separa-
quarks. The spatial component of the four-vector X speci- tions in this Lorentz frame, and q~, q;, q~, q~ for momen-
fies where the hadron is, and its time component tells how tum-energy separations.
old the hadron and the quarks become. The spatial compo- In this Lorentz frame, the uncertainty principle applica-
nents of the four-vector x specify the relative spatial separa- ble to the space-time separation of quarks is expected to be
tion between the quarks. Its time component is the time the same as the presently accepted form largely based on
interval or separation between the quarks. nonrelativistic quantum mechanics. The usual Heisenberg
Because the time-separation variable is not contained in uncertainty relation holds for each of the three spatial co-
nonrelativistic quantum mechanics," the quark model pro- ordinates:
vides an excellent testing ground to examine whether there [x",q~l =i,
exists a time-energy uncertainty relation which does not
[y",q:] =i, (5)
come from the Schriidinger equation. If there exists an un-
certainty relation along the time separation coordinate, we [z",q:] =i.
should see whether this time-energy uncertainty relation is On the other hand, the time-separation variable is a c num-
the same as the currently accepted form largely based on ber and therefore does not cause quantum excitations. The
nonrelativistic quantum mechanics.
commutator of Eq. (2) in this case takes the form
According to the currently accepted version,IO·11 the
time variable is a c number or [r",q~l = 0, (6)
[r,H] =0. (2) with
In quantum mechanics, the above commutator means that (LI r "lIL1q,l'I"'" I. (7)
there is no Hilbert space in which t and ia / at act as opera- This means that, according to the presently accepted form
tors. 12 This means that the Robertson procedure 13 applica- ofthe time-energy uncertainty relation, the c-number com-
ble to Heisenberg's position-momentum uncertainty rela- mutation relation of Eq. (6) should be accompanied by the
tion does not work here. Classically, this corresponds to the "Fourier relation"J4 ofEq.(7). The crucial question is how
fact that t and H are not canonically conjugate variables. these uncertainty relations appear to an observer in the
However, it is important to note that there still exists a laboratory frame with the space-time-separation variables
"Fourier" relation between time and energy which limits x, y, z, and r.
the precision to 14 Since the hadron moves along the z axis, the x and y
(3) coordinates are not affected by boosts, and the first two
We shall use for simplicity the convention Ii = c = I. commutation relations ofEq.(5) remain invariant:
In Sec. II, we discuss in detail the uncertainty relations [x,q,] = i,
applicable to the internal space-time separation variable in (8)
the quark model. In Sec. III, the covariant harmonic oscil- [y,q,] = i.

143 Am. 1. Phys" Vol. 53, No.2, February 1985 Hussar, Kim, and NOl 143
212 CHAPfERIII

As for the third commutation relation of Eq. (5) for the and (intrinsic spin)' of the hadron."'" As a consequence,
longitudinal direction, we have to consider the Lorentz the harmonic oscillator model constitutes a solution ofDir-
transformation of the coordinate system: ac's "Poisson bracket" equations for his "instant form"
Z= (ZO +Pl°)l(I_P,)II" quantum mechanics. 24 .25
While the exact form for the hadronic wave function is
(9) somewhat complicated, the essential element of the wave
1= (10 +pzO)/(1 _p')II'. function takes the form 16
Likewise, the transformation equations for the momen- ,pn,(xl = (I/lT2' + kn!k !)"'Hn(zO)H'(1 0)
tum-energy variables can be written as
q, = (q; + Pqt)/(1 - p')I12,
X exp{ - [(z»' + (I O)']/2}. (15)
For simplicity, we assume here that the harmonic oscilla-
(10)
tor has a unit strength. In the above expression, we have
qo = (q~ +Pq:)/(1 _p')II'. suppressed all the factors which are not affected by the
In terms of the lab-frame variables, the uncertainty rela- Lorentz transformation along the z axis. This is possible
tion [ZO,q:] = i ofEq. (5) and the TE relation ofEq. (6) can because the oscillator wave functions are separable in both
be written as the Cartesian and spherical coordinate systems.
In terms of the standard step-up and step-down opera-
[z, q,] = i/(1 -P'),
tors,
(11)
[I, qo] = iP'/(I-P').
In addition, because the Lorentz transformation is not an
orthogonal transformation, the commutation relations (16)
between z and qo and between I and q, do not vanish:
[z,qo] = iP I( I _ P '),
a~ = ~ (XI' + a~).
(12)
and the oscillator wave function of Eq. (15) satisfies the
differential equation
[I,q,] = iP/(I-P').
The commutation relations of Eqs. (8), (II), and (12) can a!a~,p(x) = (II + I)¢(x), (17)
now be combined into a covariant form 14: where the eigenvalue II, together with transverse excita-
[x~,qv]= -g~v +P~PvIM', (13) tions, determines the (mass)' of the hadron.23
The operators given in Eq. (16) satisfy the algebraic rela-
with the covariant form of Eq. (7): tion
[.<:I (P·xIM)][.<:I(P·qIM)]=eI, (14) (18)
where M is the hadronic mass. We use the convention This commutation relation is Lorentz invariant. The time-
goo= -gil = -g,,= -g33=1. like component of the above commutator is - I in every
Although the uncertainty relations can be brought to the Lorentz frame. This allows timelike excitations. Indeed, it
above convariant form, it is very difficult to give physical is possible to construct a covariant Hilbert space of har-
interpretations to them. First, in Eq. (II), the right-hand monic oscillator wave functions in which timelike excita-
side is dependent on the velocity parameter. Does this tions are allowed in all Lorentz frames.'o
mean that Planck's constant becomes dependent on the On the other hand, as we shall see in Sec. V, there is no
velocity? Second, the commutators ofEq. (12) do not van- evidence to indicate the existence of such timelike excita-
ish while there are no conjugate relations between the var- tions in the real world. 17 This is perfectly consistent with
iables involved. IS In order to resolve these puzzles, we have the fact that the basic space-time symmetry of confined
to resort to an interpretation based on wave functions. quarks is that of the o (3)-like little group of the Poincare
group."·27 We can suppress timelike excitations in the ha-
III. USE OF THE HARMONIC OSCILLATOR dronic rest frame by imposing the subsidiary condition 10.23.
FORMALISM
(19)
Traditonally, in nonrelativistic quantum mechanics, the
harmonic oscillator has been very useful in giving interpre- Then only the solutions with k = 0 are allowed, and the
tations to the uncertainty relations. It is therefore not un- commutator given in Eq. (18) is not consistent with the
reasonable to expect that a relativistic harmonic oscillator above subsidiary condition.
model may prove useful in clarifying the questions raised at How can we then construct a covariant commutator
the end of Sec. II. Is there then a model which can be used consistent with the condition of Eq. (17)1 In order to attack
for this purpose? this problem, let us divide the four-dimensional Minkow-
It has been shown that the covariant harmonic oscillator skian space-time into theone-dimensional timelike space 1 0
formalism introduced to this journal by the present authors parallel to the hadronic four-momentum and to the three-
in 1978 serves many useful purposes. IO It can explain the dimensional spacelike hyperplane perpendicular to the
basic hadronic features in the quark model, including the four-momentum spanned by x', y', and zO. 15 This leads us
mass spectrum,l7 form factors,I8 parton picture,I •. 'o and to consider the operator
the jet phenomenon. 2I In addition, the relativistic bound-
state wave functions in the oscillator formalism form a vec- b~ =a~ -(P~pvIM')av' (20)
tor space for the representations of the Poincare group di- Then b~ is perpendicular to P ~ , and satisfies the constraint
agonal in the Casimir operators corresponding to (mass)' condition

144 Am. 1. Phys., Vol. 53, No.2, February 1985 Hussar, Kim, and Noz 144
 THE TIME-ENERGY UNCERT AINTY RELATION 213

P"b
"
= PI'b:' = o. 121)
bl-' and b It, satisfy the covariant commutation relation::'H
[b",b;] = -glll.+P"P,./M'. 122)
The right-hand side ofthe above expression issymmteric in
f1 and v, and satisfies the relation
PI'I. - gill' + PpP,IM') = O. 123)
Therefore the covariant commutation relation given in Eg.
121) is consistent with the subsidiary condition of Eg. 119).
The covariant form given in Eg.I22) represents the usual
Heisenberg uncertainty relations on the three-dimensional
spacelike hypersurface perpendicular to the hadronic four-
momentum. This form enables us to treat separately the
uncertainty relation applicable to the timelike direction,
without destroying covariance. The existence of the t * dis-
tribution due to the ground-state wave function of Eg. lIS)
restricted to k = 0 by Eg.1I8) allows us to write the time- Fig. 2. The Lorentz deformation in the light-cone coordinate system. The
energy uncertainty relation in the form major (minor) axis in the uv coordinate system is conjugate to the minor
(major) axis in the q"q" coordinate system. Ifwe rotate these figures by
ILlt 'IILlqti)~ I, (24) 45~, they become Fig. 3 of Ref. 3 which explains the peculiarities observed

without postulating the commutation relation. in Feynman's parton picture.

Now let us go back to the commutators given in Sec. II.
It is not difficult to see that Eg. 122) is the harmonic oscilla-
tor realization of the covariant commutator given in Eg. I and 2 of Ref. 30. If the stationary space-time region is a
113). Since we forbid timelike excitations in the hadronic square in the uv or zt plane, then the moving region will
rest frame by imposing the subsidiary condition ofEg. (19), appear like a rectangle with the same area.
the time variable is a c number. However, we still have the The Lorentz deformation property in the q, qo plane is
time-energy uncertainty relation due to the ground-state the same as that for the zt plane, and it is possible to use the
oscillator wave function. This is the harmonic oscillator light-cone coordinate system for these variables:
realization of the relation given in Eg. 114). Indeed, the q + = Iq, + qollvL:, q _ = Iq, - qo)/vL:, 128)
oscillator wave function ofEg.IIS), together with the subsi-
diary condition of Eq. 119), constitutes a covariant realiza- with the transformation property
tion of the Eqs. 113) and 114). We shall study in Sec. IV the q+ = [I +PIIII-P)]l('q"c,
covariance property of the oscillator wave function. 129)
q_ = [II-PII(I +P)]1(2q*..
IV, LORENTZ-INVARIANT FORM OF THE The basic advantage of using the light-cone variables is
UNCERTAINTY RELATIONS that the coordinate system remains orthogonal. It is not
difficult to visualize the deformation of the regions given in
In this section, we shall complete the study of the uncer- Fig. 2 due to the boost. The circular region in the hadronic
tainty relations by studying the localization properties of rest frame will appear as an ellipse to the lab-frame observ-
the wave functions and see whether there is a manner in er.
which the uncertainty relations can be stated in a Lorentz- With this understanding, let us use the Gaussian wave
invariant manner. function which has the circular uncertainty distribution in
One of the difficulties noted in Sec. II was that the ortho- the hadronic rest frame:
gonality of the coordinate system is not preserved under
Lorentz boosts. In order to rectify the situation, we can "'Iz,t) = 11!,!;) exp{ -11!2)Uz*)' + it *)']}. 130)
consider Dirac's light-cone coordinate system which pre- This is the ground-state space-time wave function in the
serves the orthogonality relations. 24.30 The coordinate var- covariant oscillator formalism, which has been discussed
iab
es in the light-cone coordinate system are extensively in Sec. III. Since
u = (t + Z)lvL, Iz*)' + (t *)' = IU*)' + Iv*)', 131)
(25)
v = It - Z)lvL, the wave function of Eg. 130) can be written as

-..!.. (I - Plz + t)'

and the Lorentz transformation of Eq. 119) takes a simpler
form: "'Iz,t) = In !tr)II' exp[
4 I +P
u = [II +PIIII-PW 12 u*,
v = [(I - Pilil + PIl I12 v*.
126)
+ I +Plz - t)')]. (32)
I-P
Under the Lorentz transformation, one axis becomes elon- The momentum-energy wave function becomes

J
gated while the other goes through a contraction so that the
product uv will stay constant:
4> (q"qo) = C~) exp[ i(q,z - qat)] ¢(z,t)dt dz
uv = u*v*
= (t 2 _ z')/2 = [(t *)' _ (z*)']l2. (27)
=(~)exp{ - [(q:)'+(q~)']/2}, (33)
This transformation property is explained in detail in Figs.

145 Am. J. Phys., Vol. 53, No.2, February 1985 Hussar, Kim, and Noz 145
214 CHAffER III

which can also be written in a form similar to Eq. (32). QUARKS PARTONS
The Fourier relations 14 between the space-time and mo-
mentum-energy coordinates are

(34) Fig. 3. Pictures of the proton in the quark and parton models. Suppose

q, =q+ = -{fu} that a proton is sitting quietlY on the desk. According to the quark model,
it appears like a bound state of three quarks to an observer who is sitting
on the chair. However, to an observer who is on a jet plane with its speed
This means that the major and minor axes of the momen- close to thal of light, the prolon would look like a collection offree parti-
tum-energy coordinates are the "Fourier conjugates" of cles with a Wide momentum distribution. This is called Feynman's parton
the minor and major axes of the space-time coordinates, picture.
respectively. This aspect is illustrated in Fig. 2. Thus we
have the following Lorentz-invariant reiations.JI
(,ju)(,jq_) = (,ju*)(,jq*- )=1, (bl The interaction time between the quarks becomes di-
(35) lated, and the partons become free to allow an incoherent
l,jv)(,jq+) = (,jv*)(,jq*,. )=1. sum of cross sections of the constituent particles.
This is indeed a Lorentz-invariant statement of the c num- (c) The momentum distribution of partons becomes
ber time-energy uncertainty reiation combined with Hei- widespread as the hadron moves very fast.
senberg's position-momentum uncertainty relations. (d) The number of partons appears to be much larger
than that of quarks.
V. OBSERVABLE CONSEQUENCES Because the hadron is believed to be a bound state of two
In order that the time-separation variable be a c number, or three quarks, each of the above phenomena appears as a
it is essential that there be no timelike oscillations. We im- paradox, particularly Ib) and Ic) together. How can bound-
plement this concept by imposing the subsidiary condition state quarks become free while the momentum distribution
ofEq. (19), which restricts the wave functions ofEq. (15) to becomes widespread? This question has been discussed in
those with k = O. Ref. 20. Peculiarities (a) and (b) have been addressed in Ref.
On the other hand, if we allow timelike oscillations, the 19. According to Hussar's calculation," the ground-state
eigenvalue of the oscillator wave function which corre- harmonic oscillator wave function leads to a parton distri-
sponds to the (mass)' spectrum will be" bution in good agreement with the experimental data. The
parton phenomenon is therefore a direct manifestation of
A = (n - k ). (36) the time-energy uncertainty relation combined covariantly
For a given value n of the longitudinal excitation, k can with Heisenberg's position-momentum uncertainty rela-
take an arbitrarily large number. Thus (mass)' can take tion.
negative numbers with no lower bound. This does not hap-
pen in nature. Furthermore, for a given value of A, there are
infinite possible combinations of nand k, resulting in infi-
nite degeneracy. There is no evidence to indicate the exis- B)Present address: mjnDis Institute of Technology Research Institute. An-
tence of such a degeneracy in hadronic mass spectra. 17 napolis, Maryland 21401.
We noted in Sec. IV that the Lorentz deformation prop- IE. P. Wigner, in Aspects 0/ Quantum Theory, in Honour of P. A. M.
erty of the harmonic oscillator wave function enables us to Dirac's70th Birthday, edited by A. Salam and E. P. Wigner (Cambridge
define the uncertainty products in a Lorentz-invariant University, London, 1972).
'Po A. M. DIrac, Proc. R. Soc. London Ser. A 114, 234, 710 (1927).
manner. Then the question is whether the deformation
'E. P. Wigner and V. Weisskopf. Z. Phys. 63, 54(1930): 65, 18(1930).
property described in Fig. 2 manifests itself in the real 4E. P. Wigner, Phys. Rev. 10, IS. 609 (1946); E. P. Wigner and L. Etsen-
world. This question has been addressed in Refs. 19 and 20. bud, ibid. 70. 29(1947); M. Moshinsky. Rev. Mex. Fis.l. 28(1952); Phys
The point is that Lorentz deformation given in Fig. 2 can be Rev. 81, 347 (1951); 84. 525, 533 (1951); 88, 625(1952). P. T. Mathews
described in the zt and qzqo planes, which are simply 45' and A. Salam, ibid. 115, 1979 (1959); F. T. Smith, ibid. 118. 349 (1960);
rotations of the figures in Fig. 2. Figure 2 of the present Y. Aharonov and D. Bohm, ibld.1l2, 1649(1961); V. A. Fock, J. Exp.
paper and Fig. 3 of Ref. 20 are only two different forms of Thear. Phys. (U.S.S.R.) 42, 1115 (1962): Sov. Phys. JETP 15, 784(1962):
the same figure. Figure 2 is designed to explain the Lorentz Y. Aharonov and D. Bohm, Phys. Rev. 134, 1417 (1964); B. A. Lipp-
invariance of the uncertainty relation, while Fig. 3 of Ref. mann, ibid. 151, 1023 (1966); G. R. Allcock. Ann. Phys. 53, 253 (1969);
53,28611969); 53, 311 (1969); J. H. Eberly and L. P. S. Singh, Phys. Rev.
20 is designed to explain Feynman's parton picture.
D7, 359 (1973). See also articles by J. Rayski and J. M. Rayski, Jr.; by E.
It is by now a widely accepted view that hadrons such as
Recami; and by E. W. R. Papp, in The Uncertainty Principles and Quan-
nucleons and mesons are bound states of quarks, if they do tum Mechanics. edited by W. C. Price and S. S. Chissick (Wiley, New
not move rapidly. However, Feynman observed in 1969 York. 1977); M. Bauer and PA. Mello, Ann. Phys. (NY) 111, 38 (1978):
that, if a hadron moves with a velocity close to that oflight, M. Bauer, ibid. 150, 1 (1983).
it appears as a collection of partons,32 as is illustrated in 'W. Heisenberg, Z. Phys. 43,172(1927); 45,172(1927).
Fig. 3. It is also believed that partons are quarks. The par- ·W. Heisenberg, Am. J. Phys. 43, 389(1975).
ton picture, which has been a primary vehicle toward our 7The relation between the size of wave train and the linewidth was known
present understanding of high-energy hadronic interac- in classical optics long before quantum mechanics was formulated.
tions, has the following peculiarities. However. this is only the reJation between the lifetime and jinewidlh of
an unstable system mentioned in Ref. I. See W. Heitler, The Quantum
la) The picture is valid only for hadrons moving with Theory 0/ Radiation (O.ford University, London, 1954), 3rd ed.
velocity close to that of light. 'D. Han, Y. S. Kim, and M. E. Noz, Found. Phys. 11, 895 (1981).

146 Am. J. Phys., Vol. 53, No.2, February 1985 Hussar, Kim, and Noz 146
 THE TIME-ENERGY UNCERTAINTY RELATION 215

<JFor papers dealing with the time-separation variable in bound systems, 1038 (1980); 48,1043 (1980).
see H. Yukawa, Phys. Rev. 79, 416 11953); G. C. Wick, ibid. 96, 1124 I~For papers dealing with form factor behavior, see K. Fujimura, T. Ko-
(1954); M. Markov, Suppl. Nuovo Cimento3, 760 11956); T. Takabayasi, bayashi, and M. Namiki, Prog. Theor. Phys. 43, 7J (1970); R. G. Lipes,
Nuovo Cimento 33, 668 (1964); S. Ishida, Prog. Theor. Phys. 46, 1570. Phys. Rev. D 5, 2849 (1972); Y. S. Kim and M. E Noz, ibid. 8, 3521
1905 (1971); R. p, Feynman, M. Kislinger, and F. Ravndal, Phys. Rev. (1973)
D 3, 2706 1l972); G. Preparata and N. S. Craigie, Nuel. Phys. 8102,478 IQy. S. Kim and M. E. Noz, Phys. Rev. DIS, 335 (1977).
(1976); Y. S. Kim, Phys. Rev. 0 14, 273 (1976); J. Lukierski and M. "Y. S. Kim and M. E. Noz, Am. J. Phys. 51, 368 (1983).
Oziewics, Phys. Lett. 69B, 3390977); D. Dominici and G. Longhi, :' lFor papers dealing with the jet phenomenon, see T. Kitazoe and S.
Nuovo Cimento A 42, 235 (1977); T. Goto, Prog. Theor. Phys. 58, 1635 Hama, Phys. Rev. D 19, 2006 (1979); Y. S. Kim, M. E. Noz, and S. H.
(1977); H. Leutwyler and J. Stem, Phys. Lett. 73B, 75 (1978); and Nuel. Oh, Found. Phys. 9, 947 (1979); T. Kitazoe and T Morii, Phys. Rev. D
Phys. 8157, 327 (1979); I. Fujiwara, K. Wakita. and H. Yoro, Prog. 21,685(1980); Nuel. Phys. 8164, 76(1980).
Theor. Phys. 64, 363 (1980); 1. lersak and D. Rein, Z_ Phys_ C 3. 339 "E. P. Wigner, Ann. Math. 40, 149 (1939).
11980); I. Sogami and H. Yabukl, Phys. Lett. 94B, 15711980); M. Paun, "~Yo S. Kim, M. E. Noz, andS. H.Oh,J. Math. Phys.l0, 1341 (1979);Am.
in Group Theoretical Methods In Physics, Proceedings of the 9th Interna- J. Phys. 47, 892 (1979); J. Math Phys. 21, 1224 (1980).
tional Colloquium, Coeoyoe, Mexico, edited by K. B. Wolf (Springer- "P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949).
Verlag, Berlin, 1980); G. Marehesini and E. Onofri, Nuovo Cimento "D. Han and Y. S. Kim, Am. J. Phys. 49,1157 (1981).
A6S, 298 (1981); E. C. G. Sudarshan, N. Mukunda, and C. C. Chiang, ~t-.F. C. Rotbart. Phys. Rev. 023,3078 (1981). For a physical basis for
Phys. Rev. D 25,3237 (1982). Rotbart'scalculation, see L. P. Horwitz and C. Piron, Helv. Phys. Acta
IOFor a recent pedagogical paper on this problem, see C. H. Blanchard, 46,316 (1973). See also Ref. 16.
Am. J. Phys. 50, 64211982). "D. Han, M. E. Noz, and Y. S. Kim, Phys. Rev. D 25,1740(1982).
I IFor a possible departure from the accepted view, see £. Prugovecki, 2MD. Han, M. E. Noz, Y. S. Kim, and D. Son, Phys. Rev. D 27, 3032
Found. Phys. 12, 555 (1982); Phys. Rev_ Lett. 49, 1065 (1982). (1983).
12E. Prugovecki, Quantum Mechanics in Hilbert Space (Academic. New 2QThis commutator can be translated into wave-function fonnalism. For a
York, 1981). 2nd ed. wave-function description of this commutation relation. see M. 1. Ruiz,
I1H. P. Robertson, Phys. Rev. 34,163 (1929). Phys. Rev. D 10. 4306 (1974).
14The word "Fourier relation" was used earlier by Blanchard in Ref. 10. lOFor a pedagogical treatment of the light-cone coordinate system, see Y.
This word is necessary because the energy is not a variable canonically S. Kim and M. E. Noz, Am. J. Phys. 50, 721 (1982).
conjugate to the time variable "Y. S. Kim and M. E. Noz, Found Phys. 9, 375 (1979); J. Math Phys. 22,
\~G. N. Flemmg, Phys. Rev. 137, B188 (1965:1; G. N. Fleming, 1. Math 2289 (1981).
Phys.ll, 1959 (1966) '~R. P. Feynman, in High Energy Collisions, Proceedings of the Third
"Y. S. Kim and M. E. Noz, Am. J. Phys. 46, 48411978). International Conference, Stony Brook, NY, edited by C. N. Yang et al.
17For some of the latest papers on hadronic mass spectra, see N. lsgur and (Gordon and Breach, New York, 1969); Pholon Hadron Interactions
G. Karl. Phys. Rev. D 19, 2653 (1978); D. P. Stanley and D. Robson, (Benjamin, New York, 1972). See also J. D. Bjorken and E. A. Paschos,
Phys. Rev. Lett. 45. 235 (1980). For review articles written for teaching Phys. Rev. 185, 1975 (1969).
purposes, see P. E. Hussar, Y S. Kim, and M. E. Noz, Am. J. Phys. 48, "~Po E. Hussar, Phys. Rev. D 23, 2781 (1981).

147 Am. J. Phys. 53 (2), February 1985 ® 1985 American Association of Physics Teachers 147
Chapter IV

Covariant Picture of Quantum Bound States

The success of quantum field theory in the 1950's led a large number of physicists
to believe that field theory will solve all dynamical problems. Naturally, they
attempted to solve the hydrogen atom problem within the framework of field theory.
The Bethe-Salpeter equation is the most suitable field theoretic equation for
quantum bound-state problems. Although this equation offers us mathematical
challenge and generates some useful solutions, it is plagued with fundamental
difficulties, as Wick pointed out clearly in 1954. During the 1970's, field theoretic
bound state models have been proven to be ineffective in dealing with hadrons in the
quark model. We are thus fully justified to construct a relativistic model of bound
states consistent with special relativity and quantum mechanics, but not necessarily
within the framework of the mathematical framework of field theory, as is indicated
in Figure 2.
The question is then whether we can construct a relativistic bound-state model
without the fundamental difficulties contained in the Bether-Salpeter wave function.
In 1973, Kim and Noz investigated this possibility, and showed that the harmonic
oscillator model of Yukawa (1953) can satisfy this condition. Yukawa's 1953
papers are based on his earlier effort made in 1950 to formulate a field theory of
particles with space-time extension. Yukawa's aim was not different from that of
the present day string models.
The covariant harmonic oscillator model has been studied extensively by the present
authors and their associates. The orthogonality relation and Lorentz transformation
properties have also been studied by Ruiz (1974) and Rotbart (1981). In 1981, Han
and Kim showed that the covariant oscillator formalism can serve as a solution of
the Poisson-bracket equations for relativistic bound state formulated by Dirac in
1949.
The paper of Kim, Noz, and Oh in Chapter II shows that the same oscillator
formalism constitutes a representation of the Poincare group for relativistic extended
hadrons.

217
218 CHAPTER IV

SCATTERING BOUND STATE SPACE/TIME

COMET PLANET

NEWTON

><
GALILEO
BOHR

HEISENBERG, SCHRODINGER

FEYNMAN C§J)
1 1 EINSTEIN

STEP 2

FIG. 2. Histol)' of dynamical and kinematical developments. Mankind's unified

understanding of scattering and bound states has been vel)' brief. It is therefore not
unusual to expect that separate theoretical models be developed for scattering and
bound states, before a completely satisfactol)' theol)' can be developed. The
successes and limitations of the Feynman diagram approach are well known. There is
an urgent need for a relativistic bound state model. This figure is from D. Han, Y. S.
Kim, and M. E. Noz, Found. of Physics, 11,895 (1981).
COVARIANT PICTURE OF QUANTUM BOUND STATES 219

REVIEWS OF MODERN PHVSICS VOLUME 21. NUMBER J JULY, 19411

Forms of Relativistic Dynamics

P. A. M. DIRAC
St. fohn's College, Cambridge, England

For the purposes of atomic theory it is necessary to combine the restricted principle of relativity with
the Hamiltonian formulation of dynamics. This combination leads to the appearance of ten fundamental
quantities for each dynamical system, namely the total energy, the total momentum_ and the 6-vector
which has three components equal to the total angular momentum. The usual form of dynamics expresses
everything in terms of dynamical variables at one instant of time, which results in specially simple expres-
sions for six or these ten, namely the components of momentum and of angular momentum. There are
other forms for relativistic dynamics in which others of the ten are specially simple, corresponding to
various sub-groups of the inhomogeneous Lorentz group. These forms are investigated and applied to a
system of particles in interaction and to the electromagnetic field.

1. INTRODUCTION linear coordinates in space-time, shall be invariant

E INSTEIN'S great achievement, the principle of
relativity, imposes conditions which all physical
under all transfonnations of the coordinates. It brings
gravitational fields automatically into physical theory
laws have to satisfy. It profoundly influences the whole and describes correctly the influence of these fields on
of physical science, from cosmology, which deals with physical phenomena.
the very large, to the study of the atom, which deals Gravitational fields are specially important when one
with the very small. General relativity requires that is dealing with large-scale phenomena, as in cosmology,
physical laws, expressed in tenns of a system of curvi- but are quite negligible at the other extreme, the study

Reprinted from Rev. Mod. Phys. 21, 392 (1949).

220 CHAPTER IV

FOR ~I S 0 F R E L ,\ T I \. 1ST leD Y N A M I C S 393

of the atom. In the atomic world the departure of another. The imperfections may well arise from the use
space-time from flatness is so excessively small that of wrong dynamical systems to represent atomic phe-
there would be no point in taking it into account at the nomena, i.e., wrong Hamiltonians and wrong interaction
present time, when many large effects are still unex- energies. It thus becomes a mailer of great importance to
plained. Thus one naturally works with the simplest set up >lew dynamical syslems ana see if tlzey will better
kind of coordinate system, for which the tensor g"' describe the alomic world. In setting up such a new
that defines the metric has the components dynamical system one is faced at the outset by the two
requirements of special relativity and of Hamiltonian
Ff'= _gll= -g"= -Ff'= 1 equations of motion. The present paper is intended to
g"'=O for Jl;o!v. make a beginning on this work by providing the
Einstein's restricted principle of relativity is now of simplest methods for satisfying the two requirements
paramount importance, requiring that physical laws simultaneously.
shall be invariant under transformations from one such
coordinate system to another. A transformation of this 2. THE TEN FUNDAMENTAL QUANTITIES
kind is called an inhomogeneous Loren tz transforma- The theory of a dynamical system is built up in
tion. The coordinates u" transform linearly according terms of a number of algebraic quantities, called
to the equations dynamical variables, each of which is defined with
respect to a system of coordinates in space-time. The
with usual dynamical variables are the coordinates and
momenta of particles at particular times and field
quantities at particular points in space-time, but other
the a's and {3's being constants. kinds of quantities are permissible, as will appear later.
A transformation of the type (1) may involve a In order that the dynamical theory may be expres-
reflectio'l of the coordinate system in the three spacial sible in the Hamiltonian form, it is necessary that any
dimensions and it may involve a time reflection, the two dynamical variables, ~ and ~, shall have a P.b.
direction duo in space-time changing from the future (Poisson bracket) U, ~], subject to the following laws,
to the past. I do not believe there is any need for
physical laws to be invariant under these reflections,
although all the exact laws of nature so far known do
have this invariance. The restricted principle of rela-
tivity arose from the requirement that the laws of
nature should be independent of the position and A number or physical constant may be counted as a
velocity of the observer, and any change the observer special case of a dynamical variable, and has the
may make in his position and velocity, taking his property that its P.b. with anything vanishes.
coordinate system with him, will lead to a transforma- Dynamical variables change when the system of
tion (1) of a kind that can be built up from infinitesimal coordinates with respect to which they are defined
transformations and cannot involve a reflection. Thus changes, and must do so in such a way that P.b.
it appears that restricted relativity will be satisfied by relations between them remain invariant. This requires
the requirement that physical laws shall be invariant that with an infinitesimal change in the coordinate
under infinitesimal transformations of the coordinate system (2) each dynamical variable ~ shall change
system of the type (1). Such an infinitesimal transfor- according to the law
mation is given by
~'=H[~, F], (4)

with where F is some infinitesimal dynamical variable inde-

pendent of ~, depending only on the dynamical system
involved and the change in the coordinate system.
the a's and b's being infinitesimal constants.
We are thus led to associate one F with each infini-
A second general requirement for dynamical theory
tesimal transformation of coordinates.
has been brought to light through the discovery of
Let us apply two infinitesimal transformations of
quantum mechanics by Heisenberg and Schr6dinger,
coordinates in succession. Suppose the first one changes
namely the requirement that the equations of motion
the dynamical variable ~ to ~* according to
shall be expressible in the Hamiltonian form. This is
necessary for a transition to the quantum theory to be ~*= H[~, F I ],
possiblt:. In atomic theory one thus has two over-riding
requirements. The problem of fitting them together and the second one changes ~. to ~t according to
forms the subject of the present paper. ~t = ~*+[~', F,*]= ~.+[t, F,]*.
The existing theories of the interaction of elementary
particles a ld fields are all unsatisfactory in one way or The two transformations together change ~ to ~t
COVARIANT PICTURE OF QUANTUM BOUND STATES 221

394 P. A. M. DIRAC

according to a new dynamical system reduces to the problem of finding

a new solution of these equations.
;t=H[;, FI ]+[;, F,]+m, F,], FI ], An elemen tary solution is provided by the following
to the accuracy of the order FIF, (with neglect of scheme. Take the four coordinates q, of a point in
terms of order F I ' or F,'). If these two transformations space-time as dynamical coordinates and let their
are applied in the reverse order, they change ; to ;tt conjugate momenta be p.. so that
according to
[qpo q,]=O, [p" p,]=O
;tt =HU, F,]+U, F I ]+[[;, FI ], F,]. [p" q,]= gpo'
Thus The g's will transform under an infinitesimal transform-
;tt=;t+m, F I ], F,]-m, F,], F I ] ation of the coordinate system in the same way as the
=;t+[;, [FI, F,]], u's in (2). This leads to
willi the help of the first and last of Eqs. (3). This (7)
gives the change in a dynamical variable associated
with that change of the coordinate system which is the and provides a solution of the P.b. relatio.1s (6). The
commutator of the two previous changes. It is of the solution (7) does not seem to be of any practical
standard form importance, but it may be used as a basis for obtaining
other solutions that are of practical importance, as the
next three sections will show.
The foregoing discussion of the requirements for a
with an F that is the P. b. of the F's associated with
relativistic dynamical theory may be generalized some-
the two previous changes of coordinates. Thus the
what. We may work with dynamical variables that are
commutation relations between the various infinitesimal
connected by one or more relations for all states of
changes of coordinates correspond to the P.b. relations
motion that occur physically. Such relations are called
between the associated F's.
subsidiary equations. They will be written
The F associated with the transformation (2) must
depend linearly on the infinitesimal numbers a.. b" A"'O (8)
that fix this transformation. Thus we can put
to distinguish them from dynamical equations. They
F= -P'a,+!M"b,. are less strong than dynamical equations, because with
MjJ"=-~fYjJ, a dynamical equation one can take the P.b. of both
sides with any dynamical variable and get another
where P', M" are finite dynamical variables, inde- equation, while with a subsidiary equation one cannot
pendent of the transformation of coordinates.
The ten quantities p .. M" are characteristic for the
dynamical system. They will be called the ten funda-
do this in general. The lesser assumption is made,
however, that from two subsidiary equations A "" 0, B ""
one can infer a third
°
mental quantities. They determine how all dynamical
(9)
variables are affected by a change in the coordinate
system of the kind that occurs in special relativity. A subsidiary equation must remain a subsidiary
Each of them is associated with a type of infinitesimal equation under any change of coordinate system. This
transformation of the inhomogeneous Lorentz group. enables one to infer from (8)
Seven of them have simple physical interpretations,
namely, Po is the total energy of the system, P. (r= 1, [P" AJ"'O, [M", A]""O. (10)
2, 3) is the total momentum, and M" is the total A dynamical variable is of physical importance only
angular momentum about the origin. The remaining if its P.b. with any subsidiary equation gives another
three Jr.o do not correspond to any such well-known subsidiary equation, i.e., its P.b. with A in (8) must
physical quantities, but are equally important in the vanish in the subsidiary sense. Such a dynamical
general dynamical scheme. variable will be called a physical variable. The P.b. of
From the commutation relations between particular two physical variables is a physical variable. Equations
infinitesimal transformations of the coordinate system (10) show that the ten fundamental quantities are
we get at once the P.b. relations between the ten physical variables.
fundamental quantities, The physical variables are the only ones that are
really important. One could eliminate the non-physical
[P .. P.]=o } variables from the theory altogether and one could then
[M,., P,]=-gppP.+g.,P, (6)
[M,., M,.]= -gppM •• +g.,Jf,.-g,.M,.+g..M pp • make the subsidiary equations into dY,llamical equa-
tions. However, the elimination may be awkward and
To construct a theory of a dynamical system one mt'st may spoil some symmetry feature in the scheme of
obtain expressions for the ten fundamental quantitlu equations, so it is desirable to retain the p,Jssibility of
that satisfy these P.b. relations. The problem of finding subsidiary equations in the general theory.
222 CHAPTER IV

FORMS OF RELATIVISTIC DYNAMICS 395

J. THE INSTANT FORM that do not become infinitely great when one puts
The ten fundamental quantities for dynamical sys- p-p.=m' with Po>O. Since p-p.-m' has zero P.b. with
tems that occur in practice are usually such that some all the expressions (7), the modified expressions (13)
of them are specially simple and the others are compli- must still satisfy the P.b. relations (6), apart from
cated. The complicated ones will be called the Hamil- multiples of p·p.-m', with any choice of the A's. If
tonians. They play jointly the r6le of the single Ham- we now choose the X's so as to make the P" M •• given
iltonian in non-relativistic dynamics. Since the P.b. of by (13) independent of po, the P.b. relations (6) must
two simple quantities is a simple quantity, the simple be satisfied apart from terms that are independent of po
ones of the ten fundamental quantities must be those as well as being multiples of p-p.-m'. Such terms
associated with some sub-group of the inhomogeneous must vanish, so we get in this way a solution of our
Lorentz group. problem.
In the usual form of dynamics one works with The X's have the values
dynamical variables referring to physical conditions at
some instant of time, e.g., the coordinates and momenta Xo= -IPo+(p.p.+m')II-I,
X,o= -q,IPo+(p.p.+m')II-I, 1(14)
of particles at that instant. An instant in the four-
dimensional relativistic picture is a Bat three-dimen- and Eqs. (13) become
sional surface containing only directions which lie
outside the light-cone. The simplest instant referred P,=p" M"=q,p.-q.p,, (15)
to the u coordinate system is given by the equation
Po= (p.p.+m')I, Mrl)=q,(p.p.+m')I, (16)
Uo=O. (11)
with the help of (12). Equations (15) and (16) give all
The effect of working with dynamical variables referring
the ten fundamental quantities for a particle with
to physical conditions at this instant will be to make
rest-mass m. Those given by (15) are the simple ones:
specially simple those of the fundamental quantities
those given by (16) are the Hamiltonians.
associated with transformations of coordinates that
For a dynamical system composed of several particles,
leave the instant invariant, namely PI, p" p" M",
M31 , M". The remaining ones, Po, MlO , M,o, M,o, will P, and M" will be just the sum of their values for the
be complicated in general and will be the Hamiltonians. particles separately,
We get in this way a form of dynamics which is associ- P,=LP" M,,=Dg,p.-g.p,). (17)
ated with the sub-group of the inhomogeneous Lorentz
group that leaves .the instant invariant, and which may The Hamiltonians p" M rl) will be the sum of their
appropriately be called the instant Jorm. values for the particles separately plus interaction
Let us take as an example a single particle by itself. terms,
The ten fundamental quantities in this case are well
known, but they will be worked out again here to
illustrate a method that can be used also with the
Po=L(p.p.+m')'+V,
M Lg,(p.p.+m')'+ V,.
rl)=
I(18)
other forms of dynamics.
We take as dynamical coordinates the three coordi- The V's here must be chosen to make Po, M,o satisfy
nates of the particle at the instant (11). Calling these all the P.b. relations (6) in which they appear.
coordinates g" we can base our work on the scheme (7), Some of these °relations are linear in the V's and are
with the additional equation easily fulfilled. The P.b. relations for [M,,, PoJ and
[M '" M toJ are fulfilled provided V is a three-dimen-
qo=O. (12) sional scalar (in the space u" u" u,) and V, a three-
With this equation po no longer has a meaning. We dimensional vector. The P.b. relation for [P" poJ will
must therefore modify the expressions for the ten be fulfilled provided V is independent of the position
fundamental quantities given by (7) so as to eliminate of the origin in the three-dimensional space u" U" u,.
po from them, without invalidating the P.b. relations The P.b. relation for [M rl), P.] will be fulfilled provided
(6).
Let us change the expressions for the ten fundamental V,=g,V+V/, (19)
quantities by multiples of p-p.-m', where m is a where the g, are the coordinates of anyone of the
constant, i.e., let us put particles and the V,' are independent of the position
p.= P.+X.(p·p.-m') of the origin in three-dimensional space.
M•• =q.p.-g.P.+X.,(p-p.-m'), 1(13) The remaining conditions for the V's are quadratic,
involving [V, V,] or [V" V.]. These conditions are
wher~
not easily fulfilled and provide the real difficulty in the
problem of constructing a theory of a relativistic
and the coefficients X are functions of the q's and p's dynamical system in the instant formo
COVARIANT PICTURE OF QUANTUM BOUND STATES 223

396 P. A. M. DIRAC

4. THE POINT FORM so we must have

One can build up a dynamical theory in terms of
dynamical variables that refer to physical conditions and hence
on some three-dimensional surface other than an )...=q.B,
instant. The surface must satisfy the condition that the
world-line of every particle must meet it, otherwise the where R is some dynamical variable independent of 1'.
particle could not be described by variables on the Further
surface, and preferably the world-line should meet it p,p.= Ir+q·R(p'p.-m 2) I I p.+q.B(p'pp-m') I
only once, for the sake of uniqueness. =m'+ I 1+2p·q.R+q·q.B'(p·p.-m') I (p'pp-m').
To get a simple form of theory one should take the
surface to be such that it is left invariant by some In order that p.p. shall have zero P.b. with q'g" we
sub-group of the group of inhomogeneous Lorentz must take
transformations. A possible sub-group is the group of
rotations about some point, say the origin 11.=0. The
surface may then be taken to he a branch of a hyper- so that
boloid
R(p' p.- m') = (q'q,)-' I [(p'q,)'
(20) -q'q,(p'P.-m')]Lp'q,1
with K a constant. The fundamental quantities associ- The right-hand side here tends to zero as P'p.- m'->O,
ated with the infinitesimal transformations of the sub- so it is a multiple of P'p,- m', as it ought to be. We
group, namely the M '" will then be specially simple, now get finally
while the others, namely the P., will be complicated in
general and will be the Hamiltonians. A new form of p.= P.+q.K-'1 [(p'q,)'- K'(P'P.- m')]!- P'q,l} (22)
dynamics is thus obtained, which may be called the M.,=q.p,-q,p.,
point form, as it is characterized by being associated in which the expression for p. has been simplified with
with the sub-group that leaves a point invariant. the help of (21).
To illustrate the new form, let us take again the It is permissible to take K=O and so to have a light-
example of a single particle. The dynamical coordinates cone instead of a hyperboloid. The expression for B
must determine the point where the world-line of the then becomes much simpler and gives
particle meets the hyperboloid (20). Let the four
coordinates of this point in the u system of coordinates p.= P.-tq.(rq,)-l(p'P.-m'). (23)
be g•. Only three of these are independent, but instead instead of the first of Eqs. (22).
of eliminating one of them, it is more convenient to For a dynamical system composed of several particles,
work with all four and introduce the subsidiary equation the M., will be just the sum of their values for the
(21) particles separately,

It is then necessary that the ten fundamental quantities, M.,= L:.(q.P,-q,P.). (24)
and indeed all physical variables, shall have zero P.b. The Hamiltonian~ p. will be the sum of their values
with gPqp. The condition for this is that they should for the particles separately plus interaction terms,
involve the p's only through the combinations g.P,
-g,P.· p.=L:lp.+q.B(p·p.-m')I+V" (25)
The ten fundamental quantities may be obtained by The V. must be chosen so as to make the p. satisfy
a method parallel to that of the preceding section, with the correct P.b. relations. The relations for [M '" P,]
the subsidiary equation (21) taking the place of Eq. are satisfied provided the V. are the components of a
(12). We again assume Eqs. (13), and now choose the 4-vector. The remaining relations, which require the
A's so as to make their right-hand sides have zero P.b. p. to have zero P.b. with one another, lead to quadratic
with qPgp. The resulting expressions for the ten funda- conditions for the V .. These cause the real difficulty in
mental quantities will again satisfy the P.b. relations the problem of constructing a theory of a relativistic
(6), as may be inferred by a similar argument to the dynamical system in the point form.
one given in the preceding section.
We find at once 5. THE FRONT FORM

)...,=0. Consider the three-dimensional surface in space-time

formed by a plane wave front advancing with the
To obtain X., instead of arranging directly for the p.
velocity of light. Such a surface will be called a front
to have zero P .b. with g'q" it is easier to make q.p,
for brevity. An example of a front is given by the
-g,p. and p.p. have zero P.b. with gPqp. Now
equation
q.p,- q,p. = q.P,-q,P.+ (q.X,-q;A.)(p·p.-m'), 110- 11 3=0. (26)
224 CHAPTER IV

FORMS OF RELATIVISTIC DY;IIAMICS 397

We may set up a dynamical theory in which the that it shall be invariant under all transformations of
dynamical variables refer to physical conditions on a the three coordinates u" u" u+ of the front except those
front. This will make specially simple those of the for which du+ gets multiplied by a factor, and for the
fundamental quantities associated with infinitesimal latter transformations V must get multiplied by the
transformations of coordinates that leave the front same factor. The linear conditions for V; require it to
invariant, and will give a third form of dynamics, be of the form
which may be called the fro1l! form.
IC Ap is any 4-vector, put Vi=q.V+V;', (30)

A.+A,=A+, A.-A3=A_. where qi are the coordinates 1, 2 of anyone of the

particles, and V;' has the same properties as V with
We get a convenient notation by using the + and - regard to all transformations of the three coordinates
suffixes freely as tensor suffixes, together with 1 and 2. of the front except rotations associated with M 12, and
They may be raised with the help of
under these rotations it behaves like a two-dimensional
g++=g--=O, g+-=!, vector. The quadratic conditions for the V's are not
gi+=gi-=O, for i=I,2, easily fulfilled and give rise to the real difficulty in the
construction of a theory of a relativistic dynamical
as one can verify by noting that these g values lead to system in the front form.
the correct value for gP'A pA • when p. and. are summed
over 1, 2,+, -. 6. THE ELECTROMAGNETIC FffiLD
The equation of the front (26) becomes in this
notation To set up the dynamical theory of fields on the lines
1'-=0. discussed in the three preceding sections, one may take
as dynamical variables the three-fold infinity of field
The fundamental quantities PI, P2, P_, M 12 , M+_, quantities at all points on the instant, hyperboloid,
M ,_, M 2_ are associated with transformations of co-
or front, and use them in place of the discrete set of
ordinates that leave this front invariant and will be variables of particle theory. The ten fundamental
specially simple. The remaining ones P+, M '+, M 2+ quantities P PO M P' are to be constructed out of them,
will be complicated in general and will be the Hamil- satisfying the same P.b. relations as before.
tonians. For a field which allows waves moving with the
Let us again work out the example of a single particle. velocity of light, a difficulty arises with the poindorm
The dynamical coordinates are now q" q2, q+. We again of theory, because one may have a wave packet that
'assume Eqs. (13), and add to them the further equation does not meet the hyperboloid (20) at all. Thus physical
q_=O. We must now choose the A's so as to make the conditions on the hyperboloid cannot completely de-
right-hand sides of (13) independent of p+. The resulting scribe the state of the field. One must introduce some
expressions for the ten fundamental quantities will then extra dynamical variables besides the field quantities
again satisfy the required P.b. relations. on the hyperboloid. A similar difficulty arises, in a less
We find serious way, with the front form of theory. Waves
moving with the velocity of light in exactly the direction
the other A's vanishing. Thus of the front cannot be described by physical conditions
on the front, and some extra variables must be intro-
P;= pi, P_= p_, duced for dealing with them.
M 12 =q,P2-q2P.. M;_=q.p_, An alternative method of setting up the dynamical
P+= (p,'+ p,'+m')/ p_, theory of fields is obtained by working with dynamical
M,+=qi(P,'+p,'+m')/p_-q+Pi. variables that describe the Fourier components of the
Equations (27) give the simple fundamental quantities. field. This method has various advantages. It disposes
Equations (28) give the Hamiltonians. of the above difficulty of extra variables, and it usually
For a dynamical system composed of several particles. lends itself more directly to physical interpretation. It
Pi, P_, M 12, M+_, M,_ will be just the sum of their leads to expressions for the ten fundamental quantities
values for the particlc;s separately. The Hamiltonians that can be used with all three forms. For a field by
P+, M i+ will be the sum of their values for the particles itself, there is then no difference between the three
separately plus interaction terms, forms. A difference occurs, of course, if the field is in
interaction with something. The dynamical variables
P+=I:(p,'+p,'+m')/p_+V } (29) of the field are then to be understood as the Fourier
+
M i+= I: Iq,(p,'+p,'+m')/p_-q+p. I Vi. components that the field would have, if the interaction
The V's must satisfy certain conditions to make the were suddenly cut off at the instant, hyperboloid or
Hamiltonians satisfy the correct P.b. relations. front, after the cutting off.
As before, some of these conditions are linear and Let us take as an example the electromagnetic field,
some are quadratic. The linear conditions for V require first without any interaction. We may work with the
COVARIANT PICTURE OF QUANTUM BOUND STATES 225

398 P. A. M. DIRAC

four potentials A .(u) satisfying the subsidiary equation The second of Eqs. (34) then leads to
aA ./Ou. "" O.
Their Fourier resolution is
(31)
M.,=4ri fI At..(k.il/ilk'-k..a/ilk·)A.·
+At•• A .. -At..A•• }ko-'d'k. (37)

Equations (35) and (37) give the ten fundamental

quantities.
For the electromagnetic field in interaction with
with charged particles, the ten fundamental quantities will
ko= (k,'+k.'+k,')I, d'k=dk,dk.,dk,. be the sum of their values for the field alone, given by
(35) and (37), and their values for the particles, given
The factor k o-' inserted in (32) leads to simpler trans- in one of the three preceding sections, with interaction
formation laws for the Fourier coefficients A .., since the terms involving the field variables A .., At.. as well as
differential element ko-'d'k is Lorentz invariant. We the particle variables. One usually assumes that there
now take the A .., At.. as dynamical variables. is no direct interaction between the particles, only
Under the transformation of coordinates (2) the interaction between each particle and the field. The
potential A .(u) at a particular point u changes to a ten fundamental quantities then take the form
potential at the point with the same u·values in the
new coordinate system, i.e., the point with the coordi- p.=p.F+ "L.p••
nates u,-a,-b,'u, in the original coordinate system. M".=M"/+EaM,,,G,
This causes a change in A .(u) of amount where p.F, M./ are the contributions of the field alone,
- (a.+b/u.)aA./au •. given by (35) and (37), and Po", M ... are the contribu-
tions of the a-th particle, consisting of terms for the
There is a further change, of amount b.'A " owing to particle by itself and interaction terms. For point
the change in the direction of the axes. Thus, from charges, the interaction terms will involve the field
(4) and (5) variables only through the A.(q) and their derivatives
at the point q where the world-line of the particle
[A.(u), -P'a.+!M·'b,,]
meets the instant, hyperboloid or front. The expres-
=A.(u)*-A.(u)
sions for Po", M ... may easily be worked out for this
= - (a.+b/u,)aA./Ou,+b.'A"
case by a generalization of the method of the three
and hence preceding sections, as follows.
Suppose there is only one particle, for simplicity.
[A.(u), P·]=ilA./ilu., } We must replace Eqs. (13) by
[A.(u), M.,]=u.ilA./ilu'-u..aA./ilu· (33)
+g•• A,-g..A •. p.=p.F+p.+}..(,..·.... -m·) } (39)
M,.=M •• F+q.p.-q.p,+X,,(,,·,...-m'),
Taking Fourier components according to (32), we get
where
[A .., P.]=ik.A .., }
[A .. , M.,]= (kiJ/ak'-k..a/ilk·)A.. (34) 11".= p.- eA,(q),
+g•• A .. -g..A •• , and P/, M ••F are the right-hand sides of (35) and (37).
in which A .. may be considered as a function of four From (.13),
independent k's for the purpose of applying the differ-
[A.(q), p.F+p.]=O
ential operator k.a/ak'-k..a/ak· to it.
The Maxwell theory gives for the energy and mo-
[A.(q), M ••F+q.p.-q.P.]=g•• A.(q)-g•• A.(q),
mentum of the electromagaetic field and hence
[ ...., p.F+p.]=O
[ ...., M ••F+q.p.-q.P.]=g ...... -g ......·
It follows that "'11", has zero :r.b. with each of the
quantities p.F+p., M.,F+q.p.-g.P•. One can now
the - sign being needed to make the transverse
infer, by the same argument as in the case of no field,
components contribute a positive energy. In order that
that if the }.'s in (39) are chosen to make p .. M •• have
this may agree with the first of Eqs. (34), we must
zero P.b. with go, g'g. or g_, the P.b. relations (6) will
have the P.b. relations
all be satisfied. Such a choice of X's, in conjunction
[A .. , A.,.]=O } with one of the equations qo=O, q'q."".', q_=O, will
[A .. , At.,.]= -ig••/4,..' (36) provide the ten fundamental quantities for a charged
. koJ(k,- k,')J(k.- k,')J(k,- k.'). particle in interaction with the field in the instant,
226 CHAPTER IV

FOR ;vI 5 0 F R E L ,\ T I V 1ST leD Y N ,\ ;vI I C5 399

point and front forms, respectively. The subsidiary Hamiltonians. The former are the components of a
Eq. (31) must be modified when a charge is present. 6-vector, the latter are the components of a 4-vector.
The point form will be worked out as an illustration. Thus the four Hamiltonians can easily be treated as a
In this case we have at once Ap.=O. We can get Ap con- single entity. All the equations with this form can be
veniently by arranging that qp(P.- P.F)_q.(Pp_ P/) expressed neatly and concisely in four-dimensional
and !Pp-ppF-eAP(q)I!Pp-PpF-eAp(q)1 shall have tensor notation.
zero P.b. with q'q,. The first condition gives A.= q.B. The front form has the advantage that it requires
The second then gives only three Hamiltonians, instead of the four of the
other forms. This makes it mathematically the most
1+ 2"'qpB+q'q.B'("·7f.-m') =0. interesting form, and makes any problem of finding
Thus we get finally Hamiltonians substantially easier. The front form has
the further advantage that there is no square root in
Pp=p.F+P.+qpK-'![(7f·q.)' } the Hamiltonians (28), which means that one can avoid
_ F _ - K'(7f·... -m')]I- .. •q,1 (40) negative energies for particles by suitably choosing the
M •• -M•• +q.p. q,P•. values of the dynamical variables in the front, without
having to make a special convention about the sign
The above theory of point charges is subject to the
of a square root. It may then be easier to eliminate
usual difficulty that infinities will arise in the solution
negative energies from the quantum theory. This
of the equations of motion, on account of the infinite
advantage also occurs with the point form with <=0,
electromagnetic energy of a point charge. The present
there being no square root in (23).
treatment has the advantage over the usual treatment
There is no conclusive argument in favor of one or
of the electromagnetic equations that it offers simpler
other of the forms. Even if it could be decided that one
opportunities for departure from the point-charge
of them is the most convenient, this would not neces-
model for elementary particles.
sarily be the one chosen by nature, in the event that
only one of them is possible for atomic systems. Thus
8. DISCUSSION
all three forms should be studied further.
Three forms have been given in which relativistic The conditions discussed in this paper for a relativistic
dynamical theory may be put. For particles with no dynamical system are necessary but not sufficient. Some
interaction, anyone of the three is possible. For particles further condition is needed to ensure that the inter-
with interaction, it may be that all three are still action between two physical objects becomes small when
possible, or it may be that only one is possible, de- the objects become far apart. It is not clear how this
pending on the kind of interaction. If one wants to set condition can be formulated mathematiCally. Present-
up a new kind of interaction between particles in order day atomic theories involve the assumption of local-
to improve atomic theory, the way to proceed would izability, which is sufficient but is very likely too
be to take one of the three forms and try to find the stringent. The assumption requires that the theory
interaction terms V, or to find directly the Hamil- shall be built up in terms of dynamical variables that
tonians, satisfying. the required P.b. relations. The are each localized at some point in space-time, two
question arises, which is the best form to take for this variables localized at two points lying outside each
purpose. other's light-cones being assumed to have zero P.b. A
The instant form has the advantage of being the one less drastic assumption may be adequate, e.g., that
people are most familiar with, but I do not believe it there is a fundamental length A such that the P.b. of
is intrinsically any better for this reason. The four two dynamical variables must vanish if they are
Hamiltonians Po, M,o form a rather clumsy combina- localized at two points whose separation is space-like
tion. and greater than A, but need not vanish if it is less
The point form has the advantage that it makes a than X.
clean separation between those of the fundamental I hope to come back elsewhere to the transition to
quantities that are simple and those that are the the quantum theory.
COVARIANT PICTURE OF QUANTUM BOUND STATES 227

PHYSICAL REVIEW VOLUME 77, NUMBER 2 JANUARY 15, 1950

Quantum Theory of Non-Local Fields. Part 1. Free Fields

HmEIU YUKAWA·
Columl>ia Uni_sily, New York, New York
(Received September 27, 1949)

The possibility of a theory of non-local fields, which is free from the restriction that field quantities are
always point functions in the ordinary space, is investigated. Certain types of non-local fields, each satis-
fying a set of mutually compatible commutation relations, which can be ob~ned by extending familiar
field equations for local fields in conformity with the principle of reciprocity, are considered in detail. Thus
a scalar non-local field is obtained, which represents an assembly of particles with the mass, radius and spin 0,
provided that the field is quantized according to the procedure similar to the method of second quantization
in the usual field theory. Non-local vector and spinor fields corresponding to assemblies of particles with the
finite radius and the spins 1 and t respectively are obtained in the similar way.

I. INTRODUCTION to recent investigations by Umezawa and others' and by

I T has been generally believed for years that well-

known divergence difficulties in quantum theory of
Feldman,' no combination of quantized fields with
spins 0, t, and 1 can be free from all of the divergence
wave fields could be solved only by taking into account difficulties, as long as only positive energy states for
the finite size of the elementary particles consistently. bosons and real coupling constants for the interactions
Recent success of quantum electrodynamics, which took between fermions and bosons are taken into !!Ccount.
Nevertheless, the dilficulties remaining in quantum
advantage of the relativistic covariance to the utmost, I
electrodynamics are not so serious as those which
however, seemed to have weakened to some extent the
appear in meson theory. In the latter case, we know
necessity of introducing so-called universal length or
that straightforward calculations very often lead to
any substitute for it into field theory. In fact, all
divergent results for directly observable quantities such
infinities which had been familiar in previous formula-
as the probabilities of certain types of meson decay.'
tions of quantum electrodynamics were reduced to unob- Although the application of Pauli's regulators to meson
servable renormalization factors for the mass and the theory was found useful for obtaining finite results, it
electric charge in the newer formalism. Furthermore, in can hardly be considered as a satisfactory solution of the
order to get rid of the remaining difficulties that these problem for reasons mentioned above. It seems to the
renormalization factors were still either infinite or present anthor that, at least, a part of the defect of the
indefinite, main efforts were concentrated in the direc- present meson theory is due to the lack of a consistent
tion of introducing various kinds of auxiliary fields, method of dealing with the finite extension of the
either real or only formal, rather than in the direction elementary particle such as the nucleon, whereas the
of introducing explicitly the universal length or the effect of the finite extension is usually very small so far
finite radius of the elementary particles. So far as the as electrodynamical phenomena in the narrowest sense
results of the investigations in the former direction arc are concerned, except for its decisive effect on the
concerned, however, the prospect is not so encouraging. renormalizations of the mass and the electric charge.
"lamely, an ingenious method of regulators, which was Under these circumstances, it seems worth while to
investigated by Pauli extensively,' can be regarded as a investigate again the possibility of extension of the
formalistic generalization of the theory of mixed fields,' present field theory in the direction of introducing the
but cannot be replaced by a combination of neutral finite radius of the elementary particle. In this paper,
vector fields and charged spinor fields with different as the continuation of the preceding papers,' the pos-
masses, unless we admit the introduction of bosons with sibility of a theory of quantized non-local fields, which
negative energies and fermions with imaginary charges is free from the restriction that field quantities are
as pointed out by Feldman' More generally, according always point functions in 'the ordinary space, will be
discussed in detail. One may be very sceptical about the
necessity of such a drastic change in field theory, because
• On leave of absence from KYOlo University, Kyoto, Japan.
I As to the list of recent works by Tomonaga, Schwinger and
others, see V. Weisskopf, Rev. Mod. Phys. 21, 305 (1949). • Umezawa, Yukawa, and Yamada, Prog. Theor. Phys. 4. 25,
, W. Pauli aod F. Villars, Rev. ~fod. Phys.·21, 433 (1949). The 113 (1949). See also R. Jost and J. Rayski, Helv. Phys. Acta 22,
method of regulators is an extension of cut-off procedures by R. P. 457 (1949).
Feyoman, Phys. Rev. 74, 1430 (1948) aod by D. Rivier and E. C. • H. Fukuda, and Y. Miyamoto, Prog. Theor. Phys. 4, 235
G. Stueckelberg, Phys. Rev. 74, 218 (1948). (1949); Sasaki, Ooeda, and Ozaki, Prog. Theor. Phys. (to be pub-
) Field theories by Bopp, Podolsky, Dirac, and others are more li,hed); J. Steinberger, Phys. Rev. 76, 1180 (1949). See further a
formalistic in that negative energy bosons are taken into account, comprehensive survey of recent works on meson theory by
whereas those by Pais, Sakata, and Hara are more realistic. H. Yukawa, Rev. Mod. Phys. 21, 474 (1949) .
• D. Feldman, Phys. Rev. 76, 1369 (1949). Theauthoris indebted , A preliminary account of the content of this paper was pub-
to Dr. Feldman for discussing the subject before publication of lished in H. Yukawa, Phys. Rev. 76, 300 (1949), which will be
his paper. cited as 1.
219

Reprinted from Phys. Rev. 77, 219 (1950).

228 CHAPTER IV

220 HIDEKI YUKAWA.

other possibilities such as the introduction of local well-known commutation relations

fields corresponding to particles with spins higher than
1 are not yet fully investigated. However, present theory [X', p.J= ih6 ... (1)
where
of elementary particles with spins higher than 1 suffers
[A, BJ=AB-BA (2)
from the difficulty associated with the necessity of
of auxiliary conditions, and even if this is overcome by for any two,operators A and B. Usual local fields are
some revision of the formalism as proposed by Bhabha,' included as the particular case, in which the field
we can hardly expect a satisfactory solution of the whole operator U is a function of x' alone, so that it can be
problem, because the admixture of higher spin fields represented by a diagonal matrix in the representation,
may well give rise to newer t~ of divergence in return in which the operators x' themselves are diagonal. In
for the elimination of more/familiar ones. Moreover, it this particular case, it is customary to start from the
does not seem to the present author that the theory of second-order wave equation
non-local fields is necessarily contradictory to the
theory of mixed local fields. They can rather be com- 0'
plementary to each other in that a non-local field may ( ----K')U(X')=O, K=mc/fz (3)
well happen to be approximately equivalent to some
oX,ox'
mixture of local fields. The most essential point, which for the local field U(x'), in order that it can reproduce,
is in favor of the non-local field, is that the convergence when quantized, an assembly of identical particles with
of field theory might be guaranteed by introducing a a definite mass m and the spin o. Equation (3) is
new type of irreducible field instead of a mixture, which equivalent to the relation between the operator U and
is reducible. the operators p,
In this paper, as in the preceding papers, we confine
our attention to certain types of non-local field, each [P.[P', U]]+m'c'U=O (4)
satisfying a set of mutually compatible commutation for this case. We assume that the non-local scalar field
relations, which can be obtained by extending familiar U in question satisfies the commutation relatiCHl of the
field equations for local fields in conformity with the same form as (4). However, in our case, we need further
principle of reciprocity. The solutions of these operator the commutation relation between U and x', in contrast
equations can be interpreted as a field-theoretical repre- to the case of local field, in which U and x' are simply
sentation of assemblies of elementary particles, each commutative with each other. In order to guess the
having a definite mass and a definite radius. In this correct form for it, some heuristic idea is needed. The
connection, recent attempt by Born and Green' is principle of reciprocity seems to be very useful for this
interesting particularly in that they made use of the purpose. Namely, we assume that the commutation
principle of reciprocity as a postulate for determining relation between U and x' has a form
possible masses of elementary particles of various types.
However, it is not yet clear whether their method of (5)
density operators contains something essentially dif-
where ~ is a constant with the dimension of length and
ferent from the usual theory of mixture of local fields.
can be interpreted as the radius of the elementary
The most important question of the interaction of two
particle in question, as will be shown below. The rela-
or more non-local fields will be discussed in Part II of
tions (4) and (5) are not exactly the same in form, but
this paper.
differ from each other by plus and minus signs of the
II. AN EXAMPLE OF THE NON-LOCAL SCALAR last terms on the left-hand sides of (4) and (5). Thus,
FIELD the two operator equations (4) and (5) can be said to
In order to see what comes out by generalizing a be mutually reciprocal rather than perfectly sym-
field theory so as to include non-local fields, we start metrical, indicating that the radius of the elementary
from a particular case of the non-local scalar field. A particle ~ must be introduced as something reciprocal
scalar operator U, which is supposed to describe a to the mass m.
non-local scalar field, can be represented, in general, by Now the operator U can be represented by a matrix
a matrix with rows and columns, each characterized by (x: IU Ix.") in the representation, in which x. are
a set of values of space and time coordinates. Alter- diagonal matrices. The matrix elements, in turn, can be
natively, we can regard this operator U as a certain considered as a function U(X" r ,) of two sets of real
function of four space-time operators x' (x' = Xl == X, variables
x'=x,==y, x'=x.==z, x'= -xt==CI) as well as of four X.=t(x:+x:'), r,=x:-x:'. (6)
space-time displacement operators P., which satisfy
'H. J. Bbabha, Proc. Ind. Acad. Sci. A21, 241 (1945); Rev.
Accordingly, the relations (4), (5) can be replaced by
Mod. Pbys. 17, 200 (1945).
• M. Born, Nature 163, 207 (1949); H. S. Green, Nature 163, (iJ'/i)X.iJX'-K')U(X., r.)=O, (7)
208 (1949); M. Born and H. S. Green, Proc. Roy. Soc. Edinburgh
A92, 470 (1949). (,."-X')U(X" ,,)=0, (8)
COVARIANT PICTURE OF QUANTUM BOUND STATES 229
----------------------------------------------------------

QUANTUM THEORY OF NON-LOCAL FIELDS 221

respectively. Equations (7) and (8) are obviously com- Thus the most general form of U(X., '.), which satisfies
patible with each other and the former implies that all the relations (7), (8), and (14), is
U(X p, 'p) is, in general, a superposition of plane waves
of the form expik.Xp with k. satisfying the condition
(9)
U(X., '.)= J... f (dk)'u(k., ,.)o(k.k·+,')

Xo(,.,·->,,')o(k.,·) exp(ik.X·), (16)

whereas the latter implies that U(X., ,,) can be dif-
ferent from zero only for those values of '.. which where u(k., '.) is again an arbitrary function of k. and
satisfy the condition ,
p.
10

(10) Now a simple physical interpretation can be given to

the non-local field of the form (16) by considering the
Thus the most general solution of the simultaneous corresponding particle picture: Suppose that the par-
Eqs. (7) and (8) has the form ticle is at rest with respect to a certain reference system.

f··· J(dk')u(k., '.)0(-)'."->"')

In this particular case, the motion of the particle as a
U(X., r.)= whole, or the motion of its center of mass, can be repre-
sented presumably by a plane wave in X-space with the
X o(k.k·+ .') exp(ik.X·), (11) wave vector kl=k,=k.=O, k,= -K. The corresponding
form of U(X., ,,) is, apart from the factor independent
where u(k.".) is an arbitrary function of two sets of
variables k. and, p' of X"' '",
The above considerations suggest us that one set X. u(O, 0, 0, -,;,,)o(,.,·-X')o(.,,) exp(-i.X') (17)
of the real variables could be identified with the con-
which is different from zero only for those values of r.,
ventional space and time coordinates of the elementary
which satisfy the conditions
particle regarded as a material point in the .limit of
),-+0, whereas the other set '. could be interpreted as (18)
variables describing the internal motion in general case,
in which the finite extension of the elementary particle Thus, the form of U(X., r.) in this case is determined
in question could not be ignored. Thus, we might expect completely by giving u(O, 0, 0, -.; , p) as defined on the
that the field U of the above type is equivalent to an surface of the sphere with the radius>.. in ,-space. In
assembly of elementary particles with the mass m, the other words, the internal motion can be described by
radius). and the spin 0, if it is further quantized ac- the wave function u(8, '1') depending only on the polar
cording to the familiar method of second quantization. angles 8, '1', which are defined by
However, we can easily anticipate that the equivalence ',=r sinO COS'l', ',=, sinO sin '1', ,,=, cosO. (19)
is incomplete, because U(X., ,,) is different from zero
for arbitrary large values of '., so far as. they satisfy the In general, u(O, '1') can be expanded into series of
condition (10), even when only one term of the right- spherical harmonics:
hand side of (11) corresponding to a definite set of u(O, '1')= L: e(O, 0, 0, - K; I, m)p,m(o, cp), (20)
values of k. is taken into account. In other words, we I,m
need another condition for restricting the possible form
which is equivalent to decomposing the internal rotation
of U(X., r.) or u(k., '.) in order to complete the
into various states characterized by the azimuthal
equivalence ahove mentioned. For this purpose, we
quantum number I and the magnetic quantum number
introduce an auxil ary condition
m.
(12) In the case when the center of mass of' the particle is
moving with the velocity' v., v., v., it can be described
which can be said to be self-reciprocal in that the
by a plane wave in X-space with the wave vector k.,
relation which is connected with the velocit~ by the relations
. [x·[p., U]]=O (13)
V,= -klc/k" v.= -k,c/k" V,= -k,c/k"
can be deduced from (12) immediately on account of
k,= - (k,'+k,'+k.'+K')I. (21)
the commutation relation (1). Both of (12) and (13) are
equivalent to the condition In this case, U(X., ,,) has the form

au(x., '.) u(k., '.)0('."- >,,')o(k.,') exp(ik.X·), (22)

'. 0 (14)
which is different from zero only on the surface of the
spbere with the radius X in ,-space, the sphere itself
[or U(X., '.), or the restriction that u(k., ,,) should be 10 U(X", r,,) as given by the expression (6) in I was not the most
'ero unless k. and,. satisfy the condition general form in that the coefficients b(kl') were independent of I",
which corresponded to ignore the internal rotation. The author is
(15) indebted to Professor R. Serber for calling attention to this point.
230 CHAPTER IV

222 HIDEKI YUKAWA

moving with the velocity v., v" v•. Accordingly, we perform /irst tha Lorentz transformation
x,.' =a,.,x, (23)
with the transformation matrix

k1k,/K' k1k./K'
H(kJK)' k,kJK'
(24)
k,k./K' 1+ (k./l/.)'
k,/K k,/.
where K = (.(.- k,))I. Then the wave function for the m. QUAN'rIZATION OF NON-LOCAL SCALAR
internal motion can be described by a function u'(8', ",') FIELD
of the polar angle 8', ",' defined by In order to show that the non-local field above con-
,
11,=01"r,.=" , SIn
. 8',COStp," T2,=a2rT,=r , SIn 'f (25)
. 8" Stnly?, sidered represents exactly the assembly of identical
particles with the finite radius, we have to quantize the
'3 =aa,t'.. =r cosO, '. =atr1".=k,.r P/K.
field on the same lines as the method of second quan-
Incidentally, '.' as defined by the last expression in (25) tization in ordinary field theory. For this purpose, it is
is nothing but the proper time multiplied by - e for the convenient to write (16) in another form
particle, which is moving with the velocity Vx, v., v,.
Again, u'(8', ",') can be expanded into series of spherical
harmonics: U(X .. ")= f··· f (dk)'(dl)'u(k., I.)
u'{8', ",')= l: e(k .. I, m)pt M{8', ",'). (26)
~M
x 6(k.k·+ ")0(1,)'- A')o(ki')
Since the above arguments are in conformity with the X exp(ik.X·)ll. 6(,.+1.), (29)
principle of relativity perfectly, the non-local field in
question can be regarded as a field-theoretical represen- where I. is a four vector. The integrand is different from
tation of a system of identical particles, each with the zero only for those values of k., I.. which satisfy the
mass m, the radius and the spin 0, which can rotate as relations
the relativistic rigid sphere without any change in k.k'+<'=O, I,I'-A'=O, k,I'=O. (30)
shape other than the Lorentz contraction associated
Accordingly, the matrix elements for the operator U are
with the change of the proper time axis.
The non-local field U given by (16) reduces to the
ordinary local scalar field in the limit A-->O, as it should (x.' 1 U 1x.") = f··· f (dk)'(dJ)'u(k.. I,)
be, provided that the rest mass m is different from zero.
Namely, (x.' 1Ulx.") is different from zero only for X 6(k,k'+ .')6(1.1'- A') exp(ik·x.' /2)
x.' = x.", because the only possible solution of the
simultaneous Eqs. (9), (11), and (15) with m#O and xll. o(x.'-x."+I,) exp(ik'x."/2), (31)
A=O is '1=,,=,,=,,=0. On the contrary, the case of
the zero rest mass m = 0 is exceptional in that the non- which is equivalent to the relation
local field U does not necessarily reduce to the local field
in the limit A=O. This is because the simultaneous
Eqs. (9), (11), and (15) with m=O and A=O have
U= f· .. f (dk)'(dJ)'u(k.,I,) exp(ik,x'/2)
solutions of the form
Xexp(u'p./Ii) exp(ik.x·/2), (32)
,.=±(A')'k.. k,=±(kl'+k,'+k.')I, (27)
between the operators x', P. and U, where
where A' is an arbitrary constant with the dimension of
length. More generally, the simultaneous equations u(k.,I.)=u(k .. I.)6(k.k"+.')6(l.Z·- A')6(k.I"). (33)
with m=O and A#O has the general solution of the form
As the operators k.x" and I"P. in the same term on the
,.=,/±(A')'k.. k,=±(k.'+k,'+ka')I, (28) right-hand side of (32) are commutative with each other
on account of the relations (I) and (30), (32) can also
where,.' is any particular solution of the same equa- be written in the form
. tions. Thus the radius of the particle without the rest
mass cannot be defined so naturally as in the case of the
particle with the rest mass, corresponding to the cir- U= f· .. f (dk)'(dJ)'u(k.,I.) exp(ik.x·)
cumstance that there is no rest system in the former Xexp(u·p./h). (32')
case. Detailed discussions of this particular case will be
made elsewhere. Similarly the operator U·, which is the Hermitian
COVARIANT PICTURE OF QUANTUM BOUND STATES 231

QUANTUM THEORY OF NON-LOCAL FIELDS 223

conjugate of U, can be written in the form where

U·= J...J Cdk·)Cdl)·u·Ck.,I.)

uCk, El, <I»=uCkl, k" k" - CP+.')I; I.),
v*Ck, El, <I»=uC -kl, -k" -k., Ck'+.')I; -I.),
XexpC -ik,.x') expC -it'p./A). (34)
UCk, El, <I»=expCikx+iCk'+.')lx.)
Now the method of second quantization can be (41)
applied to our case in the following way: uCk., I.) and X expC,)J·p.1A),
u·Ck., I.) in Eqs. C32') and (34) are regarded as opera-
tors, which are Hermitian conjugate to each other and U·Ck, El, <I»=expC-ikx-iCk'+.')lx.)
are non-commutative in general. The fact that the
X expC - iAi'p./ 11).
operators defined by
UCk., 1.)=expCik.x') expCil'p.lh); Finally, by expanding u and v* into series of spherical
(35) harmonics, we obtain
U·Ck., I.) = expC -ik,x') expC -u'p.lh)
are unitary, i.e., satisfy the relation
UCk., 1.)u·Ck., I.) = U*Ck.. I.)UCk .. I.) = 1 (36) U= L L -
.,•••• '...
(2,,)'
L
A luCk,l,m)
4.Ck'+.')1
suggests us the commutation relations
X UCk, I, m)+v·Ck,l, m)U*Ck, 1, m) l. (42)
[uCk., I.), u*Ck/, I.') J where

= -Ik.T
k.
II• oCk.-k.')oCI.-I.')· 6Ck.k·+K') uCk, I, m)= JJ uCk, El, <1»

_ _" xo(l.I'-A')oCk.I·), (37) XP,MC8, <P) sinEldEld<l>,

[uCk.,I.), uCk., I. )J=O, (43)

[u*Ck.,I.), u*Ck.', I.')J=O, v·Ck, 1, m)= JJ v*Ck, El, <1»

which are obviously invariant with respect to the xPrCEl, 4» sinEldEld<l>,
whole group of Lorentz transformations. In order to
make the physical meaning of the relations (37) clear,
we suprse the field in a cube with the edges of the UCk, I, m)= JJ UCk, e, <1» II
length ,which is very large compared with A. Then the XPrC8, 4» sinEl d8d<l>
(44)

JJ
effects of non-Iocalizability of the field are negligible,
because they are confined to small regions very near
the surface of the cube." In this case, the integrations
U'Ck, I, m)= U*Ck, El, <1»

with respect to k. on the right-hand side of Eqs. C32') xp,mce, <1» sin8dEld<l>
and (34) are replaced by the summations with respect
to k.. which take the values assuming that the spherical harmonics p,"Ce, <1» and
their complex conjugate p,mC8, <P) are normalized
k,=(h/L)n" k,=(h/L)n" k,=C2-tr/L)n" according to the rule:

fJ
k.=±Ck,'+k,'+k,'+,,)I, (38)
where nt, nt, ns are integers, either positive or negative, p,mCEl, <I»P,"CEl, <1» sinedEld4>m= 1. (45)
including zero. The integrations with respect to I. with
fixed k. are replaced by those with respect to z.' defined Similarly, U* is transformed into the form
by
(39)
U*= I: L(~)' A IvCk, I, m)
where the coefficients a" are given by (24). Further, ., •••• 'M L 4.Ck'+ .') I
w~ intr:m~ce ~h.e polar ~ngl~ El, <1>, which are conn~cte?
With I, , I, ,I, Just as 8 , 'P are connected With T, , T, , X U(k, I, mJ+u*(k,l, m)U*Ck,l, m) I, (46)
r,' by the relations (25). Thus we obtain where

U= L
.,.,',
JJ( -2~)' A sin8dEld<l>
L 4KCk'+K')1 u'Ck, I, m)= JJ u*Ck, e, <I»P,M(El, <1»
X sin EldEld<l>,
X luCk, e, <I»UCk, El, <1» (47)
+v*Ck, El, <I»U*Ck, ~, <1»
"More precisely, L must be large compared withV(I-I!')I,
I, C4O) vCk, I, m)= JJ vCk, e, <I»P,M(El, <1»
where (3c is the maximum velocity of particles in consideration. Xsin8dEld<l>.
232 CHAPTER IV

224 HIDEKI YUKAWA

By the same transformation, we obtain from Eq. (37) instead of Eq. (37), where
the commutation relations
[A, BJ+=AB+BA (53)
[ark, I, m), a*[k', 1', m')J=o(k, k')W, l')o(m, m')'}
[b(k, I, m), bOCk', 1', m')J=o(k, k')o(/, norm, m'), (48) for any two operators A and B. However, in this case,
we arrive at the well-known contradiction in the limit
[ark, I, m), b(k', I', m')J=O, etc.
of A-+O, which prohibits the elementary particles with
for the operators defined by °
spin from obeying Fermi statistics.

2... ),
ark, I, m)= ( ( -
A )1 ·f/(k, I, m),
IV. NON-LOCAL SPINOR FIELD

L 4<(k'+ <') 1 The above considerations can easily be extended to

the non-local vector field without introducing anything
a*(k, I, m)= (( -lr)' 4K(k'+A )1 ·u*(k, m),
L <') I
I,
essentially new which needs detailed discussions. On the
contrary, the case of the non-local spinor field must be
(49) investigated from the beginning. We start from the

b(k, I, m)= (( -lr)' A )1·v(k, I, m), spinor operator", with four components, which trans-
form as the components of Dirac wave function. Each
L 4«k'+K')1 of these components can be considered as a non-local
operator just like the operator U in the case of the
b*(k,l,m)=((~)' A )1.V*(k,l,m). scalar field. As an extension of Dirac's wave equations
L 4«k'+ <')1 for the local spinor field, we assume the relations
between the operators x', p, and",:
Hence, each of the operators defined by
n+(k, I, m)=a*(k, I, m)a(k, I, m);
(50)
r'[P .. "'J+ ='"= 0, (54)

n-(k, I, m)=b*(k, I, m)b(k, I, m) fl,[x', "'J+Af=O, (55)

h:t.il eigenvalues 0, 1, 2, ... and can be interpreted as where r' are well-known Dirac matrices forming a four
the number of particles in the state characterized by vector, which satisfy the commutation relations among
the quantum numbers k, I, m with either positive or themselves:
negative charge. Thus the non-local field above con- (56)
sidered corresponds to the assembly of charged bosons We assume similar commutation relations for matrices
with the mass m, the radius A and the spin 0. It can
fl,:
easily be shown that in the limit A-->O, U reduces to the
(57)
familiar quantized local field for bosons apart from the
extra factor Then, we obtain by iteration the relations
h(Xl' -x,")o(x' - x.')o(xa' - x,")h(x.' - x/') (51) [r[P .. "'JJ+m'c'o/t=O, (58)
which must be omitted, whenever we go over from
[x.[x', "'JJ-}.''''=O, (59)
non-local to local field.
The non-local neutral field can be obtained, if we which have the same form as the relations (4) and (5)
assume that the field operator U is Hermitian, i.e., for the scalar field. However, the matrices fl. have to be
U = U*. In this case, we cannot discriminate between so chosen as to satisfy the demand that the relations
u and v, or a and b, so that we have instead of Eqs. (42) (54) and (55) are compatible with each other. Namely,
and (46) the relation from the relations
2.. ), A fl,r'[x'[P" '"JJ= }.mcy" (60)
U= E E( - 1f/(k,l,m)
., •••• ~ .. L 4«k'+<')1 r'fl.[P,[x', fJJ=}.mcy" (61)

X U(k, I, m)+u*(k, I, m)U*(k, I, m) I. (52) which can be readily obtained by considering Eqs. (54)

I
and (55), must have the same form, so that fl. must
It should be noticed, further, that we could start satisfy an additional condition:
from the commutation relations (62)
[u(k .. I.), u*(k:, I:)J+ I This condition reduces to the form
= II. o(k,-k:)o(/,-I:)· o(k,k'+<') (63)
X 0(1.1'- A')o(k,I'), (37) which is the same as the condition (12) or (13) for the
scalar field, if fl, are so chosen as to satisfy the com-
ruCk"~ I,), u(k:, I:)J+=O, mutation relations
[u*(k., I.),u*(k:, I:)J+=O. (64)
COY ARIANT PICTURE OF QUANTUM BOUND STATES 233

QUANTUM THEORY OF NON-LOCAL FIELDS 225

where C is a matrix with the determinant different where S is a matrix with four rows and four columns."
from zero. Equation (64) can be satisfied by matrices In our case, in which the spinor y, has eight components,
1", fl, which are expressed in the form we assume the same form for S in Eq. (69) except that
the numbers of rows and columns are doubled, when
)'l=ip'Pll -y2=iP'P2) -y3=ip1fTJ 1 Y'=PJ) (65)
Eq. (70) is a proper Lorentz transformation with the
determinant + 1, whereas we have to replace Eq. (69) by
fJl=P3CT l, f32=P3(f21 i33 = P3(f3! f3.=-ip'l! (66)
in terms of sets of mutually independent Pauli matrices
>/I'=w,S>/I, (71)
CThCT2, CT, and Pl, P2, p,. It is well known that the matrices
as given by (66) do not form an ordinary vector, but a when Eq. (70) is an improper Lorentz transformation
pseudovector. Thus, if we confine our attention to the with the determinant -1. This guarantees the invari-
proper Lorentz transformation, the relations (54) and ance of the relation (68) with respect to improper as
(55) are both invariant. However, if we perform the well as proper Lorentz transformations.
improper Lorentz transformation, for which the deter- However, the above procedure is unsatisfactory, par-
minant of the transformation matrix has the value -1 ticularly because it is difficult to give:a simple physical
instead of + 1, the form of the relation (55) changes into meaning to the new degree of freedom. As will be shown
in the additional remark at the end'of this paper, there
(67)
is an alternative way, in which we have no need to
whereas the relation (54) is invariant. In other words, increase the number:of components of y, from 4 to 8.
the fundamental equations for the non-local spinor Now, each component >/I; (i= 1,2,3,4) of the spinor
field, which has similar properties as the non-local scalar >/I can be represented as a matrix (x:l>/Idx:') in the
field considered in the preceding sections, can be con- representation, in which x, are diagonal. (x: Iy,d x:')
structed so as to be invariant with respect to the whole can be regarded, in turn, as a function >/I,(X., r,) of
group of Lorentz transformations including reflections, X., r., where X., r, are defined by Eq. (6). Therewith
only if both forms (55) and (67) are put together into the relations (Sol) and (68) can be represented by
one relation for one spinor field with the components
1"(a>/l(X., r,)/aX'l+i<>/I(X., r,)=O, (72)
twice as many as the four components for the usual
spinor field. This is equivalent to introduce one morc
fl,r'y,(X., r,)+ Xy,(X., r ,)= 0, (73)
independent set of Pauli matrices WI, W2, and to w, respectively, where y,(.J(" r ,) is a spinor with four com-
assume that all of the matrices 'I', fl, have each eight ponents 1';(X., r,l (i= 1,2,3, ol). The simultaneous
rows and columns characterized by eight combinations Eqs. (72), (73) for >/I(X., r,) have a particular solution
of eigenvalues of (T" P3, w,. Therewith the spinor must of the form
have eight components, tirst four components and the
remaining four corresponding respectively to the eigen-
values + 1 and - 1 of W3. where u(k., r,) is a spinor with four components satis-
In order to eStablish the invariance of fundamental fying
laws for the non-local spinor tield with respect to the
whole group of Lorentz transformations, we assume 1"k,lI+<u=O, fl,r'u+ AIL = (). (75)
further that w, and W3 change sign under improper It follows immediately from (75) that ,t must satisfy
Lorentz transformation, whereas Wl does not. ,,,re call
no\\' adopt the relation (k,k'+K')U=O, (r,r'-h')u=tJ, k,r'I'=O (76)
(68) so that " can be written in the form
in place of Eq. (55). It is clear from the above arguments U= u(k .. r ,)o(k,k'+ <')o(r ,r'- h')o(k,r'). (ii)
that the fundamental Eqs. (Sol) and (68) are invariant Each of four components of " can be expanded in the
with respect to the whole group of Lorentz transforma- same way as the scalar operator u in the preceding
tions. However, for the purpose of proving it more sections. The second quantization can be performed by
explicitly, we consider the transformation properties of assuming commutation relations of the type (37)
>/I with respect to the Lorentz transformation, whereby between field quantities, so that the non-local field
we assume that the matrices "f'" fJ,. have prescribed represents an assembly of fermions with the mass til,
forms as defined by Eqs. (65), (66) independent of the the radius X and the spin ~. Further analysis of the
coordinate system. In the usual theory, in which the non-local spinor field will be made in Part II of this
spin or field y, has four components, we have the linear paper. At any rate it is now clear that there exist non-
transforma tion local scalar, vector, and spinor fields, each corresponding
(6(») to the assembly of particles with the mass, radius, and
associated with each of the Lorentz transformations for
the spin 0, \, and ~.
the coordinates: J2 Sec, for example, \\'. Pauli, lla'ldbll{}, deT Physik 24, Part J,
(iO) 8J (\933).
234 CHAPTER IV

226 HIDEKI YUKAWA

Now the question, with which we are met first, when Tbis work was done during the author's stay at The
we go over to the case of two or more non-local fields Institute for Advanced Study, Princeton. The author is
interacting with each other, is whether we can start grateful to Professor J. R. Oppenheimer for giving him
from Schrodinger equation for the total system (or any the opportunity of staying there and also for stimulating
substitute for it), thus retaining the most essential discussions. He is also indebted to Dr. A. Pais and
feature of quantum mechanics. We know that Schro- Professor G. Uhlenbeck for fruitful conversation.
dinger equation in its simplest form is not obviously
relativistic in that it is a differential equation with the ADDITIONAL REMARKS ON NON-LOCAL
SPINOR FIELD
time variable as independent variable, space coordinates
being regarded merely as parameters. It can be extended The problem of invariance of the relation (55) with
to a relativistic form as in Dirac's many-time formalism respect to improper Lorentz transformation can be
or, more satisfactorily, in Tomonaga-Schwinger's super- solved without introducing extra components to the
many-time formalism, as long as we are dealing with spinor field. Namely, we take advantage of the anti-
local fields satisfying the infinitesimal commutation symmetric tensor of the fourth rank with the com-
relations. However, if we introduce the non-local fields ponents <,.,., which are + 1 or -1 according as
or the non-localizability in the interaction between (K, A,}J., v) are even or odd permutations of (1, 2, 3, 4)
local fields, the clean-cut distinction between space-like and 0 otherwise. Further we take into account the
and time-like directions is impossible in general. This relations
is because the interaction term in the Lagrangian or
(78)
Hamiltonian for the system of non-local fields contains
the displacement operators in the time-like directions where (K, A, lA, v) are even permutations of (1,2,3,4).
as well as those in the space-like directions. Thus, even Then (55) can be written in the form
if there exists an equation of SchrOdinger type, it cannot
be solved, in general, by giving the initial condition at i L <".-1"1"1"[1", ¢'J+iX¢,=O, (i9)
a:).I'~
a certain time in the past. Under the3e circumstances,
we must have recourse to more general formalism such which is obviously invariant with respect to the whole
as the S-matrix scheme, which was proposed by Heisen- group of Lorentz transformation. The invariance can be
berg." In other words, we had better start from the proved more explicitly by associa ting a linear trans-
integral formalism rather than the differential for- formation
malism. In locallield theory, the integral formalism such (80)
as that, which was developed by Feynman, can be
deduced from the ordinary differential formalism. H. " with each of the Lorentz transformation (70), where S
In non-local Iield theory, however, it may well happen is a matrix with four rows and columns satisfying the
that we are left only with some kind of in tegral for· relations
malism. In fact it will be shown in Part II that the non-
(81)
local fields above considered can be Iitted into the
S-matrix scheme. It should be noticed, however, that the relation (79)
is a unification of the relations (55) and (67) rather than
"W. Heisenberg, Zeits. f. Physik 120, 513, 673 (1943); Zeits. f. the simple reproduction of (55), because (79) must be
~aturforsch. 1, 608 (1946); C. M~ller, Kg!. Danske Vid. Sels. Math.
Fys. Medd. 23, Nr. 1 (1945); 22, Nr. 19 (1946) • identified witb (67) in the coordinate system, which is
.. R. P. Feynman, Phys. Rev. 76, 749, un (1949). connected with the original coordinate system by an
"F. J. Dyson, Phys. Rev. 75,486, 1736 (1949). See also man)' improper Lorent7. transformation with the determinant
papers by E. C. G. Stueckelberg. which appeared mainly in Relv.
Phys. Acta. -1.
COVARIANT PICTURE OF QUANTUM BOUND STATES 235

PHYSICAL REVIEW VOLUME 80. NUMBER 6 DECEMBER IS, 10.50

Quantum Theory of Non-Local Fields. Part II. Irreducible Fields and their Interaction·
HmEJ[1 ¥uv.wAf
Columbia U1li~sily. New York. New York
(Received August 7. 1950)

General properties of non-local OpentoB are considered in connection with the problem of invariance
with respect to the group of inhomogeneous Lorentz transformations. It is shown that irreducible fields
can be classi6ed by the eigenvalues of four invariant quantities. Three of these quantities can be interpreted.
respectively. as the mass. radius. and magnitude of the internal angular momentum of the particles ass0-
ciated with the quantized non-local field in question. Further. space-time displacement operatoB arc
introduced as a particular kind of non-local operator. As a tentative method of dealing with the interaction
of non-local fields, an invariant matrix is defined by the space-time integral of a certain invariant operator,
which is a sum of products of non-local field operatoB and displacement operatoB. It is shown that the
matrix thus constructed satisfies the requirements that it be unitary and invariant and that the matrix
elements arc different from zero only if the initial and final .tates had the same energy and momentum.
However. the remaining conditions of correspondence and convergence cannot be ful6lled simultaneously.
in general, by the S-matrix for the non-local fields. It i. yet to be investigated whether all of these require-
ments are satisfied by an appropriate change in the definition of the S-matrix.

I. ELEMENTARY NON-LOCAL SYSTEMS Lorentz transformation,

T HE notion of an elementary particle has been
intimately connected with the procedure of
x"'=a".x,, (4)

decomposing a quantized field into its irreducible parts. where x/ (1'= 1,2,3,4) denote this time the space-time
Accordingly, if the concept of the field itself is so operators in the new coordinate system. Therewith,
extended as to include the non-local field, the definition two sets of parameters, X and " are transformed into
of the elementary particle will be altered in its turn. X,.'=a",x .., r,.'c::a"", (5)
In Part I, t we confined our attention to certain types
of non-local fields which satisfied a set of operator and U(X, ,) becomes
equations and were supposed to represent assemblies of
elementary particles with finite radii. Our problem is
now to decompose more general non-local fields into
U(X'. r')= f··· f u'(k', I') exp(ik:X")
irreducible parts. Again we start from an arbitrary XII6(,.'-I.') (dk.')'(dl.')', (6)
unquantized non-local scalar field U, which can be
represented by an arbitrary matrix (x'l U I x"), where where u'(k', I')=u(k, f). k', I' are connected with k, I
x' and x" stand for x-' and x." (1'= 1, 2, 3, 4), respec- just as X', r' are connected with X, ,. In order that
tively. The matrix (x'l Ulx") can be regarded as a Eq. (6) retain the same form as Eq. (3) for an arbitrary
function U(X, ,) of two sets of real variables. Lorentz transformation (4), either one of the following
two requirements must be satisfied:
X.=i(x.'+x."), ,.=x.'-x." (1)
(i) u(k. f) is a function of k and I, which retains its
as in Part I. Then an arbitrary function U(X,') can form under an arbitrary Lorentz transformation;
be expanded in the form (ii) u(k, f) is not a mere function of k and I, but is an
ensemble of quantities, which are distinguished by the
U(X,') = f··· f u(k,') exp(ik,.x·)(dk.)' (2)
parameters k and I and which are to be subject to
second quantization.

and further in the form In the first case, it is required that

f _.. f
u(k', I') = u(k, f) (7)
U(X, ,)= u(k, f) exp(ik,.x·) for an arbitrary transformation
XII6(r.-I.) (dk.)'(dl,)', (3) (8)
where u(k,') and u(k, f) are arbitrary functions of SO that u(k, l) must be the function of invariant quan-
parameters k, ' and k, I, respectively. tities such as k,k', 1,.1> and k,.l· alone. In many cases,
Now, if we perform an arbitrary homogeneous however, we can confine our attention to the suhgroup
____ of the homogeneous Lorentz group which does not
• Publication assisted by the Ernest ~em~ton Adams Fund. include the reversal of the time, so that u(k, l) may
t On leave of absence from Kyoto Uruve.... ty. Kyoto. Japan. depend also on kJI k I provided that k is a time-like
I H. Yukawa, Phys. Rev. 77. 219 (1950). See also B. Kwal. .. " •
J. de phys. et rod. 11. 213 (195Oi. vector, and similarly for l •. Thus U(X,') can be
1047

Reprinted from Phys. Rev. 80.1047 (1950).

236 CHAPTER IV

1048 HIDEKI YUKAWA

written, in general, in the form corresponds to that which is extended to time-like

directions.
U(X, r)= f··· f w(K, L, M)8(k.k>-K)8(l,.l·-L) It should be noticed, however, that the field char-
acterized by a set of values of K, L, and M can be
, decompo~d further into irreducible parts, each of which
X8(k,.l>-M) exp(ik.X·) TI Ii(r.-I.) correspond<; to a definite value for the absolute magni-
tude of th~ internal angular momentum. Namely, as
X (dk.)'(dl.)'dKdLdM, (9) shown in Part I, the non-local scalar field U with a
where w(K, L, M) is an arbitrary function of the real given set of values K, L, M can be expanded in the form
parameters K, L, and }M, which can be positive, U= L L (2 .../L)'[>.!4K(k'+«,)IJ
negative, or zero. If we restrict the transformations to .l:14:s.l:al._
those belonging to the subgroup mentioned above, W X [u(k, I, m)U(k, I, m)+v*(k, I, m)U*(k, I, m)J, (12)
may depend also on kJlk,1 (and on /JjI,lJ. In any
case, the operator U of this type has nothing to do with provided that K is negative, L is positive, and M is
the quantized non-local field, because there is no room zero, where u, v', U, U' are defined by the expressions
for the application of the method of second quantiza- (43) and (44) in Part I. The parameter I in (12) is the
tion. However, an important family of space-time quantum number which characterizes the magnitude of
displacement operators belongs to this category as will the internal angular momentum in the coordinate sys-
be shown later on. tem moving with the particle with a given wave vector
The case (ii) is the more important, because the field k" k" k•. Since I thus defined is invariant with respect
operator U can be quantized as in the usual theory. to Lorentz transformations, each part of U with a
Namely, the coefficients u(k, /) can be regarded as an definite value of I transforms into itself and constitutes
ensemble of creation and annihilation operators for the an irreducible representation of the non-local scalar
qlJllnta associated with the field U. The requirement field. Thus, it is possible that the elementary particles
of invariance is fulfilled simply by identifying u(k, I) with the integer spins are classified by the value of four
[=u(ll, I')J with the creation or annihilation operator constants K, L, M, and I, provided that they are
for a particle in the quantum state characterized by k represented by irreducible representations of the non-
and I according as k, is positive or negative. The only local scalar field. Among these four constants, - K, L,
effect of a transformation of the type (4) or (8) is to and I can be interpreted, apart from the numerical
give a new notation u'(k', I') to the operator u(k, /), factors depending on hand c alone, as the mass, radius,
owing to the change in name for the same quantum and magnitude of the internal angular momentum of
states caused by the change of the reference system. the particle, whereas M has no immediate physical
Thus, there is the one-to-one correspondence between meaning.'
u(k, l) and u'(k', I') in two representations, (3) and (6), As was pointed out recently by Fierz,' each of these
of the same operator U. Since k.k·, 1.1', and k.I' are irreducible representations of the non-local scalar field
invariant with respect to any Lorentz transformation, finds its counterpart in the usual field theory of ele-
the one-to-one correspondence remains, even if the mentary particles with arbitrary integer spin, so far as
domain of integrations on the right-hand side of Eq. (3) the behavior with respect to Lorentz transformations is
is restricted to definite values K, L, M of these' in- concerned. The essential difference between local and
variant quantities k.k', 1.1', and k.I·, respectively. In non-local fields will be clear only when the interaction
such a case, U(X,.r) reduces to between fields is taken into account. In the case of the

f··· Ju(k, I) exp(ik.x-)l!Ii(r.-I.)

non-local spinor field, however, the situation is some-
U(X, r)= what different. Namely, a non-local spinor operator
"', (i= 1, 2, 3, 4) is equivalent to a set of four functions
X o(k.k·- K)8(IJ'- L)Ii(kJ'- M)(dk,)'(dl.)'. (10) ",,(X, r), which can be expanded in the form

It is now clear that the scalar non-local field, which

was dealt with in detail in Part I, is a particular
",,(X, r)= f··· f u,(k.,I.) exp(ik.x·)
example with XTI8(r.-I.)(dk.)'(dl')'. (13)
K=- ..., L=+>.', M=O. (11)
This can be decomposed into parts in an invariant
More generally, Land M can be either positive or manner by giving each of k.k', 1.1', k.l· a definite
negative including zero, but K can only be negative or value. Each part can further be regarded as a sum ~f
zero, because a positive K has no correspondence with operators, which differ from one another by thell"
the classical model of particles with real mass. Positive behaviors with respect to space rotations in the rest
values of L correspond to the assembly of elementary
D. Yennie, Phys. Rev., followin~ paper.
particles with a finite dimension which is extended to 1
• M. Fierz, Phys. Rev. 78, 184 (1950); Relv. Ph}". Acta 23,
space-like directions, whereas negative values of L 412 (1950).
COVARIANT PICTURE OF QUANTUM BOUND STATES 237

QUANTUM THEORY OF NON-LOCAL FIELDS 1049

system. Each of these ope !'ators thus obtained is not tation relations for u(k, I) and u*Ck, f) as given by
yet irreducible in general, because it is a mixture of Eq. (37) of Part I, for example. Of course, it must
two types of fields belonf;1ng to the same resultant always be kept in mind that the time-reversal is
(half-integral) spin. For instance, the operator corre- associated with the interchange of the annihilation
sponding to the resultant spin 1/2 may have an internal operator u(k, /) and the creation operator u*(k, l).
orbital angular momentum of eithet zero or unity. In These arguments can be applied to non-local spinor
the usual local field theory, however, a spinor field with fields without essential change. In this way we amve
the spin 1/2, for example, is already irreducible. Thus at the following suggestion: according to the non-local
the difference between the non-local spinor field and field theory it is possible that there are only two kinds
the local spinor fields with arbitrary half-integral spins of elementary particles, Bose-Einstein particles and
is apparent without taking into account the interaction Fermi-Dirac particles, which are described by a scalar
between the fields.' field and a spinor field, respectively. The customary
So far we have considered the problem of invariance discrimination of particles with spins 0, 1, 2, etc.,
of non-local operators with respect to homogeneous among Bose-Einstein particles, for instance, may well
Lorentz transformations. Now we go over to the more be reduced to the difference in the quantum number 1
general inhomogeneous Lorentz transformation of the for the internal motion of the same kind of particles.
type
x.' =a.,(x,+b,) (14) n. S-MATRIX IN NON-LOCAL FIELD THEORY'
or Now we must undertake the problem of interaction
(15) between non-local fields. In the usual field theory we
with b: = a.,b" X and r are transformed thereby into could always start from the SchrOdinger equation for
the total system. The Hamiltonian in the Schrooinger
X.'=a.,(X,+b,), r.'=a.,r,. (16) equation is derived from the Lagrangian which, in turn,
Accordingly, we have is so chosen as to give the correct field equations for
unquantized fields, when the classical variation principle
(17) was applied to the system consisting of unquantized
and fields. In the non-local field theory, however, it is
u'(k', 1')= exp( -ik.b·)u(k, I), (18) difficult to follow the same procedure as in local field
in order that U be invariant with respect to the trans- theories for two reasons. Firstly, even in the case of
formation (14). The implication of the relation (18) the free field, it is difficult to deduce all of the field
must be considered for the cases (i) and (ii) separately. equations, (4), (5), and (12), for example, for the scalar
In case (i), relation (18) is compatible with the non-local field from an invariant operator which is
assumption that u(k, I) is an invariant function of k supposed to correspond to the Lagrangian in the usual
and I, only if uCk, I) is zero for all values of k. except theory. Moreover, the procedure of variation itself is
k.=O Cp.= 1,2,3,4). This is equivalent to the following ambiguous.' Secondly, it is rather dubious whether the
statement: differentiation of the SchrOdinger function with respect
to time will play an important role in non-local field
(i)' A non-local operator U whiCh satisfies require- theory because other operators, in general, are related
ment Cil is invariant with respect to the whole group to two time instants, which differ from each other by a
of inhomogeneous Lorentz transformations only if finite amount. Even the existence of the SchrOdinger
U(X, r) is an invariant function of r alone. function in the same sense as in the local field theory is
It will be shown in the next section that some of the not at all certain.
invariant operators satisfying the requirement (iJ' will Although it is not yet clear whether these difficulties
be of importance in constructing the S-matrix for the could be overcome without renouncing the fundamental
interacting non-local fields. principles of quantum mechanics, there seems to exist
In case (ii), relation (18) reflects the situation that a tentative solution which retains many of the char-
the creation or annihilation operator u(k, I) or u*(k, I) acteristics of the present field theory. Namely, we can
is defined unambiguously except for an arbitl-ary phase start from the so-called interaction representation in
factor. In spite of this ambiguity or complication, the the usual theory, laying aside for the moment the
question of whether the free field equations in non-local
operator u*(k, I)u(k, 1)/1 k,l, which is to be identified
with the occupation operator for particles in the field theory can be deduced from the Lagrangian
quantum state characterized by k. and I. apart from formalism or not. Furthermore, we can adopt the
the purely numerical factor, is defioed uniquely and is integral formalism of the usual theory, which has been
invariant with respect to the whole group of inhomo-
• A preliminary account of the subject was published by H.
geneous Lorentz transformations. So are the commu- Yukawa, Phys. Rev. 77, 849 (1950).
• Variation principles in the non·local field theory were di!!CUssed
• Detailed di5cussioll!l of non-local spinor field will be made by C. Bloch, Kg!. Danske V.d. 50!. Math.-Fys. Medd. See also
e1aewhere. C. Gregory, Phys. Rev. 78, 67, 479 (1950).
238 CHAPTER IV

1050 HIDEKI YUKAWA

proved to be equivalent to the dilIerential forma1ism Equation (21) can be writtt'n in the form
and in which the S-matrix, instead of the SchrOdinger
wave function, came in the foreground. Then the (n'ISln")=(II'llll1")
S-matrix for local fields can be transformed in the
following manner so as to be easily extended to the +(i/,hc) f I(II', ~IL'III", %")(Ih')«Ih")<
case of non-local fields. We consider a system of local
fields, for which the interaction Hamiltonian density
H'(%, ',', I) is invariant and is equal to - L'(%, "I, I), +(i/Ac)'I· .. I E
_",•• IV
(1I',~IL'ln"',~")
where L' is the interaction part in the Lagrangian
density for the system. In,the usual one-time forma1ism,
the SchrOdinger equation' has the form X (n"', ~"I flnlv, %IV)(nIV, %IV!L'III", ~')
X (d%')«th")«d~")«lhIV)<+ . . • . (24)

."
i/ul'li(II', 1)/iJI= E(II'IO'(t) 11I")'li(II", I),

where each of II' and II" stands for a set of eigenvalues

(19)
If we define j A f for an arbitnuy non-local operator A
by
of occupation operators of various types of particles in
the system in various quantum states. 0'(1)= - L'(I) (n'ljA fill")
is the space integral of the Hamiltonian density
H'(%, ,,1,1) or - L'(%, ',', I). The dilIerential equation = f .. ·f(II',%'IAIII",%")(Ih')«d%")<, (25)
(19) can be integrated with respect to time, at least
formally, by the method of successive approximation the S-matrix with matrix elements as given by Eq. (24)
and we obtain
can be written symbolically in the form
'If(II', + co) = 'If(II', - co) S= H(i/hc)jL'f+(i/hc)'jL'eL'}
+(i/hc)'{L'eL'eL' f+ .... (26)

-
+(i/h) f+"~(II'IL'(I)III")dt''li(II'" - co)

+(i/h)'I I •• ~"'<"'I L'(I>!II'')(II''I L'(I') III"')

This could be used as the definition of the S-matrix in
non-local field theory as well as in local field theory.
Alternatively, we can define S or S-1 by
S-l={Rf, (27)
XdIdt''li(II''', - co)+, •. , (20) where

where'li(II', + 00) and 'li(II', - 00) are Schrodingerwave R=(i/hc)L'+(i/hc)'L'eL'+(i/hc)'L'eL'eL'+···. (28)

functions in the infinite future and infinite past, respec- Incidentally, the non-local operator R satisfies a linear
tively. Thus the S-matrix for -this case is given by operator equation

(II'! S! 11")= (11'1 111I")+(i/h) I (11'1 L'(I) !1I")dt

R=(i/hc)L'+(i/hc)L'eR.
The physical interpretation of the S-matrix remains
(29)

the same as in the usual theory in spite of the fact

+ (i/h)'I I :;;.(11'1 L'(I) I11"')(11"'1 L'(t') III") that the S-matrix is non-local field theory is defined
directly by Eq. (26) or by Eqs. (27) and (28) without
XdIdt'+ .. ·. (21) recourse to the SchrOdinger equation of type (19).
Thus, I(n'ISI n") I' is the probability that the system
In order to generalize this expression for the S-matrix will be in the state characterized by II' in the infinite
to the case of the system of non-local fields, we introduce future provided that it was in the state characterized
an invariant Hermitian non-local operator L' which is by II" in the infinite past. In fact, the S-matrix as
represented by a matrix (II',~!L'!II",~') and which defined by (26) satisfies two conditions:
reduces to (i) S is a unitary matrix which satisfies the relation
(II', ~IL'III", ~')= (n'IL'(~III")lIa(~.-~,.) (22) S*S=SS*= 1. (30)
• (ii) The matrix element (II'ISIII") is dilIerent from
in the limiting case of the system of local fields, wbere
each of ~ and ~' stands for a set of eigenvalues of zero only if the states characterized by II' and n",
respectively, have the same total energy and momen-
space-time operators "'-%, 1!-=" 1!-=I, x'=cI. With
the help of Eq. (22) and of another non-local operator. tum.
which is represented by the matrix Before going into the proof of these statements, we
have to take into account the third condition:
(n', ~Iflll", ~') (iii) S must be an invariant matrix.
=il[(~<-~'<)/I~<-%"·!]+I)(II'llln<). (23) In local field theories, the S-matrix defined above is
COVARIANT PICTURE OF QUANTUM BOUND STATES 239

QUANTUM THEORY OF NON-LOCAL FIELDS 1051

invariant, in spite of the fact that the operator • as for any values of rand r'. In order to prove this,
defined by Eq. (23) is not invariant with respect to we have only to multiply S as given by Eq. (31) by
Lorentz transformations. This is due to the fact that S*=l-(i/MHL'I+(i/M)'{L'D+*L'1
the Hamiltonian density B'(r) at a point r is com- -(i/M)'{L'D+*L'D+*L'I+···. (37)
mutative with the density B'(r') at any other point
r', which is located in a space-like direction with Then the condition of unitarity
respect to r. It is not so, in general, in non-local field L (n'IS*I,,",)(nIllISI,,',)
theory. An obvious way of guaranteeing the invariance .'"
of the S-matrix in such a case is to replace the operator = L (,,'ISIIIIII)(nllllS*III")=(n'llln") (38)
• in Eq. (26) by a suitable invariant non-local operator ,,",
D+ such that conditions (i) and (ii) are still fulti11ed. comes out by the help of Eq. (32) and the relation
Thus the S-matrix for the system of non-local fields {AEB}={A }{Bf, (39)
takes the form
which holds for any two non-local operators A and B.
S=H(i/MHL'f+(i/M)'{L'D~'f The operators D+* and D+ which satisfy all of these
+(i/M)'{L'D~'D~'f+"" (31) conditions are given by matrices
The actual form of the opet;ltor D+ can be determined (n', rID+III", r')=(~'llln"), 1(11'11111"), or 0;
in the following manner. If we assume that the invariant (II',rID+*ln",r')=O,l<,,'llln"), or (II'llllI"), (4O)
operator L' is a sum of products of non-local field
according as r - r' is futvre-Iike, space-like, or past-
operators, condition (ii) is satisfied for any displacement
like.
operator D+ whose matrix element (rID+lr')=D+{X,
This modification of the definition of S-matrix gives
r) is an invariant function of r. alone. The proof is simple.
rise to the new question: does it reduce to the usual
Any non-local operator A can be represented by a
definition (21) in the limit of local fields? This question
matrix (rIAlr') or a function A(X,r) and Eq. (25)
is very intimately connected with another, and probably
can be written alternatively in the form
the most important, question: is the S-matrix for non-

(n'IlA Hn")= ff (n'l A (X, r)ln")(dX)·(dr)·. (32)

local fields convergent? In order to answer these
questions, we begin with the investigation of the
particular matrix element (OISIO) of (n'ISln"), where
If the ope.rator A consists of a sum of products of L' both the initial state n" and the tina1 state II' are
and D+, {n"1 A (X, ,) In") can be expanded into a series complete vacua; i.e., all eigenvalues n' and n" are zero;'
with the typical term Now (OISIO) has the general form
(,,'I a(k.(i), '.) In") eap(iK.x.), (33) (OISIO)= H{i/M)(OI{L'f 10)
where +(i/M)·(OIlL'D~'HO)+···. (41)
K.= L. ,,"(·)k.(·)- LI ,,'(Ok.(i). (34) Let us consider a very simple case of a system consisting
Evidently 11K is the di1Ierence in momenta between the of a complex non-local scalar field V, V* and a real
initial state ,," and the final state ,,' and - licK. is the non-local scalar field U with the interaction of the form
di1Ierence in energies of the states n" and n'. If we L'=gV*UV. (42)
insert Eq. (33) into Eq. (32) and integrate with respect We have first
to X, we find that each term of (,,'I {A II ,,") contains (OIlL'1I0)=O (43)
a factor II. 8{K.' - K.") , so that (,,'I {A H ,,") is
because L' is linear in U and hence has no term which
di1Ierent from zero only if the states n' and n" have the
connects the state 0 with itself. As for the third term
same energy and momentum. It should be noticed,
however, that'we mean by the energy and momentum in Eq. (38), we have the relation

.'
of a particle the energy and momentum of its center of (OI{L'D~'f 10)=ILCOI{L'f 1,,')(11'1 {L'f 10)
mass. Thus the energy of internal motion is supposed
to be included already in the mass Ii«/c. '. In other +1(01 {L'DL'}lO) (44)
words, « must be, in general, a function' of other on account of relations (32) and (36), where the
constants such as X and l. The problem of determining operator D is defined by
the form of such a function is still completely open.
The condition (i) is also fulti11ed, if we further (45)
imply the condition with the matrix element
(35)
(11', rlDIII", r')=(II'llllI"), 0, -(n'lll""), (46)
on D+, where D+* is the Hermitian conjugate of D+ according as r - r' is future-like, space-like, or past-
and E is an invariant displacement operator with the like.' The first term on the right-hand side of (44)
matrix element
, This operator was iDtroduced by Koba independently. See Z.
(n', riEl,,", r')=(n'llln") (36) Koba, Prog. Tbeor. Phys. 5,139 (1950).
240 CHAPTER IV

1052 HIDEKI YUKAWA

vanishes on account of the fact thilt (n' I{L' flO) is directioll-.of time is not reversed. Namely, we can write
zero provided that KU < 2KY and the second term also
U=U++U_, V=V++V_, V*=V+o+V_·, (49)
vanishes for the following reason: first we expand U,
V, V·, and D in Fourier series and integrate each of where U +, U _ are positive and negative frequency parts
the terms of (OI{L'DL'f 10) with respect to all of the of U, while V +, V +0 and V _, V _0 are corresponding
space-time parameters. Actually we have eight sets of parts of V and V*. If we take the new interaction
such parameters. Then we are left with the expression operator
of the form
L'=g{V+'"'UV++ V _UV _*+ V +oUV _+ V _*UV +1 (SO)
instead of Eiq. (39), the self-energy terms for the U-type
f f ff(k.(I), k.(l), k.(I»O/(K,.K·)
particle as 'well as the V - yO_type particles are con-
vergent, although there still remains an undesirable
X (dk. (I»·(dk. (I»·(dk. (I)', (47) feature, as discussed by Yennie.'
Now, in order to remove the discrepancy between the
where K.= L; k/') and k.(I) , k.(l), k.(·) are the wave present formalism and the usual formalism in the limit
vectors of the three particles created in the intermediate of local fields, we may imagine that D-operator above
state. The first of them is a particle of U-type and the defined is a limit of the operator with the matrix
other two are particles of V - V* -type. f( .•. ) is a func- element, which is a function of r. and is different from
tion of k.''', k.(') , k.(3), which .could be determined by zero in a narrow region outside the light cone in r-space.
elementary calculations, but it is not necessary for our Then the correspondence between the present formalism
purpose to write it explicitly. 01. denotes the derivative and the usual formalism in the limit of local fields is
of the a-function with respect to the argument, which restored up to the second order, but the essential
comes from the Fourier transform of the operator D, difference between .. and D-operators remains in the
as discussed in detail by Yennie.' Thus, (oI{L'DL'fIO) third- and higher order terms. Moreover, the diver-
must be zero, unless the condition gences reappear in the case of non-local fields. It is
very difficult to construct an S-matrix which is con-
K.K·=O (48)
vergent and which reduces to the usual S-matrix in the
is fulfilled. The condition (48) can be satisfied by limit of local fields. It is not yet clear whether the
certain sets of k. (I), k/'), k. (I) only if both types of S-matrix formalism itself is not adequate for dealing
particles have the rest mass zero. with the problem of interaction of non-local fields. It
The above arguments can be applied to local fields as might be possible that the S-matrix as defined by Eq.
well as to non-local fields. According to the usual theory (24) is invariant, if the interaction operator L' has an
of local fields, the third term on the right-hand side of appropriate form, even in the case of non-local fields.
Eq. (41) must be the divergent self-energy of the However, it is more probable that the clean-cut sepa-
vacuum, whereas it is actually zero according to our ration of the free fields from their interaction is justified
formalism, except for the very particular case of only if we are dealing with the weak coupling between
particles both with the rest mass zero. The same argu- local fields. If so, we m,ust go back in search of the
ment can be applied to the case of charged particles Lagrangian formalism for the whole system of non-local
interacting with the electromagnetic field, and according fields interacting with one another. In any case, the
to our formalism the self-energy of the vacuum is zero, compatibility conditions for the field equations or the
at least up to the second order, if we assume that integrability conditions for any substitute for the
there is no charged particle with the rest mass zero. Schriidinger equation will be of fundamental importance.
Thus, the discrepancy between our formalism and the In this connection it should be noticed that so far
usual theory is already clear; they give different we have not been able to find any relation between the
answers to the same problem for local fields. mass and other constants. It is clear that a relation
Next we consider the matrix element (1ISI1) of S, which connects the mass of an elementary particle with
where only one particle of the same type in the same other constants such as the radius, the internal angular
state exists in the initial and final states. The second- momentum, and the constants of coupling with other
order term of (11 S11) corresponds to the divergent particles will be ·of vital importance in any future theory
self-energy of the particle in local field theory. As of elementary particles. Again this is closely related to
discussed by Yennie in detail,' if we start from a the problem of finding the Lagrangian operator for the
system of two non-local scalar fields of U-type and whole system or any substitute for it.
V - YO_type with the interaction operator L' as given The author wishes to express his appreciation to
by Eq. (42), the self-energy term is again divergent. Columbia University, where this work was done, for
However, the fields U and V-V· can be decomposed the hospitality shown to him, and to the Rockefeller
further into positive frequency and negative frequency Foundation for financial support. He is indebted also to
parts without destroying the invariance with respect to Mr. Yennie for his useful criticism and elucidation of
the subgroup of Lorentz transformation, in which the some of the important consequences of the formalism.
COVARIANT PICTURE OF QUANTUM BOUND STATES 241

Structure and Mass Spectrum of Elementary particles. I. General Considerations

Hideki Yukawa*
Columbia University, New York, New York
(Received May 25, 1953)
As discussed in previous papers, 1 the nonlocal field was introduced in order to
describe relativistically a system which was elementary in the sense that it could no
longer be decomposed into more elementary constituents, but was so substantial,
nevertheless, as to be able to contain implicitly a great variety of particles with
different masses, spins, and other intrinsic properties. However, the conclusions
reached so far were very unsatisfactory in many respects. 2 Among other things, the
masses of the particles associated with the irreducible nonlocal fields remained
completely arbitrary and simple and plausible assumptions concerning the
interaction between fields did not result in the expected convergence of self-
energies. It seems to the author that these disappointing consequences are not
inherent in nonlocal field theory, in general, but are rather related to the particular
type of field to which the author restricted himself. Instead, if we start anew from
less restricted nonlocal fields, a more promising aspect of possible nonlocal theories
is revealed, as shown in the following.
Let us take a scaler (or pseudoscalar) nonlocal field,

(x/I<I>lx/) == <I>(X w r,J,

where x/' xfL" (Il = 1, 2, 3,4) stand for two sets of space-time parameters and

X fL = (xfL' + xfL")/2, r fL = x/ - xfL"·

The free field equation is supposed to have a general form
F (a/aX f1' r 11' alar fL)<I>(X f1' r,J, (1)
where the operator F is a certain invariant function of a/ax f1' r f1' and alar fL and is
independent of X fL so that it is invariant under any inhomogeneous Lorentz
transformation. In particular, if we assume that F is linear in a2,ax JLaAfL and
separable, i.e.,

F =_ a a2a +F (7) [ rfLr Wa ~~]

a ,r fLa ' (2)
XfL XfL rfL rfL rfL
we have eigensolutions of the form <I> == u (X) X (r) , where u and X satisfy

Reprinted from Phys. Rev. 91,415 (1953).

242 CHAPTER IV

eei fax 11 aXil - J..L) U (X) = 0, (3)

(F(r) - J..L) X (r) = 0, (4)

J..L being the separation constant. Thus, the masses of the free particles associated
with the nonlocal field <ll are given as the eigenvalues of J..L'h in Eq. (4) for the
internal eigenfunction X . If one chooses the operator p(r) such that the eigenvalues
J..Ln ==m; are all positive and discrete, one can expand an arbitrary nonlocal field <j>
into a series of internal eigenfunctions, Xn (r) :
(5)

Now, the field equations for a scalar nonlocal field (x'I<lllx") interacting with a
local spinor field 'V(x '), for instance, can be deduced from an appropriate
Lagrangian and are

{-aa; XIlX Il
+ p(r)} <I>(X, r) = -gL'Va(X + Yzr )'Va(X - Y2r),
a
(6)

Yll ar(x:) + M'V(x') =-gJ'V(x")(x"I<lllx')dx". (7)

aXil

We insert (5) in (6), multiply both sides by the complex conjugate x:(r), and
integrate over the four-dimensional space of r 1, r 2, r 3, r 0 = -ir 4. The result is

[ ax,. ?~,," - m.'j u. (x'~ =f<il. (x', x ", x "n: .iii"(x 1",.(x "1<1x'<Ix "', (8)

where
<lln(x', x", x"') == gx:(x' -x"')o(llz(x' + x"') - x"). (9)
Similarly, we obtain from (7) the equation

Yll d'JIa(x:) + M'I' (x') = -Lni<l>n(x', x", x"') Un (x")'V(x"')dx"dx"'. (10)

X ll

If we com~are these equations with the corresponding equations (19) of Moller and
Kristensen in the theory of nonlocal interaction between a local scalar (or
pseudoscalar) field and a local spinor field, we notice that the internal eigenfunction
Xn (r) plays the role of a convergence factor. There is, however, an essential
difference between their equations and ours. Namely, in our theory, we are obliged
to take into account simultaneously all the particles with different masses mn which
were derived from an eigenvalue problem. Furthermore, the form function for each
of these particles is uniquely determined by the same eigenvalue problem.
In the following letter, the above general considerations will be illustrated and
further details will be examined.
"Now at Kyoto University, Kyoto, Japan, on leave of absence from Columbia University (July,
1953 1,H. Yukawa, Phys. Rev. 77, 219 (1950); 80,1047 (1950).
2D. R. Yennie, Phys. Rev. 80, 1053 (1950); 1. Rayski, Acta. Phys. Polonica 10, 103 (1950); Proc.
COY ARIANT PICTURE OF QUANTUM BOUND STATES 243

Phys. Soc. (London) A64, 957 (1951); M. Fierz, Helv. Phys. Acta 23, 412 (1950); Z. Tokuoka and Y.
Katayama, Progr. Theoret. Phys. 6, 132 (1951); C. Bloch, Kg!. Danske Videnskab. Selskab, MatAys.
Medd. 24, No.1 (1950); Progr. Theoret, Phys. 5, 606 (1950); O. Hara and H. Shimazu, Prog. Theoret.
Phys.5, 1055 (1950); 7, 255 (1952); 9, 137 (1953).
3 p. Kristensen and C. Moller, Kg!. Danske Videnskab. Selkab, Mat.-fys. Medd. 27, No.7 (1952); C.
Bloch, Kg!. Danske Videnskab. Seiskab, MatAys. Medd. 27, No.8 (1952). Y. Katayama, Progr.
Theoret. Phys. 8,381 (1952).
244 CHAPTER IV

Structure and Mass Spectrum of Elementary Particles n. Oscillator Model

Hideki Yukawa*
Columbia University, New York, New York
(Received May 25, 1953)
As an illustration of the general considerations on nonlocal fields in the preceding
letter, let us assume that the operator F has a very simple form
2 2
F == ax + Tt..2[ ar Jill
ax a1111 a
ar 1 J2
+ . . 4 r Ji r Ji '
/I,
(1)

where A is a small constant with the dimension of length. One may call this the
four-dimensional oscillator model for the elementary particle, which was considered
first by Born} in connection with his idea of a self-reciprocity. However, our model
differs from his model in that we have introduced internal degrees of freedom of the
particles which are related to the nonlocalizability of the field itself. The internal
eigenfunctions in our case are
Xnl n2 n3 no (r) = H,., (rill..) H,.. (r21A) H,., (r31A) H,.. (roll..)
xexp {- (rr + ri + r; + rJ) 12A2} (2)
and the corresponding eigenvalues for the mass become
1nII1 "21'13 "0 = c-fi.1A I nl + n2 + n3 - no + 1 I, (3)
where r 0 = ir 4 is a real variable and n l,n 2,n3,n 0 are quantum numbers which can
take only zero or positive integer values. H,. (x) denotes the Hermite polynomial of
x of degree n. All these eigenfunctions (2) decrease rapidly in any direction
whatsoever in the four-dimensional r space. Furthermore, the Fourier transform of
each of these eigenfunctions has exactly the same form as the original function due
to the self-reciprocity. Thus, the form function (9) in the preceding letter seems to
be sufficient to cut off high energy-momentum intermediate states in such a way that
each term corresponding to each Feynman diagram in the expansion of the nonlocal
S-matrix according to the Bloch-Kristensen-M011er formulation is convergent.
However, since we have to take into account all of infinitely many of different mass
states of the nonlocal system, the number of terms in the S matrix increases very
rapidly with the increasing power of the coupling constant, so that we can claim
nothing for the moment concerning the convergence or divergence of the S matrix as

Reprinted from Phys. Rev. 91,416 (1953).

COVARIANT PICTURE OF QUAN11JM BOUND STATES 245

a whole.
The totality of the internal eigenfunctions (2) constitutes a complete set of
orthogonal and quadratically integrable functions in the four-dimensional, space
and can be regarded as the eigenvectors for an infinite-dimensional unitary
representation of the Lorentz group. The eigenvalues (3) for the mass are all
infinitely degenerate. For instance, all those values of n's which satisfy
nl + n2 + n3 - no = 0 give the same mass, mo = fitA.. This is not a peculiar feature
of the oscillator model; it is common to all those models for which the operator F is
separable, because there can be no unitary representation of finite dimensions for the
Lorentz group. Presumably, such an undesired degeneracy could be removed either
by introducing interaction with other fields or by first introducing the coupling
between the external and internal degrees of freedom. The latter possibility can be
illustrated by the addition of the coupling term,

-~ A. - ".ax ". + A.1[ ax".

22{ [axiP]2 a]2} , 4 '".
(4)

nar~r. ~r.-.r
to the expression (1) for F, where ~ is a dimensionless real constant. The free field
equation becomes

[k l .+ +

+ ~2).2{ -[k. a~J + ;. [k.-r}] X(k., r,.J = 0, (5)

in the eight-dimensional space k". and '"., where x (k".. ,,,.) is the Fourier transfonn
of cjI (X 11' ,,,.) as defined by
J
cjI (X".. ,,,.) = exp (ik".X".) X (k".. ,,,.) (dk".).4 (6)

One can solve Eq. (5) in the coordinate system in which only one component of the
wave vector is different from zero. 2 Thus, one obtains the mass spectrum
..J2l n l +n2+ n 3- n O+ 11
InIIln2n3nO = [ In1 (7)
A. 1 - 2~2(no + Y2J

where no is restricted by the condition

no<Y2(1I~2 - 1). (8)

If we take, for instance, ~ = 1Ifi, only no =0 is allowed and the mass spectrum
reduces to
(9)
and the degree of degeneracy of the mass eigenvalues is now finite. In particular,
246 CHAPTER IV

the lowest mass, mo = 21A, is free from degeneracy and the corresponding solution
of (5) is given by

Xoooo(kw ' ~ =exp {

I [
- 2')..2 ' fl.' fI. +
2(kfl.' ~2]
mJ } (10)

in an arbitrary coordinate system, where kfl.kfl. =-mJ.

The above advantage of introducing the coupling between external and internal
degrees of freedom is offset, however, by a complication which is almost prohibitive
if we further take into account the interaction with other fields, because the general
method of reducing the theory of nonlocal fields to that of the nonlocal interaction
between local fields as discussed in the preceding letter can no longer be applied
straightforwardly to our case. On the other hand, it may well be that one could
arrive at the desired removal of the infinite degeneracy as a consequence of the
interaction between nonlocal fields without assuming the coupling between external
and internal degrees of freedom for each of the nonlocal fields. This is plausible,
because the submatrix of the S matrix corresponding to one-particle states can
always be represented by an equivalent coupling between the external and internal
variables for the particle in question, so that one can hope that a reasonable mass
spectrum which is free from the infinite degeneracy may come out even without
assuming the coupling between external and internal degrees of freedom at the
outset.
A detailed account of all these points, including the quantization of nonlocal fields,
will be given in a forthcoming paper.
·Now at Kyoto University, Kyoto, Japan, on leave of absence from Columbia University
(July, 1953).
1M. Born and H. S. Green, Proc. Roy. Soc. Edinburgh 62, 470 (1949); M. Born, Revs.
Modem Phys. 21, 463 (1949).
ZEquation (5) has no solution which is quadratical1y integral unless k is time-like, i.e., ~km
COVARIANT PICTURE OF QUANTUM BOUND STATES 247

PHYSICAL REVIEW VOLUME 96, NUMBER .. NOVEMBER 15, 1954

Properties of Bethe-Salpeter Wave Functions

G, C. WICK
Carnegie Institute of Teehnology, Pittsburgh, Pennsylvania
(Received June 30, 1954)

A boundary condition at t= ± 00 (t being the "relative" time variable) is obtained for the four-dimensional
wave function of a two-body system in a bound state. It is shown that this condition implies that the wave
function c~n ~e continued analytically to complex values of the "relative time" variable j similarly the
wave functIOn In momentum space can be continued analytically to complex values of the Urelative energy"
variable po. In particular one is allowed to consider the wave function for purely imaginary values of t or
respectively po, i.e., for real values of x4=ict and p4.=ipo. A wave equation satisfied by this functio~ is
obtained by rotation of the integration path in the complex plane of the variable Po, and it is further shown
that the formulation of the eigenvalue problem in terms of this equation presents several advantages in
that many of the ordinary mathematical methods become available.
In an especially simple case (Uladder approximation" equation for two spinless particles bound by a
scalar field of zero rest mass) an integral representation method is presented which allows one to reduce
the problem exactly (and for arbitrary values of the total energy of the bound state) to an eigenvalue
problem of the Sturm-Liouville type. A complete set of solutions for this problem is obtained in the sub-
sequent paper by Cutkosky.

1. INTRODUCTION the boundary conditions on the wave function for

T HE formulation of a completely relativistic wave

equation for two-body systems! has, in a certain
infinite values of the relative time have not been
adequately formulated.
sense, solved a long-standing problem of quantum (b) The presence of strong singularities in the
mechanics. The natural and simple way in which interaction kernel, to be avoided by special prescrip-
relativistic invariance is achieved is, of course, very real tions. Standard mathematics has practically nothing
progress, which may lead one to hope that the main to say about integral equations of this type, In partic-
features of the equation are more permanent than the ular, the prescriptions referred to imply properties of
solidity of its present field theoretic foundation might analyticity, about which one would like to know a lot
suggest. Furthermore, it is hardly necessary to recall more.
that the usefulness of the equation has been amply (c) The absence of a positive-definite nonn for the
demonstrated in several high-precision calculations of wave function and of any orthogonality theorem,
energy levels.' (d) The fact that when the coupling constant A is
Nevertheless, it is generally recognized that several set equal to zero, the equation admits obviously
serious and valid doubts remain about the significance improper solutions. Notwithstanding all that can be
and the self-consistency of the equation. Some of these said about it, this feature is a little disturbing. It is
doubts, of course, stem from the remaining unresolved connected to the other feature that the "order" of the
convergence questions of renormalized quantum electro- differential operator in the equation is higher than that
dynamics (and other similar theories). It goes without of the corresponding one-body equation. This leads to
saying, however, that these deeper questions lie the expectation that the equation may have "too many"
entirely beyond the scope of the present investigation' solu tions. On the other hand, circumstance (b) has led
The questions and doubts we shall be concerned with' some authors to suspect that there are no solutions at
arise at a less formidable level; they have to do with all!
the several unfamiliar features of the equation itself. (e) Finally, as explained by Goldstein,' we are faced
These are (and the list is probably incomplete): with the paradoxical circumstance that, owing to the
(a) The appearance of a relative time (or respectively nonrelativistic perturbation approach employed, the
a relative energy) variable, the physical role of which highly successful numerical results obtained do not
is not entirely clear; in particular, it is admitted that really offer any direct clue as to the actual properties
of the relativistic equation.
1 E. E. Salpeter and H. A. Bethe, Phys, Rev. 84, 1232 (1951)' Th' t' . d 'b d' h f II '
J. Schwinger, Proc. Natl. Acad. Sci. 37, 455 (1951). Other closely e mves IgatlOn escn e m teo owmg pages
related but more general relativistic schemes recently developed was aimed at throwing some light on these questions.
by various authors will not be discussed here. It really consists of two quite different lines of attack.
'E. E. Salpeter, Phys, Rev. 87, 328 (1952); R. Karplus and The first of these starts from the remark (Sec. 2) that
A. Klein, Phys. Rev. 87, 848 (1952) .
• In particular, expressions such as "the general structure" of an additional condition for the Bethe-Salpeter (B-S)
the equation, "the analytic properties" of the interaction kernel wave function follows from its definition' supplemented
~tc'J will be used on the assumption that such properties may b~
mferred correctly from truncated expressions of finite order in by simple stability requirements. From this, then,
the coupling constant, for example, from the lowest-order some unexpected consequences can be derived about
("ladder");approximation. ----
• See, especially, J. S. Goldstein, Phys. Rev. 91, 1516 (1953). 'M. Gell·Mann and F. Low, Phys. Rev. 84, 350 (1951).
1124

Reprinted from Phys. Rev. 96, 1124 (1954),

248 CHAPTER IV

BETHE-SALPETER WAVE FUNCTIONS 1125

the analytic continuation of the wave function to The sum extends in principle over all states, but in
complex values of the relative time (or relative energy) fact the states n giving a nonzero contribution will
variable. As far as we can tell these properties cannot belong to a rather special class. Consider for example
be obtained from the B-S equation itself. Vice versa, the case where a and b are an electron and proton,
they can be used (Sec. 3) to transform the equation, respectively. If 'fl. and 'fI bwere noninteracting fields, it
by rotation of the integration path in the complex is obvious that only one-electron states would have to
plane, to an equation in which x. = ixo (respectively be considered in the sum (3). In the presence of inter-
p. = ipo) is real. While the concept of an imaginary action, the states n may also contain photons, electron-
relative time variable does not help physical intuition, positron pairs and proton-antiproton pairs. But at any
it has mathematically several advantages. A discussion rate the fundamental integrals of the motion N.
of the eigenvalue problem in terms of the transformed (number of electrons-number of positrons) and Nb
equation will be given (Sec. 4), and the existence of (number of protons- number of antiprotons) must have
solutions will be shown to follow, under fairly general the same values,
assumptions, from considerations similar to those
commonly employed in the nonrelativistic case. No N.=l, Nb=O, (4)
claim of completeness or rigor is made for this "proof." as the one-electron states. This may be rigorously shown
Finally in Sec. 5 we shall merely itemize various from the commutation properties of N. and N b with
approximation methods that have been studied, but the field operators, (N.+l)'fI.='fI.N., etc.
will be reserved for another publication. In a similar manner, one can show that the total
The second line of attack (Sec. 6), which is the subject angular momentum quantum number J for a state n,
of a more extensive investigation in the subsequent when measured in a system of reference in which the
paper by Cutkosky,· is rather different in nature. It is total momentum p is zero, must be equal to !.
an attempt to make much more specific statements Now all states known to us in nature, and satisfying
about the exact solutions of the equation, by restricting condition (4), also satisfy the inequality,
the character of the equation to an especially simple
type. It has not been possible so far to extend this (5)
approach to any case of real practical interest. But the E. and p being the total energy and momentum in the
fact that in one case, which is not entirely artificial, state n. Furthermore, the equality sign holds true only
one can get a complete picture of all the solution (as is for one-e1ectron states.
shown more completely in the following paper') is not The inequality (5) means that among all the states
perhaps devoid of general interest. In particular the having the same values of the fundamental constants
presence of "abnormal" solutions, which do not possess of the motion p, N., N b, etc., as a one-electron state,
a nonrelativistic limit, and the circumstances under the latter is the state of lowest energy. We shall refer
which they occur may well give a qualitative indication to (5), therefore, as the stability condition for an
as to properties that will occur also in the cases of real electron. .
physical interest. In a similar way, when the relative time t is negative,
Z. THE STABILITY CONDITIONS the wave function X may be shown to depend on the sum
The relativistic wave function x(x) for a system of Lo,(OI'fl,(Xb) In')(n'I'fI.(x.) la), (3')
two particles, a and b, bound together in a state Ia)
is defined' as the matrix element, between a and the in which the contributing states n' must satisfy the
"true" vacuum state 10), of the time ordered product condition,
N.=O; Nb=l, (4')
of the Heisenberg field operators .y. and .y. describing
the two kinds of particles. If, for example, the relative and hence the inequality,
time t= t.= t. is positive,
E"2-p'~mb, (5')
x(x)=e iP . X (OI'fl.(x.)'fI,(Xb) la), (1)
which shall be called the stability condition for a proton.
where X=X.-Xb, X= (maX.+moXb)/(m.+mb), and Summing up, we have three inequalities (2), (5), and
p. X is the four-dimensional scalar product of X with (5'), which will form the basis of the following discus-
the total momentum P of the system in state a. If for sion. It should be pointed out that the above considera-
simplicity we assume tbat the compound system is at tions can be extended to other systems. If a and b were
rest, then P= (0, iE), E being the total energy. For a a neutron and proton, bound together in the ground
bound state, state a of the deuteron by a meson field, with the
E~m.+mb- B<m.+mb. (2) customary assumptions, one would then have, as
Now the matrix element in (1) can be written integrals of the motion, the number of nucleons minus
antinucleons N and the total electric charge Q. The
L.(OI'fl.(x.) In)(nl'flb(xb) la). (3) states n could be shown to have values N = 1, Q= 0
• R. Cutkosky, following paper [Phys. Rev. 96, 1135 (1954)]. and the states n' the values N = 1, Q= 1. In a theory
COVARIANT PICTURE OF QUANTUM BOUND STATES 249

1126 G. C. WICK

which neglects the ,B-decay interaction, one has the where. is an infinitesimal positive constant. We must
right to regard both neutron and proton as essentially assume, of course, that the wave function exists for
stable particles. If there were states n(n') not satisfying real values of po [i.e., that the integral (9) converges].
conditions (5) (5') the neutron (proton) could decay From the theory of Stieltjes transforms, we then infer
into those states by emission of photons, without that (9) defines an analytic function of po in the whole
viola ting any of the known conservation theorems. complex plane, in the region
Thus it is extremely reasonable to postulate that these
conditions must again be satisfied. 2.->arg(pO-WmiP);;: o. (10)
Now going back to (1) and using (3) with the Similarly q,2 is defined in the region
conditions (2) and (5), we see that for 1>0, and assum-
ing p= (O,iE), x(x) is of the form -.-<arg(po-wmox)<'-, (11)
where
x(x)= f f. .
dp
-roo

Wmln
dwl(p,w) exp(ip·x-iwt), (6)
-Wm",=BIl.+(m.'+p)!-m.>Bp..>O.
Thus ¢(p) =q,(p,po) =q"+q,, is defined in the complex
where po plane with two cuts from Wmin to + 00 and from - 00
to W mox (Fig. 1). In this case analytic continuation from
the lower to the upper half-plane is ensured through
with I'a=m./(m.+m.). Thus, when 1>0, x(x) is a the gap between the two cuts. (B>O is essential for
superposition of positive frequency terms only. the existence of the gap.) Notice also that the sense of
Similarly, from (2) and (5') it follows that, when rotation implied by (10) and (11) is the opposite of
1<0, x(x) contains negative frequencies only. Thus we that in the I plane. From the real po axis one goes
find that x (x) has properties with which we are familiar continuously into the upper half-plane if po>Wmin>O,
in the case of Feynman propagation kernels. There is, into the lower half-plane if po<Wmu<O.
of course, an analogy between the definition of these
kernels and Eq. (1). 3. TRANSFORMATION OF THE B-S EQUATION
Let us now consider I as a complex variable. Equation We shall now use the analytic properties of the
(6) shows that X(x) can be continued analytically in wave function to transform the B-S equation by a
the lower half-plane, in the region 0;;: argl> -.-. rotation of the axis of integration in the complex po
Similarly starting from the negative real axis, x(x) can (respectively xo) plane.
be con tinued in the upper half-plane, in the region The equation' may be written
.- ~ argl> 0. There is, of course, no analytic continuation
from one half-plane to the other; the two regions touch (12)
one another at one point only, 1=0.
It should be pointed out that the statements just where q, is the wave function in momentum space, i.e.,
made are not dependent on the assumption that the the Fourier transform of x (x) ; it is a function of the
sta te a is bound; they follow from well-known properties relative momentum p defined by
of the Laplace transform from the mere fact tha t w is
finite. If, however, B>O and hence Wmin>O, we can
further assert that X(x)-->O when t tends to 00 in any P, Il. and 1'. being the total momentum and the mass
direction in the lower or upper half-plane different from ratios previously defined. F. and F. are one-particle
the real axis. This suggests that the eigenvalue problem propagators, which, if one neglects radiative corrections
may take a more familiar and a simpler form if the
wave function and the wave equation are considered on
",.",+ijl
the imaginary 1 axis (i.e., for x.=il real).
fn order to examine this possibility carefully, it is
desirable to go over to momentum space. We write
x(X)=X+X2, where x,=O for 1<0 and x,=O for 1>0.
Let us calculate the Fourier transform of x,. o
'"

From (6) one easily finds

q" (p,Po) = --J

1

2n
+00

"'min
l(p,w) (w- po-i.)-'dw, (9)
FIG.!. The complex plane 01 the variable p•. The wave function
is analytic everywhere, excluding the cuts (heavy Jines) on real
axis.
250 CHAPTER IV

BETHE-SALPETER WAVE FUNCTIONS 1127

reduce to
F.=-y.p.-im., Fb=-YbPb-imb, (Dirac particles) Co}
(14)
F.= p.'+m.', Fb= Pb'+mb'. (Klein-Gordon)
Finally, I. b is the interaction operator, which has
,
different forms, depending on the kind of theory. The
\
'r P01 ••

following form' covers several cases, for the lowest pClth of Po

order ("ladder") approximation: Integrotlon I

I.III>(p) = (X/,,') J [dk]

(p-k)'+.'
paPIII>(k), (IS)
""'ne. path

where [dk] = idkodk. The various cases are obtained

from the various possible assumptions about the
"photon mass" K, and the factors PaPb(P.Pb= 1, scalar
interaction, etc.). (b)
For simplicity we shall carry out the transformation
under the assumptions (14), (1 5), but the proof can
be easily generalized to include radiative corrections
to any desired order. 7
Let us consider the right-hand side of Eq. (12), as
given by (IS) . The poles of the interaction kernel are at
ko=po±[(p-k)'+K']I. (16) I
t
Let us carry out the integration over ko first. The ~.poth
integration is along the real axis in the plane of the
complex ko variable, passing just under the cut on the
negative axis and above the cut on the positive axis.
It is also important to remember that K in (16) is
assumed to have an infinitesimal negative imaginary FIG. 2. Integration paths for the variable ko in Eq. (15 ).
part, so that the pole with the larger real part lies under
the integration path and the pole with the smaller negative real axis; it then moves to the nega tive
real part above the path. Suppose for instance po>O, imaginary axis. The net result is a counter-clockwise
then depending on the relative magnitude of the two rotation of the axis on which the wave function is used,
terms in (16) the poles wiJIlie as in Fig. 2(a) or 2(b). on both sides of the equation.
For simplicity, the cuts of Fig. 1 are not indicated in Equation (12) is thus reduced to an integral equation
Fig. 2, but they do not interfere with the following in a Euclidean vector space, with the metric
operations. First the integral path may be deformed (17)
along the dashed line [there is an assumption here,
that q,(k) tends to zero at least like ko-' when ko-> oo in One does not really have to change anything to the
any direction]. Now we move po upwards along a equation, except for the understanding that a real
circle so as to end on the positive imaginary axis. vector now has a real component p, and that, in
In Fig. 2(b) the path need not be changed. In Fig. 2(a) Eq. (15)
the left pole, around which the path is bent, moves to (18)
the left of the imaginary axis, and the path can be
straightened. In both cases we end up with po on the
the integral over k, being from - '" to 00. The fixed +
vector p= (0, iE) is now, of course, regarded as pure
positive imaginary axis, and the integral over ko along imaginary.
the imaginary axis, from -joo to +ioo. One sees at once several advantages of this trans-
A similar consideration applies when Po is on the formation. The singularities of the interaction kernel
7 A higher-order term includes, in general, anumber of integra-
[and with them the difficulties mentioned under (b)
tions over fourth components ko, ko', ko", .. . The proof is in the Introduction] are eliminated, and what is equally
most easily carried out if all these are regarded as complex important, the zeros of the Klein-Gordon factors
variables and their integration paths are rotated simultaneously.
An examination of higher-order corrections also requires a closer (14), i.e., the singularities of the inverses F.-I, Fb- I,
look at the factors FOl Fb. The analytical nature of the propagators have similarly disappeared from the space of real
Fa -1, F~-l (Le., of the Si,' AF' functions) is well understood (see
reference 14), and it is easy to show that they have no singularities p vectors. Furthermore the symmetry group of the
that stand in the way of our transformation. equation is no longer the Lorentz group, but the group
COVARIANT PICTURE OF QUANTUM BOUND STATES 251

1128 G. C. WICK

of real rotations in four dimensions. 8 This is important Eq. (20) becomes a symmetric integral equation,
in the first place, because the group determines the
polar variables, which may be used with advantage.
In the Lorentz case integrals over a surface p' = const,
v(x)=X f" K(x,y)v(y)dy, (22)
or x'= const are usually divergent; there are no •
orthogonality theorems for spherical harmonics, no with the finite kernel,
completeness theorems, etc. Here instead we have the
whole familiar machinery at our disposal. K(x,y) = 2(st)I/{s+t+<'+[ (s+t+<')'-4stN, (23)
Other advantages appear in the configuration space and the finite interval a= f( 00). Fredholm's theory can
formulation of the equation, as we shall presently see. then be applied, to conclude that (22) has a discrete
eigenvalue spectrum. The case where q, is proportional
4. DISCUSSION OF THE EIGENVALUE PROBLEM
to a four-dimensional spherical harmonic can be
We shall now examine several cases and show that similarly handled.
the transformed equation presents us with an eigenvalue It may be pointed out that if K=O, Eq. (20) can be
problem, to which many of the ordinary methods and reduced to a second order differential equation either
conclusions can be applied. by differentiating twice, or by a parametric representa-
We shall begin, like Goldstein,' with the extreme tion of the solution. Both methods will be used later,
case E=O, where the equation acquires full four- and especially in the subsequent paper by Cutkosky,·
dimensional symmetry in relative momentum space. to obtain more precise information about this case.
Unlike Goldstein, however, and for reasons to appear Let us now consider briefly Goldstein's Eq. (10),
later, we shall choose in Eq. (14) the K.G. (Klein- which applies to the case of two Dirac particles. When
Gordon) form of the factors Fa and F b. Tha t is, we written in our notation, the equation is quite similar
assume that a and b have zero spin. The equation for to (19) except that it contains only one quadratic
E=O thus has the form factor in p on the left. Goldstein manages to reduce

(p'+m.') (p'+mb').p(p) =X'Ir' f

[dkJ
(p-k)2+<,
.p(k). (19)
the equation to the one dimensional form, his Eq. (14),
in exact analogy to our Eq. (20); the transformation
in the usual frame, however, is far from trivial.'· Unlike
Eq. (20), however, Goldstein's (14) is not reducible to
We shall often use, in the following, the abbreviation
the Fredholm type. The difference in behavior is not
),,[4> for the right-hand side of (19). In particular I. an effect of our transformation, but is really due to
shall designate the interaction operator when the the different power of p' on the left-hand side. The
"photon" mass. is zero.
difficulties which Goldstein encounters in defining the
We can now, of course, separate q" using polar
eigenvalue spectrum, and which he surmounts by a
variables, and reduce the problem to a one-dimensional
special cut-off procedure, are thus not a general property
integral equation. If for example q, is a function of p' of the B-S equation, but rather of the special case
only, the integration over angular variables on the considered by him.
right-hand side of (19) is quite elementary. For simplic- For the purpose of obtaining a more general view-
ity we shall write the one-dimensional integral equation
point, let us now examine the problem in configuration
for this case only. Let p'=s, q,(p)=u(s); then space, i.e., in terms of the function xex). Consider first
again the case E=O. The Fourier transform of Eq. (19)
(s+m.')(s+mb')u(s) = 2'Af~ tu(t)dt/ is
• [(-D+m.')(-D+mb')-XV(R)Jx(x)=O, (24)
{s+t+<'+[(s+t+<')'-4stJI}. (20)
where the "potential" VCR) is
With the further change of variables (25)
x= f(s), y= f(t), K, being a modified Hankel function. The expression
for VCR) in the case K= 0,
f(s) = f ' ds'/(s'+m.') (s'+mb'), (21)
V(R)=4R-', (2Sa)
• also gives the singularity of Vat the origin in the general
sl(s+m.') (s+mb')u(s) = vex),
• Four-dimensional rotations must be applied simultaneously,
case.
of course, to the relative momentum p and to the total P. If one Goldstein's Eq. (10) becomes similarly
uses the c.m. system to begin with J so that P is pure imaginary,
it will stay pure imaginary after a real rotation. For Dirac particles, [J-m'HV(R)]x(x)=O, (26)
a linear transformation of the X or t/J function must accompany the
rotation; this can be established in the usual way. Contrary to the 10 The author is indebted to Dr. Goldstein for various interesting
Lorentz case. however, the transformation here is always unitary. conversations, and in particular for pointing out to him the
• See reference 4. Like Goldstein, we find it convenient, in peculiar "Euclidean" nature of his Eq. (14). This remark was one
general, to regard E as given, A as the eigenvalue to be found. of the early motivations for the present study.
252 CHAPTER IV

BETHE-SALPETER WAVE FUNCTIONS 1129

which presents a striking analogy to the ordinary condition on A. This will, in general, determine a
three-dimensional SchrOdinger equation. With X4 real, discrete spectrum of eigenvalues.
(26) is, of course, an elliptic difierential equation. This, We shall see later that for K=O the analysis can be
together with the boundary condition x(x)->o at carried much further. Let us now turn to the more
infinity, allows a discussion of the eigenvalue problem interesting general case E;>!O. Let us write (in the
along familiar lines. c.m. system)
A special difficulty, also encountered by Goldstein,
is presented by the boundary condition at the origin p= (O,iE)= i(m.+mb)'I, (28)
R=O, about which we have unfortunately no definite where 'I is the four vector
indication from general field-theoretic considerations.
The difficulty arises because of the Fuchsian singularity '1= (0,.), E=Ej(m.+mb).
(2Sa) j if the potential were regular everywhere, there Notice that
would be little doubt that x(x) must be regular too. ,f="<1. (29)
One can see at once, however, that the singularity The factor on the left of Eq. (19) now becomes, re-
of the potential affects (24) and (26) in a very different membering (13):
manner. Consider, for example, spherically symmetric
solutions. The radial equation corresponding to (26), (m.'+p.') (mb'+pb') = p4+ (m.'+mb') (1-,f)P'
or +4m.mbCP'l)'+m.'mb'(1-,f)'
[cl'jdW+(3jR)(djdR)-m'+AV(R)]X=0, (26a) +2i(m.-mb)(P'-m.mb)CP'I). (30)

has two solutions near the origin, of the type x =]?a It is at first sight rather puzzling that the equation now
contains an imaginary term whose presence depends
X (HcIR+· .. ) with
on m. being;>!mb. In configuration space this means
a= -1± (1-4X)I. (27) that the operator corresponding to (30) is self-adjoint
only when m.=mb. One can show that this feature is
Thus, if X<i, it is possible to make a distinction connected with the time-reversal properties of the
between the "regular" (less singular) and the "irreg- equation.
ular" solution. If X>i, it seems highly unlikely that a We shall point out, when the occasion arises, the
plausible condition to determine the right solution can difierences produced by the term in m.-mb. For the
be found. In the case K= 0, moreover, the equation can moment, we shall consider only the case m.=mb
be solved explicitly,' the "regular" solution being (=m, say). The analog of Eq. (24) then is
JrIJ.(iR), where n=+(1-4X)I. This solution, how-
ever, never satisfies the condition at infinity. We thus ([ -D+m'(1-,f)]'-4mVo'jox.')x(x)
reach the conclusion that no value X<i is an eigenvalue. =XV(R)x(x). (31)
In our opinion, for 1I> t the eigenvalue problem becomes
Since complete separation of variables is impossible, a
ill-defined. We shall not try to discuss further herell
solution must now be a superposition x=E,J.(R)Y.
whether the limiting case X= i can actually be regarded
of four-dimensional spherical harmonics Y. of difierent
as an eigenvalue.'
orders. The radial functions !. satisfy a system of
In Eq. (24), on the other hand, the singularity (2Sa)
coupled fourth-order difierential equations, and it is
does not affect the indicial equation. The radial equation
no longer possible to discuss the eigenvalue problem in
for a spherically symmetric solution, for example, has
terms of a single radial function. This is a considerable
four independent solutions near origin, say Xl, X" X" X.,
complication, but one may notice, nevertheless, that
behaving respectively like W, R", InR, and Jr'. If
the term in (31) which produces the coupling is of
there were no potential, we would clearly say that the
second order only, so that the indicial equation for
acceptable solution is a linear combination CIXI+C2X2
each radial function !. is the same as in Eq. (24). If
of the two "regular" solutions. We shall make the same
one writes !.(R)=R"(I+cIR+ ..• ) the possible
assumption when there is a potential'" Likewise we
values for a are ±n, ±(n+2)j we may assume that
can define, for large R values, four solutions behaving
only the positive values are allowed in a "regular"
respectively like Jrl exp(±p.R) and R .... exp(±p"R).
solution, just as in Eq. (24). Thus there is no qualitative
The solution CIXl+C2X2 will be a linear combination of
difierence between the two equations, with regard to
these four. In order to satisfy the condition x->O at
the behavior of solutions near R=O.
infinity, two coefficients must be zero; that is, we have
The asymptotic behavior of x(x) at infinity, on the
two conditions. One of these may be satisfied by a
other hand, is more interesting. It will be shown below
suitable choice of c';C2 j the remaining one gives a
that when x tends to infinity, X behaves asymptotically
11 It may be remarked that in reference 6 Goldstein's eigenvalue like exp[ - Rq>(8,)], i.e., it tends to zero exponentially
is also obtained from Eq. (19) in tbe limit m./m.--+O (and «=0). but with a coefficient depending on the direction,
II One can argue that x~lnR is not reaUy a solution of (24)
since it gives an additional term ~a.(x). x~l<' gives a term
more specifically on the angle 8. with the "4" axis.
[]a.(x). For our present purpose, however, it is only interest-
COVARIANT PICTURE OF QUANTUM BOUND STATES 253

1130 G. C. WICK

ing to notice that <p(O,) has a posItIve lower limit constructed in the Appendix, and it may be seen from
<p?;, 1-. so that, in a certain sense, there is again Eqs. (A7) and (AS) there that G(x) has a very weak
no fundamental difference in behavior between the singularity at the origin (it is in fact finite at x=O) and
solutions of (31) and those of (24), and we may expect tends to zero at infinity like
that in both cases the boundary conditions at R= 0
(36)
and R = 00 will determine a discrete A spectrum.
The elementary considerations developed previously where g is a factor which varies slowly compared to the
seemed of interest, because of the analogy with con- exponential and
siderations often made with regard to the ordinary
Schrodinger equation. In this sense we may say that <p(O,)=m(I-.cos8,) /cos8,/>.
(31) presents an analogy to the Schrodinger equation =mO-.')!sinO, /cosO,/ <.. (37)
for a particle in an asymmetric field, where again the If V(R)-->O sufficiently rapidly when R-->oo, the asymp-
reduction of the eigenvalue problem to a simple one- totic behavior of x (x) as given by the integral in Eq.
dimensional Sturm-Liouville problem is not feasible. (35) will reflect that of G(x), from which the conclusions
In either case, a rigorous discussion of the eigenvalue previously mentioned may be obtained. Incidentally it
problem can only be achieved by less elementary means, may be noticed that in the nonrelativistic limit. <'" 1,
such as the reduction of the problem to an integral the lower form in Eq. (37) covers almost the whole
equation. We do not wish to carry out such a study here, solid angle, and furthermore <P'" (mB)! sinO" R>p(O,)
but we may point out along what lines it could be = (mB)lr, where r=x,2+x22+ x}. We thus find the
carried out. typical exponential of the three-dimensional Schr6dinger
We already have, of course, in Eq. (19) and its function. It is indeed rather remarkable that in this
generalization for E,eO, an integral formulation of the region, i.e., with the exception of a narrow cone around
problem. In the case ma= mb corresponding to Eq. (31), the x, axis, the asymptotic form of x(x) is not time-
the equation can be reduced to the real symmetric form dependent.

<t>(p)='A f H(p,k)<I>(k)[dk], (32)

In the foregoing discussion we have, perhaps, laid
too much stress on the special case of two spinless
particles with the special interaction I. of Eg .. (19).
where It is clear that none of the conclusions we have reached
<t>(p) = f'(p)q,(p) as to discreteness of the A-spectrum, etc., must neces-
sarily remain true if we change the propagators Fa,
f(p)=[p'+m'(1-~)]'+4m~'P<' (33) F b or the interaction kernel.
H (p,k) = 'If'-'U(p) jl[ (p- k)'+K']-I[f(k) j!. If, for example, we write the analog of (32) with
Dirac propagators, the conclusion that t be equation is
Now by counting powers of p and k it is easy to see
nonsingular no longer holds true. As pointed out above,
that

f
Goldstein' already met this situation for the special
H'(p,k)[dp][dk] < 00, (34) case E= O. It is, of course, also possible to formulate
the problem in a form similar to (3S), namely,
which together with other similar inequalities, which the
mathematically inclined reader can readily discover, X(X)=Aj GD(x-X') V (R')x (X') [dx'], (38)
may be used to show that (32) is "nonsingular" and where
thus possesses a discrete 'A spectrum. Furthermore all
eigenvalues are real. Finally, one can see that the GD(x) = ['Ya(a/iJx)-ma(l +M)]
kernel is positive-definite!' so that A> O.
An alternative integral formulation can be obtained
x ['Yb(iJjiJx)+mb(Hn)]G(x). (39)

as usual in configuration space. In fact, Eq. (31) In this case the singular character of the equation comes
together with the regularity condition at the origin about because Gv(x) has a much stronger singularity
and the boundary condition x(x)-->O at infinity, can than G(x), near x=O. When this is combined with the
be replaced by an integral equation, IjR2 singularityof VCR) [Eqs. (25) and (25a)], Eq. (38)

x(x)='A J G(x-x')V(R')x(x')[dx'], (35)

becomes singular. This does not mean that discrete
eigenvalues of Awill not exist, but only that a much more
detailed study of the equa tion will be necessary. One
could, of course, also consider the possibility of less
or X='AGVX, where G is the inverse of the differential
singular potentials VCR), in which case the general
operator on the left-hand side of (31). When ma=mb,
the function G(x) is even: G(x)=G(-x), so that (35) theory of integral equations might again be applicable.
It seems pointless at present to investigate in detail
can be easily symmetrized. The function G(x) is
such possibilities. One will bear in mind, however, that
!3 G. C. Wick, Nuovo cimento (to be published). within the framework of our transformed system of
254 CHAPTER IV

BETHE-SALPETER WAVE FUNCTIONS 1131

coordinates, such questions can be attacked by ordinary One then finds easily that
mathematical methods.
10<1>= (1!2m') (p'+m')-t, (43)
s. APPROXIMATION METHODS showing that Eq. (40) is satisfied.
It is also possible to show that our transformed More generally, one can see that 10 applied to
equation has several advantages if one wants to employ (p'+2p·q+M')-', where M' and the vector q are
approximate methods of solution. We have in mind, constants, gives (p'+ 2p· q+ M')-I, apart from a
in particular; (a) a perturbation expansion in the proportionality factor. This peculiar self-reproducing
neighborhood of E=O (see also reference 4), (b) property of a quadratic form in p, under the operation
variational principles, (c) nonrelativistic approxima- I, is characteristic of the case K=O.
tions, without special restrictions as to the form of V (R). Consider now the equation for E","O. For simplicity
These questions will be discussed in a paper which the let m= 1 from now on. The equation is
author hopes to present shortly in another periodical." [p'+2ip·~+ 1-~'][p'- 2ip·~+ 1-~']q,=I 0<1>. (44)

6. EXACT SOLUTIONS FOR ,,=0 Clearly q, cannot be a function of p' alone; it must be
at least a function of p' and p.~ (for an S state). The
A comparison of Eqs. (25) and (2Sa) suggests above considerations suggest that we may be able to
that the problem of solving the B-S equation exactly generalize solution (41) by writing q, as a superposition
may be far more elementary in the latter (K=O) case. of termsl ' of the type (p'+ 2p· q+ M')-3 where q is
This is borne out by Goldstein's solution' for Eq. (26), parallel to 71, say, q=iZ71. That is

f dzdM'g(z,M')[p'+2izp·~+M'J3.
and we shall see in a moment that also Eq. (24) has
quite simple solutions if K=O and ma=m •. And, of
course, one will remember that the ordinary nonrelat-
q,(p) = (45)
ivistic Schrodinger problem is far more elementary
with a Coulomb than with a Yukawa potential. One then sees immediately that
At first, however, one would regard this analogy as
encouraging only for the special case E= 0, when the
B-S equation is separable. We were, therefore, quite
Io<I>=! f dzdM'gl(z,M')[p'+2izp·71+M']-l,

surprised when we first realized that for K=O even the (46)
g, (z,M') = g(z,M')j (M'+z''1').
nonseparable Eq. (31) can be reduced to a one-dimen-
tional integral equation, or alternatively to a one- Inserting on the right of (44) and dividing by the two
dimensional eigenvalue problem of the Sturm-Liouville quadratic factors on the left, one then tries to reduce
type. We shall explain the basic idea for the simplest the result again to the form (45) by reassembling the
type of solution and for ma = mb only. The extension to three quadratic denominators into a cube [in a similar
other cases was carried out by eu tkosky and is described way as in Eqs. (42) and (A3) in the Appendix]. One
in the accompanying paper. sees at once that if M'= 1-~' the "mass term" repro-
Choosing ma=m. (=m, say), let us first examine the duces itself. Thus we set
separable case, Eq. (24). In momentum space, the
equation has the form g(z,M')=g(z)o(M'-H~'). (47)
Carrying out the transformations indicated above and
(p'+m')'</>(p)=XI r!I>(p), (40)
writing
which is very similar to the nonrelativistic hydrogen (48)
equation in momentum space. The latter, of course, is we find
a three-dimensional equation and does not have the
square power on the left, but it will appear that the
analogy is closest when the two changes are made
q,(p)=!X J Q-'(z)g(z)dz J+1 dy {
-I 0
simultaneously. Xxdx[P'+2il"p·~+1-~']-·, (49)
In particular, the ground-state wave function of
hydrogen: </>(p) = (p'+po')-', is duplicated here by the l"=xy+(l-x)z.
solution 14 An expression of this type has a certain resemblance to the

</>(p)= (p'+m')-3 (41) parametric representations for SF' and tip' developed by M.
Gell-Mann and F. E. Low, Phys. Rev. 95, 1300 (1954). G. Kiillen
[Helv. Phys. Acta 25, 417 (1952)] has previously used similar
corresponding to the eigenvalue X= 2m'. That (41) representations for other quantities that are a little less closely
satisfies Eq. (40) can be verified most easily if one related to the B-S wave function, Eq. (I). In the case of these
first writes, a la Feynman; quantities, and of the functions SF' Ap' J it is possible as the

l'
above-mentioned authors have shown, to derive the general
form of the parametric representation from the definition of the
(k'-2k·p+p')-I(k'+m')--"= 3(1-x)'dx quantities, and from considerations of relativistic invariance.
o The author has not been able to do the same for Eq. (1). Ntver-
theless the analogy with SF' and III was used to "guess" the
X[(k-xp)'+ (l-x) (m'+xp')]-'. (42) form of Eq. (45).
COY ARIANT PICTURE OF QUANTUM BOUND STATES 255

1132 G. C. WICK

Eliminating y in favor of I, and carrying out the The lowest eigenfunction simply develops a kink at
integrations over x and Z first, (49) acquires indeed the Z= 0, while the behavior of the higher states is more
general form required by (45) and (47). Writing that the complicated; if one inserts the approximation (55)
two expressions are identical gives an integral equation into (52), one finds
for g(z).
To this end notice that if z in (45) is allowed to vary g(t)=!... (I_~2)-lg(0)A(I-ltl)' (56)
+
between -1 and 1, I will also vary between the same which requires
limits. Writing dl=xdy and noting that for given
z and \, A= (2/...)(1_~2)1. (57)

f dx=R(t,z) = I (1+\)/(Hz) if z>\

(l-t)/(I-z) ifz<l'
(SO)
This is, of course, just what one expects from the non-
relativistic Balmer formula for the lowest eigenvalue.
Clearly the limit ~2-> 1 requires a more careful
treatment for the higher eigenvalues. The reason is
one finds
that all the nodes of the eigenfunction tend to concen-
q,(p) = f+l 'Y(l")[p2+2itP'~+I-~2J-3dt, (51)
trate near Z= 0 so that the approximation (55) is not
adequate.
-1
It is easy to see that A does not tend to zero for the
where 'Y(t) is given by the right-hand side of Eq. (52) higher eigenvalues. Thus, none of the higher eigenvalues
below. The condition g='Y thus gives the integral of Eg. (54) has anything to do with the states known
equation from the nonrelativistic case. It will be shown by
Cutkosky' that the other known states are contained
in other families of solutions of the B-S Equation;
g(t)=!A f+l R(t,Z)Q-l(Z)g(z)dz. (52) each of these families, however, contains in addition
-1
"abnormal" solutions that have no nonrelativistic
This is, of course, an integral equation of Fredholm's limit.
type, and has a discrete A spectrum. We thus have We shall now examine the behavior of the "abnormal"
achieved the surprising result that the B-S equation eigenvalues of Eq. (54) when ~-> 1 and show that all
(44), although nonseparable (as far as we can tell), can these eigenvalues converge to a common limit A->t.
be reduced to a one-dimensional problem. First we can see that A< t cannot be an eigenvalue
other than (57). Consider in fact the second eigenvalue;
Further Reduction of the Problem the corresponding eigenfunction must be odd and have
a node at z=O. Hence we need only examine a solution
Equations (45), (47), and (52), of course, do not give of (54) with the boundary conditions g(O) = g(l) = O.
all the solutions; they do not even give all the S states. Assuming
The necessary generalizations, however, are natural
and will be described in the accompanying paper. Let us 1-~2«1, (58)
instead study (52) a little further. From (50) and (52)
one can see that we divide the interval 0-1 into two parts,

g(+l)=g(-l)=O. (53) O<z<zo and '0<z<1, (59)

Furthermore, differentiating (52) twice, we get choosing Zo to satisfy

g"(Z)+A(1-Z2)-IQ-l(Z)g(Z)=0. (54) (60)
These equations formulate the problem as a Sturm- In the first interval we write the equation with a slight
Liouville eigenvalue problem. Thus it is easy to predict change of variables,
qualitatively the dependence of A on ~2.
Thus consider first ~2=0; then g(z) = (l-z 2) is a x=~(1-'1')-lz,
solution, and clearly it corresponds to the lowest (61)
d'g/dx'+A(Hx')-lg=O,
eigenvalue since it has no nodes. The higher solutions
are also polynomials." The lowest eigenvalue is A= 2, neglecting terms of order::; (J _~2)1. (One can see a
as we know already. The "potential" Q-l(Z) is an poste:riori that this approximation is justified for our
increasing function of the parameter ~2. Hence every purposes.) Equation (61) is of Riemann's type, and the
eigenvalue A must decrease as ~2 increases. solution we want is
When ~2->1, Q-l(Z) develops a singularity at z=O,
in fact, g=g+-g-,
(55) g±= (Hx'),F 1(Hp, !-p; 2; W±ix», (62)
Hi See the general discussion in reference 6. p=a-A)I.
256 CHAPTER IV

BETHE-SALPETER WAVE FUNCTIONS 1133

In the second interval we write Q(z) = z', again small values of u are of the form
neglecting terms of order (J _'1/')1 at most, and write •
a=au, .B=bu, (68)
s=z', d'g/ds'+!s- I dg/ds+1>-g/(I-s)s'=0, (63) a and b being constants, whose precise value we shall
which again is of Riemann's type. The solution satisfy- not determine.
ing g=O at z= 1 is Obviously (64/1) and (67') can be joined smoothly if
a-ti-!u In(I-'I')=n1r, where n is an integer. If 1-1/'
g= (l-z')zi+ P.F,(ll+!p, t+!p; 2; l-z'). (64) is so small that -In(I-'I')>>I, the above equation
will have small roots u so that by using (68),
We will first show that if >-<1, the "internal" and
"external" solutions (62) and (64) cannot join smoothly U= (;I.-i)!=n7r/[a-b-t In(I-1/')]. (69)
at Z=Zo, i.e., x=xo=1/zo(I-1/')-I. In fact, since xo»I,
we may evaluate (62) by means of the asymptotic To an even cruder approximation, one has
formula for the hypergeometric function. One finds, q~-2...n/ln(I-1/'); >-= 1+[21rn/ln(I-1/')J'. (70)
omitting a proportionality factor,
Equation (70), for n=O, 1, 2,.·· gives an infinity of
gint~xP+I(H"')+A (P)x-P+!(H"')' (62') eigenvalues all tending to >-=1 when 1/'->1. It should
be pointed out that these correspond to odd eigenfunc-
The dots indicate expansions in powers of X-I, and since tions. In a similar way one can show, however, that the
p <! it is consistent to keep the first term of the
same formula, with 2n replaced by 2n+ 1, gives the
second expansion, while neglecting the higher terms of
eigenvalues for the even eigenfunctions.
the first expansion. Furthermore, About the possible significance of these "abnormal"
A (p) = 2'p tan(!?r-!1rp)r(2p)/r( -2p) (65) solutions we shall not try to speculate here. Since they
occur only for finite values of A (>- ~ 1), it would be
is a negative quantity which varies from 0 to -1 as unwise to assume that they are a property of the com-
>- varies from 0 to 1. plete B-S Equation. Certainly the ladder approxima-
Similarly, (64) may be evaluated for small values of tion cannot be trusted to that extent. If the theory is
z by means of the known transformation of pea, b, c, used only for small values of the coupling constant,
l-s) to hypergeometric functions of the variable s. One the abnormal solutions do not exist, in the case we have
finds studied, and no contradiction with known facts can be
g.xt~zp+I(H· .. )+ B(p)zP+I(H' .. ), (64') established. Nevertheless it would seem that these
solutions deserve further study.
where the dots now indicate expansions in powers of z',
and ACKNOWLEDGMENTS
The present work was begun while the author was a
guest of the Institute for Advanced Study, Princeton,
is a quantity which on the whole interval 0<>- < 1 New Jersey. The author is happy to acknowledge his
(O<p<!) stays quite close to -1 (and is in fact <-1). indebtedness to the Director of the Institute, Professor
Rewriting (62') in terms of the variable z and J. R. Oppenheimer, for the stimulation and encourage-
omitting again a proportionality factor, we find ment he derived from a year's stay at the Institute.
Various members of the Institute, in particular,
gint~zP+I+A (P)(1-1/')pz-P+I, (67)
Professor F. Dyson, Professor G. Kiillen, Professor
which is of the same form as (64'), but with a coefficient A. Pais, Professor W. Pauli, and Professor R. Jost gave
for the second term which is smaller than B (P) in kind encouragement and invaluable criticism. Special
absolute value, for all values of p in the stated interval. thanks are due to Dr. Murray Gell-Mann for suggesting
Hence (64') and (67) can never join smoothly. In that the analogy discussed in reference 14 might be
addition it is easy to verify that the slope g'/ g is larger of help.
for (67) than for (64'), as one expects if >- is too low
APPENDIX
to be an eigenvalue.
Let us now turn to the case >->1. One can see that We shall construct here the Green's function G(x),
essentially the same formulae will hold, except that p which is a solution of (p.'+ma')(pb'+mb')G(x)=o(x),
will be a pure imaginary, say p=iu, u= (A-i)l. One pa and Pb being defined by Eqs. (13) and (28), with
sees, then, that (64') and (67) take the respective forms p= -i Grad. We shall calculate G for the general case
ma~mb, since this involves no additional difficulty.
g•.xt~ZI sin(u Inz+.B) (64/1)
Using Fourier transforms, one sees at once that
and
gint~zl sin(u lnz-!u In(I-'1')+a), (67')
G(x) = (2'-)-'j [(Pa'+m.') (p.z+mb')J-1eipx[dPJ. (AI)
where a and .B are phases depending on u, which for
COVARIANT PICfURE OF QUANTUM BOUND STATES 257

1134 G. C. WICK

In the following we use !(m.+mb) as the unit of mass, XHO'(iz). The asymptotic behavior of (A7) when
setting • R->oo in a specified direction (i.e., keeping xJ R
m.=1+.<l, mb=1-.<l. (A2) constant) is found noting that Ko(z)~(.../2z)I.-'. The
exponential part of (A7) is then
Furthermore we transform, d fa Feynman,

[(p.'+m.')(Pb'+mb')J-1=! J +l
[p,y,.<lJ-'dy, (A3)
G(x)~· .. J dy exp[ - Rf(y)J,
-1 where
where
fey) = [(1- '1/') (1-y')+ (y+.<l)'JI- E(y+.<l)X.R-' .
[p,y,.<lJ= P'+ 2i(y+.<l) (p.~)
+(1-~')(1+2y.<l+.<l'). (A4) It is easy to see that f(Y»O in the whole interval
-l="y="+1. Hence G(x) satisfies the boundary
Furthermore, applying to Q= [p,y,.<l J the formula condition G->O as R->oo in any direction. H Ym is the
point in the interval where fey) is a minimum, then the
(!'= f'" e-aQada strongest factor in the asymptotic dependence of G(x) is
o G(x)~exp[ -Rf(ym)]. (AS)
and inserting into (Al), the integration over p may be Notice that Ym depends on the direction. Consider, for
performed, with the result example, the simplest case .<l=O. Then if Ix.1 <ER, Ym

J f
is defined by the minimum condition
+1 '"
G(x) = (3211"')-1 dy a-1da YmER=x,(l-,'+y'E')I; (A9)
-1 0
that is, writing x,/R=cosO" ymE=(1-o')lcotO,. If
Xexp[ -aU-iR'a-1+(y+.<l)(x~)J, (AS) IcosO,l > E the root (A9) is not inside the interval, so
with the minimum of fey) occurs at y=±l, according as
u= (1+2y.<l+.<l') (l-'1')+1]'(Y+.<l)'. (A6) cosO,~O; summarizing, one has

Owing to (29), U is positive for Iyl ="1; hence the IcosO.I>, f(Ym)=l-ElcosO.1
(AIO)
integral over a in (AS) is always meaningful. IcosO. I < E f(ym) = (1- E')I sinO,.
We then find that
Notice that in the latter case,

G(x) = (4'lI")-'e' dZ J-1

+1
dy G(x)~exp[ -(1-E')lrsinO.J=exp[ -(1-E')lrJ,

if r is the length of the space component of x. In the

Xe""·K o(R(1-1]'+.<l'+2y.<l+Y''I/')I), (A7) former case, instead, G(x).-v.-R+.I .. I; in particular, in
the time direction G(x) tends to zero like exp[ - (1- E)
where Ko(z) is the modified Hankel function, i(1I"/2) xlx.IJ·
258 CHAPTER IV

PHYSICAL REVIEW D VOLUME 8, NUMBER 10 15 NOVEMBER 1973

Covariant Harmonic Oscillators and the Quark Model*

Y. S. Kim
Center for Theoretical Physics, Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742

Marilyn E. Noz
Department of Physic~ Indiana University of Pennsylvanill, Indiana, Pennsylvanill 15701
(Received 22 March 1973; revised manuscript received 20 July 1973)
An attempt is made to give a physical interpretation to the phenomenological wave function o~ Yukawa,
which gives a correct nucleon form factor in the symmetric quark model. This wave function is first
compared with the Bethe-Salpeter wave function. It is shown that they have similar Lorentz-contraction
properties in the high-momentum limit. A hyperplane harmonic oscillator is then introduced. It is shown
that the Yukawa wave function, which is defined over the entire four-dimensional Euclidean space, can be
interpreted in terms of the three-dimensional hyperplane oscillators. It is shown further that this wave
function satisfies a Lorentz-invariant differential equation from which excited harmonic-oscillator states can
be constructed, and from which a gauge-invariant electromagnetic interaction can be generated.

I. INTRODUCfION world together with our field-theoretic common

sense once led us to conclude that the harmonic-
The quark,I,2 which was originally introduced to oscillator potential, which is manifestly analytic
explain SU(3) symmetry and its consequences, has at the origin, cannot be the fundamental force be-
gained considerable ground as a fundamental con- tween the quarks." However, an encouraging de-
stituent particle in hadrons. The inventors of the velopment was that the relativistic effect suitably
quark did not make any commitment to its exis- applied on the Gaussian wave function eliminates
tence. 1 In spite of this and other well-known dif- this unwanted exponential dec rease and gives the
ficulties, model calculations based on this con- desired dipole effect. 10-13
stituent particle are producing increasingly en- The above-mentioned relativistic models are
couraging results.' essentially one or another form of the Gaussian
In both the successful and the disappointing wave function multiplied by a Lorentz contraction
features of the quark model, there seems to be factor, and they do not necessarily represent a
one crucial question: What "forces" are respon- completely consistent picture of the relativistic
sible for making quarks stay together in hadrons?4 bound state. The important fact, however, is that
In the early days of the quark model, quarks were all those ''wrong'' models give a correct form fac-
put into an infinite potential purely for conve- tor. We are thus led to believe that there is some
nience" ,s and no attempts were made to assert truth in the Lorentz contraction of quantum-me-
that these simplified forces were of fundamental chanical wave functions. 14
importance. In the symmetric quark model, for We realize that there are no completely con-
instance, Greenberg used the harmonic-oscillator sistent relativistic measurement theories and that
potential in order to borrow the well-known for- we are not going to solve this difficult problem in
malism from the nuclear shell model. this paper. For this reason, we can give relativ-
As the resonance spectrum became richer, the istic interpretations only in terms of the existing
search for quantum numbers that correspond to languages that have been developed to answer this
binding forces continued.· It was Kim and Noz' ultimate question. One commonly used language
who established the existence of harmonic-oscil- is the Bethe-Salpeter equation. 14 ,IS This equation
lator-like radial modes for nonstrange baryon is well known to us and has been used extensively
resonances for which there is barely enough ex- in both low- and high-energy physics. I'
perimental data to test the linearity of the three Another important language developed for the
lowest energy levels. same purpose is the hyperplane generalization of
There are numerous calculations of decay rates the Schrlidinger equation. The concept of a space-
in the harmonic-oscillator model. 8 A more im- like hyperplane played a crucial role in the early
portant analysis seems to be that of elastic form days of quantum field theory, I, This hyperplane
factors. The first objection to the use of the har- technique was used recently by Fleming to under-
monic oscillator, that is the GaUSSian, wave func- stand the Newton-Wigner localization problem. 18 ,1"
tion is that the form factor decreases exponentially We shall use this hyperplane language in order to
for larget values. This discrepancy with the real understand Lorentz-contracted Gaussian wave

8 3521

Reprinted from Phys. Rev. D 8, 3521 (1973).

COVARIANT PICfURE OF QUANTUM BOUND STATES 259

3522 Y. S. KIM AND MARILYN E. NOZ 8

functions. by differen! mathematical techniques. However,

We are specifically interested in the covariant the inberent similarities enabled the creators of
oscillator wave function first introduced by quantum mechanics to formulate a new concept of
Yukawa'o and used by Fujimura et al. ,o .lI in their bound states in terms of the quantum superposition
successful calculation of the nucleon form factor principle. By studying the properties that are
in the symmetric quark model. In spite of their common to the Bethe-Salpeter and the Yukawa
numerical success, there does not seem to be wave functions, which have different mathematical
any physical basis for the covariant differential forms, we expect to work toward finding a possible
equation from which the wave function is de rived. new form of relativistic dynamiCS.
Thus it is fair to say that the Yukawa wave func- Since the Bethe-Salpeter equation and its wave
tion has been a purely phenomenological entity. functions are well known,15 we will only desc ribe
The purpose of this paper' is to give a physical here how Yukawa arrived at his covariant har-
meaning to this wave function in te rms of the ac- monic-oscillator model. Yukawa noticed that
cepted relativistic languages. Born's reciprocity relation21 gives an oscillator-
In Sec. II, we compare the Yukawa wave function like Hamiltonian and attempted to write down a
with the Bethe-Salpeter wave function. It is normalizable wave function in terms of the rela-
pointed out that botb wave functions are to be in- tive internal coordinates. The covariance re-
tegrated over the four-dimensional Euclidean quirement, however, forced him to introduce
space in the low-momentum region. We note that time-like excitations with negative energies. As
both the Yukawa and the Bethe-Salpeter wave a consequence, the energy levels were infinitely
functions have the same Lorentz contraction prop- degenerate. In order to eliminate this undesir-
erties in the large-momentum limit. Since the able feature, Yukawa introduced a coupling with
Bethe-Salpeter equation is a field-theoretic model, 'an external momentum. His wave function takes
we believe that tbis is the point where Yukawa's the form
nonlocal tbeory makes contact with local field
tbeory. 'l1(x,p)=exp{ -tw[ x'+2(p'x)'/m']} , (1)
In Sec. III, we introduce the hyperplane tech-
nique. The nonrelativistic harmonic oscillator where x is the -relative space-time four-vector
can be generalized to covariant hyperplanes. We and P is the total four- momentum of the bound
present a hyperplane interpretation of tbe Yukawa state. Throughout this paper we use the space-
wave function which is consistent with the Lorentz- favored metric where x' = (i)' - xo'.
invariant probability and the observed nucleon The bound-state Bethe-Salpeter Green's function
form factor. takes the form 14 ,l5
In Sec. IV, we discuss a Lorentz-invariant dif-
ferential equation which the Yukawa wave function G(x,p)=(b)' .{dClCOS(1ClP·X)
satisfies. This equation can generate a gauge-
invariant electromagnetic interaction. It is shown
tbat this harmonic-oscillator differential equation
can be separated in the normal coordinate vari- (2)
ables which are Lorentz transformations of the
space-time variables, and that the excited states This Green's function is seen to be a function of
can be constructed in this normal coordinate sys- x andP as in Yukawa's function above [Eq. (1)J.
tem. A Lorentz-invariant mass eigenvalue is The mass of the bound state is given by m. We
given. consider here the bound state of two equal-mass
In Sec. V, we discuss briefly the experimental particles whose individual mass is M. This
basis upon which the harmonic-oscillator quark Green's function is not the solution of the equation
model is built. but contains most of the features of the exact wave
function. 15
II. PROPERTIES OF THE YUKAWA AND THE We are now ready to discuss the covariance
BETHE-SALPETER WAVE FUNCTIONS properties that are common to Eq. (1) and Eq. (2).
We start from the rest frame where Ii = O. In this
In this section, we compare the covariance prop- system, Eq. (1) becomes a harmonic-oscillator
erties of the Bethe-Salpeter and tbe Yukawa wave wave function in the four-dimensional Euclidean
functions. In the early days of nonrelativistic space of i and t, and is manifestly normalizable.
quantum mechanics, the standing-wave properties We can make the Bethe-Salpeter Green's function
for the square well, the harmonic oscillator, and of Eq. (2) normalizable in the four-dimensional
the other bound-state potentials were described Euclidean space of (i, t ) by making the Wick rota-
260 CHAPTER IV

8 COVARIANT HARMONIC OSCILLATORS AND THE QUARK MODEL 3523

tion. IS
As we increase i pi , this property holds for
Eq. (2) until the kinetic energy becomes larger
than the binding energy." For I pI larger than the
'.t: - z
binding energy, the Bethe-Salpeter wave function
is no longer normalizable in the above-mentioned
four-dimensional Euclidean space. The harmonic-
oscillator wave function of Eq. (1) does not suffer
from this effect and remains normalizable for
large values of I pI . This is expected because
particles bound by an oscillator potential have in-
finite binding energy.
Let us rewrite the oscillator wave function as-
suming that p is in the z direction. We use E for
Po and p for P•. Then
'iI(x,p) ~exp [ - !w(x' + l)l
xexp{(- w/4m')[ (E - pY'(t +z f
FIG. 1. Lorentz-contracted wave func tions with two
+(E+p)'(t -d]). (3) equal and opposite momenta. The form-factor integral
of Fujimura etal. receives contributions primarily from
For large p, the small overlapping region.
w(E-PY' _~('!!)'
4m' 16 p , shrinkage is responsible for the nonexponential
(4)
decrease of the form factor.
w(E+p)' _ w(P..)' In Eq. (6), the integral is performed over Eu-
4m' m clidean space-time. We know clearly the physical
Thus meaning of the probability distribution over the
three-dimensional space, but we do not know what
"'(x,p)-exp[ -~w(x '+y')l physics, if any, the time like probability distribu-
x exp[ - t.w(m/pY'(t +zl'J tion corresponds to. We shall discuss this problem
in Sec . III.
xexp[-w(pl m)'(t - z )' J. (5)

The last factor becomes (f; Iw )(m/P)6(i- z) for Ill. HYPERPLANE FORMALISM OF
HARMONIC OSCILLATOR
large p, and the dependence on the variable (I +z)
becomes insensitive by the factor (mlp Y'. This Here we study Yukawa's phenomenological wave
contraction behavior is strikingly similar to that function from the point of view of the nonrelativis-
of the Bethe-Salpeter equation. 14 The Bethe- tic-harmonic-oscillator wave function, generalized
Salpeter wave function is a model derivable from to covariant hyperplanes .
field theory. The oscillator function is a phenom- Let us start with the non relativistic harmonic
enological wave function giving correct form fac- oscillator. The Hamiltonian is separable and the
tors. It is interesting to note that these two wave wave function is Gaussian multiplied by the ap-
functions have the same Lorentz contraction prop- propriate polynomials corresponding to excited
erties in the large-p limit. energy levels. Because the ground-state wave
We now restrict ourselves to the Yukawa wave function depends only on (x)' in the exponent, we
function . Let us analyze the form factor calcula- x
can Lorentz- generalize to the three-vector on
tion of Fujimura et al. lo in the Breit system. We the hyperplane which is perpendicular to the total
can sketch the initial and final "Lorentz-con- four-momentum of the system. We follow the
tracted" wave functions as in Fig. 1. The form- standard method of constructing this three-vector
factor integral and

F(q') ~ fd 4 x I/J i (x)I/J ,(x) exp(iq· x), (6) - -(" P.L)

xl-l- ulJ+ m2 xv' (7)

where q is the momentum transfer, receives con- When the momentum p is zero, l, becomes x.
tributions only from the small overlapping region For nonzero p,
indicated in Fig. 1. This region shrinks as the
momentum transfer increases, and this coherent
COY ARIANT PICTURE OF QUANTUM BOUND STATES 261

3524 Y. S. KIM AND MARILYN E. NOZ

and lion, it cannot give the transition probability be-

p. y\' tween two such wave functions. We believe that
fflX ::;:xiJx + (-.-- \ (8) this is one of the most pressing problems of our
';1 11 mi'
time and that we can solve this problem only by
Assume now that Ii is in the z direction. Using building models that can produce the observed ex-
{J for vic, Eq. (8) becomes
P for P, and perimental data.
(9) In order to build such a model, we go back to
our original rule that expl-~w(p, xlm)" 1 multiply
The three independent hyperplane coordinate vari- each wave function and that the integral be per-
ables are formed over the entire four-space. Then the in-
x, y, and (1_1l')-U2(z - pi). (10)
ner product becomes

The hyperplane ground-state oscillator wave

function then takes the form
(13)

There are two important differences between Here again the integration measure d'x is hyper-
the above wave function and that of Eq. (1). First, plane-independent and is good for both the Il, and
the coefficients of (p. xlm)2 are different. In Eq. the ~,plane. The above expression becomes
(1), it is 2, while it is 1 in Eq. (11). Next, Eq. (1) Eq. (12) when {3, and fl, are equal.
is integrated over the entire four-space while The next and most crucial question is whether
Eq. (11) is integrated only over the three-dimen- the above inner product produces experimentally
sional hyperplane. The purpose of this section is measurable effects. The answer is contained in
to point out that we can indeed give a hyperplane the fact that because of the additional exponential
interpretation to the Yukawa wave function of Eq. factor, the form factor calculation with this inner
(1). product becomes exactly the phenomenological
The wave function given in Eq. (11), which de- form of Fujimura el al. which we discussed in
pends explicitly on {3, is the ground-state wave Sec. II. The Single-oscillator ground-state form
function. We can excite the harmonic oscillator factor becomes in the Breit system
just as in the nonrelativistic case. If we multiply
cp or its excited form by exp[ -~w(l- jl')-'(t -{Jz)'], F(q2)~ fd 4 xexp[ii'j.x] exp[-w(x'+y')]
it does not change the hyperplane oscillator be-
cause the variable -(I-iJ')-'f'(t-pz) is perpen- xexp[ _-;,(m 2 +2q2)(t' +z')J. (14)
dicular to the three hyperplane variables given in
Eq. (9). If we perform the integration over the For large q', the time integral is like a I)-function
variable -(1- iJ,)-u'(t - pz) after this multiplica- integral, and hence this form becomes that of
tion, this certainly leaves the hyperplane oscil- Licht and Pagnamenta 12 who proposed the instant
lator intact. Therefore we can write the inner (t ~O) probability integral.
product of two wave functions belonging to the We have thus generalized the time-independent
same hyperplane as harmonic oscillator to covariant hyperplanes, and

(¢", ¢m) ~ f expl-w(l- p')-'(I - (Jz)' J

then introduced a covariant inner product. This
operation leaves the hyperplane oscillator intact,
produces the Lorentz-invariant probability for
x ¢~ (x, iJl¢m(x, Sld'x. (12)
the states belonging to the same hyperplane and
The integration measure d 4 X is invariant under gives the correct nucleon form factor.
Lorentz transformation. Because of this the above At this point, we may mention that the mathe-
quantity does not depend on the hyperplane pa- matics of the covariant harmonic oscillator is
rameter fJ. Hence, we have introduced a multi- very similar to that of the quantization of the
plication factor, exp{ -~wr ~(p·x)12}, and an inner electromagnetic field. There are two well-known
product of the form of Eq. (12), while leaving the approaches to the electromagnetic field quantiza-
hyperplane oscillator intact. By doing this we tion. One uses the Lorentz gauge, and the other
have been able to show that the hyperplane proba- uses the Coulomb gauge. The Coulomb-gauge
bility is Lorentz-invariant. method is not manifestly covariant, but its main
Let us consider next the inner product between advantage is that we do not introduce unphysical
wave functions belonging to two different hyper- photons and thus we can make quick references to
planes. 22 Since non relativistic quantum mechanics the real world.
does not say anything about Lorentz transforma- There have been many previous attempts to
262 CHAPTER IV

8 COVARIANT HARMONIC OSCILLATORS AND THE QUARK MODEL 3525

understand the covariant harmonic oscillator.23 the Y variables which contain the P dependence.
In this paper, we used the hype rplane coordinates Thus we have to use Eq. (17) to construct excited
to avoid time like excitations. The advantages states. Because of the Lorentz invariance of the
are similar to those in the Coulomb gauge case. harmonic-oscillator operator, the excited-state
By eliminating completely the burden of handling wave functions also satisfy the differential equa-
those unphysical excitations, we have been able to tion of Eq. (18).
separate clearly what can be done and what can- We now write the excited-state solution as
not be done in the framework of non relativistic
quantum mechanics. We emphasize here that a If! ,Iy) =H'I IYI)H" (y,)H, 3(Y3)H,o(Yo)
relativistic measurement theory has yet to be con- xexp[ -~w(y' +Yo')] , (19)
structed."
where
IV. COVARIANT DIFFERENTIAL EQUATION (20)
AND EXCITED STATES
The above solution is possible because the start-
In the preceding sections, we studied a possible
ing differential equation of Eq. (17) is separable
physical interpretation of the Gaussian factor
and remains separable as we change the value of
which corresponds to a ground-state harmonic the total four-momentump. The quantum numbers
oscillator. In order to construct excited states, n, are separation constants. Our Lorentz trans-
we use the Lorentz-invariant differential equation
formation therefore preserves this separability.
which is needed in generating a gauge-invariant
Because of the minus sign in front of no, the eigen-
electromagnetic interaction of the harmonic-oscil-
values of Eq. (20) are infinitely degenerate. In
lator quarks. 11
order to remove this ambiguity, we set no=O;
We rewrite here the ground-state solution
the physics of this procedure has been discussed
in Sec. III. Thus
,\=w(N+l),
as where (21)
l/Jo(X,p) = IMy)
= exp[ -~W(y,' + Y,' + Y3' + Yo')] , (15) Since the separability is preserved, the no = 0
where condition is invariant under a Lorentz transforma-
tion. The covariant harmonic oscillator now has
Yl =x 1 , Y2 =.l2, three normal excitation variables, namely, Yu
Y3 = (1- {l')-'/'(X 3 - (3t), (16) y" and y" and they are preCisely the hyperplane
variables mentioned in Sec. III. They are 0(3)-
Yo = (1- (3')-l/'(t - (3x 3)' invariant within the hyperplane and generate co-
variant excited-state wave functions in exactly the
The above linear transformation is a homoge-
same way as in the non relativistic oscillator.
neous Lorentz transformation of the original co-
The eigenvalue A can serve as the mass of the
ordinate variables. Thus ,"o(x,p) satisfies the
covariant harmonic oscillator or as its mass
equation
squared. There have been many previous attempts

{-\7/+ 8::' +w2[(Y1'-Yo'JfWo(yl=,\/fo(Yl.

to express the bound-state mass as an eigenvalue
(17) of a differential equation. 8 ,IO,'O,23 In all these at-
tempts, except possibly that of Lipes,25 the x vari-
Since the transformation of Eq, (16) is a Lorentz ables are used to excite the harmonic oscillator.
transformation, we also have Since the Gaussian factor is separable in the x
variables only in the rest frame, the mass quan-
{_\7 + f, + w'[ (x)' - t' J}~o(X'P) =,\l/Jo(x,P). (18)
2
tum numbers are good only in that frame, and an
attempt to boost will bring in an infinite number of
unphysical wave functions.
Eq. (17) and Eq. (18) represent the same equation. It has been twenty years since Yukawa intro-
The form of Eq. (18) has been discussed in the duced the Gaussian factor corresponding to the
literature and is used in constructing a gauge-in- Lorentz contraction.20 The concept of quark did
variant electromagnetic interaction. not exist at that time. It was stated in Yukawa's
The Gaussian form of Eq. (15) is not separable paper that the differential equation representing
in the x-coordinate variables. It is separable in the coupling of the total momentum to the internal
COVARIANT PICTURE OF QUANTUM BOUND STATES 263

3526 Y. S. KIM AND MARILYN E. NOZ

oscillation is so complicated that the study of the

interaction of the internal mode with the external
field is very difficult. We have shown in this pa-
~.o",quantum
Baryons Mesons

per that the differential equation is similar to the numbers

non- non-

I
Klein-Gordon equation, and that the interaction
I
stronge strange
strange strange
can be manufactured in the usual way. ;

Y. CONCLUDING REMARKS Ii n= 0 A A
I
A
I
A
I
I
In this paper, we discussed, first, Lorentz
contraction properties of the covarient Gaussian
Ii n =I A -
A I! B
- I c iI
factor. We then proposed the use of the hyper-
plane technique to study possible relativistic in-
gredients in quantum mechanics. Finally, we in- I A-
I
n=2 C D D
troduced the normal-coordinate method in solving I
l
the covariant harmonic-oscillator equation, and
showed that this method is technically equivalent n=3
to the hyperplane method.
The normal-coordinate method is the most pow-
erful weapon in attacking harmonic-oscillator
FIG. 2. Summary of the present status of the multiplet
problems. It is a convenient way of describing scheme in the symmetric quark model. A means "ex-
cQvariantly the orbital and radial quantum num- cellent", B means "good", etc.
bers. Therefore we have studied in this paper a
possible theoretical tool which can link the basic
concepts of quantum mechanics to quantities that
In this paper, we have restricted ourselves to
can be measured experimentally.
nonstrange baryons. We realize that there are
The most widely available numbers that can be
some difficulties in pionic form factors.;o As we
both calculated and measured are decay rates.'
see in the experimental summary of Fig. 2, we do
Since the decay rate calculations are not sensitive
not yet have enough experimental information from
to the exact shape of the wave function, the dec ay
which a linear mass spectrum can be derived for
rate alone does not force us to accept the harmonic-
the mesons. Therefore we cannot and do not in-
oscillator model.
sist on the simple harmonic oscillator for the
The form factor study such as the one discussed
mesons. Consequently, we do not have to explain
in this paper strengthens our assertion on the
the above-mentioned difficulty at this time.
harmonic oscillator and enables us to relate the
observed curve to Lorentz contractions."
ACKNOWLEDGMENTS
The most important characteristic of the har-
monic oscillator is, of course, the linearity of This work was started when one of us (M. E. N.)
its eigenvalues. In order to study the linearity in was visiting the UniverSity. of Maryland during the
the observed mass spectra, we need at least three summer of 1972. This visit was supported by the
radial modes. For nonstrange baryons, we barely National Science Foundation. She would like to
have these three levels, and the present authors thank her colleauges at Maryland for the hospital-
studied this linearity. 7 ity extended to her during the summer.

*Work supported in part by the National Science Founda- 175, 2024 (1968). For the latest numerical analysis
tion Grant No. NSF GP 8748. ~he N:-=- 1 andN:-o 2 multiplets, see C. T. Chen-Tsai
1M. Gell-Mann, Phys. Lett. 8, 214 (1964). and T. Y. Lee, Phys. Rev. D 6, 2459 (1972).
'G. Zweig, CERN Report Nos. TH401 and TH412, 1964 7y. S. Kim and M. E. Noz, Nuo;o Cimento I1A, 513
(unpublished) . (1972). See also T. De, Y. S. Kim, and M.'E. Noz,
3J . J. J. Kokkedee, The Quark Model (Benjamin, New ibid. 13A, 1089 (1973).
York, 1969). SR. P. Feynman, M. Kislinger. and F. Ravndal, Phys.
4For the latest attempt to keep quarks inside the hadron, Rev. D~, 2706 (1971), and the references contained
see K. Johnson, Phys. Rev. D 6,1101 (1972). therein.
'0. W .. Greenberg, Phys. Rev. Lett. 13, 598 (1964). 'So D. Drell, A. Finn, and M. Goldhaber, Phys. Rev. 157,
60. W. Greenberg and M. Resnikoff, Phys. Rev. 163, 1402 (1967). -
1844 (1967); D. R. Divgi and O. W. Greenberg, ibid. 10K. Fujimura, T. Kobayashi, and M. Namiki, Prog.
264 CHAPTER IV

8 COVARIANT HARMONIC OSCILLATORS AND THE QUARK MODEL 3527

Theor. Phys. 43, 73 (1970). hyperplanes is a dynamical question. It is of course a

uFor a 'completeand through tre8tm('nt of the form- relativistic dynamical question. See J. Kogut and
factor calcu1ation, see R. Lipes, Phys. Rev. D~, 2849 L. Susskind, Phys. Rep. 8, 75 (1973).
(1972), 23For the latest discussion-of the covariant oscillators,
12A. L. Licht and A. Pagnamenta, Phys. Rev. D 2, llf}O see S. Tshina and J. otokoz:nva, Prog. Theor. Phys
(1970); ibid. 2, 1156 (1970). - 47,2117 (1972).
13G. Cocho, C:-Fronsdal, 1. T. Grodsh.'Y. and R. White, 24~ F. Chew, Phys. Rev D 4, 2:330 (l971). In this
Phys. Hev. 162, 1662 (1967). paper, Chew states that the construction of a theoreti-
14For a discussion of the Lorentz contraction of the cal model which is demonstrably compatible both with
13ethe-Salpeter wave function, see y, S. Kim and the quantum superposition principle and with relativis-
R. Zaoui, Phys. Rev. D 4, 1764 11971). tic space-time is one of the most pressing problems.
'"G. C'. Wick, Phys. Rev. 96, 1124 (1954). We agree with him. See also Y. S. Kim and K. V. Vasa-
1\'For the latest high-energy applications, see S. D. nda, Phys. Rev. D 5, 1002 (1972).
Drell and T. D. Lee, Phys. Rev. D Q, 1738 (1972); 25Lipesll uses the xp. ~f Eq. (7) as his independent vari-
,. H. Woo, wid. G, 1127 (1972). ables. They are not linearly independent, and they do
'·'S. Tomonaga, Pr-;;g. Theor. Phys. I, 27 (1946); not form a set of variables in which the four-dimension-
J. Schwinger, Phys. Rev. 82, 914 (1951). al oscillator equation is completely separable. They
'"G. N. Fleming, Phys. Rev~37, B188 (1965); G. N. are not the y variables we use in this paper.
Fleming, J. Math. Phys. 11-:-1959 (1966). 26For the latest experimental indication of the harmonic-
"T. D. Newton and E. P. Wlgner, Rev. Mod. Phys. ~, oscillator characteristic. see P. S. Kummer, E. Ash-
400 (1949). burn, F. Foster, G. Hughes, R. Siddle, J. Allison,
20 H. Yukawa, Phys. Rev. 91, 416 (1953). B. Dickinson, E. Evangelides, M. Ibboton, R. S.
21M. Born. Rev. Mod. Phys. 21, 463 (1949). Lawson, R. S. Meahurn, H. E. Montgomery, and W. J.
22It was pointed out by Kogutand Susskind that the pro- Shuttleworth, Phys. Rev. Lett. ~. 873 (1973).
blem of physical systems belonging to two different
COVARIANT PICTURE OF QUANTUM BOUND STATES 265

Orthogonality relation for covariant harmonic-oscillator wave functions

Michael J. Ruiz
Center or Theoretical Physics. Department of Physics and Astronomy.
University of Maryland. College Park. Maryland 20742
(Received 20 May 1974)

Orthogonality relations for the Kim-Noz covariant harmonic-oscillator wave

functions are discussed. It is shown that the wave functions belonging to different
Lorentz frames satisfy an orthogonality relation. Furthermore, it is shown that for
n = m transitions there is a contraction factor of (1 - o:-)(n + 1)/2, where a is the
velocity difference between the two Lorentz frames.

The covariant harmonic-oscillator wave functions recently proposed by Kim and

Noz 1 can be applied to a wide range of hadronic processes. The harmonic-oscillator
characteristics prominently show up in hadronic mass spectra. The Lorentz-
contraction properties of the oscillator wave functions can be seen in the nucleon
elastic form factors and other electromagnetic transition amplitudes.2
In their paper, Kim and Noz are primarily concerned with the probability
interpretation of their covariant wave functions. Their oscillator wave functions
satisfy all the requirements of nonrelativistic quantum mechanics and enable us to
extend the probability concept to the relativistic region. Kim and Noz, however, did
not explicitly calculate the overlap integral of harmonic-oscillator wave functions
belonging to two different Lorentz frames. The purpose of this note is to perform
the overlap integral.
Kim and Noz start with the following differential equation:

1J_ + a22 + 0)2[ ~)2 _ ]1 'I'(x) = A'I'(X) .

J
t2
2l. (1)
V2
at
x
This harmonic-oscillator equation is separable in the and t variables. Kim and
Noz then observe that the above equation can also be written as

Reprinted from Phys. Rev. D 10,4306 (1974).

266 CHAPTERN

~-V; + :6 + ro'[ G), - Y61} VlY) ~ 1.'JI{y), (2)

where the Y variables are the Lorentz transforms of the x variables:

Yl=Xl> Y2=X2'
Y3 = (1 - p
2) -*(X3 - Pt), (3)
Yo = (1 - p2rth (t - Px3) .
Equation (2) is also separable in the Y variables. The normalizable solutions in the Y
variables are the Kim-Noz wave functions. Their wave function has the form

'I').(y) =NHnt(Yl)Hn,(Y2)Hn.(Y3)exp[-zrof.Y
1.,-+2 2
+ Yo)], (4)

where A. = co(nl + n2 + n3 + 1), and N is the normalization constant. The

elimination of timelike oscillations can be done covariantly.l Since the transverse
oscillations do not undergo Lorentz transformations, we shall assume that
n 1 =n 2 =0 in the following discussion.
The purpose of this note is to evaluate the following integral:
(5)
where the Y' variables are the Y variables of Eq. (3) with W. We can evaluate the
above integral using the generating function for Hermite polynomials. This
generating function has the form

(6)

We can now use the integral

1(, ,r) ~[: r Jd xexp<-,' + 2sy,) exp{-r' + 2ry',)

4

x exp[-.!.co<? + Y6 + yn + /6)] (7)

2
to evaluate the integral in Eq. (5). Both Y and Y' are functions of x and t. The
transverse integrals can be performed trivially. For the t and z integrals we can use
the variables ~ and 11 defined as
1 1
z = ..J2 (~+ 11), t = ..J2 (~- 11) (8)

to complete the square of the exponent of the Gaussian factor. We obtain

/(s, r)= (1- a2)IJ.. exp[2rs(1- 0.2)*], (9)
where a. =(P - W)(1 - PP'rl. Since the power-series expansion of Eq. (9) contains
only equal powers of r and s, the integral of Eq. (5) vanishes unless m = n. When
m = n, we find from the coefficient of (rs t in the expansion of Eq. (9)
COVARIANT PICTURE OF QUANTUM BOU1\'l) STATES 267

TM (~, W) = (1 - a2 )(n + 1)/2. (10)

The above results can be expressed in the following orthogonality relation?

Tnm (~, W) = (1 - a2 )(n + 1)/2onm . (11)
The above result suggests that the covariant harmonic oscillators behave like
nonrelativistic oscillators if they are in the same Lorentz frame. If two oscillators are
in different frames, the orthogonality is preserved. The ground-state oscillator wave
function, consisting of one half-wave, is contracted by (1 - 0:-)112. The nth excited
state, consisting of (n + 1) half-waves, is contracted by (1 - a2 )(n + 1)/2.
The author would like to thank Professor Y. S. Kim for suggesting this problem.
ly. S. Kim and M. E. Noz, Phys. Rev. D 8, 3521 (1973).
2R. Lipes, Phys. Rev. D 5, 2849 (1972). Lipes calculates his transition amplitudes in the Lorentz frame
where the excited-state resonance is at rest, and his wave functions coincide with those of Kim and Noz in
this particular frame. For this reason, the use of the Kim-Noz wave function will lead to Lipe's
calculation.
3The first attempt to get the Lorentz contraction factor was made by Markov, who obtained a similar
result forn = 0 and P= O. See M. Markov, NuovoCimento Supp13, 760 (1956).
268 CHAPTER IV

Complete orthogonality relations for the covariant harmonic oscillator

F.C. Rotbart
Department of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Israel
(Received 12 June 1980)

Within relativistic quantum mechanics the complete orthogonality relations for the
covariant harmonic oscillator are derived. These relations include time-axis
excitations and are valid for wave functions belonging to different Lorentz frames.
In relativistic quantum mechanics (RQM), l the covariant harmonic oscillator
appears as a natural extension of the nomelativistic case. 2 As pointed out by Kim
and Noz3 in an earlier development of a covariant oscillator wave function, these
functions can be applied to a wide range of hadronic processes. The RQM
formalism in general, and the covariant oscillator wave functions in particular, allow
the probability interpretation to be extended to the relativistic domain. This raises
the important question of orthogonality between states in different Lorentz frames.

Reprinted from Phys. Rev. D 23, 3078 (1981).

COVARIANT PICTURE OF QUANTUM BOUND STATES 269

For spatial excitations, this was investigated by Ruiz.4 However, the RQM
oscillator functions differ from those of Kim and Noz, not only in that they derive
from a more general covariant formalism, but also in that time-axis excitation states
are not excluded from the ground state as they are in the formalism by Kim and
Noz. Of course, in both formalisms these time excitation states are necessary for
completeness of the Hilbert space. The purpose of this note is to calculate the
complete orthogonality relations for states in differing Lorentz frames, including the
time-axis excitations. We shall, to some extent, be following the paper of Ruiz.
The covariant harmonic oscillator in a stationary state is described bl

[!m [- V; + a~l] + ~ k(X' - xl)] t(x) = 1CIjf(x) • (1)

where m and k are positive constants and the x in'l'(x) denotes xf1= (xO, xl, xZ, x 3) .
Under the Lorentz transformation

Y3 = 1 (x 3 - /h0)' (2)

yO = ,),(x o _(3x3) , 1= (1 - ~zr~,

the above equation becomes

1 [
[ 2m -
vz + aY5
y
iP ] + '2k<Y
l.,-+z
- YO)J 'I'(y) = K\j1(y),
2] (3)

which has the solution

'l'k(Y) = NnlnZn3noHlI,(CJyI)HlI,(CJyz)HlI,(CJy3)HlI.(CJyo)exp[- ~ cr(jl + Y6)], (4)

where K is the invariant (nl+nZ+n3-nO+1)ro, <r=..Jmro,ro=..Jklm, and

NnlnZn3nO is the normalization constant. Since the transverse directions YI and Yz
are invariant under transformation (2), we set, for convenience, n I =n z =0 in what
follows.
Since the 'l'1I(Y) [which we shall now denote as 'l'n3nO(Y)], span a closed subspace,
we can express a state in another Lorentz frame, that is, a frame characterized by W
instead of ~ in the transformation (2), as a linear sum of 'l'n' 3nO(y); namely
(5)

where for orthonormal 'l'n3no

(6)

These are the orthogonality relations we wish to evaluate.

270 CHAPTER IV

Using the generating functions for the Hermite polynomials,

.. n
e-s'+2s1; = L :, Hn@' (7)
n=o .

the calculation of expression (6) entails the integral

/(s,r,u,v)= Jd 4 x exp[-s2+2sOY3-r2+2roy'3-u2+2uoy'0-v2

+2voy'0- ~cr(?+Y6 + y-n+y'6)]. (8)

where y and y' are, by (2), functions of x. Introducing the variables ~ and 11, defined
by

(9)

this integral (8) yields

/(s, r, u, v)
= 4n
2
(1 - ri)'h exp[2sr (1 - 0.2)'12 - 2su a. + 2rv a. + 2uv (1 - a.2)lh] , (10)
0-

where a. = (~ - W)(1 - ~~'rl.

This is expanded as a power series and the problem is to compare coefficients of
powers in s , r, u, and v with those of
n s/'rn',u nov n'o ..
/(s, r, u, v) = -2
0-
L ,',,' ,
''''''''''0 n3· n 3· n O· n o· -00
Jdx 3 dxoHn,(OY3)Hn,,(OY, 3)
xHno(OYo)Hn'o(OY' 0) exp[ - ~ cr (yf + YS + y' f + Y' 6)] . (11)

A typical term in the expansion of (10) is

~ [2sr(1 - a.2)1/2]a [-2sua.t [2rva.Y [2uv(1 - a.2)1/2]d (1 _ 0.)112
0-4 a! b! c! d!
n2 2a +b +c +d
=- (1- a.2)(a+d+lY2(_I)ba.b+csa+bra+cub+dvc+d, (12)
0-4 a!b!c!d!
with a, b, c, and d integers. Comparison with (11) shows that
a +b = n3' a +c = n' 3 ,
b + d =no, c +d = n' o. (13)
These are not independent since we see that
n3 + n' 0 = a + b + c+ d = n' 3 + no
or
COY ARIANT PICTURE OF QUANTUM BOUND STATES 271

no - n3 = n' 0 - n' 3· (14)

This is to be expected from the invariance of k. Hence for a given three of

n3, n' 3' no and n' 0 the fourth is defined, and we are left with three independent
relations between the four variables a, b, c, and d. For a given n 1, no, and n' 0, say,
we can take b as equal to some integer A, a free parameter, and we find that, for
n' 0> no,
a = n3 - A, b = A, c = n' 0 - no + A, d = no - A,
as A goes from zero to the smaller of n3 or no. In this way, for the given
n3, no, and n' 0, (12) shows the coefficient to be a sum over A, and so we arrive at the
final result

J'1ft n3nO(y )'lfn' 3n' o(y' )d 4x = B:-: =~l,(1 - ( 2)(no + n, + 1)/2 a n'o- no
II" (i a)2"'(1 - a 2r"'(n3!n' 3!nO!n' 0!)1I2
XL (15)
A.=O (n3 - A)!(no - A)!(n' 0 - no + A)!A!
for n' 0 ~ no. Note that the requirement that A be summed over the smaller of
no or n3 is automatically taken care of by the factorials in the denominators. In the
case of no ~ n' 0 we simply interchange the primed quantities on the right-hand side
with the unprimed ones and replace a with (-a).
For no = n' 0 = 0, these orthogonality relations reduce to the relations obtained by
Ruiz. In addition, it can be seen from (15) that Lorentz transformations contract the
wave function not only along the z axis, but also along the time axis. This is due to
the fact that time in the wave function is on an equal footing with the space
coordinates. A hypothetical observer will measure the extent of the wave function in
time, in his frame, at equal z in essentially the same way that he would measure the
length at equal time. This extent in time is not to be confused with the time interval
between two events which occur at the same z only in one particular frame.
It is amusing to note that in evaluating the sum in expression (15), the exponent of
(1 - a) ranges from ~ (no + n3 + 1) to ~ (I no - n31 +1). This behavior is
reminiscent of angular momentum representations and suggests that a "spherical"
coordinate representation for the covariant harmonic oscillator might be useful.
This research was partially supported by the Binational Science Foundation (BSF),
Jerusalem, Israel.
lL. P. Horwitz and C. Piron, Helv. Phys. 46,316 (1973).
2L. P. Horwitz and F. C. Rotbart (unpublished).
3y . s. Kim and M. E. Noz, Phys. Rev. D 8,3521 (1973); Found. Phys. 9 375 (1979); Y. S. Kim, M. E.
Noz, and S. H. Oh, ibid 9, 947 (1979).
4M. 1. Ruiz, Phys. Rev. D 104306 (1974).
272 CHAPTER IV

Dirac's form of relativistic quantum mechanics

D.Han
Systems and Applied Sciences Corporation, Riuerdale. Maryland 20840

Y.S.Kim
Center for Theoretical Physics, Department of Physics and Astronomy, University of Mary/and, College Park.
Maryland 20742
(Received 29 November 1979; accepted 22 January 1981)

It is shown that Dirac's "instant form" dynamics provides a theoretical framework in

which models of relativistic quantum mechanics can be constructed. The convariant
harmonic oscillator formalism discussed in previous papers is shown to be such a
model. Dirac's "point" and "front" forms are shown to generate a space-time
geometry convenient for describing Lorentz deformation properties of relativistic
extended hadrons.

I. INTRODUCfION and limitations of the present form offield theory. We are

not disputing here the numerical successes and promises of
Quantum mechanics and special relativity were formu- the present form of quantum field theory. However, we
lated before most of us were born and are likely to remain as should realize also that field theory has not yet provided a
the two most important scientific languages for many years physical interpretation to the space-time boundary condi-
to come. They are, in fact, the most important and exciting tion that is essential for formulating a bound-state picture
subjects for the students who plan to become physicists. in terms of localized probability distribution. 2 .3 For this
For these reasons, it is not uncommon for students in the reason, we are allowed to construct relativistic bound-state
first-year quantum mechanics class to ask the question of wave functions from the first principles of quantum me-
how these two physical theories can be combined. chanics and relativity, in a manner consistent with what we
The usual answer to this question is that quantum field observe in the real world, particularly in high-energy
physics.'
theory takes care of the job. However, this answer cannot
prevent the students from reasoning in the following man- In Ref. 1, it was pointed out that the relativistic harmon-
ner. The basic tool for quantum mechanics is Schriidinger's ic oscillator wave function indeed satisfies the above-men-
wave function, and the mathematical apparatus for special tioned requirement. Students are then likely to ask the fol-
relativity is the coordinate transformation known as the lowing question. There are two ways of doing quantum
Lorentz transformation. Then, why can we not construct mechanics. One way is to construct wave functions, and the
wave functions that can be Lorentz transformed? other approach is to write down a system of commutators
This is indeed a good homework problem not only for that is equivalent to the algorithm based on wave functions.
students but also for "grown-up" physicists. With this If the relativistic oscillator wave function discussed in Ref.
point in mind, Kim and Noz discussed the possibility of I is to represent a form of relativistic quantum mechanics,
including the relativistic harmonic oscillator formalism in then where is its broadly based commutator form?
the first-year quantum mechanics curriculum. I The oscil- The purpose of the present paper is to show that the wave
lator wave functions discussed in Ref. I are covariant, car- function approach presented in Ref. I is equivalent to Dir-
ry a probabilistic interpretation, and are mathematically ac's "instant form" quantum mechanics. 4 Dirac's dynami-
simple enough to be included in the first-year quantum cal system consists of the ten generators of the Poincare
mechanics curriculum. This oscillator model can explain group and one constraint condition that reduces the four-
basic high-energy hadronic features associated with the dimensional space-time into a three-dimensional Euclidian
Lorentz deformation of extended objects including the ha- space in which nonrelativistic quantum mechanics is ex-
dronic form factors, mass spectra, and the peculiarities in pected to be valid. We show in this paper that the oscillator
Feynman's patron picture. It was emphasized I that it is wave functions discussed in Ref. I represent a space-time
now possible to understand some space-time symmetry solution of Dirac's "Poisson bracket" equations'
properties of high-energy hadrons without resorting to the In Sec. II, we formulate Dirac's instant form quantum
techniques of quantum field theory. mechanics for a system of two bound-state quarks. In Sec.
In Ref. I, Kim and Nozexplain why this line of teaching III, the covariant harmonic oscillator formalism discussed
physics does not do injustice to students who will eventual- in Refs. I and 5 is shown to form a space-time solution of
ly learn quantum field theory, by reviewing the successes Dirac's "Poisson bracket" equations. In Sec. IV, the con-

Reprinted from Am. J. Phys. 49, 1157 (1981).

COYARIANT PICTURE OF QUANTUM BOUND STATES 273

straint condition of Dirac and subsidiary condition of the [p".p.] =0.

oscillator formalism are discussed in detail, and they are
shown to serve the same purpose.
[Mpv. Pp] = -g", p. +g." pp. (6)
In See. V, some geometrical consequences are derived [M•• ,Mpu]
from Dirac's "point" and "front" forms. It is shown that = -g", M~ +g"" M.a -g.aMp. +g~M...
the Lorentz deformation property of the harmonic oscilla-
Dirac emphasizes in his paper that the problem offinding a
tor wave functions is derivable from this space-time geome-
new dynamical system reduces to the problem offinding a
try. It is shown further that this deformation property is
new solution of the above "Poisson brockets."
consistent with the representation of the Poincare group
discussed in Sees. II and III. The unusual feature in the above generators is that L ••
operating on the x coordinate is not affected by translations
II. DIRAC'S FORMULATION OF RELATIVISTIC and acts as the spin operator for the hadron, while operat-
THEORY OF "ATOM" ing basically on the space-time coordinates. Therefore, the
problem of finding a solution of the commutator system is
Dirac's atom in modem language is a hadron that is a to find wave functions t/> (X. x), which form a representation
bound state of quarks and/or antiquarks. Let us consider of the Poincare group. The procedure of finding such a
here a hadron consisting of two quarks whose space-time representation is to construct the wave functions that are
coordinates are x, and x,. Then the standard procedure is diagonal in the Casimir operators"
to introduce the variables P'=P·P. and W'= W·W~, (7)
X=(x, +x,)/2,
where
(I)
W. = E.~pp·MaP.
x = (x, - x,)/2v'2.
These Casimir operators commute with each other and
Both X and x have their respective spatial and time compo-
with the ten generators given in Eq. (4). Physically. p' and
nents. The spatial variable appearing in the Schrooinger
W, specify the mass and total spin of the hadron.
equation for the hydrogen atom is the distance between the
proton and electron. and thus corresponds to the spatial respectively.
component of the x coordinate. The Schrooinger equation The final step in constructing Dirac's dynamical system
does not contain the time component of x that corresponds is to make the constraint condition consistent with the gen-
to the time separation between the constituent particles. erators of the Poincare group. Dirac noted in particular
In order to control this time-separation variable. Dirac that the constraint condition of Eq, (3) can be an operator
considered in his "instant form" the condition equation. and that its "Poisson brackets" with other dyna-
mical variables should be zero or become zero in the man-
ner in which the right-hand side of Eq. (3) vanishes. We
Xo:::=:-O, (2) shall examine this point in detail in Sec. IV,
whose covariant form is
III. COVARIANT HARMONIC OSCILLATOR
xP~O, (3) FORMALISM
where P is the total four-momentum of the hadron. Equa-
In terms of the coordinate variables X and x defined in
tion (3) becomes Eq. (2) when the hadron is at rest. Dirac
Sec, II. the starting partial differential equation for the os-
did not use the exact numerical equality in writing down
cillator formalism can be written as'
the above constraint condition in order to allow further
physical interpretations consistent with quantum mechan- l [a /ax.)' + (a /ax,,)' - x; + m~]t/> (X, x) = 0, (8)
ics and relativity. In particular. Dirac had in mind the pos- with the subsidiary condition
sibility of the left-hand side becoming an operator acting on
state vectors. (a la~,la,;' '" (X, xl = 0
Dirac then points out that the relativistic dynamical sys- with
tem should consist of transformation operators that gener-
ate space-time translations. rotations. and Lorentz trans-
a.+ = x. + a/ax'.
formations. He points out further that the generators of the In Eq. (1), theX and x variables are completely separable,
Poincare group form the desired dynamical system. The and solution t/> (X, x) takes the form
generators in the present case take the form t/> (X. xl = ¢(x, P) exp( ± iPX). (9)
p,. = iwaX~) and M,,,. = L : •. + L~ •. , (4) where the internal wave function .p(x. P) satisfies the har-
where monic oscillator differential equation
L:, = i[X,. alar - X, a/ax"],
H(x)¢(x,P)=,z.p(x,P), 110)
(5) with
L •• = i[x" a/ax' - x,. ajax"~]. H (x) = l [(aJax,,)' - x; ],
The operators P" generate space-time translations. M~,. is The subsidiary condition in Eq. (8) becomes
antisymmetric under the interchange ofl' and v. ThreeMij' P"a,; ¢(x. P) = O. (1\)
with i,j = I, 2, 3, generate rotations, and three Mo. are the
The (mass)' of the hadron P 2 is constrained by the eigenval-
generators of Lorentz transformations.
ues of the above osci11ator equation:
These generators satisfy the following commutation
relations: p2=,z+m~, (12)

IIS8 Am. J. Phys., Vol. 49. No. 12, December 1981 P. Han and Y. S. Kim 11'8
274 CHAPTER IV

The harmonic oscillator equation given in Eq. (10) is sep- this requirement.
arable in many different coordinate systems. We are inter- What Dirac wanted from his "conditional" equality was
ested here in the coordinate that is most convenient for to freeze the motion along the time separation variable in a
constructing solutions that are diagonal in the Casimir op- manner consistent with quantum mechanics and relativity.
erators ofEq. (7). These operators take the simplest form in This means that we can allow a time-energy uncertainty
the Lorentz coordinate system in which the hadron is at along this timelike axis without excitations, in accordance
rest: with Dirac's own "C-number" time-energy uncertainty re-
x' = x, y' =y, lation. 7 This time-energy uncertainty without excitation is
widely observed in the relation between the decay lifetime
z'=(z-/1I)/(I-/1'),I2, (13) and energy width of unstable systems.' The C number in
I' = (t -/1z)l(1 _ /1')'/'. the matrix language is one-by-one matrix, and is the
ground state with no excitations in the harmonic oscillator
We assume here that the hadron moves along the z direc-
system.
tion with velocity parameter /1.
In te.-ms of the above coordinate variables, we can write We have observed in Sec. III that the subsidiary condi-
the Casimir operators as tion of Eq. (II) becomes Eq. (17) in terms of the coordinate
variables in which the hadron is at rest. Eq. (17) restricts the
P'= -(alaX~)' (14) I' dependence to that of the ground state. Equation (II)
and therefore eliminates all timelike excitations in the Lorentz
frame where the hadron is at rest, and makes the uncertain-
w, = M '(L')', (15)
ty associated with the t ' direction a C-number uncertainty
where relation. We can therefore conclude that the subsidiary
L: =- iEij,X;(alax'). condition ofEq. (II) is a quantum-mechanical form of Dir-
ac's "instant form" constraint given in Eq. (3).
We also have to take into account the fact that the hadron In order that the dynamical system be completely consis-
(mass)' operators is constrained to take the eigenvalues de- tent, the subsidiary condition should commute with the
termined by the oscillator equation for the internal wave generators of the Poincare group:
function and that we have to consider p' of the form
[Pa , P"a,; 1= 0,
p' = m~ + H(x'). (16) (19)
In addition, we have to consider the form of the subsid-
iary condition given in Eq. (II). In terms of this moving
[Mao' P"a"+ 1= o.
coordinate system, the subsidiary condition takes the form The above equations follow immediately from the fact that
the operator P"a"+ is invariant under translations and Lo-
(I' +alal')¢(x,P) =0. (17)
rentz transformations.
By using the moving coordinate system, we have achieved Since the Casimir operators are constructed from the
considerable simplification in the expressions for the Casi- generators of the Poincare group, we are tempted to con-
mir operator W' and the subsidiary condition, without clude that the constraint operator commutes also with the
complicating the forms for p' given in Eqs. (14) and (16). invariant Casimir operators. However, we have to note
This subsidiary condition restricts the t ' dependence to that that the operator p' also takes the form of Eq. (16). There-
of the ground state and forbids excitations along the time- fore, it should commute with H (x) given in Eq. (10). How-
separation variable in the Lorentz frame where the hadron ever, a simple calculation gives
is at rest.
As was discussed in Ref. I, the subsidiary condition of [H(x), P"a,; 1= P"a"+. (20)
Eq. (17) forbids timelike excitations that contribute nega- This means that the right-hand side is not identically zero,
tively to the total eigenvalue. This condition therefore but vanishes when applied to the wave functions satisfying
guarantees the existence of the lower limit in the (mass)' the subsidiary condition of Eq. (II).
spectrum. The absence of such timelike excitations are per- In his paper, Dirac considered also the commutation re-
fectly consistent with what we observe in the real world. lations between dynamical quantities and the constraint
After these preparations, it is a simple matter to write condition that is only "approximately" zero. He asserted
down the solutions which form the desired representation that the resulting "Poisson bracket" should also vanish in
of the Poincare group': the same "approximate" sense. The commutatorofEq. (20)
¢(X,x) = ifJ(x, P)exp( ± iPX), (18) indeed vanishes in the manner prescribed by Dirac'
It is well known that Dirac concludes his paper by noting
with some difficulties associated with the potential term in mak-
ifJ(x, P) = (1/1T)'I2[exp( - t "/2)]Rn(r')yrn(O', ¢ 'I, ing his system of "Poisson brackets" completely consistent.
where r', 0', ¢' are the spherical coordinate variables in a The crucial question is whether the harmonic oscillator
three-dimensional Euclidian space spanned by x, y, and z'. formalism can resolve Dirac's "real difficulty." We shall
attack this problem here by carrying out some explicit
IV. FURTHER CONSIDERATIONS OF THE calculations.
In formulating his scheme to solve the commutator
CONSTRAINT CONDITION
equations for the generators of the Poincare group, Dirac
As was pointed out in Sec. II, Dirac was interested in a chose to adopt the view that each constituent particle in
possible quantum-mechanical form of his "instant form" "atom" (bound or confined state) is on its mass shell, and
constraint of Eq. (3). The key question at this point is that the total energy is the sum of all the free-particle ener-
whether the subsidiary condition given in Eq. (II) meets gies and the potential energy. This potential term indeed

1159 Am. J. Phys., Vol. 49, No. 12, December 1981 D. Han and Y. S. Kim 1159
 COVARIANT PICfURE OF QUANTUM BOUND STATES 275

causes the real difficulty in making the commutator system and considered the constraint
self-consisten t. v = O. (26)
In the covariant oscillator formalism, we observe that
The basic advantage of using the light-cone variables is
the Casimir operators of the Poincare group clearly indi-
that the Lorentz transformation takes a very simple form.
cate that the mass of the hadron is a Poincare-invariant
For instance, the transformation given in Eq. 113) can be
constant, but they do not tell anything about the masses of
written as
constituent particles. Let us write down the four-momen-
tum operators for the constituents in terms of the X and x (27)
variables:
v' = [(l-P)/[1 +P)l'!2 v.

121) In discussing detailed geometrical properties, we can ig-

nore the transverse coordinates that are not affected by
Lorentz transformations. Let us first look at the point form
In order that the constituent mass be a Poincare-invariant constraint ofEq. (23) using the coordinate variables for the
constant, pi and pj have to commute with the Casimir front form. Eq. (23) then becomes
operators ofEq. (7) and with the harmonic oscillator opera- uv = u'u' = (t 2 - Z')/2 = A 14, 128)
tor H (x) of Eq. (10). The constituent (mass)' operators de-
where A is a constant. We are accustomed to associate the
rivable from the above forms do not commute with H (x)
above form with a hyperbola in thezt plane. It is interesting
due to its "potential" term. We have therefore translated
to note that this equation also represents a rectangle in the
Dirac's "real difficulty" into
uu plane as is specified in Fig. 1. The area of this rectangle
[p;,Hlx)]"cO, remains invariant under Lorentz transformations.
(22)
[Pi. Nix)] "cO. Let us next look at the space-time geometry of the covar-
iant oscillator wave function. Because of the oscillator
The above commutators indicate that the (mass)' of the equation in Eq. (10) is separable also in the Cartesian varia-
constituent quark is not a Poincare-invariant quantity. In bles x, y, z', and t', the solution given in Eg. 118) can be
1949, when Dirac's paper was written, the idea of an off- written as a linear combination of the Cartesian solutions.
mass-shell particle was not accepted, and therefore the The portion of the wave function that is affected by the
non vanishing of the commutators was regarded as a prob- Lorentz transformation is'i·'O
lem. Because of our experience with, among other things,
quantum field theory, we no longer regard the necessity of l/J(x, P) = H,,(z')exp[ -Iz" + 1")l2J. (29)
a bound particle being off-mass shell as a problem. Only the The localization property of this wave function is dictated
full bound system must be on the mass shell as is specified by its Gaussian form. If we write this exponential function
by the condition given in Eq. 116). in terms of the light-cone variables u and v,"
We note further that the concept of virtual off-mass-shell
particle is derivable from the violation of causality allowed l/Jlx,p)-ex p[ _ ~(I -P u' + 1+(3 u2 )]. (30)
by the time-energy uncertainty relation. 9 It is interesting to 21+P 1-P
see that Dirac's own work on this subject resolves the diffi- When the hadron is at rest and P = 0, the above wave
culty he mentioned in his 1949 paper4 function is concentrated within a circular region around
the origin. As the hadron moves and Pincreases, the circle
V, FURTHER GEOMETRICAL CONSIDERATIONS becomes an ellipse whose area remains constant. This spa-
In addition to the i~stant form, Dirac considered two cetime geometry is basically the same as that of the square
other kinematical constraints that generate three-dimen- and rectangle given in Fig. 1. The harmonic oscillator mod-
sional Euclidian subspaces of the four-dimensional Min- el is therefore a specific example of the more general space-
kowskian space-time. They arc "point" and "front" forms. time geometry derivable from Dirac's "point" and "front"
We do not know how to take advantage of these two forms forms.
in constructing representations of the Poincare group,lO It is important to note here that this form of hadronic
but they seem to provide a convenient space-time geometry Lorentz deformation is consistent with what we observe in
for discussing Lorentz deformation properties of the har- high-energy laboratories. 12-15 In particular, it is possible, in
monic oscillator wave functions, and thus of relativistic
extended hadrons.
In his point form, Dirac imposes the constraint
X,lX II = C = canst. (23)
This condition is not unlike the mathematics of the mass-
shell condition for free particles, and the resulting Eucli-
dian space consists of x, y, and z, with the time separation
variable constraint to take the values
t= ±[C+x'+y'+z,],n 124)
In his front form, Dirac introduced the light-cone
Fig. 1, Space-time geometry derivable from Dirac's point and front forms.
variables
The hyperbola represents the point-form geometry where ;r'x l , = cons!.
u= It + z)/v'2, The area of the rectangle is also Lorentz invariant. This "front form"
125) geometry is very convenient in describing Lorentz deformation properties
u = It - z)/v'2, of relativistic extended hadrons.

1160 Am. 1. Phys., Vol. 49, No. 12. December 1981 D. Han and Y. S. Kim 1160
 276 CHAPTER IV

special relativity.
'The most successful bound-state model in field theory is of course the
Bethe-&lpeter equation. However, the Bethe-Salpeter wave function
does not yet have proper quantum-mechanical interpretation. See Sec. I
" ofG. C. Wick, Phys. Rev. 96, 1124 (1954). The difficulty in giving a
physical interpretation to the relative time-separation variable between
two bound-state particles was mentioned earlier by Karplus and Klein.
" See R. Karplu. and A. Klein, Phys. Rev. 87, 848 (1952).
'Po A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949).
'Yo S. Kim, M. E. Noz, and S. H. Oh, Am. J. Phys. 47, 892 (1979); 1.
Math. Phys.20, 1341(1979).
Fig. 2. Space-time geometry for the instant form in the light--cone coordi· 'E. P. Wigner, Ann. Math. 40, 149(1939).
nate system. The usual quantum excitations take place along the z' direc- 'P. A. M. Dirac, Proc. R. Soc. wndon A 114, 243, 710 (1927).
tion. The "C-number" uncertainty relation holds along the I' axis, and HE. P. Wigner, in Aspects o!Quantum Field Theory, in HonourofP.A. M.
there are no excitations along this timelike direction. Dirac's 70th Birthday, edited by A. Salam and E. P. Wigner (Cambridge
University, wndon, 1972).
terms of thisLorentz deformation picture, to explain Feyn- 9W. Heitler, The Quantum Theory of Radiation, 3rd ed. (Oxford Univer-
man's parton phenomenon I. both qualitatively'7 and sity, wndon, 1954). See also D. Han, Y. S. Kim. and M. E. Noz, Found.
quantitatively. '8 Phys. (to be published).
'''Yo S. Kim, M. E. NOl, and S. H. Oh, J. Math. Phys. 21, 1224(1980).
Finally, let us describe the "instant form" quantum me- 11This exponential form is also derivable from Yukawa's work. See Eq.
chanics discussed in Secs. II, III, and IV using the light- (10) ofH. Yukawa, Phys. Rev. 91, 416(1953). For an interpretation of
cone coordinate system. As is seen in Eq. (29), the Gaussian this original paper, see D. Han and Y. S. Kim, Prog. Theor. Phys. 64,
form is diagonal also in the instant-form coordinate varia- 1852(19801·
bles. The z' and t' axes in the uv coordinate system are '2The fact that the proton (one ofhadrons) is not a point particle and has a
shown in Fig. 2. There are usual quantum excitations along space-time extension was discovered by Hofstadter. See R. Hofstadter,
the z' axis. Along the t ' direction, Dirac's C-number time- Rev. Mod. Phys. 28, 214(1956).
I 'Since Hofstadter's discovery, there have been many attempts to con-
energy uncertainty relation holds.
struct theoretical models for relativistic extended hadrons. See. for in-
stance, V. N. Gribov, D. L. loWe, and I. Ya. Pomeranchuk, J. Nuel.
VI. CONCLUDING REMARKS Phys. (USSR) 2, 768 (1965) or5ov. J. Nucl. Phys. 2, 549(19661; N. Byers
As we stated in Sec. I, quantum mechanics and relativity and C. N. Yang. Phys. Rev. 142. 796 (1966); J. D. Bjorken and E. A.
Paschos, ibid. 185, 1975 (1969); B. L. Iolfee, Phys. Lett. B 30, 123
are two of the most important subjects in the physics cur-
/1969); K. Fujimura, T. Kobayashi, and M. Namiki, Prog. Theor. Phys.
riculum. Because of its mathematical simplicity, the har- 43,73(1970); A. L. Licht and A. Pagnamenla, Phys. Rev. D 2, 1150,
monic oscillator is one of the most effective teaching instru- 1156(1970); S. D. Drell and T. M. Yan, Ann. Phys. (NYI60, 578 (19711.
ments. In their recent papers,"s Kim et al. emphasized that Y. S. Kim and R. Zaoui, Phys. Rev. D 4,1764 (1971); R. G. Lipes, ibid.
the oscillator model can serve an effective purpose in teach- 5.2849(1972); S. IshidaandJ. Otokozawa, Prog. Theor. Phys. 47, 21\7
ing relativistic quantum mechanics and high-energy phys- (19721; T. D. Lee, Phys. Rev. D 5,1738 (19721; G. Feldman, T. Fulton,
ics at the level of the first- and second-year graduate andJ. Townsend, ibid. 7,1814(1973). Y. S. Kim and M. E. Noz. ibid. 8.
curriculum. 3521 (1973). See also Refs. 1,2, and the references contained therein.
The wave functions in the oscillator formalism are com- '''Perhaps one of the current models of extended hadrons is the "MIT bag
patible with the known principles of quantum mechanics model," as is explained by K. Johnson in Sci. Am. 241 (1),112 (July
1979). One interesting question in this model is how "bags" would look
and relativity. However, what was missing in the past has
to moving observers.
been a broader theoretical base from which this specific '·'The quark confinement problem is regarded as one of the most impor-
model is derivable. In this paper, we have shown that this tant current problems in the particle theory front. The ultimate goal of
theoretical base had already been given by Dirac in his this program is to find a potential that confines the quarks Inside ha-
"instant" form quantum mechanics. drons within the field theoretic framework ofQCD (quantum chromo-
Since the appearance of Driac's original paper in 1949," dynamics). The batiic question is then this. What are we going to do with
this confining potential? The next step is naturally to construct bound-
many authors have made and are still making attempts to
state wave functions, which eventually leads to the question of their
construct solutions of the "Poisson brackets" given in Eq. Lorentz transformation properties. As was noted in QED (quantum
(6).'9 The point is that the commutator equations are basi- electrodynamics),l this does not as yet appear to be an easy problem.
cally differential equations without any specific form for For an introductory review article on QCD, see W. Marciano and H.
potentials. The final form of solutions therefore depends on Pagels, Phys. Rep. 36 C, 138(19781.
boundary conditions andlor forms of potentials. It is thus II'We have to say that the most important observation made on Lorentz-
possible to end up with solutions that are not covariant. 2" deformed hadronswas Feynman's parton model. See R. P. Feynman, in
In the present paper, we discussed a solution with a covar- High Energy Collisions, Proceedings of the 3rd International Confer-
iant form for potential satisfying a covariant space-time ence, Stony Brook, New York, edited by C. N. Yangetal. (Gordon and
boundary condition. Breach. New York, 1969); Photon-Hadron intcractions {Benjamin,
Reading, MA, 19721.
'Yo S. Kim and M. E. Noz, Am. J. Phys. 46, 484(1978). '1For an explanation of the peculiarities in Feynman's parton picture, see
2R. P. Feynman, M. Kislinger. and F. Ravndal. Phys. Rev. D 3, 2706 Y. S. Kim and M. E. Noz, Phys. Rev. DIS, 335(19771. For a graphical
(1971). The point of this paper is that the inventor of Feynman diagrams interpretation of the formulas in this paper, see Y. S. Kim and M. E. Noz,
stated that it is not practical, if not impossible, to use Feynman dia- Found. Phys. 9, 375 (19791.
grams for relativistic bound-state problems. Feynman et al. suggested 1KFor a calculation of the proton structure function, see P. E. Hussar.
that the relativistic harmonic oscillator model, even if it is not totally Phys. Rev. D 23. 2781(1981).
consistent, can serve useful purposes. The point of Ref. I is that the I~For one of the most re(..'Cnt papers on this subject, see A. Kihlberg, R.
oscillator model does not have to be imperfect, and therefore that it can Marnelius, and N. Mukunda, Phyc;. Rev. D 23, 2201 (1981).
be made consistent with the known rules of quantum mechanics and 1HSee, for instance, R. Fong and J. Sucher, J. Math. Phys. 5,456 (19641.

1161 Am. J. Phys., Vol. 49, No. 12, December 1981 D. Han and Y. S. Kim 1161
Chapter V

Lorentz-Dirac Deformation in High Energy Physics

In 1955, Hofstadter and McAllister observed that the proton is not a point particle.
Although several field theoretic approaches had been made immediately after this
discovery to explain the space-time extension of the proton, a satisfactory answer to
this question can be found in the quark model, in which the proton is a bound state
of three quarks.
In order to explain the high momentum-transfer behavior in the Hofstadter
experiment, we need a wave function for the proton which can be Lorentz boosted.
The covariant harmonic wave function discussed in the papers of Chapter IV is a
suitable wave function for this purpose. In 1970, Fujimura, Kobayashi, and Namiki
calculated the form factor of the proton, and showed that the asymptotic behavior of
the form factor is due to the Lorentz deformation of the wave function.
The most peculiar behavior in high-energy physics is Feynman's parton picture. In
1969, Feynman observed that a rapidly moving proton can be regarded as a
collection of an infinite number of partons whose properties appear quite different
from those of quarks. This model is clearly spelled out in the paper of Bjorken and
Paschos (1969). In 1977, using the covariant oscillator formalism, Kim and Noz
showed that the static quark model and Feynman's parton picture are two different
limiting cases of one covariant physics. Hussar in 1981 calculated the parton
distribution for the rapidly moving proton using the covariant harmonic oscillator
wave function.
LORENlZ-DIRAC DEFORMATION IN HIGH ENERGY PHYSICS 279

Reprinted from: Physical Review. Volume 98.1955. pp. 183-184

Electron Scattering from the Proton *f*

Robert Hofstadter and Robert W. McAllister

Department of Physics and High-Energy Physics Laboratory.

Stanford University. Stanford. California
(Received January 24, 1955)

With apparatus previously described1,2 we have studied the elastic scattering of

electrons of energies 100, 188, and 236 MeV from protons initially at rest. At 100
MeV and 188 MeV, the angular distributions of scattered electrons have been
examined in the ranges 60 0 _138 0 and 35 0 -138 0 , respectively, in the laboratory
frame. At 236 MeV, because of an inability of the analyzing magnet to bend
electrons of energies larger than 192 MeV, we have studied the angular distribution
between 90 0 and 138 0 in the laboratory frame. In all cases a gaseous hydrogen
target was used.
We have found that deviations in excess of Mott scattering are readily apparent at
large scattering angles. The early results (reported at the Seattle meeting, July,
1954) at smaller angles showed the expected agreement with the Mott formula
within experimental error. Deviations from the Mott formula such as we have found
may be anticipated at large angles because of additional scattering from the
magnetic moment of the proton. 3 We have observed this additional scattering but in
an amount smaller than predicted by theory.
The experimental curve at 188 MeV is given in Fig 1. It may be observed that the
experimental points do not fit either the Mott curve or the theoretical curve of
Rosenbluth? computed for a point charge and point (anomalous) magnetic moment
of the proton. Furthermore, the experimental curve does not fit a Rosenbluth curve
with the Dirac magnetic moment and a point charge. The latter curve would lie
close to the Mott curve and slightly above it. Similar behavior is observed at 236
MeV.
The correct interpretation of these results will require a more elaborate explanation
(probably involving a good meson theory) than can be given at the moment,
although Rosenbluth already has made weak-couplin~ calculations in meson theory
which predict an effect of the kind we have observed.

Reprinted from Phys. Rev. 98, 183 (1953).

280 CHAPTER V

Nevertheless, if we make the naive assumption that the proton charge cloud and its
magnetic moment are both spread out in the same proportions we can calculate
simple form factors for various values of the proton "size". When these calculations
are carried out we find that the experimental curves can be represented very well by
the following choices of size. At 188 MeV, the data are fitted accurately by an rms
radius of (7.0±2.4lX10-14 cm. At 236 MeV, the data are well fitted by an rms radius
of (7.8±2.4)X10-1 cm. At 100 MeV the data are relatively insensitive to the radius
but the experimental results are fitted by both choices given above. The 100-MeV
data serve therefore as a valuable check of the app'aratus. A compromise value
fitting all the experimental results is (7.4±2.4)XlO-14 cm. If the proton were a
spherical ball of charge, this rms radius would indicate a true radius of 9.5xlO-14
cm, or in round numbers 1.0xlO-13 cm. It is to be noted that if our interpretation is
correct the Coulomb law of force has not been violated at distances as small as
7xlO- 14 cm.

\ E~ECTRONI SCATT~RING

,
FROM HYDROGEN r--
\ 188 MEV (LAB)

\ -- - -

r--
~ ANOMALOUS MOMENT
CURVE ___

-~

I MOTT CURVE r ~
'~
, ~
EXPE~IMENTAL CURV!>?' ~ I
\
\

-- ~
\

2
50 70 90 110 130 150
LABORATORY ANGLE OF SCATTERING (IN DEGREES)

Fig. 1. The figure shows the experimental curve, the Mou curve, and the point-charge, point-magnetic-
moment curve. The experimental curve passes through the points with the auached margins of elTOr. The
margins of elTOr are not statistical; statistical error would be much smaller than the elTOrs shown. The
limits of elTOr are, rather, the largest deviations observed in the many complete and partial runs taken over
a period of several months. Absolute cross sections given in the ordinate scale were not measured
experimentally but were taken from theory. The radiative corrections of Schwinger have been ignored
since they affect the angular distribution hardly at all. The radiative corrections do influence the absolute
cross sections. Experimental points in the figure refer to areas under the elastic peaks taken over an
LORENIZ-DIRAC DEFORMATION IN HIGH ENERGY PHYSICS 281

energy interval of ± 1.5 MeV centering about the peale. The data at the various points are unchanged in
relation to each other when the energy interval is increased to ± 2.5 MeV about the peak; the latter widths
include essentially all the area under the peale.

These results will be reported in more detail in a paper now in preparation.

We wish to thank Dr. D. R. Yennie for his generous aid in discussions of the theory.
We wish to thank Mr. E. E. Chambers for assistance with several phases of the
work. In the early phases of this research, the late Miss Eva Wiener made important
contributions.
·The research reported in this document was supported jointly by the U. S. Navy (Office of Naval
Research) and the U. S. Atomic Energy Commission, and the U. S. Air Force through the Office of
Scientific Research of the Air Research and Development Command.
t Aided by a grant from the Research Corporation.
*Early results were presented at the Seattle Meeting of the American Physical Society [Phys. Rev. 96,
854(A) (1954)]. More recent results were presented at the Berkeley meeting of the American Physical
Society [Bull Am. Phys. Soc. 29, No.8, 29 (1954)].
IHofstadter, Fechter, and McIntyre, Phys. Rev. 92,978 (1953).
2Hofstadter, Hahn, Knudsen, and McIntyre, Phys. Rev. 95, 512 (1954).
3M.N. Rosenbluth, Phys. Rev. 79, 615 (1950).
4See also the classical calculation of L. 1. Schiff reported in Rosenbluth's paper.
282 CHAPTER V

73

Progress of Theoretical Physics, Vol. 43, No.1, January 1970

Nucleon Electromagnetic Form Factors at High Momentum

Transfers in an Extended Particle Model Based
on the Quark Model

Kimio FUJIMURA, Tsunehiro KOBAYASHI and Mikio NAMIKI

Department of Physics, Waseda University, Tokyo

(Received August 21, 19(9)

Taking account of the Lorentz contraction effect of the extended nucleon core as a nucleon
but not as a quark, it is shown that the Gaussian inner orbital wave function can produce
the form factor very close to the dipole formula.

Recent experiments show that the nucleon electromagnetic form factor are
empirically described by the "scaling law" e-1GEP=/1p-1GMP=IL,.-IGMn (~F) and
GEn=o and by the "dipole formula" F= (l+K-'[t[)-', where we have followed
the usual notations and K'=0.71 (GeV/c)'. The scaling law was already dis-
cussed on the theoretical basis of the nonrelativistic urbaryon (quark) model. 1).')
Ishida et a1.') and Drell et a1. 3) attempted to extract information about the inner
orbital wave function at short distances from the [t[-dependence of F in a wide
region of [t[ over M' (At being the nucleon mass), using nonrelativistic for-
mulas. In this note we show that if possible relativistic effects as a nucleon
(not as a quark), especially the Lorentz contraction of the nucleon core, are taken
into account in a proper way, their conclusions become never true but the simple
Gaussian inner orbital wave function can produce the form factor very close to
the dipole formula.
Those who are working with the nonrelativistic quark model have believed
that if It I<;Mq' (Mq being the quark mass), nonrelativistic formulas can be used
for everything. As for the form factor, therefore, they have used

F= W NR (Drell et a1. and others), (la)

F=(I+ltl/m~)-JWNR (Ishida et a1.), (1b)

with the nonrelativistic formula

(2)

where m~ is the mean square mass of p and (J) mesons and q the momentum
transfer. ¢ (x, ... ) stands for the inner orbital wave function, where independent
inner coordinates are denoted by x and Assuming the simple Gaussian func-
tion for ¢, we have got

Reprinted from Prog. Theor. Phys. 43,73 (1970).

LORENTZ-DIRAC DEFORMATION IN mGH ENERGY PHYSICS 283

74 K. Fujimura, T. Kokayashi and M. Namiki

(3)

where (r'), is the mean square radius of the nucleon core. It is evident that
the simple Gaussian function never gives us the form factor consistent with the
dipole formula for It I?:..M'. This is the reason why Ishida et a1. introduced a
singular wave function and Drell et a1. discussed singular potentials among con-
stituent particles. It is, however, to be noted that Eq. (2) is a nonrelativistic
formula to be verified not only for It I<Mq' but also for It I<M'. Here we want
to emphasize that relativistic effects as a nucleon (but not as a quark) become
very important for ItI2M'. Indeed, we can see that the Lorents contraction ef-
fect as a nucleon for Itl>M' should reduce <r'), in Eq. (3) by the Lorentz
factor, r- l, approximately proportional to M'ltl- l • Hence Eq. (3) must be
modified essentially in its It I-dependence in the following way:

(4)

where OFT (q') is the overlap integral defined by (¢F, ¢iI)'

Hence we cannot exclude the Gaussian inner wave function. Furthermore
note that the region Mq'>ltl?:..M' covers a wide range from one (GeV/c)' to
several ten (GeV/c)' if Mq~(5 to 10) xM.
In order to take properly the relativistic effect into account we must in·
evitably use the four-dimensional inner orbital wave function. In the quark model,
the nucleon is assumed to be a composite particle of three quarks. Suppose that
the three quarks have, respectively, four.position coordinates Xl> x, aed x 3 • After
separating the center-of-mass coordinate X = (Xl + X, + xs) /3, we keep two inde-
pendent relative coordinates r= (x, -Xs) /../6 and s= (-2XI + X,+ xs)/3../2. Now,
as the simplest example, we can choose the four-dimensional Gaussian function

cfJ(r s' P) = (a/a)' exp[a

, , 2
{I"+S'-~
M' (P.r)'--'l-
M' (P.s)·}] (5)

for the inner orbital wave function, *) where P stands for the center-of-mass
momentum of the composite system, i.e. the nucleon momentum, and (a/a)' is
the normalization costant determined by !! IcfJI'd'rd's"= 1. The constant a is related
to the mean square radius of the nucleon core through a-I = <r').l3. It may be
worth while to emphasize another reason why Eq. (5) is used here: Equation
(5) represents the ground state eigenfunction of the Hamiltonian of a four-
dimensional harmonic oscillator consistent with the famous linearly raising tra-
jectory in an extended particle modeL') Our procedure should be regarded as
one theoretical attempt in an extended particle model represented by a trilocal field
based on the quark model, rather than one in the naive relativistic quark model.
284 CHAPTER V

Nucleon Electromagnetic Form Factors at High Momentum Transfers 75

The relativistic form factor is given by the formula*)

lVR = SS¢*(r, s; P},)e1q.(aT+b·')¢(r, s; Pr)d'rd's, (6)

or symbolically
(6')
where Pr and PF are, respectively, the initial and final momenta of the nucleon
and q = PF - Pr . a and b is one of the following pairs; (0, - -/'2,), (-/3/2, 1/ -/'2,)
and (- -/3/2, 1/ -/'2,). Note here that a' + b' = 2 for every pair. Inserting Eq. (5)
into Eq. (6), one obtains

(7)

Here the first factor is not other than the overlap integral

with (8)

T=I+~ and B=4,

2M'
where B=4 means the number of dimensions giving the Lorentz contraction,
namely, the longitudinal space-like inner coordinates and two time-like inner
coordinates. As mentioned above we can see in Eq. (7) that -/ behaves just r
like the effective Lorentz contraction factor and the exponential function in Eq.
(7) goes to a constant as It I increases over M'. The It I-dependence of the form
factor for j\1q '>
It I:2;M' is, therefore, governed mainly by the overlapping-effect
factor (l/-/T)B which goes to (Itl/2M')-BI'. We can remark that power B is
nothing other than the number of independent relative coordinates. Thus the
dipole-like behavior of W R can be obtained from B=4, namely, the inner freedom
of motion equivalent to four indepencent relative coordinates. Thus the dipole-
like behavior of W R can be obtained from B=4, namely, the inner freedom of
motion equivalent to four independent relative coordinates.
Equations (la) and (lb) should, respectively, be replaced with
F= W R , (9a)
F=(1+i<r')vT-'ltl)-'WR, (9b)
where <r')v = 6mv2 is the mean square radius of the vector meson cloud. Both
formulas (9a) and (9b) together with Eq. (7) behave like the dipole formula

*) It is to be noted that most of the infinite component field theories have identified the form

factor with the overlap function (¢F, ¢I) but never with (¢F, e iqx ¢1) itself. In fact, some authors
have derived our later result, (¢ F, ¢ 1) = (1 + Itl/2M2) -2 as the form factor, using the infinite component
field theory. See A. O. Barut's lecture given at the Colorado Summer School in 1967. We must em-
phasize here that the form factor should not be given by (¢r, ¢,).
LORENTZ-DIRAC DEFORMATION IN HIGH ENERGY PHYSICS 285

76 K. Fujimura, T. Kobayashi and M. Namiki

modified by a constant factor for M q '>ltl2:M". Note that they contain only
one free parameter <r'), to be adjusted. Let us first compare the theoretical
form factor given by Eq. (9a) together with Eq. (7)--call it Case (i)--
with experiment') in Fig. 1, in which we have used <r')o=7.50 (GeV/c)~'. From
them one can see that the theoretical curve given by Eqs. (9a) and (7) is not
inconsistent with the experimental plot but quite different from the nonrelativistic
Gaussian form factor Woo Next we examine Case (ii) in which Eq. (9b) is
combined with Eq. (7), namely, each quark has the vector meson cloud. Choos-
ing <r'), = 1.82 (Ge V / c)~', we see in Fig. 2 that the theoretical curve can
reproduce the experimental plot in a wide range of It I from zero to about 25
(Ge V / c)'. It is repeatedly noted that this fit has been obtained by adjusting only
one parameter <r')" and that the nonrelativistic Gaussian form factor is strongly
modified in its essence.

o
It I (GeVid
Fig. 1. Comparison of the theoretical form factors with experiments in Case (i)
with <r')c=7.50 (GeV/c) -'. The dipole formula and the nonrelativistic Gaussian
form factor are, respectively, shown by the broken and chain lines.
286 CHAPTER V

Nucleon Electromagnetic Form Factors at High Momentum Transfers 77

0.9

0.8
I ci'

0.7

os
-2 0~~0.~~~O~I~0--~O~15~~~~--~~~~
10
II I (GeV/d

Case(iil

16 Case (iv)

o 10 20 30
III {GeVid

Fig. 2. Comparison of the theoretical form factors with experiments in Case (ii)
with <r
2 ), =1.82 (GeV/c) - 2, in Case (iii) with <r» , =8.81 (GeV/c) - 2 and

.1=1.4, and in Case (iv) with <r» , =1.20 (GeV/c) - 2 and .1=0.9. The broken
line shows the dipole formula.

Here we want to introduce a new parameter, say A, into the lllner orbital
wave function as follows:

(a)" - -
¢,(r, s ; P) = ,- ;; v2A -1 exp [a{
"2 r' + s'- M'
2A (P·r)'- M'
2A (p·s)' }] . (10)

It is easy to see that the parameter A distinguishes the time-like extension from
the space-like one of the inner orbital motion, and that A= 1 gives us the original
one Eq. (5). Using ¢., we have got

W~J =r, ~L)\ - lex p [_.l <r'>,r.-1It l ]

-1(1 + 2A_ A_-12M' (11)
6 '
where
LORENTZ-DIRAC DEFORMATION IN HIGH ENERGY PHYSICS 287

78 K. Fujimura, T. Kobayashi and M. Namiki

(12)

The form factor F, is obtained by Eq. (11) together with the modified formulas

(13a)

(13b)

In case (iii) Eq. (13a) is combined with Eq. (11), and Case (iv) is given by
Eq. (13b) together with Eq. (11). In both cases the form factor goes to one
proportional to (22\,1'ltl- 1)' like the dipole formula. Figure 2 shows us that the
experimental plot can be fitted by the theoretical curves with <r'), = 8.81 (Ge V / c)-'
and A=1.4 in Case (iii) and with <r'),=1.20 (GeV/ct' and A=O.9 in Case (iv).
The theoretical curves are in good agreement with experiment. Needless to say,
Cases (iii) and (iv) include Cases (i) and (ii), respectively, as their special cases
with l= l.
From the above arguments we have inferred that the Lorentz contraction of
the extended nucleon core can be a possible origin of the "dipole formula".
The same effects will appear also in inelastic electron proton collisions leading
to the isobar excitation. In the nonrelativistic quark model we have got the
differential cross section for the inelastic collision in the following form :6)

(14)

using the simple Gaussian wave function, where L is an integral number deter-
mined by the type of transition, and A a numerical factor. If we take the
Lorentz contraction factor into account, then we can infer that Eq. (14) should
be replaced with

(15)
where Vr and OF! (q') are, respectively, the effective Lorentz contraction factor
and the onrlap integral in the inelastic collision. The similar structure of the
wave function suggests us that r ~q' and OF! (q') ~ (q')-' as q' goes over M', and
then that the inelastic cross section would behave like the dipole formula squared
for q'?:J1,1'. Indeed, it seems to us that recent experiments indicate such a be-
havior for the cross section.') Detailed discussions will be given in a forth-
coming paper in which the full nucleon and isobar wave functions and the quark
current to be valid for .~lq'';P It I?M' will be formulated.
The earlier form of this work was done wheh one of the authors (M. N.) was
working in the Niels Bohr Institute in Copenhagen. He would like to express
his sincere gratitude to Professor A. Bohr for his kind hospitality and to Professor
z. Koba for many discussions. Be is also much indepted to Professor T. Taka-
bayashi for helpful discussions.
288 CHAPTER V

Nucleon Electromagnetic Form Factors at High Momentum Transfers 79

References

1) Y. Kinoshita, T. Kobayashi, S. Machida and M. Namiki, Prog. Theor. Phys. 36 (1966), 107.
In the first several sections they derived the scaling law, assuming that a virtual photon
couples directly to one point quark in the nucleon core subject to the 56-dimensional re-
presentation of the SU(6) symmetry.
2) S. Ishida, K. Konno and H. Shimodaira, Prog. Theor. Phys. 36 (1966),1243. They first derived
the scaling law in a semi phenomenological way, assuming that a virtual photon couples to
the nucleon only through p and (J) mesons. They have also got the quark-theoretical form
factor, Eq. (lb), together with Eq. (2), assuming each quark to have a vector meson cloud.
3) S. D. Drell, A. C. Finn and M. II. Goldhaber, Phys. Rev. 157 (1967), Bl57.
4) T. Takabayasi, Phys. Rev. 139 (1965), B138L
5) D. H. Coward, H. DeStalbler, R. A. Early, J. Litt, A. Minten, L. W. Mo, W. K. H. Panofsky,
R. E. Taylor, M. Breidenbach, ]. I. Friedman, H. W. Kendall, P. N. Kirk, B. C. Barish,
.r. Mar and J. Pine, Phys. Rev. Letters 20 (1968), 292.
L. N. Hand, D. G. Miller and R. Wilson, Rev. Mod. Phys. 35 (1963), 335.
T. Jansens, R. Hofstadter, E. B. Hughes and M. R. Yearian, Phys. Rev. 142 (1966), 922.
W. Albrecht, H. ]. Behrend, W. Flauger, H. Hultshig and K. G. Steffen, Phys. Rev. Letters
17 (1966). 1192.
W. Albrecht, H. J. Behrend, H. Dorner, W. Flauger and H. Hultshig, Phys. Rev. Letters
18 (1967), 1014.
6) K. Fujimura, Ts. Kobayashi, Te. Kobayashi and M. Namiki, Prog. Theor. Phys. 38 (1967),
210.
7) For example, see W. K. H_ Panofsky's report presented at the XIVth International Con-
ference on High Energy Physics in Vienna in 1968.
LORENTZ-DIRAC DEFORMATION IN lllGH ENERGY PHYSICS 289

The Behavior of Hadron Collisions

at Extreme Energies

RICHARD P. FEYNMAN

California Institute of Technology, Pasadena, California

Talk given at

Third International Conference

on High Energy Collisions

State University of New York, Stony Brook

September 5-6, 1969

C.N. Yang et al. Eds.

Gordon and Breach
(New York, 1969)

Reprinted from High Energy Collisions (1969).

290 CHAPTER V

There are several reasons to be interested in this problem of very high

energy hadron scattering. Firstly, most theoretical inventions are based on
analysis of simple collisions, in which· only a small number of particles
come out. But it is at once realized that questions of unitarity, the
asymptotic behavior for high energy in dispersion integrals, etc, require
some ansatz be made for the higher energy collisions, in order to close the
infinite hierarchy of equations which result. Secondly, experiments at high
energies usually yield many particles, and only by selecting the rare
collision can we find those about which the theorist has been speaking.
For the highly multiple inelastic collisions (to which the major part of the
inelastic cross section is due) so many variables are involved that it is not
known how to organize or present this data. Any theoretical suggestion
(even if it proves to be not quite right) suggests a way that this vast amount
of data may be analyzed. For this reason I shall present here some
preliminary speculations on how these collisions might behave even
though I have not yet analyzed them as fully as I would like.

Hadron phenomena possess a number of remarkably simple properties.

Besides the well-known agreements with relativity, analyticity, unitarity,
etc., there are of course the conservation rules of isospin and strangeness
and an approximate agreement with SU3 symmetry. In addition to these,
however, we have some special regularities which appear to be true
empirically which apply to very high-energy behavior and which may
point the way to an ultimate dynamical theory. A partial list of these
regularities at very high energy are:
1. Total cross sections are constant. The elastic part appears to
be a constant fraction of this.
2. Exchange reaction cross sections fall with a power of the
energy s2a.-2.
3. This power, a, seems to be t-dependent, and for those t = m2
where a(t) is an appropriate integer (or half-integer) there are
resonant particles of mass m (t is the square of the four-
vector momentum transfer in the collision).
4. <x(t) varies with t as a straight line a(t) = a o +)'1:. Although
<Xo varies with the quantum numbers which must exchanged,
the value ofris the same (0.95 per (GeV)2).
5. Cross sections fall very rapidly with transverse momentum
transfer. The average transverse momenta of the particles in
inelastic collisions are limited (to about 0.35 GeV).
LORENTZ-DIRAC DEFORMATION IN mGH ENERGY PHYSICS 291

There are, of course, very many rough approximations in this brief

summary. For example, we do not know if the total x-section might not
vary very slowly (for example, like In s); all the slopes of Regge
trajectories are not exactly equal; whether the pion nonet is on the
trajectory of such a slope is unknown; there are corrections to the simple
"regge expectations" (2), (3), presumably for absorption; some of the
inelastic cross sections can be associated with diffraction dissociation of
the elastic part; etc. Nevertheless, the list above contains a number of
main phenomena whose general behavior must ultimately be understood.
It will be noticed that in discussing the power laws associated with Regge
behavior, I have explicitly separated the behavior of the total x-section
which is often described as the Pomeranchuk trajectory, for it is not cenain
that this is a typical trajectory. In trying to understand the meaning of
these regularities of high energy behavior, I have been led to suggest
cenain further regularities accompanying them. I should like to present
these guesses here to see if they are possibly true, or, if some of them are
obviously in disagreement with experiment, to learn where I may have
already gone off the track in my thinking.

I shall, for completeness, first explain how I am trying to go about

analyzing these things; second, describe some special considerations
dealing with Regge exchange; and finally, present my suggestions for the
limiting behavior of cross sections at high energies.
I. FIELD THEORY AT fiGH ENERGY
In order to think about these questions I wish to use concepts which will
immediately insure that the most fundamental properties of relativistic
invariance, quantum superposition, unitarity, etc., will automatically be
satisfied. the only theoretical structure I know which has a chance of
doing this is a quantum field theory (I say, a chance, because we are not
sure if unrenormalized field theories can give finite answers, or if
renormalized theories are still unitary). It is not that I believe that the
observed high-energy phenomena are necessarily a consequence of field
theory. Even less do I know what specific field theory could yield them.
But, rather, I believe that they share some of the properties of field theory,
so they might share others. Therefore, I wish to study the behavior
expected from field theory for collisions of very high energy. What field
theory shall I choose? What shall be the fundamental bare particles that
the theory begins with? I do not know and perhaps I do not care. I shall
try at first to get results which are more general and characteristic of a
wide class of theories and which can be stated in a way independent of the
field theory which served as a logical crutch for their discovery. Only
292 CHAPTER V

later, possibly, might it be worth trying to see if certain special

experimental details imply something about a special theory which
underlies all these phenomena.

In the meantime I call the fundamental bare particles of my underlying

field theory "partons" (which may be of several kinds, of course). For
example, in quantum electrodynamics the partons are bare electrons and
bare photons. Imagine this theory to have a Hamiltonian H which may be
separated into two pieces H = Hfree + Hint> one to represent free partons
and the other the interaction between them. This H is, at first, expressed in
terms of creation and annihilation operators (a*,a) of these partons (or, if
you prefer, local field operators of the parton fields in space). Next, for
special application to collisions of extremely high energy, W, incoming in
the z-direction in the center-of-mass system, only the operators of finite
transverse momenta (Le., x,y components) are kept as W ~ 00. The ones
with positive z-component of momentum of order W (Le., P z = xW, where
x is a finite quantity as we take a limit as W ~ 00) are separated from those
of negative z-component. The first are called right movers (aR) and the
second left movers (aL)' If this is done, and the Hamiltonian reexpressed,
we get H = HR + HL + HI where HR is the Hamiltonian involving right
movers (aR *, aR) only (containing, of course, interaction terms among
these right movers coming from Hint, HL involves left movers (aL*' aU
and HI contains both aR and aL' and represents an interaction between the
objects moving [to] the left and those moving to the right. But HI' as W ~
00, becomes a very simple expression (depending on the theory of course).
For example, a collision of a right moving proton with a left moving
neutron yielding right and left moving particles can then be analyzed in a
simple way. The proton is an eigenfunction of HR, the neutron of HL .
Neglecting HI' the system of states before and after collision are complete
eigenfunctions (not simple partons) of HR + HL . The operator HI makes
the transition between them (not necessarily in first order, of course).

I leave to a later publication a more detailed description of how this might

be carried out, but here I need only make some remarks about the variables
on which things appear to depend in the limit as W ~ 00.

To describe a proton of momentum Po' energy Eo' ordinary field theory

gives a state function or wave function giving the amplitude that a number
of partons of 3-momentum 131,132, ... etc., are to be found in it. The total
momentum of these partons L?i' equals Po' the momentum of the proton,
LORENlZ-DIRAC DEFORMATION IN HIGH ENERGY PHYSICS 293

but their total energy LiEi (where each energy Ei is calculated from the

mass ~ of the parton via Ei = "';~2 + Pi . Pi is not equal to that of the final
proton Eo' In fact, the amplitude to find this state contains, among many
other factors, one which is inversely proportional to this energy difference
A - (Eo- ~E)-l (1)
i
Knowing this wave function completely for some Po' say at rest, how can
we find it at some other momentum? It is very difficult to do, and in fact
requires knowledge of the entire Hamiltonian operator H, for the wave
function is not a relativistic invariant. This is emphasized by the point that
the momentum is the sum of the momenta of the parts but the energy is
not. The wave functions that should be useful for us are those in which Po
is very large in the z-direction and finite in the directions perpendicular to
that. If we take Poz = Xo Wand measure the parton's momentum in the
z-direction in the same scale Piz = ~ W, then the wave function has a
definite limiting form as W ---7 00 for xo' Xi finite. (xo ' of course, is
arbitrary; it may be taken to be unity, for example.) We have
xo = k.J
"" x·I (2)

Let the components of momentum perpendicular to z be called Q a two-

dimensional vector; then

(3)

Finally we can see how A varies (insofar as its denominator behaves) by

writing it as
A - «Eo-Poz )- ~(Ei-Piz))-l
which equals (1) since ithe total z-momentum is the sum of its parts.
However, for large z-momentum,

E - P = (m 2 + p2 + p2)1/2 _ P
z z z
= (m2 + Q2 + x 2WZ)1I2 - xW
,., I m2 +Q2
2W x
where m is the mass of the particle. Therefore, the amplitude becomes
2 2 2 2
mo + Qo "" mi + Qi ]-1 (4)
A - 2W[-- k.J
Xo i Xi
294 CHAPTER V

When the numerator as well as the denominator are expressed in these

variables x, Q only a simple power of W appears in a which can be
removed by a renormalization of the scale of Pz in phase space integrals.
The conclusion that I wish to remark for our present purposes is: when
expressed in terms of .Q. the transverse momentum in absolute scale, and b
the longitudinal momentum in relative scale, the wave functions have
definite limits as ~ the energy scale of the longitudinal momentum of the
state goes to 00.1 In this limit the values of x are positive only. They
represent the fractions of the momentum Xo which each parton has (thus, if
Xo = 1, all xi run from 0 to 1). The reason xi must be positive is that if a
parton has a negative momentum P z = Wx with x negative or Pz = -Ixl W
the energy Ei is approximately +Ixl W so E - P z is 21xl W which is of order
W2 larger than the E - P z of positively moving partons. Appearing in the
denominator of amplitudes, such amplitudes are of order (for W ~ 00)
when Ixl is as small as order c/W. The masses and transverse momenta
seem to be of the order of 1 Ge V so we shall call an Ixl of order (1 Ge V)/w
a "wee" x. For example, for Po = W = 100 GeV in the center-of-mass
system (taking Xo = 1), x = 1110 is a small x, but it is not wee; a wee x
would be x = 0.005 say. The behavior of the amplitude for wee x is not
without interest, as we shall see, but for the present we shall discuss only x
which is not wee. Then the amplitude as W ~ 00 appears to be a function
of the Q's and x's of the parts.

In a high-energy collision, the initial state consists of one of these groups

of partons moving to the right interacting with another similar group
moving to the left. What is the interaction? It will not be enough to just
naively say that one parton has a cross section for colliding with another,
for, in field theory, interaction is represented by mediation of a field; in
fact, by the exchange of just another parton coupled by the piece HI of the
Hamiltonian. But this HI in its form, is not entirely independent of the
form of HR for they both come from operations on the original
Hamiltonian H. Thus the amplitude to interact via the exchange [of] a
parton is closely related to the amplitude that there is some parton in the
right moving system in the initial state that can be "mistakenly" considered
as really being a parton belonging to the left moving system. (Just as two
electrons interact in first order by exchange of a photon, so it is also true
that a right moving physical electron of Xo = 1 has a first order amplitude

IThe statement is not precisely correct. What is meant is the density matrix has definite
limits.
LORENn-DIRAC DEFORMATION IN HIGH ENERGY PHYSICS 295

to be an ideal electron of longitudinal momentum I-x (times W) and a

photon of momentum x. Interaction results if we consider that the left
moving electron (insofar as it is bare) has some amplitude to be a left
moving outgoing physical electron if we make it up of a left bare electron
and the aforementioned photon.) However, a parton of momentum xW to
the right would be moving backwards in the left system and would have
practically no amplitude (as W ~ 00) to be "mistakenly" considered as
belonging to this left system. This is true, of course, only if x is not wee (
of order 1 Ge Vrw). If x is wee, a right moving parton and a left moving
parton are very similar in appearance. Thus interaction occurs only
through exchange of partons or systems of partons of wee longitudinal
momentum.

The energy dependence of reactions thus depends upon the probability of

finding wee partons of a certain nature. A great deal of information on
wee partons can be gotten from a knowledge of the partons where x is not
wee, but only small. For the small x and wee x behavior must join in a
continuous fashion. For example, suppose, for small x the amplitude to
find a parton system with x small varies as xadx where x[a?] is some
constant (a < 1). Then the amplitude to find a wee parton of momentum -
lrw in dx would, to fit on, have to vary as (l1W)-a, but the range of x that
such wee partons occupy is of order lIW so that the amplitude to find a
wee parton must vary as Wa-l, if W is the momentum of the right moving
object. If El is the energy of the right mover, this amplitude is (E1)a-l. If
this is to exchange with a similar system moving to the left with energy E2
the amplitude that this parton system is acceptable to the other system is
(E2)a-l. The amplitude for exchange therefore is (El)a-l(~)a-l or varies
as Sa-l since S = El E2. The cross section (there is always a problem of
convention of the normalization of amplitudes) varies as the square of this,
or S2a-2. A constant cross section means a = 1 or the amplitude to find
partons of small x must vary as dxlx. The amplitude to find wee parton is
JO?
not llW dx/x because this dxlx law fails below lrw. The amplitude to find
a wee parton is just constant, independent of energy since the curve lIx
cuts off for x below lrw at a value of order W and the integral below x =
lIW is finite in this event. Since cross sections (such as the total x section)
are constant, we see that this must actually happen. It is, therefore, not
strictly true that as W ~ 00 if we keep all x and Q constant there is a
definite limiting wave function (as we said earlier) for there is always a
finite amplitude for wee partons. However, the finite x part of the
probability distribution of partons has a definite limit, in this limit there are
probabilities of finding partons varying as dxlx. The apparently diverging
296 CHAPTER V

character of this distribution for small x is cut off at wee x (that is at x of

order 1/w). (A complete theory would have to describe this cutoff region
in detail. We shall say more about it later.)

The equations for x not wee simplify if one concentrates on the small x
part. It is then seen that there is an approximate scaling law for small x
(the approximation improving as x decreases) so that solutions with special
distributions of partons with a power law scale dependence (x-<X) are
eigenfunctions natural to field theory.

It may help to give a few, nearly trivial examples. First, according to first
order perturbation theory in the expression (4) for the amplitude, the
numerator does not depend on x, Q for scalar partons (couplings involve
no momenta). If one of the partons has an especially low x, the the term
(J..l2 + ~)/x belonging to it dominates and we get an amplitude
proportional to x (times the scale dP/E, or dx/x, of relativistic phase space).
This corresponds to <X = 0 for the scalar meson. Likewise, it can be shown
that the amplitude for (longitudinally polarized) vector partons varies as
constant (times dxlx). In this case a factor 1/x comes from the numerator
couplings. For spin 1/2 particles coupled in the simplest ways, the
amplitude varies as xll2 (times dx/x). In general, <X equals the spin of the
particle. In perturbation theory, these agree with well-known results for
the energy dependence of x sections, in particular that vector meson
exchange as in electrodynamics lead to constant cross sections in
perturbation theory.

The deep inelastic behavior of electron-proton scattering has been looked

at from the point of view described here. It can be argued that the curve of
J.l.2 W2 vs. Q2/2Mv (in the variables of Bjorken2 ) is the distribution in x of
Q2
charged partons (each weighed by the square of its charge). A behavior
like dxlx for the low momentum partons is indicated by experiment.

ll. REGGE BEHAVIOR

By "Regge behavior" is meant the second item of our list of regularities,
that the cross section for exchange reactions vary as an (inverse) power of
the energy, which power depends on the momentum transfer.

2SLAC publication No. 571

LORENTZ-DIRAC DEFORMATION IN HIGH ENERGY PHYSICS 297

The original expectation of Regge behavior were the results of a brilliant

induction from Regge's non-relativistic studies by Gell-Mann and
Frautschi. Now, however, I should like to consider it as an established
empirical fact and to try (in this section) to understand physically why it
should be so.

Let us consider a typical exchange reaction, for example, a charge

exchange reaction like p + x- ~ n + rr.O, in which the x-, Xo are right
moving, and the p, n left moving. The easiest view of this is that a
negative charge has been exchanged from the x system to the nucleon
system. The cross section falls about as s-1 or according to the best
estimates as s2ao-2 with no = 0.43 or s-1.14. It might at ftrst be thought that
an exchange via a vector meson such as a p- would lead (as it does in
perturbation theory) to a constant cross section. However, it is to be noted
that an important current density (the 3 component of isotopic spin) has
been suddenly reversed. Initially (x-,p) this current density had fast
moving components of -1 to the right, +112 to the left. Afterwards (XO, n)
it has 0 to the right, -112 to the left. Although the total 3 component of
isospin is not changed, a motion of a part of it (-1 unit) is suddenly
changed from right to left motion. In electrodynamics we are aware that a
sudden reversal of electric current density induces a copious
Bremsstrahlung -- a sudden reversal of the direction of an electron carrying
a photon fteld to the right, leaves the fteld coasting on to the right in the
form of photons (and, of course, the new motion of the charge to the left
generates left moving photon Bremsstrahlung).

The hadrons may act similarly. These currents are of considerable

importance in our present theories and in fact we believe there are particles
(p mesons in fact) strongly coupled to them. Thus the strong current
reversal in a charge exchange will tend to shed p mesons. Perhaps we can
guess some of the behavior of the high-energy inelastic collisions by
working by analogy to Bremsstrahlung.

In studying the pure reaction p + x- ~ n + XO at high energy, we insist,

first that a current be suddenly reversed and, second, that no
Bremsstrahlung actually occur. This latter is because we insist that the
reaction have only the two particles n, XO in the ftxed state. We can
interpret the fact that in such an exchange the cross section falls (relative to
the main behavior of the majority of inelastic scatterings -- for which a
constant cross section is empirically more appropriate) as the energy rises,
by the observation that as the energy rises it becomes increasingly less
likely that the current reversal can be accomplished without
298 CHAPTER V

Bremsstrahlung.

The theory of Bremsstrahlung with strong coupling and with the "photons"
of the field carrying the very type of currents which are sources of further
Bremsstrahlung has not been worked out in detail. Nevertheless, we may
boldly try to guess. that certain analogies to electromagnetic weak
interaction Bremsstrahlung exist. Some hope for sense here comes from
noting that many features can be be seen from a classical view which takes
h ~ 0 so e 21f:c large. Therefore some properties are understandable both
for e2lf:clarge, and for e2{fc small, may have more general validity. This is
especially likely if we understand the reasons for them clearly.

First, the spectrum of the particles in longitudinal momentum is dPiE (or

dxlx for x's which are not wee). This is because the Lorentz contracted
field is so sharp in z that the energy in it is distributed uniformly in P z (the
Fourier transform of a pulse being a constant). The energy distribution of
the radiated particle is therefore dPz' or if the individual particles have
energies E their longitudinal momenta are distributed as dPz"E. Also, such
a distribution is stable under further disintegration of the particles, or of
interaction between the particles. If we write tanh w + PiE for z
component of the velocity of a particle, the relativistic rule for the addition
of velocities becomes, as is well known, simply the addition of rapidities.
Suppose a particle in its rest system can disintegrate or yield a new particle
with rapidity u. Then if the old particle has rapidity w, instead, the new
particle appears with rapidity w' =w+u. Therefore, if the old particles are
distributed uniformly in w (as dw) the new particles appear with a
distribution also uniform in rapidity because dw'=dw. Thus this feature (a
spectrum dw = dPz"E) is to be expected independently of whether we are
seeing what was originally radiated (say, p mesons) or are observing some
other secondary particles that these may have changed into (e.g., if p's go
into kions). Hence I believe we should expect it for our inelastic
distribution of particles of small (and wee) x in our strongly interacting
systems also. (We cannot expect this to be valid for large x, say x = 1/2
also, because if that much energy is taken from the primary particle by
radiation of one emitted particle, subsequent emissions are severely
affected. The dPz"E spectrum for photons in electrodynamics is precisely
valid only for smaller values of x.)

Next, the energy in the field this radiated is some fraction of the energy of
the particle which radiates. Thus the particle may be found after the
radiation to have lost on the average some fixed fraction of its energy.
LOREN1Z-DlRAC DEFORMATION IN HIGH ENERGY PHYSICS 299

This is found experimentally, in some cases. For example, in pp collisions

which yield a forward proton, its average momentum is about 0.60 of the
incident momentum3 (in individual collisions, its value fluctuates widely).

For weak coupling electrodynamics, the vector field particles are emitted
independently into a Poisson distribution with mean number n emitted.
The probability that none are emitted is e-n. The sum of the chance of
emitting none, one, two, etc., (that is, the total cross section) is much like it
would be without coupling to the photons. Here we know the total x
section is constant, and so can try to interpret the energy fall- off s2cxo-2 of
the pure two body charge exchange reaction as the factor e-n for the
probability of no emission, where i1 is the expected mean number of
primary particles emitted. This multiplicity n must rise logarithmically
with energy then as n= (2-20.0 ) In s. The particles we observe are not, of
course, the primary field particles emitted, but rather the observed particles
are secondary disintegration products of these unknown primaries. But if
each primary produces on the average a fixed number of secondaries, we
see that the expectation is that the multiplicity grows logarithmically with
E.
This is necessary if our various ideas are to fit together. Because we have
already suggested that the mean number of any kind of particle emitted is
to vary with x as c dxlx for small x and, for a given x, not to vary
otherwise with the energy W of the collision (so that c is a constant). The
J
total mean number emitted, then, is c dxlx. The upper limit of x is of
finite order (for the formula fails as x ~ 1 and x cannot exceed 1) but the
lower limit is of order of wee x, (i.e., order trw) where the dxlx fails.
Thus the mean number emitted to the right must vary as c(ln W + const).
(Actually we can do the integral all the way to zero, for we expect the
integrand to be c dP.J..J~2 + OZ + p/ where ~ is the mass and Q is the
transverse momentum of a typical particle. Putting P z = xW, this is

J X!
c
dx
= c in 2
2Wx1
2 112
o ..Jx2 + (~2 + Q2)/WZ (J.L + Q )

for finite Xl. To go further, we should have to know the transverse

3Report on the Topical Conference on High Energy Collisions, CERN 68-7, February
1968, Turkot, p.316.
300 CHAPTER V

momentum distribution.)

The one respect in which the electrodynamic analogy leads us astray is in

the transverse momentum distribution of· the £hotons. These large
dPrJ Q
transverse momenta fall off slowly (like --2 for sudden current
E Q
reversals) but in the strong collisions this is empirically not true. Some
characteristics of the distribution of charge across the face of the of the
interacting particle is involved here.

I have not yet studied the regularities involving the transverse momenta
(items (4) and (5) in our Introduction) from the viewpoint being developed
here. In the meantime, we can take these as empirical facts to be included
in any expectations. In the same way we leave for further research
strangeness and isospin character of these effects. We should notice,
however, that, although we discussed a charge exchange arising from an
exchange of a particle of the quantum numbers of the p-, the exchange of
any current of the usual octet would have analogous effects on the possible
Bremsstrahlung of particles coupled to other (non-commuting) currents.

In a pure two-body exchange reaction, since the probability of not

Bremsstrahlung depends on exactly what currents are reversed, the value
of a o will, from this point of view, depend on the quantum numbers of the
particle exchanged (which, of course, it does). The a o here referred to is
evidently only the largest for a given set of quantum numbers exchanged.

The quantum numbers exchanged may involve not only currents of unitary
symmetry, for baryon number (and possibly spin) may be exchanged, and
we do not know if there are special couplings to baryon currents (or spin
currents) which are also involved in determining ao' However, I should
like to hazard the guess that baryon number cannot be exchanged without
the transfer of a fundamental part of spin 112. Such an ideal part already
has a = 112 which would imply a l/Vs behavior of amplitudes before
corrections to Bremsstrahlung. Therefore, if we do an experiment which
freely allows the emission of wee mesons, except that the quantum
numbers are controlled so that a baryon must be exchanged between right
and left systems this cross section probably approaches a 1/s behavior,
instead of the constant expected for similar experiments in which no
baryons need be exchanged.

A final question is that of the distribution of correlations among the

various emitted particles in the average inclusive collision. In the
LORENTZ-DIRAC DEFORMATION IN mGH ENERGY PHYSICS 301

perturbation theory these are emitted independently and at random in a

Poisson distribution so the probability that there will be k of them of
momenta Xl' x2' ... xk is just (cdxl/xl)' (cdx2/x2)' ... (cdxk"xk)e-Ir'k! where il
is the mean number emitted. I have not yet found a good reason whether
this would be true in our non-perturbative case or not, but if one insists on
comparing the experimental distributions and correlations to some theory,
perhaps this is the first thing to try: that the pion distribution results from
an original Poisson distribution of p' s, each distributed for small and wee X
as c dP .JEz with c near 1.1 or 1.2 (c is 2-20.0 for the p trajectory). In fact,
if we suppose two pions for every p, the multiplicity of p's would be c In s,
and the multiplicity of the pions twice this, or about 2.3 In ELAB(Ge V).
Surprisingly, this fits observations very well (see Table 1). As an
additional coincidence, this value c is what one gets from perturbation
theory if one uses the coupling constant determined for p nucleon
coupling. It must be admitted that in this paragraph we have gone much
further than we should -- for our precise numerical result depends upon a
choice of which particles are fundamental and which they disintegrate into.
All our other predictions were of those features which were independent of
such specific choices.

The probability that the total momentum of all the emitted right moving
p 's is less than y is proportional to yC so that (aside from diffraction
dissociation) the momentum distribution of the ongoing particle, when it
takes a fraction of momentum x close to 1 should vary as (l-x)C where I-x
is small.

In Section I we discussed the longitudinal momentum distribution of

partons expected in a hadron wave function, but we have not seen how this
might lead us to expectations for the distributions of momenta of real
hadrons in a collision, for partons are not real hadrons. Nevertheless, we
shall suppose that when a hadron is disturbed via interaction, so that its
distribution of partons is no longer exactly that of a single real hadron
state, it must be compounded of a series of real hadron states, but the
distribution of longitudinal momentum of these real hadrons is
qualitatively like those of the partons which we described before (in
Section I). I have no way to see why this must be true, but the features of
the distributions discussed in that section seem, firstly, to rely mainly only
on qualities of relativistic transformation; secondly, the principle is right in
perturbation theory; and thirdly, the results of assuming this fit very well
with the qualitative predictions of the Bremsstrahlung analogy. Finally,
for one reason or another -- empirical or theoretical, good or bad -- I
suspect that the high energy collisions to have a number of features which
302 CHAPTER V

we summarize in the next section.

Ill. EXPECTED BEHAVIOR AT

HIGH ENERGY COLLISIONS
In a collision of very high energy between two hadrons each of momentum
W, in the center-of-mass system, the outgoing particles of the collision
should be described by two variables Q, the transverse momenta in
absolute units and x the longitudinal momentum as a proportion of W (so
that P z = xW). We intend to describe cross sections for various processes
as W increases without limit, keeping the x, Q's of the particles constant.
If for large W an x is of the order 1 GeV/w (so that its momentum in the
C.M. system is in the BeV range or less) we say the particle has a wee
momentum. Small x simply means a value of x much below one, but
higher than order 1 GeV/W for extremely large W. Finally we should like
to characterize experiments as being of two classes -- exclusive and
inclusive.

An exclusive experiment is one in which it is asked that only certain

particular particles of fixed Q, x, and character be found in the final state,
and no others. In particular, it excludes the emission of particles with wee
momenta in the limit. Examples are: two-body reactions AB ~ CD; an
experiment which uses missing mass to try to select events which are
virtually two-body reaction, etc. For such collisions, the cross sections
should fall off inversely as a power of s = 2W2, the power being 2a(t) - 2
where a(t) is the a appropriate to the highest Regge trajectory capable of
carrying the necessary exchange of quantum numbers between the right
moving and left moving system, and where t is the negative square of the
transverse momentum which must be exchanged. (With these x, Q {these}
variables fixed, the longitudinal momentum transfer and the energy
transfer go to zero inversely {as W} as W ~ 00.) In the special case that
no quantum numbers need to be exchanged, the cross section is constant
(empirically, at least, ift = 0); it does not fall as W ~ 00. This phenomena
is called diffraction dissociation, and is sometimes expressed as the
exchange of the "Pomeranchuk trajectory" .

An inclusive experiment is one in which certain particles are looked for at

given Q, x, but one also permits any number of additional particles. More
precisely, it does not in any way exclude the emission of arbitrary numbers
and kinds of particles with wee momenta. Examples are the total inelastic
cross section, the mean number of Q mesons emitted with momentum x in
range dx, the probability that no single particle is moving right nor left in
the (center-of-mass system) with more than 112 the original momentum W,
LORENTZ-DIRAC DEFORM AnON IN HIGH ENERGY PHYSICS 303

etc. Such cross sections should approach constant finite values as W ~ ~.

The mean number of mesons of a given kind formed with small x in a

high-energy inclusive experiment should vary as dx/x. This should
extrapolate right through the wee x region in the dP jE where E is the
energy -Y)l2 + Q2 + Pz2 of the meson momentum PZ' at fixed Q. (This
suggests that more appropriate variables for the small x region, would be
w, Q where w is the z-component rapidity w = tanh- 1 (Pz/E). The
distribution should be uniform in dw (for each Q) and ultimately
independent of W.) As a consequence, the multiplicity of a given kind of
hadron should rise logarithmically with W.

It is this dxlx distribution with its logarithmically divergent character for

small x which makes it possible that the probability of finding any specific
set of particles with given x, Q values (except the elastic or diffraction
dissociation ones) falls with energy as a power, and yet the total cross
section can be constant.

In an inclusive experiment of A colliding from the right, with B from the

left, the probability that some particle C comes out moving to the rilht
with an x close to unity should vary as (1_x)2-2a(t), as long as (l-x)lI is
not wee. Here aCt) is the value appropriate to the trajectory of highest a
(excluding the Pomeranchon) which could, upon emission, carry away the
quantum numbers and transverse momentum needed to turn A to C.

In a special kind of partially exclusive process in which a baryon must be

exchanged to get the reactants of finite x, but no wee baryon{s} appears
among the particles of wee momentum, then I believe the cross section
will vary as llW but this is not on as firm a basis as the other suggestions.
304 CHAPTER V

TABLEt

Multiplicity of Pions in High Energy Collisions

ELAB (BeV) Mult. 2.3lnELAB

30 7 7.8
470 13 ± 1 14.1
1500 18 ± 2 16.8
12300 24±4 21.6
LORENTZ-DIRAC DEFORMATION IN HIGH ENERGY PHYSICS 305

PHYSICAL REVIEW VOLUME lS5. NUMBER 5 25 SRPTEMBER 196()

Inelastic Electron-Proton and ,,(-Proton Scattering and the

Structure of the NUcleon*
.J. D. BjORKEN ANn E. A. PASCHOS
StanJ{l/'d Uneal' A cce!era-i(lJ' Center) Stanford lInive1sify, Stonford, ('abjornia. 94305
(Received 10 April 1909)

A model for highly inelastic electron-nucleon scattering at high energies is studied and compared with
existing data. This model envisages the proton to be composed of pointlike constituents ("partons")
from which the electron scatters incoherently. We propose that the model be tested by observing 'Y rays
scattered inelastically in a similar way from the nucleon. The magnitude of this inelastic Compton-scat-
tering cross section can be predicted from existing electron-scattering data, indicating that the experiment
is feasible, but difficult, at presently available energies.

I. INTRODUCTION curves, and we conclude that such experiments may be

O NE of the most interesting results emerging from

the study of inelastic lepton-hadron scattering at
feasible at energies available to SI,AC.

high energies and large momentum transfers is the II. INELASTIC e-p SCATTERING
possibility of obtaining detailed information about the The basic idea in the model is to represent the in-
structure, and about any fundamental constituents, of elastic scattering as quasifree scattering from pointlike
hadrons. We discuss here ,m intuitive but powerful constituents within the proton, when viewed from a
model, in which the nucleon is built of fundamental frame in which the proton has infmite momentum. Tbe
pointlike constituents. The important feature of this electron-proton center-of-mass frame is, at high energies,
model, as developed by Feynman, is its emphasis on a good approximation of such a frame. In the infinite-
the infinite-momentum frame of reference. momentum frame, the proton is Lorentz-contracted
It is argued that when the inelastic scattering process into a thin pancake, and the lepton scatters instan-
is viewed from this frame, the proper motion of the taneously. Furthermore, the proper motion of the con-
constituents of the proton is slowed down by the stituents, of partons, within the proton is slowed down
relativistic time dilatation, and the proton charge dis- by time dilatation. We can estimate the interaction
tribution is Lorentz-contracted as well. Then, under time and the lifetime of the virtual 5t:ttes within the
appropriate experimental conditions, the incident lepton proton. By using the notation of Fig. 1, we find the
scatters instantaneously and incoherently from the following.
individual constituents of the proton, assuming such a Time of interaction:
concept makes sense.
We were greatly motivated in this investigation by r'" 1j qo= 4P j(2M v-Q'), (2.1)
Fcynman, who put the above ideas into a highly work-
where 4" was uLlculatcd in the lepton-proton center-of-
able form. In Sec. II, we discuss the basic ideas and
mass frame.
equations for the model as they apply to electron-proton
Lifetime of virtual states:
scattering. Two models are then discussed in detail,
with interesting consequences for the ratio of clectron- T={[(XP)2+!'12J1I 2
proton and electron-neutron scattering. For a broad
class of such models, we lInd a sum rule which indicates +[ (1_x)2P2+!'22],!2_ [P'+ M p2Jl12) ~l
that, although it is not difficult to fit the data within
~50%, it is more difficult to do beller; the observed
21'
(2.2)
cross section is uncomfortably small.
In Sec. III, we look for stringent tests of F eynman's
picture. We propose that, under similar experimental If we now require that
conditions, inelastic Compton scattering can also be r«T, (2.3)
calculated within the model. It is shown that the ratio
of inelastic electron-proton to inelastic 'Y-proton scatter- then we can consider the partons, contained in the
ing, under identical kinematical conditions, is model- proton, as free during the interaction. Furthermore, if
independent and of order unity, provided the proton we consider large momentum transfers -rf»M2, then
constituents (which Feynman calls "partons") possess we expect the scattering from tbe individual partons to
unit charge and spin 0 or !. We propose experiments be incoherent. The above conditions appear to be
which can measure inelastic Compton scattering. To satisfied in the high-energy, large-mom en tum-transfer
this end we have estimated the yield and background experiments at SLAC.
The kinematics for e-p inelastic scattering have been
* Work supported by the U. S. Atomic Energy Commission. discussed in many places, in as many different nota-
185 1975

Reprinted from Phys. Rev. 185, 1975 (1969).

306 CHAPTER V

1976 J. D. BJORKEN AND E. A. PASCHOS 185

FINAL -STATE
INTERACTION
At infinite momentum, we visualize the intermediate
state from which the electron scatters as follows:
(a) It consists of a certain number N of free partons
_lg~§l~II~E~1 HAORONS
(with probability PN).
(b) The longitudinal momentum of the ith parton
is" fraction Xi of the totallllomcntUI1l of the proton:
p,=X,P. (2.10)
e, k e,k' (c) The mass of the parton, before and after the
FIG. 1. Kinematics of lepton-nucleon scattering collision, is small (or does not significantly change).
in the parton model. (d) The transverse momentum of the parton before
the collision can be neglected, in comparison with
tions. I We recall here that V(Q"), the transverse momentum imparted as p --+00.
EE' du du With these assumptions, it should be a good approAi-
mation to write, at infinite momentum,
.. drfd. dfldE' 4E'sin4 (!O)
(2.11)
X[W, cos'(!O)+2W, sin'(!O)], (2.4)
The question of corrections to this approximation has
where W, and W, are functions of the two invariants been studied by Drell, Levy, and Yan.' We shall not
v= (E-E')=q·PIM, consider them further here.
The contribution to W, from a single parton of mo-
Q"= -rf=4EE' sin'aO), (2.5) mentum xP' and charge Qi is then
evaluated in the laboratory frame. The ratio W ,IW, is
W,(')=x.Q;'Ma(q·x;P-!Q")
bounded. Using
W'=(H":'~_~ (2.6) =Q.'Ma(q.p- Q")=Q,'a(v-~). (2.12)
W2 Q";;;';;' , 2x. 2Mx
and the approximation ~O«l, we can write The factor Xi in front is necessary to ensure that

du ~ 0.2 W'(rf'V)[H(~~J, (2.7) du(')

lim --=Q. - - ,
\4..0.2)
(2.13)
dfldE' 4E'sin'(!O) u,+u1hEE' E_ drf q'
where u, and UI (~O) are the absorption cross sections
for transverse and longitudinal photons.' consistent with the Rutherford formula. For a general
It remains to calculate the invariant functions W, distribution of partons in the proton, we bave
and W" especially W,. Within the model, the virtual W,(v,rf)=L P(N)(L Q")N
photon interacts with one of the partons, while the N

xJ' ax jN(X)a(v-~).
rest remain undisturbed during the interaction. The
interaction with the parton is as if the parton were a (2.14)
free, structureless particle. The cross section du I dlliE' o 2xM
is then a sum over individual electron-parton interac-
tions appropriately weighted by the parton charge and Here peN) is the probability of finding a configuration
momentum. For a free particle of any spin and unit of N partons in the proton, (L, Q")N equals the average
charge, elementary calculation yields value of L' Ql in such configurations, and jN(X) is the
probability of finding in such configurations a parton
W,(v,rf) = a(v-Q"12M) = Ma(q·p-!Q') , (2.8) with longitudinal fraction X of the proton's momentum,
while for WI, we have that is, with four-momentum xP'.
Upon integrating over x, we find
u,=O for spinO,
vW,(v,rf) =L P(N)(L Q,')NXjN(X) = F(x) , (2.15)
0",=0 for spin L (2.9) N

with an indetenninate result for higher spins. with

1 s. Drell and J. Walecka~ Ann. Phys. (N. Y.) 28, 18 (1964);
x=Q'/2Mv. (2.16)
J. D. Bjorken, pJiys. Rev. 1/9, 1547 (1969).
• L. Hand, in bocetdings of Ihe Third International Symposi1t,. Therefore vW, is predicted to be a function of a single
on Eleclron and Pholon Inleradion at High Energies, Stanford
Line(Jl" Accelerator Center, 1%7 (Clearing House of Federal • s. D. Drell, D. J. Levy, and T.-M. Yan (private commu-
Scientific and Technical Information, Washington, D. c., 1968). nication),
LORENTZ-DIRAC DEFORMATION IN HIGH ENERGY PHYSICS 307

185 I NE LA ST IC e-p A )lD ~-p CATTER I NG , AND STR CT RE OF N 1977

0.4

0.3

]0.2
2'
(M/dSl) I+R 02
2
Flw)' /I d (TI SldE 11+2 ....!.... (I+~ )Ton 2 8
2
r I

"- MOTT
1.20
·
·+
R '(T, I"'l
.. R 1.40 v

0.1
e, 6° 2 (8eV/c)2 0
LGO
1.80 4

··
2.00

.
0.70 0
()'80 ~ 2.20
1.00 2.30

0
0 2 3 4 5 6 7
W' /1/ 0 2 in (BeV) - 1

Fro. 2. Plot of the data as a function of ./0'.

variable v/Q', a feature apparently satisfied by the Numerically,

data.' Furthermore, the model provides an interpreta-
tion of the nature of the function F(x)/x: It is the mean -Q' jdV
-W2= j dxF(x)",,0.16, (2.20)
square of the charge of partons with four-momentum 2M v
xP'. The experimental' determination of F(x) is shown
in Fig. 2. yielding a rather small mean-square charge per parton.
Before going into detailed models for jN(X), we In the following models, out of ignorance, we shall
notice that choose one-dimensional phase space for the distribution
function j N(Xl' .. XN); that is,
jN(XI' . 'XN)= const. (2.21)
(2.17)
1.I
An elementary calculation yields
dX,jN(XI)=l, jN (x) = (N -I)(I-x) N-' . (2.22)

where jN(XI, "', XN) is the joint probability of finding A. Three-Quark Model
partons (irrespective of charge) with longitudinal frac-
tions X " • • . , XN . It follows that IN is a symmetric Assuming that the proton is made up of three quarks
function of its arguments. Therefore, with the usual charges,' we obtain
vW,=j,(x)=2x(1-x) , x=Q'/2Mv. (2 .23)
[' xldx,jN(XI)=~ j dXI" ·dXN(L, Xi)
Jo N ' While th e data support vW, -> const as p ->c<> (or
x -> 0), the model predicts that pW, should vanish, a
XjN(.~ I, "',xN)o(l-L, Xi)
result not dependent on the specific choice of j,(x), but
=1/N. (2.18) only on the fact that ja is normalizable. In fact, within
our one-dimensional model, if the number of partons
Putting together (2 .18) and (2. 15), we obtain a sum rule is held finite, then the cross section vanishes as x -> O.

j dx F(x) =~ P(N)(li Q" )N/ N

If, and only if,

lim N'P(N)=const".<O (2.24)

N'..",
= mean-square charge per parton. (2.19)
4 See the rapporteur talk ofW. H. K. Panofsky, in Proceedings of will vW, approach a constant as X->O (v IQ' -""" ),
the Fourteenth International Conference on High-Energy Physics, This is shown in the Appendix.
Vienna, 1968 (CERN, Geneva, 1968), pp. 36-37, based on the
work of E. Bloom, D. Coward, H. DeStaebIer, J. Drees, J. Litl,
G. Miller, L. Mo, R. Taylor, M. Breidenbach, J. Friedman, G. • M. Gell-Mann, Phys. Letters 8, 214 (1964) ; c. Zweig, CER:-I
Hartmann, H . Kendall, and S. Loken. Report Nos. TH 401 , 402, 1964 (unpublished).
308 CHAPTER V

1978 J. D. BJORKEN AND E. A. PASCHOS 185

The normalization constant C is determined from the

condition
1
0.'
1= r:
N-3.'''·
PN(x)=C L: ---=C(I-ln2).
N - 3,5··· N (N -1)
(2.30)

For the neutron, the term in square brackets is omitted;

thus the final expressions are

1 (I-x) I
F.(x)=-- ---+---
12
x
1-lnl 9 (Z-xl 6 (I-x)'

X[lnC:
02 O~ ~ ~ I~
1.'11l Mv
FIG. 3. Plot of the results for a model of three quarks in a sca
X
)-2(I-xl Jl, (2.31)
of quark-antiquark pairs. The dashed line is visual fit through the
experimental points of Ref. 4.

B. Three Quarks in a Background of

F (Xl-=(_1
" - 9
)(l-X)
I-ln2 2-x .
Quark-Antiquark Pairs
It is clear that the results are model-dependent and
In order to try to improve the model, we assume that not to be taken too seriously. There is a need for a
in addition to the three quarks there is a distribution model-independent check of the basic assumptions.
of quark-antiquark pairs (the "pion cloud"?). The In Sec. III we discuss the corresponding process with
mean-square charge of the cloud we take to be statistical electrons replaced by 'Y rays as a test of the basic idea
a: Q?-)doud/N =![W'+ (!)'+ (i)'J= 2/9. (2.25)
of the calculation.
In Fig. 3, we compare Eqs. (2.31) with the experi-
Therefore, ment. The shape of the curve is in fair agreement, and
a: Q")N= H2/9(N -3)=2/9N+! for the proton
could be improved by suppressing the contribution of
three-parton configurations. On the other hand, the
over-all normalization is off, as discussed below [(2.28)]'
=H2/9(N-3)=2/9N for the neutron. Readjustment of the coefficients P N or the distributions
(2.26) /N(X) of longitudinal fraction cannot improve this
feature. The ratio of neutron cross section to proton
For peN), we choose remains nearly constant and about 0.8 over a large
P(N)=C/N(N-1), (2.27) range of x, although the ratio approaches 1 as x ..... o.
According to (2.9), if the partons all have spin ~, we
on the grounds that it is simple and has the asymptotic expect u,/u, ..... 0; if they are spinless, we find instead
behavior which makes vW, .... canst as x .... 0. Before u,iu, ..... O. A finite ratio would indicate that both kinds
we begin, we note, from (2.26) and (2.19), that

l'
are present.

dx F(x) = l' dx vW,=2/ 9+Hl/N»0.22 (proton)

=2/ 9=0.22 (neutron)
",0.16 (expt.). (2.28) PROTON
~YJt PARTON S

Thus we cannot expect a fit better than ~ 50% to the

data. Inserting (2.27), (2.26), and (2.22) into the ex-
la~
pression (2.15) for F(x), we can perform the sum. For
the proton,
F(x)=x r:
N-3,5···
P(N) (N-l)(I-x)N-2(2/9N+i) I~ FIG. 4. Diagrams contribu-
ting to inelastic Compton
scattering.

=c r:
N-a,5 .. ·
(=+":")X(I-X)lV-'
9 3.Y
IC~

'1 ---+--- -2(1-1') JlI'

=C
2 (1-:1') 1 x [ (2-:1')
'J (2 -x)
In -
6 (I-x)' x
\d)

g@\; \~~~

~~~ -~~~- -~-

(2.29) lei
LORENTZ-DIRAC DEFORMATION IN HIGH ENERGY PHYSICS 309

185 INELASTIC e-p AND "'{-p SCATTERING. AND STRUCTURE OF N 1979

III. INELASTIC COMPTON SCATTERING

In this section, we suggest that a good way to find
out about the internal structme of the proton is to
look at it. That is, one should measure the inelastic
scattering of photons from protons, yielding a photon
plus anything else. The yield can be predicted in terms ) HADRONS

of this model. We visualize the inelastic scattering again

as incoherent scattering from the partons in the proton FIG. 5. Kinematics of Compton scattering.
according to the point cross section. The elementary
photon-parton scattering goes by the diagrams shown
in Fig. 4. Diagram (c) in Fig. 4, does not occur for with
spin-! partons. Its contribution is evidently very similar R=1 for spin!
to the case of electron scattering. We argue that
=0 for spin O. (3.4)
exchange-terms such as in diagrams (d) and (e) can
be ignored on the basis that the lifetime of the inter-
If the parton has longitudinal fraction x, we make the
mediate states, between absorption and emission of
replacements (we are here in the infinite-momentum
the photon, is of order Eo. m.-1 in the "Y-proton center-
frame)
of-mass ,,'stem and much less than the lifetime of the
virtual-p;rton states of the proton estimated in (2.2). S~XS, U-4XU, t-4t,
Fmthermore, because we require the momentum trans- 6(v+I/2M) -->o(v+I/2Mx). (3.5)
fer between the photons (0 be large (. 1 BeV/c), the
parton will necessarily have transverse momentum We then multiply by the distribution function fN(X)
much larger than the average momentum within the and by peN), integrate over x, and sum over N [d.
proton and the probability that it interacts with another Eqs. (2.12)-(2.15)] to obtain
parton within the proton at such high Pi is very small.
The kinematics for the process is illustrated in Fig. 5,
with the value
dlT
-=L:
/.1 dxP(N)fN(x)-
471"'"
dldv N 0 x's'
s= (k+ p)'",,2k· p= (2Mk)lab»M' ,
1= (II-k')'= -2k·k'= -[4kk' sin'(!8)]lab, (3.1) X[1-R(S+U)'JIl(~+-I-)(L: Q,'JN. (3.6)
Mv= (k-k')·p=M(k-k')lab, 2su . 2Mx •
,,""-s-I. Integrating over x and expressing the result in labora-
We require - I to be large, as we already men- tory variables, Eq. (3.1) gives the final result:
tioned, so that the process is incoherent, and only
dlT
Figs. 4(a)-4(c) (for integer spin) need be taken into 01.' V[HR~J
accoun t. The method for calculation is the same as in drldk' 411' sin'(!8) kk' 2kk'
Sec. IT: We take the Compton cross section from point
partons and average over the parton momentum dis- xL: P(N)XjN(X)(L: Q,')N, (3.7)
tributions and proton configurations. This elementary N ,
photon-parton interaction is given (for spin-l partons) with
bv the Klein-Nishina formula written in terms of the x=-1/2Mv. (3.8)
lVi:andelstam variables s, I, u. If the parton carries the
full momentum P(x= 1), we have Making the identification
k <-> E, k' +-> E', -I"'" Q', R <-> 1T,/(IT,+lTr), (3.9)
dlT 471"'" ( I)
- = - 6 v+- Q,' for spin 0
dldv s' 2M we find a remarkable correspondence between (3.7)
and (2.14), (2.9), and (2.7), the corresponding cross
= - 2~OI.'(~+~)6(v+~)Qi' for spin!, (3.2) section for electrons. The only changes are the addi-
tional factor v'/kk' and the replacement Q,' --> Q,' .
.I' u s 2M
Even the factor in square brackets dependent on parton
which can be combined intu the fortll spin is the same. We conclude that within the validity
of this parton model, for par tons of unit charge (Q'=Q')
tilT
-=-
471"<"( 1-R---- I)Q,',
(s+U)') 6( v+- (3.3) and spin 0 or t. the ratio of electron scattering to--:r
dldv s' 2su 2M scattering is a model-independent number of order
310 CHAPTER V

1980 .I, D, BJORKEN AND E. A, PASCHOS 185

Icf'" \'
I"
- PHOTONS fROM I
CONPTOH \
\
- - - A«:ITONS FROM
10- 31 .". DECAv \-_. r;~~5 fR ~ r·
... J8 rAWe. \
\
.\
I
MI~
Ee>
u :.
10"
I

~• b r~c:
'D

" 'D
lOll

10"

100 J)
0 2 6 lO '2 14 16 16 20 FIG. 8. Inelastic Compton scattering for a 22-GeV incident
k# GeV/( bremsstrahlung spectrum. The background curves are the same
as in Fig. 6.
FIG. 6. Double-difJerentiallaboratory cross section for inelastic
Compton scattering for an 18-GeV incident bremsstrahlung spec-
trum. The solid curve corresponds to the signal. The dotted curve where for any operator O(N) we have
is the background of 'Y's from "'s using the data of Ref, 8 as dis-
cussed in the text. (0)= EN P(N)O (N)fN (x) , (3.11)
For our model of three quarks in a cloud of quark-
unity, In general,'
antiquark pairs, there exist upper and lower limits for

d") £j'd")
(tKldE'
- (LQ;')/(EQ,'),
kk' tKldE' '. '
7P
=
,
- (3,10)
(L Q')! (E Q'), We note from (2,25) and (2,26), for
the proton, that
11 2 1 2
(E Q;')N = - +-(N - 3) =-(E Q;')N+-
IV
27 27 3 , 27

5 4
= -(~ Q" )N--N. (3.12)
·. t 9 ,
Therefore, for identical kinematical regions' we have
81

Mel~ lOll
u :.
1 -,,'j dU) ( du) 5.' ( du )
.
~I~"
b "
~"
10'''' "3 £i?,.,dfldE' '. < dfldE' >P <"9 EE' dfldE' '.
(3,13)

In principle it should be possible, if the model is

correct, to distinguish between fractional and integer-
charged partons,
K) 12 14 16 18 20
IV. BACKGROUND AND RATE ESTIMATES FOR
k'GeWc
INELASTIC -r-RAY EXPERIMENT
FIG, 7, Inelastic Compton scattering for a 20-GeV incident
bremsstrahlung spectrum. The background curves are the same We consider an experiment in which a bremsstrahlung
as in Fig. 6. beam is sent through hydrogen and the inelastically
scattered 'Y ray is detected with momentum k' at angle
6 We point Qut that the argument used in Sec. II in deriving
the sum rule can be repeated here, giving e,The main problem, in principle, is to differentiate the
Compton'Y rays from the 'Y rays coming from the decays
(!' J( d")
~~r,
uual',
kk'
T-<i.='LP(N)(LQ;')N/N,
-,p V U R N of photoproduced 'JI'0\. Here, we calculate the effective
where <TR=[a'/ 4k' sin'(!O)](1+R",/ 2kk'), We have also assumed 7 These bounds depend on the implicit assumptions of Eq.
the same momentum distribution !N(X) for the spin-O and spin-l (2.25). A smaller lower bound of ~ is obtained if we assume that
partons. all the pairs are made up of charges! and - t.
LORENTZ-DIRAC DEFORMATION IN HIGH ENERGY PHYSICS 311

185 INELASTIC e-p AND -y-p SCATTERING, AND STRUCTURE OF N 1981

It :0 25 GeV/c
- - PHOTONS FROM COMPTON
---- PHOTONS FROM DECAYS .,,0

FIG. 9. Inelastic Compton scattering

for a 25-GeV incident bremsstrahlung
spectrum. The background curves are the
same as in Fig. 6.

5 10 15 20 25
k' GeV/c

cross section for the production of Compton-scattered per nucleon,

"Y rays by folding the Compton cross section over the
bremsstrahlung spectrum_ We then calculate for com-
parison the corresponding background"y rays, estimated ( dur,) = (yield)r+/O.7 t'ldt(gfradlengthofBe)
d'Jdk ,r: } '-0
at SLAC from th e beam survey experiments' for

-l
charged ".'s_ We define X (Avogadro No.)
d u eff E electron dk du'tp =8.2X10--"(yield) r+ cm'/ sr BeV. (4.4)
-- -- (4.1)
d'Jdk' k'+I ' I/2M k d'Jdk ' - The terms in the denominator have the following
origin: The thin-target bremsstrahlung spectrum is
We assume, optimistically, that the partons have unit tdk / k, where t is the thickness in radiation lengths (r.l.)
charge and spin!; from (3.3) and (4_1) we obtain (the target was 0.3 r.!. Be). The factor 0.7 is a thick.-
target correction calculated by Tsai and Van Whitis.'
dqef f 4a2 J .\!(E -k')/ 4Ek' ,;in2 (!9)
- =- XdXF(X) (Yield)r+ is taken from the SLAC User's Handbook'
d'Jdk' M'k' ' I' and the 'Y-ray flux is obtained by folding the ,,-'-decay
spectrum into (4.4):
x[ 1 4k'X ~'GO) +8k"X';'(!8)] , (4_2)
( -dU)'
- =2 lE(dur)
-- dk' 21~(dur)
-""- - - dk'
dfldk r' k, dfldk' ,H k' k, k, dfldk' 'If
with
'" (2Eo/ k,8) (8.2 X 10--") (yield)r+
X=M. / Q'. (4.3)
(as function of k,), (4.5)
We have calculated this expression for several in-
cident electron energies as a function of k' and 8, using where in the last step we have used the empirical ob-
for F(,,) the values given in Ref. 2. The results are servation that
shown as Figs. 6-9. In the same figures are shown our dUr/df!dk'~e-k'''EO , (4.6)
estimates of the corresponding background from the with
decay of photoproduced ".o's into "Y rays. The estimate E o",0.154 BeV.
was made by assuming that the yield of ".+ measured
in the SLAC beam survey experiment' equals the ,,-' In Figs. 6-9, the background is that from an 18-BeV
yield. We thereby obtain, for the effective cross section bremsstrahlung beam. It is expected that this back-
ground increases slowly with beam energy, and keeps
8 SLAC User's Handbook, Sec. 0.1, Figs. 1 and 2 (unpublished). • Y. S. Tsai and Van Whitis, Phys. Rev. 149, 1948 (1966).
312 CHAPTER V

1982 J. D. BjORKEN AND E. A. PASCHOS 185

1m'
Q.fjtO

I 2 3 • FIG. 11. Contour of integra-

--l~~~---I--R.. tion for;tbe Sommerfeld-Wat-
son transform.

FIG. 10. Muon·pair production by inelastic Compton scattering.

a-iO'J

the same shape, in particular, the exponential depend-

ence on transverse momentum. However, as the primary
electron energy increases, the 'Y-ray spectrum from the over the contour shown in Fig. 11 :

1
inelastic Compton process, if it exists, is displaced up-
ward in energy, so that for 20-BeV electrons and above, (l-x),N-'e"N
there is a region where the Compton signal dominates
F(x)=2x NP(2N+l)(L: Q")(2N+l). dN.
t: i sln1rN
the ,..0 noise. In any case, it will certainly be necessary
to compare the 'Y-ray spectrum with that of the ,..+ (AI)
under the same conditions as reassurance that Compton
'Y rays are indeed being seen. The contribution of the semicircle at infinity is negli-
A variation of this experiment is to consider the gible. For what remains in the integral, we can use
inelastic Compton terms in I'-pair photoproduction (for x --70)
(see Fig. 10). We have not analyzed this process in x(l-x)'N-''''xe-('N-')x= _!M-('N-l)x/ON, (A2)
detail. The rate is diminished by a factor roughly'O
~ (2a/3,..)[In (Emax/m,) -3.5J~ 1/350 but if the charged and integrate by parts:
pions are absorbed immediately downstream from the
iJ
l
·+i~
target, the background muon flux from 7r± decay can F(x) = .('N-l)x_
be reduced by a factor ~ 1/700 as well. Furthermore, a-ioa aN
the two muons are strongly correlated in angle, pro-
viding a quite unique signature. All this is encourage-
ment that perhaps the background is manageable. The (A3)
"singles" background from the Bethe-Heitler diagrams,
for which the undetected muon predominantly goes in In the limit of x --7 0, we have
the forward direction, is interesting, as well, and very
likely exceeds the singles rate from Compton I' pairs. N P(2N +1) (L Q;)(2N+lle'rNI·+i~
But this is also of interest in testing I'-e universality at F(x) ~ --=----=-=--=-=--
sinn-N
very high '1'. The "Bethe-Heitler" muons probably 4-iao

dominate the background muons from ,..+ decay, but (M)

the necessary estimates have not yet been made.
Therefore F(x) approaches a constant if and only if
APPENDIX peN) -+ c/NeL Ql)N. For eLi Ql)N linear in N, this
We attain a condition between peN) and (L: Q")N reduces to the condition mentioned in the text.
which guarantees that F(x) is analytic at x=O and
equal to a nonzero constant. Using the Sommerfeld- ACKNOWLEDGMENTS
Watson transformation, we rewrite (2.15) as an integral
We thank R. P. Feynman, J. Weyers, and our
10 R. H. Dalitz, Proc. Phys. Soc. (London) A64, 667 (1951). colleagues at SLAC for many helpful conversations.
LORENTZ-DIRAC DEFORMATION IN HIGH ENERGY PHYSICS 313

PHYSICAL REVIEW D VOLUME IS, NUMBER I 1 JANUARY 1977

Covariant harmonic oscillators and the parton picture

Y. S. Kim
Center for Theoretical Physics. Department of Physics and Astronomy. University of Maryland, College Park, Maryland 20742

Marilyn E. Noz
Department of Radiology, New York University Medical Center. New York. New York 10016
(Received 23 February 1976)
It is shown that the covariant~harmonic·oscillator wave function exhibits the peculiarities of the Feynman
parton picture in the infinite· momentum frame.

In our previous publications",2 we discussed both singularities which cause deviations from Bjorken
the conceptual and phenomenological aspects of scaling. lO Our oscillator model does not contra-
the covariant-harmonic-oscillator formalism. dict this physical picture.
Based on the Lorentz-invariant differential equa- Perhaps the most puzzling and irritating ques-
tion proposed by Yukawa in connection with Born's tions in Feynman's parton picture' have been the
reciprocity hypothesis,' our starting point was a following problems:
technical innovation over the work of Feynman (a) The picture is valid only in the infinite-mo-
et ai.' Our solutions to the same oscillator equa- mentum frame.
tion satisfy all the requirements of nonrelativistic (b) Partons behave as free independent particles.
quantum mechanics in a given Lorentz frame, and (c) While the hadron moves fast, there are wee
satisfy the requirement of Lorentz-contracted partons.
probability interpretation for different Lorentz (d) The longitudinal parton momenta are light-
frames. We contend that our oscillator model is like.
the first formalism since the invention of quantum (e) The number of partons seemS much larger
mechanics in which the wave functions carry a than the number of quarks inside the hadron.
covariant probability interpretation. 3 The purpose of this paper is to provide qualita-
The real strength of our oscillator model lies in tive answers to all of the above questions. Our
the fact that one and the same wave function can starting point is the system of two bound quarks
provide the languages for both slow and fast had- in the rest frame which can be described by a
rons. Our formalism can be applied to the quark- covariant-harmonie-oscillator wave function. We
model calculations til the low-, intermediate-, and shall then boost this covariant bound system to an
high-energy regions.I.,2,6 However, one of the infinite-momentum frame and show that the pecu-
most challenging questions in high-energy physics liarities of the covariant oscillator coincide ex-
has been how to explain Feynman's parton" 8 pic- actly with the parton properties mentioned above.
ture in terms of a formalism Which can also de- Following Feynman et al.' we call these two
scribe the static properties of the hadron. quarks a and b. In the harmonic-oscillator for-
Another approach to this problem has been to malism,'·2,3 the quark momenta P. and p, are not
explain Bjorken scaling in terms of the light-cone on the mass shell, but the total hadronic momen-
commutators and the initial hadron in its rest tum
frame. 9 Here, one promising line of reasoning
(1)
has been that the hadron is a composite particle
and that its distribution function eliminates all the is on the mass shell. It is convenient to use the

15 335

Reprinted from Phys. Rev. D 15, 335 (1977).

314 CHAPTER V

336 Y. S. KIM AND MARILYN E. NOZ 15

four-momentum difference (V)(q _2) =i , (9)

q = /2(P, -P.) . (2) (L 2 )(q/) =t .
We can also assign space-time coordinates to Let us go back to the wave functions of Eqs. (4)
these quarks. Let us denote their coordinates by and (5). If fJ =0, the wave functions correspond to
x. and x" and introduce the relative coordinate those in the rest frame. As the momentum of the
hadron becomes large,
1
2T2 (x, -x.) (3)

G:~j-(~r '
x= .
(10)
The transverse variables play only trivial roles
in the harmonic-oscillator formalism and also in where M is the mass of the hadron. As P o -"",
the parton picture. For this reason, we shall the width of the L (and q J distribution becomes
omit the transverse part of the wave function in the vanishingly small. Consequently,
following discussion.
c=O and q_=O. (11)
If the hadron moves along the z axis with veloc-
ity fl, the ground-state wave function for this two- This means that both ~ and q are lightlike vectors,
quark system can be written as'·ll and

- 2" "i+i3 ~+ 2
Wexp {W[(1-fJ) + (1+/'
1=131) L 2l}
~+=l2z=/2t ,
iJ!(x,ll) = 2iT J ' q+=/2q.=/2qo·
(12)

(4) In the infinite-momentum limit, the effective

spring constant associated with the ~+ motion be-
where
comes vanishingly small. The motion along the ~ +
1 1 axis therefore becomes like that of a free lightlike
~+ = 72«( +z), L =J2(t-z). particle.
The behavior of the q _ distribution and that of
W is the" spring constant" of the oscillator system. the q + distribution are illustrated in Fig. 1. The
We can construct the momentum wave function by width of the q + distribution becomes large when
taking the Fourier transform of the above expres- Po becomes large. This may appear as a violation
sion, of the uncertainty relation, but it is not. q + and ~ +
are not conjugate variables. The precise uncer-
<!>(q)=(-!;)' f d'xe-'q·Xq;(x,fJ) . (5) tainty relation was derived in Ref. 11 and is stated
in Eq. (9) of the present paper.
We can now associate the above-mentioned
This momentum wave function takes the form pecularity with the puzzling features of Feynman's
_ 1 exp {1
<!>(q,/'I )-2uw 11(1-1l) 2 (1+fJ\ 2J} ,
-2wL~q+ + I-fijq-
parton picture. Let us first observe that the ha-
dronic four-momentum P becomes lightlike in the
infinite-momentum limit, and consider the four-
(6)

where

According to Eq. (5), we have

(7)

Because of the above asymmetry in sign,

. a (8)
q+=zac·
FIG. 1. Momentum wave flUlctions in the rest frame
and in a large-momentum frame. In the rest frame
This means that q + is conjugate to L, and q _ is
where f3 '= O. the q + and q_ distributions take the same
conjugate to L. In terms of these variables, the
fJ
above -dependent wave functions generate the fol-
form. When the hadron moves fast and {3-1, the q_
distribution becomes narrower while the q + distribu-
lowing Il-independent (Lorentz-invariant) mini- tion becomes wide spread. This wide-spread q + dis-
mum-uncertainty product. l1 tribution corresponds to the parton distribUtion.
LORENTZ-DIRAC DEFORMATION IN HIGH ENERGY PHYSICS 315

15 COVARIANT HARMONIC OSCILLATORS AND THE PARTON ... 337

momentum of the constituent quark particles. According to Eq. (12), the ~+ axis is
1 1
also the f2t axis. Therefore, the time duration is
P'='iP-m q (13) of the order of (P';M rw). This interval increases
·
Since the four-vector q is lightlike, and we are
as Po becomes large. If this interval is much
larger than the characteristic time of electro-
considering here only longitudinal momenta, P. is magnetic interaction, then the partons of the pres-
also lightlike. ent paper will indeed behave as Feynman's partons.
Considering the width of the Gaussian function We have shown above qualitatively how the co-
for the q + distribution, which is also the f2 q 0 variant oscillator produces Feynman's parton pic-
distribution, we can say that the momentum of the ture in the infinite-momentum limit. The next
constituent quark mostly lies in the interval de- question then is how we can use this formalism to
fined by the following limits: carry out the parton-model calculations.

Pm~ =PoG +
In order to answer this question, we note first
;) , of all that the above two-body formalism can be
(14) easily generalized to the three-quark nucleon sys-
Pmm =Po(~ - ;). tem. 4 In performing the parton-model calcula-
tions, we have to square the wave function to get
The quantity (rw/2M) is of the same order of the probability-density function. The Gaussian
magnitude as ~. For this reason, the lightlike form remains Gaussian during the squaring pro-
four-momentum P. can be written as cess. The Lorentz -contraction property of the
Gaussian probability distribution is identical to
p.=aP, (15)
that of the wave function except for the factor of 2
with a ranging apprOXimately from zero to one. in the exponent. In fact, the width quoted in Eq.
This wide-spread distribution and division of the (14) is derived from the width of the probability
four-momentum are exactly like those of the par- function.
ton model. As was noted earlier in this paper, the proba-
Let us go back to the C distribution, which is bility function exhibits a 6 function in the (q. - qo)
also the f2z distribution. We noted above that the variable in the infinite-momentum limit. We can
motion along this axis should be almost free. Then now eliminate the q 0 dependence by integrating
the momentum has to be sharply defined, and the over this variable. The resulting function be-
momentum cannot have a wide-spread distribution. comes the parton distribution function in the three-
Therefore the momentum distribution we noted in dimensional space.
Eqs. (14) and (15) should be regarded as a distri- The immediate calculations we can do using the
bution of free particles which are lightlike. This above-mentioned procedure have already been
is exactly what we have in the original form of carried out by Le Yaouanc e tal. 12 Starting from
Feynman's parton model, as well as being charac- the three-dimensional parton distribution function
teristic of the quantum-mechanical picture of which we could obtain by following the procedure
blackbody radiation. In both cases, the number of outlined above, Le Yaouanc et al. indeed carried
lightlike particles is not conserved. out a comprehensive phenomenological analysis of
Finally, let us consider the time interval during all interesting physical quantities in the inelastic
which the above-mentioned partons behave as free electron-nucleon scattering.

!Y. S. Kim and M. E. Noz, Phys. Rev. D 8,3521 (1973). 7R. P. Feynman. in High Energy ColliSions, proceedings
'Yo S. Kim and M. E. Noz, Phys. Rev. D 12, 129 (1975). of the Third International Conference, Stony Brook,
For orthogonality and Lorentz-contractiOn properties New York, edited by C. N. Yanget ai. (Gordon and
of the harmonic-oscillator wave functions, see M. J. Breach, New York, 1969).
Rulz, Phys. Rev. D 10, 4306 (1974). 8J. D. Bjorken and E. A. Paschos, Phys. Rev. 185, 1975
'H. Yukawa, Phys. Re-;'- 91, 416 (1953). (1969). -
4R. P. Feynman, M. Kisllnger, and F. Ravndal, Phys. 'R. A. Brandt, Phys. Rev. Lett. 22, 1149 (1969); 23,
Rev. D 3, 2706 (1971). 1260 (1969). For a review article, see Y. Frisb;;;an,
sp. A. M:-Dirac, The Development of Quantum Theory in Proceedings of the XVI Internaiional Co1!ference on
(Gordon and Breach, New York, 1971). High Energy Physics, Chicago-Batavia, m., 1972,
'Yo S. Kim and M. E. Noz, Phys. Rev. D 12,122 (1975); edited by J. D. Jackson and A. Roberts (NAL, Batavia,
M. J. Ruiz, ibid. 12,2922 (1975); Y. S. Kim, ibid. 14, Ill., 1973), Vol. 4, p. 119.
273 (1976); Y. S. Kim and M. E. Noz, ibid. (to be p~_ lOS. D. Drell and T. D. Lee, Phys. Rev. D 5, 1738 (1972);
Jished). C.lI. Woo, Phys. Rev. D.!!., 1128 (1972).-We would
316 CHAPTER V

338 Y. S. KIM AND MARILYN E. NOZ 15

like to thank C. H. Woo for explaining the content of wave ftmctions. The contraction property is explained
his paper. in terms of the step-up operator which transforms
lIy. S. Kim, Univ. of Maryland CTP Tech. Report No. like the longitudinal coordinate.
76-008, 1975 (unpublished). This paper contains also 12A. Le Yaouanc, L. Oliver, O. P~ne, and J.-C. Raynal,
a more precise explanation of the Lorentz-contrac- Phys. Rev. D g, 2137 (1975).
tlon property of excited-state barmonlc-oscillator
LORENTZ-DIRAC DEFORMATION IN HIGH ENERGY PHYSICS 317

PHYSICAL REVIEW D VOLUME 23, NUMBER 11 1 JUNE 1981

Valons and harmonic oscillators

Paul E. Hussar
Center for Theoretical Physics. Department of Physics and Astronomy. University of Maryland. College Park. Maryland 20742
(Received 19 January 1981)
The valon distribution derived by Hwa is compared with the valence~quark distribution from the covariant·
harmonic-oscillator wave function which correctly describes the proton-form-factor behavior, and which provides a
covariant representation of the hadron mass spectra. It is shown that the harmonic-oscillator curve closely
approximates the valon distribution for x > 0.25. For 0 < x < 0.25, the agreement is reasonable.

Since the introduction of the parton model,' one smeared by the momentum-fraction distribution
of the central issues in high-energy physics has of the valons. The assumption is that the inter-
been the relationship between the partons and the action which confines the valons, that is, the
valence quarks which seem responsible for other gluons which are exchanged among them, will not
high-energy properties such as mass spectra play so large a role in scattering processes as
and form factors. It was once naively believed to make the above analysis insufficient. The con-
that the proton structure function could be cal- sistency with which the valon picture appears to
culated from the valence-quark distributions inside model the actual behavior of nucleon structure
the hadron.' However, it is by now firmly esta- functions' is a persuasive argument that the
blished that quantum-chromodynamics (QeD) assumption made here is a good one.
processes stand between the valence quarks and The nucleon structure functions in the valon
the observed structure functions. model take the form'
With this point in mind, Hwa recently developed
an appealing method for dealing with nucleon FN(X,Q2)~ ~ { dY GVIN (y)FV(x/y,Q 2), (1)
structure functions. Hwa's approach separates
out a component of the structure functions which where FN(x, Q2) is a nucleon structure function
is completely determined by QeD renormalization, (either F 2 or xF ,), P(x, Q2) is the corresponding
and uses the data to calculate a momentum-fraction function for a valon v, and GVIN(y)dy is the prob-
distribution for three constituent quark clusters ability of the valon having momentum fraction
or "valons".' The purpose of this paper is to point between y and y + dy. The sum is taken over the
out that Hwa's valon distribution is close to the three valons which constitute the nucleon. From
valence-quark distribution derivable from the the definition, we must have
covariant harmonic-oscillator model which has
been effective in explaining the nucleon mass spec- (2)
tra and form factors.
The valon picture'·' is basically an attempt to and
establish a connection between the quark model
in which hadrons are bound states of their con- (3)
stituent quarks and the parton model which seems
necessary if we are to explain the observed struc- In the expression of Eq. (1) the nucleon structure
ture functions. When probed at high Q2, each of functions FN(x, Q') are well known from experi-
the valence quarks will itself be resolved into ment. For high-Q' processes, the renormalization-
infinitely many constituents due to the fact that group methods in QeD allow a description of
each of the valence quarks will be accompanied FV(x, Q2) in terms of its moments. Making use of
by a cloud of quarks, antiquarks, and gluons data from neutrino and muon scattering, and known
produced in ongoing QeD processes. We can cal- results from QeD, Hwa has obtained
culate the evolution of the valence quarks in-
(4)
volved here to leading order in QeD using the
renormalization-group methods. s The nucleon
structure functions in the valon model are then (5)
given by the corresponding functions for each of
(6)
the valons (the valence quark plus its cloud)

23 2781 © 1981 The American Physical Society

Reprinted from Phys. Rev. D 23,2781 (1981),

318 CHAPTER V

2782 COMMENTS 23

where G./~(y) and G./~(y) are the momentum- function can explain the peculiarities in the orig-
fraction distributions for the u valon and the d inal version of Feynman's parton modeL'
valon in a proton, respectively, and Go/~(y) is the The three-particle kinematics associated with
momentum-fraction distribution obtained assuming the proton wave function has been studied exten-
that these distributions are flavor independent. sively.' It is, then, a straightforward matter to
The G./~ and Go/~ functions are plotted in Fig. 1. derive the momentum-fraction distribution func-
In spite of QCD's effectiveness in dealing with tion in this scheme.'o The result is
the Q2 evolution of the hadronic structure func-
tions, it has not yet been helpful in determining G.se(Y) =[3m/(21rw)'/2] exp[ -(m 2 /2w){l_ 3y)"j.
what should really be the starting point of this (7)
evolution, that Is, the distribution of the valence
quarks inside the hadron. In the meantime, we In the above expreSSion, m represents the nucleon
mass, while w is the oscillator spring constant.
are allowed to consider other models which are
G•• eIY) is also plotted in Fig. 1. The value of w
consistent with existing rules of quantum mech-
anics and special relativity. The covariant har- is taken to be m 2/2, which is the most acceptable
monic oscillator is such a model. value in the calculations of the mass spectrum 11
The relatiVistic oscillator model existed long and the gAlg. ratio.'2
before the quark model was invented." The early Noteworthy, perhaps, is the fact that G•• eIY) is
applications of the osc illator model in the quark normalized over the whole real line, while Hwa's
picture of hadrons include the study of hadronic distribution extended only from y =0 to y =1. The
mass spectra. Its effectiveness in the relativistic oscillator wave function is not restricted to the
domain was demonstrated first by Fujimura et region of physically observable constituent en-
al. in their successful calculation of the proton ergies. However, this is nothing strange, inas-
form factor.7 Fujimura et al. used normalizable much as the nonrelativistic oscillator exhibits the
relativistic wave functions for three valence quarks. same property. It should also be pointed out that
We propose to use the same relativistic wave the integral given in Eq .. (1) which determines the
structure functions is taken only over values from
function to determine the valence-quark distri-
bution in the parton regime.
o to 1 so that a nonzero value of G outside of this
region will play no role. At the present time,
The covariant harmonic oscillator has been
we do not have enough experimental accuracy to
discussed extensively in the literature. In par-
decide whether the curvature is of the polynomial
ticular, it was shown by Kim and Noz· that a
type given by Hwa or of the GauSSian form.
rapidly moving hadron in this model has a broad
The agreement between G (y) from the data and
longitudinal-momentum distribution while the
the momentum-fraction distribution function for
spring constant of the oscillator becomes weak
the constituent quarks in the covariant oscillator
to the laboratory-frame observer. It was pointed
model is surprisingly good. Clearly for y > 0.25,
out in Ref. 8 that this behavior of the oscillator
the oscillator distribution is very close to the
phenomenological curve given by Hwa. For y
< 0.25, the numerical agreement is not as good
as in the larger-y region. However, we have to
accept the fact that there are still large experi-
2.0 mental uncertainties in this small-y region, and
it would be difficult to trust at this time any closer
agreement than that given in Fig. 1.
1.5 As for the flavor dependence indicated by Hwa's
valon curves, it is not yet clear to us to what
1.0
extent the difference between G./~ and Go/, is
actually required by experimental evidence since
this feature is observed mostly at smaller values
0.5 of y where, again, the experimental uncertainties
are the greatest. In any case, the covariant oscil-
lator is not designed to account for such a dif-
1.0 J ference. If the difference really exists, there
must be additional dynamical effects, beyond
FIG. 1. The experimental and calculated GIY) func-
tions. The G function calculated in the covariant os- what we can account for with the oscillator model
cUlator model is compared with the valon distribution and QCD, to explain the flavor dependence. This
functions obtained by Hwa from experimental data. is beyond the scope of this paper.
LORENTZ-DIRAC DEFORMATION IN HIGH ENERGY PHYSICS 319

23 COMMENTS 2783

This paper is based on a part of the anthor's losophy. The author would like to thank Professor
dissertation to be submitted to the faculty of the Y. S. Kim for suggesting this research. The serv-
University of Maryland in partial fulfillment of ices of the University of Maryland Computer Sci-
the requirements for a degree of Doctor of Phi- ence Center are also appreciated.

IR• P. Feynman, in Third Topical Conference on High 'i K.Fujimura, T. Kobayashi, and M. Namiki, Prog.
Energy Collisions, edited by C. N. Yang (Gordon and Theor. Phys. 43, 73 (1970). See also R. G. Llpes,
Breach, New York, 1969). Phys. Rev. D5. 2849 (1972).
'J. D. Bjorken and E. A. Paschos, Phys. Rev. 185, 1975
8
Y. S. Kim and -M. E. Noz, Phys. Rev. D 15, 335 (1977).
(1969). - See also Y. S. Kim and M. E. Noz, Fou;;;). Phys.!!,
3R• C. Hwa, Phys. Rev. D 22, 759 (1980). 375 (1979).
'R. C. Hwa and M. S. Zahlr,- Phys. Rev. D 23, 2539 9R. P. Feynman, M. Kislinger. and F. Ravndal, Phys.
(1981). - Rev. D 3, 2706 (1971).
10 -
('. A. DeGrand, Nucl. Phys. B151, 485 (1970). Y. S. Kim and M. E. Noz, Prog. Theor. Phys. 60, 801
H. Yukawa, Phys. Rev. 91, 461 (1953); M. Markov, Qn~ -
Nuovo eimento Suppl. 3~760 (1956). For the latest Up. E. HUBsar, Y. S. Kim. and M. E. Noz, Am. J. Phys.
reinterpretation of these early papers, see D. Han 48, 1043 (1980).
and Y. S. Kim, Prog. Theor. Phys. &i, 1852 (1980). I'M. Ruiz, Phys. Rev. D 11, 2922 (1975).
Chapter VI

Massless Particles and Gauge Transformations

It was shown in Wigner's 1939 paper that the little group for massless particles is
isomorphic to the two-dimensional Euclidean group consisting of rotations around
the origin and translations along the two perpendicular directions. It is not difficult
to associate the rotation with the helicity. However, the physieal interpretation of
the translation-like degrees of freedom had been an unsolved problem. Fortunately,
two of Weinberg's 1964 papers started breaking ground on this problem, and led to
the clue that the translation-like transformations are gauge transformations.
In 1982, Han, Kim, and Son used the Lorentz condition to reduce the complicated
transformation matrix of the little group into the transpose of the coordinate
transformation matrix on the two-dimensional plane. They confirmed therefore, that
the translation-like transformations are gauge transformations. These authors
extended their study to the symmetry of massless particles with spin 1/2 and
concluded that the polarization of neutrinos is due the requirement of gauge
invariance.

321
MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 323

PHYSICAL REVIEW VOLUME 134, NUMBER 4B 25 MAY 1964

Feynman Rules for Any Spin. II. Massless Partic1es*

STEVEN WEINBERGt
Department of Physics, University of California, Berkeley, California
(Received 13 January 1964)

The Feynman rules are derived for massless particles of arbitrary spin j. The rules are the same as those
presented in an earlier article for m>O, provided that we let m --+ 0 in propagators and wave functions,'and
provided that we keep to the (2j+1)-component formalism [with fields of the (j,O) or (O,j) type] or the
2 (2j+ I)-component formalism [with (j,O) Ell (O,j) fields]' But there are other field types which cannot be
constructed for m=O; these include the (j/2,j/2) tensor fields, and in particular the vector potential for
j = 1. This restriction arises from the non-semi-sirnple structure of the little group for m = O. Some other
subjects discussed include: T, C, and P for massless particles and fields; the extent to which chirality con-
servation implies zero physical mass; and the Feynman rules for massive particles in the helicity formalism.
Our approach is based on the assumption that the S matrix is Lorentz invariant, and makes no use of
Lagrangians or the canonical formalism.

I. INTRODUCTION These assumptions will be sufficient for all our pur-

T HIS article will develop the relativistic field theory

of massless particles with general spin, along the
poses. In particular, we will have no need of Lagrangians
and the canonical formalism, nor will we need to start
lines followed in an earlier work' on massive particles. with any preconceptions about the form or even the
Our chief aim is, again, to derive the Feynman rules. existence of the field equations.
We assume that the S matrix can be calculated from We begin in Sec. II with a review of the transforma-
Dyson's formula tion properties of massless particle states and creation
and annihilation operators. This information is used in
(-i)nj Sees. III and IV to construct (2j+l)-component fields
I:--
00

S= d4x,···d4xnT{:lC(xJ)···3C(xn)}. (1.1) transforming according to the (j,O) and (O,j) repre-

n=O n! sentations. Condition (1.4) is used in Sec. V to complete
the construction of the fields, and to prove the spin-
Here, 3C(x) is the interaction energy density in the
statistics theorem and crossing symmetry. The Feynman
interaction representation. In general, it would be the
rules are presented in Secs. VI, VII, and VIII. The
00 component :I"OO(x) of a tensor :I""'(x), but in order
inversions P, C, and T are discussed in Sec. IX.
that S be Lorentz-invariant it is necessary that :I""'(x)
In Sec. X we attack a separate problem: To what
be of the form
extent does chirality conservation guarantee the exist-
:I""'(x) = -g"'3C(x) , (1.2) ence of a particle of zero physical mass? Our conclusion
[for general j~JJ is that this theorem can probably
with 3C(x) a scalar. Lorentz invariance also dictates that
only be proved in the context of perturbation theory.
3C(x) commute with 3C(y) for x-y space-like, in order
But if parity as well as chirality is conserved, then it is
that the 0 functions implicit in the time-ordered product
possible to prove the nonexistence of a nondegenerate
in (1.1) not destroy the Lorentz invariance of S.
We also assume that 3C(x) is built out of the creation particle of finite mass.
The chief conclusion of this work is that the Fevnman
and annihilation operators of the free particles appear-
rules for massless particles in the (2j+ 1)-compo~ent or
ing in the unperturbed Hamiltonian. In order that 3C(x)
2(2j+l)-component formalisms are precisely the same
transform properly we construct it as an invariant
polynomial in various free fields "'n(x), which behave as
usual under translations, and which transform according °
as for m>O, except, of course, that we must pass to the
limit m -> in wave functions and propagators.' In this
limit it becomes impossible to produce or destroy
to various representations of the homogeneous Lorentz
particles with helicity other than ±j.
groap
But there is still one important qualitative distinction
between m=O and m>O. We prove in Sec. III that not
all of the field types which can be constructed out of the
creation and annihilation operators for m>O can be so
In order that JC(x) commute with itself outside the constructed for m= 0. Specifically, the annihilation
light cone, we require that the "'n(x) have causal operator for a massless particle of helicity A and the
commutation or anticommutation rules: for x-y space-
2 This conclusion is in agreement with the theorem that the
like, decomposition of the S matrix into invariant amplitudes takes the
(1.4) same form for m=O and m>O, proven by D. Zwanziger, Phys.
Rev. 133, B1036 (1964). Neither Zwanziger's work nor the present
* Research supported in part by the U. S. Air Force Office of article offer any understanding of the fact that photons and
Scientific Research, Grant No. AF-AFOSR-232-63. gravitons interact with conserved quantities at zero-momentum
t Alfred P. Sloan Foundation Fellow. transfer. This point will be the subject of further articles) to be
1 S. Weinberg, Phys. Rev. 133, B1318 (1964). published in Phys. Letters and in Phys. Rev.
B882

Reprinted from Phys. Rev. 134, B882 (1964).

324 CHAPTER VI

FEY N MAN R U L E S FOR ANY S PIN. I I. MAS S L E SSP ART I C L E S B883

creation operator for the antiparticle with helicity - >. The states 1>.) must furnish a representation of the
can only be used to form a field transforming as in (1.3) little group. That is, the unitary operator U[ ffi] corre-
under those representations (A,B) of the homogeneous sponding to ffi'. does not change the momentum of the
Lorentz group such that >.=B-A. This limitation states 1>'), and thus must just induce a linear trans-
arises purely because of the non-semi-simple structure of formation:
the little group for m = 0. The difficulties (indefinite
metric, negative energies, etc.) encountered in previous U[ffi]I>')=L dA'A[ffi]1 >.'), (2.3)
A'
attempts to represent the photon by a quantized vector
potential A ,(x) can therefore now be understood as due with
to the fact that such a field transforms according to the
(t,!) representation, which is not one of the repre-
sentations allowed by the theorem of Sec. III for Therefore, we can catalog the various possible spin
helicity X= ±1. On the other hand, the (j,O) and (O,j) states I>.) by studying the representations d[ffi] of the
representations used in this article (corresponding for little group.
j = 1 to the field strengths) are allowed by our theorem, This is most easily accomplished by examining the
and they cause no trouble.' In a future article we shall infinitesimal transformations of the little group. They
show that it is in fact possible to evade our theorem, take the form
and that the Lorentz invariance of the S matrix then (2.5)
forces us to the principle of extended gauge invariance.
In Ref. 1 we gave the Feynman rules for initial and where !l'. is infinitesimal and annihilates k:
final states specified by the z components of the massive
particle spins. In order to facilitate the comparison with (2.6)
the case of zero mass, and for the sake of completeness,
we present in Sec. VIII the corresponding F eynman In order that (2.5) be a Lorentz transformation we must
rules in the helicity formalism of Jacob and Wick! The also require that
external-line wave functions are much simpler, though (2.7)
of course the propagators are the same.
the index v being raised in the usual way with the metric
tensor g", defined here to have nonzero components:
II. TRANSFORMATION OF STATES
gl1=g"=g"'=l, g"0=-l. (2.8)
The starting point in our approach is a statement of
the Lorentz transformation properties of massless par- Inspection of (2.6) and (2.7) shows that the general !l"
ticle states. The transformation rules have been com- is a function of three parameters 8, Xl, X2, with nonzero
pletely worked out by Wigner,' but it will be convenient components given by
to review them here, particularly as there are" some little
known but extremely important peculiarities that are !l12= _(l21=8, (2.9)
special to the case of zero mass.
Consider a massless particle moving in the z direction
with energy K. It may have several possible spin states,
which we denote 1>'), the significance of the label>. to be !J20 = -!l"'=!J'3= -!J'2= X2 • (2.11)
determined by examining the transformation properties
of these states. Wigner defines the "little group" as the The Lie algebra generated by these transformations can
subgroup of the Lorentz group consisting of all homo- be determined by recalling the algebra generated by the
geneous proper Lorentz transformations ffi', which do full homogeneous Lorentz group, of which the little
not alter the four-momentum k' of our particle. group is a subgroup. An infinitesimal Lorentz trans-
formationA'. can be written as in (2.5), with(l'. subject
(2.1) only to (2.7). The corresponding unitary operator takes
the form
kl=k'=O; k'=ko=K. (2.2) U[H!l]=H(i/2)(l'·J." (2.12)
• As • case in point, there does not seem to be any obstacle to the
construction of field theories for massless charged particles of J,.= -J.,=J,.I. (2.13)
arbitrary spin j, provided that we use only proper field types, like
(j,0) or (O,j). The trouble encountered for j~ 1 by K. M. Case
and S. G. Gasiorowicz [Phys. Rev. 125, 1055 (1962)], can be
It is conventional to group the six components of J.,
ascribed to their use of improper field types, such as (!.t). We plan into two three-vectors:
to discuss this in more detail in a later article on the electro-
magnetic interactions of particles of any spin.
'M. Jacob and G. C. Wick, Ann. Phys. (N. Y.) 7, 404 (1959).
(2.14)
• E. P. Wigner, in Tlreorelical Physics (International Atomic
Energy Agency, Vienna, 1963), p. 59. K;=J;o=-Jo;, (2.15)
MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 325

B884 STEVEN WEINBERG

with commutation rules +

spin j!i;;O as one with helicity ~ equal to j or - j,
respectively.
[1;,J;J=iE;;dk, (2.16) It is, of course, very well known that a spinning
[l;,K;J = iE;jkK k, (2.17) massless particle need not occur in more than one spin
state (or two, if parity is conserved). The restriction
[K;,KjJ=-iE;jdk. (2.18) (2.25) is much less familiar, but we shall see that it is
We see that the unitary operator corresponding to the responsible for the dynamical peculiarities of massless
general infinitesimal transformation (2.9)-(2.11) of the particle field theories.
little group is A particle of general momentum p and helicity ~ may
now be defined by a Lorentz transformation
U[ <R(0,XI,X2)]= 1+i8J,+iX I L I +iX2L 2 , (2.19)
Ip,X)=[K/ Ipi ],/'U[£(p)J IX), (2.31)
where
L I =K I -J 2 , (2.20) where U[£(p)J is the unitary operator corresponding to
the Lorentz transformation £.'(p) which takes our
L 2=K 2+J I • (2.21)
"standard" four-momentum k' into P':
The commutation rules for the three generators of the
P'=£',(p)k' ,
little group are given by (2.16)-(2.18) as (2.32)
p'={p,lpl); k'={O,O,K,K,).
[J"L IJ=iL2, (2.22)
There are various ways of making the definition of £(p)
[1"L2J= -iLl, (2.23) unambiguous, but we will find it convenient to define £
[LI,L2J=0. (2.24) as
(2.33)
We can now find all the representations of the little Here, B ( Ipi) is a "boost" along the z axis with nonzero
group by finding the representations of this Lie algebra. components
But it strikes one immediately that this algebra is not
semi-simple because the elements LI and L2 form an BII(lpl)=B22(lpl)= 1,
invariant Abelian subalgebra. [In fact, Wigner' points B'.(lpl )=BOo(lpl )=cosh4>(lpl), (2.34)
out that (2.22)-(2.24) identify this algebra as that of all
rotations and translations in two-dimensions, a fact of B'o(1 pi )=BO,(I pi )=sinh4>(1 pi),
no known physical significance.J In order that the states 4>(lp[)=ln(lpl/K). (2.35)
I~) form a finite set, it is necessary to represent the
Since B', takes k' into {O,O,lpl ,Ipl}, we choose R(jj) as
"translations" by zero, i.e., the rotation (say, in the plane containing p and the z
Ld~)=L21~)=0 . (2.25) axis) which takes the z axis into the unit vector
p=p/lpl. The factor [<llpl],/2 is inserted in (2.31) to
Therefore, a general <R'. in the little group transforms keep the normalization conventional,
I~) into
U[<RJI~)=exp{iS[<RJJ,) I~), (2.26) (p' ,~' Ip,~) = 8'(p- p')8n' . (2.36)
the angle EJ[ <RJ being some more or less complicated Having defined helicity states of arbitrary momentum
real function of the <R'" which is given for infinitesimal in terms of states I~) of a fixed standard four-momentum
<R by (2.19) as k', it is now quite easy to find their transformation
S[<R(0,XI,X2)J ->0. (2.27) properties. A general Lorentz transformation A'" repre-
sented on Hilbert space by a unitary operator U[AJ,
If we now identify the states I~) as eigenstates with will transform Ip,~) into
definite helicity ~,
U[AJ Ip,X)
J,I~)=~I~), (2.28)
=[<11 pi J1I'U[AJU[£(p)JI~)
we see that the physically permissible irreducible repre- = [<I Ipi ]'12U[£ (Ap)JU[£-1 (Ap)A£ (p)J IX). (2.37)
sentations of the little group are all one dimensional:
But the transformation £-I(Ap)A£(p) leaves k' un-
U[<RJI~)=exp{fAS[<RJ) I~). (2.29) changed, and hence belongs to the little group. Equation
Comparing with (2.3) and (2.4) shows that S must (2.29) then lets us write (2.37) as
satisfy the group property U[AJI p,X)=[<llpl J1I'
S[ <RI]+S[<R2J= EJ[<RI<R2J. (2.30) Xexp{iXEJ[£-I(Ap)A£(p)J)U[£(Ap)JI~),
and finally
For global reasons it is necessary to restrict the
helicity ~ to be a positive or negative integer or half- U[AJI p,X) = [IAPI/lpl]112
integer ±j. We define a right- or left-handed particle of X exp{fAS[.c-1(Ap)A.c(p)]) IAp,X). (2.38)
 326 CHAPTER VI

FEY N MAN R U L E S FOR ANY S PIN. I I. MAS S L E SSP ART I C L E S B885

A general state containing several free particles will tation D[AJ of the homogeneous proper orthochronous
transform like (2.38), with a factor [I pi I/ Ipi J1I'e ilH for Lorentz group:
each particle. These states can be built up by acting on
the bare vacuum with creation operators a*(p,>') which ULA]1/In l+) (x; X)U[AJ-l
satisfy either the usual Bose or Fermi rules: = L Dnm[A-l]1/Im l +) (Ax; >.). (3.2)
(2.39)
so the general transformation law can be summarized It is well known that the various representations
in the statement D[AJ can be cataloged by writing the matrices J and K,
which represent the rotation generator J and the boost
U[AJa*(p,>.)U-l[AJ= [I Api / Ipi J1I2 generator K as
Xexp{i>'El[£-l(Ap)A£(p)])a*(Ap,>.). (2.40)
J=A+B; K=-i(A-B). (3.3)
Taking the adjoint and using the property [see (2.30)J
Since J and K satisfy the same commutation rules
El[R] = - El[R-IJ (2.41) (2.16)-(2.18) as J and K, the A and B satisfy decoupled
gives the transformation rule of the annihilation commutation rules
operator AXA=iA; BXB=iB,
(3.4)
U[AJa(p,>')U-I[AJ= [IAPI/ Ipi JI2 [cti,lBjJ=O.
X exp{i>.El[£-l(p)A-l£(Ap)J)a(Ap,>') . (2.42)
The general (2A + 1)(2B+ l)-dimensional irreducible
We speak of one massless particle as being the anti- representation (A ,B) is conventionally defined for inte-
particle of another if their spins j are the same, while ger values of 2A and 2B by
all their charges, baryon numbers, etc., are equal and
Aab.a'b' = ~bb'] aa,(A) ,
opposi teo Whether or not every massless particle has (3.5)
such an antiparticle is an open question, to be answered Bab.a'b' = Oaa'] bb,CB) ,
affirmatively in Sec. V. But if an antiparticle exists,
then its creation operator b*(p,>.) will transform just where a and b run by unit steps from -A to +A and
like a*(p,>'), and b*(p, ->.) will transform just like from - B to + B, respectively, and JI>l is the usual
a(p,>.) : 2j+1-dimensional representation of the angular mo-
mentum
U[AJb*(p, ->')U-l[AJ=[IAPl/lpIJU'
[Jl(i)±iJ,!i)Jq'q=Oq'.q±l[U'FCT) (j±CT+ 1»)112,
Xexp{i>.El[£-l(p)A-l£(Ap)J)b*(Ap, ->.). (2.43) (3.6)
[JaliJ].'q=CTOq'q.
If a particle is its own antiparticle, 6 then we just set
b(p,>') = a(p,>.). For massive particles of spin j, we have already seen
in Sec. VIII of Ref. 1 that a field 1/1<+) (x) can be Con-
III. A THEOREM ON GENERAL FIELDS structed out of the 2j+l annihilation operators a(p,CT),
which will satisfy the transformation requirements (3.1)
As a first step, let us try to construct the "annihilation
and (3.2), for any representation (A,B) that "contains"
fields" 1/In l+) (x; >'), as linear combinations of the annihi-
j, i.e., such that
lation operators a(p,>'), with fixed helicity >.. We require
that the 1/I n I+J transform as usual under translations j=A+BorA+B-10r .. · or IA-BI. (3.7)
i[P .,1/In(+) (x; >.)J= 0.1/In(+) (x; >.) (3.1) [A spin-one field could be a four-vector a,!), a tensor
and transform according to some irreducible represen- (1,0) or (0,1), etc.J We might expect the same to be true
for mass zero, but this is not the case. We will prove in
\I It is not so obvious what is meant by a massless particle being this section that a massless particle operator a(p,X) of
its own antiparticle. If charge conjugation were conserved, then helicity >. can only be used to construct fields which
we would call a particle purely neutral if it were invariant (up to a
phase) under C. But if we take weak interactions into account then transform according to representations (A ,B) such that
only CP and CPT are available, and they convert a particle into
the antiparticle with opposite helicity, For massless particles there B-A=>'. (3.8)
is no way of deciding whether a particle is the "same" as another
of opposite he1icity, since one cannot be converted into the other
by a rotation. This point has been thoroughly explored with regard For instance, a left-circularly polarized photon with
to the neutrino by J. A. McLennan, Phys. Rev. 106, 821 (1957) >.= -1 can be associated with (1,0), (!.t), (2,1), ...
and K. M. Case, ibid. 107, 307 (1957). See also C. Ryan and S. fields but not with the vector potential (!,!), at least
Okubo, Rochester Preprint URPA-3 (to be published). Even if a
massless particle carries some quantum number (like lepton until we broaden our notion of what we mean by a
number), we can still call it purely neutral if we let its quantum Lorentz transformation. It will be seen that the restric-
number depend on the helicitYi however, in this case it seems more tion (3.8) arises because of the non-semi-simple structure
natural to adopt the convention that the particle is different from
its antiparticle, with b(p,X) ;o'a(p,X). of the little group.
MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 327

B886 STEVEN WEINBERG

The condition (3.1) requires that ,pn(+) be constructed matrix representatives J and K:
as a Fourier transform
D[ <R(O,x"x,)] = 1+iO,9,+ix,(XI- ,92)
1 +ix,(X2+,9I) , (3.16)
,p.(+)(Xj X)=--
(2,..)3/' or, using (3.3),

X f d'p
--e,p··a(p,X)un(p,X) , (3.9)
[2Ipl]'/'
D[ <R(O,X"x,)]= 1+iO( Cl 3+tB,)+ (x,+iX.) (Cl,-iCl.)
+ (x,-iX,) (tB,+iCB,). (3.17)

Recalling from (2.27) that El--> 0, our condition (3.13)

the factor (2,..)--312[2Ipl]-'/' being extracted from the is now split into three independent conditions:
"wave function" U.(p,A) for later convenience. The
condition (3.2) together with the transformation rule [a,+tB,]U(A) = AU (X) , (3.18)
(2.42) then requires that U.(p,A) satisfy
[a,-ia,]U(A)=O, (3.19)
exp{ iXEl[£-1 (p)A-l £ (Ap) ]}U n (p,X)
[CB1+iCB 2]u(A) =0. (3.20)
= L: Dnm[A-l]um(Ap,A). (3.10)
Of these three conditions, (3.18) could certainly have
been anticipated as necessary to a field of helicity A. The
We will now show that this determines Um(p,A} uniquely. other two arise from the detailed structure of the little
In particular (3.10) must be satisfied if we choose group, but are equally important, for they force u(X) to
be an eigenvector of a 3 and CB" with
p=k .. {O,O,K} j A=£(q),
Cl3U(X) = -AU(A) , (3.21)
where q is some arbitrary momentum. In this case (3.10)
reads CB,u(X)=+Bu(X) , (3.22)
Un(q,X)=L: Dnm[£(q)]um(X) , (3.11)
or more explicitly
(3.23)
where Um(A) is the wave function for our "standard"
momentumk Using (3.18) now gives the promised restriction on A
(3.12) andB:
-A+B=A. (3.8)
Insertion of (3.11) into both sides of (3.10) shows that
(3.10) is satisfied by (3.11) if and only if the um(X) For a left-handed particle with A= - j, the various
satisfy possible fields are

exp{ iXEl[£-' (p)A-I£ (Ap) ]}L: Dnm[£(p)]um(h) [left] (j,O), (jH, !), (jH, 1), ... , (3.24)

while a right-handed particle with A= + j can be as-

sociated with a field transforming like

or in other words, if and only if

[right] (O,j) , (t, jH), (1, Hl), .... (3.25)
If parity is conserved, then the particle must exist in
L: Dnm[ <R]um(X) = exp{ ihEl[ <R]}un(X) (3.13) both states A=±j, and the field must then transform
reducibly, for example, like (j,O)(£) (O,j).
for any Lorentz transformation <R of the form Our theorem certainly applies to the in and out fields,
since they are constructed just like free fields. It must
<R= £-1 (p)A-'£ (Ap) . (3.14) then also apply to the Heisenberg representation field
that interpolates between in and out fields if we insist
But these <R's, for general p and A, just constitute the that they all behave in the same way under Lorentz
little group discussed in Sec. II. In order that (3.13) be transformations. Furthermore, the only "M functions'"
satisfied for all such <R it is necessary and sufficient that that can generally be formed from the S matrix are those
it be satisfied for all infinitesimal transformations corresponding to the representations (3.24) and (3.25).
In a forthcoming article we shall see what goes wrong
when we try to construct a field with A and B vio-
the nonvanishing components of 0 being given by lating (3.8).
(2.9)-(2.11). The matrix D[<R] corresponding to (3.15) 7 H. Stapp, Phys. Rev. 125, 2139 (1962); A. O. Barnt, I.
is obtained by replacing J and K in (2.19) by their Muzinich, and D. N. Williams, ibid. 130, 442 (1963).
328 CHAPTER VI

FEYNMAN RULES FOR ANY SPIN. II. MASSLESS PARTICLES B887

IV. (2j+l)-COMPONENT FIELDS 2j+ i-dimensional unitary representation' of the ordi-

For a left- or right-handed particle with A= - j or nary rotation group. [Note, also, that if we tried to
A= + j, the simplest field type listed in (3.24) or (3.25) construct a (j,0) field for a right-handed particle, or a
is, respectively, (j,0) or (O,j). The corresponding (O,j) field for a left-handed particle, we would not only
(2j+l)-component annihilation fields will be called fail to get the desired Lorentz transformation property,
I'.<+)(x) and x.<+) (x). They are given by (3.9), (3.11), but we would also find a catastrophic factor Ip I-I in the
and (3.23), as wave function.]

j
Using the wave functions (4.8) and (4.9) in (4.1) and
1 d'p (4.2), the annihilation fields now take the form
9'.<+)(x)=-- ---
(2.-)'/2 [21 pi ]'/2
9'/+)(x)=_1_ jd'P[2 Ipl ]/-l/2
XeiND •. _;w[.e(p)Ja(p, - j), (4.1) (2.-)'/2

x.<+)(x)=--
1 j ---
d'p XeiP·'D •._Pl[R(p)]a(p, - j), (4.10)
(2.-)'/2 [21 pi Jl/2
x.<+)(x)=_1_ jd'P[2IpIJH/2
XeipxD./"[.e(p)Ja(p,j), (4.2) (2.-)'12

and they transform according to XeiP'D.}iJ[R(p)Ja(p,j). (4.11)

We have redefined their normalization by replacing the
.'
U[AJ9'.<+) (x) U-l[AJ= L D..,W[A-1J9'.,C+) (Ax), (4.3)
factor K- I by 21. We see that only the ordinary unitary

.
rotation matrices' are needed; R(p) is the rotation that
U[AJX. C
+)(X)U-l[AJ=L D ..,W[A-1JX.,C+) (Ax) . (4.4) carries the z axis into the direction of p.
'
If our particle has an antiparticle (perhaps itself), then
Here DW[AJ and DW[AJ are the nonunitary (2j+l) there is available another operator b*(p, -A) which
X (2j+l)-dimensional matrices corresponding to A in transforms just like a(p,A) [see (2.43)J, and which
the (j,O) and (O,j) representations, respectively. They carries the same charge, baryon number, etc. It is then
are the same as used in Ref. 1, and can be defined by possible to define creation fields
taking Cl3=0 or a=o, or, equivalently, by representing
the generators J, K with 9'.H(X)=_l_ jd'P[2IpIJ/-l/2
(211')'/2
DC1l: J=JC1l, K=-iJC1l, (4.5)
Xe-iPXD •._P)[R(p)Jb*(p,j), (4.12)
fjw: l=JW, K=+iJW, (4.6)
where JW is the usual spin- j representation of the x/-)(x)=_1_ jd3P[2Ipl]H/2
angular momentum, defined by (3.6). In particular, the (2.-)'12
transformation .e(p) defined by (2.33) is represented on Xe-ip,xD •. Pl[RCP)]b*(p, - j), (4.13)
Hilbert space by
which satisfy (3.1), which transform according to (4.3)
U[.e(p)J= U[RCP)] exp{ -i¢(1 pi )K,} , (4.7) and (4.4), respectively, and which also transform like
9'<+) and x c+) under gauge transformations of the first
</>(lpl)=ln[lpl/K], (2.35)
kind. [For a "purely neutral" particle,' b*(p,A) is to be
and therefore the wave functions appearing in (4.1) and replaced by a*(p,A).]
(4.2) are The most general fields satisfying all these conditions
are linear combinations of creation and annihilation
D•._Pl[.e(p) ] fields.
1'. (x) = hl'.<+)(X)+~RI'.H (x), (4.14)
.'
= L D ...</J[R(p)][exp{ -q,(lpl)J,W}].,._i
x.(x) = ~RX.C+)(X)+~LX.C-)(x). (4.15)
They again transform as in (4.3) and (4.4):
D•. pl[.e(p)]
U[AJ9'.(x) U-l[A] = L D..,W[A-1J9'., (Ax) ,
.'
(4.16)
L D..,W[R (p)][exp{</> (I p I)J,W}].,.i
=
.'
.'
=D •. PJ[R(P)](lpl/Kl/. (4.9) U[AJX.(x) U-l[AJ= L D.. ,W[A]X., (Ax) . (4.17)

Note that the matrices DW[R] and fjW[R] for a pure IJ See, for example, M. E. Rose 1 Elementary Theory of Angular
rotation R are both equal, being given by the familiar Momentum (J. Wiley & Sons, Inc., New York, 1957), p. 48 If.
MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 329

B888 STEVEN WEINBERG

If these particles have no antiparticles (including them- The matrices .. and if can be easily calculated by use
selves), then we have to take 'IL='1Il=O. We will see in of the obvious formulas
the next section that, instead, requirement (1.4) (and
hence the Lorentz invariance of the S matrix) dictates 1 ;
full crossing symmetry, with 1'I1l1 = Ihi, I'ILI = 1~1l1·
a•._,1i"._i=-[ IT (X-J,)] ••" (5.5)
(2j)! X-i+1
The fields obviously obey the Klein-Gordon equation
1 H
D'<p.(x) = 0; D'x.(x) = O. (4.18) a.".a.,,J=-[ IT (J,-A)] ••,. (5.6)
(2j)! X-i
However, they are (2j+ l)-component objects con-
structed out of just two independent operators a(p,A), Applying the rotation matrix DW[R(P)] and multi-
b*(p, -A), and so they have a chance of obeying other plying by 12pI'i gives
field equations as well. It is not hard to see from
(4.10)-(4.13) that they do indeed satisfy the additional 22j i
.. (p)=- IT (XP-p·J) (5.7)
field equations (2 j) ! X-;+l '

[JW·v- j (ajat)].p (x) =0 , (4.19) 2'i ;-1

if(P)=- II (p·J-XP)· (5.8)
[Jw·V+ j(ajat)]x(x) =0. (4.20) (2j)! X-i
For j =! these are the Weyl equations for the left- and These are monomials of order 2j in the light-like four-
right-handed neutrino fields, while for j = 1 they are vector p', so (5.1) and (5.2) now become

J d'p
just Maxwell's free-space equations for left- and right-
circularly polarized radiation: 1
[<p.(x) ,<p.,t (y)]±=---1I' ••' ( -ia) -
vX[E-iB]+i(O/at)[E-iB]=O, (4.21) (2".)3 21pI

vX[E+iB]-i(ajat)[E+iB]=O. (4.22) X[I h I'e"'(-')± (- )'il '71l!'e-"'(-U)] , (5.9)

The fact that these field equations are of first order for
any spin seems to me to be of no great significance, since
1
[x.(x),x.,t(y)]±=-if.., (-ia)
J d'p
-
in the case of massive particles we can get along per- (2".)3 21pI
fectly well with (2j+ 1)-component fields which satisfy XC Itill 'e'P'(x-u)± (- )'i!'1LI 'e-"'(%-u)]. (5.10)
only the Klein-Gordon equation.
In order that (5.9) and (5.10) vanish for x-y space-
V. CROSSING AND STATISTICS like, it is necessary and sufficient that exp[ip· (x-y)]
and exp[ -ip· (x-y)] have equal and opposite coeffi-
We are assuming that the a's and b's satisfy the usual cients
commutation (or anticommutation) rules (2.39), so it is Ihl'='F(_)'il'7IlI', (5.11)
easy to work out the commutators or anticommutators
of the fields 'P. and X. defined by (4.10)-(4.15): 1~1l1'='F(_)'il'7LI2. (5.12)

['P.(x),'P.,t(y)]±=-
1 J d'p
--1I' •• ,(p)
So we must have the usual connection between spin and
statistics
(2 ..)' 21pI
(±)=_(_)2i, (5.13)
xCI hl'e"'(~u)± 1'1IlI'e-"'(~')], (5.1)

[x.(x),x.,t(y)]±=-
(2..)3
1 J d'p
- i f•• ,(P)
21pI
and furthermore, every left- or right-handed particle
must be associated, respectively, with a right- or left-
handed antiparticle (perhaps itself) which enters into
interactions with equal strength:
x[1 til I'ei"(~u)± I'1L I'e-"'(~')], (5.2)
where Ihl=I'7IlI,
(5.14)
....,(p) = I2pl'iD •._iW[R(p)]D.,._/I1'[R(p)] , (5.3) Itlll =1'7LI·
if •• '(p) = 12pl'iD.ji)[R(P)]D.,./,l'[R(p)]' (5.4) By redefining the phases of the a's and b's, and the
normalization of 'P and x, we can therefore set
These are the only nonvanishing commutators (or
anticommutators) among the 'P, <pI, X, and Xl (except
for a "purely neutral" particle, in which case X is
proportional to 'P I; see Sec. IX). with no loss of generality. The fields are now in their
330 CHAPTER VI

FEY N MAN R U L E S FOR ANY S PIN. I I. MAS S L E SSP ART I C L E S B889

final form: with properties:

(al II and ii are scalars, in the sense that
'P.(X)=_l_ !d'P[2IpIJi-1/2D•. _PJ[R<P>]
(211-)31' DW[A]II (q)DW[AJI= II (Aq) , (6.3)
DW[A]ii(q)DW[A]f= ii(Aq). (6.4)
X[a(p, - j)e;P"+b*(p,j)e-;"'] , (S.16)
(b) 1 and i are symmetric and traceless in PlJl..·· 'P2i'
X.(X)=_l_ !d'P[2IpIJi-1/2D•. ;'i)[R{P)J (c) II and ii are related by an inversion
(2,..)'1' ii( -q, qO) = II(q). (6.5)
X[a(p,j)e;"'+b*(p, - j)e-;P"]. (5.17) (d) II and ii" are related by a similarity transformation

The commutator or anticommutators are ii*(q) = CII (q)C- 1 , (6.6)

where
['P.(x),'P.,I(y)]±=i,.. •• ,( -ia)t:.(x-y) , (S.18) -Jw·=CJ WC- 1 • (6.7)
(e) II and ii are further related by
[x.(x),x.,I(y)]±=i;r•• ,( -ia)t:.(x-y) , (S.19)
II (q)ii(q) = ii(q)II(q) = (_q')'i. (6.8)
where it:.(x-y) is the commutator for zero mass and
j=O. (f) If q is in the forward light cone then
II(q) = (-q')iexp[ -28(q)4·JW] , (6.9)
ii(q) = (_q')i exp[28(q)q·JW] , (6.10)
sinh8(q)"[1 ql'/ _q']1I'. (6.11)
(g) For integer j and arbitrary q
If a particle has no additive quantum numbers like
the photon, we must' set b(p,X) equal to a(p,X), and IIW (q) = (_q')i+[( -q')i-1/2!J(2q.JW)(2q.JW-2qO)
"causality" then tells us through (5.14) that the particle +[( -q')i-2/4!](2q·JW)[(2q.JW)2- (2q)']
must exist in both left- and right-handed helicity states.
X[2q .JW-4qoJ+[ (-q')i-2/6!J(2q.JW)
Both fields 'P.(x) and x.(x) can be constructed, and in
fact we shall see in Sec. IX that 'P is just proportional X[(2q.JW)'- (2q)2][(2q·JW)2- (4q)'J
to Xl. X[2q.JW-6qO]+ ... , (6.12)
On the other hand, a particle which carries some the series cutting itself off autornaticallyafter j+1 terms.
additive quantum number that distinguishes it from its (h) For half-integer j and arbitrary q
antiparticle can possibly exist in only the left- or the
right-handed helicity state, and "causality" only re- IIW(q) = (-q')i-1I2[qO-2q.JW]+(l/3!)( _q')i-212
quires that it has an antiparticle of opposite helicity. X [(2q·JW)2_ q2J[3qo-2q.JW]
(A familiar example is the neutrino.) In this case only + (1/5!) (_q2)HI2[ (2q .JU)2_q2]
one of the fields 'P. and x. can be constructed. Of course, X[(2q.JW)2- (3q)2J
if parity of charge conjugation are conserved, then both
particle and antiparticle must exist in both left- and X[5qO-2q.JWJ+"" (6.13)
right-handed states, and both 'P. and x. exist. the series cutting itself off automatically after j+!
terms.
~.LORENTZINV~NCE It follows from (6.12), (6.13), and (6.S) [or, more
directly, from (6.9) and (6.10)J that for a light-like
Our formulas (S.18) and (S.19) for the commutators vector p' the monomials II and IT simplify to
or anticommutators were derived in a Lorentz invariant
manner, but they do not look like invariant equations. 22i ;
It will be necessary to see how their invariance comes II(p)=-. II (Xfl-p·JW), (6.14)
about before we are able to derive the Feynman rules. (2)! '-i+l
It was shown in Appendix A of Ref. 1 that the 22i j

familiar angular momentum matrices JW can be used to ii(P) = - . II (Xfl+p·J(i), (6.15)

construct a pair of scalar (2j+ l)X (2j+1) matrices II (2)! '-i+l
and ii, as monomials in a general four-vector q':
or in terms of the matrices (S.7), (S.8)
II •• , (q) = ( - ) 2il •• ,"'''·· ,.,iq"q., .. q.,p (6.1) II .. , (p) =,.. •• ' (p) [p light-like], (6.16)
ii •• , (q) = (- )'ii•• ,"'''·· '''iq"qPO' .. qPOi' (6.2) ii •• ,(p)=;r•• ,(p) [p light-like]. (6.17)
MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 331

B890 STEVEN WEINBERG

The Lorentz invariance of formulas (5.18) and (5.19)

for the commutators or anticommutators now follows
immediately from (6.3) and (6.4). (2'l1")-"'(2Ipl )H"Dd"W[Rcp)]e ip "
[particle destroyed], (7.7)
VII. THE FEYNMAN RULES (2,..)-"'(21 pi )H"Dd" W[RCP) ]*e- iP ' x
The Hamiltonian density JC(x) is to be constructed as [particle created]. (7.8)
an invariant polynomial in the (2j+ I)-component fields (2,..)-'1'(21 pi) HI'D d,_,W[ R(p)]e- iP "
'I',(x) and Xd(x), without any distinction made between
zero and nonzero mass. In each term of JC(x) all rI [antiparticle created], (7.9)
indices on the 'l'd(X) are to be coupled together to form a (2,..)-'1'(21 pi )HI'Dd,_,UJ[R(p)]*e iP "
scalar, using Clebsch-Gordan coefficients in the familiar [antiparticle destroyed]' (7.10)
way. The same is to be done independently with the
indices on the Xd(X). If adjoint fields enter in JC(x) then We remind the reader that DUJ[R] is the usual (2j+ 1)
C",-'xd,t(x) is to be treated like 'I',(x) andC",-l'l'd,(x)t X (2j+ 1) unitary matrix' corresponding to an ordinary
is to be treated like x,(x); the matrix C is defined by rotation R, and that RCP) is the rotation that carries the
z axis into the direction of p.
DW[A]*=CDUl[A]C-', (7.1) The "raw" propagator corresponding to an internal
or more specifically, massless particle line running from x to y is
-]W'=C]WC-l. (7.2) (T{ 'l'd(X),'I'd,t(ymo
=O(x-y)( 'I'd (x) 'l'd,t (y»o
[We use an asterisk for the ordinary complex conjugate
of a matrix.] If derivatives appear they will enter as a
+(- )'iO(y-X)('I'd,t (y) 'I'd (x»o , (7.11)
2X2 matrix: or
(T{ Xd(X),Xd,t (y)})o
iJ"'=rI",i(iJjiJxi)-o,,,(ajiJt) , (7.3)
=O(x-y)(Xd(X)Xd,t(y»o
where (Ii are the usual Pauli spin matrices; the indices rI
and rI' are to be treated as if they appeared respectively
+(-
)'iO(y-x)(Xd,t(y)Xd(x»o. (7.12)
on j =! fields '1', and X," An elementary calculation using (5.16), (5.17), (5.3),
We list below some typical examples of possible in- (5.4), (6.16), and (6.17) gives the vacuum expectation
varian t terms in JC (x) : values as
('Pd(X)'I'd,t(Y»o=iII",( -ia)~(x-y), (7.13)
(7.4)
(- )'i('I'.,t(y) 'I'.(x»o= iIl", (-iiJ)Ll+(y-x) , (7.14)
and
(X.(x)X.,t(y»o=ifi", (-iiJ)~(x-y), (7.15)
(- )'i(Xd,t (y)x.(x»o= ifi",( -iiJ)Ll+(y-x) , (7.16)
X '1',,(;') (x) '1',/,,) (x)C"",-'x".(j,) (x)t, (7.S)
where

X '1',,(;') (X)'I'd,(i,) (x)iJ",X,,<I) (x), (7.6)

etc. The fields '1', and Xd appearing here may be either

=~[~-i"'O(X'l«X)J. (7.17)
411"' x'
of zero or of nonzero mass.
The 5 matrix can be calcnlated from JC(x) by using As discussed in Ref. 1, the presence of the 0 functions
Wick's theorem to derive the Feynman rules, as we did in (7.11) and (7.12) makes these propagators non-
in Sec. V of Ref. 1. The only additional information covariant at the point x=y, for spins jf;, 1. In order that
needed here is a statement of the wave functions the 5 matrix be Lorentz invariant, it is necessary to
corresponding to external mass zero lines, and a formula assume that noncovariant contact interactions appear
for the propagators corresponding to internal mass zero in JC(x) which cancel the noncovariant terms in (7.11)
lines. and (7.12). (The Coulomb interaction in Coulomb
The factor arising from the destruction or creation at gauge is such a contact interaction, made necessary by
x of a massless particle or antiparticle of helicity A= ±j the unit spin rather than by the zero mass of the
can be detcrmined from (5.16) and (5.17) as the coeffi- photon.) With this understanding, we can move the
cient of the appropriate creation or annihilation opera- derivative operators Il( -iiJ) and fie -iiJ) in (7.13)-
332 CHAPTER VI

FEY N MAN R U L E S FOR ANY S PIN. I I. MAS S L E SSP ART I C L E S BS91

(7.16) to the left of the 0 functions in (7.11) and (7.12), (It should be kept in mind that the index 0', which is of
obtaining the propagators no direct physical significance, will appear on some
other wave function or propagator, and eventually be
S ••' (x-y)= -ill", (-iil)a'(x-y)
summed over.) The corresponding wave function for a
= -i 2 i+ 1ttTl1 ,J.llJ.l2· "J.l 2j dp./J iJ2 , •• particle of definite helicity A is
X il"ia' (x-y) , (7.18)
U.(x j p,A) = L Dp,lil[R(P)]u.(Xj p,p.), (8.3)
S •• , (x-y)= -ifi ••, (-iil)a'(x-y)
= -i 2 i+ 1lur1 ,fJIJ.12" . J.l 2id J.ll°J.l2 , ••
where R(p), as always, is the rotation that carries the z
X il p2i a'(x-y) , (7.19) axis into the direction of p. Using (8.1) in (8.3) gives
where -ia'(x-y) is the usual propagator for spin zero
U .(Xj p,A) = (2W)-I!'(21r)-'!'
and mass zero
X {exp( -p·](i)O)Dli)[R(p)]}.,e'p,
-ia'(x) =i8(x)a+(x)+iO( -x)~( -x) = (2W)-I!'(21r)-'!'
= +[l/47r'(x'+i<)]. (7.20) X{DW[R(p)] exp(-],WO)}.,e'p,
Equations (6.3) and (6.4) show that these propagators = (2w)-I!'(21r)-3I2D.,Ii)[R(p)]e-"e'P·'. (8.4)
are covariant in the sense that
Furthermore we see from (8.2) that
D(i)[A]S(x)DW[A]t=S(Ax) , (7.21)
e-A8=[w(p)+ Ipl/m]-'. (8.5)
D(i)[A]S(x)DW[A]t=S(Ax). (7.22)
In order to avoid m's appearing in the denominator of
The propagators in momentum space are given by the U. for negative helicity, it will be convenient to re-
Fourier transforms of (7.18) and (7.19) normalize all fields of mass m by multiplying them with

S(q) = f d'xe"'S(x) = -iII (q)/q'-i< , (7.23)

a factor mi. With this understanding, the wave function
for a particle of spin j, helicity A, momentum p, and
mass m, destroyed by 'P.(x), is

S(q) = f d'xe-'"'S (x) = -ifi(q)/q'-k (7.24)

U.(Xj p,A) = (2W)-I!'(21r)-'!2D.,W[R(P)]
Xmi+A(w+lpl)-'e'P". (8.6)
The wave function for the creation of the same particle
Explicit formulas for the monomials II(q) and fi(q) are by 'P.t(x) is just the complex conjugate
given in Eqs. (6.12), (6.13), and (6.5), or for j;;;'3 in
Table I of Ref. 1. U. '(Xj p,A)= (2W)-I!'(21r)-'!'D.,W'[R(p)]
Xmi+A(w+lpl)-'e-'p,. (8.7)
VIII. GENERAL HELICITY AMPLITUDES
AND THE LIMIT m - 0 The wave function for the creation by 'P.(x) of the
The Feynman rules were given in Ref. 1 for incoming antiparticle with helicity A and spin p can be easily
and outgoing massive particles having prescribed values obtained in the same way from Eq. (5.4) of Ref. 1, by
for the z components of their spins. It turns out, how- using the relations
ever, that the external-line wave functions are much DW,[R(P)]=CDli)[R(p)]C-l,
simpler in the Jacob-Wick formalism,' where initial and Cp,-1 = (_ )-i+A~ •. _,.
final states are labeled instead by the particle helicities.
For m=O, of course, we have had no choice, since only We find that the antiparticle creation wave function is
the helicity amplitudes are physically meaningful. We
will first derive the helicity wave functions for m>O, and V.(Xj p,A) = (2W)-1!2(21r)-'!2 ( - )-i+'D•. _,W[R(p)]
Ipi jAe'P",
then use them to show how the Feynman rules given
here for m=O can be obtained by taking the limit m ->
of the Feynman rules for positive m.
° XmH(w+
and the wave function for destruction of the same
(8.8)

According to the Feynman rules of Ref. 1, the wave antiparticle by 'P.t(x) is the complex conjugate
function for a particle of spin j, ],=p., momentum p, V. '(Xj p,A) = (2W)-I!' (21r)-'!2 (- )-i+AD •. _,W'[R(p)]
and mass m, destroyed by 'P.(x), is XmH(w+lpl)'e+'p,. (8.9)
u.(Xj p,p.) = (2w)-I/'(21r)-3I'
A massive particle can be created or destroyed in any
X[exp(-p.]WO)].pe'P" , (8.1) helicity state by either the (j,O) field 'P.(x) or the (O,j)
where field x.(x). Inspection of the field x.(x) given in Eq.
w= [p'+m']'!' , (6.9) of Ref. 1 shows that the wave functions corre-
(S.2)
sinhO= Ipl/m. sponding to (8.6)-(8.9) are given by replacing 0 by -0,
MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 333

BS92 STEVE" WEINBERG

and supplying a sign (- )'i for antiparticles: IX. T, C, AND P

O,(x; p,A) = (2W)-1/'(21T)-3I2D"W[R(p)] Time-reversal (T) and space inversion (P) are classi-
cally defined as transforming a particle of momentum p
XmH(w+ Ipi Ye ipx
and helicity A into
[particle destroyed], (S.10)
Tlp,A)a: I-p, A), (9.1)
O,*(x; p,A) = (2W)-1/'(21T)-3I2D"W'[R(P)]
XmH(w+1pI )'e- ipx Plp,A)a:I-p,-A), (9.2)
[particle created], (S.11) while charge conjugation (e) just changes all particles
V,(x; PA)= (2w)-1/2(2rr)-3/2( - )i+'D,._,U)[R(Pl] into antiparticles, with no change in p and A. However,
in quantum mechanics there appear phases in (9.1) and
XmiH(w+ Ipi )-'e- ip . x
(9.2), which we shall see are necessarily momentum-
[antiparticle created], (8.12) dependent for massless particles. In order to get these
V.*(x; p,A)= (2W)-1/2(2rr)-3/2( - )iHD,,_,U)'[R(P)] phases right it is necessary first to define the action of T
and P on our standard states IA) of momentum
XmiH(w+ Ipl )-Ae+ ip . x
k={O,O,K}, and then use the definition (2.31) of Ip,A).
[antiparticle destroyed]. (8.13) We will define "standard phases" ~A(T) and ~A(P) by
Now suppose that m -> 0, or, more precisely, that TIA)=~,*(T)U[R,]IA), (9.3)
Ipl/m-toc. The only wave functions among (8.6)-
(8.13) that survive in this limit are (8.6), (8.7), (8.12), PI;\)= (- )i+A~A'(P)U[R,]I-A), (9.4)
and (8.13) for A= - j, and (8.8), (8.9), (8.10), and where R, is some fixed but arbitrary rotation such that
(8.11) for A= + j. This agrees with the situation for
m=O, in which case we know that <p, and <p/ can only R,{O,O,l) = to, 0, -1), (9.5)
create and destroy particles with A= - j and antipar-
so that U[R,] 1;\) is a state of momentum {O,O, -K).
ticles with A= + j, while x, and x,t only create and
[The factor (- )i+A is extracted from ~A *(P) for con-
destroy particles with A= + j and antiparticles with
venience later.] In order to calculate the effect of T and
A= - j. Furthermore, if we set A= - j in (8.6) or A= + j
P on Ip,A) we need the well-known formulas
in (8.10) we see that these wave functions reduce for

°
Ipl/m-toc to the particle destruction wave function
given for m = by (7.7). The same agreement is ob-
tained on comparison of (8.7) and (8.11) with (7.8),
TJ iT-l=-J"
TKiT-l=K"
(9.6)
(9.7)
(8.8), and (8.12) with (7.9), and (S.9), and (S.13) with PJ,P-'=J" (9.8)
(7.10). [The observation that particles described only
by <p,(x) are difficult to create or destroy for Ipi »m in PK,P-'=-K,. (9.9)
any helicity state other than A= - j is very familiar for [It is easy to check that (9.6)-(9.9) are consistent with
electrons in beta decay.] the commutation relations (2.16)-(2.18), if we recall
The propagators for an internal <p or X line are given that T is antiunitary.] According to (2.31) and (4.7),
in Ref. 1 as the state Ip,;\) is
S ... (x-y)= -iTI ... ( -ia)Ll'(X-y; m), (8.14) [p,A)=TK/[p[ JI/'U[R(P)] exp[-iq,([ p[ ) K a][;\) , (9.10)
,~ ... (x-y)= -ifi ... (-iJW(x-y; m). (S.1S) so therefore
[Recall that we are now using fields renormalized by a TI p,;\)=~, * (T)[</ Ip I ]'/'U[R(P) ]
factor mi, so the factor m- 2i in Eq. (5.7) of Ref. 1 is
Xexp[i</>(I pi )K,]U[R,]IA) ,
absent here.] We see that the propagators given for
m=O by (7.1S) and (7.19) are the limits respectively of Plp,A)= (- )iH~, *(P)[K/lpl ]1/'U[R(P)]
(S.14) and (8.15) as m -t 0. For m~O there is also a Xexp[i</>(I pi )K,]U[R,]I-A).
"transition propagator" between <p, and X,.t, but it is But

° °
proportional to m'i and disappears as m -> 0.
In contrast, the Feynman rules for m = could not be and thus
obtained as the limit as m -> of the corresponding
TI p,A)=~' *(T)[</I pi ]'/2U[R(p)R,]
rules for m>O, if we used one of the field types like
(j/2,j/2) which are forbidden by the theorem of Sec. Xexp[ -i</>(lpl)K,]IA), (9.11)
III. For example, it is well known that the propagator Plp,A)= (- )iH~A'(P)[K/lpl ]'/'U[R(P)R,]
for a vector field has a longitudinal part which blows up
Xexp[ -i</>(lp[)K,]I-A). (9.12)
as m-' for m -t 0; this is just our punishment for
attempting to use the forbidden (t.!) field type for j= 1 The rotation R(p)R, carries the z axis into the direc-
particles of zero mass.' tion of -p, and must therefore be the product of R( -p)
334 CHAPTER VI

FEY N MAN R U L E S FOR ANY S PIN. I I. MAS S L E SSP ART I C L E S B893

times a rotation of ip(P) degrees about the z axis (9.25):

.'
U[R(p)R,] = U[R( -p)] exp[iip(p)J,J. (9.13) T\".(x)T-'=~_;(T)E C"'\"., (x, -x") , (9.26)
The angle ip (P) depends on how we standardize R, and

.'
R(P), but we will fortunately not need to calculate it, as TX.(x)T-'=~;(T)E C ••,X., (x, -x") , (9.27)
it will cancel in the field transformation laws. Using

.'
(9.13) in (9.11) and (9.12), and recalling that J. C\".(x)C-l=~_;(C)E C••,-lX.,I(x), (9.28)
commutes with K" we have at last

.'
CX.(X)C-I=~j(C)( - )2; E C•• ,-l\".,I(X) , (9.29)
Tlp,A)=~,*(T) exp[iAip(P)]I-p, A), (9.14)
Plp,A)= (- );H~,*(P) exp[ -iAip(P)]I-p, -xl. (9.15) P\".(X)P-I=~_j(P)X.(-X, x"), (9.30)
These one-particle transformation equations can be
pX.(X)P-l=~j(P)\".(-x, x"). (9.31)
translated immediately into transformation rules for the
annihilation operator: In deriving (9.26)-(9.31) it is necessary to fix the
Ta(p,X)T-'=~,(T) exp[ -iAip(p)]a( -p, A), (9.16) antiparticle inversion phases as
Pa(p,X)P-I= (- );H~,(P) ij,(T)=~_,*(T) , (9.32)
Xexp[iAip(p)]a(-p, -X). (9.17) ij,(C)=~-,*(C) , (9.33)
The antiparticle operators will transform similarly, but ij,(P) = (- )2;~_,'(P), (9.34)
perhaps with different "standard" phases ij,(T) and
ij,(p) : because any other choice of the ij, would result in the
creation and annihilation parts of the field transforming
Tb(p,A)T-'=ij,(T) exp[ -iAip(P)]b( -p, A), (9.18) with different phases, and would therefore destroy the
Pb (p,A)P-1= (- );Hij,(P)
possibility of simple transformation laws.
It is interesting that the tran~formation rules (9.26)-
X exp[t"Aip(p)]b ( -p, -A). (9.19) (9.31) tum out to be identical with those derived in
And, of course, C just changes a's into b's and vice versa. Sec. 6 of Ref. 1 for the case of massive particles, though
the derivation has been different in many respects. The
Ca(p,A)C-'=~,(C)b(p,X), (9.20) same is true of the phase relations (9.32)-(9.34), except
Cb(p,X)C-'= ij,(C)a(p,X). (9.21) that the only correlated particle and antiparticle in-
version phases are those of opposite helicity. In par-
The phases ~,(T,C,P), ij,(T,C,P) are partly arbitrary,' ticular, (9.34) tells us that a left- or right-handed
partly determined by the structure of the Hamiltonian, particle plus a right- or left-handed antiparticle together
and partly fixed by the specifically field-theoretic con- have intrinsic parity
sidera tions below.
In order to calculate the effect of T, C, and P on the ~_,(P)ij,(P)= (- )';, (9.35)
fields \".(x) and x.(x), it will be necessary to use the while the intrinsic parity of a massless particle anti-
well-known reality property of the rotation matrices particle pair of the same helicity is not fixed by these
DW[R]*=CDW[R]C-1 , (9.22) general field-theoretic arguments.
If a particle is its own antiparticle' then we must set
where, with the usual phase conventions,
b(p,X)=a(p,X). (9.36)
C".= (- );+'6., ,_.= [exp(i.-J2(;»].,.. (9.23)
In this special case, the (j,0) and (O,j) fields are related
We shall fix the rotation R, introduced in Eq. (9.5) as a by
rotation of 1800 about the y axis, such that
DW[R,]=C-'= (- )2;C. (9.24)
.'
X.t(x)=E C•• ,\"., (x) ,

\".t(x) = (- )2; E C••,x.,(x).

(9.37)

.'
(9.38)
Another needed relation then follows from (9.13).
Also (9.36) requires that the antiparticle inversion
D.,W[R(p)] phases ij, be equal to the corresponding ~" and therefore
= (- );H exp[ -iAip(P)]D•. _,W[R( -P)]. (9.25) (9.32)-(9.34) provide relations between ~, and ~_,:
The effect of T, C, and P on the fields (5.16) and ~,(T)=~_,'(T) , (9.39)
(5.17) can now be easily determined by using (9.16)-
~'(C)=~-' "(C), (9.40)
9 For a general discussion, see G. Feinberg and S. Weinberg,
Nuovo Cimento 14, 571 (1959). ~,(P)= (- )2;~_,*(P). (9.41)
MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 335

B894 STEVEN WEINBERG

However, there is still no necessity for any of these and we write the transformation (10.1) as
phases to be real.
Observe that (9.17) and (9.19)-(9.21) make sense "'(x) --> exp(ie/,6)"'(X). (10.3)

/'6=[1 0].
only if both the particle and its antiparticle each exist in
both helicity states h= ±j. For a particle not identical
with its antiparticle, this is now a part of the assumption
of C or P invariance, whereas in the case of massive
° -1
particles it followed directly from the Lorentz invariance There are other possible discrete or continuous chirality
of the S matrix. transformations, but our discussion will apply equally
In contrast, T conservation leaves open the possibility to all of them.
that the particle exists in only one of the two helicity The question, of whether chirality conservation im-
states, with an antiparticle of the opposite helicity. plies zero physical mass, can be asked on two different
This is consistent with (9.26) and (9.27), which show levels:
that T does not mix 'P. and x •. The same is true of the (1) Suppose that Ho is chosen so the interaction
combined inversion CP. representation fields 'P.(x) and/or x.(x) describe free
particles of zero mass, and suppose that the interaction
CP'P.(X)P-1C-l density JC(x) is invariant under the transformation
(10.1). Is the renormalized mass then zero in each order
=~i(C)~-j(P)L C..,-l'P.' t( -x, x"), (9.42) of perturbation theory?
.' (2) Suppose that there exists a unitary operator
which induces the transformation (10.1) on the Heisen-
berg representation fields, and which leaves the physical
vacuum invariant. Can we then prove anything about

.'
=~-j(C)~j(P)L C..,-lx"t( -x, x"), (9.43) the physical mass spectrum?
Our answers to these two questions are (1) yes, and
and of course it is also true of CPT. (2) not necessarily. Let us consider perturbation theory
first. The bare momentum-space propagator of the 'P.
field is given by (7.23) as
X. CHIRALITY AND RENORMALIZED MASS
We have not made any distinction, either here or in S(g)= -iII(g)/ (q'-ie). (10.4)
Ref. 1, between the mass characterizing the free field
The exact propagator is
and the mass of the physical particles. This was
purposeful, because it is always possible and preferable S' (q) = S(q)+S(g)2:(*) (q)S' (q)
to arrange that the unperturbed and the full Hamil-
= [S-l(q)-2:(*) (q)J-l. (10.5)
tonians have the same spectrum. But there still remains
the question: Under what circumstances will the physi- The (2j+l)X(2j+l) matrix 2: 1*) (g) is the sum of all
cal particle mass in fact be zero? The classic conditions proper diagrams with one 'P. line coming in and one
are gauge invariance or chirality [i.e., "/,6"J conserva- going out, with no propagators on these lines. Stripping
tion. Gauge invariance is without content for the (j,O) away its external propagators changes the Lorentz
and (O,j) fields discussed in this article, so we are led to transformation behavior of 2: ..,1*) from that of 'P.'P.' * to
consider the implications of chirality conservation. Our that of X.X., *, so Lorentz invariance dictates its form as
work in this section is entirely academic except for
j=t, but even in this familiar case our conclusions are 2: ..,1*) (g) =ifi.., (q)F( _q2). (10.6)
not quite in accord with public opinion.
For definiteness we will understand chirality conser- Using (6.8) now gives the exact propagator (10.5) as
vation as invariance under a continuous transformation
-iII(q)
S'(q)=------- (10.7)
[1- (_q2)iF( _q2)J[q2-ie]

In the 2(2j+l)-component formalism lO we unite the We have not used chirality yet. In general the self-
(j,O) and (O,j) fields 'P.(x) and x.(x) into a (j,0)(f) (O,j) energy parUI*) (q), and hence the function F( _q2), may
field t(x): have a pole at q'=0, due to graphs with one intermedi-
ate X.line. But under any form of chirality conservation
'P(X)]
t(x)= [ (10.2) such graphs are forbidden. (For example, there is no
X (x) neutrino X. field.) Hence F( -q') has no pole at q2=0,
10 See Ref. 1. Many features of this formalism have been worked
and therefore S' (q) does have such a pole, corresponding
out independently in unpublished work by D. N. Williams. to a particle of zero renormalized mass.
336 CHAPTER VI

FEY N MAN R U L E S FOR ANY S PIN. 11. MAS S L E SSP ART I C L E S B895

Of course there may also be another particle with non- Unfortunately this theorem offers no proof that the
zero mass m given by accepted chirality-conserving weak interactions do not
give a massive neutrino, with a distinct massive
1=m2iF(m2).
antineutrino. It should be kept in mind that we cannot
But such a particle would have to be unstable so m decide just by looking at a Lagrangian whether the
would lie off the physical sheet. physical one-particle states will be purely neutral or not.
Now let us turn to the second question. We assume Of course, any massless particle can be called purely
that there exists a unitary chirality operator X(o) which neutral, but this is not relevant if what we want is to
transforms the Heisenberg representation fields into prove the absence of massive particles.
We can say somewhat more about the mass spectrum
X(O)I".H(X)X-l(o)=ei'I".H(X) , (10.8) if we are willing to assume parity conservation [which
X(o)x.H(x)X-l(o)=e-i·X.H(x) , (10.9) links I".(x) with x.(x) by (9.30) and (9.31)J as well as
chirality conservation. In this case the propagator of
and which leaves the physical vacuum invariant. It is I".(x) or x.(x) can receive no contribution from any
certain that this assumption alone is not sufficient, in massive one-particle state that has no degeneracy, be-
itself, to allow us to prove anything about physical yond the (2j+ I)-fold degeneracy associated with its
particle masses, because we have not yet said anything spin, and an additional 2-fold degeneracy if it happens
to connect the fields I".(x) and x.(x) with each other. to have a distinct antiparticle. For it would then be
For instance, we might choose I".(x) as (1+1'6)/2 times possible to form a one-particle chirality eigenstate 1p,!,):
the electron field, and x.(x) as (1-1',)/2 times the
muon field. Then (10.8) and (10.9) are obviously X(o) 1 p,!,) = exp(io~) 1 p,!,) (10.14)
satisfied if we choose the chirality operator as by taking 1 P,I') as either the one-particle state itself or
X(o)=exp{io [electron number some linear combination of it and its charge conjugate.
-muon numberJ). (10.10) Lorentz invariance requires that

But we can hardly conclude from this that the electron (OII".H(X) 1 p,I')=N .(2w)-1/2D,.W[L(p)Je ip .%, (10.15)
or muon is massless. (01 X.H (x) 1 p,I')=N x(2w)-1/2.ii,.W[L(p)Je iP ·%. (10.16)
Clearly, the only information that can be gleaned
solely from the existence of X (0) is just what would Parity conservation tells us further that
follow from any ordinary additive conservation law. (10.17)
Namely, the propagator of I".(x) or x.(x) can receive no
contribution from any massive purely neutral one- This is just to say that the matrix element of the
particle state that has no degeneracy beyond the 2(2j+1)-component field ,!(x) satisfies the generalized
(2 j+ 1)-fold degeneracy associated with its spinY For Dirac equation [Eq. (7.19) of Ref. 1J, which is to be
any such state Ip,!,) would have to be a chirality expected under the assumption of parity conservation.
eigenstate But (10.8) and (10.14) give N=O unless ~=+1, while
(10.9) and (10.14) give N =0 unless ~= -1, so we may
X(o) 1 p,!,)=eil'l p,l') (I'=-j, "',j), (1O.11) conclude that N = O. Again, this proof does not apply for
and thus zero mass, because the two helicity states are uncon-
nected by space rotations and hence may have differ-
(10.12) ent es.
(OII".Ht(x)lp,!,)=O unless ~=-1. (10.13) [It might at first sight appear that the free fields
constructed in Ref. 1 provide a counter-example to this
But CP or CPT conservation tells us that these two proof. In the absence of interactions they certainly
matrix elements are proportional to each other, and describe nondegenerate particles with nonvanishing bare
hence must both vanish. [Observe that we cannot and physical masses, and yet there is no coupling that
forbid a massless purely neutral particle from contrib- violates either parity or chirality. The trouble with this
uting to the propagator of I".(x) or x.(x), since CP and argument is that no operator X (0) can be constructed;
CPT reverse its helicity, and its two helicity states in fact Eqs. (7.23) and (7.25) of Ref. 1 show that
might have opposite chirality. This is consistent with
(T{ 1".(x),x.,t(y»))o,.oO. (10.18)
the remark' that it is only a matter of convention
whether we call a massless particle purely neutral or This point is more transparent in the conventional
not.J language in which we would just say that the free-field
11 This is an abbreviated version of a proof given by B. Touschek,
Lagrangian does not conserve chirality. As m -> 0,
in Lectures on Field Theory and the Many·Body Problem, edited by (10.18) vanishes as m2i, and for m=O it is easy to con-
E. R. Caianiello (Academic Press Inc., New York, 1961), p. 173. struct X (0) explicitly. J
It is not clear from Touschek's article whetber he feels that this
theorem implies that the neutrino cannot have finite mass. As The last proof is of some interest, because it shows
indicated herein, I do not. that unless the vacuum or electron is degenerate, the
MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 337

B896 STEVEN WEINBERG

mass of the electron cannot arise entirely from electro- field ",(x), which transforms according to the reducible
magnetic interactions, which conserve both parity and (j,O)(f) (O,j) representation; for j =! this yields the
chirality. But it is useless for the neutrino, and we are Dirac formalism, while for j = 1 it corresponds to the
forced to conclude that only perturbation theory can union of the irreducible fields E±iB into a six-vector
account for its zero mass. {E,B}. Here again there is no distinction to be made
between zero and nonzero mass, so we need not repeat
XI. CONCLUSIONS here the details of the 2(2j+ 1)-component formalism lO
The Feynman rules for massless particles in the constructed in Ref. 1.
(2 j+ 1)-component formalism are identical with those We have seen no hint of anything like gauge invari-
derived in Ref. 1 for particles of mass m>O. It is only ance in our work so far. In fact, the really significant
necessary to pass to the limit m -. 0 to obtain the cor- distinctions between field theories for zero and nonzero
rect propagators for internal lines, and wave functions mass arise when we try to go beyond the (2j+ 1)- or
for external lines. Also, the various possible invariant 2(2j+ 1)-component formalisms. In particular, for
Hamiltonians JC(x) can be constructed out of the fields m>O there is no difficulty in constructing tensor fields
",.(x) and x.(x), with no distinction between massive transforming according to the (j/2,j/2) representations,
and massless particle fields. while for m=O this is strictly forbidden by the theorem
Furthermore, the transformation properties of ",,(x) proven in Sec. III. We will see in a forthcoming article
and x.(x) under T, C, and P are the same for m>O and that the attempt to evade this prohibition and yet keep
m=O. If P and/or C are conserved it is very convenient the S matrix Lorentz-invariant yields all the results
to unite ",.(x) and x.(x) into a 2(2j+1).component usually associated with gauge invariance.

PHYSICAL REVIEW VOLUME !34, NUMBER 4B 25 MAY 1964

Possible Effects of Strong Interactions in Feinberg-Pais Theory

of Weak Interactions. II
N. P. CHANG'
Institute for Advanced Study, Princeton, New Jersey
AND

H. S. MANIt
Physics Department, Columbia Uni'lJersity, New York, New York
(Received 23 December 1963)

In a previous paper, a simplified model was used to study the effects of strong interactions on the weak
interaction theory of Feinberg and Pais. In this paper, we use a more general argument, a power count
based upon the \Vard-Takahashi-Nishijima multimeson vertex function identity, to show that the same
conclusion remains valid even when crossed ladder graphs are included. Our conclusion may not apply, how-
ever, to the modified program of peratization where W - W scattering plays an essential role.

I. INTRODUCTION an argument which shows that the same power counting

I N a previous paper,! the possible effects of strong

interactions on the peratization theory of Feinberg
conclusion holds when all possible effects of strong
interactions, within the framework of peratization
theory, are taken into account. Furthermore, the very
and Pais' were studied in a simplified model where the
nature of our argument shows that the same conclusion
strong interactions acted through modifications only of
holds even when one includes, in peratization theory,
the baryon vertices and propagators. It was shown
the sum over the crossed ladder graphs so long as power
there that the final "peratized" nuclear vector i1-decay
counting is valid. That is to say, if we define the
coupling strength Gp" is no longer equal to the "pera-
peratized (crossed+uncrossed ladder graphs) wdecay
tized" I'-decay coupling strength, G. if the vector
constant by G.= (g2/m') (1-~), then the corresponding
current is conserved. In this paper, we wish to present
peratized nuclear vector i1-decay constant is Gp'
* Supported in part by a grant from the National Science = (g'/m')(1-Z~), where Z is the strong interaction
Foundation. nucleon renormalization factor. Thus, unless peratiza-
t Boese predoctoral fellow. tion vanishes (~=O) when all graphs are included, the
1 N. P. Chang, Phys. Rev. 133, B454 (1964).
2 G. Feinberg and A. Pais, Ph),s. Rev. 131, 2724 (1963); 133,
situation remains that Gp;cGp' when the vector current
B477 (1964). is conserved, This makes it hard to understand the
338 CHAPTER VI

PHYSICAL REVIEW VOLUME 135. NUMBER 4B 24 AUGUST 1964

Photons and Gravitons in S-Matrix Theory: Derivation of Charge Conservation

and Equality of Gravitational and Inertial Mass*
STEVEN WEINBERct
Physics Department, University of California, Berkeley, California
(Received 13 April 1964)

We ~ve a pu~ely S-matrix-theoretic proof of the conservation of charge (defined by the strength of soft
phot?D m~eractIons) and the equality of gravitational and inertial mass. Our only assumptions are the Lor-
entz .mvanance and pole structure of the S matrix, and the zero mass and spins 1 and 2 of the photon and
gravlton. We also prove that Lorentz invariance alone requires the S matrix for emission of a massless
particle of arbitraryint~ger spin to satisfy a "mass-shell gauge invariance" condition, and we explain why
there are no macroscopIc fields corresponding to particles of spin 3 or higher.

I. INTRODUCTION for all nonrelativistic particles (and is twice the total

I T is not yet clear whether field theory will continue

to playa role in particle physics, or whether it will
energy for relativistic or massless particles).
Property (1) is actually a straightforward conse-
ultimately be supplanted by a pure S-matrix theory. quence of the well-known'" Lorentz transformation
However, most physicists would probably agree that properties of massless particle states, and is proven in
the place of local fields is nowhere so secure as in the Sec. II for massless particles of arbitrary integer spin.
theory of photons and gravitons, whose properties seem (It has already been proven for photons by D.
indissolubly linked with the space-time concepts of Zwanziger.')
gauge invariance (of the second kind) and/or Einstein's Property (2) does not at first sight appear to be
equivalence principle. derivable from property (1). Even in field theory (1)
The purpose of this article is to bring into question does not prove that the photon and graviton "currents"
the need for field theory in understanding electro- ] .(x) and B.,(x) are conserved, but only that their
magnetism and gravitation. We shall show that there matrix elements are conserved for light-like momentum
are no general properties of photons and gravitons transfer, so we cannot use the usnal argument that
which have been explained by field theory, which canno~ fd'xfO(x) and fd'xBo·(x) are time-independent. And
also be understood as consequences of the Lorentz in pure S-matrix theory it is not even possible to define
invariance and pole structure of the S matrix for mass- what we mean by the operators P(x) and O·'(x).
less particles of spin 1 or 2,1 We will also show why there We overcome these obstacles by a trick, which re-
can be no macroscopic fields whose quanta carry spin 3 places the operator calculus of field theory with a little
or higher. simple polology. After defining charge and gravitational
What are the special properties of the photon or mass as soft photon and graviton coupling constants in
graviton S matrix, which might be supposed to reflect Sec. III, we prove in Sec. IV that if a reaction violates
specifically field-theoretic assumptions? Of course, the charge conservation, then the same process with inner
usual version of gauge invariance and the equivalence bremsstrahlung of a soft extra photon would have an
principle cannot even be stated, much less proved, in S matrix which does not satisfy property (1), and hence
terms of the S matrix alone. (We decline to turn on would not be Lorentz invariant; similarly, the inner
external fields.) But there are two striking properties of bremstrahlung of a soft graviton would violate Lorentz
the S matrix which seem to require the assumption of invariance if any particle taking part in the reaction
gauge invariance and the equivalence principle: has an anomalous ratio of gravitational to inertial mass.
Appendices A, B, and C are devoted to some technical
(1) The S matrix for emission of a photon or graviton problems: (A) the transformation properties of polariza-
can be written as the product of a polarization "vector" tion vectors, (B) the construction of tensor amplitudes
f/J or "tensor" f~fJ' with a covariant vector or tensor
for massless particles of general integer spin, and (C) the
amplitude, and it vanishes if any .. is replaced by the presence of kinematic singularities in the conventional
photon or graviton momentum q•. (2j+l)-component "M functions."
(2) Charge, defined dynamically by the strength of A word may be needed about our use of S-matrix
soft-photon interactions, is additively conserved in all theory for particles of zero mass. We do not know
reactions. Gravitational mass, defined by the strength whether it will ever be possible to formulate S-matrix
of soft graviton interactions, is equal to inertial mass
2 E. P. Wigner, in Theoretical Physics (International Atomic
• Research supported by the U. S. Air Force Office of Scientific
Research, Grant No. AF-AFOSR-232-63. Energy Agency, Vienna, 1963), p. 59. We have repeated Wigner's
t Alfred P. Sloan Foundation Fellow. work in Ref. 3.
1 Some of the material of this article was discussed briefly in a 3 S. Weinberg, Phys. Rev. 134, B882 (1964).
recent letter [So Weinberg !'hys. Letters 9,357 (1964)]. We will 'D. Zwanzig~r, Phys. Rev. Il3, B1036 (1964). Zwanziger
repeat a few pomts here, m order that the present article be omits some straightforward details, which are presented here in
completely self·contained. Appendix B.
B1049

Reprinted from Phys. Rev. 135, BI049 (1964).

MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 339

Bl050 STEVEN WEI:-.IBERG

theory as a complete dynamical theory even for strong The polarization '±'(q) is defined by
interactions alone, and the presence of massless particles
'±'(q)=R(q)",,±' , (2.4)
will certainly add a formidable technical difficulty, since
every pole sits at the beginning of an infinite nnmber of where R(q) is a standard rotation that carries the z axis
branch cuts. All such "infrared" problems are outside into the direction of q, and '±' is the polarization for
the scope of the present work. We shall simply make momentum in the z direction:
believe that there does exist an S-matrix theory, and
that one of its consequences is that the S matrix has ,±'=(1, ±i, 0, O}/V1. (2.5)
the same poles that it has in perturbation theory, with Some properties of '±'(q) are obvious:
residues that factor in the same way as in perturbation
theory. (We will lapse into the language of Feynman ,±:(q)'±'(q) = 1, (2.6)
diagrams when we do our 2.. bookkeeping in Sec. IV, ,±,(q)'±'(q) = 0, (2.7)
but the reader will recognize in this the effects of our
,±,* (<1) = ''I'' (<1) , (2.8)
childhood training, rather than any essential dependence
on field theory.) ,±O(q)=O, (2.9)
When we refer to the "photon" or the "graviton" in q,,±'(tj)=O, (2.10)
this article, we assume no properties beyond their zero
mass and spin 1 or 2. We will not attempt to explain why L± f±'(q)'±'*(q) = 11"(<1)= g"'+ (ij'q'Hl'q')/ iqi 2 ,
there should exist such massless particles, but may guess [Ii'={-q,iqi}], (2.11)
from perturbation theory that zero mass has a special L± f±"(q)'±"(q)f±"*(q)'±'2*(<1)
kind of dynamical self-consistency for spins 1 and 2,
which it would not have for spin O. = H II"" (<1)II""'(<1)+ II""(q)II""(q)
Most of our work in the present article has a counter- - II""(q)II"'2(<1)}. (2.12)
part in Feynrnan-Dyson perturbation theory. In a We also note the very important transformation rule,
future paper we will show how the Lorentz invariance proved in Appendix A,
of the S matrix forces the coupling of the photon and
graviton "potentials" to take the same fonn as required (A,"-q"M/ i qi ),±'(A<1) =exp{±i8[q,A]}'±'(q) , (2.13)
by gauge invariance and the equivalence principle. with 8 the same angle as in (2.1).
II. TENSOR AMPLITUDES FOR MASSLESS
If it were not for the q' term in (2.13), the polarization
PARTICLES OF INTEGER SPIN "tensor" f±Pl.. "I::I./ i would be a true tensor, and the
tensor transformation law (2.3) for M±""""i would be
Let us consider a process in which a massless particle sufficient to ensure the correct behavior (2.1) of the
is emitted with momentum q and helicity ±j. We shall S matrix. But '±' is not a vector,' and (2.3) and (2.13)
call theS-matrix element simply S±j(q,p), letting p stand give the S-matrix transformation rule
for the momenta and helici ties of all other particles
participating in the reaction. The Lorentz transforma- S±i(q,P) = (21 q I)-1/2 exp{±ij8 (q,A)}
tion property of S can be inferred from the well-known X[,±"(Aq)- (Aq)"A,O,±'(Aq)/ iqi]*' ..
transformation law for one-particle states'; we find that X [,±'i(Aq)- (Aq)"iA,',±'(Aq)/ i q i]*
S±j(q,p)~ (IAql/lql)li2 XM ±..,...,,(Aq,Ap). (2.14)
Xexp[±ij8 (q,A)]S±i(Aq,Ap). (2.1) For an infinitesimal Lorentz transformation A',= 0',
The angle 8 is given in Appendix A as a function of the +w"" we can use (2.2) and the symmetry of M to put
momentum q and the Lorentz transformation A',. (2.14) in the form
We prove in Appendix n that, in consequence of S±Jeq,p) = (IAq I/ I ql )1/2 exp{ ±ij8 (q,A) }S±i(Aq,Ap)
(2.1), it is always possible for integer j to write S±i as
the scalar product of a "polarization tensor" and what - je21 q 1')-'1/2(w,O,±'*eq))q'le±"*eq)' ..
Stapp' would call an "M function": X,±'i*(<1)M±." ... ,,(q,p). (2.15)
S±i(q,P) = e2[ q I)-1/2'±"*(q) ... Hence the necessary and sufficient condition that (2.14)
X,,,ti*(q)M±"""i(q,P) (2.2) agree with the correct Lorentz transformation property
(2.1), is that S± vanish when one of the '±" is replaced
with M a symmetric tensor,' in the sense that with q':
M ±'" ·,'i(q,p)=A,,'1. .. .\,j"iM ±'i"·'i(Aq,Ap). (2.3) q",±"'*(q)' .. 'x'i* (<1)k! ±." ... ,,(q,p) = O. (2.16)
Ii functions for massive particles \verc introduced by H.
jf For .i = 1 this may be expressed 'IS the conservation
Stapp, Ph),s. Rev. 125, 2139 (1962). See also A. O. Barut, 1.
Muzinich, and D. N. Williams, Ph),s. Rev. 130,442 (1963). 7The transformation rule (2.13) shows that EJ.IJ(q) transforms
+ + -}.
e \Ve use a real metric, with signature {+ Indices are according to one of the infinite-dimensional representations of the
raised and lowered in the usual way. The inverse of the Lorentz Lorentz group discllssed by V. Bargmann and E. P. \Vigner,
transformation AJJ.~ is [A-l]"p=A/'. Proc. Nat!. Acad. Sci. 34, 211 (1948).
340 CHAPTER VI

PHOTONS AND GRAVITONS IN S-MATRIX THEORY B10S1

condition or with the related question of whether m=O poles can

q.M,/(q,p) =0. (2.17) really be separated from the branch cuts on which they
lie. Our purpose is to explore the implications of the
For j=2 we conclude that
generally accepted ideas about the pole structure.)
q.M±·'(q,p)e<q'. (2.18) Let us first consider the vertex amplitude for a very-
However, (2.7) shows that the subtraction of a term low-energy massless particle of integer helicity ±j,
proportional to g" from M ±" will not alter the S matrix emitted by a particle of spin J =0, mass m (perhaps
(2.2), so M ±" can always be defined in such a way that zero), and momentum p'= {p,E}, with E= (p'+m')l!'.
(2.18) becomes (We are restricting ourselves here to very soft photons
(2.19) and gravitons, because we only want to define the
charge and gravitational mass, and not the other electro-
The condition (2.16) may look empty, since it can magnetic and gravitational multipole moments.) The
always be satisfied by a suitable adjustment of only tensor which can be used to form M±" .. ··; is
M ± 0.... ";, which in light of (2.9) will have no effect on P'" .. pM [note that terms involving g'" do not con-
the S matrix. But we cannot play with the time-like tribute to the S matrix, because of (2.7)] so the tensor
components of M±" .. ·.; and still keep it a tensor in the character of M ±"" "; dictates the form of the vertex
sense of (2.3). Neither (2.3) nor (2.16) is alone sufficient amplitude as
for Lorentz invariance, and together they constitute a
nontrivial condition on M ±., ....;.
P.,· .. P.;<±·'*(1)· .. <±';*W/ 2E(p) (21 q 1)1/2. (3.1)
Condition (2.16) may, if we wish, be described as If the emitting particle has spin J>O, with initial and
"mass-shell gauge invariance," because it implies that final heIicities (I and (If then (3.1) still gives a tensor M
the S matrix is invariant under a regauging of the function if we multiply it by ~ •• , ; this is because the unit
polarization vector matrix has the Lorentz transformation property
(2.20) ~ .., -+D •• "(J)(p,A)D.,.",(J)*(p,A)~.".",=~ •• ,, (3.2)
with A±(q) arbitrary. It was purely for convenience where D(J) (p,A) is the unitary spin-J representation of
that we started with the "Coulomb gauge" in (2.4), the Wigner rotation' (or its analog! if m=O) associated
(2.5). [However, the theorem in Sec. III of Ref. 3 shows with momentum p and Lorentz transformation A. How-
that it is not possible to construct an <±'(1) which would ever, the vertex amplitude so obtained is not unique.
satisfy (2.13) without any q' term.] For instance if J=! and m>O then we get (3.1) times
The S matrix for emission and absorption of several Ot1f1' if we use a "current"9
massless particles can be treated in the same way,
except that <.* is replaced by <' when a massless particle
if;{-y.,p.,. .. p.;+permutationslf, (3.3)
is absorbed. while using 'Y,'Y. in place of 'Y. would give a helicity-flip
vertex amplitude.
III. DYNAMIC DEFINITION OF CHARGE At the end of the next section we will see that these
AND GRAVITATIONAL MASS
other possibilities are prohibited by the Lorentz in-
We are going to define the charge and gravitational variance of the total S matrix. Indeed, the only allowed
mass of a particle as its coupling constants to very-Iow- vertex functions for soft massless particles of spin j are
energy photons and gravitons, with "coupliug constant" of the form (3.1) times ~ ••, for j= 1 and j= 2 (and none
understood in the same sense as the Watson-Lepore at all for js3). We may therefore define the soft photon
pion-nucleon coupling constant. In general, such defini- coupling constant e, by the statement that the j = 1
tions are based on the fact that the S matrix has poles, vertex amplitude is lO
corresponding to Feynman diagrams in which a virtual 2ie(21l')4~ ••. P.,±.* (fj)
particle is exchanged between two sets of A and B of (3.4)
incoming and outgoing particles, with four-momentum (21l')9/2[2E(p)J(2Iql)l/2 '
nearly on its mass shell. The residue at the pole factors 'E. P. Wigner, Ann. Math. 40, 149 (1939). For a review, see
into r A and r B, the two "vertex amplitudes" r A and S. Weinberg, Phys. Rev. 133, B1318 (1964).
II For j=2, see 1. Y. Kobsarev and L. B. Okun, Dubna
r B depending respectively only upon the quantum (unpublished) .
numbers of the particles in sets A and B, and of the 10 Proper Lorentz invariance alone would allow different charges

exchanged particle. Hence it is possible to give a purely e± for photon helicities ±1. Parity conservation would normally
require that e+ = L (with an appropriate convention for the photon
S-matrix-theoretic definition of the vertex amplitude r parity). However if space inversion takes some particle into its
for any set of physical particles, as a function of their antiparticle then its "right chargeJJ e+ will be equal to the "left
charge" e_ of its antiparticle, and we will see in the next section
momenta and heIicities; the coupling constant or con- that this gives e+=t_= - L . In this case we speak of a magnetic
stants define the magnitude of r. (As discussed in the monopole rather than a charge. The same conclusions can be
introduction, we will not be concerned in this article drawn from CP conservation. We will not consider magnetic
monopoles in this paper, though in fact none of our work in Sec. IV
with whether the above remarks can be proven rigor- will depend on any relation between e+ and e_. Time-reversal
ously in S-matrix theories involving massless particles, iuvarianl;e allows 1:IS to take e as real.
MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 341

B1052 STEVEN WEINBERG

the factors 2, i, and .. being separated from e in obedi- On the other hand, if a is massless or extremely rela-
ence to convention. And in the same way we may define tivistic, then Ea»ma and (3.S) gives
a "gravitational charge" f, by the statement that the
j=2 vertex amplitude is" (3.10)

2if(s..G) [/'(2..)'<l .., (p,'±'*(tJJ' [Formulas (3.8) or (3.10) should not of course be under-
(3.5) stood to mean anything more than already stated in
(2 .. )9/2[2E(p)J(2 Iq i )1/ 2 (3.7). However, they serve to remind us that the re-
sponse of a massless particle to a static gravitational
the extra factor (s..G) [/2 (where G is Newton's constant)
field is finite, and proportional to J. J
being inserted to make f dimensionless.
The presence of massless particles in the initial or
In order to see howe and f are related to the usual
final state will also generate poles in the S matrix,
charge and gravitational mss, let us consider the near
which, like that in (3.7), lie on the edge of the physical
forward scattering of two particles with masses ma and
region. It is therefore possible to measure the coupling
mb, spins J a and h, photon coupling constants eaand eb, constants e and J in a variety of process, such as
and graviton coupling constants fa and J.. As the in-
Thomson scattering or soft bremsstrahlung, or their
variant momentum transfer 1= ~ (pa~ pa')2 goes to
analogs for gravitons. All these different experiments
zero, the S matrix becomes dominated by its one-
will give the same value for any given particle's e or f,
photon-exchange and one-graviton-exchange poles. An
for purely S-matrix-theoretic reasons. The task before
elementary calculation" using (2.11) and (2.12) shows
us is to show how the e's and 1's are related for different
that for t --+ 0, the S matrix becomes
particles.
Ol1al1a'OUb"b'
---[eaeb(Pa' Pb) IV. CONSERVATION OF e AND UNIVERSALITY OF f
4..'E aE bt
+S..GfaJ.{ (pa' pb)'-ma'mb2/2}]. (3.6) Let Spa be the S matrix for some reaction a --> /3, the
states a and /3 consisting of various charged and un-
If particle b is at rest, this gives charged particles, perhaps including gravitons and
photons. The same reaction can also occur with emission
(3.7) of a very soft extra photon or graviton of momentum
q and helicity ± 1, or ±2, and we will denote the corre-
sponding S-matrix element as Spa±l(q) or SPa±2(q).
Hence we may identify ea as the charge of particle a, These emission matrix elements will have poles at
while its effective gravitational mass is q = 0, corresponding to the Feynman diagrams in which
ma= Ja{2Ea~ (m.','Ea)}. (3.8) the extra photon or graviton is emitted by one of the
incoming or outgoing particles in statesa or /3. The poles
If particle a is nonrelalivistic, then Ea~ma, and (3.8) arise because the virtual particle line connecting the
gives its gravitational rest mass as photon or graviton vertex with the rest of the diagram
gives a vanishing denominator
(3.9)
1I Proper Lorentz invariance alone would not rule out different 1/[(pn+q)'+mn2J= 1j2pn·q
values for the gravitational charges f ± for gravitons of helicity ±2. (particle n outgoing) ,
Parity conservation (with an appropriate convention for the (4.1)
graviton parity) requires that i+= f-. This conclusion holds even 1/[(pn~q)2+mn2J= ~ 1/2pn·q
for the magnetic monopole case discussed in footnote 10, since
then f+=J-, and we will see in Sec. IV that the antiparticle has (particle n incoming) .
"left gravitational charge" f- equal to f-. The same conclusions
can be drawn from CP conservation. Time-reversal invariance For Iql sufficiently small, these poles will completely
allows us to take f as real. dominate the emission-matrix element. The singular
12 The residue of the pole at t=Ocan be mut easily calculated by
adopting a coordinate system in which q=pa,-pa=Pb-Pb' is a factor (4.1) will be multiplied by a factor ~i(2 .. )-'
finite real light-like four-vector, while p", Pl., pal, Pb' are on their associated with the extra internal line, a factor
mass shells, and hence necessarily complex. Then the gradient
terms in (2.11) and (2.12) do not contribute, because q,p"~q'P'
~ 0, so that II" may be replaced by g,,,yielding (3.6). We are 2ie[pn' '±* (q) J(2 .. )'
justified in using (3.6) in the physical region (where pa, Ph, Pal, Ph' (·U)
are real and q is small, though not in the direction of the light cone) (2 .. )3/2(2 Iql )[/2
because Lorentz invariance tells us that the matrix element or
depends only upon sand t. Lorentz invariance is actually far from
trivial in a perturbation theory based on physical photons and 2if(S..G) [/2[pn' <± *(<1) J'(211")'
gravitons, since then the Coulomb force and Newtonian attraction (4.3)
must be explicitly introduced into the interaction in order to get (2 .. )'/2(2Iql)1/2
the invariant S matrix (3.6). (Such a perturbation theory 'will be
discussed in an article now in preparation.) The Lorentz-invariant arising frolll the vertices (3.4) or (3.5), and a factor Spa
extrapolation of (3.6) into the physical region of small t is the
analog, in S-matrix theory, of the introduction of the Coulomb for the rest of the diagram. Hence the S matrix for soft
and Newton forces in perturbation theory. photon or graviton emission is given in the limit
342 CHAPTER VI

PHOTONS AND GRAVITONS IN S-MATRIX THEORY BI053

q --> 0 by l3-l5 here takes the simpler form (2.19)

S~.±I(q) --> (2".)-3/'(21 ql )-1/2 0= qpM"'(q, a --> (j)
= (2..-)-8/2(8wG)1/2[L, 'l/ofopo·JS~.. (4.12)
(4.4)
But the po' are arbitrary four-momenta, subject only
or to the condition of energy momentum conservation:
S~.±2(q) --> (2".)-3/'(21 q 1)-1/2 (8"c) 1/2 E'l/op.p=O. (4.13)

X[ r:.• 'I/.f.[P"E±*(q)J']
(p.-q)
S~., (4.5)
The requirement that (4.12) vanish for all such po', can
be met if and only if all particles have the same gravita-
tional charge. The conventional definition of Newton's
the sign '1/. being + 1 or -1 according to whether constant G is such as to make the cornmon value of the
particle n is outgoing or incoming. f. unity, so
These emission matrices are of the general form (2.2), f.=1 (alln) '(4.14)
i.e., and (3.8) then tells us that any particle with inertial
(4.6) mass m and energy E has effective gravitational mass
iii=2E-m'/E. (4.15)
where M. and M •• are tensor M functions In particular, a particle at rest has gravitational mass
iii equal to its inertial mass m.
It seems worth emphasizing that our proof also
applies when some particle n in the initial or final state
M"(q, a --> fJ) = (2".)-3/2(&rG)'/2 is itself a graviton. Hence the graviton must emit and
absorb single soft gravitons (and therefore respond to a
X[r:. 'I/.f.Pn"Pn'/(p.·q)JS~.. (4.9) uniform gravitational field) with gravitational mass 2E.
It would be conceivable to have a universe in which all
However, we have learned in Sec. II that the covariance f. vanish, but since we know that soft gravitonsinteract
of M. and M •• is not sufficient by itself to guarantee the with matter, they must also interact with gravitons.
Lorentz invariance of the S matrix; Lorentz invariance Having reached our goal, we may look back, and see
also requires the vanishing of (2.2) when anyone E±'(q) that no other vertex amplitudes could have been used
is replaced with g'. For photons this implies (2.17), i.e., for q --> 0 except (3.4) and (3.5). A helicity-flip or
helicity-dependent vertex amplitude could never give
O=q'M,(q, a --> (j)= (2".)-"'[r:. 'I/.e.]S~., (4.10) rise to the cancellations between different poles [as in
(4.10) and (4.12) ] needed to satisfy the Lorentz in-
variance conditions (2.17) and (2.19). It is also interest-
so if S~. is not to vanish, the transition a --> {j must
ing that such cancellations cannot occur for massless
conserve charge, with particles of integer spin higher than 2. For suppose we
(4.11) take the vertex amplitude for emission of a soft massless
particle of helicity ±j (j=3, 4, ... ) as
For gravitons Lorentz invariance requires (2.18), which 2igW(2".)4(E± *(q). p);a ••,
(4.16)
"Formula (4.4) is well known to hold to all orders in quantum (2".)'/2[2E(p)J(2J ql )1/2
electrodynamic perturbation theory. See, for example, J. M. Jauch
and F. Rohrlich, Theory of Photons and Electrons (Addison- in analogy with (3.4) and (3.5), the S matrix S~«±;(q)
Wesley Publishing Company, Inc., Reading, Massachusetts, for emission of this particle in a reaction a --> {j will be
1955), p. 392, and F. E. Low, Ref. 14.
If It has been shown by F. E. Low, Phys. Rev. 110, 974 (1958),
given in the limit q --> 0 by
that the next term in an expansion of the S matrix in powers of
Iq I is uniquely determined by the electromagnetic multipole S~.±;(q) .... (2".)-3/'(21 ql )-1/2
moments of the participating particles and by S~a.. However, this x[r:. 'I/.g.W[P,·E±*(q)Ji/(po·q)]S~.. (4.17)
next (zeroth-order) term is Lorentz-invariant for any values of the
multipole moments .
.. Relations like (4.4) and (4.5) are also valid if S~a±l(q), This is only Lorentz invariant if it vanishes when any
S,a",±2(q), and Sfja are interpreted as the effective matrix: elements
for the transition a ~ p, respectively, with or without one extra one E±' is replaced with g', so we must have
soft photon or graviton of momentum q, plus any number of un-
observed soft photons or gravitons with total energy less than some (4.18)
small resolution .1.E. [For a proof in quantum-electrodynamic
perturbation theory, see, for example, D. R. Yennie and H. Suura,
Phys. Rev. 105, 1378 (1957). The same is undoubtedly true also But there is no way that this can be satisfied for all
for gravitons, and in pure S-matrix theory.] momenta p. obeying (4.13), unless j=1 or j=2. This
MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 343

Bl054 STEVEN WEINBERG

is not to say that massless particles of spin 3 or higher ACKNOWLEDGMENTS

cannot exist, but only that they cannot interact at zero I am very grateful for helpful conversations with
frequency, and hence cannot generate macroscopic N. Cabibbo, E. Leader, S. Mandelstam, H. Stapp,
fields. And similarly, the uniqueness of the vertex ampli- E. Wichmann, and C. Zemach.
tudes (3.4) and (3.5) does not show that electromag-
netism and gravitation conserve parity, but only that
parity must be conserved by zero-frequency photons APPENDIX A: POLAlUZATION VECTORS
and gravitons. AND THE LITTLE GROUP
The crucial point in our proof is that the emission of
soft photons or gravitons generates poles which in- In this Appendix we shall discuss the "little group'"
for massless particles, with the aim of defining the
dividually make non-Lorentz-invariant contributions
angle e(q,A), and of determining the transformation
to the S matrix. Only the sum of the poles is Lorentz-
properties of the polarization vectors E±@.
invariant, and then only if e is conserved and f is
The little group is defined as consisting of all Lorentz
universal. Just as the universality of f can be expressed
transformations (lV" which leave invariant a standard
as the equality of gravitational and inertial mass, the
conservation of e can be stated as the equality of charge light-like four-vector K":
defined dynamically, with a quantum number defined 6I",K'=K" , (Al)
by an additive conservation law. But, however, we state
them, these two facts are the outstanding dynamical KI=K'=O, K8=Ko=K>0. (A2)
peculiarities of photons and gravitons, which until now
have been proven only under the a priori assumption of It is a matter of simple algebra to show that the most
a gauge-invariant or generally covariant Lagrangian general such 61", can be written as a function of three
density. parameters El, Xl, X':

cosEl
61" = [ -sinEl
sinEl
cose
-XlcosEl-X.sine
XlsinEl-X,cosEl
Xl cosEl+X, sine
-XlsinEl+X,cosEl
1
(A3)
, Xl X, l-X'j2 X'j2'
Xl X, -X'j2 1+X'j2
X'""XI'+X,'.
(The rows and columns are in order 1, 2, 3, 0.) Wigner' has noted that this group is isomorphic to the group of
rotations (by angle e) and translations (by vector (XI,X,}) in the Euclidean plane. In particular the "transla-
tions" form an invariant Abelian subgroup, defined by the condition El=O, and are represented on the physical
Hilbert space by unity. It is possible to factor any 61", into
cose sinEl -Xl
1
61"-
,-
[ -sine cose
0
o
o
o o
~ ~l[i 1.
1
-X.
l-X'j2
X,
Xl
X'j2
Xl X, -X'j2 1+X'j2
. (A4)

The representation of 61", on physical Hilbert space is components

determined solely by the first factor, so BII=B'.=l,
U[6IJ=exp(ie[6IJI,) . (AS) B8.=Boo=coshl" ,
(AS)
B'o=BO.=sinhl",
In discussing the transformation rules for massless
particles it is necessary to consider members of the little 'I'""log(lql/K) ,
group defined by and R(fj) is the rotation introduced in (2.4), which takes
the z axis into the direction of q. The transformation
6I(q,A)=.c- I (q)A-I£(Aq). (A6)
£(q) takes the standard four-momentum K" [see (A2)J
Here A is an arbitrary Lorentz transformation, and into q"""{q,lql}:
.c(q) is the Lorentz transformation: (A9)
so therefore,
6I",(q,A)K'= [£-1 (q)A-I]",(Aq)'
where B( Iqi) is a "boost" along the z axis, with nonzero = [£-1 (q)]",q'=K". (A 10)
344 CHAPTER VI

PHOTONS AND GRAVITONS IN S-MATRIX THEORY BI055

Hence CR(q,A) does belong to the little group. or, recalling that E,:'=O,
It was shown in Ref. 3 that, as a consequence of (AS),
the S matrix obeys the transformation rule (2.1), with (A/-AN.')e,:'(Aq)=exp[±i8(q,A)J<,:;(q).
EJ(q,A) given as the EJ angle of CR(q,A): This also incidentally shows that EJ(q,A) does not
depend on Iql.
EJ(q,A)= El[.e-1(q)A-1.e (Aq)]. (AU)
We have not had to define the rotation R(~) any
We now tum to the polarization "vectors" E;,t(~), further than by just specifying that it carries the z axis
defined in Sec. II by into the direction of q. However, the reader may wish to
see explicit expressions for the polarization vectors, so
E,:'(q)=R'.(~)E,:·, (A12) we will consider one particular standardization of R(q).
<,:'",,{l, ±i, 0, 0}/v1. (A 13) Write ~ in the form
<1= {-sin~ cos'Y, si~ sin'Y, co~} (A22)
Observe that we could just as well write (A12) as
and let R(~) be the rotation with Euler angles 0, fj, 'Y:
E,:'(q)=.e'.(q)<,:· (A14)
co~ cos'y sin'Y - si~ cOS'Y 0]
since B ( Iq I) has no effect on E,:.
An arbitrary CR'. of the form (A3) will transform E,:' R"(~)= [ -C~?in'Y cr siU:i~n'Y ~. (A23)
into
CR''',:'=exp(±iEJ[CR]<,:'+ X,:[CR]K', (A15)
Then (2.4) and (2.5) give
where
.,:.(~)= (co~ cos'Y±i sin'Y,
(A16) -co~ sin'Y±i cos'y, sinfj, 0}/v1
CI'= 1, 2, 3,0). (A24)
If we let CR be the transformation (A6), and use (A14), We can easily check (2.6)-(2.12) explicitly for (A24).
then (AIS) gives
APPENDIX B: CONSTRUCTION OF
[.e-1(q)A-l]''',:·(Aq)=exp[±iEJ(q,A)},:· TENSOR AMPLITUDES
+X,:(q,A)K', (A17) We consider a reaction in which is emitted a massless
where particle of momentum q andintegerhelicity±j,allother
X,: (q,A) particle variables being collected in the single symbol p.
Let us first divide the set of all possible {q,p} into dis-
X1[.e-1(q)A-1.e (Aq)]±iX,[.e-1(q)A-1.e (Aq)] joint equivalence classes, {q,p} being equivalent to
{q',p'} if one can be transformed into the other by a
J(I/l
(AIS) Lorentz transformation. (This is an equivalence rela-
tion, because the Lorentz group is a group.) The axiom
Multiplying (AI7) by .e(q), we have the desired result of choice allows us to make an arbitrary selection of one
set of standard values {q,.P,} from each equivalence
A.·E,:·(Aq) = exp[±iEJ(q,A) J<,:'(q) + X,:(q,A)q·. (AI9) class. so any {q.p} determines a unique standard {q,.P,}.
Note that it is the "translations" which at the same such that for some Lorentz transformation L', we have
time make the little group non-semi-simple, and which q=Lq" p=Lp,. (BI)
yield the gradient term in (AI9).
The quantity X,:(p,A) may be found in terms of It will invariably be the case in physical processes that
E,:(q) by setting 1'=0 in (AI9): the only AP, leaving both q and p invariant is the identity
a••• so the L', in (BI) is uniquely determined by q and p.
X,:(q,A) Iql =A,'e,:'(Aq). (A20) (This is true. for instance. if p stands for two or more
general four-momenta.) Hence the arguments {q.p}
Hence we may rewrite (AI9) as a homogeneous trans-
stand in one-to-one relation to the variables {q"p,.L}.
formation rule: Now let us construct an M,J,"l···.;(q"p,) satisfying
(A.'-A,'q'/lql)<,:(Aq)=exp[±i8(q,A)h'(~) (A21) (2.2) for each standard {q"p,}. A suitable choice is

M,:·l·····;(q,.P,) = (2Iq,1 )1/2',:'1(~,) .. · ••,t;(<1,)S,:;(q"p,). (B2)

which satisfies (2.2) because of (2.6). The tensor amplitude for a general q. p is then defined by
M ,:.1· ...i(q.p)=L.l.1(q.p)· .. L·;.;(q.p)M,:·l·"';(q,.P,). (B3)
where q" p" and L(q.p) are the standard variables and Lorentz transformation defined by (Bl). With this definition
MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 345

BlOS6 STEVEN WEINBERG

we can easily show that M ±"," '.1 is a tensor, because

M±"" "I(q,p) = Yl'l (q,p) ... L.I'i(q,P )L,r'l (Aq,Ap)' .. Lp/' (Aq,Ap)M ±Pl'" p'(Aq,Ap)
=Apl·l.· ·Ap/iM±Pl"··PI(Aq,Ap), (B4)
the latter equality holding because L(q,p)L-I(Aq,Ap) induces the transformation {Aq,Ap} ~ {q"p,} ~ {q,p} and
hence must be just A-I.
We must now show that (B.3) satisfies (2.2) for all {q,p}. The Lorentz transformation property (2.13) of '±'
can be wri t ten as
'±'(q) =exp{=Fi0(q,L-l(q,p»}[L',(q,p)-q'LO,(q,p)! Iq Ih'(q,j·
Hence, (B.3) gives
,±'l*(q) . .. '±.i*(q)M.l .... Jq,p) = exp{ ±i j0(q,L-l(q,p»} [,±.l(q,) - q,'l'±"(q,)Lo,,(q,p)! Iq IJ* ...
X [,±'I(q,)-q,'i'±"(q,)LO'I(q,P)/ Iq IJ*M " .... I(q"P,). (BS)
But (B2) and (2.10) show that all q,' terms may be dropped, because
(B6)
so (BS) simplifies to
,±'l*(q) . .. ,±.I*(q)M.l ....I(q'P) = exp{ ±i j0(q,L-1 (q,p» }'±"'*(q,)' .. ,±PI*(q,)M" ....I(q"P,) (B7)
or, using (B2) and (2.6),
(21 ql )-l/2,±'I*(q) . .. '±'I' (q)M.l ....1(q,p) = (I q,l! Iq I)1/2 exp{±ij0(q,L-l(q,p»)S±;(q"p,). (BS)

The right-hand side is just the formula for S±Aq,p)

obtained by setting A=L-l(q,p) in (2.1), so (BS) gives
finally
(C1)
S±;(q,p) = (21 ql)-l/2,±.l'(q) ...
X,±·I*(q)M.l .... i(q,p). (B9) where p and p' are the initial and fmal charged-partide
momenta. This may be rewritten as
It should be noted that (B2) is not valid for all q, p,
S~.±(q) ---+ (21 q I )-1/2M[e.,] (q, a ---+ (3)
since then M±o ..... "(q,p) would vanish in all Lorentz
frames, and M ± could hardly then be a tensor. X {q'*'±' (q) - q"±'* (q)}, (C2)
where M[p] is a (1,0)$ (0,1) M function
APPENDIX C: (2j+l)-COMPONENT M FUNCTIONS e[p.p',- p,p'.JSp.
M[ •. ,](q,a---+j3)= . (C3)
It has become customary' to write the S matrix for (27r)3/2(p.q) (p' .q)
massive particles of spin j in terms of 2j+ 1-component It can be shown that S~.+ and S~a- receive contribu-
M functions, which transform under the (j,O) or (O,j) tions, respectively, only from the self-dual and anti-
representation of the homogeneous Lorentz group. In self-dual parts of M[..,], which transform according to
contrast, the symmetric-tensor M functions used here the three-component (0,1) and (1,0) representations.
transform according to the (j/Z, j/2) representation. But (C3) shows that these conventional M functions
The massless-particle S matrix could also have been have a double pole, arising simultaneously from the in-
written in terms of a conventional (2j+ I)-component coming and ou tgoing charged particle propagators. This
M function, but only at the price of giving the M func- singularity is partly kinematic, since the S matrix (C1)
tion a very peculiar pole structure. involves a sum of single poles, but certainly no double
To see what sort of peculiarities can occur for zero pole. The presence of kinematic singularities in M[..,]
mass, let us consider the emission of a very soft photon makes it an inappropriate covariant photon amplitude.
in a reaction like Compton scattering, in which there is Similar remarks apply to gravitons, but not to any
only one charged particle in the initial state a and in the other massless particles like the neutrino, for which
final state j3. The S-matrix element is then given by there is no analog to charge.
346 CHAPTER VI

PHYSICAL REVIEW D VOLUME 26, NUMBER 12 15 DECEMBER 1982

E(2)-like little group for massless particles and neutrino polarization

as a consequence of gauge invariance

D. Han
Systems and Applied Sciences Corporation, Riverdale, Maryland 20737

Y. S. Kim and D. Son

Center for Theoretical Physics, Department of Physics and Astronomy, University of Maryland,
College Park, Maryland 20742
(Received 18 January 1982)
The content of the isomorphism between the two-dimensional Euclidean group and the
E(2)-like little group for massless particles is studied in detail. Representations of the
E(2) group are explicitly constructed. The finite-dimensional representations which corre-
spond to physical massless particles are discussed in detail, particularly for the cases of
spin I and +. For photons, the little-group transformation matrix is reduced to the

+
transpose of the coordinate transformation matrix in the E(2) plane. In the case of spin-
particles, it is shown that the polarization of neutrinos is a consequence of the require-
ment of gauge invariance.

I. INTRODUCTION tion relations for the E(2) group. Wigner notes

also that there is an E(2)-like subgroup of SL(2,C)
The basic spacetime symmetry for relativistic which is isomorphic to the group of Lorentz
particles is that of the inhomogeneous Lorentz transformations. He then attempted to construct
group. 1 In constructing representations of this E(2) eigenfunctions in a two-dimensional space
group, we usually write do·.vn its ten generators, spanned by the parameters of the translational de-
and then construct the Casimir operators which grees of freedom in the E(2) group. However, for
commute with all of the ten generators. When the E(2) representatives diagonal in the helicity
studying symmetry properties of free particles, we operator, Wigner gives only the 0(2)-symmetric
specify first the four-momentum of the particle, part in his Eq. (93).1
and then ask which operators commute with the In our previous paper,' we gave an E(2)
four-momentum operator. representative which contains Wigner's 0(2) func-
The above-mentioned procedure is known as the tion.] It is a solution of Laplace's equation in the
method based on "little groupS.,,1 The little group two~dimensional space spanned by the parameters
is defined to be a group whose transformations do of the translational degrees of freedom, which, in
not change the four-momentum. The little groups the case of photons, correspond to gauge transfor-
for massive and massless particles are known to be mation parameters.',4 However, in Ref. 2, we
isomorphic to 0(3) and E(2), respectively. The gave only a general form of the E(2) eigenfunction,
study of the isomorphism between the 0(3)-)ike lit- and made no attempt to exploit physical implica-
tle group for massive particles and the rotation tions of the result.
group is straightforward, and has thus been The purpose of the present paper is to study the
thoroughly examined in the literature. However, representations of the E(2) group and those for the
for the E(2)-like little group for massless particles, little group for massless particles more carefully.
there are still problems which can and should be We shall show first that the polar-coordinate form
examined. given in Ref. 2 is applicable only to massless par-
The study of massless particles starts from Secs. ticles with integer spin, and that the E(2) genera-
6-D and 7-B of Wigner's paper. 1 In these sections, tors applicable to the coordinate variables should
Wigner shows that the generators of the little be supplemented by the elements of the E{2}-like
group for massless particles satisfy the commuta- subgroup of the SL( 2, C) group, as in the case of

26 3717 © 1982 The American Physical Society

Reprinted from Phys. Rev. D 26, 3717 (1982).

MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 347

3718 D. HAN, Y. S. KIM, AND D. SON

the rotation group where the rotation operator con- pZ=Pf'P,... W z= Wf'Wf' ' (2)
sists of the orbital and spin parts.
We shall then show that the SL(2,C) part corre- where
sponds to spin-T massless particles. From our Wf'=TEf'",ppvMaJ/ .
analysis, we conclude that neutrino polarization is
a consequence of the requirement of the invariance We are considering in this paper massless parti-
under the translationlike transformation of the cles, and assume without loss of generality that the
E(2)-like little group, which in the case of photons momentum of a given particle is along the z direc-
is a gauge transformation. tion. Then
In Sec. II, we reorganize Wigner's work l on (3)
massless particles into a form suitable for studying
the content of the isomorphism between the E(2) where PI' are the eigenvalues of the operators Pf'"
and E(2)-like little groups. It is emphasized in Sec. The generators of the little group which commute
m that the case of the O(3)-like little group for with PI' in this case are
massive particles will be helpful in understanding
NI =K1-JZ, Nz=Kz+JI> JJ' (4)
the relation between the little group and E(2). Sec-
tion IV contains a detailed discussion of the E(2) where
group. It is pointed out that the E(2) representa-
tive given in our previous papei1 is adequate only J j = TEjjkMik, K j =Mjo •

for integer-spin particles, and that the E(2)-like These operators satisfy the commutation relations
subgroup of SL(2,C) should be used for spin-T
massless particles. [JJ,Nil=iN z ,
In Sec. V, the representation of the E(2}-1ike lit- [h,Nz)=-iN 1 , (5)
tle group is studied in detail. The four-by-four
little-group transformation matrix is reduced to a [NI>Nz)=O,
form similar to the three-by-three regular represen-
tation of the E(2) group. The relationship between which are like those for the generators of the two-
these two matrices is worked out in detail. In Sec. dimensional Euclidean group which is often called
VI, the algorithm developed for photons is applied the E(2) group. 1 The little group for massless par-
to the case of neutrinos. It is shown that the re- ticles is therefore locally isomorphic to the E(2)
quirement of gauge invariance leads to the polari- group.
zation of neutrinos. The study of isomorphism does not stop at the
In Appendix A, it is shown that the gauge commutation relations. As in the case of the
transformation on photon or neutrino wave func- O(3)-like little group for massive particles, the
tions is a transformation within an equivalence study should include explicit construction of repre-
class defined by a given rotation angle in the E(2) sentations, and this construction starts with the
plane. In Appendix B, the connection between the choice of commuting operators.
photon wave function and the E(2) coordinate is The above generators of the little group com-
discussed in detail. It is pointed out that, although mute with N Z, where
there is a one-to-one correspondence between these NZ=Nlz+Nzz, (6)
two quantities, one cannot be transformed to the
other through a linear transformation. and the Casimir operator W Z of Eq. (2) takes the
form

II. E(2)-LIKE LITTLE GROUP

The set of commuting operators will therefore con-
Let us go back to 1939 1 and write down the ten sist of either
generators of the inhomogeneous Lorentz group

PI' and Mf'v, (I)

or
which generate translations and Lorentz transfor- PO,P3,WZ,J3 • (8)
mations, respectively. Their commutation relations
are well known. The Casimir operators are It is important to keep in mind that the above
348 CHAPTER VI

26 E(2)-LIKE LITILE GROUP FOR MASSLESS PARTICLES AND ... 3719

operators are still the generators of Lorentz (12) are given in the literature. 4,7-10 Compared
transformations. However, since the commutation with the 0(3) case, this matrix appears to be com-
relations of Eg. (5) are exactly like those for the plicated, and this probably was the reason why not
generators of the E(2) group, we can learn lessons many authors were encouraged in the past to study
from this simpler group.l the E(2) problem for massless particles. In the fol-
If we use the four-vector convention l ,4 lowing sections, we shall examine whether this ma-
trix can be reduced to a transformation matrix in a
x~=(x,y,z,t) , (9)
two-dimensional Euclidean space.
then the generators of the coordinate transforma-
tion take the form
III. LESSONS FROM THE O(3)-LIKE
o
0 -i LITILE GROUP FOR MASSIVE PARTICLES
0 0 0 0
Nl= There are enough books and papers on the
0 0 0 three-dimensional rotation group, and we are guite
0 0 0 familiar with the language developed for studying
this group. Therefore, the most effective way to
0 0 0 0 study the E(2) group and its isomorphism with the
o
0 -i little group for massless particles is to organize the
N z= (10)
0 0 0 material in a way parallel to the case of the 0(3)-
0 0 0 like little group for massive particles.
Also for the case of massive particles, the little
o -i 0 0 group consists of four-by-four Lorentz transforma-
o 0 0
tion matrices. However, in the Lorentz frame
JJ= 0 0 0 0 where the particle is at rest, the four-by-four ma-
trix reduces to a three-by-three rotation matrix and
o 0 0 0 a one-by-one unit matrix. From this reduced ex-
pression, we can immediately see the content of the
The above generators lead to the transformation
isomorphism between the 0(3) group and the little
matrices
group. In the E(2) case, with the form given in
(II) Eg. (12), it is not easy to see the correspondence.
The physical quantities associated with the 0(3)
where degrees of freedom are well known. The 0(3)
Dl(u)=exp( -iuNl) ' group has three parameters. One of them is used
for the amount of rotation around a given axis,
Dz(v)=exp(-ivN z ) , and two of them are for the orientation of the axis.
DJ(O)=exp(-iOJ J ) . The direction of the rotation axis is the direction
of the spin. All rotations with the same amount of
After a straightforward algebra, we can write the rotation, but with different axis orientations, be-
D matrix as long to the same equivalence class. Therefore, the
reorientation of the axis, without changing the
cosO -sinO -u u amount of the rotation, is a transformation within
sinO cosO -v v an equivalence class. s Is there this kind of reason-
D(u,v,O)=
u· v· l-rz/2 r Z/2 ing for the E(2)-like little group?
u· v· -r2/2 l+r z/2 Again for the 0(3)-like little group, both 0(3)
and SU(2) groups are needed. The SU(2) group is
(12) needed for specifying particles with half-integer
spin, particularly the electron. The 0(3) group is
needed for the description of orbital motion of
quarks inside an extended hadron. 6 In studying

[u. [u J the E(2Hike little group, it is reasonable to expect

J [COSO sinO J (13)
v· = -sinO cosO V both single- and double-valued representations.
Indeed, in view of the recent progress made on
Similar explicit forms for the big matrix of Eg. understanding the little groups of the Poincare
MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 349

3720 D. HAN, Y. S. KIM, AND D. SON

TABLE I. Table of the little groups for massive and massless particles. Both the 0(3)
and E(2)-like little groups are subgroups of the 0(3,1) or SL(2,C) groups depending on the
spin. The little groups for electrons and hadrons have been studied in Refs. I and 6, respec-
tively. The little group for photons has also been studied in Refs. 1-4 and 7 -10, but there
is enough room for further investigation. The little group for neutrinos is expected to be a
subgroup of SL( 2, C)

0(3,1) SL(2,C)

Massive: O(3)-like subgroup SU(2)-1ike subgroup

p2>O of 0(3,1): hadrons of SL( 2, C): electrons

Massless: E(2)-like subgroup E(2Hike subgroup of

p2=O of 0(3,\): photons SL(2,C): neutrinos

group, we can summarize what has been done and mutes with all three of the above generators. Thus
what to expect in Table I. In the following sec- we have to solve the eigenvalue equation
tions, we shall use the above-mentioned parallelism
(15)
with the familiar 0(3) group to exploit the contents
of the E(2)-like little group for massless particles. where

IV. REPRESENTATIONS OF THE E(2) GROUP

Reference 2 contains a detailed discussion of solu-
As was pointed out in Sec. II, the four-by-four tions of the above differential equation for both
matrix of Eq. (12) is too complicated to manage, zero and nonzero values of b', and the result is
although its algebra is known to be isomorphic to summarized in Table II. As is well known, the
that of the E(2) group. We are therefore led to b'=O solutions correspond to physical states. If
look for a simpler way to approach the internal de- b'=O, the E(2) representative becomes
grees of freedom for massless particles, and to
study the E(2) group carefully.
1/J(r,O)=[Ar m +B(l/r)mjexp(±imO) . (16)

For this purpose, let us consider a two- The expression given in Eq. (16) leads us to the
dimensional Cartesian plane with coordinate vari- temptation to say that m should take either integer
able u and v. In Ref. 2, we used the following or half-integer values, 2 in view of the form given
forms as the generators of the E(2) group: in Eq. (93) of Wigner's paper. I,ll However, the
continuity of the transformation requires that
NI = -ia/au ,
group representatives be analytic. For this reason,
N 2 =-ia/av, (14) we have to write the b'=O solution as

J) = -i(ua/av-va/au) . (17)

We noted in Ref. 2 that the operator N' com- only for integer values of m. Then, where are the

TABLE II. Solutions of Laplace's equation in two-dimensional space for E(2) representa-
tives. Physical particles correspond to the finite-dimensional representation diagonal in J 3
with b'=O.
Commuting set Infinite-dimensional Finite-dimensional
of operators representation representation
exp[i(blu +b 2v)]

Jm(br)exp( ±imO) (Arm+Br-m)exp(±imO)

with b'=O
350 CHAPTER VI

26 E(2)-UKE UTILE GROUP FOR MASSLESS PARTICLES AND ... 3721

E(2) representatives for spin-T or half-integer-spin discussed in Appendix A. In Appendix B, we

particles? study the vector spaces in which the E(2) transfor-
In order to answer this question, we should look mations are performed.
at the case of the 0(3) little group for massive par- For neutrinos, only the two-by-two matrix parts
ticles, in which the SU(2) group is needed in addi- are needed for the E(2) generators given in Eq.
tion to the orbital group, and this SU(2) group is a (19). The representations for neutrinos are dis-
subgroup of SL(2,C) which is isomorphic to the cussed in detail in Sec. VI.
group of Lorentz transformations. Therefore we
are led to ask the question of whether there is an
E(2)-like little subgroup in the SL(2,C) group. The V. PHOTONS
answer to this question is definitely "Yes." The
E(2}-like subgroup, which was discussed in If we use the four-vector form for the photon
Wigner's paper,' is generated by wave function, the little-group transformation ma-
trix is the four-by-four matrix given in Eqs. (II)
T, = TUa,-a2) , and (12), The matrix performs both gauge and
T2 =T Ua 2+ a ,) , (18) Lorentz transformations while leaving the four-
momentum invariant. 1O Its algebraic properties are
isomorphic to those of the E(2) group.
However, the explicit form of Eq, (12) is rather
which satisfy the commutation relations for the unattractive and does not appear like any of the
E(2) group given in Eq. (5). Thus the generators of standard form of E(2) transformation matrices, in
the E(2)-like group should in general take the form contrast to the case of massive particles where the
N,=-i3/3u+T 1 , identification of the little group with the 0(3)
group is straightforward.,,6 This is the issue we
N 2 =-i3/3v+T2 , (19) would like to address in this section,
The E(2) representative for photons is given in
J 3 = -i(u3/3v-v3/3u )+S3 .
Eq. (17). The generators of the E(2) group in this
We should note here that J J in the above expres- case are the differential operators given in Eq. (14)
sion is identical to that for the rotation group. or those in Eq. (19) without the SL(2,C) part.
The above generators lead to the following Since m in Eq, (17) is I for the photon case, the
transformation matrices: transformation matrix is a linear coordinate
transformation matrix. In this case, the E(2)
E1(u)=exp( -iuN,) , transformation matrix acting on the column vector

I
E 2 (v)=exp(-ivN 2 ) , (20) (u 0, Vo, 1) takes the following matrix form:

E 3 (1I)=exp( -iIlJ 3 ) • COSII - sinll u

E(u,v,II)= [ sinll cosll v . (22)
The most general form of the transformation ma-
trix then is
o 0 I

(21)
It is easy to show, if not well known, that this ma-
trix is the three-parameter regular representation of
whose algebraic properties have been discussed in the E(2) group. The geometrical properties of this
the literature." 7,8,10 E(2) matrix are discussed in AppendixA. Appen-
For photons, only the "orbital" parts are needed dix B contains a discussion of the vector spaces to
in Eq. (19), and we can still use the function given which this matrix is to be applied.
in Eq. (17) as the state vector. Since m = I, ,p(r,lI) It is also easy to calculate the inverse of the
of Eq. (17) is the coordinate variable in the two- above form:
dimensional Euclidean space. The E(2) transfor-
mation in this case can be achieved through the cose sinll - u'
three-parameter three-by-three matrix for the regu- E-1(u,v,lI)= -sinll cosll -v' (23)
lar representation of the E(2) group. o 0
The explicit form of this regular representation
is given in Sec. V. The similarity between the E(2) where u' and v· are given in Eq. (13). This ma-
geometry and that of the familiar 0(3) group is trix also has vanishing elements in the lower left
MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 351

3722 D. HAN, Y. S. KIM, AND D. SON 26

corner. Unlike the case of the 0(3) group, the in-

verse of the E(2) matrix is not its transpose, and
the third row of E - I is identical to that of E.
cosO - sinO 0
]5(u,v,O)= sinO cosO 0
I . (27)

o The problem is whether the D matrix of Eq. (12) u' v· I

can be reduced to a form similar to the E matrix. We have used ]5, instead of D, to specify that the
The first obstacle we have to face is that the D matrix is for antiphotons. The above expression is
matrix given in Eq. (12) is quadratic while the E identical to the form given in Eq. (25) for photons.
matrix of Eq. (22) is linear in the u and v parame- Therefore, photons and antiphotons have the same
ters. In order to overcome this obstacle, let us go gauge transformation property.
back to the Lorentz condition on the photon four- The D matrix is to be applied to photon polari-
vector, which imposes the restriction that the third zation vectors while the E matrix is for the two-
and fourth components be identical. We note fur- dimensional Cartesian plane. Then, how are these
ther that, in the expression for the D matrix of Eq. two vector spaces related? As is well known, the
(12), the nontrivial parts of the third and fourth translation subgroup of the E(2) group is Abelian
columns have opposite signs, and they vanish when and invariant. Therefore E(2) is neither simple
the matrix is applied to the photon four-vector. nor semisimple. As was pointed out by Racah,13
We can therefore write the D matrix as groups containing Abelian invariant subgroups are
"most troublesome," and require a careful exam-
cosO -sinO o0 ination. In Appendix B, we shall see how this
sinO cosO o0 "trouble" affects the relation between the photon
D(u,v,O)= (24)
u' v· I 0 wave function and the E(2) tranformation in the
u' v· 0 I Cartesian plane,

This form is linear in u and v which are gauge

transformation parameters, Z,4, 10 and is similar to VI. NEUTRINOS
the E matrix of Eq. (20),IZ We can reduce this
matrix further by noting that the fourth column In the case of neutrinos, we should discard the
and fourth row are redundant. Thus the reduced differential operators in Eq. (19), and retain only
D matrix can be written as the SL( 2, C) part, If we choose the representation
given in Ref. 1 for neutrinos, the E(2) transforma-
cosO -sinO tion matrix takes the form
D(u,v,O)= [sinO cosO (25)
u· v·
_ [exp( -iO/2) (u -iv )expUO/2) 1
E(u,v,O)- 0 expUO/2)'

We are using the same notation for the four-by- (28)

four matrix of Eq. (24) and for the three-by-three
According to the isomorphism established for pho-
matrix of Eq. (25), but this should not cause any
tons in Eq. (26), we have to obtain the D matrix by
confusion. We can now identify the above expres-
sion with the E- I matrix given in Eq. (22): D(u,v,O)=[E-I(-u,-v,O)jt. (29)

l
(26) Thus

exp(~O/2) ].
exp( -iO/2)
Let us next consider antiparticles. Wigner's ori- D(u,v,O)= (u+iv)exp(-iO/2)
ginal work I includes discussions of the little groups
for particles with negative energies. If the energy (30)
is negative, NI and N z of Eq. (10) should be re- If this matrix is applied to the spin-up and
placed by their respective Hermitian conjugates. spin-down states we get
J 3 is Hermitian. This replacement does not change
the E(2) commutation relations given in Eq. (5).
Thus the little group for antiphotons is also iso-
D(u,v,O) [~ 1
= l(u :~~)~:;;~~2) 1' (31)

morphic to the E(2) group. We can then construct

the four-by-four D matrix,s and then reduce it to a
three-by-three form, The result is D(u,v,O) [~ 1 [exp(~o/2) 1·
= (32)
352 CHAPTER VI

26 E(2)-LIKE LITILE GROUP FOR MASSLESS PARTICLES AND ... 3723

The spin-down state remains invariant under the u theory based on a definite eigenvalue of rs is called
and v transformations, while the spin-up state un- the "two-component theory of neutrinos." It is in-
dergoes spin flips. teresting to note that this two-component theory is
In the case of photons, the parameters u and v a gauge-invariant theory.
generate gauge transformations. 2,4,10 Then, accord-
ing to Eq. (3 I), the gauge transformation changes
VII. CONCLUDING REMARKS
the spin orientation if the spin is parallel to the
direction of momentum. However, according to In this paper, it was noted first that the little-
Eq. (32), the gauge transformation does not change
group transformation matrix applicable to the pho-
the spin state if the spin is anti parallel to the
ton four-vector is somewhat complicated. We have
momentum. reduced this unattractive form into a three-by-three
The above analysis therefore leads us to the con-
matrix which can be compared with the regular
clusion that the spin of the spin-+ massless par- representation of the E(2) group. This reduced
ticle should be antiparallel to the momentum in or- form allows us to compare the little-group parame-
der that the spin state be gauge invariant. For ters for massless particles with those in the well-
the case of spin- + antiparticles, we construct the E known O(3)-like little group for massive particles.
matrix using the Hermitian conjugates of Tl> T 2 , The explicit construction of the isomorphism be-
and S3 given in Eq. (18), and calculate the D ma- tween the little group and E(2) allows us to study
trix using again Eq. (29), Consequently, the spin internal space-time parameters for neutrinos. We
of the antiparticle has to be parallel to the momen- have shown in this paper that the polarization of
tum. neutrinos is a consequence of the requirement of
However, it is important to realize that the invariance under the translationlike transformation
above conclusions on the directions of neutrino and of the little group of the Poincare group which, in
antineutrino spins depend on the choice of the E(2) the case of photons, is a gauge transformation.
representation. The E(2) matrix which was given As is well known, the subject of neutrino polari-
in Ref. 1 and which we used in the above analysis zation has a stormy history. It was Weyl who first
has a vanishing element in the lower left comer, proposed the two-component theory of neutrinos,
similar to the regular representation of the E(2) but this suggestion was rejected by Pauli on the
group given in Eq. (22). However, this is not the grounds that the theory does not preserve parity in-
only E(2) for which can be constructed as a sub- variance. IS Since 1956,19 we have understood neu-
group of SL(2,C).14 We can also consider trino polarization as a manifestation of parity
E'(u,v,O)=[E-I(u,v,O)jt. (33) violation. The time has come for us to ask what
space-time invariance principle is responsible for
If we use the matrix in Eq. (28) and follow the the polarization of neutrinos. In this paper, we
same reasoning as before, the spins of neutrinos have provided an answer to this question.
and antineutrinos would be parallel and anti parallel
to the momentum, respectively. ACKNOWLEDGMENT
It is clear in either case that the polarization of
neutrinos is a conseqnence of the requirement of We would like to thank Professor George A.
gauge invariance. Let us translate this conclusion Snow for very helpful criticisms and stimulating
into the familiar language of the Dirac equation. discussions.
It is easy to construct the NI and N2 operators ap-
plicable to the Dirac spinors for massless particles.
It then turns out that the Dirac spinors are invari- APPENDIX A
ant under the N I and N 2 transformations for both
polarizations. However, rs commutes with the In this appendix, we discuss the similarity be-
Hamiltonian, and this allows us to choose a defin- tween the E(2) and 0(3) groups using the concept
ite eigenvalue of rs. If the eigenvalue is -I, neu- of equivalence class. s
trinos and antineutrinos are left- and right-handed, Both the 0(3) and E(2) groups are thref-
respectively.ls,16 If the eigenvalue of rs is + 1, parameter groups. We can obtain the E(2) group
then the polarizations are opposite to those for the from 0(3) through a process of group contrac-
rs= -I case. Experimentally, the eigenvalue of rs tion. 2o The group contraction process goes as fol-
is known to be - 1.17 As is well known, neutrino lows. Every rotation can be regarded as a rotation
MASSLESS PARTICLES AND GAUGE TRANSFORMATIONS 353

3724 D. HAN, Y. S. KIM, AND D. SON

around a given axis. S The orientation of this rota- effect of the E matrix given in Eq. (22) on the
tion axis can be specified by two angular variables. column vector (uo,vo, 1) is well known. However,
This can also be achieved through the coordinate an interesting case here is the action of E -I in
specification on a spherical surface. The reorienta- view of Eq. (26):
tion of the rotation axis in this case can be speci-
fied by a movement of a point on the spherical E-1(-u,-v,IJ)=E(u·,v·,-0). (BI)
surface. If the radius of this sphere becomes suffi-
ciently large, and if the reorientation is sufficiently In order to see the effect of the above matrix on
localized, the axis reorientation would appear like the column vector (uo,vo, 1) let us carry out expli-
a motion of a point on a flat surface. As is citly the following matrix multiplication:
described in the Iiterature,20 the translation on the

:: I[~ ~ ::
E(2) plane is the limiting case of the axis reorienta-
tion in the 0(3) group.
u' cosO sinlJ o
The amount of rotation around a given axis is v' -sinO cosO o
an independent quantity. All rotations with the o 0 I 0 0
same amount of rotation, but not necessarily
(B2)
around the same axis, belong to the same
"equivalence c1ass."s The traces of transformation From this matrix algebra, u' and v' can be written
matrices belonging to the same equivalence class out as
are known to be the same. Likewise, for photons
and neutrinos, it is easy to see from the expression u'=(uo+u)coslJ+(vo+v)sinO,
given in this paper that the trace of the transfor- (B3)
v'=-(uo+u)sinO+(vo+v)cosO.
mation matrix is independent of the u and v vari-
able which only change the location of the rotation The above E(2) geometry is easy to understand,
axis on the E(2) plane. Since u and v are the gauge and does not require any further explanation.
transformation parameters, the gauge transforma- If we insist on doing the same matrix algebra us-
tion is a transformation within the same equiva- ing the D matrix, applicable to the photon polari-
lence class. zation vector,
Let us translate what we said above into formu-
las. As is well known, every rotation matrix R can
-sinO
be brought to the form
cosO
R =A exp(-iaJz)A -I, (AI)
v·
where a is the rotation angle, and A is the two-
parameter matrix which brings the rotation axis to (84)
the desired direction from the z axis. The trace R
is independent of the parameters of the A matrix.
In the E(2) case, it is always possible to write For convenience, we shall hereafter call the
column vectors, to which the E and D matrices are
E (u,v,O)=E(5, 1/,O)E(O,O,O)E -1(5,1/,0) . applicable, the E and D vectors, respectively. The
(A2) geometry of the E vector is well known. Its third
component is trivial. Its first two components
E (0,0,0) is a rotation around the origin. The specify the coordinate position in the two-
E(5,1/,0) in the above expression moves the rota- dimensional space spanned by the u and v vari-
tion axis from the origin to (5,1/). The trace of the ables.
above matrix depends only on the rotation parame- The physics of the D vector is well known. Its
ter. The translation of the axis to (5,1/) is a first two components specify the photon polariza-
transformation within the same equivalence class. tion state, and its third component is parallel to
the direction of the momentum and is an un-
measurable gauge parameter. The question then is
APPENDIX B how the D vector is related to the geometry of the
E vector.
We study in this appendix the vector spaces to The matrix algebras of Eqs. (82) and (B4) allow
which the D and E matrices are to be applied. The us to see the correspondence between the D and E
354 CHAPTER VI

E(2)-LIKE LITTLE GROUP FOR MASSLESS PARTICLES ANO ... 3725

vectors. The matrix algebra for D given in Eq. Hermitian conjugate, and this is one of the compli-
(B3) is simply the Hermitian conjugate of the alge- cations in groups containing Abelian invariant sub-
bra of Eq. (B4) for the E matrix. However, it is groupS.13 Therefore, the D vector is not a linear
not possible to construct a similarity transforma- transformation of the E vector, although there is a
tion matrix which will bring the E matrix to its one-to-one correspondence between them.

IE. P. Wigner, Ann. Math. ~, 149 (1939). of the present paper. Our statement in Ref. 2 about
20. Han, Y. S. Kim, and O. Son, Phys. Rev. 0 ~, 461 the integer values of m is correct. However, what we
(1982). said there about the half-integer values is incorrect
3The fact that the compact 0(2) group is only a sub- 12The D matrix of Eq. (12) performs both Lorentz and
group of the full noncom pact E(2) group was pointed gauge transformations. Once reduced to the form of
out by Gottlieb. See H. P. W. Gottlieb, Proc. R. Soc. Eq. (24), D is nO longer a Lorentz transformation ma-
London A368, 429 (1979). trix. It only performs gauge transformations.
4S. Weinberg, Phys. Rev. 134, B882 (1964); 135, BI049 13G. Racah, CERN Report No. 1961-8 (unpUblished).
(1964). 14M. Flato and P. Hillion, Phys. Rev. 0 I, 1667 (1970).
5E. P. Wigner, Group Theory and Atomic Spectra, 1ST. O. Lee and C. N. Yang, Phys. Rev. 105, 1671
translated by J. J. Griffin (Academic, New York, (1957); A. Salam, Nuovo Cimento ~, 299 (1957); L. O.
1959). Landau, Nuc!. Phys. 1, 127 (1957).
6y. S. Kim, M. E. Noz, and S. H. Oh, J. Math Phys. 16For a density matrix formulation of this problem, see
2Q, 1341 (1979); O. Han and Y. S. Kim, Am. J. Phys. A. S. Wightman, in Dispersion Relations and Elemen-
~, 1157 (1981); O. Han, M. E. Noz, Y. S. Kim, and tary Particles, edited by C. OeWitt and R. Omnes
O. Son, Phys. Rev. O~, 1740 (1982). (Hermann, Paris, 1960).
7E. P. Wigner, Z. Phys. 124, 665 (1948). 17M. Goldhaber, L. Grodzins, and A. W. Sunyar, Phys.
8F. R. Halpern, Special Relativity and Quantum Rev. li)2, 1015 (1958),
Mechanics (Prentice-Hall, Englewood Cliffs, New Jer- 18W. Pauli, Ann. Inst. Henri Poi~care Q, 137 (1936).
sey, 1968). 19The violation of parity invariance was discovered in
9J. Kupersztych, Nuovo Cimento ill, I (1976). 1956. See C. S. Wu, E. Ambler, R. W. Hayward, O.
100. Han and Y. S. Kim, Am. J. Phys. ~,348 (1981). O. Hoppes, and R. P. Hudson, Phys. Rev. lill., 1413
IIWhat we did in Ref. 2 was to replace Wigner's Eq. (1957).
(93) (Ref. D, which takes the form exp( tim f)) for 2oR. Gilmore, Lie Groups and Lie Algebras, and Some
both integer and half-integer values of m, by Eq. (16) oj Their Applications (Wiley, New York, 1974).
Chapter VII

Group Contractions

While the earth is like a sphere, its surface is like a flat plane in a reasonably
confined area. This raises the question of whether the symmetry governing the
sphere can be made that of a flat plane in certain limits. Indeed, in 1953, Inonu and
Wigner formulated this problem as the contraction of the three-dimensional rotation
group into the two- dimensional Euclidean group. In this case, the limiting
parameter is the radius of the sphere. When the radius becomes very large, the area
element on the surface becomes flat.
Since the little groups of massive and massless particles are locally isomorphic to
the three-dimensional rotation group and the two-dimensional Euclidean group
respectively, it is natural to suspect that the little group for massless particles is a
limiting case of that of massive particles. Then, what is the limiting parameter.
This has been shown to be the momentum/mass. The 1984 paper of Han, Kim, Noz,
and Son explains why the momentum/mass acts as the radius of the sphere in the
limiting process.
While the above-mentioned procedures constitute applications of the group
contraction, it is possible to achieve the same purpose by obtaining the little group
for massless particles from kinematical considerations. In 1986, Han, Kim, and Son
observed that the little group transformation is the transformation which does not
change the momentum, and constructed a set of non-colinear transformations whose
net effect is to leave the momentum invariant. This matrix is analytic in the (mass)2
variable in the neighborhood of (mass)2 = O. Therefore, it is possible to obtain the
zero-mass limit using the explicit expression for the little group transformation
matrix. The role of Wigner's little groups is summarized in Figure 3.

355
356 CHAPTERVll

I I
Massive Massless
between
Slow Fast
I
Energy 2 I Einstein's
E=-P- E=p
Momentum 2m E=.Jm2+p2

Spin, Gauge S3 Wigner's S3

Helicity SI S2
I Little Group I Gauge Trans.

FIG. 3. Physical implications of Wigner's little groups. Einstein's E = me2 unifies

the energy-momentum relation for massive and slow particles and that for massless
particles. Wigner's little group unifies the internal space-time symmetries of massive
and massless particles.
GROUP CON1RACTIONS 357

510 MATHEMATICS: INONU AND WIGNER PROC. N. A. S.

ON THE CONTRACTION OF GROUPS AND THEIR

REPRESENTATIONS
By E. INONU AND E. P. WIGNER

PALMER PHYSICAL LABORATORY, PRINCSTON UNIVBRSITY

Communicated April 21, 1953

Introduction.--Classical mechanics is a limiting case of relativistic

mechanics. Hence the group of the former, the Galilei group, must be
in some sense a limiting case of the relativistic mechanics' group, the
representations of the former must be limiting cases of the latter's repre-
sentations. There are other examples for similar relations between
groups. Thus, the inhomogeneous Lorentz group must be, in the same
sense, a limiting case of the de Sitter groups. The purpose of the present
note is to investigate, in some generality, in which sense groups can be
limiting caseS of other groups (Section I), and how their representations
can be Qbtained from the representations of the groups of which they
appear as limits (Section II). Section III deals briefly with the transition
from inhomogeneous Lorentz group to Galilei group. It shows in which
way the representation up to a factor of the Galilei group, embo·died in

Reprinted from Proc. Nat. Acad. Sci. (U.S.A.) 39, 510 (1953).
358 CHAPTER VII

VOL. 39, 1953 MATHEMATICS: INONU AND WIGNER 511

the SchrOdinger equation, appears as a limit of a representation of the

inhomogeneous Lorentz group and also gives the reason why no physical
interpretation is possible for the real representations of that group.

I. CONTRACTION OF GROUPS

Let us consider an arbitrary Lie group with n parameters at and tn-

finitesimal operators It. These shall be given, as usual by

It = lim g(he t) - g(O) (1)

h -+0 h
where 0 are the parameters of the unit element and et differs from 0 by a
unit increase of at. The If are skew herrnitean matrices if the group
consists of unitary matrices; instead of them one often uses the herrnitean
quantities ilf • However, all our equations remain somewhat simpler if
expressed in terms of the If. The structure constants C are defined by
(2)

If we subject the It to a linear homogeneous non-singular transformation,

the C will be replaced by other constants. These are obtained from the
C by contragradient transformations of its upper and lower indices. How-
ever, such a transformation has, naturally, no effect on the structure of
the group. Let us denote the transformation in question by
(3)

I t corresponds to the transformation

at = I: Ui,b' (3a)

according to which the J are obtained by the same equation (1) as the
I, except that the et have to be replaced in it by a similar quantity, defined
with respect to the b.
The above transformation may lead to a new group only if the matrix
U of (3) is singular. We shad call the operation of obtaining a new group
by a singular transformation of the infinitesimal elements of the old group
a contraction of the latter. The reason for this term will become clear
below. The singular matrix will be a limiting case of a non-singular matrix.
The latter will depend linearly on a parameter fi which will tend to zero:
(4)
For 0 < E < EO the determinant of (4) is different from zero, it vanishes for
E = O.

We shall transform (4) into a normal form by a non-singular and E

GROUP CON1RACTIONS 359

512 MATHEMATICS: INONU AND WIGNER PROC. N. A. S.

independent transformation of both the I, and It. If the matrices of the

corresponding transformations are denoted by a and {3, respectively, u
and w will be replaced by {3ua- 1 and {3wa- 1• It is possible, by such a
transformation, to give u and w the form

1 0 v 0
u = w = (5)
o 0 o 1

The number of rows and columns in the unit matrix in u, and in v, is equal
to the rank r of u. It is advantageous to label the transformed I and I
with a pair of indices, the first referring to the subdivision of u given in
(5), the second specifying the various I and I within that subdivision.
Hence, (3) assumes the form
r

II, Ib + L E V,Jlp (II 1,2•... , r)

I
I' -
(6)
1 2, EI2,. (II 1, 2, ... , n - r)

The corresponding transformation of the group parameters is

+ E L v,pb lp
T

ab b1, (II = 1,2. ... , r)

I' - 1
(6a)
a2, Eb2 ,. (II = 1.2. ... , n - r)

It is well to remember that the parameters a lead to the infinitesimal

elements I. the parameters b to the 1. The last equation shows that a
given set of parameters b correspond, with decreasing E, to smaller and
smaller values of the parameters a2. In the limit E = 0 (if such a limit
exists), one will have contracted the whole group to an infinitesimally
small neighborhood of the group defined by the air alone. This justifies
the name given to the process considered.
The transformation of the infinitesimal elements which we carried out
also changes the structure constants and we shall write for (2)
r " - T

[Ia.. 1/11'1 = L Ca •. flplKIIK + L Ca•. {Jp 2.cI2K (7)

«""'1 «--1

wherein a and (3 can assume the values 1 and 2. This gives for

[Ib' 111'] + EL(V.,,!5pp' + !5,.,vpp, + EfI..,f)pp,)[Ib " III"]

L CI ,. IplK lie +! L Cir. 1/" 1 2" + 0(1).
" E "
/

Hence, if the commutator of 110 and Ill' is to converge, as E- 0, to a linear

('ombihation of the I, the structure constants
360 CHAPTER VII

VOL. 39, 1953 MATHEMATICS: INONU AND WIGNER 513

(8)

i.e., the Ib must span a subgroup. On the other hand, if this happens
to be the case, the structure constants will converge to definite values
Ca •• 11/1"Y· as E - 0
h 1. 2. 2c
Cb.i/l Cb.i/l Cb.i/l C b . l ,. 0
1. 2.
Cb.2/1 0 Cb.2/1 C b • 2/12. (9)
C2o.2/1
1.
C2 •• 2/1
2.
o.
These structure constants satisfy Jacobi's identities since the structure
constants for the J do this for non-vanishing E. We shall say that the
above operation is a contraction of the group with respect to the infinitesi-
mal elements Ib or that the infinitesimal elements 12/1 are contracted. We
then have, from (9).
THEOREM 1. Every Lie group can be contracted with respect to any of its
continuous subgroups and only with respect to these. The subgroup with
respect to which the contraction is undertaken will be called- S. The con-
tracted infinite#mal elements form an abelian invariant subgroup of the con-
tracted group. The subgroup S with respect to which the contraction was under-
taken is isomorphic with the factor group of this invari.ant subgroup. Con-
versely, the existence of an abelian invariant subgroup and the possibility to
choose from each of its cosets an element so that these fot"m a subgroup S, is a
necessary condition for the possibility to obtain th, group from another group
by contraction.
It is easy to visualize now the effect of the contraction oil the whole
group. The subgroup S with respect to which the contraction is undj!I"-
taken remains unchanged and it is advantageous to choose the group
parameters in such a way that afp = 0 throughout S. Then (6a) can be
replaced by
(6b)

and this can be assumed to be valid throughout the whole group, not only
in the neighborhood of the unit element. As E decreases, a fixed range of
the parameter b will describe an increasingly small surrounding of S. As
E tends to 0, the range of the bfp will become infinite and describe only
those group elements which differ infinitesimally from the elements of S.
The elements which are in the neighborhood of the unit element of the
original group but have finite parameters b20 will commute and form the
aforementioned commutative invariant subgroup. Naturally, the ele-
ments of this invariant subgroup will not commute, in general, with the
elements of the subgroup S: the change of the parameters a20 = Ebb will
be, upon transformation by finite elements of S, of the same order of
magnitUde as these parameters themselves. Naturally, the convergence
GROUP CONTRACTIONS 361

514 MATHEMA TICS; INONU AND WIGNER PRoc. N. A. S.

of the original gTOUp toward the contracted group is a typically non-

unifonn convergence.
Every Lie group can be contracted with respect to any of its one para-
metric subgroups. If the three dimensional rotation group is contracted
in this way, one obt:;tins the Euclidean group for two dimensions. Con-
traction of the hom(,igeneous Lorentz group with respect to the subgroup
which leaves the time coordinate invariant yields the homogeneous Galilei
group, contraction of the inhomogeneous Lorentz group with respect to
the group generated by spatial rotations and time displacements yields the
full Galiki group. Contraction of the de Sitter groups yields the inhomo-
geneous Lorentz group. It should be remarked, finally, that if a group,
obtained by contraction of another group with respect to the subgroup S,
is contracted again with respect to S, the second contraction remains with-
out effect.
The above considerations show a certain similarity with those of 1. E.
Segal. 1 However, Segal's considerations are more general than ours as
he considers a sequence of Lie groups the structure constants of which
converge toward the structure constants of a non-isomorphic group. In
the above, we have considered only one Lie group but have introduced
a sequence of coordinate systems therein and investigated the limiting
case of these coordinate systems becoming singular. As a result of our
problem being more restricted, we could arrive at more specific results.

II. CONTRACTION OF REPRESENTATIONS

If one applies the transformation (6) to the infinitesimal elements of

a representation of the group to be contracted, and lets E tend to zero, the
J 2• will also tend to zero. The representation will become isomorphic to
the representation of the subgroup S, i.e., will be a representation of the
factor group of the invariant subgroup. In order to obtain a faithful
representation, one must either subject the J b to an E dependent transforma-
tion, or consider the J,. which correspond to different representations, e.g.,
go to higher and higher dimensional representations as E decreases. We
shall give examples for both procedures.
(a) Representations of the Contracted Group by Means of E Dt>pendent
Transformations.-The first procedure is applicable only if the infinitesimal
elements are not bounded operators, i.e., as far as irreducible representa-
tions are concerned, only if the group is not compact. The simplest non-
compact non-commutative ~oup is that of the linear transformations
x' = e'x + fJ. The general group element is 0 ...8 = T(fJ)R .. with the group
relations T(fJ)T{fJ') = T{fJ + fJ'); RaR .. , = R",+"" and R",T{fJ) = T(e"'fJ)R",.
The only faithful irreducible unitary representation of this group can be
given in the Hilbert space of square integrable functions of 0 < x < co
362 CHAPTER VII

VOL. 39, 1953 MATHEMATICS: INONU AND WIGNER 515

(10)

The infinitesimal operators are

(lOa)

with the structure relati.on

(lOb)
Contraction with respect to the group of transformations x' e'>X leaves
the group unchanged: the only non-vanishing structure constant is C)22
and (9) shows that this does not change. We can try, therefore, to trans-
form II and El2 with an E dependent unitary matrix so that II remam
unchanged, the transformed d 2 converge to 12
(11)

This is indeed possible: one has to choose S. = RID.. This commutes

with It. It follows from the group relation that S, -'T({3)S, = R -In .T({3)
R ln , = T({3/f) and one has, hence, as h -+ 0
ES, -II2 S, = ES, - I lim h-I(T(h) - I)S,
= lim Eh-I(T(h/E) - 1) = 12 (lla)

Hence ES. -II2S. not only converges to 12 = 12 but remains equal to it for
all E.
It is more surprising, perhaps, to see that the same device is possible
also if one contracts the group with respect to the subgroup of the T({3).
The contracted group is, in this case, the two parametric abelian group.
We demand, in this case, that S, commute with I{J and hence, by (lOa),
that it be multiplication with a function of x. Because of Sf unitary na-
ture, we can give it the form exp (if(x, E)). Transformation of the Ela of
(lOa) with this gives
ES,-IIaS, = Ee-;/(z, ')('/2 +
x-d/dx)e;/(z, ,)
= E('/2 +x d/dx) +
Eix d/dx f(x, E). (12)
The first part of this converges to 0 as it should since (12) should converge
to an operator which commutes with ix. The second part converges to
12 = ixJ'(x) = ig(x) if one sets
f(x, E) = clf(x). (12a)
Hence, the transformations of the contracted group corresponding to the
parameters cr, {3 is mUltiplication with
(13)
GROUP CON1RACTIONS 363

516 'MATHEMATICS: INONU AND WIGNER PROC. N. A. S.

which is again a faithful (reducible) representation of the contracted

group.
The operators corresponding to finite group elements could have been
obtained directly by transforming 0a.l/J = T(E{J)Ra with R 1n in the case
I

of contraction with respect to Ra. Similarly, (13) could have been ob-
tained directly by trljnsforming O.a. (J = T({3)R.a with exp (t-1if(x».
It is not clear how generally one can obtain a faithful representation
of the contracted group as a limit of an E dependent transform of a faithful
representation of the original group and the substitution (6b) of its param-
eters. Certainly, the procedure is not applicable to irreducible repre-
sentations of compact groups or, more generally, if the infinitesimal oper-
ators are bounded.
(b) Representations of the Contracted Group from a Sequence of Repre-
sentations.-We shall now give a few examples for the second procedure,
i.e., obtaining a representation of the contracted group by choosing a
sequence of unitary representations D(I), D(2), ... , D(I), ... so that each
of the operators
(II = 1,2, .. , ri J.I = 1,2, ... , n - r) (15)

converge to a finite operator as E - 0 and 1 - (x). Alternately, we can

ask that the transformation corresponding to finite group elements
(15a)
converge to a unitary representation of the contracted group as E - 0,
1- (x). The b are the parameters of the contracted group; the operator
(15) corresponds, in the representation 1, to the group element of the
original group the parameters a of which are given by (6b). The formula-
tion makiijg use of the finite group elements (15a) is unambiguous because
it deals with the convergence of unitary operators; the first one is usually
easier to attack directly.
The convergence of the sequences of (15) and (15a) will depend not
only on the values which E assumes and on the corresponding representa.-
tions D(I>, it will also depend in which form that representation is assumed.
Hence, method (a) can be considered as a special case of the present method
in which all D(I) are unitary equivalent.
The contracted group is always an open group because the variability
domain of the b is infinite. Hence its representations are, as a rule,
infinite dimensional. . If the D(l) are finite dimensional, they should be
considered to affect only a finite number of the coordinates of Hilbert
space. /
The" simplest non-commutative compact group is the three-dimensional
rotation group. All its su~groups are one parametric, we shall contract
it with respect to the rotations about the z axis of a rectangular coordinate
364 CHAPTER VII

VOL. 39. 1953 MATHEMATICS: INONU AND WIGNER 517

system of ordinary space. The contracted group is the Euclidean group

of the plane, i.e., the inhomogeneous two-dimensional rotation gr6up.
We shall choose for D(I) the representation which is usually denoted 2 by
D(I): it is 2l +
1 dimensional (t being any integer) and is usually described
in a space the coordinate a)l:es of which are labeled with m = -t, -I I, +
... , t - 1, t. Hence, we label the coordinate axes of Hilbert space with
all integers m from - 00 to ro. In keeping with our previous notation,
we call the infinitesimal element which corresponds to rotations about the
z axis M. = h One can then write for m t, m' I I ::; I I ::;
I
(16)

(12%(1»)....... = (M% (I)) ...... , = - ~ V (l - m)(l + tn') 0.......+1 +

~ V(l - m')(t + m)om'm-I
; V (l - m)(l + m') 0"""'+1 +
; V(l - m')(l + m) O""m-I.

All matrix elements vanish if either m or I I Im' I is larger than t. As

t- 00, the II converge to a definite operator

( - co < m, m' < ro) (17a)

In fact, the convergence is strong in the sense that II<p converges strongly
to JI<P if <p is in the definition domain of Jr. If E and't converge to zero
and infinity, respectively, in such a way that it -+- :::, the other two in-
finitesimal elements will also convetge to J 2r , J 2v , respectively, where

(J2r)mm' = ~ :::(Om'm-I - O""m+l) (17b)

It follows from the fact that there are only a finite number (two) non-
vanishing matrix elements in both I 2r and 12, that the operators satisfy
the commutation relations of the contracted group
[J2X1 J 2v ] = 0
It further follows from
(11(1»)2 + (12r(l»)2 + (/2,(/»)2 = -let + 1)
by multiplication with E2 and going to the limit in the above way that
(18a)
GROUP CONTRACTIONS 365

518 MATHEMATICS: INONU AND WIGNER !>Roc. N. A. S.

It was important, for obtaining convergent sequences of 1,(1) and J 2 (1),

to have assumed the D(I) in the fonn given by (16). This form was
reduced out with respect to the subgroup S and this caused, in this case,
the convergence of the sequence I, (l). The convergence of the El2 (I) does
not actually follow from the convergence of the 1,(/), and hence from the
reduced out fonn of the representations D(l), but was made at least possible
by this circumstance.
Before going over to the investigation of the Lorentz groups, it may
be worth while to make a final remark about the above contraction, even
though it has little to do with our subject. We shall determine, first,
the matrices which correspond to finite group elements. For this purpose,
it is useful to consider that form of the above representation in which the
Hilbert space consists of functions of x and y and the infinitesimal operators
have the natural form for infinitesimal operators of the Euclidean group
(a and r are polar coordinates x = r cos a, y = r sin a)

J; = -Oloa = y%x - x%y (19)

(19a)

One should keep i~ mind that J~z arose from 12z which is the infinitesimal
rotation about the x axis and corresponds to a displacement in the - y
direction. Similarly J~. corresponds to a displacement in the x direction.
The function 'P(a, r) corresponds in the new Hilbert space to the vector
which has the components 'Pm in the Hilbert space of (l7a), (17b). It
further follows from (l8a) that

(20)

whence one can write

f e -iE(x cos a' + y sin a') g(a')da' (20a)
f e- iEr cos (a - a') g(a')da'
f p -iEr cos a' g( a - a ')d a.
'

All integrations are from 0 to 211". Expanding g(a - a') into a Fourier
series of a - a', one finds
'P(x,y) = fe-jErcosa',£gme-jm(a-a')da' (20b)

m runs from - to to to. The last fonn makes it easy to calculate J; by

(19), the first fonn permits one to calculate J~z and J;v easily. Comparison
of ,the expressions obtained in this way with (17a) and (17b) shows that
gm = 'Pm. In order to remain in keeping with the usu31 notation, we
define 3
366 CHAPTER VII

VOL. 39. 1953 MATHEMATICS: INONU AND WIGNER 519

Jm is then the ordinary Bessel function of order m. This permits one to

write for (20b)
(21)
Because of (19a), one can write down at once the finite group operations.
In particular, for the displacement T(t, '1) by t and '1 in the x and y direc-
tions one has
T(t, '1)\O(x, y) = \O(x - t, y - '1). (22)
Hence, denoting the matrix for the same operation in the original Hilbert
space by T(t, '1)mm" one has
\O(x - t, Y - '1) = 2r :E :E T(t, '1)mm'\Om,i- me- ima Jm(Zr). (22a)
m m

°
This permits an explicit determination of the T(t, '1)mm'. We shall not
carry this out completely but set only r = in (22a). Since ali Jm(O) =
except Jo(O) = 1, the summation over m disappears on the right side.
°
The left side becomes, at the same time by (21)
\O(-t, -'1) = 2r:E\Omi -m e -im(Il+-) Jm(Zp) (22b)
where tJ, p are the polar coordinates for t '1. Comparing (22a) and (22b)
one finds, with t = 0, tJ = i/27f, '1 = P
J m(:~P) = T(O, P),Om. (23)

The group relations and the form of the infinitesimal operators (19a) gives
at once the most important relations for Bessel functions, such as the addi-
tion theorem, differential equation (d. (20», etc. Up to this point the
argument is not new but merely a repetition, for the two dimensional
Euclidean group, of a similar reasoning given before 4 for the rotation
group. This led to the equation 4
_
DOl(O, tJO)Om = pm(cos {:J) =
(I +-
(l
m)!)11 2
m)! P7 (cos (3) (24)

in which P~ is Legendre's associated function of the first kind, p~ is normal-

ized to the same value as P,O = P,. Furthermore, 0, (3, is the rotation
about the x axis so that, by the definition of T(O, p)
°
T(O, P)""m = (- )",' lim D(I) (25)

This, together with (23), gives the asymptotic expression for the associated
Legendre functions 5
lim P~(cos (pll») = Jm(P). (25a)
,-~
GROUP CON1RACTIONS 367

520 MATHEMATICS: INONU AND WIGNER PROC. N. A. S.

III. CONTRACTION OF LORENTZ GROUPS

Let us consider. first. the inhomogeneous Lorentz group with one space-
like. one time-like dimension. It is given by the transformations
x' x cosh X + t sinh X + ar (26)
t' = x sinh X+ t cosh X + a I.
We wish to contract it with respect to the subgroup of time displacements
t' = t + a l • The infinitesimal elements of (26) are: time displacement
It. space displacement I2r and "rotation" in space-time /2)". Their com-
mutation relations read

Hence. by (9). the commutation relations of the contracted group are

r1 2r • 12),.1 = o. (27a)
The "rotations" in space-time. together with the displacements in space.
form a commutative invariant subgroup.
The matrices
cosh X sinh A
sinh X cosh X (26a)
o o 1

form a natural. though not unitary. representation of the group of trans-

formations (26). We can carry out the contraction by setting at = bl>
A = fV, a% = fb r or A = vic, ax = bxlc and letting f converge to 0,
or c converge to infinity. If we do this directly in (26a), the representation
will not remain faithful for the contracted group. We shall transform
therefore (268.) with a suitable E (or c) dependent matrix: mUltiply the
first row with c. the first column with l/c. If c goes to infinity in the
matrix obtained in this way, one obtains the transformations of the con-
tracted group
x' x + vt + b% (27a)
t' t + bl.
It is the inhomogeneous Galilei group with one spatial dimension. The
transformations x' = x vt + +
br • t' = t form the commutative invariant
subgroup. .
The same contraction can be carried out for an inhomogeneous Lorentz
group with an arbitrary number of spatial dimensions. The only difference
is that the subgroup S, with respect to which the contraction is carried
out, cqntains not only the displacements in time as in the above example.
but also all purely spatial rotations. i.e .• all homogeneous transformations
368 CHAPTER VII

VOL. 39. 1953 MATHEMATICS: INONU AND WIGNER 521

which leave t invariant. The invariant subgroup of the contracted group

consists of aU spatial displacements and Galilei transformations:
, b '
x; =
x; + v,t + " t = t. ;I

Contraction of the Unitary Representations of the Lorentz Groufls.-We

shaU be principally concerned here with the group of the special theory
of relativity, i.e., the inhomogeneous Lorentz group with three space-like
and one time-like dimension. The subgroup S with respect to which we
shall co~tract it contains the displacements in time, the spatial rotations,
and the products of these operations. The contracted group is the ordinary
Galilei group, i.e., the group of classical mechanics.
We shall denote the displacement operators in the direction of the three
space-like axes by I" (k = 1, 2, 3), the displacement in the direction of the
time axis by 10• The rotations in the kl plane will be denoted by 11;1, the
acceleration in the direction of the k axis by 1,,0, 10 and Itl span the sub-
group S.
The quantity
(28)

is a constant in every irreducible representation and the irreducible repre-

sentations can be divided into three classes according to the value of this
constant.' In the first class, P < 0 and the momenta (which are - i times
the infinitesimal operators) are space-like. In the second class P = 0 and
the momenta form a null vector, P > 0 in the third class.
It is generally admitted that the representations of the first class have
no physical significance because the momenta of aU observed particles are
time-like or null vectors. Hence we shaH investigate only the simplest
one of these representations. Its operators are most easily given in the
Hilbert space of functions of three variables PI. Pt, p, which are restricted
to the outside of a sphere of radius v' -Po The expressions for the in-
finitesimal operators are
(28a)
This last equation also shows the reason for the variables PI: to be restricted
by p~ + P~ + p~ > - P: if this inequality is not fulfilled, 10 ceases to be
skew hermitean and, hence, the representation is not unitary. Further
(28b)

and
1"0 = - (p~ + p~ + pi + P)1/z ()/fJPt. (2&)
It is useful to introduce new variables instead of the Pl both in order to
simplify the definition domain of the variables and also to bring the opera-
tions of the subgroup S into a form which is independent of P. This can
GROUP CONTRACTIONS 369

522 MATHEMATICS: INONU AND WIGNER !>Roc. N. A. S.

be done, most simply, by introducing as new variables

The Ot are restricted to a unit sphere, po changes from 0 to 00. In these

variables, the scalar prpduct between IP and y, reduces to
(IP, y,) = f dof dpo(p! - P)I/2iPy, (29a)
dO being the surface element of the unit sphere over which the integration
with respect to the U is to be extended. In terms of the new variables, the
infinitesimal operators assume the form
(30)

and
(30a)

ho = - (p! - P)I/20ol: c)/fJpo - ( 2 POp)1/2 ~)l,(G, c)/c)O.l: - G.I: c)/c)G 1).

Po - I

If we now set 1.1: = EIh the 1t will converge to zero unless -P becomes
inversely proportional to E2, i.e., unless - f2p converges to a definite limit
P. If this is assumed, the second term of UtO will converge to zero and
the infinitesimal elements of the representation of the contracted group
become
1tl = 0 1 C)/C)O.l: - Gt ()/()G 1 10 = -ipo (31)

J.l:O = - PGt ?>/()Po J.I: = i PGt • (3Ia)

These operators indeed span a unitary representation of the Galilei group:
they correspond to case II with m = 0 of a recent determination of these
representations. 7 The Pt of this article correspond to our PO.I:, the variable
s is given by i P ?>/?>po. One also understands now why it was impossible
to find a physical interpretation to this representation: it is the limiting
case of a representation of the relativistic group with imaginary mass pl/!.
The same is probably true of the other true representations7 of the Galilei
group.
We go over now to the investigation of the simplest representation with
positive P, i.e., the representation of the Klein-Gordon equation. The
infinitesimal operators are again given by (28a), (28b),. (2&). However,
since P is positive, the variability domain of the p extends over the whole
three dimensional space. The scalar product of two functions IP and y, is
now given' by
(IP, y,) = f f f dPI dP2 dp3 PO-I iP'/I (32)
in which po is still given by (29).
370 CHAPTER vn

VOL. 39. 1953 MATHEMATICS: INONU AND WIGNER 523

If we set, in the sense of (6),

Jl; = fIl; = Eipl; (33)
the operation of time displacement becomes

i10 = ilo = (- I~ - Ii - Ii + P) 1/2 =

f- l (-J~ - Ji - Ji + lp)I/2 (33a)

Since the Jl; are skew hermitean, their squares are positive definite
hermitean operators. Since E2P is also positive the expectation value
('P, iJo'P) of ilo is, for any state 'P, greater than E- l times the expectation
value of ill' It follows that if 11'P converges to a vector in Hilbert space
as E -+ 0, the vector Jo'P must grow beyond all limits. The same is true,
of course, for the other ll;. It follows that the representations considered
cannot be contracted in the sense discussed in the previous sections and
the same is true of all representations of the classes P ~ O.
It is possible, however, to contract these representations to representa-
tions up to a factor of the Galilei group. The commutation relations of
the infinitesimal elements of representations up to a factor differ from the
commutation relations of real representations by the appearance of a
constant in the structural relations. Hence
(34)

where c~/J are the structure constants of the group to be represented (in
our case, the inhomogeneous Galilei group) and the balll are multiples·of
the unit operator. One will, therefore, obtain infinitesimal elements of
representations up to a factor if one sets, instead of (6)
(34a)
in which all a may depend on f. Since the additional terms in (34a)
commute with all other operators, these additional terms will not affect
the left side of (34). Hence, they must be compensated also on the right
side and this is done by the additional terms baill. The point of introduc-
ing the terms al in (34a), which then necessitates the introduction of the
bin (34), is that the right sides of (34a) may converge to finite non-vanish-
ing operators even if the I b , d 2• cannot be made to converge.
The above generalization of the concept of contraction indeed allows
a contraction of the representations given by (28a), (28b), (2&) also for
p > O. As we let P go to infinity, 10 will also tend to infinity «Io'P, Io'P)
converges to infinity for all 'to). However, subtracting _ipl/2l from 10 ,
it will converge to
10 lim -i( -Ii - n - Ii + P)I/2 + ip l/2

= (i/2p l / 2 )(Ii + Ii + Ii) = (i/2pl/2 f 2)(1; + 1; + l~). (35)

GROUP CONTRACTIONS 371

524 MATHEMATICS: INONU AND WIGNER Paoc. N. A. S.

This shows that J o will converge to a finite operator if the J" = El" do and
if P·"E 2 converges to a finite constant m as P - to, E - O. Both can be
accomplished by assuming the representation in such a form that, instead
of (28)

(36)
This is indeed possibl~ because the variability domain of the PIt is un-
restricted and the abov~ form of the infinitesimal elements can be obtained
by unitary transformation of the operators given in (28a). Such a trans-
formation leaves the Iu of (28b) unchanged but transforms the ItO of
(2&) into

Hence we shall have

J o = lim -i(P + (Pi + p~ + Pi)/E2)1/2 + ip = l/ 2

- (i/2m)(pi + p~ + p!)
Ju = Pi()/()Pt - P,,()/()P.l (37.1)

J" lim E(ip,,/ E) = ip"

Jk.O lim -E(P + (p~ + p~ + P~)/E2l/2~/()p"

= lim - (PE 4 + i(pi + p~ + p:»1/2()/()p" = -m()/()Pt. (37.2)

The reader familiar with the transition from the Klein-Gordon to the
SchrOdinger equation will recognize the increase of the rest mass with
increasing c and the elimination of this rest mass by the subtraction of
_iPI/21 from the infinitesimal operator of the time-displacement operator.
The infinitesimal operators (37.1), (37.2) for the contracted group are in
fact those of SchrOdinger's theory. It is likely that a similar contraction
is possible also for the other representations with positive rest mass (i.e.,
P> 0) but this and the behavior of the representations with P = 0 will
not be further discussed here.
J Segal, I. E., Duke Math. J., 18,221 (1951).
2 Cf. e.g. Wigner, E., Gruppentheorie und ihre Auwendiengen etc., Friedr. Vieweg,
Braunschweig (1931) and Edwards Brothers, Ann Arbor (1944). Chapter XV.
I Cf. Jahnke, E., and Emde, F., Tables of Functions, Dover Publications, 1943, p. 149;

or Watson, G. N., Treatise on Bessel Functions, Cambridge Univ. Press, 1922, p. 19ft.
• Reference 2, Chapter XIX, particularly p. 230, 232. Cf. also Wigner, E. P., J.
Franklin Inst., 250, 477 (1950), and Godement, R., Trans. Amer. Math. Soc. 73, 496
(1952).
I For m = 0 this is given on p. 65, of Watson's Bessel Functions (ref. 3).

• Cf. e.g. Bargmann, V., and Wigner, E. P., these PROCEEDINGS, 34, 211 (1948) and
further literature quoted there.
7 Inonu, .;;., and Wigner, E. P., NuOfJO cimento, 9, 705 (1952).
 CHAPfERVII
372

Internal space-time symmetries of massive and massless particles

D. Han
Systems and Applied Sciences Corporation. Hyattsville. Maryland 20784
V.S. Kim
Centerfor Theoretical Physics. Department ofPhysics and Astronomy. University ofMaryland. College Park,
Maryland 20742
Marilyn E. Noz
Department ofRadiology, New York University, New York. New York 1(}()16
D. Son
Department ofPhysics. Columbia University, New York, New York ](}()27

(Received 20 October 1983; accepted for publication 20 February 1984)

A unified description is given for internal space-time symmetries for massive and massless
particles. It is noted that the little groups governing the internal symmetries of massive and
massless particles are locally isomorphic to the three-dimensional rotation group or 0 (3) and the
two-dimensional rotation group or E (2), respectively. It is noted also that the E (2) group can be
obtained from a large-radius/llat-surface approximation of 0 (3). This procedure is then shown to
be directly applicable to that of Obtaining the E (2 )-like little group for massless particles from the
o (3)-like little group for massive particles in the infinite-momentum/zero-mass limit.

I.INTRODUCfION massive and massless particles through

One of the beauties of Einstein's special relativity is the E= [(cP)2+(MC)2J'12. (I)

unified description of the energy-momentum relation for In addition to mass, energy, and momentum, relativistic

@ 1984 American Association of Physics Teachers 1031

Reprinted from Am. J. Phys. 52, 1037 (1984).
GROUP CONTRACTIONS 373

Table I little groups for massive ami ma!.sless partlde~ Why can we not formulate a group theory based on this
daily experience?
P: fOUf-momentum Subgroup nf 0 13.11 Subgroup of SL (2,c) (cl Strictly speaking, when we travel on the surface of the
Earth, we are performing rotations around the center of the
\1a:'S1H' o (JI-Ilke ~uhgrour SL' 121-hkt: ~ubgloup of
Earth. Can the E (21 group be regarded as a limiting case of
P ,ll nf 0 iJ.11 hadron\ Sr:2,(1 c/ecrron\
the rotation group"
Massle"s El21-hke ~ubgroup EI2Htke subgroup of We shall therefore start this paper with a discussion of the
p 'll ofO:3.1I.phOlonl SL 12.cl neurnnor; E (21 group.
In Sec. II, transformations on the two-dimensional Eu-
clidean plane are discussed. It is shown that solutions of the
two-dimensional Laplace equation form the basis for finite-
dimensional representations of the E (2) group. The 3 X 3
particles have internal space-time degrees offreedom, For matrices representing coordinate transformations on the
instance, a massive particle has rotational degrees of free- E (2) plane are discussed in detail. In Sec. III, the E (21 group
dom in the Lorentz frame in which the particle is at rest On is discussed as a contracted form of 0 (3). It is noted that we
the other hand, free massless particles have the helicity and can achieve this purpose by looking into a small and almost
gauge degrees offreedom, We are therefore led to the ques- flat portion of a spherical surface whose radius becomes
tion of whether the internal symmetry for massless parti- large.
cles can be obtained as an infinite-momentumlzero-mass We discuss in Sec. IV the internal space-time symmetries
limit of the space-time symmetry for massive particles, as of massive and massless particles. It is shown that the E (2)-
in the case of the energy-momentum relation. like symmetry of massless particles may be regarded as a
In order to study internal space-time symmetries of rela- limiting case of the 0 (3)-like symmetry for massive parti-
tivistic particles, Wigner in 1939 formulated a method cles. In Sec. V, we discuss further applications of group
based on the little groups of the Poincare group. ' The little contractions. It is shown in the Appendix that the E (21
group is a subgroup of the Lorentz group which leaves the group can serve as a useful example for illustrating the dif-
four-momentum of a given particle invariant. The little ference between active and passive transformations.
groups for massive and massless particles are locally iso-
morphic to the three-dimensIOnal rotation group and the
two-dimensional Euclidean group, respectively. For con- II. WHAT IS THE E (2) GROUP?
venience, we shall use the word "like" in order to indicate
that two groups have the same algebraic properties. The The two-dimensional Euclidean group, often called E (2),
internal symmetries of massive and massless parhcJes are consists of rotations and translations on a two-dimensional
dictated by the 0 (31-like and E (2)-like little groups, respec- Euclidean plane. The coordinate transformation takes the
tively. form
The first step in obtaining a unified picture of both mas- x' = x cos IJ - y sin IJ -t u,
sive and massless particles is to gain thorough understand-
(2)
ing of each of the four cases listed in Table I. The represen-
tation suitable for electrons and positrons was discussed in
y' = x sin IJ + Y cos IJ + u.
Wigner's original paper. I The representations for relativis~ This transformation can be written in matrix form as
tic extended hadrons in the quark model have been dis- - sin B
cussed extensively in the literature. '.l The representations
suitable for photons and neutrinos have also been worked
(~'X') = (COS IJ
Si~ B cos B (3)
OUt,4.5 o
The purpose of the present paper is to discuss in detail The algebraic properties of the above transformation ma-
the problem of showing that the 0 131-like little group for trix have been discussed in Ref. 4.

=
massive particles becomes the E 12)-like little group for The 3 X 3 matrix in Eq. (3) can be exponentiated as
massless particles in the infinite-momentum/zero-mass D(B,u,v) exp[ - i(uP, + vP2 )]expl- iBL ,), 14)
limit. In order to deal with this problem, we have to show
first that the E 12) group can be regarded as a limiting case of The generators in this case are
0(3). We are quite familiar with the three-dimensional ro- -i
tation group. However, the E (21 group is largely unknown
to us, in spite of the fact that this group can serve as a good
illustrative example for many important aspects of geome-
LJ=C 0
0 ~).
try and group theory. 15)
0 0
=(~ ~} P2=(~ D
Indeed, from a pedagogical point of view, the E (2) group
has its own merit. When we study the three-dimensional 0 0
rotation group, it is quite natural for us to ask the following
P,
0 0
questions.
These generators satisfy the following commutation rela-
(a) We discuss 0 (3) repeatedly in the established curricu-
tions:
lum, because it describes an important aspect of physics,
and because it generates a beautiful mathematics. Then, is [P"P,] =0,
the rotation group the only interesting example in the exist- [L"P,] =iP2, (6)
ing curriculum?
(b) When we commute from home to school, we are mak- [L"P 2 ] = - iP,.
ing translations and rotations on a two-dimensional plane. The transformation described in Eqs.12) and (3) and gener-

1038 Am. J. Phys., Vol. 52, No. 11, November 1984 Han et af. 1038
374 CHAPfERVIl

ated by the matrices ofEq. (5) is "active" in the sense that it and is suitable for exercise problems even in the undergrad-
transforms the object, as is described in Eq. (2). uate curriculum.
Let us next consider transformations of functions of x If k ' = 0, the differential equation of Eq. (9) takes the
and y, and continue to use P, and P, as the generators of form
translations and L, as the generator of rotations. These
generators take the form [( ~)' + (~),I1b(xJ') = O. (10)
P . a This is a two-dimensional Laplace equation, and its solu-
I = -I ax'
tions are quite familiar to us. The analytic solution of this
.a
P
,= -lay' (7)
equation takes the form
1b = r'" exp[ ± imq,] = (x ± IJr, (II)
L, = -i(X!!...-y~), where
ay ax <6=tan- l (ylx).
The above operators satisfy the commutation relations of
This is an eigenstate of L, or a rotation around the origin.
Eq. (6). The transformation using these differential opera-
The effect of the rotation operator
tors is "passive" in the sense that it is achieved through a
coordinate transformation which is the inverse of that giv- R(O)=exp[-iOL,] (12)
en in Eq. (2). Although they achieve the same purpose, the
active and passive transformations result in coordinate
1b
on ofEq. (10) is well known. Ifwe translate the expression
of Eq. (11) by applying the operator
transformations in the opposite directions. In the Appen-
dix, the effects of active and' passive transformations are T(u,u) = exp[ - i(uP, + uP,)], (13)
discussed in detail. then the translated form becomes
As in the case of the rotation group, the standard method T(u,u),p(x,y) = [(x - u) ± i(y - u)]m. (14)
of studying this group is to find an operator which com-
mutes with all three of the above generators. It is easy to This is an eigenstate of a rotation around the point x = u
check that p', defined as andy = u.
If m = I, the basis vector for the representation diagonal
P'=P;+Pi, (8) inL, is
commutes with all three generators. Thus one way to con-
struct representations of the E (2) group is to solve theequa- X + iY )
tion W, = ( X~iY . (15)

(9)
The generators of the E (2) transformation matrices take the
using the differential forms of P, and P, given in Eq. (7). form

-I ~).
This partial differential equation can be separated in the
0
polar, Cartesian, parabolic, or elliptic coordinate system.·
If we are interested in constructing representations diag- L,=G
onal in p' and P, and P" we use the Cartesian coordinate 0
system. On the other hand, if we are interested in represen-

-)
(16)
tations diagonal in p, and L" the polar coordinate system 0 0
is appropriate. These possibilities have been considered in
the past as summarized in Table II.
Inonu and Wigner constructed infinite-dimensional uni-
PI=G 0
0
-I ,

0
P2=G 0
0
~I).
tary representations by solving the differential equation of There is a nonsingular matrix which transforms the col-
Eq. (9) with nonvanishing k '.7 On the other hand, the finite- umn vector in Eq. (2) to W, of Eq. (15). The above 3 X 3
dimensional non unitary representations are based on the matrices are applicable to functions and not to the coordi-
solutions with k' = O. We are interested here in the repre- nates. For this reason, each of them is related to the nega-
sentations diagonal in L, with k 2 = 0, because they de- tive of its counterpart in Eq. (5) through a similarity trans-
scribe the wave functions for massless particles observed in formation.
the real world. 4 •s In addition, the mathematics required for The matrices of Eq. (16) satisfy the commutation rela-
studying this case is much easier than that for the general tions for the E (2) group given in Eq. (6), and
case discussed in the original paper of Inonu and Wigner, 7
P;+P;=O, (17)
which is a reflection ofEq. (5). In addition, P, andP, satisfy
P; =P; =P,P,=O, (18)
Table II. Representations of the E(2) group.
because
Diagonal
in
Unitary infinite
dimensional
Nonunitary finite
dimensional
(! Y(X ± iy) = (~)'tx ± iy)
P, and P, Wigner in 1939 (Ref. I) Trivial =bx±iy)=O. (19)
L) Ioonu and Wigner in 1953 Han et 01. in 1982 axiJy
(Ref. 7) (Ref. 5) We expect that the procedure of constructing represen-
tations for larger values of m will be similar to the m = 1

1039 Am. 1. Phys., Vol. 52, No. 11, November 1984 HanetaJ. 1039
GROUP CONTRACTIONS 375

case. It would be an interesting exercise to construct explic- For the present purpose, we can consider the case where
it matrices for an arbitrary integer value of m.' The basic z is large and approximately equal to the radius of the
vector W, given in Eq. (15)is not unlike the spherical vector sphere, and write
whose components are (x ± iy) and z in which z is replaced
by I.
(27)
III. E (2) GROUP AS A LIMITING CASE OF 0 (3)
The column vectors on the left- and right-hand sides are,
The discussion given in Sec. II on E (2) is quite similar to
respectively, the coordinate vectors on which the 0 (3) and
the case ofO (3). Like 0 (3). the E (2) group has three genera-
E (2) transformations are applicable. We shall use A for the
tors. and its coordinate transformation matrices are 3 X 3.
3 X 3 matrix on the right-hand side. Then, in the limit of
Then how are these two groups related? This fundamental
large R,
question was addressed by Inonu and Wigner in their the-
ory of group contraction. 7.9 L, =A -IL,A,
One way to define this problem is to consider a sphere P, = (IIR IA -IL,A. (28)
with a large radius. Imagine a football field on the surface
ofthe Earth covering the north pole. A player can run from
P, = - (IIR IA-IL,A,
east to west, and from north to south. He can also turn where L,. P" and P, are given in Eq. (5). This limiting
around at any point in the field. Indeed, the player can procedure is called the contraction of 0 (3) to E(2).
performE (2) transformations on himself. Strictly speaking, During the contraction process, L, remains invariant.
however, these translations and rotations are all rotations For L, and L" the upper-right parts of the above matrices
on the spherical surface of the Earth. remain unchanged, except for a sign change in L, due to
Let us start with the familiar 0 (3) rotation operator Eq. (24). However, the lower-left parts become zero. In
which can be written as terms of the spherical harmonics Y;"(O,,p ), the above limit-
exp[ - i(aL , + (3L, + OL,)]. (20) ing procedure is the same as replacing cos 0 by I, and
(e ± '. sin 0 ) by (x ± iy).
We are interested in the effect of this rotation on the Hat
Another interesting property of E (2) which is inherited
football field. L, generates rotations around the north pole. from 0 (3) is the concept of equivalence class. 10.11 In 0(3),
L" which generates rotation around the y axis, takes the rotations by the same angle around different axes belong to
form the same equivalence class. This notion is translated to ro-

- -i(Z~-xi.).
L,= (21) tations by the same angle around different points on the xy
ax az plane forming an equivalence class. It is not difficult to
form a geometrical visualization of the concept of equiv-
Therefore, for large values of the radius R, z = Rand
alence classes applicable to 0 (3) and E(2).
L = - iR ~ = RP, or P, = IIIR )L. (22)
- ax -
IV. INTERNAL SPACE· TIME SYMMETRIES OF
If we rotate the system by angle {3 around the y axis, the RELATIVISTIC PARTICLES
resulting translation on the E (2) plane is
In describing a free relativistic particle, we specify first
{3L, = - i{3R ~ = uP,. (23) its mass, momentum, and energy. After determining its
ax four-momentum, we should ask what other space-time de-
with grees of freedom the particle has. This question was sys-
tematically formulated by Wigner in his 1939 paper on re-
u =(3R or{3= uiR.
presentations of the inhomogeneous Lorentz group or the
Likewise Poincare group.' The subgroups of the Poincare group
L, = - RP,. and a = viR. (24) governing internal space-time symmetries are called the
little groups. '
The parameters u and v are discussed in Sec. II.
The little group is generated by a maximal subset of J,
Ifwe write the commutation relations for the 0 (3) group
and K, which leaves the four-momentum invariant, where
as
J, is the generator of rotations around the ith axis. and K, is
[L"IIIR )L,] = i(IIR )L" the boost generator along the ith axis. These generators
[L,.IIIR )L,] = - i(IIR )L ,• (25) satisfy the commutation relations
III1R )L ,,(IIR )L,] = illlR )'L,. [J,.J;) =iEijJ,.
it is easy to see that these expressions, in the large R limit, [J"K}) =iE.,K" (29)
become the commutation relations for the E (2) group given
[K"K}) = -iE,},J,.
in Eq.12).
Let us translate the above limiting procedure into the The little groups for massive and massless particles are
language of matrices. 3 X 3 rotation matrices applicable to locally isomorphic to 0 (3) and E (2), respectively, and they
coordinate variables (x.y,z) are well known. They are gener- have been discussed in separate papers in this Journal. 2.'
After studying the procedure of obtaining the E (2) group as

J
ated by L, of Eq. (5) and
a limiting case of 0 (3), we are naturally led to consider the

G~ ~ ~ ~ D
internal symmetry group of massless particles as a limiting
L, = L, =( i (26) case of the o (3)-like little group for massless particles.
If a massive particle is at rest, the symmetry group is

1040 Am. J. Phys., Vol. 52, No. II, November 1984 Haneto[. 1040
376 CHAITER VlI

generated by the angular momentum operators J" J" and Let us start with a massive particle at rest with its mass
J,.' These operators do not change the four-momentum of M. Then the little group is isomorphic to the 0(3) group
the particle at rest. If this particle moves along the Z direc- generated by J" J" and J,. Ifwe boost this massive particle
tion, J l remains invariant, and its eigenvalue is the helicity. along the z direction, its momentum and energy will be-
However, we have been avoiding in the past the question of come Pand E = [P' + M ']'1', respectively. The boost ma-

. (I~
what happens to J, and J" particularly in the infinite-mo- trix is
mentum limit.
There are no Lorentz frames in which massless particles
o o
are at rest. The little group for a massless particle moving I o (34)
along the z direction is generated by fl' N" and N,,' where
B(P) = o E/M
N, =K,-J" o PIM
(30) Under this boost operation, J, given in Eq. (33) remains
N,=K 2 +J,· invariant:
The four-momentum of the massless particle remains in- J; = BJ,B -, =J,. 1351
variant under transformations generated by these opera- However, the boosted J, and J, become
tors. These generators satisfy the commutation relations
J; =IEIMiJ,-IPfM)K"
[N"N,] =0, (36)
[f"N,]=iN2 , (31) J; =IEIMiJ, + (P/M)K,.
Because the Lorentz boosts in Eqs. (35) and (36) are similar-
[J 3 ,N,] = - iN"
ity transformations, the J' operators still satisfy the 0 (3)
which are identical to those for the E (2) group given in Eq. commutation relations:
(6). J 3 is like the generator of rotation while N, and N2 are
(37)
like the generators of translations in the two-dimensional
plane. These translationlike operators are known to gener- Since the quantities in Eq. (36) become very large as the
ate gauge transformations' momentum increases. we introduce new operators:
Einstein's energy-momentum relation of Eq. (I) clearly
G,= -(MfEiJ;,
indicates that a massive particle becomes like a massless
138)
particle as the momentum/mass ratio becomes infinite. We G,=IM/EiJ;.
are thus led to the suspicion that the 0 (3)-like internal sym-
In terms of these new operators, wecan write the 0 (3) com-
metry for massive particles will become the E (2)-like sym-
mutation relations of Eq. 137) as
metry for massless particles, and that this limiting proce-
dure will be like the group contraction procedure discussed [J"G,] = - iG"
in Sec. III. [J"G,] =iG" (39)
If we boost the massive particle along the z direction, J,
will remain invariant. The analysis of Sec. III leads us to [G"G,] = - (M /E)'J,.
expect thatJ, andJ, for massive particle at rest will become The quantity (M IE)' becomes vanishingly small if the
N, and - N" respectively, in the infinite-momentum/ mass becomes small or the momentum becomes very large.
zero-mass limit, just like the contraction of 0 (3) resulting in In this limit,
E(2)." G,~N, and G,~N" 140)
Let us carry out an explicit calculation to justify the
above expectation starting with a massive particle at rest. If where
we use the four-vector convention 0 -i
X e = Ix,y,z,t), x e = (x,y,z, - II,
with c = I, the generators of Lorentz transformations ap-
plicable to this coordinate space are
(32)
N'~(~ 0
0
0
0
0
0 1} (41)

)
0 0 0 0

,~(! ~ K'~U 1)
0 0

N'~(:
0 -i 0 0
0 -i
0 0 0
0 o .

,,{ )
0 0 0 0
0 0 0
0 0 0

~,
1
The fact that the above N matrices generate gauge transfor-
0 0 o K _ 0 0 0 mations has been extensively discussed in the litera-
,- ("
(33) ture. 4 ,5,13
-1 0 0 ~' ~ 0 0
0 0 0 0 Indeed, rotations around the axes perpendicular to the
momentum become gauge transformations in the infinite-
-i 0 0 0

"~(l V K'~(:
momentum/zero-mass limit.
0 0 0 0
0
0
where J i and K,
0
0
0 0
6 V
generate rotations and boosts, respective-
V. FURTHER PHYSICAL APPLICATIONS OF
GROUP CONTRACfIONS
The purpose of this section is to indicate that there are
ly. many other interesting applications of group contractions.

1041 Am. J. Phys., Vol. 52, No. ll, November 1984 Han etal. 1041
GROUP CONTRAcrIONS 377

In Sec. III, we studied the contraction ofO (3) toE(2) using let
the notion of a plane tangent to a spherical surface. The
f(l+ 1)"",(f+I)2 (47)
concept of this tangent plane plays a very important role in
many branches of physics and engineering dealing with In their 1976 paper,15 Misra and Maharana made an
curved surfaces. interesting observation that, when the scattering angle is
For example, let us consider the surface of the hyperbola very small, we can replace (L ~ + L ;) by
in a three-dimensional space spanned by x, y, and t: R'(P~ +P;)=(/+!)', (48)
(et)' - x' - y' = const, (42) in the spirit of Eqs. (22) and (24) with suitable redefinitions
where e is a constant and may become very large. This is a for R and Pi' In view of the discussion given in Sec. IV, we
description of the 0 (2,1) group consisting of Lorentz boosts can readily letR be (Po/M). 15 Thus, for a given value of Po,
along the x and y directions and rotations on the xy plane. the eigenvalue of
The pedagogical value of this group has been amply dis-
cussed in Ref. 14.
(P~ + Pi) (49)
We can now consider a plane tangent to the surface at will give a measure of f. In view of the discussion given in
x = y = O. As the constant e becomes very large, the por- Sec. II, thi~ new parameter will be that of the Bessel func-
tion of surface in which (x' + y') is finite becomes fiat and tion. Thus, for large values of f, the Legendre polynomial
coincides with the tangent plane. It is then not difficult to becomes the Bessel function"·:
imagine that transformations on this tangent plane are Ga- PI (cos 0 )--Jo\aqO), (50)
lilean transformations.
In order to see this point, let us start with coordinate where
transformations of the group 0 (2,1) generated by L, ofEq. a=Po/M.
(5), and The parameter q now measures f, and becomes continuous
o for large values of PD'

KI=G ~i o
o
(43) The above Bessel-function form is commonly used for
studying high-energy data. 15.1. It is interesting to note that
the transition from the use of the Legendre polynomials for
applicable to the column vector (x,y,et). 14 K, andK, are the low-energy processes to that ofthe Bessel functions in high-
generators of Lorentz boosts along the x and y directions, energy scattering is a group contraction of 0(3) to E(2).
respectively.
The column vector (x,y,et) can be written as ACKNOWLEDGMENTS

C)=G : DG)'
We are grateful to O. W. Greenberg for providing the
(44) following information. In 1962, Wigner gave a series of
lectures on the representations of the Poincare group at
Then, as e becomes very large, the circumstance is identical Trieste and Istanbul. At one of his lectures, the problem of
to the case of Eqs. (26) and (28). The resulting transforma- obtaining the E (2)-like little group for massless particles as
tion matrix becomes a limiting case of the 0 (3)-like little group for massive parti-
cles was informally discussed as an unsolved problem, al-

(y'X') = (COSO
- sinO though this discussion was not included in Wigner's lecture
smO cos 0 (45) notes published in Ref. 17. We would like to thank M.
t' 0 o Parida for bringing Ref. IS to our attention and for explain-
This form is a rotation on the xy plane followed by Galilean ing its content to us.
boosts along the x andy directions. Indeed, special relativi-
ty becomes Galilean relativity in the limit of large e.'·· APPENDIX
Let us consider another example. In scattering processes We study in this Appendix active and passive transfor-
in which two incoming particles collide with each other mations on the E (2) plane. Let us define the transforma-
resulting in two particles moving in different directions, we tions given in Eqs. (2) and (3) to be active. This transforma-
commonly use the Legendre polynomials Plicos 0 ) to de- tion first rotates the coordinate point (x,y) by angle 0
scribe the dependence on the scattering angle. The quan- around the origin. It then translates the rotated point by u
tum number f is the angular momentum around the scatter- and v along the x and y directions, respectively.
ing center, and can be regarded as a measure of the On the other hand, if we perform the same rotation on
incoming momentum multiplied by the impact parameter. the function
When particles move slowly, it is sufficient to consider g(x,y) = (x + iyr = r"'e im "', (AI)
only two or three lowest values f. On the other hand, when
the particles move with speed very close to that oflight, the using L, given in Eq. (7),
scattering becomes predominantly forward and becomes (e ~ i8L')g(X,y) = r"'eimIO ~ 81. (A2)
like the Fraunhofer diffraction. In this case, we have to deal
with large values of f. One way to approach this problem is If we apply the translation operators on the above expres-
sion,
to start from the operator
(e ~ il"p, + 'P,le ~ iOL')g(X,y) = (x" + iy")m = g(x" ,y"), (A3)
(46)
where
with the eigenvalue f (f + I). For large values of f, we can x· = (x - u)cos 0 + (y - v)sin 0,
ignoreL, whose eigenvalue is usually not larger than I, and y" = - (x - u)sin 0 + (y - v)cos O.

1042 Am. J. Phys., Vol. 52, No. 11, November 1984 Hanetal. 1042
378 CHAPTER VII

The above linear transformation can also be written as 'E. Innnu and E. P. Wigner. Proc. Natl. Acad. Sci. U. S. A. 39. 510119531.
"d. Han, y, S. Kim. and D. Son. University of Maryland Physics Publi-
sin 0
cosO ~U(:i~O; ~: ~O~~ ~)~) .
cation No. 83-141 (19831.
"'For a pedagogical reformulation of the theory of group contraction.
o including a discussion of the E(2) group as a contraction of the 0(3)
group, see R. Gilmore, Lie Groups and Lie Algebras. and Some o/Their
(A4)
Applications (Wiley. New York. 1974).
The matrix in this expression is precisely the inverse of that lOA. S. Wightman, in Dispersion Relations and Elementary Particles, edit-
of the active transformation matrix of Eq. (3). ed by C. De Witt and R. Omnes (Hermann, Paris. 1960).
liThe concept of equivalence class was discussed in detail for the 0 (3) case
by Wigner. This concept survives in £12) after contraction. See E. P.
Wigner, Group Theory, and Its Applications to the Quantum Theory of
IE. P. Wigner, Ann. Math. 149.40(1939). See also V. Bargmann and E. P. AlOmicSpectra (Academic, New York, 19591.
Wign". Proc. Natl. Acad. Sci. U. S. A. 34. 211 (1946). "D. Han. Y. S. Kim. and D. Son. Phy,. Lett. 131B. 327119831.
'V. S. Kim. M. E. Noz. and S. H. Dh. Am. J. Phy •. 47. 892 (1979); D. Han "S. Weinberg. Phy,. Rev. 134. B882 11964); 135. 81049119641.
and Y. S. Kim. ibid. 49.115711981). "Y. S. Kim and M. E. Noz. Am. J. Phys. 51. 368119831.
'V. S. Kim. M. E. Noz. and S. H. Dh. J. Math. Phy•. 20.1341 (1979); D. I~S. P. Misra and J. Maharana. Phys. Rev. D 14.1330976).
Han. M. E. Noz. Y. S. Kim. and D. Son. Phy•. Rev. D 25.1740 11982). IhR. Blankenbecler and M. L. Goldberger, Phys. Rev. 126, 766( 1962). See
'D. Han and Y. S. Kim. Am. J. Phy •. 49. 348 11981). a1,o S. 1. Wallace. Phys. Rev. D 8. 1846119731 and D 9. 40611974).
'D. Han. Y. S. Kim. and D. Son. Phy,. Rev. D 25. 46111982); 26. 3717 17E. P. Wlgner, in Group Theoretical Concepts and Methods in Elementary
(1982). Particle Physics. edited by F. Glirsey (Gordon and Breach. New York.
'Po Wintemitz and I. Fri,. Yad. Fiz.l. 88911965) [Sov. J. Nucl. Phy,. I. 1962); and in Theoretical Physics, edited by A. Salam (International
63611965)). Atomic Energy Agency, Vienna. 19621.

Am. J. Phys. S2 (11), November 1984 @ 1984 American Association of Physics Teachers 1043
GROUP CONTRACfIONS 379

Eulerian parametrization of Wigner's little groups and gauge transformations

in terms of rotations in two-component spinors
D. Han
SASC Technologies. Inc.. 5809 Annapolis Road. Hyattsville. Maryland 20784
Y.S. Kim
Department of Physics and Astronomy. University ofMaryland. College Park. Maryland 20742
D. Son
Department of Physics. Kyungpook National University. Daegu 635. Korea

(Received 20 February 1986; accepted for publication 30 April 1986 )

A set of rotations and Lorentz boosts is presented for studying the three-parameter little groups of
the Poincare group. This set constitutes a Lorentz generalization of the Euler angles for the
description of classical rigid bodies. The concept of Lorentz-generalized Euler rotations is then
extended to the parametrization ofthe E(2)-like little group and the 0(2,1 )-Iike little group for
massless and imaginary-mass particles, respectively. It is shown that the E(2)-like little group for
massless particles is a limiting case of the 0 (3) -like or 0 (2,1 ) -like little group. A detailed analysis
is carried out for the two-component SL(2,c) spinors. It is shown that the gauge degrees of
freedom associated with the translationlike transformation of the E(2)-like little group can be
traced to the SL( 2,c) spins thatfail to align themselves to their respective momenta in the limit of
large momentum and/or vanishing mass.

I_ INTRODUCTION both massive and massless particles, we observe that the

transformation that changes a given four-momentum can be
The Euler angles constitute a convenient parametriza- carried out in many different ways. However, as Wigner ob-
tion of the three-dimensional rotation group. The Euler served in 1957, the resulting spin orientation depends on the
kinematics consists of two rotations around the z axis with way in which the transformation is performed and on the
one rotation around the y axis between them. The first ques- mass ofthe particle. 2 Forinstance, when a particle with posi-
tion we would like to address in this paper is what happens if tive helicity is rotated, the helicity remains unchanged. As
we add a Lorentz boost along the z direction to this tradition- far as the momentum is concerned, we can achieve the same
al procedure. Since the rotation around the z axis is not af- purpose by performing a simple boost. However, this boost
fected by the boost along the same axis, we are asking what is does not leave the helicity invariant. Furthermore, the
the Lorentz-generalized form of the rotation around the y change in the direction of spin depends on the mass.
axis. Indeed, the difference ·between the rotation and boost
Since the publication ofWigner's fundamental paper on was studied for massless photons by Kupersztych, 3 who ob-
the Poincare group in 1939,' a number of mathematical served that this difference amounts to a gauge transforma-
techniques have been developed to deal with the three-pa- tion. In this paper, we extend the kinematics of Kupersztych
rameter little groups that leave a given four-momentum in- to include massive and imaginary-mass particles. We shall
variant. Our second question is why we do not yet have a show that this extended kinematics con.titutes the above-
standard set of transformations for Wigner's little groups. mentioned Lorentz generalization of the Euler rotations.
In this paper, we combine the first and second questions. We then study the extended Kupersztych kinematics
One ofWigner's little groups is locally isomorphic to O( 3). using the SL(2,c) spinors. Among the four two-component
Furthermore, the Euler angles constitute the natural lan- SL(2,c) spinors, two of them preserve the helicity under
guage for spinning tops in classical mechanics, while boosts in the zero-mass limit, as was noted by Wigner in
Wigner's little groups describe the internal space-time sym- 1957. However, the remaining two do not preserve the heli-
metries of relativistic particles, including spins. It is thus city in the same limit. We show that these helicity nonpre-
quite natural for us to look for a possible Eulerian parametri- serving spinors are responsible for gauge degrees of freedom
zation of the three-parameter little groups. contained in the E(2)-like little group for photons.
As far as massive particles are concerned, the traditional In Sec. II, we work out the Kupersztych kinematics for
approach to this problem is to go to the Lorentz frame in massive particles. It is pointed out that this new kinematics is
which the particle is at rest, and then perform rotations equivalent to the traditional 0(3)-like kinematics in which
there. ' Then, its four-momentum is not affected, but the di- the particle is rotated in its rest frame. We show in Sec. III
rection of its spin becomes changed. This operation, how- that the E(2)-like little group for massless particles is the
ever, is not possible for massless or imaginary-mass parti- infinite-momentum/zero-mass limit of the O( 3) -like little
cles. group discussed in Sec. II. In Sec. IV, we discuss the continu-
In order to construct a Lorentz kinematics that includes ation of the transformation matrices for the 0 (3 ) -like little

@ 1986 American Institute of Physics 2228

Reprinted from J. Math. Phys. 27,2228 (1986).

380 CHAPTER VII

II. KINEMATICS OF THE O(3)-LlKE LITTLE GROUP

The Euler rotation consists of a rotation around the y
axis preceded and followed by rotations around the z axis. If
the boost is made along the z axis, the rotations around the z
axis are not affected. In this section, we discuss a Lorentz
generalization of the rotation around the y axis and its rela-
tion to the O(3)-like little group for massive particles.
Let us start with a massive particle at rest whose four-
momentum is
(O,O,O,m) . (I)

We use the four-vector convention: X " = (x,y, z, I). We can

boost the above four-momentum along the z direction with
0- __"...,....- -8
velocity parameter a:
P=m(O,O,a/(I-a')I!',I/(l-a') ' /' ). (2)
FIG. I. Lorentz-generalized Euler rotations. The traditional Euler parame-
trization consists of two rotations around the z axis with one rotation The four-by-four matrix which transforms the four-vector of
around the yaxis between them. If we add a Lorentz boost along thez axis, Eq. (I) to that of Eq. (2) is
the two rotations around Ihez axis are not affected. The rotation around the
y axis can be Lorentz-generalized in the following manner. Ifwe boost the

(0~1 001 ~ ~a')I/2)'

system along the z direction , we are dealing with the system with a nonzero
four-momentum along the same direction. The four-momentum p can be A(a) = (3)
rotated around the y axis by angle fJ. The same result can be achieved by 1/(l-a' )' /2 a/(l
boost S - I . However. these two transformations do not produce the same
effect on the spin. The most effective way of studying this difference is to
o a/(I-a')'/2 1/(l -a')'/2
study the transformation SR, which leaves the initial four-momentum in- Let us next rotate the four-vector of Eq. (2) using the
variant. rotation matrix:
COS 0 o sin 0

group to the case of imaginary-mass particles. R(O) =

(
0 0
o (4)
In Sec. V, we study the transformation properties of the - sm o cos 0
four two-component spinors in the SL (2,e) regime. It is o o 0
shown that in the limit of infinite momentum and/or zero This rotation does not alter the helicity of the particle.'
mass, two of the SL(2,e) spinors preserve their respective As is specified in Fig. I, we can achieve the same result
helicities, while the remaining two do not. We note, in Sec. on the four-momentum by applying a boost matrix. How-
VI, that four-vectors can be constructed from the four two- ever, unlike the rotation ofEq. (4), this boost is not a heli-
component SL (2,e) spinors. It is shown that the origin of the city-preserving transformation.' We can study the differ-
gauge degrees of freedom for photons can be traced to the ence between these two transformations by taking the
spinors that refuse to align themselves to the momentum in product of the rotation and the inverse of the boost. This
the infinite-momentum/zero-mass limit. inverse boost is illustrated in Fig. I, and is represented by

_ ( I + 2(sinh(A ~2)COS(0 /2»)' o - (sinh(A /2»)' sin 0 - (sinh A~COS(O /2»)

I o (5)
S- _ (sinh(A / 2»)' sin 0 o 1+ 2(sinh(A /2)sin(0 /2»)' (sinh A)sin(O / 2) ,
- (sinh A) COS(O /2) o (sinh A)sin(O / 2) cosh A

where

A = 2Itanh - l (a sin(O /2»)] . (6)

This matrix depends on the rotation angle 0 and the velocity parameter a, and becomes an identity matrix when the particle is
at rest with a = O.
Indeed, the rotation R(O) followed by the boost S(a,O) leaves the four-momentum p ofEq. (2) invariant:

P = D(a,O)P, (7)

where
D(a ,O) = S(a ,O) R(O) .
The multiplication of the two matrices is straightforward, and the result is

2229 J . Math. Phys" Vol. 27. No. 9. September 1986 Han , Kim, and Son 2229
GROUP CONTRACfIONS 381

_(I -(I - ;')U'/ZT o -u/T

o
au/T
o
)
D(a,e) - u/T (8)
o 1+ u2 /2T au'/2T '
au/T o - au'/2T 1+ au 2 /2T

where convenient for studying the relation between the Euler an-
gles and the parameters of the O(3)-like little group.
u = - 2(tan(e12)) and T= I + (I-a 2 )(tan(e/2))'.
We have so far discussed the transformations in the x-z
This complicated expression leaves the four-momentum Pof plane. It is quite clear that the same analysis can be carried
Eq. (Z) invariant. Indeed, if the particle is at rest with van- out in the y-z plane or any other plane containing the z axis,
ishing velocity parameter a, the above expression becomes a This means that we can perform rotations R,(¢» and
rotation matrix. As the velocity parameter a increases, this R, (,p), respectively, before and after carrying out the trans-
D matrix performs a combination of rotation and boost, but formations in the x-z plane. Indeed, together with the veloc-
leaves the four-momentum invariant. e,
ity parameter a, the three parameters ¢>, and ,p constitute
Let us approach this problem in the traditional frame- the Eulerian parametrization of the 0(3)-like little group.
work. 1 The above transformation is clearly an element of the
0(3 )-like little group that leaves the four-momentum P in-
variant. Then we can boost the particle with its four-momen- 111. E(2)-LlKE LITTLE GROUP FOR MASSLESS
tum Pby A-I until the four-momentum becomes that ofEq. PARTICLES
(1), rotate it around the y axis, and then boost it by A until
Let us study in this section the D matrix ofEq. (8) as the
the four-momentum becomes PofEq. (2). It is appropriate
particle mass becomes vanishingly small, by taking the limit
to call this rotation in the rest frame the Wigner rotation."
of a~l. In this limit, the D matrix ofEq. (8) becomes
The transformation of the O( 3 )-like little group constructed
in this manner should take the form o -u
D(a,e) =A(a)W(e')[A(a)] 1
(9) I o ou ) (12)
o 1- u'/2 u'/2 .

ceo
where W is the Wigner rotation matrix
o -u'/2 1+ u2 /2

:)
0 sin e*
0
Wee') =
e* o .
0 (10)
- Slll e' 0 cos
0 0 0 I
8'
e
We may call * the Wigner angle. The question then is ISoof----------
whether D ofEq. (9) isthesameasDofEq. (8). In order to 170 0 1 - - - -_ __
answer this question, we first take the trace of the expression
given in Eq. (9). The similarity transformation of Eq. (9)
assures us that the trace of Wbe equal to that of D. This leads
to
e* _
- cos
-I (1-
I+
(l-a )(tan(e/2))')
2

a')(tan(e 12))2 .
(I _
(II)

It is then a matter of matrix algebra to confirm that D ofEq.

(9) and that ofEq. (8) are identical.
We have plotted in Fig. 2 the Wigner rotation angle e'
asa function of the velocity parametera. Here * becomes e e
when a = 0, and remains approximately equal to when a is e
smaller than 0.4. Then e'
vanishes when a~ I. Indeed, for a
given value of e, it is possible to determine the value of e'
that is the rotation angle in the Lorentz frame in which the
particle is at rest.
The D matrix in the traditional form ofEq. (9) is well 1 0 ° t - - - - -_ _ __
known. 1 However, the fact that it can also be derived from
theciosed-loopR( e) and S(a,e) suggests that it has a richer o 0.2 04 06 O.S 1.0 a
content. For instance, the closed-loop kinematics does not
FIG. 2. Wigner rotation angle ver'iUS lab-frame rotation angle. We have
have to be unique. There is at least one other closed-loop
plotted {} * as a function of a for various \aluesoffl using Eq. ( 11 ).0 = 0 * at
kinematics that leaves the four-momentum invariant.' The (1 = o. ();' IS nearly equal tn (, for moderate values of a, but it rapidly ap-
Kupersztych kinematics, which we are using in this paper, is rroacht.'~ 0 as It becomes 1

2230 J. Math. Phys., Vol. 27. No.9, September 1986 Han, Kim, and Son 2230
382 CHAPTER VII

After losing the memory of how the zero-mass limit was invariant, with a greater than I. Although particles with
taken, it is impossible to transform this matrix into a rotation imaginary mass are not observed in the real world, the trans-
matrix. There is no Lorentz frame in which the particle is at formation group that leaves the above four-momentum in-
rest. If we boost this expression along the z direction using variant is locally isomorphic to 0(2, I) and plays a pivotal
the boost matrix role in studying noncompact groups and their applications
0 0 in physics. This group has been discussed extensively in the

PI(I~P""}
literature. II
0
We are interested here in the question of whether the D
0 1/(1 _ P 2) 1/2
matrix constructed in Sees. II and III can be analytically
0 P/(I_P 2)1/2 1/(I_P2)1/2 continued to a > I, Indeed, we can perform the rotation and
(13) boost of Fig. I to obtain the D matrix of the form given in Eq,
D remains form-invariant: (8), if a is smaller than a o where
D'(u) =B(P)D(u)[B(P)]-1 =D(u'), (14) a~ = [I + (tan(0/2»)2]/(tan(0 /2»)2. (17)
where As a increases, some elements ofthe D matrix become singu-
u' = [(I + P)/(I - Pll 112 u . lar when T vanishes or a = ao' Mathematically, this is a
simple pole that can be avoided either clockwise or counter-
The matrix of Eq. (12) is the case where the Kupersz-
clockwise, However, the physics of this continuation process
tych kinematics is performed in the x-z plane. This kinema- requires a more careful investigation,

)
tics also can be performed in the y-z plane. Thus the most
One way to study the D transformation more effectively
general form for the D matrix is is to boost the spacelike four-vector of Eq, (16) along the z
0 -u direction to a simpler vector
-v ,v
D'' '-O
(O,O,im,O) , (18)
1- (u 2 + v2)/2 (u 2+ v2)/2 .
v· _ (u 2 + v2)/2 I + (u 2 + v2)/2
using the boost matrix ofEq. (13) with the boost parameter
P = I/a, Consequently, the D matrix is a Lorentz-boosted
(15)
form of a simpler matrix F:
The algebraic property of this expression has been discussed D=B(l/a)F(A)[B(I/a)j-I. (19)
extensively in the literature. I.'" If applied to the photon
four-potential, this matrix performs a gauge transforma- Here F is a boost matrix along the x direction:
tion.'" The reduction of the above matrix into the three-by-
cosh A o 0
three matrix representing a finite-dimensional representa-
I 0
Sin~A)
tion of the two-dimensional Euclidean group has also been
discussed in the literature.'
F(A) =
(
~ o I o ' (20)

Let us go back to Eq. (9). We have obtained the above sinhA o 0 cosh A
gauge transformation by boosting the rotation matrix W giv- where
en in Eq. (10). This means that the Lorentz-boosted rota-
-2(a'-I)I/'tan(e/2)
tion becomes a gauge transformation in the infinite-momen- tanh'! = ,
tum and/or zero-mass limit. This observation was made I + (a' - I) (tan(e /2»)'
I + (a: - I) (tan(e /2»)' ,
(21)
earlier in terms of the group contraction of 0 (3) to E (2),9.10 cosh'! =
which is a singular transformation. We are then led to the 1- (a- - 1) (tan(e /2»)'
question of how the method used in this section can be ana-
lytic, while the traditional method is singular. Ifwe add the rotational degree offreedom around the z axis,
The answer to this question is very simple. The group the above result is perfectly consistent with Wigner's origi-
contraction is a language of Lie groups.9.10 The parameter a nal observation that the little group for imaginary-mass par-
we use in this paperis not a parameter of the Lie group. If we ticles is locally isomorphic to 0(2, I). I
use TJ as the Lie-group parameter for boost along the z direc- We have observed earlier that Ihe D matrix ofEq, (8)
tion, it is related toa by sinh TJ = a/(I - a 2) II'. However, can be analytically continued from a = I to I < a < ao, At
this expression is singular at a = ± I. Therefore, the con- a = ao, some of its elements are singular, If a> ao, cosh'! in
tinuation in a is not necessarily singular. We shall continue Eqs. (20) and (21) become negative, and this is not accepta-
the discussion of this limiting process in terms of the SL(2,c) ble,
spinors in Sec. VI. One way to deal with this problem is to take advantage
of the fact that the expression for tanh A in Eq, (21) is never
singular for real a greater than I. This is possible if we
IV. 0(2,1)-LlKE LITTLE GROUP FOR IMAGINARY-MASS change the signs of both sinh'! and cosh A when we jump
PARTICLES from a < a o to a > a". Indeed, the continuation is possible if
it is accompanied by the reflection of x and t coordinates,
We are now interested in transformations that leave the After taking into account the reflection of the x and t coordi-
four-vector of the form nates, we can construct the D matrix by boosting F of Eq,
P=im(0,0,a/(a'-1)1:2,I/(a'-I)1/2) (16) (20). The expression for the D matrix for a> a" becomes

2231 J. Math. Phys .• Vol. 27, NO.9, September 1986 Han, Kim, and 50n 2231
GROUP CONTRACfIONS 383

o ulT
D=(I_~/T I o
-aulT )

-uiT o I +2/[(a'-I)T] 2al[(a~-l)T] .

\22)

-aulT o -2a/[(a'-I)T] 1- 2a'[(a' -l)T]

This expression cannot be used for the a-> I limit, but can be corresponding to WofEq. (10) is
used for the a-> 00 limit. In the limit a-> 00, P of Eq. (16)
0* _ (COS(O*/2) - Sin(O"/2»)
(25)
becomes identical to Eq. (18), and the above expression be- W( ) - sin(O*/2) cos(O*/2) •
comes an identity matrix. As for the question of whether D of
where the rotation angle 0 * is given in Eq. (II).
Eq. (22) isananalyticcontinuationofEq. (8), the answer is
Using the formula of Eq. (9), we can calculate the D
"no," because the transition from Eq. (22) to Eq. (8) re-
matrix for the SL(2,e) spinors. The D matrix applicable to
quires the reflection of the x and taxes.
the undotted spinors is

V. PARTICLES WITH SPIN-l (1 + a)UI2ff) ,

lIff
The purpose of this section is to study the D kinematics
(26)
ofspin-! particles within the framework ofSL(2,e). Let us
study the Lie algebra of SL (2,e) (see Refs. 12 and 13): where Tand u are given in Eq. (8). The D matrix applicable
to the dotted spinors is
[S"Sj I = iEijkSk, [S, ,Kj I = iEijk Kk ,
(I-a)U/2ff) .
[K"Kj I = - iE"kSk , (23)
I Iff
where S, and K, are the generators of rotations and boosts, (27)
respectively. The above commutation relations are not in- We can obtain D'-' from D'+' by changing the sign of a.
variant under the sign change in S" but they remain invar- Both Dt+' andDH become WofEq. (25) when a = O.
iant under the sign change in K,. For this reason, while the If the D transformation is applied to the X ± and ± X
generators of rotations are S, = \0'" the boost generators can spinors,
take two different signs K, = ( ± ) (i/2)O',.
(28)
Let us start with a massive particle at rest, and the usual
normalized Pauli spinors X + and X _ for the spin in the posi- the angle between the momentum and the directions of the
tive and negative z directions, respectively. If we take into spins represented by X + and X_ is
account Lorentz boosts, there are four spinors. We shall use O'=tan- t ((I-a)tan(OI2»), (29)
the notation X + to which theboostgeneratorsK, = (i/2)eT,
areapplicable,-andx ± towhichK, = - (i/2)O', areappli- which becomes zero as a->1. On the other hand. in the case
cable. There are therefore four independent SL(2,e) spin- of X _ and X+' the angle becomes
ors."·" In the conventional four-component Dirac equa- 0" =tan-'((I +a)tan(11/2»). (30)
tion, only two of them are independent, because the Dirac
equation relates the dotted spin Drs to the undotted counter-
parts. However, the recent deve10pment in supersymmetric
theories, 14 as well as some of more traditional approaches, I' Momentum
indicates that both physics and mathematics become richer
in the world where all four of SL (2,e) 'pinars are indepen-
dent. In the Appendix, we examine the nature of the restric-
tion the Dirac equation imposes on the four SL(2,e) spinors.
As Wignerdid in 1957,' we start with a massive particle
whose spin is initially along the direction of the momentum.
Boost

.+x;
The boost matrix, which brings the SL (2,e) spinors from the
zero-momentum state to that of p, is
A' ± '(a)
x_
= ((( I ± a)/ (\ =ta»)"· 0 ) x,
O
((I =t a)/(l ± a»)"· ' X~

(24)
FIG. 3. Lorentz-boosted rotation~ or the four SL( 2.e) spillors. Ifthe part i-
where the superscripts ( + ) and ( - ) are applicable to the deveiocilY is zero, all rlw!o.pinors rolate likc the Pauli !'Ipinors. A!'I the parti-
clespeed approaches that nflight. two of the !o.pin .. linc up with the momen-
undotted and dotted spinars, respectively. In the Lorentz
tum, while the remaining two refu!o.e 10 do!.o. Tho!.e !.pincmth4t line up are
frame in which the particle is at rest, there is only one rota- gauge-invariant spinOfs. Tho!o.e that do 1I0t are not gauge imariant. and they
tion applicable to both sets of spinors. The rotation matrix form the origin of the gauge degrees of freedom for photon four-potentials.

2232 J. Math. Phys .• Vol. 27. No.9. September t986 Han, Kim, and Son 2232
384 CHAPTER VII

In the limit of a--+I, this angle becomes 0" where pure boost along the x axis:
0, = tan-'(2(tan(0/2»)). (31)
F' ± '(A) = ( cosh(A 12) ± sinh (A 12»)
Indeed, the spins represented by X _ and X+ refuse to align ±sinh(AI2) cosh(A 12) , (32)
themselves with the momentum. This result is illustrated in
Fig. 3. where A is given in Eq. (21).
There are D transformations for the a > I case. In the Fora<ao,wecancontinuetouse D'+'and D'-'given
special Lorentz frame in which the four-momentum takes in Eq. (26) and Eq. (27), respectively. However, fora >ao,
the form ofEq. ( 18), the D transformation becomes that of a the D matrix is
I

D '±' (a,u)
£I _ ( (a'-I)/'(tan(OI2WFT
-
±(a± 1)/(a=F I»)'/'/FT\
(33)
± (a=F I)/(a ± I)))/'/~ - T (a' - I) '/2(tan(O 12WFT ) .

The above expression becomes an identity matrix when The four-potentials are gauge dependent, while the spinors
a-+ 00, as is expected from the result of Sec. IV. The D matri- allowed in the Dirac equation are gauge invariant. There-
ces of Eq. (33) are not analytic continuations of their coun- fore, it is not possible to construct four-potentials from the
terparts given in Eqs. (26) and (27), because the continu- Dirac spinors.
ation procedure, which we adopted in Sec. IV and used in On the other hand, there are gauge-dependent SL(2,c)
this section, involves reflections in the x and t coordinates. spinors, which are given in Eq. (37). They disappear from
the Dirac spinors because N _ vanishes in the a-+ I limit.
VI_ GAUGE TRANSFORMATIONS IN TERMS OF However, these spinors can still play an important role if
ROTATIONS OF SPINORS they are multiplied by N +, which neutralizes N _. Indeed,
It is clear from the discussions of Secs. III-V that the we can construct unit vectors in the Minkowskian space by
limit a-+ I can be defined from both directions, namely from taking the direct products of two SL(2,c) spinors
a< I and from a > I. In thelimita-+I,D1+landD'-'ofEq. - X +X + = (l,i,O,O), X-X- = (I, -i,O,O),
(26) and Eq. (27) become (38)
X+X- = (0,0,1,1), X.X+ = (0,0,1,-1).

1
u), D'.'=( I
-u . ~) (34) These unit vectors in one Lorentz frame are not the unit
vectors in other frames. For instance, if we boost a massive
After going through the same procedure as that from Eq. particle initially at rest along the z direction, Ix +X +) and
(12) to Eq. (15), we arrive at the gauge transformation ma- Ix -X -) remain invariant. However, Ix +X _) and Ix -X +)
trices· acquire the constant factors [( I + a)/(1 - a) ]'/2 and
I iV) [(I-a)/(1 +a)]'/',respectively. We can therefore drop
D1+'(u,v) = (0
U -
I ' Ix-X +) when we go through the renormalization process of

°
(35) replacing the coefficient [(1 +a)/(1-a)])/'by I for par-
D'-'(u,v) = ( I . 1), ticles moving with the speed of light.
-u-w The D(u,v) matrix for the above spinor combinations
applicable to the SL (2,c) spinors, where the D ' ± , are appli- should take the form
cable to undotted and dotted spinors, respectively.
(39)
The SL(2,c) spinors are gauge invariant in the sense
that +'
where D' and D 1-' are applicable to the first and second
spinors ofEq. (38), respectively. Then
D'+'(u,v)X + = X +, D'-'(u,v)X _ = X _ . (36)
On the other hand, the SL( 2,c) spinors are gauge dependent
D(u,v)( -Ix+x+» = lX+x+) + (u+iv)lx+x_) ,
in the sense that D(u,v)lx_x_) = Ix-x-'> + (u -iv)lX+x-) , (40)

D1+'(u,v)X _ = X _ + (u - iv)X + , D(u,v)lx+x_) = Ix+x-'>·

(37)
D1-'(u,v)X + = X ~ - (u + iV)X .. The first two equations of the above expression correspond
to the gauge transformations on the photon polarization vec-
The gauge-invariant spinors of Eq. (36) appear as polarized
tors. The third equation describes the effect of the D trans-
neutrinos in the real world. However, where do the above
formation on the four-momentum, confirming the fact that
gauge-dependent spinors stand in the physics of spin-l parti-
D( u,v) is an element of the little group. The above operation
cles? Are they really responsible for the gauge dependence of
is identical to that of the four-by-four D matrix ofEq. (15)
electromagnetic four-potentials when we construct a four-
on photon polarization vectors.
vector by taking a bilinear combination of spinors?
The relation between the SL(2,c) spinors and the four-
vectors has been discussed for massive particles. However, it VII. CONCLUDING REMARKS
is not yet known whether the same holds true for the mass- We studied in this paper Wigner's little groups by con-
less case. The central issue is again the gauge transformation. structing a Lorentz kinematics that leaves the four-momen-

2233 J. Math. Phys. Vol. 27. No.9. September '986 Han, Kim, and Son 2233
GROUP CONTRACfIONS 385

I I rotation and boost generators take the form

Massive Massless
between
Slow Fast la,
( 0 o ) ,K,. = (U/2)aj o )
I I s=
, 0 (AI)
Energy I Einstein's I !U j
- (i/2)a j ,

E=1 E=p These generators accommodate both signs of the boost gen-
2m
Momentum
I E=.Jm2+p2 I erators for the SL(2,c) spinors. In this representation, y, is

Spin, Gouge S3 I Wigner's I S3 diagonal, and its eigenvalue determines the sign of the boost
generators,

I little Group I Gouge Trons.

In the Weyl representation, the D matrix should take the
Helicity SI S2
form
D1+I(U,U)
FIG. 4. Significance of the concept ofWigner's little groups. The beauty of D(u,u) = ( 0 (A2)
Einstein's special relativity is that the energy-momentum relation for mas-
sive and slow particles and that for massless particles can be unified.
applicable to the Dirac spinors, which, for the particle mov-
Wigner's concept of the little groups unifies the internal space-time symme-
tries of massive and massless particles. ing along the z direction with four-momentum p, are

U(p) = ( N +X -:- ), V(p) = (±N-;-X_) , (A3)

±N_X+ N+x-
where the + and - signs in the above expression specify
tum of a particle invariant. This kinematics consists of one positive and negative energy states, respectively. Here N +
rotation followed by one boost. Although the net transfor- and N _ are the normalization constants, and
mation leaves the four-momentum invariant, the particle
spin does not remain unchanged. The departure from the (A4)
original spin orientation is studied in detaiL As the momentum/mass becomes very large, N _/N +
For a massive particle, this departure can be interpreted becomes very small. From Eqs, (36) and (37), we can see
as a rotation in the Lorentz frame in which the particle is at that the large components are gauge invariant while the
rest. For massless particles with spin-I, the net result is a small components are gauge dependent. The gauge-depen-
gauge transformation. For a spin-l particle, there are four dent component of the Dirac spinor disappears in the a--> I
independent spinors as the Dirac equation indicates. As the limit; the Dirac equation becomes a pair of the Weyl equa-
particle mass approaches zero, the spin orientations of two of tions. If we renormalize the Dirac spinors of Eq. (A3) by
the spinors remain invariant. However, the remaining two dividing them by N +, they become
spinors do not. It is shown that this noninvariance is the
cause of the gauge degrees of freedom massless particles with
(AS)
spin-I.
In 1957,2 Wigner considered the possibility of unifying
the internal space-time symmetries of massive and massless For y, = ± 1, respectively. The gauge-dependent spinors
particles by noting the difference between rotations and disappear in the large-momentum/zero-mass limit. This is
boosts. Wigner considered the scheme of obtaining the inter- precisely why we do not talk about gauge transformations on
nal symmetry by taking the massless limit of the internal neutrinos in the two-component neutrino theory.
space-time symmetry groups for massive particles. In the The important point is that we can obtain the above
present paper, we have added the gauge degrees of freedom decoupled form of spinors immediately from the most gen-
and spinors that refuse to align themselves to the momentum eral form of spinors by imposing the gauge invariance. This
in the massless limit. The result of the present paper can be means that the requirement of gauge invariance is equivalent
summarized in Fig, 4, While Einstein's special relativity uni- to y, = \, as was suspected in Ref. 8.
fies the energy-momentum relations for massive and mass-
less particles, Wigner's little group unifies the internal space-
time symmetries of massive and massless particles. IE. P. Wigner, Ann. Math. 40,149 (1939); V. Bargmann and E. P. Wigner,
Proe. Natl. Acad. ScL (U.S.A.) 34, 211 (1946); E. P. Wigner, in Theoreti-
ACKNOWLEDGMENTS cal Physics, edited by A. Salam (I.A.E.A .. Vienna, 1963).
2E. P. Wigner, Rev. Mod. Phys. 29, 255 (1957). See also C. Kuang-Chao
We are grateful to Professor Eugene P. Wigner for a and L. G. Zastavenco, Zh. Exp. Tear. Fiz. 35,1417 (1958) [SOy. Phys.
very illuminating discussion on his 1957 paper2 on transfor- JETP8, 990 (1959)]; M. Jacob and G. C. Wick, Ann. Phys. (NY) 7, 404
( 1959).
mations that preserve helicity and those that do not. We 1J. Kupersztych, Nuovo Cimento B 31, I (1976); Phys. Rev. D 17, 629
would like to thank Dr. Avi I. Hauser for explaining to us the (1978).
content of his paper on possible imaginary-mass neutrinos. 14 .lThe Wigner rotation is frequently mentioned in the literature because two
successive boosts result in a boost preceded or followed by a rotation. We
believe, however, that the Wigner rotation should be defined in the Lor-
APPENDIX: SL(2,c) SPINORS IN THE DIRAC SPINORS
entz frame In which the particle is at rest, in view of the fact that theO(3)-
We pointed out in Sec. V that the four-component Dirac like little group is the rotation group in the rest frame. See R. Gilmore, Lie
Gruups, Lie Algebras, and Some of Their Applications ill Physics (Wiley,
equation puts a restriction on the SL( 2,c) spinors, Let us see
New York, 1974); A. Le Yaouanc, L. Oliver, O. Pene, and J. C. Raynal,
how this restriction manifests itself in the limit procedure of Phys. Rev. D 12,2137 (1975); A. Ben-Menahem, Am. J. Phys. 53,62
a~l. In the Weyl representation oflhe Dirac equation, the ( 1985). The concept of rotations in the rest frame played an important

2234 J. Math. Phys., Vol. 27, No.9, September t 986 Han, Kim, and Son 2234
386 CHAPTER VII

role in the development of quantum mechanics and atomic spectra. See L. 100. Han, y'S. Kim,andD. Son,Phys. Lett. B131. 327 (1983); D. Han, Y.
H.Thomas, Nature 117, 514 (1926); Philos. Mag. 3, I (1927). S. Kim, M. E. Noz, and D. Son, Am. I. Phys. 52, 1037 (1984).
'D. Han and Y. S. Kim, Am. I. Phys. 49, 348 (1981); D. Han, Y. S. Kim, lly. Bargmann, Ann. Math. 48, 568 (1947); L. Pukanszky, Trans. Am.

and D. Son, Phys. Rev. D 31, 328 (1985). Math. Soc. 100, 116 (1961); L. Serterio and M. Toller, Nuovo Cimento
(,E. P. Wigner, Z. Phys. 124, 665 (1948); A. S. Wightman, in Dispersion 33,413 (1964); A. O. Barnt and C. Fronsdal, Proc. R. Soc. London Ser. A
Relations and Elementary Particles, edited by C. De Witt and R. Dmnes 287,532 (1965); M. Toller, Nuovo Cimento 37, 631 (1968); W. I. Hol-
(Hermann, Paris,1960); M. Hamermesh, Group Theory (Addison-Wes- man and L. C. Biedenharn, Ann. Phys. (NY) 39, I (1966); 47, 205
ley, Reading, MA, 1962); E. P. Wigner, in Theoretical Physics, edited by (1968); N. Makunda, I. Math. Phys. 9, 50,417 (1968); 10, 2068, 2092
A. Salam (LA.E.A., Vienna, 1962); A. Janoer and T. Jenssen, Physica 53, (1973); K. B. Wolf, I. Math. Phys. 15, 1295, 2102 (1974); S. Lang,
I (1971); 60, 292 (1972);1. L. Richard, NuovoCimento A8,485 (1972); SL(2,r) (Addison-Wesley, Reading, MA, 1975).
H. P. W. Gottlieb, Proc. R. Soc. London Ser. A 368, 429 (1979). 12M. A. Naimark, Am. Math. Soc. Transl. 6, 379 (1957); I. M.Gel'fand, R.
'So Weinberg, Phys. Rev. 134, B 882 (1964); 135, BI049 (1964). A. Minlos, and Z. Va. Shapiro, Representations o/the Rotation and Lor-
liD. Han, Y. S. Kim, and D. Son, Phys. Rev. D 26,3717 (1982). entz Groups and their Applications (MacMillan, New Yark, 1963).
9E. Inonu and E. P. Wigner, Proc. Nat!. Acad. Sci. (U.S.A.) 39, 510 nyu. V. Novozhilov, introduction to Elementary Particle Theory (Perga-
(1953); D. W. Robinson, Helv. Phys. Acta 35, 98 (1962); D. Korff, I. mon, Oxford, 1975).
Math. Phys. 5, 869 ( 1964); S. Weinberg, in Lectureson Particles and Field 14S.1. Gates, M. T. Grisaru, M. Rocek, and W. Siegel, SuperJpaces (Benja-
Theory, Brandeis 1964, Yol. 2, edited by S. Deser and K. W. Ford (Pren- min/Cummings, Reading, MA, 1983). See also A. Chados, A. I. Hauser,
tice-Hail, Englewood Cliffs, NJ, 1965); J. D. Talman, Special Functions, and y. A. Kostelecky, Phys. Lett. B ISO, 431 (1985); H. van Dam, Y. J.
A Group Theoretical Approach Based on Lectures by E. P. Wigner (Benja- Ng, and L. C. Biedenharn, ibid. 158,227 (1985).
mm, New York, 1968); S. P. MisraandJ. Maharana, Phys. Rev. 0 14,133 !5L C. Biedenharn, M. Y. Han, and H. van Dam, Phys. Rev. D 6, 500
(1976). ( 1972).

2235 J. Math. Phys., Vol. 27, NO.9, September 1986 Han, Kim, and Son 2235
GROUP CONTRACTIONS 387

Cylindrical group and massless particles

Y.S. Kim
Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742
E. P. Wigner
Joseph Henry Laboratories, Princeton University. Princeton. New Jersey 08544
(Received 30 September 1986; accepted for publication 31 December 1986)
It is shown that the representation of the E(2)-like little group for photons can be reduced to
the coordinate transformation matrix of the cylindrical group, which describes movement of a
point on a cylindrical surface. The cylindrical group is isomorphic to the two-dimensional
Euclidean group. As in the case ofE(2), the cylindrical group can be regarded as a contraction
of the three-dimensional rotation group. It is pointed out that the E(2)-like little group is the
Lorentz-boosted O( 3) -like little group for massive particles in the infinite-momentum/zero-
mass limit. This limiting process is shown to be identical to that of the contraction of 0(3) to
the cylindrical group. Gauge transformations for free massless particles can thus be regarded
as Lorentz-boosted rotations.

I. INTRODUCTION approximation of a spherical surface with large radius,' the

In their 1953 paper,' Inonu and Wigner discussed the contraction to the cylindrical group is a tangent-cylinder
contraction of the three-dimensional rotation group [or approximation. Using this result, together with the fact that
0(3)] to the two-dimensional Euclidean group [or E(2)]. the representation of the E(2)-\ike little group reduces to
Since the little groups governing the internal space-time that of the cylindrical group, we show that the gauge degree
symmetries of massive and massless particles are locally iso- of freedom for massless particles comes from Lorentz-boost-
morphic to 0(3) and E(2) respectively,' it is quite natural ed rotations.
for us to expect that the E(2)-like little group is a limiting In Sec. II, we discuss the cylindrical group and its iso-
case of the 0(3)-like little group.' morphism to the two-dimensional Euclidean group. Section
The kinematics of the O( 3 )-like little group for a mas- III deals with the E(2 )-like little group for photons and its
sive particle is well understood. The identification of this isomorphism to the cylindrical group. It is shown in Sec. IV
little group with 0 (3) can best be achieved in the Lorentz that the cylindrical group can be regarded as all equatorial-
frame in which the particle is at rest.' In this frame, we can belt approximation ofthe three-dimensional rotation group,
rotate the direction of thespin without changing the momen- while E(2) can be regarded as a north-pole approximation.
tum. Indeed, for a massive particle, the little group is for the In Sec. V, we combine the conclusions of Sec. III and Sec. IV
description of the spin orientation in the rest frame. to show that the gauge degrees of freedom for free massless
The kinematics of the E(2)-like little group has been particles are Lorentz-boosted rotational degrees offreedom.
somewhat less transparent, because there is no Lorentz
frame in which the particle is at rest. While the geometry of 11. TWO-DIMENSIONAL EUCLIDEAN GROUP AND
E(2) can best be understood in terms of rotations and trans- CYLINDRICAL GROUP
lations in two-dimensional space, there is no physical reason
The two-dimensional Euclidean group, often called
to expect that the translationlike degress of freedom in the
E( 2), consists of rotations and translations on a two-dimen-
E(2)-Jike little group represent translations in an observable
sional Euclidian plane. The coordinate transfonnation takes
space. In fact, the translationlike degrees of freedom in the
the fonn
little group are the gauge degrees of freedom. 4 Therefore, in
the past, the correspondence between the E(2)-like little x' =xcosa - ysina + u, (2.1 )
group and the two-dimensional Euclidean group has been y' = x sin a + y cos a + u.
strictly algebraic.
This transformation can be written in the matrix fonn as
In this paper, we fonnulate a group theory of a point
-sina
[~.X'] = [COS
~][~]
moving on the surface of a circular cylinder. This group is
a
locally isomorphic to the two-dimensional Euclidean group. si~a cos a (2.2)
.
We show that the transformation matrix of the little group o
for photons reduces to that ofthe coordinate transformation The three-by-three matrix in the above expression can be
matrix of the cylindrical group. The cylindrical group there- exponentiated as
fore bridges the gap between E(2) and the E(2)-like little
group. E(u, u, a) = exp[ - i(uP, + uP2 ) ]exp( - iaL 3 ),
As in the case of E(2), we can obtain the cylindrical (2.3)
group by contracting the three-dimensional rotation group. whereL 3 is the generator of rotations, and P, and P2 generate
While the contraction of 0(3) to E(2) is a tangent-plane translations. These generators take the form

© 1987 American Institute of Physics 1175

Reprinted from J. Math. Phys. 28,1175 (\986).

388 CHAPI'ER VII

0]
[0 00,
then x and y can be written as
-i
L3=i x = cos ,p, y = sin ,p, (2.14)

i]
000 and the transformation ofEq. (2.12) takes the form

].
(2.4)

oo 0] a
-:~:a ~][~:~]=[ :~:~~:;~
COS
0 0 [0
P, = [ 0 0 0 , P,O i, [ sma
o 0 0 0 o 0 u I z z+ucos,p+vsin,p
and satisfy the commutation relaiions (2.15)
[P"P,] = 0, [L 3 ,P, ] = iP" [L 3 ,P,] = - iP" (2.5) We shall see in the following sections how this cylindrical
which form the Lie algebra for E(2). group describes gauge transformations for massless parti-
The above commutation relations are invariant under cles.
the sign change in P, and P,. They are also invariant under
Hermitian conjugation. Since L3 is Hermitian, we can re-
place P, and P, by III. E(2)-L1KE LITTLE GROUP FOR PHOTONS
Q, =- (P,)', Q, = - (P,)t, (2.6) Let us consider a single free photon moving along the z
direction. Then we can write the four-potential as
respectively, to obtain
A "'(x) =APeiw(z-t), (3.1)
[Q"Q,J = 0, [L 3 ,Q'] = iQ" [L 3 ,Q,J = - iQ,.
(2.7) where
These commutation relations are identical to those for E (2) A ,. = (A I.A ,.A ,.A o).
given in Eq. (2.5). However, Q, and Q, are not the genera- The momentum four-vector is clearly
tors of Euclidean translations in the two-dimensional space.
Let us write their matrix forms: pi' = (O,O,UJ,UJ). (3.2)
Then, the little group applicable to the photon four-potential

Q, = [~ ~ ~],
i 0 0
Q, = [~ ~ ~].
0 i 0
(2.8)
is generated by
-i 0
HereL 3 is given in Eq. (2.4). As in the case ofE(2), we can
consider the transformation matrix J,- [1 0
0
0
0
0
0 11
C{u,v,a) = C(O,O,a)C{u,v,Q), (2.9)

il
(3.3)
where C{ O,O,a) is the rotation matrix and takes the form
0 -i 0 0

C(O.O,a) = exp( - iaL 3 ) = [

COS

Si~ a
a -sina
cos a
o ~] ,
N'~[1 0
0
0
0
0
0
N,~ [1 0
i
i
-i
0
0 11
(2.10)
These matrices satisfy the commutation relations:
[J,.N,] = iN" [J"N,] = - iN" [N,.N,] = 0,
C(u,V,Q) = exp[ - i(uQ, + vQ,) J = [~ ~ ~]. which are identical to those for E(2). From these genera-
(3.4)

(2.11) tors, we can construct the transformation matrix:

The multiplication of the above two matrices results in the D(u,v,a) = D(O,O,a)D(u,v,O), (3.5)
most general form of C( u,v,a). If this matrix is applied to
where
the column vector (x,y,z) , the result is
D(u,v,O) = exp[ - i(uN , + vN,) J,

[ c~sa
sma :o::n a ~][;] = [: :~:: ~; ::::] . D(O,O,a) =R(a) =exp[ -iaJ,].
u v I z z + ux + vy We can now expand the above formulas in power series, and
(2.12)
the results are
This transformation leaves (x' + y') invariant, while z can
vary from - 00 to + 00. For this reason, it is quite appro-
priate to call the group of the above linear transformation the COS a -sina
cylindrical group. This group is locally isomorphic to E(2). [ sin a cos a
~ (3.6)
If, for convenience, we set the radius of the cylinder to be R(a)= o
unity, o
(x'+y') = I, (2.13) and

,,76 J. Math. Phys .• Vol. 28, No.5, May 1987 Y. S. Kim and E. P. Wign., 1176
GROUP CONTRACTIONS 389

D(u,v,O) o o
0]o [0
0 0

+~V')/2]' N~ ~
o -u 0 0 N o oI 0

1 - (u'
_
-v
+ v')/2
(u' + v')/2
(u'
1 + (u' + v')/2
[: 0
0
0
0
~' 2=V2~ o
o o 11
(3.12)
(3.7) As a consequence, D( u,v) takes the form
When applied to the four-potential, the above D matrix per-
0 0

D("D){~,
forms a gauge transformation: while R(a) is the rotation
matrix around the momentum. 0

1]
(113 )
The D matrices of Eq. (3.5) have the same algebraic vlV2 I
property as that for the Ematrices discussed in Sec. II. Why, 0 0
then, do they look so different? In the case of the 0(3)-like andR (a) remains the same as before. It is now clear that the
little group, the four-by-four matrices of the little group can four-by-four representation of the little group is reduced to
be reduced to a block diagonal form consisting of the three- one three-by-three matrix and one trivial one-by-one matrix.
by-three rotation matrix and one-by-one unit matrix.' Is it If we use )3' lV" and lV, for the three-by-three portion of the
then possible to reduce the D matrices to the form which can four-by-four J" N" and N, matrices, respectively, then
be directly compared with the three-by-three E or C matrices
discussed in Sec. II?
.I, = L" lV, = (lIV2)Q" lV, = (lN2)Q,. (3.14)
One major problem in bringing the D matrix to the form Now the identification ofE(2)-like little group with the cy-
of the E matrix is that theD matrix is quadratic in the u and v lindrical group is complete.
variables. In order to attack this problem, let us impose the
Lorentz condition on the four-potential:
IV. THE CYLINDRICAL GROUP AS A CONTRACTION OF
~ (A "(x)) =P"A"
axil
(x) = 0, (3.8) 0(3)
The contraction of 0(3) to E(2) is well known and
resulting in A, = Ao. Since the third and fourth components
discussed widely in the literature. ' The easiest way to under-
are identical, the N, and ;V, matrices of Eq. (3.3) can be
stand this procedure is to consider a sphere with large radius,
replaced, respectively, by
and a small area around the north pole. This area would
appear likea flat surface. We can then make Euclidean trans-
o o formations on this surface, consisting of translations along
o o the x andy directions and rotations around any point within
o o this area. Strictly speaking, however, these Euclidean trans-
formations are SO (3) rotations around the x axis, y axis, and
o o
around the axis which makes a very small angle with the z
axis.
At the same time, the D(u,v,O) ofEq. (3.7) becomes
Let us start with the generators of 0(3), which satisfy

Di.,D) ~ [~ : ~ ;]
the commutation relations:
(4.1)
(310)
Here L3 generates rotations around the north pole, and its
matrix form is given in Eq. (2.4). Also, L, and L, take the
This matrix has some resemblance to the representation of form
the cylindrical group given in Eq. (2.11).'
o
In order to make the above form identical to Eq. (2.11),
we use the light cone coordinate system in which the combi- L, = [~ o ~ /], L, = [~ ~ ~]. (4.2)
nationsx,y, (z + t)1V2, and (z - t)/V2 are used as the coor- o 0 0 -/
dinate variables." In this system the four-potential of Eq. For the present purpose, we can restrict ourselves to a small
(3.1) is written as region near the north pole, where z is large and is equal to the
radius of the sphere R, and x and yare much smaller than the
A I' = (A,A,,(A, + A o )/V2,(A, - Ao)IV2). (3.11 )
radius. We can then write
The linear transformation from the four-vector of Eq. (3.1)
to the above expression is straightforward. According to the
(4.3)
Lorentz condition, the fourth component of the above
expression vanishes. We are thus left with the first three
components. The column vectors on the left- and right-hand sides are,
During the transformation into the light-cone coordi- respectively, the coordinate vectors on which the E(2) and
nate system, J, remains the same. Ifwe take into account the o (3) transformations are applicable. We shall use the nota-
fact that the fourth component of A I' vanishes, N, and N, tion A for the three-by-three matrix on the right-hand side.
become In the limit oflarge R,

1177 J. Math. Phys .. Vol. 28. No.5. May 1987 Y. S. Kim and E. P. Wigner 1177
390 CHAPTER VII

L3=ALy4 -',
P, = (IIR)AL,A -', (4.4)
P, = - (IIR)AL,A -'.
This procedure leaves L, invariant. However, L, and L, be-
come the P, and P, matrices discussed in Sec. II. Further-
more, in terms of P" P, and L" the commutation relations
for 0(3) given in Eq. (4.1) become
[L 3 ,P,] = iP" [L 3 'p,] = - iP"
(4.5)
[P"P,] = -i(IIR)'L3.
In the large-R limit, the commutator [P" P,] vanishes, and
the above set of commutators becomes the Lie algebra for
E(2).
We have so far considered the area near the north pole
FIG. 1. Contraction of the three-dimensional rotation group to the two-
wherezis much larger than (x' + y') 1/2. Let us next consid- dimensional Euclidean group and to the cylindrical group. The rotation
er the opposite case, in which (x' + y') ,/, is much larger around the z axis remains unchanged as the radius becomes large. In the
than z. This is the equatorial belt of the sphere. Around this case of E( 2), rotations around they and x axes become translations in the x
and - y directions. respectively, within a flat area near the north pole. In
belt, x and y can be written as
the case of the cylindrical group, the rotations around they andx axes result
x=Rcos</J, y=Rsin</J. (4.6) in translations in the negative and positive z directions, respectively, within
a cylindrical belt around the equator.
We can now write
o
[C~s</J]
SID</J =
[IIR
0 IIR (4.7)
z 0 o mation. Therefore if the boost matrix takes a diagonal form
as in the case ofEq. (4.3) or Eq. (4.7), we should be able 10
to obtain the vector space for the cylindrical group discussed
in Sec. II. The three-by-three matrix on the right-hand side obtain N, and N, by boosting J, and J" respectively, along
the z direction. 7
of the above expression is proportional to the inverse of the
matrix A given in Eq. (4.3). Thus in the limit oflarge R, Indeed, in the light-cone coordinate system, the boost
matrix takes the form
L3 =A -'Ly4,
Q, = - (lIR)A -'L,A, (4.8) o o
o
oo 1'
Q, = (lIR)A -'L,A. o (5.1 )
In terms of L 3 , Q" and Q" the commutation relations for o R
0(3) given in Eq. (4.1) become o o 11R
[L 3 ,Q,] = iQ" [L 3 ,Q,] = - iQ"
[Q"Q,] = - i(lIR)'L 3, (4.9) _( 1+11)'/'
R- 1_11 '
which become the Lie algebra for E(2) in the large-R limit.
The contraction of 0(3) to E(2) and to the cylindrical where 11 is the velocity parameter of the particle. Under this
group is illustrated in Fig. I. boost, J 3 will remain invariant:
Ji=BJ3B-'=J3 • (5.2)
Here J, and J, in the light-cone coordinate system take the
form
V. E(2)-LIKE LITTLE GROUP AS AN INFINITE-

il
MOMENTUM/ZERO-MASS LIMIT OF THE O(3)-LIKE 0 0

J,- ~ [:
LITTLE GROUP FOR MASSIVE PARTICLES
0 -i
If a massive particle is at rest, the symmetry group is
0
generatedbytheangularmomenlumoperatorsJ"J"andJ3 •
0 -i 0
If this particle moves along thez direction, J 3 remains invar-
iant, and its eigenvalue is the helicity. However, what hap- (5.3)

J'-~[ :
0

~']o .
pens to J, andJ" particularly in the infinite-momentum lim-
it? 0 0
In orJer to tackle this problem, let us summarize the v2 -I 0 0
results of the preceding sections. The generators of the E (2)- i 0 0 0
like little group can be reduced to those of the cylindrical
group. The cylindrical group can be obtained from the three- If we boost this massive particle along the z direction, the
dimensional rotation group through a large-radius approxi- boosted J, and J, become

1178 J. Math. Phys., Vol. 28, No.5, May 1987 Y. S. Kim and E. P. Wign., 1178
 GROUP CONTRACTIONS 391

North Pole VI. CONCLUDING REMARKS

The isomorphism between the two-dimensional Euclid-
Equatorial
ean group and the little group for massless particles is well
Belt
known and well understood. However, the isomorphism in
this case does not mean that they are identicaL We have
shown in this paper that the E(2)-like little group can be
reduced to the identity group and the cylindrical group
which is isomorphic to E(2). As in the case ofE(2), we can
Identical obtain the cylindrical group by contracting the three-dimen-
sional rotation group. This contraction procedure is identi-
cal to the LorentL boost of the O(3)-like little group for a
R=~
1-{3
massive particle at rest to the E(2)-like little group for a
massless particle. The result of the present paper is summar-
FIG. 2. Here are E(2), the E(2)-ilke lutle group for mas~less particles, and ized in Fig. 2.
the cylindncal group. The corre~pondence between E( 2) and the E( 2 )-like
little group i~ isomorphic but not identicaL The cylindrical group IS identI-
cal to the E{2 )-like little group. Both E( 2) and the cylindncal group can be IE. Inonu and E. P. Wlgner, Prm:. Natl. Acad. Sci. USA 39,5\0 (1953); J
regarded a~ contractlOI1!-> of 0(3) in the large-radlU~ limIt. The Lorentz D. Talman, Special Functio//I, A Group Theorelicil/ Approach Based on
boost of the 0(3 )-like little group for a ma~'>lve panicle at re~t 10 the E(2)- Lectures by E. P. WiXflef (BCTlpnIlTl, New York, 1(68). See abo R. Gil-
like little group for a massless partick IS exactly the ~ame as the contraction more, Lie GrouP\' Lie A/gehrt/I, and Some o/Thcir Applications in Phy~io
orO( 3) 10 the cyhndrical group. The radiu'> of the ~phere In thi~ ca~e can be (Wiley, New York, 1974).
identifieda,!!1 j (3)/(1 -(3))'" 2E. P. Wigner, Ann Math 40, 149 (19J9)~ V. Bargmann and E. P. Wigncr,
Proc. NaiL Acad. SCI. USA 34, 211 (1<)46); E. P. Wigner, Z. Phys.124, 665
( 194R); A. S. WIghtman. in Di.\pasion Relations and Elementary Particle~,
edited by C. Oe Witt and R. Omnes (Hermann, Paris, 1960); M. Hamer-
mc~h, Group Theory (Addb,on-Wesley, Reading, MA, 1962); E. P
Wlgner, in Theoretical Physin, edited by A. Salam (IAEA, Vienna, 1962);
A. Janner and r. Jenssen, Phy~u..:a 53, I (1971): 60, 292 (1972); J. L. RI-
0 0 chard, Nuovo Omento A 8, 485 ( 1(72); H. P. w. Gottlieb, Proc. R. Soc.

J; =BJ,B ,~ ,'2 1 [:
0
0
iR
i/R
0
,;o1' London Ser. A 368, 429 (1979); H. van Dam, Y. J Ng, and L. C. Bieden-
ham. Phy,- Lett B 158, 227 ( 1Qg5) For a recent textbook on thiS subject,
see Y. S. Kim and M. E. Noz, Theory and AppiicatlOnx oj/he Poincare
Group (Reidel, Dordlecht, Holland, 1(86).
0 - ilR 0 0 lE. P. W!gner, Rev. Mod. Phy~. 29, 255 (1957). See also D. W. Robinson,
(SA) He\v. Phys. Acta 35, 98 ( 1962); D. Korff,!. Math. Phy,. 5. 869 ( 1964); S.

Tl
Weinberg, in LeClurewn Particles and Field Theory, Brandeis 1964, edited
0 ilR

~~PR
by S. Deser and K. W. Ford (Prentice-Hall, Englewood Cliffs, NJ, 1965)
0 0 Vol 2; S. P. MisraandJ. Maharana, Phys. Rev. D 14,133 (1976); O. Han,
J, =BJ,B Y. S. Kim. and D. Son, J. Math. Phys. 27, 2228 (1986).
0 0 4S. Weinberg, Phys Rev. B 134, 882 (1964); B 135, 1049 (1964); 1. Ku-
ilR 0 0 perzstych, N uovo Omento B 31, I (1976); D. Han and Y> S. Kim, Am. 1.
Phys. 49, 348 (1(81); 1. J. van der Bij, H. van Dam, and Y. 1. Ng, Physica
A \16. 307 (1982). D. Han. Y. S. Kim, and D. Son, Phys. Rev. D 31, 328
Because of the Lorentz condition, the iR terms in the fourth ( 1985).
column of the above matrices can be dropped. Therefore, in ~D. Han, Y. S Kim, and D. Son, Phys. Rev. D 26,3717 (1982). For an

the large-R limit which is the limit of large momentum, earlier effort to study the E( 2 )-like little group in terms of the cylindrical
group, sec L. J. Boya andJ. A. de Azcarraga, An. R. Soc. Esp. Fis. Quim. A
N,= -(1IR)J;, N 2 =I1IR)J;, (5.5) 63, 143 (1967). We are grateful to Professor Azcarraga for bringing this
paper to our attention.
where N, and N, are given in Eq. (3.12). This completes the {'P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949); L. P. Parker and O. M.
proof that the gauge degrees of freedom in the E(2)-like Schmieg, Am. J. Phys. 38, 218,1298 (1970); Y. S. Kim and M. E. Noz, J.
little group for photons are Lorentz-boosted rotational de- Math. Phy,. 22, 2289 (1981).
70. Han, Y. S. Kim, and D. Son, Phy~. Lett. B 131, 327 (1983); D. Han, Y.
grees of freedom. The limiting process is the same as the
S. Kim, M. E. Noz, and D. Son, Am J. Phys. 52, 1037 (1984). These
contraction of the three-dimensional rotation group to the authors studied the correspondence between the contraction of O( 3) to
cylindrical group. E(2) and the Lorentz boost of the O(3)-like little group.

1179 J. Math. Phys., Vol. 28, No.5, May 1987 Y. S. Kim and E. P. Wigner 1179
Chapter VIII

Localization Problems

In quantum mechanics, solutions of the SchrOdinger equation carry a localizable

probability interpretation. On the other hand, solutions of Maxwell's equations do
not necessarily have a probability interpretation. It was in fact pointed out by
Newton and Wigner in 1949 that there is no covariant Hermitian position operator
for photons. In 1962, Wightman established this as a mathematical theorem. This is
known as the photon localization problem.
On the other hand, from oscilloscope screens, we observe what can be described as
localized photons. Are they really photons? The answer to this question is "No."
They are localized light waves. It was shown by Han, Kim, and Noz in 1987 that it
is possible to localize light waves in a covariant manner with a probability
interpretation. However, this light wave cannot be given a particle interpretation in
terms of the creation and annihilation operators. Two different mathematical
algorithms are employed for photons and light waves, and the difference can be seen
very easily.

393
LOCALIZATION PROBLEMS 395

Reprinted from REVIEWS OF MODERN PHYSICS, Vol. 21, No.3, pp. 400-406, July. 1949
Printed in U. S. A.

Localized States for Elementary Systems

T. D. NEWTON AND E. P. WIGNER
Palmer Physical Lahoratory, Princeton University, Princeton, New Jersey

It is attempted to formulate the properties of localized states on the basis of natural invariance require-
ments. Chief of these is that a state, localized at a certain point, becomes, after a translation, orthogonal
to all the undisplaced states localized at that point. It is found that the required properties uniquely define
the set of localized s~tes for elementary systems of nOD-zero mass and arbitrary spin. The localized func-
tions belong to a continuous spectrum of an operator which it is natural to call the position operator. This
operator has automatically the property of preserving the positive energy character of the wave function
to which it is applied (and it should be applied only to such wave functions). It is believed that the develop-
ment here presented may have applications in the theory of elementary particles and of the collision matrix.

INTRODUCTION case of a 1I"-meson. Qualitatively, a ".-meson differs in

I T is well known that invariance arguments suffice to
obtain an enumeration of the relativistic equations
no way from a very sharp resonance state, formed by
the collision of a wmeson and a neutrino. Strictly
for elementary systems.' The concept of an "elementary speaking, the states of a 1I"-meson do not form an
system" is, however, not quite identical with the elementary system because, after a sufficiently long
intuitive concept of an elementary particle. Intuitively, time, it can be either in the dissociated or in the
we consider a particle "elementary" if it does not appear undissociated state and the distinction between these
to be useful to attribute a structure to it. The definition is surely invariant relativistically. Nevertheless, the
under which the aforementioned enumeration can be life time of the 1I"-meson is very long as compared with
made is a more explicit one: it requires that all states any relevant unit of time (such as h/mc') and within
of the system be obtainable from the relativistic this time interval its states do form an elementary
transforms of any state by superpositions. In other system. On the other hand, the properties of a 1I"-meson
words, there must be no relativistically invariant are very different from what one would expect from a
distinction between the various states of the system compound consisting of a I'-meson and a neutrino.
which would allow for the principle of superposition. Thus the second condition for an elementary particle is
This condition is often referred to as irreducibility fulfilled. It is this condition which has no counterpart
condition. Relativistic transform is meant to include in the detinition of an elementary system. As a result
in the above connection not only the customary of this circumstance the concept of an elementary
Lorentz transformations but also rotations and dis- system is much broader than that of an elementary
placements in space and time. particle; as was mentioned above, a hydrogen atom in
The role of elementary systems as initial and tinal its normal state forms an elementary system.
states of collision phenomena, and hence their connec- Every system, even one consisting of an arbitrary
tion with the theory of the collision matrix, will be number of particles, can be decomposed into elementary
discussed at the end of this article. We wish to turn systems. These elementary systems can be specified in
now to the connection of elementary systems with a relativistically invariant manner, as containing only
elementary particles. certain states. Thus, the restriction to the normal state
Two conditions seem to play the most important of the hydrogen atom selected an elementary system
role in the concept of an elementary particle. The first from all the states of the hydrogen atom, which,
one is that its states shall form an elementary system in together, do not form an elementary system. The
the sense given above. This condition is quite unam- usefulness of· the decomposition into elementary sys-
biguous. The second condition is less clear cut: it is tems depends on how often one has to deal with linear
that it should not be useful to consider the particle as combinations containing several elementary systems.
a union of other particles. In the case of an electron or The great drawback of using the elementary systems
a proton both conditions are fulfilled and there is no as a basis of the theory is that their existence follows by
question as to the elementary nature of these particles. a rather abstract argument from the principles of
Only the first condition is fulfilled for a hydrogen atom quantum mechanics. As a result, the expressions for
in its normal state and we do not consider it to be an some of the most important operators "get lost" in the
elementary particle. process. The only physical quantities for which the
The situation is more ambiguous, for instance, in the theory directly provides expressions are the basic
'E. P. Wigner, Ann. of Math. 40, 149 (1939). The concept of quantities of the components of the momentum-energy
an elementary system, which will be explained below, is a de- vector and the six components of the relativistic angular
scription of a set of states which forms, in mathematical language,
an irreducible representation space for the inhomogeneous momentum tensor. The subject of the present article is
Lorentz group. an attempt to find general, invariant theoretic principles
400

Reprinted from Rev. Mod. Phys. 21,400 (1949).

396 CHAfYfER VIII

401 LOCALIZED STATES FOR ELEMENTARY SYSTEMS

on the basis of which operators for the position coordi- the wave functions of those states which are, at time
nates can be found. 1=0, localized at the origin of the coordinate system.
If we restrict ourselves to an elementary system, the We postulate that the states which represent a
physical interpretation of the operators to be found is system localized at time 1=0 at x=y=z=O: (a) form
unique: they will correspond to the position of the a linear set So, i.e., that the superposition of two such
particle if we deal with an elementary particle. Other- localized states be again localized in the same manner;
wise they may correspond to the center of mass of the (b) that the set So be invariant under rotations about
system. If the system is not elementary, the interpreta- the origin and reflections both of the spatial and of the
tion will not be unique and neither will our postulates time coordinate; (c) that if a state'" is localized as
lead us to a uniquely determined set of operators. above, a spatial displacement of'" shall make it orthog-
Before proceeding with our argument, we wish to onal to all states of So; (d) certain regularity conditions,
refer to other investigations with somewhat similar amounting essentially to the requirement that all the
objectives. The problem of the center of mass in infinitesimal operators of the Lorentz group be appli-
relativity theory has been treated particularly by cable to the localized states, will be introduced later.
Eddington' and by Fokker' on the basis of non-quantum It is to be expected that the states localized at a
mechanics. Their work was evaluated and a quantum certain point have the same properties as characteristic
mechanical generalization thereto given by Pryce.' We functions of a continuous spectrum, i.e., they will not
shall have frequent occasion to refer to his results. be square integrable but the limits of square integrable
Ideas related to Pryce's work have been first put functions. It seems to us that the above postulates are
forward by Schrodinger' and, more recently, by a reasonable expression for the localization of the system
Finkelstein' and also by Mpller. 7 to the extent that one would naturally call a system
The present paper arose from a reinvestigation of the unlocalizable if it should prove to be impossible to
irreducible representations' of de Sitter space which satisfy these requirements.
was undertaken by one of us.' These representations We shall carry out our calculations in the realization
are in a one to one correspondence with relativistically of the elementary systems which was described by
invariant wave equations for elementary systems in de Bargmann and WignerlO and will proceed with the
Sitter space. At the conclusion of the investigation it calculation.
appeared that the physical content of the equations
which were obtained could be understood much more Particle with no spin (Klein-Gordon particle)
readily if position operators could be defined on an
The determination of the localized state is particu-
invariant theoretic basis. As an introduction to this, larly simple in this case. It will be carried out in some
a similar investigation was undertaken in flat space detail in spite of this, because the same steps occur in
with the results given in the following sections. the consideration of systems with spin.
The wave functions are defined, in this case, on the
POSTULATES FOR LOCALIZED STATES AND
POSITION OPERATORS positive shell of a hyperboloid PO'=P,'+P,'+p,'+J.l'
and we shall use P', p2, P' as independent variables. In
The position operator could easily be written down any formula, po is an abbreviation for (P,'+P,'+p,'
if the wave function of the state (or the states) were +J.l')I. The invariant scalar product is
known for which the three space coordinates are zero
at t=O. If '" is such a function and T(a) the operator
of displacement by ax, a., a" a" the wave function
T(a)-l", represents a state for which the space coordi-
nates are ax, au, a, at time a,. Thus the knowledge of The wave function <I> in coordinate space becomes
the wave functions corresponding to the state X= y= Z
=0 at t=O (and the knowledge of the displacement
operators) entails the knowledge of all localized states,
<I>(x" x', x', x')= (2.,..)-1 f q,(p" P" p,)
i.e., of all characteristic functions of the position Xexp( -i{x, p})dp,dp,dp,/ Po, (2)
operators. From these, the position operators are easily where
obtained. For this reason we concentrated on obtaining
{x, p} =XOP'-X'p'-x'p2_x'p2
'A. S. Eddington, Fundamental Theory (Cambridge University =XOPO-XlP,-X,P,-x,p" (3)
Press, London, 1946).
'A. D. Fokker, Relativitatstheorie (Groningen, Noordhoff, 1929). is the Lorentz invariant scalar product. Throughout
• M. H. L. Pryce, Proe. Roy. Soc. 195A, 62 (1948). this paper, the covariant and contravariant components
'E. Sehr5dinger, Ber!' Ber. 418 (1930); 63 (1931). are equal for the time (0) coordinate, oppositely equal
'R. J. Finkelstein, Phys. Rev. 74, 1563A (1948).
7 Chr. M~ller, Comm. Dublin Inst. for Adv. Studies A, No.5
for the space (1, 2, 3) coordinates. This governs the
(1949); also A. Papapetrou, Acad. Athens 14, 540 (1939).
8 L. H. Thomas, Ann. of Math. 42, 113 (1941). 10 V. Bargmann and E. P. Wigner, Proc. Nat. Acad. Sci. 34,
9 T. D. Newton, Princeton Dissertation (1949). 211 (1948).
LOCALIZATION PROBLEMS 397

T. D. NEWTON AND E. P. WIGNER 402

raising and lowering of all indices. Occasionally, we As far as postulates (a), (b), (e) are concerned, I{; could
shall use for the scalar product of two space-like vectors be a discontinuous function, being +pol= (P'+I") I for
the notation (x,p) so that, e.g., {x,pi=xop'-(x·p). some p, and -(P'+I")I for the remaining p. However,
The linear manifolds which are invariant with respect no matter how I{; is chosen, consistently with (7), there
to rotations about P.=P2=P,=0 are, for any integer j, is, in this case, only one state localized at the origin
the 2j+ 1 functions because if there were two, say I{;1, and I{;" the I{;1 would
have to be orthogonal not only to I{;1 exp( -i(a, p)) but
Pmi((}, cp)j(p) (m= -j, -HI, .. oj-l,}), (4)
also to I{;, exp( -i(a, p)) from which not only 1I{;1'~Po
where p, tJ and cp are polar coordinates for p., P" pa and but also I{; 1*I{;,~ po and hence the proportionality of I{;1
j is an arbitrary function. The Pmi are the well-known to I{;, follows.
spherical harmonics. The sets (4) are also invariant In order to eliminate the discontinuous I{; as localized
with respect to inversion, i.e., replacement of PI, p" pa state, we introduce the further regularity condition that
by -PI, -P" -pa. Naturally, not only a single set
(8)
(4) has these properties of invariance but the sum of
an arbitrary number of such sets as long as one includes shall remain finite as the normalizable wave functions
with one function (4) all 2j+ 1 functions and their I{;n approach I{;. The Mo. is the infinitesimal operator of
linear combinations. The j(p) could be different for a proper Lorentz transformation in the x!'x' plane, its
different j. operator is lO
Under time reversal I{;(p.p,pa) goes over into\!
MOk = ip°i)/ iJp•. (8a)
EJV;(p1, p" pa)=I{;(-p., -P" -palO. (5)
This further postulate eliminates all discontinuous I{;
We understand by time reversal the operation which and we obtain for the wave function of the only state
makes out of a wave function I{; the wave function IJif; which is localized at the origin
on which every experiment, if carried out at - t, yields
same results as the same experiment carried out on I{; (9)
at time t. Because of (5) and our postulate (b), if The regularity requirement (d) actually asks for the
the Pmi(O, cp)j(p) are localized at the origin, the set finiteness of (8) for all Mkl. However, if one substitutes
P ~i(O, cp)j(p)*, i.e., the P+mi(O, cp)j(p)* are also local- M23, M31 or M12 for MOk in (8), the resulting expression
ized. The same is then true for the sum and difference is automatically bounded-in fact their sum is j(j+ 1).
of the corresponding pairs of functions which shows Hence requiring the applicability of the M", M31, M12
that the f(p) can be assumed to be real without loss to I{; does not introduce a new condition.
of generality. The localized wave function in coordinate space is
The displacement operator in momentum space is obtained by (2). It is, apart from a constant"
simply multiplication with exp( -i{a, pi);
(9a)
T(a)I{;= exp( -if a, pi )1{;. (6)
It goes to zero at r= 00 as e-.', at r= 0 it becomes
We shall have to consider purely space-like displace-
infinite as r- 5I'. It is, of course, not square integrable
ments, i.e., assume that aO=O. It then follows from
our postulate (e) that, in particular, exp(i(a, P))f is since it is part of a continuous spectrum.
Applying the operator of displacement to (9) we
orthogonal to I{; if f is localized, or that
obtain for the wave function of the state which is

fff I f(p1p,pa) I' expi(aIP1+ a,P,

localized at Xl, x', x' at time 1=0
T( -x)I{;= (27r)lpol exp-i(p1x1+p'x'+p'x')
= (211-)-lpo'e- i (p.x).
+aaPa)dP1dp,dpa/ po= 0, (6a)
This must be a characteristic function of the operator
for any non-vanishing vector a. This shows that only rt for the k-coordinate with the characteristic value x'.
the zero wave number part occurs in the expansion of The operator rt is therefore defined by
II{;I'/po into a Fourier integral. Hence II{;I'/po is a
constant, II{;I proportional to Po!. Comparing this with
(4), we see that only the set j=O can be chosen.
Since, furthermore, we saw that j(p) can be assumed
to be real, we have X ei(r' ,x)cp(p')dxdp'jpo'; (10)
(7)
dx and dp' stand for dx 1dx'dx' and dP1' dp,' dpa'. One
As was anticipated, (f, I{;) is infinite, the localized
function is part of a continuous spectrum. 12 G. A. Campbell and R. M. Foster, "Fourier Integrals for
Practical Applications," American Telephone and Telegraph
11 E. P. Wigner, Gattinger Nachrichten 546 (1932). Company (1931).
398 CHAPTER VIII

403 LOCALIZED STATES FOR ELEMENTARY SYSTEMS

can transform (10) in a well-known fashion The consistency of these was shown before. to The y.

P')
apply to ~., two y with different first indices commute,
qkq,(p)= ( i a
- -i- - (21r)-' with the same first index they satisfy the well-known
iifJ' 2 po' relations
(13a)
The great difference between the present case and that
of zero spin consists in the limitation (13) of the
permissible wave functions, in addition to the limitation
= -i(~+.f..)q,(P)' (11)
to the positive hyperboloid. This latter limitation can
be taken care of by using only the PI, P" p. as inde-
ap. 2po'
pendent variables, the former limitation cannot be
These expressions are valid for finite as well as for taken care of in an equally simple fashion. We shall
vanishing rest mass. It is remarkable that the operator make extensive use, however, of a device, most success-
rf can be transformed into coordinate space and retains fully employed by Schrodinger' and define operators
a relatively simple form
E.=~(pO)-I(LY.'p,+I')y.o. (14)
1 fexP(-1'1 x-yl) a<J>(y) This is a projection operator: E.'=E. and E.¥t auto-
qk<J>(x) = x'<J>(x)+- -dy. (12)
8.. Ix-yl oy. matically satisfies the corresponding Eq. (13). Denoting
the product of all the E. by E
x and y stand for the spatial part of the four vectors x·
and y. and dy indicates integration over yl, t,
y'. The
(14a)
customary rf operator contains only' the first term of any E¥t is a permissible wave function, satisfying all
(11), Eqs. (13).
It may be well to remember at this point that the For the scalar product, we shall use the expression
position operators to which our postulates lead neces-
sarily commute with each other so that only Pryce's
case (e) can be used for comparison. In fact, our rf is
identical with his 'i/. It may be pointed out, second,
that a state which is localized at the origin in one It follows from this at once, because of our postulate
coordinate system, is not localized in a moving coordi- (e), and since (6) is valid in this case also, that every
nate system, even if the origins coincide at t=O. wave function which is localized at the origin satisfies
Hence our operators rf have no simple covariant mean- the analogon of (i):
ing under relativistic transfarmations. This is not the
case for the customary operators rf either. Further- Lfl.yI'~(2")-'PoMl. (16)
more, even though it appears that <J>(x) = <I (x) is in- The operator for time reversal is
variant under relativistic transformations which leave
the origin unchanged, this is not much more than a
mathematical quirk. One sees this best by transforming where C is a matrix which operates on the ~ coordinates
the <I-function to momentum space through the inver- and satisfies the equations
sion of (2). The result, po, seems to have a simple
covariant meaning. However, it does not represent a Cy"* = y.oC ; (Ol= 1,2, .. ',25)
square integrable function and if one approximates it Cy.'*=-y.'C. (0l=1, 2, ···,25;k=1,2,3). (lia)
by one, say by ¥t.= po exp( -a'po') , the Lorentz trans- If yO, y', y' are real, yl imaginary
form of ¥t. will not approach ¥t. with decreasing Ol. In
fact, as soon as 0l1'«1, the scalar product of ¥t. and its ,.
transform will be independent of Ol and smaller than C= II j'h.'; C'=(-)2.. (lib)
the norm of ¥t•.
~I

Particles with spin and finite mass Since C, as defined above, is a real matrix we also have
8'= (- )" which is true independently of the chcice of
We again use the description given in reference 10, the y-matrices. The operator for the inversion of the
i.e., define wave functions on the positive hyperboloid space coordinates is
po'= PI'+P,'+P,'+I" and use in addition to PI, P" P.
the 25 spin variables tl, t" ... t,. all of .them four- N(pI, P" p,)=.yt'y,o .. 'y,N( -PI, -P" -p,); (Iii)
valued. The wave functions which describe the possible it commutes with the E. of (14).
states of the system will be symmetric functions of the In order to determine the sets of wave functions
t and satisfy the 25 equations which are invariant under rotations, we first define the
2:" y.'P,¥t = I'¥t Ol= 1, 2, ,,·25. (13) analogue of the pure spin function for the relativistic
LOCALIZATION PROBLEMS 399

T. D. NEWTON AND E. P. WIGNER 404

Eqs. (13). For this purpose we define auxiliary functions giving a totalj from wave functions with given "orbital
Vm which are independent of PI, P" p, and functions of momentum" / and "spin momentum" s.
the ~ only. They satisfy the equations Since the polar angles tJ, q, are indeterminate for p=O,
the fl(P) must vanish for p=O unless /=0. Otherwise,
y.oVm=Vm (a=l, 2,' ", 2s) (19)
and the !/tjm would become singular at p=O and the MO.
could not be applied to them in the sense of the bound-
!i~a 'Yct1"Ya2Vm=mvm edness of (8). (Actually it is necessary to postulate this
(m= -s, -s+ 1, .. " s-l, s). (19a) equation for the square of MOk instead only for MOk.)
It follows that !/tim vanishes at p= 0 unless the series
Since the yO and the iy'y' commute, it is possible to
(21) contains a term with /=0. However, (16) shows
assume temporarily that they are all diagonal. Equation
that !/t cannot vanish at this point if the rest mass is
(19) then demands that they all belong to the character-
finite and that, hence, every localizable wave function
istic value + 1 of each yO; there are 2" such functions.
must have an /=0 term in its expansion. This happens
However, we are interested only in symmetric functions
only if j = s and the wave function 'has even parity.
of the ~ and there are only 2s+ 1 of these. They are
If the parity of !/t;m is odd, only /I, f3, etc., enter (21)
distinguished by the index m: the Vm has non-zero
and these still vanish at p=O. It follows that the wave
components only for those t for which s+m of the
functions which are localized at the origin all have
iyly' are +1, the remaining s-m are -1. For these ~
angular momentum j = s and the form
the value of Vm is «s+m) !(s-m) !/(2s) !)I2-' so that
Vm is normalized in the sense
,.
"'m= L L S(/, S) •. m_m'.m'P'm-m,(tJ, q,)h Vm'. (21a)
l=O m'
(19b)
Physically, m corresponds to the spin angular momen- We now skip the part of the calculation which deals
with the determination of the fl and give only the
tum about the ~ axis, the parity of Vm is even because
of (19) and (18). result: The wave functions localized at the origin are
The v.. are not permissible wave functions because the 2s+ 1 functions
they do not satisfy the wave Eqs. (13). We therefore !/tm= (211")-3"2'Po"+!(Po+IL)-'
define as spin functions X Vm(PIP,P,; h,'" .~.) (21b)

V..(PI, P" p" ~l, "', t,.)=Ev,. (m= -s, "', s). (20) (i.e., the /=0 term alone remains from (21a)). Actually,
this result is far from being surprising."
They are permissible wave functions of even parity The operator for the position coordinate can be
and Vm represents a state of angular momentum mk calculated in exactly the same way as this was done in
about the third coordinate axis. Their normalization the case of zero spin, and gives
is, instead of (19b)
Lt 1Vml'= «Po+IL)/2po)". (20a)
,. Po"+! ( Po-I
<t=EII (1+1'.0)-- - i - - - E . (22)
a)
The most general solution of (13) is a linear combi-
~l (Po+IL)' ap. (Po+IL)'
nation of the V.. multiplied with arbitrary functions of For s=! this again agrees with Pryce's result' for his
the PI, P" p,. A set of wave functions which is invariant case (el, i.e., for his operator q.
under rotations and reflections contains wave functions The significance of the projection operators E in
of the form (22) is only to annihilate any negative energy part of
the wave function to which it is applied and to produce
I/;;m= L S(l, S)I.~m'.m'P'~m,(tJ, q,)f,(P)Vm,. (21)
a purely positive energy wave function. Since q' is the
l,m'
position operator only for wave functions which are
The p, tJ, q, are again polar coordinates for pI, po, p3;
the f' are arbitrary unknown functions of the length of 14 The proof runs as follows. One first shows, by considering
p. However, if one function of the form (21) occurs in >Pm±e.y-m that the J, of (21a) can all be assumed to be real. One
then subdivides >Pm into two parts: the 1= 0 part of the sum (21a)
the set, all others with different m but the same fl also will be denoted by >P', tbe rest >p'. As we have seen, ",,"is finite at
occur. The summation over / is to be extended over all p=O, while ",r vanishes at that point. The proof then consists in
even values between [j-sl and j+s if the parity of showing that there can be no region in which ",r is finite but very
much smaller than >p'. It then follows from the continuity of botb
the!/tl is to be even, over all odd values of / if the parity >P' and ,p' that the latter vanishes everywhere. Inserting ,p'+,p'
of I/;i is odd. The SCI, s) are the customary coefficients" for >P in (16), one can neglect in the aforementioned region the
square of 1/Ir as compared with the other terms. The right side, as
13 See e.g., E. P. Wigner, Gruppentheorie, etc. (V. Vieweg & Sobn, well as the term from the square of 1/10, are independent of iJ and Ip.
Braunschweig, 1931). The composition of the V and the spherical This must be true, therefore, also for the term arising from the
harmonics pi to the 1/11 is the same operation as the composition of cross product of 1/10 and t/lr, This term is, however, a sum of expres-
the spin functions with a definite S and the space coordinate func- sions Pm'({J,,,)J,J. whicb cannot be independent of {J,,, except if
tions with a definite L, to functions of both, with definite J. This allJ. with 1>0 vanish (j, is finite by assumption). It tben follows
composition is explained in Chapter XXII. The coefficients of that tbe J. vanish everywhere and (21a) reduces to a single term.
the composition, i.e., our S(l, s) are calculated p. 202 ff. (tbey are This can be obtained from (16) by taking the square root on both
denoted by ,ILS'). sides.
400 CHAPfER VIII

405 LOCALIZED STATES FOR ELEMENTARY SYSTEMS

defined on the positive hyperboloid alone, the E on (22) transforms positive energy functions into positive
the right could logically be omitted. Both E can be energy functions.
omitted' if one calculates a matrix element between It is often stated that a measurement of the position
two purely positive energy wave functions. The factors of a particle, such as an electron, if carried out with a
involving po are necessary in order to make iiJ!iJp; greater precision than the Compton wave-length, would
hermitian: because of the factor PO-·_l in the volume lead to pair production and that it is, therefore, natural
element (15), an operator is hermitian if it looks that the position operators do not preserve the positive
hermitian after multiplication with po<+1 on the right energy nature of a wave function. Since a position
and division with the same factor on the left. The measurement on a particle should result in a particle
z, at a definite position, and not in a particle and some
operator II !(1+YaO) is a projection operator, i.e., it is pairs, this consideration really denies the possibility of
0.=1
identical with its square and could therefore be inserted the measurement of the position of the particle. If this is
into (22) once more before the second E, thus making accepted it still remains strange that pair creation ren-
(22) somewhat more symmetric. The position operator ders the position measurement impossible to the same
(11) for the Klein-Gordon particle is a special case of degree in such widely different systems as an electron, a
(22) and can be obtained from (22) by setting s=O. neutron and even a neutriono. The calculations given
If one displaces a state by a and measures its x· above prove, at any rate, that there is nothing absurd in
coordinate afterwards, the result will be greater by a" assuming the measurability of the position, and the
than the x" coordinate measured on the undisplaced existence of localized states, of elementary systems of
state. This leads to the relation non-zero mass. Moreover, the postulates (a), (b), (c) and
(d), which are based on considerations of invariance, de-
T(-a)q'T(a) = q'+a' (23) fine the localized states and position operators uniquely
for aO=O. Inserting the expression (6) for T(a) and for all non-zero mass elementary systems.
going to the limit of very small ak, one obtains No similarly unique definition of localized states is
possible for composite systems. Although it remains
(q'pl_ plqk)q,= -io.!t/> (23a) easy to show that definite total angular momenta j can
where q, is any permissible wave function. Actually be attributed to localized states, one soon runs into
one obtains by direct calculation, using in particular difficulties with the rest of the argument. In particular,
the identity the summation in (16) must be extended not only over
Ea(1+'YaO)E.= Po-l(po+p.)Ea (24) the spin coordinates ~ but also over all states with
different total rest mass and different intrinsic spin.
the commutation relation As a result, one can, e.g., find states which can coexist
(25) as localized states in the sense of our axioms even
though their j values are different. This is also what
The commutation relations of the q' with po are natu- one would expect on ordinary reasoning since, if the
rally also the usual ones as po is a function of the pk system contains several particles, the states in which
alone. Since the q" are the components of a vector anyone of them is localized at the origin satisfy our
operator in three-dimensional space, their commutation postulates. This holds also for the states in which
relations with the spatial components of Mk! are also another one of the particles is so localized or for states
the usual ones. in which an arbitrary linear combination of the coordi-
We wish to remark, finally, that a consideration, nates is· zero. Asa result, not only is the number of
similar to the above, has been carried out also for the localized states greatly augmented but, further, one
equations with zero mass. In the case of spin 0 and t, must expect to find many such large sets for which our
we were led back to the expressions for localized systems postulates hold, although no two sets can be considered
which were given in (9) and (21b). However, for higher to be localized simultaneously. In oth·er words, each
but finite s, beginning with $= 1 (i.e., Maxwell's set of localized states is not only much larger for
equations), we found that no localized states in the composite systems but one also has to make a choice
above sense exist. This is an unsatisfactory, if not between many sets all of which satisfy our postulates
unexpected, feature of our work. The situation is not by themselves. It does not appear that one can proceed
entirely satisfactory for infinite spin either. much further in the definition of localized states for
composite systems without making much more specific
DISCUSSION
assumptions. Naturally, one can define as localized
One might wonder, first, what the reason is that our states those, which, in any of the elementary parts of
localized states are not the o-functions in coordinate the composite system, appear localizable. It appears
space which are usually considered to represent localized reasonable to assume that this definition corresponds
states. The reason is, naturally, that all our wave to the center of mass of the whole system.
functions represent pure positive energy states. This One may wonder, even in the case of elementary
is not true of the o-function. Similarly, our operator particles, whether the determination of the localized
LOCALIZATION PROBLEMS 401

T. D. NEWTON AND E. P. WIGNER 406

states and positIOn operators has much significance. had gone straight to the scattering center and then
Such doubts might arise particularly strongly if one is continued in the new direction without any delay." In
inclined to consider the collision matrix as the future order to answer such questions in the relativistic region,
form of the theory. One must not forget, however, one will need some definition of localized states for
that the customary exposition of this theory refers elementary systems. From this point of view it is
only to questions about cross sections. There is another satisfactory that the localized states could be defined
interesting set of questions referring to the position of without ambiguity just for these systems.
the scattered particles: how much further back (i.e.,
closer to the scattering center) are they than if they 15 L. Eisenbud, Princeton Dissertation (1948),
402 CHAPTER VIII

REVIEWS OF MODERN PHYSICS VOLUME 34, NUMBER 4 OCTOBER 1962

On the Localizability of Quantum

Mechanical Systems*
A. S. WIGHTMAN
Princeton University, Princeton, New Jersey

inevitable limitations of the nonrelativistic theory,

1. INTRODUCTION
may be regarded as completely satisfactory.
F ROM the very beginning of quantum mechanics,
the notion of the position of a particle has been
Historically, confusion reigned in the relativistic
case, because situations requiring a description in
much discussed. In the nonrelativistic case, the proof terms of many particles were squeezed into a for-
of the equivalence of matrix and wave mechanics, malism built to describe a single particle. I have in
the discovery of the uncertainty relations, and the mind the difficulties with wave functions for a single
development of the statistical interpretation of the particle which seem to yield nonzero probability for
theory led to an understanding which, within the finding it in a state of negative energy. Soon attention
shifted to the problems of the quantum theory of
• Dedicated to Eugene Wigner on his sixtieth birthday. fields and the question of the status of position

Reprinted from Rev. Mod. Phys. 34, 845 (1962).

LOCALIZATION PROBLEMS 403

846 A. S. WIGHTMAN

operators for relativistic particles was left without a sional space at a given time, must satisfy the follow-
clear resolution. That does not mean that papers ing axioms:
were not written on the subject, but that those Ca) S. is a linear manifold;
papers had completely different objectives in mind: (b) S. is invariant under rotations about a, re-
They permitted the particles in question to be in flections in a, and time inversions;
nonphysical (negative energy) states or they studied (c) S. is orthogonal to all its space translates;
operators which could not serve as position observa- (d) certain regularity conditions.
bles since their three components did not commute. The solutions of (a) ... Cd) for elementary systems,
In the opinion of the present author, the decisive i.e., for systems whose states transform according to
clarification of the relativistic case occurs in a paper an irreducible representation of the inhomogeneous
of Newton and Wigner.' These authors show that, Lorentz group, turn out to be continuum wave func-
if the notion of localized state satisfies certain nearly tions when they exist at all, i.e., according to the
inevitable requirements, for a single free particle it usual definitions of Hilbert space, there is no mani-
is uniquely determined by the transformation law fold S•. However, it is physically and mathematically
of the wave function under inhomogeneous Lorentz clear that Newton and Wigner's formulation ought
transformations. The resulting position observables to be regarded as the limiting case of a notion of
turn out, in the case of spin-!, to be identical with localizability in a region.
the Foldy-Wouthuysen "mean position" operators.' In the present paper, I propose a reformulation of
An analogous investigation for the case of Galilean the physical ideas of (a) ... Cd) in terms of a notion
relativity was carried out by Inonti and Wigner.' of localizability in a region. When the ideas are so
The essential result of Newton and Wigner is that formulated, one sees that the existence and unique-
for single particles a notion of localizability and a ness of a notion of localizability for a physical system
corresponding position observable are uniquely de- are properties which depend only on the transforma-
termined by relativistic kinematics when they exist tion law of the system under the Euclidean group,
at all. Whether, in fact, the position of such a particle i.e., the group of all space translations and rotations.
is observable in the sense of the quantum theory of The analysis of localizability in the Lorentz and
measurement is, of course, a much deeper problem; Galilei invariant cases is then just a matter of dis-
that probably can only be decided within the context cussing what representations of the Euclidean group
of a specific consequent dynamical theory of parti- can arise there. To obtain uniqueness, one must add
cles. All investigations of localizability for relativistic invariance under time inversion and an analogy of
particles up to now, including the present one, must Newton and Wigner's regularity assumption. As
be regarded as preliminary from this point of view: would be expected, all the results obtained earlier in
They construct position observables consistent with the old formulation come out. One can ask what is
a·given transformation law. It remains to construct the point of the present extended footnote to Newton
complete dynamical theories consistent with a given and Wigner's paper. First, it seems worthwhile to me
transformation law and then to investigate whether to have a math~matically rigorous proof of the
the position observables are indeed observable with fundamental result of Newton and Wigner that a
the apparatus that the dynamical theories them- single photon is not localizable. Second, the work of
selves predict. Newton and Wigner can be regarded as a contribu-
In Newton and Wigner's formulation, the set S. tion to the general problem of determining what
of states localized at a point a of the three-dimen- physical characteristics of a quantum mechanical
system are consistent with a given relativistic
transformation law. In this connection, it is inter-
1 T. D. Newton and E. P. Wigner, Revs. Modern Phys. 21,
400 (1949).
esting to regard the axioms I ... V below for localiza-
2 L. Foldy and S. Wouthuysen, Phrs. Rev. 78, 29 (1950). bility in a region as a very special case of the notion
This paper was widely read because 0 its exceptional clarity. of particle observables for a quantum theory. Else-
The mean position operators themselves were discussed before
by A. Papapetrou, Acad. Athens 14, 540 (1939); R. Becker, where' I gave a set of axioms for the notion of a
Gott. Nach. p. 39 (1945); and M. H. L. Pryce, Proc. Roy. Soc. particle interpretation which yield I ... V when
(London) A150, 166 (1935); A195, 62 (1948). For further
references and discussion see A. S. Wightman and S. Schweber, specialized to the case of a single particle. One of the
Phys. Rev. 98, 812 (1955). main reasons for giving full mathematical detail in
'E. Inonii and E. Wigner, Nuovo cimento 9, 705 (1952).
The main point of this paper is that laws of transformation
of the states of a particle under the inhomogeneous Galilei
group other than those in the ordinary Schrodinger mechanics 4 See, Les problbMs mathbruJtiques de la theoTie quantique
are inconsistent with localizability. des champs, (CNRS, Paris, 1959), especially pages 36-38.
404 CHAFfER VIII

LOCALIZABILITY IN QUANTUM MECHANICS 847

the present simple case is in preparation for the prob- III. E(S, uS.) = E(S,) + E(S.) - E(S, n S.).
lem of determining particle interpretations.
It turns out that the natural mathematical tool for II S;,i = 1, 2, ... are disjoint Borel sets then
the analysis of localizability as understood here is the
theory of imprimitive representations of the Euclid-
E(U S;) = L;-' E(S;).
ean group. The notion of imprimitivity was intro- IV. E(R3 ) = 1.
duced for finite groups early in the history of group
theory. It was generalized to the case of a large class V~ E(RS + a) = U(a,R)E(S)U(a,R)-',
of topological groups by Mackey.' From a mathe-
matical point of view, the present paper merely where RS + a is the set obtained from S by carrying
writes out Mackey's theory in detail for the case of out the rotation R followed by the translation a, and
the Euclidean group. However, I decided to make U(a,R) is the unitary operator whose application
the exposition as self-contained as possible, and to yields the wave function rotated by R and translated
incorporate certain elegant ideas of Loomis in the bya.
proofs.· The purpose of this expository account is to The notation S, n S. and S, u S. is used to
make it possible for the reader to understand how indicate the common part and union, respectively,
the mathematical arguments go for the Euclidean of the sets S, and S.. U S; is the union of the
group without having to work through the general sets 8..
case, however character building that experience The physical significance of these axioms is as
might be. follows.
The Borel sets form the smallest family of sets
2. MATHEMATICAL FORMULATION OF THE AXIOMS which includes cubes and is closed under the opera-
AND PRELIMINARY HEURISTIC DISCUSSION tions of forming complements and denumerable
unions. One might try to replace the Borel sets by
The axioms are formulated in terms of projection all sets obtained by forming complements and finite
operators E(S), where S is some subset of Euclidean unions starting from cubes and require III only for
space at a given time. The E(S) are supposed to be finite sums. However, it can be shown that any such
observables. They must be projection operators be- E(S) could be extended to one defined on the Borel
cause they are supposed to describe a prope:rty of the sets and satisfying III as it stands. (See Appendix I
system, the property of being localized in S. That for further discussion of this point.) In fact, E(S)
is, if of> is a vector in a separable Hilbert space, x, can be extended even further to all Lebesgue measur-
describing a state in which the system lies in S, then able sets, but this extension will not be needed here. 8
E(S)of> = of>. If the system does not lie in S then II states that a system which is in both S, and S.
E(S)if> = O. E(S) can therefore only have proper is in S, n S •. It is immediately clear from II th"\t
values one or zero and, as an observable, must be E(S,)E(S.) = E(S.)E(Sl).
self-adjoint. Thus, it is a projection operator.' III states that the set of states of the system for
The axioms are: which it is localized in S, u S. is the closed linear
I. For every Borel set, S, of three-dimensional manifold spanned by the states localized in S, and
Euclidean space, R3, there is a projection operator those localized in S2.
E(S) whose expectation value is the probability of IV says that the system has probability one of
finding the system in S. being somewhere.
II. E(S, n S.) = E(S,)E(S.). V' says that if of> is a state in which the system is
localized in S, then U(a,R) of> is a state in which the
• G. W. Mackey, Proc. Nat!. Acad. Sci. U.S. 35, 537 (1949); system is localized in RS + a.
Ann. Math. 55, 101 (1952); 58, 193 (1953); Acta Math. 99,
265 (1958). That Mackey's theory applies to localizability in I venture to say that any notion of localizability
quantum mechanics was independently realized by Mackey in three-dimensional space which does not satisfy
himself. I thank Professor Mackey for correspondence on the
subject. Mackey's treatment is summarized in his Colloquium I . . . V' will represent a radical departure from
Lectures to the American Mathematical Society, Stillwater, present physical ideas.
Oklahoma Aug. 29-Sept. 1, 1961. It is a part of a coherent
axiomatic treatment of quantum mechanics given in his un- The E(S) define a set of commuting coordinate
published Harvard lectures 1960-61.
6 L. H. Loomis, Duke Math. J. 27, 569 (1960).
7 For a general discussion of observable. describing a
property see J. von Neumann, Mathematical Foundntitm. of 8 An argument that the Lebesgue measurable set. form a
Quantum Mechanics (Princeton University Press, Princeton, physically natural class is contained in J. von Neumann, Ann.
New Jersey, 1955), pp. 247-254. Math. 33, 595 (1932).
LOCALIZATION PROBLEMS 405

848 A. S. WIGHTMAN

operators q"q.,q. which form a vector in 3-space. In write A,a,. This will be done throughout the follow-
fact, ing. Thus, V' is replaced by

q, = J_:AdE({X, ..; All , (2.1) +

V. U(a,A)E(S)U(a,A)-' = E(AS a) for all Borel
sets S of R', and all {a,A} E 8,. Here AS a is +
where E({x, ..; A}) is the projection operator for the set obtained from S by the transformation
the set {x,"; A} of all points of three-space whose {a,A} and {a,A}--> U(a,A) is the representation of
ith coordinates satisfy x, ..; A. Of course, (2.1) has 8, belonging to the physical system in question.
to be interpreted as meaning that the Stieltjes
integral In the terminology of Mackey, I ... V state that
the set of operators {E(S)} are a system of imprimi-
(<I>,q,v) = J_:Ad(<I>,E({X,"; A})V) tivity for the representation U(a,A) of 8, with base
R'. In order to see the present problem in the context
holds for all <I> and all v on which q, can be defined. of Mackey's general theory, recall that he considers
Thus, any set of E(S) uniquely determines a position a topological group G and two continuous realiza-
operator q. Conversely, one can regard the require- tions of G, one a representation by homeomorphisms
ment that the E(S) exist as a precise way of stating of a topological space M: x -> h(g)x (homeomor-
that q exists and its components are simultaneously phism means one-to-one mapping continuous both
observable. A notion of localizability for which ways) and the other a unitary representation of G
[q"q;] -.t- 0 does not fall under the above scheme if, in a Hilbert space X: g --> U(g). Then a system of
indeed, such a notion makes sense at all. imprimitivity with base M is a family of projection
Axiom V' has been stated in terms of the unitary operators in X which satisfy I, II, III, IV with the
operators U(a,R). It is well known that without loss sets S interpreted as Borel sets of M, and, in addi-
of physical generality these can be assumed to form tion, the appropriate modification of V:
a representation up to a ± sign,' i.e.,
U(g)E(S)U(g)-' = E(h(g)S) .
U(a"R,)U(a2,R2) A representation U(g) which has at least one system
= w(a"R,;a,R,)U(a, + R,a"R,R,) , of imprimitivity (with respect to M) is said to be
where w = ± 1. It is more convenient, from a imprimitive (with respect to M). A system of im-
mathematical point of view, to deal with a true primitivity is transitive if the group of homeomor-
representation for which w = + 1. It is also well phisms g -> h(g) is, i.e., if each point x is carried into
known that this can be arranged by passing to the every other by a suitable h.
two-sheeted covering group of the Euclidean group In the case of a transitive system of imprimitivity,
8,.'0 It may be defined as the set of pairs a, A, where the space M can be replaced by a coset space as
a is again a three-dimensional translation vector and follows. Let Gz be the subgroup of all g E G such that
A is a 2 X 2 unitary matrix of determinant one. The h(g)x = x. Notice that if h(g,)x = y = h(g,)x then
matrices ± A determine the same rotation given by h(g,'g.)x = x, so g,g = O. where g E G•. The set of
all elements of the form g.g, g E G. is denoted g,G.
Ax· ..A* = (R(±A)x)· .. and called the left coset of G. belonging to g,. Thus,
Here .. stands for the Pauli matrices each left coset corresponds to a point of M, distinct
cosets corresponding to distinct points, and by a

T
, [0 1J
= 1 0 ' T' = [? -iJo
, ,T
3 = [1 OJ
0 -1 .
mere change of names M can be replaced by the
space of left cosets, usually denoted GIG•. In the more
The mUltiplication law of 8, is general case of a nontransitive system the space M
will split into orbits and the points of an orbit can
{a,A,) {a"A,} = {a, + A,a.,A,A,} . be labeled by the points of GIG. where x is any point
Here, for brevity, instead of writing R(A,)a. we of the orbit.
In the problem of localizability considered here,
the system of imprimitivity is transitive but for
9 The argument (originally due to E. P. Wigner) is outlined momentum observables and particle observables, in
in Disp<TSicm Relaticms and Ekmentary PaTticles (John Wiley
& Sons, Inc., 1961), pp. 176-18l. general, the system of imprimitivity is not transitive.
10 The argument (originally due to E. P. Wigner for the
rotation group and Poincare group) is given for the Euclidean
Mackey's theory shows that the transitive system
group in V. Bargmann, Ann. Math. 59, 1 (1954). of imprimitivity and its associated representation
406 CHAPTER VIII

LOCALIZABILITY IN QUANTUM MECHANICS 849

can be brought into a standard form by a suitably To obtain V, one may note first that the operator
chosen unitary transformation, V: T(a) defined by

{E(S),U(g)} -> {VE(S)V-"VU(g)V-'} . (TCa)<I>)(x"x,) = <I>(ax, + (1 - a)x,,(a - l)x,

In this standard form the VU(g)V-' becomes a so-

+ (2 - a)x,)

called induced representation associated with a is unitary and satisfies

unitary representation of G. where x is some fixed
T(a)x~PT(a)-' = X(·) •
point of M. Two pairs {E,(S),U,(g)} and {E.(S),
U.(g)} are unitary equivalent:
Then V is given by
E,(S) = VE.(S)V-',U,(g) = VU2(g)V-', V = TC~)Tc.)-' .

if and only if the unitary representations of G. are Clearly, the kind of nonuniqueness appearing in this
equivalent. example may be expected to be absent only when one
Detailed proofs of these assertions of Mackey's is dealing with a single particle. Theorem 4 obtained
theory for the special case of 8. will be offered in the below gives a precise criterion for uniqueness and a
following sections. For the moment, the results will parametrization of the possible answers when more
be taken for granted and used to discuss the unique- than one exists.
ness of E(S) for given U(g). Clearly, for U(g) given The uniqueness of the notion of localizability for
the only unitary transformations, V, which can give given representation of the Euclidean group has
new VE(S) V-' -F- E(S) are ones which commute been discussed assuming Mackey's theory. Now I
with the given U(g) but not with the E(S). attempt to give an intuitive idea of the circumstances
That this possibility is actually realized in simple in which a notion of localizability exists.
physical examples can be seen by considering a com- Since all the E(S) commute, diagonalize them.
pound system of two free spinless Schriidinger parti- Then the state vectors are represented by quantities
cles with wave function ",(x"x,). Let the correspond- <I>(x) defined on space and with a number of compo-
ing representation of the Euclidean group be U(a,R): nents which may vary with x. [In fact, these 4>(x)
for x = a are just Newton and Wigner's linear mani-
",(x"x,) -> (U(a,R)",)(x"x,)
fold S •. ] In this realization the scalar product of
= ",(R-'(x, - a),R-'(X2 - a» . two vectors 4> and'll is
Define the operators XCa) by
(<1>,'1') = ! dx(<I>(x),'1'(x» ,
X Ca ) = ax~P + (1 - a)x~P
where the scalar product appearing under the
where a is any real number, and by definition integral sign is in the components of 4>(x) and ,y(x)
for fixed x. The operators E(S) take the form
(x?4»(y"y,) = y,4>(y"y,)
Y24>(y"y,) . (E(S)4>)(x) = x.(x)4>(x) ,

°
(x~P4»(y"Y2) =

Then, for each a, XCa) defines a possible position where x.(x) = 1 if xES, if x $ S. From the
operator (the spectral representation of x/a), j = 1, transformation law of E(S) it is plausible that by a
2, 3 yields the projections appearing in (2.1), and suitable choice of basis it can be arranged that
the general E(S) can be found from these). In par- (U(a,l)4>)(x) = 4>(x - a) .
ticular, X CO) = X~P and XC') = xlP, are possible posi-
tion operators. From this equation, it follows that the number of
Now there exists a unitary operator, V, which components of 4>(x) is the same for all x. It is also
commutes with the representation of the Euclidean plausible that by a suitable choice of basis the
group transformation law under rotation can be made to
look the same for each x:
[V,U(a,R)l_ = 0
and carries X(a) into X(~) (U(O,A)4»(x) = ~(A)4>(A -'x),

VX(a) V-' = X(~) • where ~(A) acts on the components of 4>CA-'x) at

LOCALIZATION PROBLEMS 407

850 A. S. WIGHTMAN

each point. Once these results are accepted, one can quence of this simple kinematical fact." For spin-O,
pass by Fourier transform to momentum space (iii) is satisfied and so the phonon is localizable. l'
amplitudes. There one has It is an oddity that the same is not true for Wigner's
particles of infinite spin, I' as will be seen in Sec. 5,
(U(a,A)<I»(p) = e-,p'a:n(A)<I>(A-lp) (2.2) even though in that case each angular momentum
with the scalar product along p appears just once.
There is one paradox to which the preceding dis-
(<I>;w) = f dp(<I>(p),w(p)) . (2.3)
cussion might appear to give rise. Suppose one
describes a photon by a real-valued three-component
field B(x) satisfying
The canonical form (2.2) is to be compared with
divB=O, (2.5)
(U(a,A)<I>)(p) = e-'P'aQ(P,A)<I>(A-lp), (2.4)
defines a scalar product (this is a real Hilbert space)
where by
Q(P,A)Q(A -lp,B) = Q(p,AB) ,
and the scalar product is

(<I>,w) = f dp.(p)(<I>(p),w(p)) ,
and a representation of the Euclidean group
(U(a,R)B)(x) = RB(R-l(x - a)) .
a form which will be derived in Sec. 3. Attempt to define projection operators by the equa-
The comparison shows: tion
(i) When the representation is in the canonical
form (2.4) the measure dp.(P) on momentum space is (E(S)B)(x) = xs(x)B(x) .
just Lebesgue measure dp. Why does not this describe the photon as a localiza-
(ii) The dimension of the vectors <I>(P) is the same ble system? The answer is that the E(S) carry vectors
for all p. satisfying the condition (2.5) into vectors which do
(iii) The operators Q(p,A) are of the form :n(A), not satisfy it, so E(S) is not a well-defined operator
where A ----> :n(A) is a representation of the unitary in the manifold of states and the x in B(x) has noth-
unimodular group. ing to do with localizability.
Intuitively (i) and (ii) are accounted for because, The notion of localizability discussed here is con-
if one makes any state whose x dependence is a cerned with states localized in space at a given time.
a function one gets all momenta. Thus, one would It is natural to inquire whether there exists a corre-
expect to have the same number of linearly inde- sponding property in space-time. Then the E(S)
pendent states for each p. (iii) is essentially a conse- would satisfy
quence of the rotational invariance of the states
localized at a point. U(a,A)E(S)U(a,A)-l = E(AS + a) ,
All three restrictions are nontrivial if applied to an where S is a Borel set of space-time and {a,A I is an
arbitrary representation of 83 • However, as will be inhomogeneous Lorentz transformation of space-
seen in Sees. 5 and 6, (i) and (ii) are always satisfied time translation, a, and homogeneous Lorentz trans-
in any relativistic theory (provided one leaves out formation, A. However, a requirement analogous to
the vacuum state). (iii) excludes a very important (i) follows from Mackey's theory: All four-momenta
physical system, the single photon. One can see this must occur in the theory. This is in flat violation of
immediately by looking at the Q(p,A) for those A the physical requirement that there be a lowest
which leave p invariant. Such Q's have two eigen-
11 That the photon was nonlocalizable was stated and be-
vectors corresponding to right-circularly and left- lieved long before reference 1 was written. See, for example,
circularly polarized photons having angular mo- L. Landau and R. Peierls Z. Physik 62, 188 (1930); 69, 56
mentum along p, ± h, respectively. On the other (1931); especially p. 67 of the latter. While the arguments
given could possibly be regarded as plausible, they do not make
hand, in :n(A) one cannot have states with angular clear what is the heart of the problem.
12 If the neutrino had turned out to possess states of both
momentum ± h along p without also having states helicities, i.e., states with components ±!n. of the component
with zero component of angular momentum along p. of angular momentum along p, then it too would be localizable.
The nonlocalizability of the photon (and all other A neutrino of definite helicity is not localizable.
13 E. P. Wigner, Ann. Math. 40, 149 (1939); Z. Physik 124,
particles of spin >! and mass zero) is a conse- 665 (1947-8).
408 CHAPrER VIII

LOCALIZABILITY IN QUA",TUM MECHANICS 851

energy state. Thus, a sensible notion of localizability of the three-dimensional translation group :r, is
in space-time does not exist. unitary equivalent to one of the following form:

3. RECAPITULATION OF THE UNITARY

REPRESENTATIONS OF 8, THE UNIVERSAL where <I> is an element of a direct integral,
COVERING GROUP OF THE EUCLIDEAN GROUP f!:dp.(p)X p , over :r:
with measure p. and multi-
In this section a canonical form of the representa- plicity function p(P) = dim XP.
tions of 83 will be derived in which the translation A bounded operator, B, which commutes with the
subgroup is diagonalized. operators of the representation can be written in the
Any continuous unitary representation of 83 : form
{a,A} -> U(a,A) gives rise to a continuous unitary (B<I>)(p) = B(p)<I>(p) ,
representation of the translation group :r3: a
-> U(a,I). The first step in the analysis is to de- where B(p) is a bounded operator in Xp and such
scribe all such representations. By a unitary trans-
that for all <1>, and <1>, E f~:dp.(p)Xp,
formation the U(a,l) are to be diagonalized, i.e.,
brought into the form (<I>,(p), B(p) <I>,(p)) is measurable in p.
(U(a,I)<I» (p) = e-;p.a<l>(p) . (3.1) Two such representations a -> U,(a) and a
-> U,(a), with measures /L, and p., and multiplicity
(The minus sign in the exponent is a matter of con-
vention; it is adopted to conform with custom in functions p,(p) and .,(p), respectively, are unitary
quantum mechanics.) equivalent if and only if
For this purpose, the notion of direct integral of (1) p., == p." i.e., p., and /L' give zero measure for
Hilbert spaces and representations is needed. It will the same sets of :r:.
be described briefly in the present special context." (2) ., (P) = ., (p) except, perhaps, in a set of /L'
Let p. be a positive measure on three-dimensional measure zero.
(momentum) space :ri. For each point p of :r:, let
there be given a Hilbert space XP whose dimension For a sketch of a proof of Theorem l, the reader
p(p) is a p.-measurable function of p. Then the direct is referred to Appendix II, and the references quoted
integral of the XP with respect to p. is a Hilbert space there.
denoted f~:dp.(p)Xp whose elements are functions The measures p. and multiplicity functions •
defined on :r:, with values satisfying <I>(p) EX •. appearing in a general representation of the transla-
Furthermore, the elements must satisfy ( \1>, (p), \I>,(p)) tion group are completely arbitrary. Those which
is a p.-measurable function of p for any two can appear in a representation of :r3 obtained by

<1>,,<1>, E J; dp.(p)xp (3.2)

restriction from a representation of 83 are quite
special. This comes about because a -> U(Aa,l)
defines a representation of :r3
which is unitary
[here (<I>'(P),<I>2(P)) is the scalar product in ;!Cp], and equivalent to a -> U(a,l) as a consequence of

J (<I>(p),<I>(p))dp.(p) < 00 • (3.3) U(O,A)U(a,l)U(O,A)-' = U(Aa,l) .

Now when U(a,l) is brought into the diagonal form
The scalar product in f!:dp.(p)Xp is defined by (3.1) by an appropriate unitary transformation, the

(<1>,,<1>,) = J dp.(p)(<I>, (p),<I>, (p)) . (3.4)

representation a -> U(Aa,l) takes the form
(U(Aa,l)<I»(p) = e-i(A-,p).a<l>(p)
With this notation, the following theorem holds.
and this in turn can be brought into the standard
form by the unitary transformation
Theorem 1. Every continuous unitary representation
(W<I>)(p) = <I>(Ap)
14 For a full account of the notion of direct integral see J.
Dixmier, Les algebres d'ope.rateurs dans l'espace Hilbertien which yields
(Gauthier-Villars, Paris, 1957). The theory gives a precise
mathematical meaning to the Dirac formalism of Hrepresen-
tations" in quantum mechanics. See P. A. M. Dirac, The (WU(Aa,l)W-')(W<I»)(p) = e-ip.a(W<I>)(p)
Principles of QWIntum Mechanics (Oxford University Press,
New York, 1947), 3rd ed., Chap. III. and carries the direct integral f~:d/L(p)X, into
LOCALIZATION PROBLEMS 409

852 A. S. WIGHTMAN

J~:dl'(P)XA' where dI'A(P) = dl'(Ap). The unitary Since Q(A) is unitary Q(p,A) must be unitary for
equivalence criterion given in Theorem 1 then almost all p. Furthermore, the group multiplication
implies law implies
I' == P.A (3.5) Q(A)T(A)Q(B)T(B) = Q(AB)T(AB) ,
pep) = p(Ap) for all p except possibly on a set of p. which yields
measure zero. (3.6)
Q(p,A)Q(A-'p,B) = Q(p,AB) (3.10)
Now in Appendix 2, it is shown that the only
measures on :t:
satisfying (3.5) are equivalent to for each A and B and almost all p.
At this point a measure-theoretic technicality
ones of the form
arises. It is possible a priori, that the set of measure
p.o5(p) + dp(ipi)d",(p) , (3.7) zero on which (3.10) does not hold could depend on
where P.o ;;. 0, dw(p) is the area on the sphere of A and B in such a way that when one took the union
radius Ipi and dp is a measure on the positive real over all such sets one would get a set of measure
==
axis. Since, if p. P.I, the unitary mapping greater than zero. Actually, one can show that one
can alter Q(p,A) on a set of measure zero in p so
(W<p) (p) = <p(p) [ dp.(p)JI/2 that Q(A) is unaffected, but (3.10) holds for all
dp.,(p) p,A,B and Q(P,A) is measurable in both variables.
carries the direct integral J~:dp.(P)Xp into This argument is deferred to Appendix IV, because
J~:dl"(P)X" one may for convenience choose p. in of its technical character. The result will be assumed
the form (3.7)." Later on I' will be taken in this form in what follows.
but for the moment a general p. satisfying (3.5) will The representation has now been reduced to the
be carried along. Furthermore since any two Hilbert standard form
space of the same dimension can be mapped on one (U(a,A)<p)(p) = e-,p'aQ(p,A)
another by unitary transformation, there is no loss
in generality in taking X. = XAP for all A. X <p(A -1 ) [dl'(A -l p )JI/2 (3.11)
The next task is to put the operators U(O,A) in P dp.(p)
standard form. They will be written as a product To understand the physical meaning of the Q(p,A)
U(O,A) = Q(A)T(A) where T(A) is defined by it is helpful to consider some elementary examples.
For a single free particle in Schr6dinger theory, the
(T(A)<p)(p) = <p(A-'p) [dp.(A- l p)JI 1 2 wave function may be taken as a complex-valued
dp.(p)
function of p, the scalar product is

f
(Here the convention X. = X Ap has made it possible
to equate vectors from two different Hilbert spaces.) (<p,'Y) = dp <p(p)*'Y(p) (3.12)
It is easy to verify that T(A) is unitary, with an
adjoint given by and the representation of the Euclidean group is

(T(A)*<p) (p) = <p(Ap) [dp.(AP)JI /2 (3.8) (U(a,A)<p)(p) = e-,p'a<p(A -Ip). (3.13)

dp.(p)
Thus, for a single free particle of spin zero Q(p,A)
An elementary computation shows that = 1. On the other hand, for a single free particle of
T(A)U(a,I)T(A)-l = U(Aa,l) , spin-! (Pauli theory), <P has two components, the
integrand in the formula (3.4) for the scalar product
so U(O,A) and T(A) satisfy the same commutation is
relation with U(a,I). Therefore Q(A) commutes with 2
U(a,1). Thus, by Theorem 1, Q(A) can be written L <p,(p)*'Y,(p) ,
in the form i-l

and the transformation law (3.11) becomes

(Q(A)<p)(p) = Q(p,A)<p(p) (3.9)
(U(a,A)<p)(p) = e-'··aA <P (A -I p ).
15 We here use the Radon-Nikodym theorem which asserts
that if two measures 1-'1 and J.l2 are equivalent, i.e., take the Here, evidently Q(p,A) = A and describes the
value zero for the same sets, then there exists a positive
measurable function pep) such that d!'l(p) ~ P(P)d!,2(P)' transformation properties of the spin degree of
P(p) is customarily denoted (d!'I(d!',Xp). See, for example, P. freedom. In this case Q(p,A) is independent of p. To
R. Halmos, Measure Theory (D. Van Nostrand Company, Inc.,
Princeten, New Jersey, 1950), p. 128. get an example in which Q(p,A) cannot be brought
410 CHAPfER VIII

LOCALIZABILITY IN QUANTUM MECHANICS 853

by unitary transformation of U(a,A) to a form in [pi such that for all other p and all A, (3.17) holds.
independent of p one can consider the case of a single Next, it will be shown that, if there exists a V(P)
photon described in Sec. 5. [One of the results of for a single p which satisfies
section 4 is that for a localizable system Q(p,A) can
always be chosen independent of p.] Clearly, in all Ql(P,A) = V(P)Q.(p,A)V(p)* (3.18)
these examples the Q(p,A) gives the transformation for all A in the little group of p, then V(q) can be
law of the internal degrees of freedom of the system extended to all q with [q[ = [pi so that (3.17) holds.
under rotations. (The statement holds trivially for p = 0 so p ~ 0 is
A detailed analysis of the consequences of the assumed.) Solved for V(A-lp), (3.17) reads
multiplication law of the Q's, Eq. (3.10), will be
undertaken shortly. For the moment, only the fact V(A-lp) = Ql(p,A)-lV(p)Q,(p,A). (3.19)
that for those A which satisfy Ap = p, (3.10) This will be consistent as a definition of V at A -lp
implies only if the right-hand side is constant on right cosets
of the little group of p, i.e., only if Al' = A.lA s'
Q(P,A)Q(p,B) = Q(p,AB) (3.14) with A,l in the little group of p implies that the
is needed. Such A form a group called the litae grQUp right-hand side of (3.19) takes the same value for
of p, and (3.14) means that A -> Q(p,A) defines a A = Al and A,:
continuous unitary representation of the little group
of p. (Again see Appendix IV for a proof that every Ql (p,Al)-lV(p)Q.(p,A ,)
measurable unitary representation is continuous.) = [Ql (P,A.)Ql (A;lp,A.WlV(p)
Evidently, when p = 0 the little group of p is the
X [Q,(P,A.)Q.(A;lp,A.)]
group of all A, i.e., the unitary unimodular group
itself. On the other hand, when p ~ 0, the little = Ql (P,A,) -l[Ql (p,A.) -IV (P) Q,(P,A 3 )]Q. (P,A.)
group is the two sheeted covering group of the group = Ql(p,A,)-lV(p)Q.(p,A.) .
of rotations around a fixed axis. It is therefore
This defines V(q) for all q with [q[ = [pl. Next, it
isomorphic to the multiplicative group of the com-
has to be verified that V so defined satisfies
plex numbers e"/', 0 '" 8 < 471'.
The problem of determining when two representa- Ql(q, A) = V(q)Q,(q,A)V(A-lq)-l. (3.20)
tions of &. are unitary equivalent can now be reduced Suppose that q = B-lp. Then, the right-hand side
to a related problem for their Q(p,A). For, if {a,A} of (3.20) is
-> Ul(a,A) and {a,A}-> U.(a,A) are equivalent
representations, Theorem 1 implies PI == P. and [Ql (p,B) -1 V (p) Q. (p,B)] Q. (B-lp,A)
" =" almost everywhere. Thus, by a unitary X [Q,(p,BA)-lV(p)Q.(p,BA)r l
transformation one can bring Ul(a,A) into a form
where Ul(a,l) = U.(a,1). Then Ul and U. differ = Ql(P,B)-lQl(P,BA) = Ql(q,A) ,
only in their Q(P,A). If where, in the last step, the identity Ql (p,B)-l
= Ql(B-lp,B-l) which follows from (3.10), has been
Ul(a,A) = VU.(a,A) V-I , (3.15)
used.
where V is a unitary operator, then, applying Therefore, a necessary and sufficient condition
Theorem 1, one finds that V is of the form that Ul be unitary equivalent to U, is PI = P" '1
= " almost everywhere and the representations of
(V<J»(P) = V(P)<J>(p) (3.16)
the little groups A -> Ql(p,A), A -> Q,(p,A) be
and (3.15) reduces to unitary equivalent for almost all [pi and at least one
p for each [pl.
Ql(p,A) = V(P)Q.(p,A)V(A-lp)-l. (3.17)
Incidentally, in the course of the argument, it has
If p rather than A -lp occurred in the last factor, this been established that the little groups for p and q
would describe unitary equivalence of Ql(P,A) and have unitary equivalent representations if [pi = [q[.
Q.(p,A). When A belongs to the little group of p, Explicitly, if q = Bp and Aq = q, then B-lABp = P
A -lp = P and that is indeed the case. and
Again at this point a measure-theoretic technicality
arises. Equation (3.17) holds for almost all p, for Q(q,A) = Q(p,B-l)-lQ(p,B-lAB)Q(p,B-l ). (3.21)
each A. Again the reader is referred to Appendix IV The mapping A -> B-lAB is an isomorphism be-
for a proof that there is a fixed set of measure zero tween the little groups of q and p and (3.21) displays
LOCALIZATION PROBLEMS 411

854 A. S. WIGHTMAN

the unitary equivalence of the corresponding repre- and almost all Ipl
sentations.
The classification of the unitary inequivalent no/') = noP) for all j = 0, " 1, j,' ...
representations of the little groups is well known. Any Euclidean invariant theory has a manifold
For p = 0, they are labeled by giving an intf~r­ of states whose transformation law is unitary equiva-
valued multiplicity function no; for j = 0, + " lent to one of this form. It is to be expected (and may
l,j,' . '. no; is the number of times the irreducible be seen in detail from the discussion of Sees. 6 and 7)

°
representation of angular momentum j appears. For
p rf. the unitary inequivalent representations are
labeled by an integer or +
infinity valued function,
that the imposition of requirements of relativistic
invariance will eliminate some of these representa-
tions.
n"".,m = 0, ± t, ± 1" ., where npm is the number Up to this point, the only assumption that has
of times the one-dimensional irreducible representa- been made about the quantum mechanical system
tion </> -> e'm. occurs. under consideration is its invariance under the
All these results are collected in Theorem 2. Euclidean group. Now the operation of time in-
version I, will be adjoined. It is well known that I,
Theorem 2. Every continuous unitary representa- has to be represented by an antiunitary operator,
tion of ea, the universal covering group of the U(I,), whose square is w(I,) = ± 1, and that by
Euclidean group, is unitary equivalent to one of the suitable choice of phase it can be arranged that ' •
following form.
Let Xp be a family of Hilbert spaces, one for each U(I,)U(a,A)U(I,)-' = U(a,A)
pE :r:
r
identical for all p with the same Ipl. Let U(a,A)U(l,) = U({a,A}I,)

X = dp(lpl)dw(p)3Cp$ Xu f U ({ a"A.} I ,)U ({ a.,A.} I,)

= w(l,)U({a.,A}I,{a.,A}I,)
where dp is a non-negative measure on the positive U({a"AdI,)U({a.,A.}) = U({a.,AdI,{a.,A.}).
real axis, such that p(p) = dim X P is measurable,
Notice that if w(I,) = -1, this is a representation
and dw(p) is the measure on the unit sphere of the
only up to a sign.
vectors p/lpl, invariant under rotations. Xu, the
To get a standard form for U(1,) when U(a,A)
contribution from p = 0, mayor may not occur.
is in the form (3.12), an extension of Theorem 1 to
The representation is defined by
the case of antiunitary operators is needed. It will
(U(a,A)<l»(p) = e-,p·aQ(p,A)<l>(A-'p) (3.22) be assumed here. The result is
where Q(p,A) is a unitary operator in XP satisfying (U(l,)</»(p) = Q(p,l,)4>(-p)*, (3.23)
Q(p,l) = 1 and with the Q unitary operators satisfying
Q(p,A)Q(A-'p,B) = Q(p,AB) Q(p,I.)Q(-p,I.)* = w(l.)
for all A, B. and
Two representations U. and U. are unitary equiva-
lent if and only if Q(P,I,)Q( -p,A)* = Q(P,A)Q(A -'p,l,) .
(1) p. = P" i.e., the measures p. and p. have the The full analysis of these equations yields a proof
same null sets as measures on the positive real axis. that two representations including time inversion
(2) Xo either occurs or not in both representations are unitary equivalent if and only if their measures
(3) p.(p) = P.(p) for almost ail p. p. are equivalent, they have the same multiplicity
(4) the representations of the little groups whose functions, and their representations of the little
elements are all A such that Ap = p given by group extended by time inversion:
A -> Q.(p,A) A -> Q.(p,A) A -> Q(p,A) Ap = p
are unitary equivalent for almost allipl. I, -> Q(p,I,)K
The conditions (2), (3), and (4) are satisfied if the
are equivalent for at least one p and almost all Ipl.
multiplicity functions of the representations of the
little groups satisfy 16 See E. P. Wigner, Group Theory and Its Application to
the Quantum Mechanics of Atomic Spectra (Academic Press
nlp!m (1) = nlpl m(2) for all m = 0, ± !, ± 1" .. Inc., 1959), Chap. 26.
412 CHAPTER VIII

LOCALIZABILITY IN QUANTUM MECHANICS 855

Here K stands for complex conjugation. Only a spe- (b), (c) also satisfies (a), (b), (c) so these functions
cial case will be considered here, namely, that in form a vector space. If a scalar product of eI> and w
which Q(P,A) = :D(A) where A -> :D(A) is a continu- is defined

f
ous unitary representation of the unimodular group.
As will be shown in the next section, for localizable (eI>,w) = da (eI> (a,A) w(a,A»
systems this can always be arranged. A second spe-
cialization will be made. Only time inversion trans- the vector space becomes a Hilbert space :IC.'. The
formation laws for which representation [jD is defined in :IC by

Q(P,I.) = :D(/) (3.24) (U(a,A)eI» (b,B) = eI>(b + Ba,BA). (4.2)

will be considered. This amounts to considering the This representation possesses a transitive system of
case of ordinary type." Time inversion invariance imprimitivity defined by
will be used only to get Theorem 4 on the unique- (E(S)eI>)(a,A) = xs(a)eI>(a,A)
ness of the position observables.
defined for Borel sets S of :to where, as usual, Xs is
the characteristic function of S: xs(a) = 1 if a
4. REPRESENTATIONS OF 8. WHICH POSSESS A
E S, 0 if a Et: S. It is easy to verify using (4.2) that
TRANSITIVE SYSTEM OF IMPRIMITIVITY'
the E(S) transform correctly under U(a,A), i.e.,
The discussion of this section is in three parts. satisfy V.
First, Mackey's standard form of an imprimitive Because of the smooth fashion in which A acts
representation is given and shown to be equivalent, on a this representation can be put in a simpler form.
in the special case at hand, to a simpler form which If, for the moment, attention is restricted to con-
will be more convenient for present purposes. Second, tinuous functions eI>(a,A), Eq. (4.1) can be used to
for a given imprimitive representation a unitary write
transformation is found which brings it into Mackey's
form. Third, the unitary transformations which eI>(a,A) = :D (A) eI>(A -'a,l) (4.3)
commute with an imprimitive representation U(a,A) which expresses eI>(a,A) for general values of A in
but not with its system of imprimitivity E(S) are terms of its value for A = 1. Conversely, given any
parametrized. This yields a parametrization of the continuous function eI>(a) with values in :IC(:D), one
nonuniqueness in the definition of a position opera- can define a continuous eI>(a,A) by (4.3) and it will
tor. then satisfy (4.1). The scalar product of two such
Suppose there is given a continuous unitary eI>(a) and w(a), Jda(q;(a),'l'(a» is equal to that of
representation A -> :D(A) of the 2 X 2 unitary the corresponding eI>(a,A),w(a,A) so the one to one
unimodular group in a Hilbert space :IC(:D). Then correspondence can be extended by continuity to a
the representation of 8, iruiuced by :D(A) is denoted unitary mapping between the Hilbert space :IC and
[jD and constructed as follows. Consider functions the Hilbert space of the measurable square integrable
eI>(a,A) which are defined on 83, whose values lie eI>(a).
in :IC(:D), and which satisfy The representation (4.2) determines a correspond-
(a) (eI>(a,A),x) is a measurable function of .Ia,A}, ing representation on the eI>(a) given by
for every X E :IC(:D). [The indicated scalar product
is in :IC(:D).] :D(B) (U (a,A)eI» (B-'b)

(b) eI>(Ab,AB) = :D(A)eI>(b,B) (4.1)

= :D(BA)eI>«BA)-'(b + Ba»
or
(c) flleI>(a,A)II'da < '" (U(a,A)eI>)(b) = :D(A)eI>(A-'(b + a».
Notice that (b) implies Now this looks just like the standard form of
Euclidean transformation appearing in SchrOdinger
(eI>(a,A),w(a,A» = (eI>(Ba,BA),w(Ba,BA»
so the integral in (c) is independent of A. Clearly, 18 The 'deta.ils of the proof involve identifying functions
which differ only on " set of m....ure zero and establiehing
any linear combination of functions satisfying (a), that the space is closed. For a proof which is easily adapted to
the present circumstances see M. H. Stone, Linear Tram!tN-
mations in Hilbert Space (American Mathematical Society,
17 See reference 16, especially pp. 343-344. Providence, Rhode Island, 1932), pp. 23-32.
LOCALIZATION PROBLEMS 413

856 A. S. WIGHTMAN

theory except that there one has -a instead of a on since G is a group. The positivity of the quadratic
the right-hand side. That just means that one uses as form then implies that the determinant of its matrix
representative of the function cI>( - b) instead of is positive, i.e.,
cI>(b). This will be done from this point on. Thus, in
the present context, Mackey's form of the imprimi-
1<p(g)1 < <p(e) .
tive representation induced by ~ may be taken as Any unitary representation of G, g --t U(g), yields
examples of positive definite functions'·
(U(a,A)cI>)(b) = ~(A)cI>(A-'(b - a)) (4.4)
<p(g) = (tJ.>,u(g)cI»
(E(S)cI>)(b) = xs(b)cI>(b) (4.5)
because, in this case,
with the scalar product
I: a,*ak<P(g,'g.) = III: a,U(g,)cI>ll':> O.
(cI>,'l1) = Jdb (cI>(b),'l1(b)) . (4.6) If the representation is continuous then <p(g) is con-
tinuous.
Now, the second step of the argument is under- Conversely, given a continuous positive definite
taken; it is to be shown that for each pair consisting function one can construct a continuous representa-
of a continuous unitary representation {a,A I tion of G. Let r, s be complex-valued functions on G
--t U(a,A) and a system of imprimitivity E(S), there which are different from zero only at a finite number
exists a unitary operator V such that VU(a,A)V-' of points. (Such functions form a vector space.)
and VE(S)V-' are of the form (4.4) and (4.5), re- Introduce the form
spectively. Available to show this are several lines of
argument, not one of them trivial. Here the elegant (r,s) = I: reg) <p(g -'h)s(h) (4.9)
(J,hEG
proof of Loomis" will be written out for the present
(r,s) is sesqui-linear, i.e.,
simple case.
The first step in the argument is to express the (r,s! + S2) = (r,s,) + (r,s,) , (r,as) = a(r,s) (4.10)
problem in terms of certain complex-valued functions
defined on the group. This is quite analogous to the (r,s) = (s,r) (4.11)
study of general unitary representations in terms of by virtue of (4.8), and
positive definite functions on the group. To motivate
Loomis' method, a brief sketch will first be given of (r,r) :> O. (4.12)
the relation of positive definite functions and repre- Now it may happen that there are some r for which
sentations. (r,r) = O.
A function <p defined on a group G is positive
definite if for each n = 1,2" .. and all complex If so, it is easy to see that they form a linear subspace
numbers a, ... Un and g, ... gn E G and the components orthogonal to this linear sub-
space form a vector space on which (r,s) again
satisfies (4.10), (4.11), and (4.12) but, in addition,
(r,r) = 0 implies r = O. This space mayor may not
Clearly, taking n = 1, one gets be complete. If not, complete it and get a Hilbert
space H•. To get a continuous representation of G in
<p(e):> O. (4.7) H., define, first on functions with only a finite num-
For n = 2, ber of values different from zero,

(lad' + la,I')<p(e) + a,*a,<p(g~'g,) (U(g)r)(h) = r(g-'h) (4.13)

+ a,*a,<p(g;'g,) :;;. 0 . with the inverse U(g-').
So defined U(g) leaves the scalar product invariant
From the reality of the left-hand side one concludes
<p(g,'g,) = <p(g;:'g,)* which is equivalent to (U(g)r,u(g)s) = I:•.• r(g-'h)*<p(h-'k)s(g-'k)
<p(g-') = <p(g)* all g E G, (4.8)
= I: r(h')*<p«gh')-' (gk'))s(k') = (r,s)

,. See reference 6. One of the main virtues of Loomis' treat- ce:;:r~~~v~:.,~:. f~fc~:h:.~re .!"dd ~J:~l-R~~o~ }~~
ment is that it applies to nonseparable HilbertsJlaces. Since Abelian and locally-eompact groups, respectively. A system-
separability is assumed here th,s advantage will not be ap- atic account of their properties is found in R. Godement, Trans.
parent. Am. Math. Soc. 63, 1 (1948).
414 CHAPfER VIII

LOCALIZABILITY IN QUANTUM MECHANICS 857

so the subspace of those r for which (r,r) = 0 is left is a cyclic vector for the representation defined above.
invariant by U(g). Therefore, so is its orthogonal Thus, what has been established in the preceding
complement. Because U(g) is therefore defined and paragraphs is that all cyclic representations are
continuous on a dense subset of H. it can be extended unitary equivalent to those of the form (4.9) and
by continuity to be a unitary operator in H•. Clearly, (4.13). Since any representation can be written as a
on the original functions direct sum of cyclic representations, it suffices for
many purposes to study cyclic representations.
U(gdU(g,) = U(g,g,) , In the present case, there is a system of imprimi-
so by continuity, g -> U(g) defines a representation. tivity E(S) in addition to the group representation
To prove U(g) is continuous in g, consider U (g) so one has to consider cyclic vectors and repre-
sentations of E(S) and U(g) together. This suggests
II(Ug) - U(g'))rll' = II(U(g-'g') - 1)rll' studying the function (E(S)cf!,U(g)cf!) = 'Po (g) and
= 2[(r,r) - Re (U(g-'g')r,r)]. using it to construct a pair unitary equivalent to
{E(S),U(g) I and in Mackey's form.
Clearly, this equation implies that it suffices to Now return to the special case of 8•. When the
verify (U(g)r,r) is continuous in g at g = e for all
representation and system of imprimitivity is in
r E H•. For r of the special kind appearing in (4.9), Mackey's form (4.4) and (4.5), the function 'Po(a,A)
which only take values different from zero at a finite is
number of points the continuity is easy to verify:

(U(g)r,r) = Lo .• reg -'h)*'P(h-'k)r(k) 'Po(a,A) = f db((E(S)cf!)(b),(U(a,A)cf!)(b))

= L •.• r(h)*'P(gh)-'k)r(k) ,
which clearly converges to (r,r) as g -> e because 'P
= f db(cf!(b),~(A)cf!(A-'(b
o
- a))) (4.14)

is continuous and there is only a finite number of The next task is to show that 'Po(a,A) has a form
terms in the sum. For a general r, there always exists closely related to this for any representation and
an 8 of the above form so that Ilr - 811 < ./3. By transitive system of imprimitivity.
the above argument a neighborhood of e can be Before the discussion can begin a preliminary re-
found so that II U(g)s - sll < ./3. Then mark is necessary. Extensive use is going to be made
of the part of the Radon-Nikodym theorem which
IIU(g)r - Til < IIU(g)r - U(g)sll + IIU(g)s - sll says that if, for two measures p" and p", p" (S) = 0
+ lis - rll, implies P,.(S) = 0, then there exists a measurable
function p(x) such that dp.(x) = p(x)dp, (x). To make
which completes the proof that g U(g) is a continuous these applications it is essential to know that E(S)
unitary representation of G. = 0 for all Borel sets S of Lebesgue measure zero.
Actually, if the continuous positive definite func- To obtain this result, it is convenient to use the fact
tion from which one starts is of the form (cf!, V(g)cf!), that the E(S) possess a separating vector, i.e., a vector
the representation constructed by the above process cf! such that E(S)cf! = 0 implies E(S) = O. Although
will be closely related to V itself. For, if the subspace this is a standard result" a proof will be outlined.
(of the Hilbert space X in which 'P lies) spanned by Choose an arbitrary unit vector cf!" and let X, be
vectors of the form V(g)cf! is denoted X, the con- the subspace spanned by the E(S)cf!,. Choose a unit
structed representation as unitary equivalent to the vector cf!. orthogonal to X, and let X, be the sub-
restriction of V to X. The required unitary equiva- space spanned by the E(S) cf!•. Continuing in this
lence is obtained by making ~ r(g)V(g) correspond way one gets a family of orthogonal subspaces such
to r, for r differing from zero only at a finite number that X is the direct sum of the X, and cf!, is a cyclic
of points. Equation (4.9) is just arranged to make unit vector for X,. Take as separating vector cf!
scalar products correspond. Clearly (U(g)r) corre- = ~.2-·cf!•. Clearly, if E(T)cf! = 0 then E(T)cf!, = 0
sponds to V(g) ~ r(h) V(h) cf!. The correspondence for all i. Consequently, E(T) yields zero when
can be extended by continuity to yield the required applied to a dense set of vectors, the linear combina-
unitary equivalence. tions of the E(S) cf!,. It is therefore zero and cf! is a
A representation V for which there is a vector cf! separating vector. Note first that if E(S) = 0, then
such that the V(g) cf! span the representation space E(AS + a) = U(a,A)E(S)U(a,A)-' = O. Thus if cf!
is called cyclic and cf! is then a cyclic vector. Note that
the function which is one at g = e and zero elsewhere 21 See reference 14, p. 20.
LOCALIZATION PROBLEMS 415

858 A. S. WIGHTMAN

is a separating vector, (<JI,E(S)<JI) = IIE(S)<JIII' is absolute value less than or equal to 1 such that
quasi-invariant under Euclidean transformation, i.e.,
for all {a,A), (<JI,E(S)<JI) = oif and only if (<JI,E{AS
+ a)<JI) = O. Furthermore, (<JI,E(S)<JI) defines a (1-
fdadA\08(a,A) = ffT s
dadAdbq(a,A;b)p(b). (4.17)
additive positive measure on the Borel sets S of R'. From (4.17), it follows that
Now in Appendix II it is shown that any measure
defined on the Borel sets of R' and quasi-invariant
under translations is equivalent to Lebesgue measure. \Os(a,A) = f s
dbq(a,A;b)p(b)
That implies in particular that (<JI,E(S) <JI) and there-
fore E(S) = 0 whenever S is a Borel set of measure for almost all {a,A} which begins to look like (4.14).
zero. Thus, the Radon-Nikodymn theorem implies This completes the first stage of the proof.
that if <JI is any vector there exists a non-negative The next stage is the construction of the Hilbert
measurable function p such that space of the <JI(a) which appears in (4:4) ... (4.6).
This is done in close analogy with the construction
(<JI,E(S)<JI) = fs
p(b)db (4.15)
carried out in connection with (4.9) but for technical
reasons which will appear in the proof it is convenient
to consider continuous functions of compact support
p(b) is clearly integrable over all space. on &. rather than the functions differing from zero
This equation can be used to get an expression only at a finite number of points, which were used
for \Os(a,A) which is the first step in proving that it there. Therefore let f and g be continuous complex-
can always be arranged to have the form (4.14). valued functions of compact support on &, and define
Note that

l\Os(a,A) I = I(E(S)<JI,U(a,A)<JI) I < IIE(S)<JIII UU) = f dbdBf(b,B)U(b,B)

where for convenience it has been assumed that II <JIll
= 1. It therefore follows that if T is any Borel set U(g) = f dcdCg(c,C)U(c,C) . (4.18)
of &. Then

f•
\Os(a,A)dadA < f
•
dadA f s
p(b)db. (4.16) (E(S)UU)<JI,U(a,A)U(g)<JI) = f dbdB f dcdCf(b,B)*

Now the left-hand side as a function of Sand Tis X g(c,C) (E(S)U(b,B)<JI,U(a,A)U(c,C)<JI)

initially defined for products of rectangles S X T,
and is bounded by the product p.(S)v(T) where p.(S) = f dbdB f dcdCf(b,B)*g(c,C) (E(B-'S - B-'b)<JI
= Jsp(b)db and veT) is the Lebesgue measure of T.
It is finitely additive on such products in the sense X U({b,B}-'{a,A}{c,C})<JI) = fdrfdbdB
that if S X T = US, X T, where (S, X T,) n (S;
X T;) = 0 for i ... j, then its value on S X T is the
sum of its values on the S. X T,. Consequently, it
X fB-IS-B-'b
dcdCf(b,B)*g(c,C)
has a unique extension to a Borel measure on R' X q({b,B}-'{a,A}{c,C} ;r)p(r) ,
X &,. [The necessary argument first shows that it
can be extended to be finitely additive on sets which
are arbitrary finite unions of disjoint products
= f s
dr f dbdB f dcdCf(b,B)*g(c,C)

S, X T,. Second, it shows that the boundedness X q({b,Br'{a,A}{c,C} ;B-'(r - b»

described by (4.16) implies that this extension is X p(B-'(r - b» . (4.19)
actually (1 additive. Finally, it uses a standard ex- For {a,A} = {O,l} this reduces to
tension theorem" to assert that the resulting set
function has a unique extension to be a Borel
measure.] Clearly, this measure is bounded by the (E(S)U(f)<JI,u(g)<JI) = f 8
dr f dbdB f dcdC
product measure p. X v. Thus again by the Radon X feb + r,B)*g(c + r,C)q({b,B}-'{c,C} ;-B-'b)
Nikodym theorem this time for complex valued
measures there exists a measurable function q of X p( -B-'b) , (4.20)
which suggests introducing (U(f)<JI)(r) as the func-
22 See reference 15, p. 54, Theorem A. tionf(b + r,B) of band B.
416 CHAPTER VIII

LOCALIZABILITY IN QUANTUM MECHANICS 85U

Then the form in the subspace spanned by the U(f)~ into a dense

f f
«U(f)~)(r),(U(g)~)(r)) = dbdB dedC
set of vectors in the Hilbert space spanned by the
functions of r: (U(f)~)(r) and preserves scalar prod-
ucts it can be extended by continuity to become a
Xf(b + r,B)*g(e + r,C)q({b,Bj-' {e,C}, -B-'b) unitary transformation V.
X p( -B-'b) (4.21) All this discussion is collected in Theorem 3.
is suggested as the scalar product appearing in the
Theorem 3. Let {a,Aj-- U(a,A) be a continuous
integrand of (4.14).
unitary representation of s. with a transitive system
With these definitions, one has
of imprimitivity, E(S), based on R'. Then there
(E(S)U(f)~,U(a,A)U(g)~) = f8
dr
exists a unitary transformation V, such that
VU(a,A) V-' = W(a,A) and VE(S) V-' = F(S), .re-
X «U(f)~)(r),W(a,A)(U(f)~))(r)), (4.22)
spectively, given by

where W(a,A) is the operator defined by (W(a,A)~)(b) = !D(A)~(A-'(b - a)) (4.26)

feb + r,B) --f(r'b + A-'(r - a),A-'B). (F(S)~)(b) = xs(b)~(b) . (4.27)

Notice that if linear transformation !D(A) is defined Here A -> :O(A) is a continuous unitary representa-
by the correspondence tion of the 2 X 2 unitary unimodular group in a sep-
arable Hilbert space X and the ~(b) are functions on
feb + r,B) --f(r'b + r,A-'B). (4.23)
R3 with values in X which are measurable in the sense
Then W may be written that for all pairs of such functions (~(b),ir(b)) is a
measurable function of b. In symbols
(W(a,A)U(f)~)(r) = !D(A)(U(f)~)(A-'(r - a))
so that W is precisely of the form (4.4). It is obvious
(by a simple change of variable) that !D(A) leaves the
X = til
R'
dbX p with x p = X .

scalar product (4.21) invariant. The remaining task of this section is to examine
Now it has to be verified that (4.21) does indeed the arbitrariness in the definition of the position
define a scalar product. First note that it is linear in observable. For this purpose, one can bring the pair
g and conjugate linear in f. Furthermore, because {E(S),U(a,A) j into the form (4.26) and (4.27), and
(4.20) holds for every Borel set S, then determine all unitary operators which commute
(E(S)U(f)<I>,U(y)<I» = [(E(S)U(y)<I>,U(f)<I»] with U(a,A) but not with E(S). It is convenient for
this purpose to rewrite (4.26) in momentum space
and
(U(a,A)~)(p) = e-'P··:O(A)~(A-'p).
(E(S)U(f)~,U(f)~) >0
imply If B is a unitary operator such that [B,U(a,l)] = 0,
Theorem 1 shows that B can be written in the form
«U(f)~) (r), (U(g)~) (r))
(B~)(p) = B(p)~(p)
= [«U(g)~)(r),(U(f)~)(r))]* (4.24)
where B(P) is a unitary operator in XP = X. The
and
commutativity with U(O,A) then implies
«U(f)~)(r),(U(f)~)(r)) >0 (4.25)
B(o):O(A) = :o(A)B(A -'p) (4.28)
for almost all r. However, since f and yare continuous
and of compact support the integral appearing in r for almost all p.
is continuous in r. Therefore (4.24) and (4.25) hold This equation can be discussed along lines familiar
for all r. Now, just as in the case of (4.12), one can from Sec. 3 and· Appendix IV. For those A which
introduce components of vectors orthogonal to the satisfy Ap = p, i.e., for A in the little group of p,
subspace for which (4.25) is an equality, and com- (4.28) reduces to
plete the resulting space to get a Hilbert space X
B(p):o(A) = :o(A)B(p) . (4.29)
the same for each r. A -- !D(A) is then a continuous
unitary representation inX. Since the correspondence The set of all B(p) satisfying this equation is easy
U(f)~ -- (U(f)~)(r) carries a dense set of vectors to compute. Supposing them known one gets the
LOCALIZATION PROBLEMS 417

860 A. S. WIGHTMAN

general solution of (4.28) by using it as a definition: Here

B(p) = ~(Aq"prl B(q)~(Aq+p) . (4.30)
Here the Aq+p satisfy Aq+pp = q and parametrize
the cosets of the little group of q. By virtue of (4.29)
at q any parametrization yields the same B(p). An and B(q) is a solution of
argument just like that following equation (3.20)
[B(q),~(A)l = 0
shows that the B(p) defined for a fixed q/lql and all
p by (4.30) satisfies (4.28). for all A satisfying Aq = q.
To obtain all solutions of (4.29), one can decom-
pose A -> ~(A) into irreducible representations of In the discussion up to this point, symmetry under
the 2 X 2 unitary unimodular group, and these in time inversion and any analog of the regularity
turn into irreducible representations of the little assumption of Newton and Wigner have been
group ignored. This is natural in the case of Theorem 3
~ = L n;'J/;) , because the canonical form of a transitive system of
;=0,1/2,1,'" imprimitivity can be obtained without the use of
~(;) = L
m--i.-j-l,···j
e'm•. these additional assumptions. However, for Theorem
4, they are of decisive importance. Even in this case
Thus A -> ~(A) restricted to the little group of p ~(A) one dimensional, (4.33) would give a wide
is unitary equivalent to variety of distinct position observable (B(q)) is then
'" 'm. (4.31)
a complex-valued function of the form B(q)
£.....nme
= e"<I.I>,7} real. In this case, the effect of the assump-
with
tion of time inversion invariance is to force B to be
nm = L: n;. real, and therefore to be equal to + 1. However, it
j~lml

Here the summation over j is over integers if m is could be + 1 for some Ipi and -1 for others without
integral and half-odd integers if m is half an odd violating either Euclidean or time inversion in-
integer. variance. It is here that Newton and Wigner's
TheB(p) corresponding to a given set {nm},m = 0, assumption of regularity has the effect of making
± !, ± 1,.·· is a direct sum of unitary operators B a constant and F(S) = E(S). They require (in a
acting in the subspaces of vectors with a definite Lorentz invariant theory) that the infinitesimal
value of m, and any such defines a possible B(p). Lorentz transformation operators be applicable to
The number of real parameters free in an arbitrary localized states in the sense that if <I>,.is a sequence of
nm X n" unitary matrix is n!. so that B(p) contains vectors which converge to a state localized at a point
Lm n!. arbitrary real parameters, each of which could a, as n -> 00, then lim.~. 11M,,<1>.11/11 <1>.11 < 00.
be a function of Ipl. Since M."i = 1,2,3 are essentially differentiation op-
Collecting the information acquired in the preced- erators this forces continuity on the momentum space
ing discussion one has Theorem 4. representation of Newton and Wigner's localized
(continuum) state. An analogous requirement in the
Theorem 4. If E(S) is a system of imprimitivity present formulation has an analogous consequence.
for the unitary representation {a,A} -> U(a,A) of The details are as follows.
03 in the standard form (4.26), (4.27), then all other According to (3.14), the transformation law of
systems of imprimitivity consistent with U are given states under time inversion is of the form
by
(U(I.)<I>)(p) = ~(/)<I>( -p)*
F(S) = BE(S)B- 1 ,
where B is a unitary operator given by The requirement that B commute with U(I,) then
forces
(B<I»(p) = ~(Aq+p)-lB(q)~(Aq+p)<I>(p) (4.32)
so that
(F(S)<I»(p) = ~(Aq.p)-lB(q)-l~(A.+p) which is
X (2 .. - 3 /2 jxs(p - r)ddl(A ••,)-'B(q)-' ~(/)~(Aq+ .... )-lB(q)~(Aq+ ....)

X ~(A.+,)<I>(r) . (4.33) = ~(A.+p)-lB(q)~(Aq+p)~(/)

418 CHAPTER VIII

LOCALIZABILITY IN QUANTUM MECHANICS 861

or using to be differentiable at p -q. In summary, we

have Theorem 5.
:D(A) = :D(r')-':D(A):D(r'),
:D(Aq+ -.,A;~ _p):D(r')B(q) Theorem 5. In a Euclidean invariant system with
= B(q):D(Aq+.,A~~p):D(r') . (4.35) time inversion symmetry the possible observables
F(S) which describe localization are given by (4.33)
For suitably chosen p the factors in :D cancel and one with B(q) a real unitary operator.
gets If localized states are differentiable in p and A
B(q) = B(q) , (4.36) -> :D(A) is an irreducible representation of the 2 X 2
unitary unimodular group then B(p) is a constant
provided that q is not along the 2 axis as will be multiple of the identity and F(S) = E(S). Converse-
assumed. The remaining condition on B(q) says that ly, if the F(S) are unique A -> :D(A) must be irre-
it commutes with all :D(Aq+pAq~-p ir'). This will be ducible and localized states differentiable.
no further restriction since we may for convenience
choose q along the 3-axis and then every such trans- In the following in Theorems 6 and 7, the regu-
formation is an element of the little group of q, and larity and time inversion invariance requirements
B(q) already commutes with them. [To see this it assumed in Theorem 5 will always be taken for
is convenient to choose a particular form for the granted.
Ap~q:

5. REPRESENTATIONS OF &, WHICH ARE

( q.p )-'/'
A q+p = 2 1 + IPllql RESTRICTIONS OF REPRESENTATIONS OF THE
COVERING GROUP OF THE INHOMOGENEOUS

X[1 + I;I'I~I - ifp~q~ .~ J. LORENTZ GROUP

It is well known that every continuous unitary
This is well defined for all p ~ -q. representation of the covering group of the Poincare
Then group is unitary equivalent to one of the form
r'
A q+pn.q+_p . (p X q) (U(a,A)ol'>)(p) = e,p·aQ (p,A) ol'>(A -lp)
= -t~.~.

in X,
It is easy to choose p so that (p X q/lpllql'~) r' = 1;
then (4.36) follows. However p is chosen provided
q is along the 3 axis (q X p/lqllpl) '~T' leaves q in-
variant. This proves the second statement.] with
A comparison of these statements with the discus-
sion just before Theorem 4 shows that the effect of dp.(p) = p.oo(p)dp + dp+(m)dQm+(p)
time inversion invariance on the arbitrariness of
B(q) is to reduce the number of arbitrary real
+ dp-(m)dQm-(p)
parameters from L n;' to L nm(n m - 1) each of + dp(im)dQ'm(P) ,
which could be on a function of Iql. It is clear that dQm~(p) = d p/[m' + p']'/' being the invariant meas-
the position observable will be nonunique as long as ure on the hyperboloids p' = m', pO ::: 0, respectively.
:D(A) is not irreducible. If :D(A) is irreducible and the dQ'm(p) is the invariant measure on the hyperboloid
elements of the little group have :D(A) reduced to p' = -m'. Q(p,A) is unitary and satisfies
diagonal form B(q) is diagonal with diagonal ele-
ments which are real functions of Iql of square 1; the Q(p,A)Q(A -'p,B) = Q(p,AB) .
position observable is still not unique. However,
unless B(q) is the constant matrix ±1, the formula
For the subrepresentations with m' > 0, Q(p,A) can
be chosen in the form
(4.33) will yield discontinuous functions of p. [Take
a compact set S, then the integral in (4.33) is dif-
ferentiable, so discontinuities in the function outside
the integral are discontinuities of (F(S)ol'»(p).] Such where k = (m,O,O,O) and A,+k is given by
discontinuities will appear at any value of q where
B(lql) jumps so B([q[) must be constant in [qt. It
A,+k = [2(qO + m)mr'l2[m1 + q] , q = qO + q.~
must be a constant multiple of the identity if it is and A -> Q(A) is a continuous unitary representa-
LOCALIZATION PROBLEMS 419

862 A. S. WIGHTMAN

tion of the unitary unimodular group. For m = 0, system is localizable if the representation of the little
the Q(P,A) are a direct sum of two parts, the first of group A -> Q, (k,O,A) is the restriction of a repre-
which contains all the finite spin constituents while sentation of the unitary unimodular group. This
the second contains all infinite spin constituents. For happens for the spin-zero case but for no other ir-
both of these (5.1) again holds but k is some standard reducible representation. For the case of mass zero
light-like vector, say (1,0,0,1), and A ... , is a parame- and S ,e 0, the representation of the little group is a
trization of the cosets of the little group of k. That direct integral over irreducible representations which
little group is isomorphic to the two-sheeted covering are determined by the value of [sf and the represen-
group of the Euclidean group of the plane and tation of the little group of the little group, A = ± 1.
A -> Q(A) is a continuous unitary representation of The representatives of the state vectors, <I>(k,S) can
it. For the finite spin part this representation is trivial be expanded in Fourier series on the circle [s [
for the "translations" while for the infinite spin part = const. This corresponds to a decomposition into
it is not. The subspace of the mass zero representa- irreducible representations of the subgroup of the
tions can be written as a direct integral over two- unitary unimodular group that leaves k fixed. In case
dimensional S space the little group of the little group is trivially repre-
sented, each integer angular momentum along k
3Cp = J'" dIT(S)3C z,
p p' = 0, appears exactly once. In case it is nontrivially repre-
sented, each half odd integer angular momentum
with the scalar product along k appears twice. Such representations can

J J
never be the restriction of a representation of the
(<I>,>It)m-o =

where
dflo(p) dIT(S) (<I>(p,S),>It(p,S))
°
full unimodular group. Thus elementary systems
with S ,e are never localizable. Reducible systems
are localizable only if each representation [S [ ap-
pears with infinite mUltiplicity or not at all.
dIT(S) = ITo5(S)ds + dIT, ([s[)d<p
with S = Z, + iZ, = Isle"~ and Theorem 6. Lorentz invariant systems of m' >
are always localizable. Their position observables
°
(Q(k,A)<I>)(k,S) = exp (is.t)Q,(k,S,A)<I>(k,e- i6 S)
are unique if the systems are elementary, i.e., their
for representations are irreducible.
A = [1 +! tee, + ie')'~l For m = 0, the only localizable elementary ~;y~tem
has spin zero. For a reducible system to be localizable
X [cos 0/2 - i sin (0/2)(k/kobl
°
with ei = 1 = el, e,'e, = = e,·k = e.·k, t = t,
+ it,. Here Q,(k,O,A) may be expressed in terms of a
it is necessary and sufficient that each irreducible
representation of infinite spin appear with zero or
infinite multiplicity, and the finite spin parts con-
representation Q, of the above A leaving k fixed. tribute states of angular momentum along a fixed
direction whose multiplicities coincide with those of
Q, (k,O,A) = Q,[cos 0/2 - i sin (0/2)(k/ko) ."']
°
Q,(k,S,A), S ,e may be expressed in terms of a
representation Qof the two element groups A = ± 1,
the restriction of a representation of the unimodular
group.
The identity representation for which p = can °
which is the subgroup of those unitary unimodular A not appear in the transformation law of any localiza-
which leave k and some s, say S, fixed: ble system.
Q,(k,S,A) = Q(Az+z,AAA-'Z"z.) 6. REPRESENTATIONS OF 8. ARISING
where A is a transformation of the form cos 0/2 - i IN GALILEI-INVARIANT SYSTEMS
sin (0/2) (k/kO).~ carrying S, into S. Unlike the case of Lorentz invariance where all
The representations of imaginary mass and null representations up to a factor are physically equiva-
four-momentum (apart from the identity represen- lent to representations of the covering group, Galilei
tation) will be ignored here as being irrelevant to the invariance leads to factors which cannot be got rid of
transformation properties of physical systems. by passing to the covering group. However, as Barg-
Clearly, when {a,A I is restricted to lie in 8., the
subrepresentation which comes from mass is in
precisely the form (2.2) and Theorems 4 and 5 apply
° mann showed," one can regard them as true repre-

23 V. Bargmann, Ann. Math. 59, 1 (1954), especially pp.

directly. For the case of mass zero and S = 0, the 38-43.
420 CHAPfER VIII

LOCALIZABILITY IN QUANTUM MECHANICS 863

sentations of a certain extension of the covering direct integral over the character group whose ele-
group of the Galilei group. The first task of this sec- ments are exp i[qt1 + h-'(ET - p·a)]. The states are
tion is to express this statement in explicit formulas then functions <I>(q,p) labeled by integers q and a real
and summarize the classification of the representa- four-component p = (E/h,p/h). The scalar product is

f
tions.
The Galilei transformations will be denoted (a,r) (<I>,if) = dp.(q,p) (<I>(q,p),if(q,p))
or in more detail (T,a,v,R) where (O,r) = (O,O,v,R)
(a,l) = (T,a,O,l) and and

(T,a) {~} = {~ : :}, (v,R) {~} = {kx + vt} .

(U(exp i8,a,1)<I» (q,p)
= expi[qt1 + h-'(ET - p.a)]<I>(q,p).
R is a rotation of the three space of the x. The group
The action of r on the subgroup (exp i8,T,a,1) is
multiplication law is
(l,O,r) (exp i8,a,l) (l,O,r)-'
(T"a"v"R,) (T"a"v"R,)
= (T' + T" a, + V,T" V, + R,v" R,R,) .
= (exp i[e + (M/h)(v·A·a + ! v'T)],ra,l) .
The covering group is obtained by replacing R by It induces a corresponding transformation of the
A, a 2 X 2 unitary matrix of determinant 1, just as characters
in (2.1). For simplicity, la,r) will be written for the q->q E->E - p·v+ lqMv'
group elements in this case also.
Bargmann showed that by physically inessential p->A-'(p - qMv).
changes of phase, one could bring all the factors into From this, it follows that
the following form:
dp.(q,E,p) '" dp.(q,E + v·Ap + f Mv',Ap + qMv).
w(a"r,;a.,r,) = exp i(M/h)
X (v,·A,·a. + ! V~T') . To yield a representation of the form (6.1), dp. must
be a product of a" with a measure in (E,p) above,
Here M/h is a constant of the dimensions
d.(E,p), satisfying
time/[length]·, which has arbitrarily been written
as a ratio in order that its interpretation shall come
out automatically when applied to SchrOdinger dv(E,p) == d.(E + v·Ap + ~ v',Ap + my) .
theory (h is Planck's constant divided by 2...).
Furthermore, Bargmann pointed out that every This in tum implies that d. is equivalent to a measure
representation up to a factor with this factor arises constant on parabolas
from a representation of the group whose elements dv(E,p) = dp(Eo)dNB,(p)
are (exp i8,a,r) 0::; 8 < 2,.. and whose multiplica-
tion law is where dN B. (p) is the measure dp on the parabola

(exp i8"a"r,)· (exp i8"a"r,) E = Eo + 2~

= (exp i[8, 8,+ +(M/h) (v,·R,·a, + ! ViT')],
a, + r,a" r,r.) and dp(Eo) is a measure on the real axis describing
the spectrum of rest energy of the system. Again just
via the formula in the case of the Euclidean group, there is a canoni-
cal form
U(exp i8,a,r) = e"U(a,r) . (6.1)
For the case M = 0, this refinement is unnecessary. U(exp i8,a,r)<I>)(Eo,p) = exp (i8)
That case will be discussed later. X exp [(i/h)(ET - p'a)]Q(Eo,p,r)
Now the elements of the group of the (exp i8,a,r)
which are of the form (exp i8,a,l) form a normal sub- X <I>(Eo,A-'(p - Mv)) (6.2)
group and the group is the semi-direct product of where
this Abelian normal subgroup and the subgroup of
Q(Eo,p,v"A,)Q(Eo,A~'(p - Mv,),v.,A.)
the (,p,O,r). Just as in the case of the Euclidean group
one diagonalizes the Abelian subgroup in terms of a = Q(E.,p,v, + A,v"A,A.) (6.3)
LOCALIZATION PROBLEMS 421

864 A. S. WIGHTMAN

and the scalar product is but the measure p. now satisfies

dp.(E,p) == diJ.(E + v·A,p,Ap)
and this implies that diJ. is equivalent to
The little group of a vector q consists of all (v,A) of
the form (M-'(q - Aq),A) and is isomorphic to the dw(p)dp(!pi)dE + iJ.oo(p)dEdp,
unitary unimodular group. where dw(P) is the area on the sphere of radius Ipl.
The Q's can be brought into the canonical form (The fact that the energy spectrum of the system
runs to - 00 , makes these representations of dubious
Q(Eo,p,v,A) = Q(Eo,O, - pIM,l)-'Q(Eo,O,O,A)
physical interest, but does not exclude their being
X Q(Eo,O,A-' (v - pIM,l) (6.4) localizable.) The full transformation law is of the
where form

A ---7 Q(A) = Q(Eo,O,O,A) (U(a,r)<p)(E,p) = exp [i(ET - p'a)lh]Q(E,p,r)

is a continuous unitary representation of the group X <p(E - v'p,A -'p)

of unitary unimodular matrices. Evidently, (6.4) where Q(E,p,r) satisfies
just describes a superposition of Schrodinger parti-
cles of mass M, and various rest energies (described Q(E,p,v"A,)Q(E - v,.p,A~'P,V2,A2)
by Eo), and spins (described by the irreducible con- = Q(E,p,v, + A,V2,A,A2) .
stituents of Q).
It is clear that the representation of 63 that is
°
The little group of p, p ~ consists of all ("',A) such
that Ap = P and v·p = 0. This is a group isomorphic
obtained from (6.4)
to the two-sheeted covering group of the Euclidean
(U(a,A)<I>(p» = e-i"'Q(A)<I>(A-'p) group of the plane. For p = 0, the little group is the
full unitary unimodular group. There can be no
with the scalar product contribution of this latter kind in any localizable

(<I>,'Ji) = f dp (<I>(p) ,'Ji (p»

system because the criterion (i) of Sec. 2 is not satis-
fied so only the former case will be considered. There,
the criterion (i) forces dp(lpll to be equivalent to
and Ipl2dlpl. From this, it is clear that no irreducible rep-
resentation is localizable because an irreducible rep-
(<I>(p),'Ji(p» = jdp(Eo) (<I>(Eo,p),'Ji(Eo,p» . resentation has dll concentrated on an orbit ipi
= const."
Thus for M > 0, the situation is essentially identical
The general Q(E,p,v,A) is expressed in terms of
with that in the Lorentz invariant case. There is
the representation of the little group of the vector
always a position operator and the arbitrariness in
(O,q), where Ipi = Iql, in the following way:
it is that associated with the representation of the
unimodular group which describes the transforma- Q(E,p,r) = Q(O,q,r((E,p) <- (O,q)-'
tion properties of the system under rotations in the
rest system. X Q(O,q,r«(E,p) <- (O,q))-'rr((r-' (E,p)) <- (O,q))
For M < 0, localizability still makes sense but X Q(o,q,r((r-' (E,p) <-- (O,q))).
such representations are rejected on the physical
ground that the kinetic energy of a particle is nega- Here r((E,p) <- (O,q») is a Galilei transformation
tive. which carries (O,q) into (E,p), so
For representations with M = 0, the preceding r((E,p) <-- (O,q»-'rr((r-'(E,p») <--- (O,q)
argument has to be reexamined. There is no need to
introduce 8 as in (6.1). The diagonalization of U(a,l) belongs to the little group of (O,q).
leads to The same procedure can be applied to analyze the
representation (v,A) ---7 Q(O,q,v,A) , v·q = 0, Aq = q,
(U(a,l)<I»(E,p) = exp [i(ET - p·a)lh]<I>(E,p) of the little group of (O,q) as was applied to 63 itself.
with a scalar product One diagonalizes the "translations" v. Then the

(<I>,'Ji) = jdp.(E,P) (<I> (E,p) ,'Ji (E,p))

24 This result agrees with that of Inonii and Wigner, reference 3.
422 CHAPTER VIII

LOCALIZABILITY IN QUANTUM MECHANICS 865

representation takes the form (a). The measure on momentum space is equiva-
lent to Lebesgue measure;
Q(O,q,v,A)<I»(q,n) = e;v'DQI(q,n,A)<I>(q,A -'n) . (b). The subrepresentation of the little group of
Here n is a two-component vector in the plane (O,q) for which the pure Galilei transformations r
perpendicular to q which labels the characters of the = (v,1) are trivially represented is, for almost all
"translation" subgroup. The scalar product is Iql, the restriction to the group of A such that Aq

f
= q of a fixed representation of the 2 X 2 unitary
(<I>(q),v(q)) = du(n)(<I>(q,n),v(q,n)) , unimodular group;
(c). The subrepresentation of the little group of
where the measure u is equivalent to one of the form (O,q) for which the pure Galilei transformations are
non-trivially represented contains each irreducible
du(n) = u0 8(n)dn + du, (inlld<p, n, + in. = Inle;' . with multiplicity zero or infinity, the same for almost
The little group of the little group is the little group all [q[.
itself if n = 0, while it is the two-element group:
ACKNOWLEDGMENTS
A = ±1 if n "e 0. In the former case A ...... Q,(q,O,A)
is any continuous unitary representation of the little A substantial part of this paper was written in
group of q. In the latter case, ±1 ...... Q,(q,n, ± 1) is 1952, when the author was a National Research
any unitary representation of the 2-element group Council Post Doctoral Fellow in Copenhagen. He
and the QI of general argument is expressed in terms thanks Professor Niels Bohr for the hospitality of
of the elements of the little group by the Institut for Theoretisk Fysik and Professor Lars
Garding for the hospitality of Lunds Matematiska
Q,(q,n,A) = [QI(q,Ro,AD<-D,-')r' Institution. The paper was completed in 1962 with
X QI (q,Ro,AD+D,AAA-'D<-D.)QI (q,Ro,AA-'D+D. ) the support of the National Science Foundation. The
author thanks Professor Robert Oppenheimer for the
The irreducible representations of the little group
of the little group have either Uo > 0, du, = or Uo
= 0, du,(lnl) = 8(lnl - a)dlnl, for some a> 0. The
° hospitality of the Institute for Advanced Study dur-
ing the later period.

corresponding Q, are one dimensional. APPENDIX I. FINITE ADDITIVITY ON FINITE

In the case Uo > 0, Q,(q,O,A) is just Q(O,q,v,A) for
°
v·q = Aq = q, so the system will be localizable if
A ...... Q(O,q,O,A) for Aq = q defines a representation
UNIONS OF CUBES
In connection with axioms I, II, III, it was re-
which is a restriction of a representation of the full marked that it might appear more natural from a
unimodular group. In the case [n[ "e 0, Q,(q,ft, ± 1) physical point of view to weaken the axioms so that
= +1 yields a Q(O,q,O,A), Aq = q which contains the existence of the observable E(8) is required only
each integer angular momentum along q just once, for 8 a finite union of cubes, and finite additivity is
so it is not localizable. A necessary condition for required:
localizability is that the representation QI (q, ft, ± 1)
= +1 have zero or infinite multiplicity. The ir-
E(8, u 8.) = E(S,) + E(S.) if S, n S. = °
reducible representation Q,(q,n, ± 1) = ±1 yields instead of the complete or u-additivity described
a Q(O,q,O,A), Aq = q which contains each half-odd in III.
integer angular momentum along q just once so it is In this Appendix, it is shown that such a weaken-
not localizable. A necessary condition for localiza- ing of the axioms is only apparent because any E(S)
bility is that Q,(q,n, ± 1) = ±1 appear with zero satisfying the weakened axioms can be extended
or infinite multiplicity. uniquely so as to satisfy I, II, III, as they stand.
All this is summarized in Theorem 7. Consider the family A of all sets of R3 which are
finite unions of half-open intervals. By a half-open

°
Theorem 7. Every Galilei invariant system with interval is meant a set [a,b) of the form
M > is localizable. {y; al";; YI < b" a.";; y. < b., a3";; y. < b.j ,
For M = 0, no elementary system is localizable

= const. Systems with M = °

because such a system has momentum satisfying Ipi
and a reducible repre-
sentation of the Galilei group are localizable if and
that is, the set of all y satisfying the listed inequali-
ties. By assumption, the cases a; = - <Xl and bi
= + <Xl are also included; in the former case, the
only if: equality sign in a; ..;; y; should be ignored. A is re-
LOCALIZATION PROBLEMS 423

866 A. S. WIGHTM AN

ferred to as an algebra of sets because it is closed If F is real valued it is said to be positively monotonic
under the operations of taking the complement of a if t.F[a,b) ;;. 0 for all [a,b)" If the values of Fare
set and taking the union of a finite number of sets. commuting projections the analogous requirement
A cr algebra of sets is one closed under complemen- is that t.F[a,b) be a projection for all intervals [a,b).
tation and denumerable unions. A projection-valued Notice that if E(S) is any finitely additive projection
finitely-additive measure on A is a function, E, with valued measure defined on A, it yields such an F
values which are projections in a Hilbert space X, from the definition
defined for all sets of A and satisfying II, and
F(X"X2,X3) = E({Y;Yl < Xl,Y, < X2,Y3 < X31l . (AI)
III' E(S, uS,) = E(S,) + E(S,) - E(S, n S,)
Conversely, the following theorem holds.
for any S" S, E A. Theorem Ai. Let F be a positively monotonic
A projection-valued finitely additive measure that function defined on R3 with values which are com-
satisfies in addition mutative projections. Suppose

E(US,) LE(Si) F( - <XI ,X2,X3) = F(Xl, - <XI ,X3)

i-I

for any sequence of S, E A,i = 1,2,' " such that = F(Xl,X" - <XI) = O. (A2)
S, n Si = 0, i rf j and US, E A is called completely Then there exists a finitely additive projection
additive or cr additive. The precise statement of the valued measure E on A satisfying (AI).
result of this Appendix is The proof is completely elementary and will be
Theorem A5. Any finitely-additive projection- omitted.
valued measure on A which satisfies Now consider the increasing sequence of projec-
E(S + a) = U(a)E(S)U(a)-l
tions

for some continuous unitary representation of the F(Xl - 11k", 'X3 - 11k) k = 1,2" ...
translation group a -> U(a) is necessarily completely It converges to a projection F-(Xl,·· 'X3) which
additive on A. It then possesses a unique completely mayor may not be F(Xl," ·X3).
additive extension to the (J algebra of all Borel sets Example. Consider the function Et defined on A
on R3. which is the projection E '" 0 for a set S if there is
Variants of the last statement of the theorem are an interval of the form {y;ll - • -< y, < 1"t2 - •
quite standard in various contexts in measure theory,
so it will not be proved here. (In Halmos' book,
-< Y, < t,,13 - • -< Y3 < t31 which lies in S and zero
otherwise. It is easy to see that E, is a finitely addi-
reference 15, p. 54, the theorem is stated: "If J.I is a tive projection valued measure on A. It is not com-
cr finite measure on a ring R, then there is a unique pletely additive because the interval Iy;tl - 1 -< y,
measure )l on the cr ring, S(R), generated by R such < Il,t, - 1 -< y, < 12,t3 - 1 -< y3 < t31 can be
that for E in R, )l (E) = J.I(E); the measure )l is written as a denumerable union of intervals for which
q finite." The assumptions of the present Appendix
the coordinates Yi lie in intervals where right-hand
are more general in that one has a projection-valued end points are lcss than ti . For each such interval
measure rather than a real-valued measure, but E,(S) = 0 but for the union Et(S) = E. Clearly,
otherwise everything is more special: The ring of sets, the F corresponding to Et does not satisfy F(tl' .. t3)
R, is an algebra because the whole space is in R, the = L(tl" ·t3).
measure is finite rather than only (J finite.) The first If for each x E R3, F(x) = F _(x), then the phe-
part of the theorem is a consequence of the following nomenon occurring in the example cannot happen
chain of four theorems. The argument is a straight- and the projection valued measure defined by F is
forward generalization of one due to Hewitt.25 (J' additive on A.
If F is any function on R 3 whose values can be Theorem A2. Every projection valued positively
added and subtracted and [a,b) is an interval, define monotonic function F on R3 which satisfies (A2) and
Lh[a,b) = F(bl,b"b3) - F(al,b"b3) - F(b " a2,b3) lim F(xl - 11k" . 'X3 - 11k) = F(Xl,' . ·X3). (A3)
- F(bl,b"a,,) + F(a " a2,b3) + F(al,b"a3) k~w

:r F(b " a."a3) - F(al,a"a3) . 26 A detailed discussion of positively monotonic fUIlctioIl!:!

is given in, E. J. McShane l Integratian (Princeton University
25 E. Hewitt, Mat. Tiusskrift (1951B), pp. 81-94. Press, Princeton, New Jersey, 1947), pp. 242-274.
424 CHAPTER VIII

LOCALIZABILITY IN QUANTUM MECHANICS 867

defines a projection valued measure E on A which is of discontinuity unless F(!) = O. Thus, there are no
(J additive. nontrivial purely finitely-additive projection valued
Proof. Since each element of A is a finite union of measures quasi-invariant under translations.
disjoint intervals and E is finitely additive according Theorem A4. Every finitely additive projection
to Theorem A2, it suffices to consider the case of a valued measure on A which is quasi-invariant under
denumerable union of sets in A whose union is an translations is (T additive.
interval. But such a union defines a monotonically From Theorem A4 and the result already cited
increasing sequence of projections which converges that (J additive projection-valued measures on A
to the projection belonging to the interval by virtue have unique extensions to the Borel sets of R',
of (A3). Therefore E is complctely additive. Theorem A5 follows.
A finitely-additive projection-valued measure E While the results \if this Appendix make it clear
is called purely finitely additive if there is no nontrivial that the assumptions of I to V can be weakened
(J additive projection-valued measure which is zero without impairing the results of the paper, it should
on every set S for which E(S) = o. (It is not difficult be noted that the particular weakened assumptions
to see that the example E, is purely finitely additive.) used have been chosen primarily for reasons of
Theorem AS. Every finitely additive projection mathematical elegance. A deeper physical analysis
valued measure on A is the sum of a purely finitely would ask whether the existence of some kind of
additive part and a (J additive part. This decomposi- approximate position measurement implied the exist-
tion is unique. ence of precise position measurements in the sense
Proof. The difference F(x) - F _(x) is a projection, of I to V.
and two such, corresponding to distinct points x are
orthogonal. Because the Hilbert space is separable, APPENDIX II. SKETCH OF THE DERIVATION OF
there can be at most a denumerable set of points x THE CONTINUOUS UNITARY REPRESENTATIONS
where F(x) - F_(x) ;c 0; call them t(k). Let E,(k)(S) OF THE TRANSLATION GROUP
be the projection-valued measure given in the The result of Theorem 1 which describes all unitary
example above with E = F(tC'» - F_(t(k». Then representations of the translation groups has been
E(S) - Lk E,(k,(S) used in physics since the beginning of quantum
mechanics, but explicit mathematical statements and
defines a finitely-additive projection-valued measure proofs of it are relatively recent. The purpose of
whose F satisfies (A3) for all points x and so by this Appendix is to outline some of the ideas involved
Theorem A2 is (J additive. Thus in the proofs.
E(S) = L,E,(k)(S) + E(2)(S) The translation group of n-dimensional real
Euclidean space Rn will here be denoted ;t with
defines a decomposition into a purely finitely additive elements a. (The whole machinery works in the same
part and a (J additive part. For the case in which way for any dimension n so the assumption n = 3 is
E(S) is purely finitely additive, E(2)(S) = 0 because dropped.) The derivation of Theorem 1 can be
otherwise E(2)(S) would be a (J additive projection- divided into three parts:
valued measure vanishing whenever E(S) does in (1) Determination of the character group ;to of ;t,
contradiction with the definition of a purely finitely- (2) Derivation of the spectral representation
additive measure. This shows that the purely finitely-
additive part of any E is uniquely determined by
the discontinuities of the corresponding F.
U(a) = r e-i•. adF(p) ,
lx·
Now note that if E(S) is quasi-invariant under
translations in the sense that E(S + a) = 0 if and
(3) Spectral multiplicity theory for the projection
valued measure F on ;t*.
only if E(S) = 0, then the same applies to the purely
These stages actually reflect the historical develop-
finitely-additive part, E(J)(S), of E(S) and the
ment of the theorem and I will follow them here at
(J additive part of S. [E(S) is surely quasi-invariant
least in part.
if there exists a representation a -> U(a) of the transla-
A character of X is a one-dimensional continuous
tion group such that E(S + a) = U(a)E(S)U(a)-11
unitary representation of X, i.e., a complex-valued
Furthermore, if F(!) has a nonzero discontinuity
F(I)(X) - F _(!)(x) at x = t, it must also have a continuous function X of modulus one, which
satisfies
nonzero discontinuity at x = t + a. This statement
is in conflict with the dcnumerability of the points x(a + b) = x(a)x(b} . (A4)
LOCALIZATION PROBLEMS 425

868 A. S. WIGHTMAN

It is well known that any such X is of the form XJ" for self-adjoint operators H = f:~pdF(p) so (A6)
where can be written

x.(a) = exp -i(p·a) and p·a = L pia; .

i-I
(A7)

[The argument goes as follows. From (A4), x(O) = 1 Here F defines a projection valued measure via
and x(a) can be written F(S) = f sdF(p). The extension of (A7) to arbitrary
Abelian groups was carried out by a number of
x(a) = x(a',O,···Olx(O,a',O,·· ·0)· ··x(O,O,,· ·an ) , authors." Since the step from the one-dimensional
to n-dimensional translation group is easy, and excel-
where x(O·· ·a;···) is a charnci.er of thp, one-dimen- lent textbook accounts of Stone's theorem are
sional translation group of ai . Thus the problem is available," no more details of (2) will be given here.
reduced to finding all characters for the translation The problem of determining when two represen-
group of the real line. By introducing i In X = f one tations are unitary equivalent is reduced by the
reduces the problem to that of finding all real con- SNAG theorem to the corresponding problem for
tinuous f(a) defined mod 2.. such that their F's. A solution of this problem is provided by
f(a) + f(b) = f(a + b) mod 2.. (A.5) (3), the theory of spectral multiplicity. It shows that
the unitary equivalence class of an F can be charac-
To complete the proof it is convenient to specify terized by two objects, a measure class on :to and a
f(a) completely instead of mod 2... Because X is con- multiplicity function on :t*, which described, re-
tinuous, a unique specification is obtained in some spectively (and roughly), tell which irreducible
neighborhood of a = 0 by requiring f(O) = 0 and representations of :t occur in a --> Ural and how
f(a) continuous in the neighborhood. From (A5), one often. This theory is to the theory of (2) what the
then derives qf(rr'e) = f(e) for any c in the neighbor- Hellinger-Hahn theory of a self-adjoint operator30 is
hood and any integer q. Thus, again using (A5), to the spectral resolution of a self-adjoint operator.
f«p/q)c) = (p/q)f(c) for any rational number p/q There are available nearly as many approaches to
< 1. The continuity of f then implies frye) = yf(c) the theory of spectral multiplicity as there are
for every real number < 1, i.e., f(y) = yf(e)/e for y authors who have written on the subject. One may
in the neighborhood. Finally, using (A5) again, one make a direct analysis of the commutative algebra
getsf(y) = yf(e)/e mod 2dor all y. Q.E.D.] of projections." This leads to a decomposition of the
The characters clearly form a group under multipli- Hilbert space into orthogonal subspaces X; on which
cation the projections are uniformly j-dimensional. That
means that X; is a direct sum of j subspaces X( .. ·x;
X., (alx., (a) = X., +', (a) such that the projections E take of the form
and, if the usual topology of Euclidean space is
E(ip,,···ip;) = (E,ip,,···E;ip;)
introduced for the p's, the group operations are con-
tinuous. The set of all characters (or equivalently the and on xi the E, are uniformly one dimensional.
set of all p' s) is denoted :to and called the character Finally, a uniformly one-dimensional algebra of
group of :t". projections is one which is maximal Abelian, i.e.,
The step (2) alone can be regarded as a decompo- any projection which commutes with all the given
sition of an arbitrary continuous unitary represen- projections is one of them. It is shown that a uni-
tation into irreducibles. This operation is familiar in
quantum mechanics for the one dimensional transla- 28 Stone's original paper is Ann. Math. 33, 643 (1932). The
~xten8ion :0 any locally compact Abelian group is contained
tion group as Stone's theorem: Anyone-parameter m M. Naumark, Izvest. Akad. Nauk U.S.S.R. 7, 237 (1943);
continuous unitary group is of the form W. Ambrose, Duke Math. J. 11, 589 (1944); R. Godement,
Compt. rend. 218, 901 (1944). It is sometimes referred to as
the SNAG theorem.
U(a) = exp -iaH , (A6) ,. See for example F. Riesz and B. Sz.-Nagy, Le~tm8 d'analyse
fonctionelle (Budapest, 1953), p. 377.
where H is self-adjoint. Then by the spectral theorem 30 See M. H. Stone, Linear Transformations in Hilbert Space
(American Mathematical Society, Providence, Rhode Island,
1932), Chap. VII.
31 See, for example, H. Nakano, Ann. Math. 42, 657 (1941);
27 This construction of the character group can be carried I. E. Segal, Memoirs Am. Math. Soc. 9 (1951), Secs. I and II;
out for an arbitrary locally compact Abelian group. See, for P. R. Halmos, Introduction to Hilbert Space and the Theary of
example, L. Pontrjagin, Topological Groups (Princeton Uni- Spectral Multiplicity (Chelsea Publishing Company, New
versity Press, Princeton, New Jersey, 1939), Chap. V. York,1951).
426 CHAPTER VIII

LOCALIZABILITY IN QUANTUM MECHANICS 869

formly one-dimensional algebra of projections is functions of S' and S are related by

unitary equivalent to one in which the projections
have the form Xs'(x,y) = x.(x - y)

(E(8)<I>)(x) = xs(x)<I>(x) , and so because xs is positive

where <I>(x) are complex-valued functions square

integrable with respect to a measure /J. Reassembling
JJ xs'(x,y)dxd/J(Y) J(J dxxs(x - y)) d/J(Y)
=

J(J d/J(ylxs(x - y) ) dx.

the Hilbert space, one just gets a form of the projec-
tion operators just like that indicated for the projec- =
tion operators on momentum space given in Theorem
1. Because dx is invariant under translation the second
Alternatively, one can imbed the projection opera- expression is (f sdx) I d/J(Y). On the other hand,
tors in an appropriately chosen commutative algebra xs(x - y) = x-s+«Y) so the last equality becomes
of bounded operators and then use the spectral theory
of such commutative algebras to obtain the required
canonical form. 32

APPENDIX III. QUASI-INVARIANT MEASURES From this equality, the required equivalence can be
deduced as follows. Note that I sdx = 0 if and only
In this Appendix, the structure of quasi-invariant
if I _sdx = O. Thus, from (A9) , I sdx = 0 implies
measures defined on the Borel sets of R' is determined
for two different situations. In the first, the group
/J(S+ x) = 0 for almost all x. By the quasi-invari-
ance of /J, this, in turn, implies /J(S) = O. Conversely,
acting on R' is R' itself. Then, every finite quasi-
invariant measure is equivalent to Lebesgue measure.
+
if /J(S) = 0 and therefore /J(S x) = 0, (A9) implies
Isdx = O.
In the second, the group acting on R' is the rotation
This completes the proof of the equivalence of /J
group. Then the most general finite quasi-invariant
with Lebesgue measure. The Radon-Nikodym theo-
measure is equivalent to a measure of the form

r
rem guarantees that d/J(x) = p(x)dx where p(x) is

/J(8) = /JoXs(O) + o
dp(a) J
d",.(p) , (A8)
positive and measurable.
In the second situation, one has a finite measure /J
sn~p;lpl-ar on the Borel sets of R' such that for every Borel set S
and every rotation R, /J(S) = 0 if and only if p.(RS)
where 1'0 ;;. O,Xs is the characteristic function of the
= O.
set 8, d",.(p) is the invariant surface element on the
It is easy to see that any such quasi-invariant
sphere Ipi = a, and dp(a) is a measure on the positive
measure is equivalent to an invariant measure. In
real axis.
fact, consider the non-negative set function
The result for the first situation is a special case
of the general result that any Borel measure on a
locally compact group quasi-invariant with respect ji(S) = JdRp.(RS) ,
to the action of the group on itself is equivalent to
Haar measure" The proof of Loomis given in" is so where the integration is over all the rotation group
simple that it will be repeated here in the special and dR is the invariant measure on the rotation group
context of R'. for which I dR = 1. It is not difficult to verify that
Let S be any Borel set in R'. Denote the finite ji is (f additive. Furthermore, it is equivalent to /J,
measure quasi-invariant with respect to Lebesgue because ji(S) = 0 implies /J(RS) = 0 for almost all
measure by /J. Let S' be the set in R' defined by R, which, because of the quasi-invariance of /J, yields
x - yES. (It is a Borel set because x - y is a con- /J(S) = O. Conversely, /J(S) = 0 implies /J(RS) = 0,
tinuous function of x and y. Then the characteristic which implies ji(S) = O. Thus, it suffices to consider
invariant /J.
32 R. Godement, Ann. Math. 53, 68 (1951); J. Dixmier, It is convenient in completing the proof to use an
"Les Algebres d'Operateurs dans l'Espace Hilbertien," alternative characterization of a finite measure on R'
Algebres de von Neumann (Gauthier-Villars, Paris, 1957), Chap.
II; see especially pp. 216-224. as a non-negative bounded linear functional on the
" See, for example, G. W. Mackey, Duke Math. J. 16, 313 continuous functions of compact support, e(R').
(1949), Lemma 33; J. von Neumann, Bull. Amer. Math. Soc.
42, 343 (1936). That the functional, fI, is non-negative means p.(f)
LOCALIZATION PROBLEMS 427

870 A. S. WIGHTMAN

; ;. ° f ;;;.
for 0, fEe. That 110 is bounded means necessary, be altered on a set of p.-measure zero so
that it becomes measurable in both variables relative
sup 1110 (f) I < cx), where If I = sup If(x) I . to the measure 110 X a, where a is the invariant
Ifl"" .ER·
measure on the 2 X 2 unitary unimodular group.
The relation between the functional 110 and the corre- Let ~;,j = 1,2,··· be a complete orthonormal
sponding measure 110 is simply set in X. Then it suffices to treat the functions
(~j(P), Q(P,A)~.(p)) separately because the general
p.(f) = jf(X)dp.(x) . case then follows by the expansions

Since the measure is uniquely determined by the 'It, = L aj~" 'It, = L bj~j, and
functional, to verify the equality of two measures it ('It, (p),Q(p,A)'It,(p)) = L a1b,
suffices to verify the equality of the corresponding i,k-I

functionals. 34 X (~j(p),Q(p,A)~.(p)) .
Now, for an invariant measure An ugly little lemma is necessary.
Lemma. Let f(p,A) be a complex-valued function
on R' X G which is 110 measurable and p. essentially
bounded on R' for each A E G, the 2 X 2 unitary
because the approximating sums to the integral unimodular group. Suppose J f(p,A)xE(p)dp.(p) is a
f (Rf)(x)dR converge uniformly in x, and p.(f) is measurable on G for each p. measurable subset E of
continuous for uniform convergence of its argument. R' of finite measure. Here, a-measurability on Gis
Butf(x) ---> f (Rf)(x)dR = f f(Rx)dR maps the con- with respect to the invariant measure dA.
tinuous functions of compact support on R' onto the
continuous functions of compact support on
" Ixl < and convergence in e(R') implies con-
CX)
° Then there exists a function, g, 110 X a measurable
on R' X G, and such that for a certain p.-measurable
subset N of R' of zero measure
vergence in e([O, ro )). Thus, the functional 110 re-
garded as defined on e([O, defines a finite measure
CX) ) )
f(p,A) = g(p,A) for all A E G and pEEN.
on the non-negative real axis. Splitting it into a con- This lemma is a special case of Lemma 3.1 of ref-
tribution with support at 0, and the rest, one has erence 33, and will not be proved here.
just the 1100 and dp of (AS). In fact, (AS) is just an The lemma shows that by a suitable redefinition
explicit form in terms of measure of of Q(p,A) which does not affect the corresponding op-
erator Q(A), one can have Q(P,A),p. X A measurable.
p.(f) = p.URfdR ) . The next step in the argument is to show that in the
equation
APPENDIX IV. SOME MEASURE-THEORETIC L1 (~j(p),Q(P,A)~I(P))(~'(P),Q(A -1p,B)~.(p))
NICETIES CONNECTED WITH EQS. (3.10) AND (3.17)
= (~j(p),Q(p,AB)~.(p)) (AI0)
This Appendix is devoted to some fine points which
arise in the otherwise elementary derivation of which holds for each A,B E G and pER' such that
Sec. 3. pEE N , (A,B),A-'p EE N,(A,B) where N, and N, are
Recall that Theorem 1 states that if p.-measurable sets of p'-measure zero, the right- and
left-hand sides are p. X a X a measurable on R' X iG
[Q(A),U(a,l)] = 0, X G. Because a Borel-measurable function of a Borel-
measurable function is Borel measurable, it suffices to
then Q(A) is of the form
prove that the mappings T , :{p,A,B} ---> {p,AB} and
(Q(A)~)(p) = Q(p,A)~(p) , T,:[p,A,B} ---> {A-1p,B} are Borel-measurable func-
tions.
where for each unitary unimodular A, Q(P,A) is
Now T, and T, are continuous, and a set F which is
measurable in p in the sense that for each 'It " 'It,
110 X a measurable in R' X G differs from a Borel set
EX, ('It, (p), Q(p,A)'It,(p)) is 110 measurable. The first
by a subset of a Borel set of zero p. X a measure."
step in the argument is to prove that Q(p,A) can, if
Furthermore, a continuous function has the prop-
erty that the antecedent of any Borel set of its range
34 See, for example, P. R. Halmos, Measure Theory (D. van
Nostrand Company, Inc., Princeton, New Jersey, 1950), pp.
243-9. .. See reference 15, pp. 55-56.
428 CHAPTER VIII

LOCALIZABILITY IN QUANTUM MECHANICS 871

is a Borel set of its domain. Thus to prove the Because

I-' X a X a measurability of T, and T" it suffices to
show that for any Borel set F of R' X G of zero 1!(U(x) - U(Y))1>!I' = II(U(y-'x) - 1)1>[1'
I-' X a measure Tl'(F) and T,'(F) have zero I-' X a = 2(1),1>) - 2Re (U(y-'x)1>,1»
X a measure. Consider T" the proof for T, being the strong continuity of U(x), i.e., the requirement
similar. that for each 1> E X and y E G the first of these ex-
Let Fp denote IA;lp,AI E Fl. Clearly, Ip,ABI pressions be small when x is close to y, is implied by
E P if and only if AB E Fp (or A E F pB-') for some the weak continuity of U(x) at the identity, i.e., the
p. Now requirement that for each 1>,\[1 E X, (1),U(x)\[I) is
close to (1), \[I) for x close to the identity. The c@-
I-' X a(F) = 0 = j dl-'(p)dAxFP(A) = j dl-'(p)a(Fp) tinuity of (1),U(x)\[I) at the identity for all 1>, \[I IS
implied by the continuity of (x,U(xlx) for all X as one
This implies a(Fp) = 0 for I-' almost all p. By the in- sees by considering X = 1> + \[1,1> + i\[l,1>, \[I in turn,
variance of a, a(FpB-') = 0 for each B and I-' almost and taking appropriate linear combinations. Because
all p. Because Ip,A,BI E T,'(F) if and only if U(x) is unitary, it suffices to verify the weak con-
A E FpB-' for some p and B, tinuity for the elements of any dense set of vectors in
X, say 1>,. To see this one can look at the identity
(I-' X a X ex)(T,'F) = j d(1-' X ex X a) (p,A,B)
(1), (U(x) - 1)1» = (1) - 1>,,(U(x) - 1)<1»
X XT,_'(F)(p,A,B) + (1),,(U(x) - 1)(1) - 1>,)

= jd(1-' X a) (p,B)a(FpS-') = o. + (1),,(U(x) - 1)1>,) ,

which yields the estimate
The preceding argument shows that (A1O) is a re-
lation between I-' X a X a measurable functions, (<I>,(U(x) - 1)1» ~ 21:<1>11111> - 1>,1:
which, for fixed A,B can fail to hold only on a set of + 2111>,1111<1> - 1>,11 + I(<I>,,(U(x) - 1)1>,)1·
p's of I-'-measure zero. However, the union of these
The first two terms on the r(ght-hand side can be
null sets as A and B run over G could, a priori, be a
made small by appropriate choice of 1>,.1>, having
set of measure greater than zero. That this is, in fact,
been chosen, the last term can be made small by an
not the case is seen as follows. Since the set of
appropriate choice of x according to the assumed con-
Ip,A,BI where (A10) fails is of (I-' X ex X a)-measure
tinuityof (<I>"U(x)1>,).
zero its section for p and B fixed is of a measure zero.
Since U(x) is measurable and unitary (<I>,U(x) <1» is
But as A runs over a set of a measure zero, A -'p runs
a bounded measurable function for each 1>. Thus, for
ove; a set of I-' measure zero. [Here the equivalence of
each continuous function of compact support, cp, it
I-' to a measure of the form (AS) is being used.] Thus,
makes sense to talk about
the set of A -'p where
Q(A-'p,B) = Q(p,A)-'Q(p,AB) (All) (<I>,U(cp)1» = j cp(x)dx(1>,U(x)<I»
fails, p and B being fixed, is a set of I-' measure zero. and if cpy is defined by cpy(x) = cp(y-1x),
By redefining Q(A -'p,B) at those A -'p by (All), one
obtains a new family of Q(q,B) measurable in Iq,B},
1·(<I>,(U(cpy) - U(cp»)<I» I .;; jlcp,(x) - cp(x)ldxll<l>II'·
which still yield the old Q(B) but for which (All) [or
equivalently (A1O)] is always valid. This completes (A12)
the justification of the statement just after Eq. (3.10).
Here dx is the left invariant integral on G. The right-
A second measure theoretic point arises in connec-
hand side of this inequality is small for y sufficiently
tion with Eq. (3.14). Using the Q(p,A) whose exist-
close to the identity. Now X is a direct sum of sub-
ence has just been established, one gets a measurable
spaces in which there is a vector 1> such that vectors
but, a priori, not necessarily continuous unitary rep-
of the form U(cp)1> are dense, cp being continuous and
resentation of the little group. In fact, every measur-
of compact support so to prove U(x) continuous it
able unitary representation of any locally compact
suffices to verify that (U(cp)1>,U(x)U(cp)<I» is con-
group G is continuous, as will now be shown by a well-
tinuous for any such cp and <1>. But
known argument which has not yet crept into the
text books.
LOCALIZATION PROBLEMS 429

872 A. S. WIGHTMAN

so that the required continuity follows from (A12) sides of this equality are (p. X a) measurable func-
and the proof is complete. tions of p and A and the set on which the equality
Finally, there is the matter of sets of measure zero fails is of p. X a measure zero. It then follows that, for
in the criterion for unitary equivalence (3.17). Solved fixed p, the set of A on which it fails is of a measure
for V(A-lp) it reads zero. That in turn implies that the set of A -lp for
which it fails is of p. measure zero. Picking one p from
each orbit and altering yep) on the corresponding set
of measure zero one gets a new family V (p) which is
By an argument just like that used in the first few also measurable andyields the same V but for which
paragraphs of this Appendix, one concludes that both (3.17) always holds.
430 CHAPTER VIII

PHYSICAL REVIEW A VOLUME 35, NUMBER 4 FEBRUARY 15, 1987

Uncertainty relations for light waves and the concept of photons

D.Han
National Aeronautics and Space Administration, Goddard Space Flight Center (Code 636), Greenbelt, Maryland 20771

Y.S.Kim
Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742

Marilyn E. Noz
Department of Radiology, New York University, New York, New York 10016
(Received 25 September 1986)
A Lorentz-covariant localization for light waves is presented. The unitary representation for the
electromagnetic four-potential is constructed for a monochromatic light wave. A model for covari-
ant superposition is constructed for light waves with different frequencies. It is therefore possible to
construct a wave function for light waves carrying a covariant probability interpretation. It is
shown that the time-energy uncertainty relation (~t)( ~w b 1 for light waves is a Lorentz-invariant
relation. The connection between photons and localized light waves is examined critically.

I. INTRODUCTION is not Lorentz invariant, and that Planck's constant de-

pends on the Lorentz frame in which the measurement is
For light waves, the Fourier relation (M)(aw) was taken. This is not correct, and we need a better under-
known before the present form of quantum mechanics standing of the transformation properties of M and a",.
was formulated. I ,2 However, the question of whether this This problem is related to another fundamental prob-
is a Lorentz-invariant relation has not yet been properly lem in physics. We are tempted to say that the above-
addressed.) Let us consider a blinking traffic light. A mentioned Fourier relation is a time-energy uncertainty
stationary observer will insist on (at)( aw b I. An ob- relation. However, in order that it be an uncertainty rela-
server in an automobile moving toward the light will see tion, the wave function for the light wave should carry a
the same blinking light. This observer will also insist on probability interpretation. This problem has a stormy his-
(at*)(aw*b,l on his or her coordinate system. Howev- tory and is commonly known as the photon localization
er, these observers may not agree with each other because problem. 4 - 6 The traditional way of stating this problem
neither at nor I!wJ is a Lorentz-invariant variable. The is that there is no self-adjoint position operator for mass-
product of two noninvariant quantities does not always less particles including photons.
lead to an invariant quantity. In spite of this theoretical difficulty, it is becoming in-
Let us assume that the automobile is moving in the neg- creasingly clear that single photons can be localized by
ative z direction with velocity parameter {3. Since both t detectors in laboratories. The question then is whether it
and w are the timelike components of four-vectors (x,t) is possible to construct the language of the photon locali-
and (k,ro), respectively, a Lorentz boost along the z direc- zation which we observe through oscilloscopes.
tion will lead to new variables Throughout the history of this localization problem, the
main issue has been and still is how to construct localized
photon wave functions consistent with special relativity.
where the light wave is assumed to travel along the z axis We do not propose to solve this difficult problem in
with k = w. In the above transformation, the light wave is this paper. We shall instead approach this problem by
boosted along the positive z direction. If the light passes constructing covariant localized light waves and compar-
through the point z =0 at t =0, then t =z on the light ing them with photon field operators. First, we construct
front, and the transformations of Eq. (1) become a unitary representation for Lorentz transformations for a
monochromatic light wave. It is shown then that a
t*= [ 1+{3 ]112t, ",*=
[ 1+{3 ]112w. (2)
Lorentz-covariant superposition of light waves is possible
for different frequencies. After constructing the covariant
1-{3 1-{3
light wave, we shall observe that there is a gap between
These equations will formally lead us to the concept of photons and that of localized waves. From
the physical point of view, this gap is not significant.
(at*)(aw*)= [ :~~ ](M)(aW) , (3) However, there is a definite distinction between the
mathematics of photons and that of light waves.
In approaching the problem of the covariant superposi-
which indicates that the time-energy uncertainty relation tion of light waves, we shall start with the uncertainty re-

35 1682 © 1987 The American Physical Society

Reprinted from Phys. Rev. A 5 1682 (1987).
LOCALIZATION PROBLEMS 431

35 UNCERTAINTY RELATIONS FOR LIGHT WAVES AND THE ... 1683

lation applicable to nonrelativistic quantum mechanics.

We shall then borrow the techniques from the covariant
harmonic oscillator model which provides a quantifica-
tion of the uncertainty relations observed in the relativis-
tic quark model. 7 - 1O Since the uncertainty principle is (0)
universal, the uncertainty relation applicable to one specif-
ic physical example should be consistent with those for

#
other physical phenomena.
In Sec. II, we start with the motion of free-particle
wave packets in the Schriidinger picture of nonrelativistic
quantum mechanics. For localized light waves, there is
no difficulty in giving a probability interpretation if 1_ BO#5T
1
Lorentz boosts are not considered. It is pointed out that
the basic problem for light waves is how to make the
probability interpretation Lorentz covariant. ib) z z
In Sec. III, we discuss Lorentz-transformation proper-
ties of the four-vector representation for photons. Section
-- -

IV examines the time-energy uncertainty relation applic-

FIG. I. The time dependence of the wave packets. lal shows
able to the relativistic quark model. It is noted that the the spread of the Schriidinger wave function. (bl shows the
uncertainty relation applicable to the time separation vari- behavior of the light wave which does not spread. However, for
able between the quarks confined in a hadron can be com- an observer moving in the negative z direction, the SchrOdinger
bined cavariantly with the position-momentum uncertain- wave function is boosted according to the Galilei transforma-
ty relation. tion. The quantum probability interpretation is consistent with
In Sec. V, based on the lessons we learned in Secs. II, the Galilean world. On the other hand, the light wave carries
III, and IV, we construct a model of Lorentz-covariant lo- the burden of being consistent with the Lorentzian world.
calization of light waves. Finally, in Sec. VI, we examine
closely how the concept of photons can emerge from lo-
calized light waves. center of the wave function moves along the positive z
direction as is specified also in Fig. lib). The transformed
11. LIGHT WAVES AND WAVE PACKETS wave function takes the form
IN NONRELATIVISTIC QUANTUM MECHANICS
1/!u (z,t) = exp[ -imlvz- +v'ti]
In this paper we are concerned with the possibility of
constructing wave functions with quantum probability in-
X Jg(k _mv)eilk,-k"l2mldk , (71
terpretation for relativistic massless particles. The natural where" is the boost velocity. This expression is different
starting point for tackling this problem is a free-particle from the usual expression in textbooks by an exponential
wave packet in nonrelativistic quantum mechanics which factor in front of the integral sign. The origin of this
we pretend to understand. Let us write down the time- phase factor is well understood."
dependent Schriidinger equation for a free particle moving In nonrelativistic quantum mechanics, 1/!lz,tl has a
in the z direction: probability interpretation, and there is no difficulty in giv-
.ata1/!lz,/} = -2m
I-
- I [a-a .I-
Z
1/!(z,t) . (41
ing an interpretation for the transformed wave function in
spite of the above-mentioned phase factor. The basic un-
solved problem is whether the probabilistic interpretation
The Hamiltonian commutes with the momentum opera- can be extended into the Lorentzian regime. This has
tor. If the momentum is sharply defined, the solution of been a fundamental unsolved problem for decades, and we
the above differential equation is do not propose to solve all the problems in this paper. A
reasonable starting point for approaching this problem is
1/!iz,L)= exp[iipz-p't/2m)] . (51 to see whether a covariant probability interpretation can

r'
be given to light waves.
If the momentum is not sharply defined, we have to
For light waves, let us start with the usual expression
take the linear superposition
1/llz,l)= Jgik)exp[ilkz -k't!2m)jdk i6)
/lz,O= 12~ f g(k)ei,h-wtldk . (8)

The width of the wave function becomes wider as the time

variable increases, as is illustrated in Fig. 1(al. This is Unlike the case of the Schriidinger wave, '" is equal to k,
known as the wave-packet spread. and there is no spread of wave packet. The velocity of
Let us study transformation properties of this wave propagation is always that of light, as is illustrated in Fig.
function. Rotation and translation properties are trivial. ](bl. We might therefore be led to think that the problem
In order to study the boost property within the framework for light waves is simpler than that for nonrelativistic
of the Galilei kinematics, let us imagine an observer mov- Schriidinger waves. This is not the case for the following
ing in the negative z direction. To thi~ ohserver, the reasons
432 CHAPTER VIII

1684 D. HAN, Y. S. KIM, AND MARILYN E. NOZ 35

(I) We like to have a quantal wave function for light called the photon polarization vectors.
waves. However, it is not clear which component of the In order that the four-vector be a helicity state, it is
Maxwell wave should be identified with the quantal wave essential that the time.like and longitudinal components
whose absolute square gives a probability distribution. vanish:
Should this be the electric or magnetic field, or should it
(13)
be the four-potential?
(2) The expression given in Eq. (8) is valid in a given This condition is equivalent to the combined effect of the
Lorentz frame. What form does this equation take for an Lorentz condition
observer in a different frame?
(3) Even if we are able to construct localized light _a_[A"(x)]=O (14)
waves, does this solve the photon localization problem? ax" '
(4) The photon has spin I either parallel or anti parallel and the transversality condition
to its momentum. The photon also has gauge degrees of
freedom. How are these related to the above-mentioned V·A(x)=O. (15)
problems?
As before, we call this combined condition the helicity
Indeed, the burden on Eq. (8) is the Lorentz covariance.
gauge. IS
It is not difficult to carry out a spectral analysis and
While the Lorentz condition of Eq. (14) is Lorentz in-
therefore to give a probability interpretation for the ex-
variant, the transversality condition of Eq. (15) is not.
pression of Eq. (8) in a given Lorentz frame. However,
However, both conditions are invariant under rotations
this interpretation has to be covariant. This is precisely
and under boosts along the direction of momentum. We
the problem we are addressing in the present paper.
call these helicity preserving transformations. The boost
along an arbitrary direction is illustrated in Fig. 2. This is
III. UNITARY REPRESENTATION
not a helicity preserving transformation. However, ac-
FOR FOUR-POTENTIALS
cording to Ref. 15, we can express this in terms of helicity
One of the difficulties in dealing with the photon prob- preserving transformations preceded by a gauge transfor-
lem has been that the electromagnetic four-potential could mation.
not be identified with a unitary irreducible representation Let us consider in detail a boost along the arbitrary
of the Poincare group.12-15 The purpose of this section is direction specified in Fig. 2. This boost will transform
to resolve this problem. In Ref. 15 we studied unitary the momentum to p', as is illustrated in Fig. 2;
transformations associated with Lorentz boosts along the (]61
direction perpendicular to the momentum. In this section
we shall deal with the most general case of boosting along However, this is not the only way in which p can be
an arbitrary direction. transformed to p'. We can boost p along the z direction
Let us consider a monochromatic light wave traveling and rotate it around the y axis as is shown in Fig. 2. The
along the z axis with four-momentum p. The four- application of the transformation [R(&IB,(S-I] on the
potential takes the form four-momentum gives the same effect as that of the appli-
cation of B¢( 71). Indeed, the matrix
(9)
D( 71)= [B.( 71 I]-IR W)B,(S) 1171
with

We use the metric convention xll=(x,y,z,tl. The momen-

tum four-vector in this convention is
p"=(O,O"u,w) . (10)

Among many possible forms of the gauge-dependent

four-vector A", we are interested in the eigenstates of the I
/B4>I~)
\R181
helicity operator
!

o -i 0 0
o 0 0
S]= 0 0 0 0 (Ill
o 0 0 0 ----------__ p-6p
FIG. 2. Lorentz boost along an arbitrary direction of the
The four-vectors satisfying this condition are light wave. The four-momentum can be boosted either directly
Ai=(I,±i,O,O) , (121 by B, or through the rotation Ry preceded by B, along the l

direction. Th~~e operations produce two Jiffen:ut four·vector~

where the subscripts + and - specify the positive and when applied to the polari7ation vector. However, they are con-
negative helicity states, respectively. These are commonly nected by a gauge transformation.
LOCALIZATION PROBLEMS 433

35 UNCERTAINTY RELATIONS FOR LIGHT WAVES AND THE. 1685

leaves the four-momentum invariant, and is therefore an course, is a limitation of the model we present. However,
element of the E(2)-like little group for photons. our apology is limited in view of the fact that laser beams
The effect of the above D matrix on the polarization these days can go to the moon and come back after reflec-
vectors A,± has been calculated in the Appendix, and the tionYl
result is With this point in mind, we note first that the above-
mentioned unitary transformation preserves the photon
(18)
polarization. This means that we can drop the polariza-
where tion index from A ~ assuming that the photon has either
positive or negative polarization. A ~(x) can now be re-
placed by A (x).
Next, the transformation matrices discussed in this sec-
tion depend only on the direction and the magnitude of
uly/,I)I= [
cos~ 1 [cost 1J + [[ cos~cosh-i ]'
-I
]112 the boost but not on the photon energy. This is due to the
fact that the photon is a massless partic1e 17 Indeed, the
matrices in Sec. III remain invariant even if w in Eq. (91 is
replaced by a different value. This means that for the su-
Thus D 1y/) applied to the polarization vector results in perposition of two different frequency states,
the addition of a term which is proportional to the four-
(241
momentum. D (y/ 1 therefore performs a gauge transfor-
mation on A~. 14,15 a Lorentz boost along an arbitrary direction results in a
With this preparation, let us boost the photon polariza- rotation preceded by a boost along the z direction. Since
tion vector neither the rotation nor the boost along the z axis changes
the magnitude of Aili = 1,2), the quantity
(19)
(25)
The four-vector A~ satisfies the Lorentz condition
p'~ A±~ ~ 0, but its fourth component will not vanish. remains invariant under the Lorentz transformation. This
The four-vector .4~ does not satisfy the helicity condi- result can be generalized to the superposition of many dif-
tion. ferent frequencies:
On the other hand, if we boost the four-vector after A'ie A(x)=~AkelWklz-t) ,
performing the gauge transformation D 11/ I, (26)
k
A'f =B"Iy/IA,± with
~B"IY/)I [B"IY/I]-IRII)IB,(#;) IA,± I A I'=.ll Ak 12 . (27)
k
=R(I)IB,(#;IA'± (201
The norm I A I' remains invariant under the Lorentz
Since B,(#;) leaves A,± invariant, we arrive at the con- transformation in the sense that it is invariant under rota-
clusion that tions and is invariant under the boost along the z direc-
tion.
(21)
Can this sum be transformed into an integral form of
This means Eq. (8)? From the physical point of view, the answer
should be yes. Mathematically, the problem is how to
A'±=Bx(y/JD(y/)A~ =lcose,+i,-sinl),O), (22) construct a Lorentz-invariant integral measure. It is not
which satisfies the helicity condition difficult to see that the norm of Eq. (27) remains invariant
under rotations, which perform unitary transformations
A';'=O on the system. The problem is how to construct a mea-
sure invariant under the boost along the z direction.
and (231
For this purpose, we shall borrow the techniques
p'·A·,=O. developed for the covariant harmonic-oscillator formalism
which has been very effective in explaining the basic rela-
The Lorentz boost B (y/ 1 on A ~ preceded by the gauge tivistic features in the quark model,lO, 18-20 and which en-
transformation Diy/I leads to the pure rotation R(el. ables us to combine covariantly Heisenberg'S position-
This rotation j:.-, a finite-dimensional unitary transforma- momentum uncertainty relations and the c-number time-
tion. energy uncertainty relation.),9,21
The abme result indicates, for a monochromatic wave,
that all we have to know is how to rotate. If, however, IV. LOCALIZATION PROBLEMS
rhe photol! momentum has a distribution, we have to deal IN THE RELA TIVISTIC QUARK MODEL
Wit il a linear superposition of waves with different nl()-
melaa. The phot~)n momentum can have both longitudi- We ~hall di~cu~.., in thi,;,; section the :.lSp~Ch d' ~he co-
nal anO tJ,lIl~\~r...,c distributions. In thi~) paper we shall Q\- vuriant harmunic-t)~cillator formalism whi..:h a:'(..: ~[seful in
.\utf/e t!!at th~re [~onl)' longitudinal distribution. Thh, or converting the :-,lllll ,1f Eg. (26) into an llltefld form
434 CHAPTER VIII

1686 D. HAN , Y. S. KIM, AND MARILYN E. NOZ 3S

while preserving the invariance of the sum of Eq. (27)

under the Lorentz boost along the z direction. The co- Z = [l + /3 112z' z = [ 1 112z'
l -Jl (32)
variant oscillator formalism has been extensively dis-
+ 1-/3 1 + , - 1+ /3 -
cussed in the literature. What we need is here to review
z + and z _ are called the light-cone variables, and the
its property under the boost along the z direction.
product z+z_ remains invariant under the boost:
Let us use Xo and Xb for the space-time coordinates of
the two quarks bound together in a hadron. Then it is (33)
more convenient to use the four-vectors
In the light-cone coordinate system, the ground-state
(28) wave function of Eq. (29) takes the form
It is not difficult to write down the uncertainty relations
for space-time separation variables and to define the re- tb(z/) = ~ exp
'VlT
I I - [~~z
1-/3 2 +1+ /3 211
- -z
1+ /3 + 1-/3 -
(34)

gion within which the hadronic wave function is locali zed

in the Lorentz frame where the hadron is at rest. Howev- This wave function or the probability density is locali zed
er, the crucial question is how these uncertainty relations in a circular region centered around the origin in the
appear to an observer in the laboratory frame. z+z _ plane when /3 = 0. As the hadron moves, the region
The uncertainty principle applicable to the space-time becomes elliptic. This elliptic deformation property is il-
separation of quarks in the hadronic rest frame is the lustrated in Fig. 3.
same as the currently accepted form based on the existing Let us next consider the momentum-energy wave func-
theories and observations. The usual Heisenberg uncer- tion. If the quarks have four-momenta Po and Ph , respec-
tainty relation holds for each of the three spatial coordi-
nates. The time-separation variable is a c number and
therefore does not cause quantum excitations 3 . 21 The
question is then whether this peculiar time-energy uncer-
tainty can be combined with Heisenberg'S position-
momentum uncertainty relation to a covariant form"
QUARKS - PARTONS
Such a combination is possible within the framework of
the covariant harmonic-oscillator formalism which can
explain the basic hadronic features including the mass
spectrum,lS proton form factors,I9 parton picture,9.22 and

-,
jet phenomenon 2o ,,·0 BOOS1' ,'Oil

We assume throughout this section that the hadron

moves along the z axis, and ignore and x and y coordi-
nates which are not affected by the boost along the z
~- cp
direction . If we consider only the ground-state wave
~
function, then the localization dictated by the uncertainty
relations associated with both space and time separation ~
.:( W'::::T::;~ )
O"ofh et<_
variables will lead to a distribution centered around the ~ 1.1"'01,1 11ft

origin in the hadron-rest frame with the z' and /' vari-
ables. The ground-state harmonic-oscillator wave func-
~
-
------------------------------
Cl O ~o
tion takes the form

The question then is how this region appears . to the

laboratory-frame observer, while the coordinates of the
two different frames are related by the Lorentz transfor-
mation
z=(z' +/3/' )/(1_/32 )1/2,
(30)
/ =(I'+/3z')/(I - /32 )1/2,
FIG. 3. The Lorentz deformation of a relativistic extended
where /3 is the velocity
parameter of the hadron. hadron in both the space-time and momentum-energy coordi-
nate systems. Because the Lorentz transformation property of
In order to approach this problem, let us employ
momentum-energy four- vector is the same as that of the space-
Dirac's form of Lorentz transformation. In his 1949 pa-
time four-vector, the Lorentz deformation in the momentum-
per,2l Dirac introduced the light-cone coordinate system
energy plane is expected to be the same as that in the zt plane.
in which the coordinate variables are
This figure summarizes the content of the earlier paper on the
(31) parton picture. (Refs. 9 and 10) in which the hadron, while be-
ing a bound state of quarks in its rest frame, appears as a collec-
In terms of these variables, the Lorentz transformation of tion of partons to an observer who moves with a speed close to
Eq. (30) takes the form Ihat of light.
LOCALIZATION PROBLEMS 435

35 UNCERTAINTY RELATIONS FOR LIGHT WAVES AND THE. 1687

tively, then the standard procedure is to introduce P and q is independent of {3. However, dz ~ or dz + alone is not.
The integration over z+ gIves the factor
(35)
[i 1+f3I/( 1_{3)]'12, and this factor is compensated by its
where P is the four-momentum of the hadron, and q is inverse [( l-f3I/( I +{3)]'12 coming from the z~ integra-
the momentum-energy separation between the quarks. tion.
We are concerned here with the uncertainty relations be- We used in this section the Gaussian form of the wave
tween the x variables of Eq. (28) and the above q vari- function purely for convenience. The above reasoning is
ables. The momentum-energy wave function is valid for all forms of distributions having the same
space-time boundary condition as that of the Gaussian
<!>iq"qo)= [5.; 1f exp[i(q,z-qot)]1/J(z,t)dtdz
function. Indeed, if we give up the localization along the
z + axis, then the integration measure along the z ~ axis
should be compensated by the contraction or elongation
=(l/V;)expl-[(q:)'+(q~ )']/21, (36) along the z + direction. If the system is boosted along the
where q, and qo are the momentum and energy separation
z direction, dz + and dz ~ are transformed as

[:~~ r'dZ+, dz~~ [:~~ ]'!2dL'

variables, respectively. Their Lorentz transformation
property is the same as that for z and t. The form of this dz+-. (42)
wave function is identical to that of the space-time wave
function. In terms of the light-cone variables: We can give the same reasoning for the momentum-
q~ =(q,+qo)/v':!, q~ =(q,-qo)/v'2, (37)
energy measures dq + and dq ~ . This transformation
property will play an important role in constructing local-
the momentum wave function of Eq. (36) takes the form ized light waves.

V. COVARIANT LOCALIZATION
OF LIGHT WAVES

(38) We discussed in Sec. IV the relativistic quark model in

which the overall hadron four-momentum is well defined,
The Lorentz deformation of this wave function is also il-
and the internal coordinate system has a momentum-
lustrated in Fig. 3.24
energy distribution. In the case of light waves, the fre-
The basic advantage of using the light-cone variables is
quency or momentum is not sharply defined, but has a
that the coordinate system remains orthogonal, and z +
distribution. In this case, we can take the average value of
and z ~ do not become linearly mixed when the system is
the momentum, and the distribution around this average
boosted. The Fouricr relations between the space-time
value. We can treat the average momentum like the ha-
and momentum-energy coordinates are
dronic momentum, and the distribution around the aver-

r'
(39) age value like the internal momentum distribution.
With this point in mind, let us rewrite Eq. (8) as
This means that the major and minor axes of the
momentum-energy
of the minor and
coordinates are the Fourier conjugates
major axes of the space-time coordi-
/(z,I)= [2~ f g(k)e;(k'~wt)dk . (43)
nates, respectively. Thus we have the following Lorentz- We shall approach this problem using Dirac's light-cone
invariant uncertainty relations. 25 coordinate system discussed in Sec. IV. For convenience,
(<lz+ i(f!.q~ )=(<lz: i(f!.q"- bl , we shall define here the light-cone variables as
(40)
(<lz~)(f!.q+)=(<lz*-)(f!.q:bl . s=(z+I)/2, u=(z-t). (44)

These uncertainty relations are well understood when sand u are different from z + and z _ of Eq. (31) by a
the hadron is at rest with {3=0. On the other hand, the factor of Vl. but their Lorentz transformation property
limit {3~ I can teach us many interesting lessons. The remains the same. We shall also define the new momen-
connection between this limit and Feynman's original tum variables as
form of the parton model" has been discussed repeatedly ku =(k +w)/2, k, =(k -wi . (45)
in the literature!' 10 As far as the localization of massless
particles is concerned, the distribution along one of the In the case of light waves, k, vanishes and k. becomes k

r'
light-cone axes becomes so widespread that it loses its lo- or w. In terms of the light-cone variables, the expression
calization along the axis. of Eq. (43) becomes

[2~
In Sec. V, we shall "give up" the localization along one
of the light-cone axes in order to study photons and light /(u)= f g(k)eikUdk . (46)
waves. In so doing, we will have the problem of normal-
izing the wave function by integration. The integration For a massive particle, the most convenient Lorentz
measure dz + dz is a boost-invariant ·quantity. This frame is the frame in which the particle is at rest, as was
means that the normalization integral, noted in Sec. IV. For a massless particle, as our study in
Sec. III suggests,26 we can start with a specific Lorentz
f ¥,(z,ti,'dz +dz~ , (41) frame in which the photon momentum has a given magni-
436 CHAPTER VIII

1688 D. HAN, Y. S. KIM, AND MARILYN E. NOZ 35

tude along the z direction. In this Lorentz frame, we as- this in order to make the system covariant. The net result
sume that the average photon frequency is 0 0 is that
'!4
(47)
!(u)=
[
:~~ ]
A(u) (57)

with
and
f
Il-W-]'!2n 1I
N= I golk,Oo) I'dk (48)

It is important to note that the introduction of this specif- 1-{3 (J

ic Lorentz frame does not cause any loss of generality.26

We can obtain the most general form by boosting the pho- (58)
ton along the z direction. If we use (3 as the boost param-
The velocity parameter (3 is zero when the average photon
eter, the new photon frequency is
frequency is no.
'!' Let us examine the problem using a concrete form of
0= [ 1-{3
1+{3 ] 0
0,
(491 glk,n). The covariant oscillator model discussed in Sec.
IV suggests the following normalized Gaussian form for
and this frequency should be the average value calculated the frequency distribution:
from the new and most general distribution function
g(k,O):

0=*fklglk,0)12dk, ISO)
g( k)= I I
7Tb
1'/4 [ I ]'12 [1-.{3
no
r
]:1(3 J
4

where N is given in Eq. (48) of golk,Oo). N is a

xexp [ ;bl Ik _n)2] , (59)

Lorentz-invariant quantity. In order that unitarity be

preserved, glk,!1) should satisfy the normalization condi- where b is a constant and specifies the width of the distri-
tion bution. The above form describes the distribution in k or
N= f Iglk,O)I'dk. (51) k, around 0, and there is no localization in the k, vari-
ables. In view of the discussion given in Sec. IV, it is not
The basic problem here is that the integral measure dk is difficult to understand the origin of the factor
not Lorentz invariant. One form for g(k,O) which meets [( 1-(3)/(I +(3)]'!2 in Eq. (59). The space-time wave
the invariance requirement is function f( u) takes the form

glk,!1)=(I/O)'12a (k -!1), (52)

!Iu)= I~ r4lt~ r 4exp [ifllZ-tJ-1(z-tI']

where alk -0) is a scalar function depending only on (60)

(k-O). The (1/0)'12 factor is proportional to
[(I -(3)/( I +(3) ]l!<, and assures the Lorentz invariance of This function has a distribution along the u = (z - tJ axis,
the normalization integral. This makes (1 /Q )dk a but has no localization along the saxis.
Lorentz-invariant measure. Let us go back to the question mentioned in Sec. I. Is
Let us next consider the left-hand side of Eq. (8). If we the time-frequency uncertainty relation a Lorentz-
insist on the Lorentz invariance of the normalization in- invariant relation? The wave function of Eq. (59) and
tegral Eq. (60) constitute the quantification of the Lorentz-
f If(u) 1 2du , (53)
invariant uncertainty relation

(Jlu)(Jlkbl. (61)
then 10/Oo)du is a Lorentz-invariant measure, and f( u)
can take the form From the definition given in Eq. (45), Jlk = Jlu). From
Eq. (44), Jlu = - JlI for a fixed value of z. This relation
(54)
becomes Jlu = JlI when the symbol tJ. means the width of
where A lu) is a scalar quantity. The integral form of Eq. distribution. Thus the time-frequency relation IJlw)(JltJ
(26)is is a Lorentz-invariant relation.

ISS)
VI. THE CONCEPT OF PHOTONS
with
We discussed in this paper Lorentz-covariant wave
(!1!!1o) L~~ I Alu) 1
2du =(I!!1) L~~ Ia Ik -0) 2dk .
1
functions for light waves. It is possible to construct a lo-
(56) calized wave function for light waves with a Lorentz-
invariant normalization. The mathematics of this pro-
Indeed, in Eq. (56), we have to multiply and divide the cedure is not complicated. We are then led to the ques-
right- and left-hand sides, respectively, by VQ. We did tion of why the photon localization is so difficult, while it
LOCALIZATION PROBLEMS 437

35 UNCERTAINTY RELATIONS FOR LIGHT WAVES AND THE .. 1689

is possible to produce photons in states narrowly confined when we make the transition from localized Maxwell
in space and time.2J waves to photons through second quantization.
Let us see how the mathematics for the light-wave lo-
calization is different from that of quantum electro- ACKNOWLEDGMENT
dynamics (QED) where photons acquire a particle inter-
pretation through second quantization. In QED, we start We are grateful to Professor Eugene P. Wigner for ex-
with the Klein-Gordon equation with its normalization plaining to us the background of the photon localization
procedure. As a consequence, we use the expression 28 problem.

g(k)=(l/Vk)a(k) , (62) APPENDIX: UNITARY TRANSFORMATIONS

OF PHOTON POLARIZATION VECTORS
instead of the form given in Eq. (52). The Lorentz-
transformation property of this quantity is the same as Let us work out the kinetics of Fig. 2. If we use the
that for g(k) of Eq. (52). four-vector convention xP=lx,y,z,t), the matrix which
However, the basic difference between the above expres- boosts p to p' is
sion and that of Eq. (52) is that the kinematical factor in
front of a (k) is (I/vk) in Eq. (62), while that for Eq. c 2+s 2Icosh1)) 0 sclcosh1) -1) slsinh1))
(52) is 11 vQ). There is no concept of the average 0 0 0
momentum in quantum field theory, while it was essential B.(1))=
sclcosh1) - I ) 0 s2+c 2Icosh1)) clsinh1) )
for the localized light wave discussed in Sec. V. Numeri-
cally, the above expression becomes equal to glk) of Sec. s Isinh1)) 0 clsinh1) ) cosh1)
V when a (k) of Eq. (62) represents a sharp distribution IAI)
around a fixed value of k. This is why the photon can ap-
pear as a light pulse on oscilloscope screens. where c = cos</> and s = sin</>. The parameters 1) and </>
On the other hand, the normalization property of Eq. specify the magnitude and direction of the boost, respec-
(62) is quite different. In quantum field theory, it is possi- tively. On the other hand, we can achieve the same pur-
ble to give a particle interpretation in terms of creation pose on the four-momentum by boosting p along the z
and annihilation operators by second-quantizing a Ik). In direction first and rotating the boosted four-momentum
the light-wave normalization, it is very difficult, if not as is specified in Fig. 2. The boost matrix takes the form
impossible, to give a particle interpretation. This means
that, from the mathematical point of view, the gap be- 0 0 0
tween photons and localized light waves is real and very 0 0 0
serious. B,ls)= (A2)
As for the space-time distribution I( u) of Eq. (46), we 0 0 coshS" sinhS
use the form 0 0 sinbS coshS"

Ilu)=A(u) (63)
with e'=cosh1)+lcosd>lsinh1). The rotation matrix is
in QED, without the factor 1fl/110)1/2 discussed in Sec.
V. As a consequence, the normalization condition is that cose 0 sine 0
the integral 0 I 0 0
R(8)= IA3)
f [A'(U)~A(U)-A(U)~A'IU) ]dZ
-sine 0 cose 0
i (64)
0 0 0 1

be Lorentz invariant. If we use this form of normaliza-

tion, the total probability is not always positive.'"
The rotation angle e is related to the boost parameters 1)
and</> by
We can summarize the discussion of this section in
Table I. There definitely is a gap between the concept of sine= (sinp)[lcosh1) -I \cosp + sinh'll
localized waves and that of photons. Numerically this cosh1) + (cos</! )sinh1)
gap is not serious. However, we have to cross this gap
IA4)
cose= I +lcosp)sinh'l+lcosc/»'lcosh'l-I)
TABLE I. Light waves and photons. Light waves can be lo- cosh1) +lcos</»sinh1)
calized, but we still do not know how to localize photons. The
difference between these two cases is not serious from the physi- The key question then is what is the difference between
cal point of view, The mathematical difference is still a serious these two transformations which produce the same result
problem. on the four-momentum. The best way to examine this
Probability Particle problem is to examine the closed-loop transformation
interpretation interpretation
lAS)
Light waves yes no
Photons no yes The matrix algebra is somewhat complicated, but is
straightforward. The result is
438 CHAPTER VIII

1690 D. HAN, Y. S. KIM, AND MARILYN E. NOZ

0 -u u We can now write sin", and cos'" in terms of e, and arrive

at the expression given in Eq. (18).
0 I 0 0
D(u)= (A6) Similar calculations exist in the literature, but the previ-
u 0 l-u'/2 u'/2 ous calculations are only for specified values of "'. In Ref.
u 0 -u'/2 l+u'/2 15 the calculation was made for", = 90°. In a recent paper
by Han et a/. 17 a similar calculation was carried out for
where the angle which will make 5=0. Here we carried out the
-(sinp)sinh?] calculation for the most general case.
u
cosh?] + (cos'" )sinh?]

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E. Noz, ibid. 8,3521 (19731. See also Ref. 10. Pagnamenta, Phys. Rev. D 2,1150, (1970); 2,1156 (1970). V.
2oFor papers dealing with the jet phenomenon, see T. Kitazoe N. Gribov, B. L. Ioffe, and I. Ya. Pomeranchuk, J. Nucl.
and S. Hama, Phys. Rev. D 19,2006 ([979); Y. S. Kim, M. Phys. (USSR) J 2, 768 (1965) [Sov. J. Nucl. Phys. 2, 549
E. Noz, and S. H. Oh, Found. Phys. 9,947 (1979); T. Kita- (1966)]; B. L. Ioffe, Phys. Lett. B 30. 123 (19691; Y. S. Kim
zoe and T. Morii, Phys. Rev. D 21, 685 ([980); Nucl. Phys. B and R. Zaoui, Phys. Rev. D 4,1738119711; S. D. Drell and T.
164, 76 ([980). M. Yan, Ann. Phys. (N.Y.) 60, 578 (1971).
21For continuing debates on the time-energy uncertainty rela- 2Sy. S. Kim and M. E. Noz, Found. Phys. 9, 375 ([979).
tion, see Y. Aharonov and D. Bohm, Phys. Rev. 122, 1649 26Th is point has been extensively discussed in the literature. See
(19611; V. A. Fock, Zh. Eksp. Teor. Fiz. 42, 1135 ([962) Refs. 10, 12, 13, 14, 15, and 17.
[Sov. Phys.-JETP 15,784119621]; J. H. Eberly and L. P. S. 27For a pedagogical discussion of the connection between pho-
Singh, Phys. Rev. D 7, 359 (1973); M. Bauer and P. A. Mel- tons and light waves, see E. Gordin, Waves and Photons (Wi-
lo, Ann. Phys. (N.Y.) 11,38 ([978); M. Bauer, ibid. 150, I ley, New Yack, 1982).
(1975), For some review papers on the time-energy uncertain- 2RFor the role of the Klein-Gordon equation in the development
ty relation, see articles by J. Rayski and J. M. Rayski, Jr., E. of Schrooinger's form of quantum mechanics, see P. A. M.
Recami, and E. W. R. Papp, in The Uncertainty Principles Dirac, Development of Quantum Theory (Gordon and Breach,
and Quantum Mechanics, edited by W. C. Price and S. S. New York, 1972).
Chissick (Wiley, New York, 1977); C. H. Blanchard, Am. J.
Chapter IX

Lorentz Transformations

Let us start with a massive particle at rest with its spin along the x direction. If we
boost this particle along the x axis, it will gain a momentum along the same
direction. If we boost this moving particle along the y direction, the direction and
the magnitude of the momentum will be changed. The resulting transformation will
be a Lorentz boost preceded by a rotation. This rotation does not change the
momentum of the particle at rest, but will change the direction of the spin. This is
called the Wigner rotation and, as was carefully analyzed by Han, Kim and Son in
1987, manifests itself as the Thomas precession in atomic physics.
The Lorentz group is useful also in studying charged particles in electromagnetic
fields. In 1959, Bargmann, Michel, and Telegdi studied the precession of the spin of
a charged particle in a homogeneous magnetic field. In 1976, Kuperzstych studied a
charged electron in a plane-wave electromagnetic field. He discussed the possible
origin of the Lorentz force in terms of the little group for photons.
The light-cone coordinate system is very useful in many physical applications. In
this system, Lorentz boosts are scale transformations in the light-cone variables. In
InO, Parker and Schmieg studied the fundamental hypotheses of special relativity
in terms of the light-cone variables.
The group of Lorentz transformations, while being the basic language for special
relativity, is becoming an indispensable theoretical tool many other branches of
physics. The (2 + I)-dimensional Lorentz group is locally isomorphic to the group
of homogeneous linear canonical transformations in phase space. As was discussed
by Han, Kim, and Noz in 1988, the group of canonical transformations is very
useful in studying coherent and squeezed states in terms of the Wigner distribution
function. In their 1986 paper Yurke, McCall and Klauder (1986) used the (2 + 1)-
dimensional Lorentz group very effectively in their discussion of a new
interferometer. Figure 4 illustrates the point that the Lorentz group is useful both in
special relativity and modern optics.

441
442 CHAPTER IX

Fo COSwt

mg

Special Modern
Relativity Optics

FIG. 4. Analogy of analogies. The analogy between the forced hannonic oscillator
and the driven LCR. circuit is well known. Since the Lorentz group is rapidly
becoming one of the standard languages in optical sciences, there will be many
instances in which one fonnuIa in the Lorentz group will describe one physics in
optics and another physics in special relativity. This figure is from D. Han and Y.S.
Kim, Univ. of Maryland P.P. #88-92 (to be published in Phys. Rev. A in 1988).
LORENTZ TRANSFORMATIONS 443

Reprinted from: Physical Review Letters. Volume 2.1959. pp. 435-436

PRECESSION OF THE POLARIZATION OF PARTICLES MOVING IN

A HOMOGENEOUS ELECTROMAGNETIC FIELD'"

V. Bargmann
Princeton University, Princeton, New Jersey
Louis Michel
Ecole Poly technique, Paris, France
and
V.L. Telegdi
University of Chicago, Chicago, Illinois
(Received April 27, 1959)

The problem of the precession of the "spin" of a particle moving in a homogeneous

electromagnetic field - a problem which has recently acquired considerable
experimental interest - has already been investigated for spin liz particles in some
particular cases. 1 In the literature the results were derived by explicit use of the
Dirac equation, with the occasional inclusion of a Pauli term to account for an
anomalous magnetic moment. On the other hand, following a remark of Bloch2 in
connection with the nonrelativistic case, the expectation value of the vector operator
representing the "spin" will necessarily follow the same time dependence as one
would obtain from a classical equation of motion. To solve the problem for
arbitrary spin in the relativistic case, it will thus suffice to produce a consistent set of
covariant classical equations of motion. Such equations have been indicated a long
time ago by Frenkel3 and are discussed by Kramers. 4 These authors use an
antisymmetric tensor M as the relativistic generalization of the intrinsic angular
momentum observed in the rest-frame of the particle. A formulation in terms of the
(axial) four-vector s which describes the polarization in a covariant fashion 5 -
though basically equivalent - is however much more convenient for our problem.
We shall therefore derive first the equations of motion directly in terms of this four-
vector s.
Let the spin of the particle be represented6 in the rest-frame (R) by -r. We assume
(a) that there exists a four-vector s such that in (R) it coincides with-r:
in (R ), s =(0, S) . (1)
Denoting the four-velocity of the particle by u = (uo, It) = ,),(1, V) [where V is the
ordinary velocity, and ')'(v) = (1 -v2rIl2], one has in every frame

Reprinted from Phys. Rev. Lett. 2,435 (1959).

444 CHAPTER IX

(2)
We further assume (b) that st obeys in (R) the customary equation of motion
Uld'C = (ge 12m )(stxii), (R) (3)

where it, e, and m have their standard meanings, while the gyromagnetic ratio g is
defined by this very equation. While sO vanishes by hypothesis in any instantaneous
rest-frame, ds Old 'C need not. In fact, (2) implies
ds°ld'C =st·(tNld'C), (R) (4)

for such frames. In general, duld'C =/Im (where / = four-force), while in a

homogeneous external electromagnetic field specified by P = -(E, it)
duld'C = (elm)P·u . (5)
The immediate generalization of Eqs. (3) and (4) to arbitrary frames is
dsld'C = (geI2m) [p·s + (s·p·u)u] - [(duld'C)·s]u, (6)

as can be checked by reducing to the rest-frame. With (5), one has for
homogeneous fields
dsld'C = (elm)[(gl2)F·s + (gl2 - 1) (s·p·u)u]. (7)
(5) and (7) constitute, for any value of g and arbitrary spin S, a consistent set of
equations of motion; they imply that s·s and s·u are constant, so that condition (2)
is maintained? For experiments of current interest, the main use of (7) is in the
computation of the rate n at which longitudinal polarization is transformed into a
transverse one (and vice versa). For this, we express s in the laboratory frame (L) in
terms of two unit polarization four-vectors, el and et:
sIS = e,coscj) + e,sincj),
where
S = (-s·s)~,
el='Y(v,V/v)='Y(v, v), e,=(O,Ii),

Ii·Ii = 1, Ii·v =0 . (L) (8)

Clearly, n =dcj)ldt =dcj)/'Yd'C. Introducing (8) into (7), and expressing all quantities
as ordinary vectors, we find
n = (elm) (fA ·Iilv)[(gI2 - 1) - gl2r] + (v·ilXJi)(gI2 - I)} . (9)
The relevant "anomaly" of spin -1f2 particles, (g12 - 1), is clearly exhibited in (9)
although our derivation was classical throughout.
We now specialize (9) to some cases of practical interest (the references are to
experiments):

(A) E x 'it = i l x 'it = 0; n = O. The character of the polarization does not

LOREN1Z 1RANSFORMATIONS 445

change, but the transverse polarization precesses around V in longitudinal fields with
an angular frequency ro = (ge 12m y)H = (g 12)roL' as follows readily from (8).

(B)8 ir,; x v = H, E = 0; Q = roL<g 12 - l)y, where roL is the Larmor frequency

defined in (A) above.

(C)9 E''; = E, iI = 0; Q = rop [-g 12y+ (g 12 - l)y], where rop=eE Imyv is the
angular frequency of the particle's motion in the laboratory.

(D)1O E·fI = 0, rectilinear motion: E = -V x fI; fl·,; x V = H, Q = roL<g l2y).

(E)ll E·iI = 0, fl·,; x V = H, E,'; = -Ev,Jv, trochoidal motion: E IH« 1;

Q = (e 1m )[(g 12 - 1)(H - Ev,,) + Ev"/(r - 1)],
(A<j>/21t) per loop = y(E /H)y (v)[1 - (E IH)v,,](g 12 - 1)
= Y(v')(gI2 - 1),

where v' is velocity in a frame where E' = 0.

The generalization of (6) to cover particles having an intrinsic electric dipole
moment"£ = (g'e 12m )~ m~ be of interest. In the (R) frame, the effect of"£ is taken
into account by adding exE to the right-hand side of (3), while leaving (4)
unchanged. Thus the required change in the right-hand side of (6) is the addition of
a term -(g'el2m)[(F· 's) + (s·F· 'u)u], denoting by F· the dual of F, i.e.,
F· = -(if, -1n For the experiment (B) above, one obtains then
IQ I = roL yx[g 12 - 1)2 + (g'v 12)2]'h.
Discussions leading to this note were initiated when the two last-named authors
were visiting Princeton University. They are indebted to J. R. Oppenheimer for
making the facilities of the Institute for Advanced Study available to them.
"Research partly assisted by the Office of Scientific Research, Air Research and Development
Command, and by the French Service des Poudres.
IH. A. Tolhoek and S. R. de Groot, Physica 17, 17 (1951); H. Mendlowitz and K. M. Case,
Phys. Rev. 97, 33 (1955); L. M. Carrassi, Nuovo cimento 7, 524 (1958).
2F. Bloch, Phys. Rev. 70,460 (1946).
3J. Frenkel, Z. Physik 37, 243 (1926).
4H. A. Kramers, Quantwn Mechanics (North Holland Publishing Company, Amsterdam,
1957), p. 226 et seq. Kramers' Eq. (4) (p. 229) does not correspond to our Eq. (6). His conclusion that
already classically g = 2 is implied for an electron, based on the relativistic equation he uses, stems from
the fact that that equation corresponds in the rest-frame to ds°ldt = (gel2m)3'·I1. Comparing with our
(4), with dv'ldt = (e Im)I1, one sees that the "derived" result is, in fact, built into the theory from the
start. A more general equation is mentioned by Kramers on p. 231, in fine print, and attributed to Frenkel.
The inconsistencies arising in Kramers' discussion of what he calls "spin-omit" forces [i.e., of the form
(VHVit in the rest-frame1are connected with the fact that neither of his equations of motion applies when
the field is inhomogeneous. In that case, du Id t is not given by (5) alone, but has to include an additional
term (the covariant analog of the gradient force just mentioned) before being introduced into (6).
sThe covariant polarization four-vector s is essentially the expectation value of the operator w
used by Bargmann and Wigner [Proc. Nat!. Acad. Sci. U. S. 34, 211 (1948») to characterize
representations of the inhomogeneous Lorentz group: <w·w > =-s (S + l)m 2, S = spin. s can be
446 CHAPTER IX

expressed in tenns of the skew tensor M of Frenkel (which satisfies M·1l = 0), and vice versa:
a =M- ·Il,M- =aXll, i.e.,M- iIc =a i Ilk - akll i • For the quantum-mechanical applications of a see, e.g.,
C. Bouchiat and L. Michel, Phys. Rev. 106, 170 (1955).
60ur notation is: c - I, It - 1 throughout; coordinate four-vector of components
xO=1 ,Xl, xl, x 3: x = (%0,1'>, ~ = {xa.} (11 = 1,2,3); metric of signature (+ --); t - proper time; a dot
between symbols, contraction of neighboring indices with the metric tensor, e.g., x·x = (x~l - rz;skew
tensor of components Til; indicated as T = ct', f'), 1" = {TOm}, f· = {Tftr}, 11, p, Y= I, 2, 3; its
dual by T- = ct" - 1,).
7Equations (5) and (I) can be integrated explicitly by reference to four orthonormal four-
vectors Il (i) such that each of them obeys (5), and 1l(0l:u.
8Crane, Pidd, and Louisell, Bull. Am. Phys. Soc. Ser. 11,3,369 (1958).
9H. Frauenfelder III al., Phys. Rev. 106, 386 (1957).
lOp. E. Cavanagh III al., Phil. Mag. 2, 1105 (1957).

lip. S. Farago, Proc Phys. Soc. (London) 72, 891 (1958).

LORENTZ TRANSFORMATIONS 447

Is there a Link between Gauge Invariance, Relativistic Invariance

and Electron Spin?

J. KUPERSZTYCH

Commissariat a l'Energie Atomique, Oentre d'Etudes de Limeil

B.P. 27, 94190, Villeneuve· Saint· Georges, France

(ricevuto il 28 Luglio 1975; manoscritto revisionato ricevuto 1'8 Ottobre 1975)

Summary. - Without any assumptions other than the theory of classical

clectrodynamics applied to the problem of a charged particle interacting
with the field of an electromagnetic plane wave, an operator is derived
which has no other significance than the motion of the spin of a Dirac
particle, given by the solution of the Bargmann·Michel.Telegdi equation.
It is shown that the motion of an electron and the motion of its spin in
a plane wave are given by an operator of Lorentz type which has the
remarkable property of being equivalent to a gauge transformation.

Introduction.

It is well known that the notion of spin is a notion which has no classical
equivalent. Historically, the spin appeared as a supplementary degree of freedom
(an intrinsic angular moment) necessary to explain the Zeeman effect. One of
the most remarkable results of Dirac's electron theory is, perhaps, the theoretical
derivation of the correct value of the electron magnetic moment, initially
postulated by KRONIG, UHLENBECK and GOUDSMIT from experimental data (1).
The fact that the existence of spin can be revealed by seeking a manifestly
Lorentz-invariant theory seems to show that the spin is an essentially relativistic
notion. This idea is reinforced by the fact that electron spin occurs when

(1) M. JAMMER: The Conceptual Development of Quantum Mechanics (New York,

N. Y., 1966).

Reprinted from Nuovo Cimento 31B, 1 (1976).

448 CHAPTER IX

2 J. KUPERSZTYCH

electromagnetic interactions take place, interactions which do not satisfy

Galilean invariance.
In spite of the spin appearing to be a purely quantum mechanical and re-
lativistic notion, it seems reasonable to ask how the spin of a particle which
is moving classically according to the Lorentz force law, behaves.
The answer to this question is given, in the case of slowly varying electro-
magnetic fields, by the Bargmann-Michel-Telegdi (BMT) equation (2). In ad-
dition, this equation takes into account the influence of the « anomalous part »
in the magnetic moment of the particle. The BMT equation can be derived
either by calculating the classical limit of the generalized Dirac equation, or
more simply, by using relativistic invariance considerations.
Thus, in classical electrodynamics, the spin of a particle seems to be an ex-
traneous concept. Its origin seems only to be justified by the existence of the
relativistic quantum theory.
In this paper, using only the laws of classical electrodynamics (Maxwell
equations, Lorentz force law) and invariance properties (gauge invariance, re-
lativistic invariance) in the problem of an electron interacting with the field
of an electromagnetic plane wave, we shall show that it is possible to derive
an operator which cannot be physically understood without introducing the
notion of electron spin.
We shall begin by considering only the electromagnetic field and we shall
show that, in the case of electromagnetic plane waves, it is possible to find
an operator of Lorentz transformation which is equivalent to a gauge trans-
formation.
We shall next consider an electron interacting with that field and we shall
show that the above-mentioned operator allows the passage from the frame
where the electron was at rest before the field was switched on, to a frame
where the particle is at rest at each moment of time (an instantaneous rest
frame of the particle).
This operator of Lorentz type will be constructed from the product of two op-
erators if and f1A which will be shown each to have a precise physical significance.
We shall show that the operator if represents the motion of a charged particle
in the field and, giving the exact solution of the Bargmann-Michel-Telegdi
equation, that the operator f1A represents the motion of spin of a particle having
a magnetic moment exactly equal to that of a Dirac particle.

1. - Derivation of a Lorentz transformation equivalent to a gauge transformation

in the case of electromagnetic plane waves.

In this section it will be shown that it is possible to find a Lorentz trans-

formation which will leave the fields of a plane wave unaltered.

(2) V. BARGMANN, L. MICHEL and V. L. TELEGDI: Phys. Rev. Lett., 2, 435 (1959).
LOREN1Z TRANSFORMATIONS 449

IS THERE A LINK BETWEEN GAUGE INVARIANCE ETC. 3

Let us consider an electromagnetic plane wave travelling along the x-axis

of a frame L which will be called the laboratory frame. The fields of the plane
wave are functions of the relativistic invariant r: = nr = ct.- x, where nand r
are respectively the four-vectors (!) (c;).
and n is the unit vector of the x-axis.
A unit system where n= C = 1 will henceforth be used.
The problem is to find a Lorentz transformation, characterized by a matrix
..It, such that it will yield the following:

(1) I E'(r:')

H'(r:')
= E'(r:) = E(r:) ,
= H'(r:) = H(r:) ,

where E, Hand E', H' are respectively the electric and magnetic fields of the
plane wave, derived from the four-potentials (~) and (~:) in the frame Land
in the transformed frame L'.
If we use the definition of fields from potentials, the system of eqs. (1)
involves the following equation:

(2)

where A is an unknown function of time and space.

Equation (2) is obviously significant: while the right member of this equation
is a definite gauge transformation of the four-potential (~), the left member
is the transformation of this four-potential by the desired Lorentz operator.
In order to derive the operator ..It, it is necessary to know the form of the
most general transformation of the proper Lorentz group.
It is known that a general Lorentz transformation (3), generated by an
operator ..It, can be constructed from a special Lorentz transformation .P
(without change in direction of the space co-ordinate axes, but with an arbitrary
direction of the velocity (3), followed by an arbitrary rotation ~ of the axes
in space.
Consequently

(3) c. MOLLER: The Theory of Relativity, Second Edition, Subsect. 2.4 (London, 1972).
450 CHAPTER IX

4. J. KUPERSZTYCH

where

Because of the axial symmetry around the x-axis, we can choose ~ in the
(x, y)-plane without prejudicing the generality of the calculation. Let a: be the
angle between n and ~.
A priori, the operator &I is a function of three independent parameters
(for instance the three Euler angles). However, in order to shorten the cal-
culation, the following must be kept in mind: since the Lorentz transformation
..It in question would be equivalent to a gauge transformation, the direction n
of propagation of the plane wave must not, of course, be modified. Now, owing
to the ~-transformation, the a:-angle will change in the (n, ~)-plane. Therefore
to compensate for this change, we can immediately conclude that it is neces-
sary for the desired rotation to be in the (n, ~)-plane, that is, around the z-axis.
Calling "P the angle of rotation, we can then write the operator ..It in the fol-
lowing form:
..It = &I("P) ~(P, a:),

that is

o 0

cos"P -sin "P

(4) ..It =
sin "P cos "P
0 0 0

I' -I'P cos a: -I'P sin a:

-I'P cos a: 1 + (1'-1) cos 2 a: (I' -1) sin a: cos a:

-I'P sin a: (I' - 1) sin a: cos a: 1+ (1'-1) sin 2 a: 0

0 0 0 1

The Lorentz transformation ..It which will leave the fields of the plane
wave unaltered is obviously gauge invariant. We can therefore require the
potentials cp, A, cp', A', and the gauge function A(t, r) to be functions of 7: only.
Then, these requirements involves the following relations:

aA = OA = 0 cp =
ay oz ' A~,
LORENTZ TRANSFORMATIONS 451

IS THERE A LINK BETWEEN GAUGE INV.ARlANCE ETC. ;;

which, inserted in the basic equation (2) with .A given by (4), now give the
system of two equations

[sin 1p + (I' -1) cos oe sin (oe + 1p) - yfJ sin (oe + 1p)] rp +

+ [cos 1p-1 + (1'-1) sinoe sin (oe + 1p)]A,,= 0,

[cos 1p + (I' -1) cos oe cos (oe + 1p) + yfJ cosoe-yfJ cos (oe + 1p) -y]rp+

+ [- sin 1p + (I' -1) sin oe cos (oe + 1p) + yfJ sin oe] All = 0 .

Since rp and All are linearly independent, all their coefficients must be equal
to zero. We finally obtain the solution to our problem in the form of the two
following relations:

1
(5) oe = a.rc cos I' fJ1 = arc cos JL or 1 - fJ cos oe = - ,
I' 1'+1 I'

(6) 1p=n-2oe.

The matrix.A of the sought Lorentz transformation is finally

I' - (1'-1) - [2(1'-1)]'

(1'-1) (2-1') - [2(1'-1)]1

(7) .A(y) =
- [2(1'-1)]1 [2(1' -1)]' 1

0 0 0

and is dependent on the arbitrary parameter 1';;;.1. We can now obtain the
gauge function A(T) such that eq. (2) has a solution. It is

I
T

(8) A(T, 1') = [2(1' _1)]' AI/(T') dT' •

Therefore, when the potentials are functions of the retarded time T like the
fields of the plane wave, the Lorentz transformation .A(y) given by (7) is equi-
valent to a gauge transformation whose gauge function is given by (8).
Thus, we have shown that it is possible to find a Lorentz transformation
which will leave both the electric and magnetic fields of a plane wave unaltered.
452 CHAPTER IX

J. KUPERSZTYCU

2. - On the motion of a charged particle in the field of an electromagnetic

plane wave.

Let us consider the equations of motion of a charged particle in the field

of a plane wave (supposed to be linearly polarized along the y-axis) ip. the la-
boratory frame L where the particle was at rest before the field was switched on.
They are (')

e(l') = m(l- v2('l') )-t = m (1 + 112;1')) ,

(9)
P(1') = mll(1')j+ ~m1l2(1')n,

where e(1') represents the energy of the particle in the field, P(1') is its momentum,
v is its velocity, m is its rest mass and - e (e> 0) is its charge. j is the unit
vector of the y-axis, 11 = (e/m)(- AJ' A,i is a dimensionless parameter.
If we use the same device as that employed for the definition of proper time
of a moving particle (6), the motion of a charged particle in the field of a plane
wave may be considered as uniform at each moment of time.
Thus, at each moment of time we can use a Lorentz transformation in order
to pass from the frame L to the particle instantaneous rest frame R. It will
now be shown that the frame R can be deduced from the frame L using pre-
cisely a Lorentz transformation of type .A(y) which is equivalent to a gauge
transformation.
In order to do this, the parameter y of the operator .A(y) given by (7), has
now to be determined. We have consequently to solve the following equation:

e(1'))
(10) .A(y) ( P(1') = (m)
0 '

where (;~;)) is the four-momentum of the particle in the frame L given by

eqs. (9) and where (:) is this four-momentum in the rest frame R.
From eq. (10) we have immediately

(11) y(1') = 1 + 112(1')

2
,

(4) E. S. SARACHIK and G. T. SHAPPERT: PhY8. Rev. D, 1, 2738 (1970); L. D. LANDAU

and E. M. LIFSHITZ: The Olassioal Theory of Field8, Second Edition, Sect. 47, Problem 2
(Oxford, 1962).
(5) L. D. LANDAU and E. M. LIFSHITZ: The Ola8sioal Theory of Field8, Second Edition,
Sect. 3 (Oxford, 1962).
LOREN1Z TRANSFORMATIONS 453

IS THERE A LINK BETWEEN GAUGE INVARIANCE ETC. 7

and the matrix .#(y) now takes the following form:

')12
1+~2 2
-')I 0

')I" ')12
1-- -')I 0
2 2
(12) .#(')I(T)) =
-')I ')I 1 0

0 0 0 1

In other words, the equations of motion (9) of a charged particle in the field
of a plane wave can be written in the simple following form:

(13)

Therefore, it may be asserted that in the particle instantaneous rest frame R,

the fields E and H of the plane wave are the same as in the laboratory L, where the
particle was at rest before the field was switched on.
Now we may observe that eq. (13) can also be written in the form

It is clear that the operator 2(')1) is sufficient to bring about the passage
from the laboratory L to a frame where the particle is at rest at each moment
of time.
We remember that 2 is a Lorentz transformation (without change in direc-
tion of the space co-ordinate axes) characterized by the following relation:

(5')

It is easy to check that the condition (5') is satisfied by the velocity v of

the particle. As a matter of fact, we have from (9)

I-v(T)·n = (1 + ')12)-1
2 '

which is exactly the relation (5). Moreover, it follows from this above relation
that T is also the proper time of the particle.
454 CHAPTER IX

8 J. KUPERSZTYCH

The physical significance of the operator 2' is therefore obvious: it shows

the motion of the particle in the field.
On the other hand, the physical significance of the operator of rotation fJ?
of angle 1p submitted to relation (6) does not seem at all evident. In order to
clear up this problem, it is convenient to have the operator fJ?{v) in an explicit
form. The matrices fJ? and 2', written as functions of v, if we start from expres-
sion (4) and take into account the relations (5), (6) and (11), becomes

1 0 0 0
1-'1'2/4 v
0 0
1 + '1'2/4 1+'1'2/4
(14) fJ?(V(i)) =
v 1-'1'2/4
0 0
1 + '1'2/4 1+ '1'2/4
0 0 0 1

'1'2
1+~ -v 0
2 2
'1'2 '1'4 '1'3
0
(15) 2'(V(T)) = 2 1 + 8(1 + '1'2/4) 4(1 + '1'2/4)
'1'3 '1'2
-v 0
4(1 + '1'2/4) 1 + 2(1 + '1'2[4)
0 0 0 1

So, if the transformation 2'{v) is applied to the four-vector (~~;)) given

by (9), we have, as mentioned above

,13(T))
2'(V(T)\P(i)
(m)
= 0 .

Thus, the particle is also at rest in the frame K which is transformed from
the frame L by the Lorentz transformation 2'{v).
We arrive now at the crucial point of the paper. The operator fJ?{v) given
by (14), which stems from the calculation, has been derived without any as-
sumption further than the theory of classical electrodynamics.
In order to find its physical significance, we shall now consider a dynamic
quantity for the particle other than its four-momentum, namely its intrinsic
angular moment, that is to say, its spin (if it exists). In what follows it will
be shown that the motion of the spin of a charged particle, the gyromagnetic
ratio of which is g = 2 as for a Dirac particle, is shown by just the operator
fJ?{v) in question.
LORENTZ TRANSFORMATIONS 455

IS THERE A LINK BETWEEN GAUGE INVARIANCE ETC. 9

3. - Motion of the spin of an electron with an anomalous magnetic moment

in the field of an electromagnetic plane wave.

Let us consider the particle in its instantaneous rest frame K. We will now
look for the solution to the problem of the behaviour of the spin of a charged
particle which is executing a given classical motion in the field of a plane wave.
This solution will be given by the solution of the following equation (6),
which is derived from the BMT equation:

(16) d~ = 2f-lm+ 2f-l'(e-m) ~xH + 2f-l' e (v.H)v x~+

dt e e+ m

where ~ is the spin vector of the particle in its instantaneous rest frame K, and
where f-l' = f-l + e/2m is the anomalous part of the magnetic moment f-l of the
particle.
If we take into account eqs. (9) and after a straightforward calculation,
the eq. (16) can be written in the form of the following equations:

dCx dv
(17) dr = e(v) dr CII ,

dCy dv
(18) dr = - e(v) dr Cx ,

(19) dC. _ 0
dr - ,

where
V2)-1 m
e{v) = ( 1 + 4" -2 ef-l' .

If we define now the complex quantity

eqs. (17) and (18) can be written in the form

dX . dv
- = -te(v)- X.
dr dr

(6) V. B. BERESTETSKII, E. M. LIFSHITZ and L. P. PITAEVSKII: Relativistic Quantum

Theory, Part I, Sect. 41, eq. (41.9) (Oxford, 1971).
456 CHAPTER IX

10 J. KUPERSZTYCH

The solution of the above equation is obviously

(20)

where
v m
(21) "P'(r) = 2 arctg--2 _,u' v,
2 e

and where Co,., Co", Co. were the components of the vector t( r) when the potentials
were put equal to zero, that is before the field was switched on.
Obviously eq. (19) gives
(22)

while eq. (20) yields

(23) I CAr) = Re x(r) = Co,. cos "P'(-r)

C,,(r) = 1m x(r) = - Co,. sin "P'(r)
+ Co" sin "P'(r) ,
+ Co. cos "P'(r).
The relations (22) and (23) show clearly that the particle spin rotates in the
plane (x, y), around the vector H(r) with angle - "P'(r) given by (21).
Let us now neglect the anomalous part ,u' of the magnetic moment ,u of
the particle, i.e. let us now consider a charged particle (sayan electron) with
a gyromagnetic ratio g = 2 (as for a Dirac particle) (,u = - ge/4m).
In this case, from (21) we have

and •
sm"P r
'()
= 1 +V'1'2/4 •

The motion of spin of a classical Dirac particle in the field of a plane wave is
then given by
t(r) = 8i(- v) to = 8i- 1 (v) to ,

or
to = 8i(v)t(r),

where 8t'(v) is just the operator given by (14).

One can therefore conclude that the passage from the frame where an electron
was at rest before the field of a plane wave was switched on, to the frame where the
electron and its spin are at rest at each moment of time, is made by an operator
of Lorentz transformation which has the remarkable property of being a gauge
transformation.
LORENn TRANSFORMATIONS 457

IS THERE A LINK BETWEEN GAUGE INVARIANCE ETC. 11

4. - Conclusion.

We have therefore shown that for the problem of a charged particle clas-
sically interacting with the field of a plane wave, it is possible to derive an
operator which cannot be physically understood without introducing the notion
of electron spin. The operator in question &?(v) represents exactly the motion
of the spin ?f a particle the magnetic moment of which is precisely that of a
Dirac particle. As was shown, another value of the magnetic moment would
provide another operator than the one which was previously derived.
This result seems to indicate that there is a link between relativistic in-
variance, gauge invariance and electron spin on the classical level.
However, we must not be too optimistic: the results presented in this paper
do not imply that the spin « must exist ». The Lorentz force law is a priori
valid for any charged particle (neglecting radiation reaction) and can therefore
be used in the problem of a charged pion which is classically interacting with the
field of a plane wave. But since this particle has no spin, a physical interpre-
tation of the operator ~(v) does not exist in the case of the pion.
For such a particle, from an aesthetic point of view we can only come to
an irritating conclusion .

• RIASSUNTO (.)

Senza alcuna ipotesi oltre la teo ria dell'elettrodinamica classica applicata al problema
di una particella carica interagente con il campo di un'onda elettromagnetica piana,
si deduce un operatore che non ha altro significato che il moto dello spin di una par-
ticella di Dirac, dato dalla soluzione dell'equazione di Bargmann-Michel-Telegdi. Si
dimostra che il moto di un elettrone ed il motodel suo spin in un'onda piana sono dati
da un operatore del tipo di Lorentz che ha la lodevole propriet1t di essere equivalente
a trasformazioni di gauge.
('J Tmduzione a cum della Redazione.

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HHBapHaHTHOCTbIO H CUHHOM 3JIeKTpoua?

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iKeHHoil: '1aCTRUbI, BJaIlMo,n:eil:cTBYIOIIJ;eil: C nOJIeM IIJIOCKOil: 3JIeKTpOMarHHTHoil: BOJIHbI,
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3KBHBaJIeHTeH KaJIH6pOBO'IHOMY npeo6pa30BaHmo.

(') IIepe6eoeno peoaKlIueii.

458 CHAPIERIX

Reprinted from AMERICAN JOURNAL OF PHYSICS, Vol. 38, No, 2, 218-222, February 1970
Printed in U. S. A.

Special Relativity and Diagonal Transformations

LEONARD PARKER AND GLENN M, SCHMIEG

Department of Physics, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201

(Received 13 June 1969; revision received 7 September 1969)

We discuss the form of the special Lorentz transformation, and the corresponding transforma-
tion of the electromagnetic field, in which the transformation matrix is diagonal. We derive
the diagonal form of the special Lorentz transformation directly, in a simple way, and show
that it is sometimes more convenient to apply than the algebraically equivalent conventional
form of the transformation. The convenience i, especially evident in deriving the linear
Doppler effect, and the relativistic addition of more than two parallel velocities. By writing
Maxwell's equations in terms of linear combinations of coordinates which have simple
transformation properties, we arrive at the transformation eqnations of the Maxwell fields
in a diagonal form, as well as at the plane wave solutions, in a natural manner. The derivations
and applications described above should be of use in a comse on relativity because of their
simplicity and directness.

INTRODUCTION I. THE LORENTZ TRANSFORMATION

We wish to present here an alternate derivation The special Lorentz transformation is the
,of the special Lorentz transformation, and several transformation connecting the space-time co-
of its consequences, These derivations are ap- ordinates of two inertial frames Sand S' in
propriate for use in a course on special relativity standard configuration, The velocity of the origin
because of their simplicity and directness, The of S' with respect to S is v, and it is directed along
authors have used these techniques in the class- the positive x axis. The axes of S' are parallel to
room, and it is hoped that they will prove useful those of S, and the spatial origins of the two
to others, systems coincide when the clocks there read zero
The derivation of the special Lorentz trans- in each system, By assuming the invariance of the
formation given here has, to the best of our velocity of light, the linearity of the transforma-
knowledge, not appeared before, It arrives tion (or homogeneity), and isotropy, many
directly at what we shall call the diagonal form of authors arrive at the following equations':
the special Lorentz transformation, that is, the (1)
y'=y, z'=z,
form in which the transformation matrix is
diagonaL That form of the Lorentz transformation (x-ct) (x+ct) = (x'-el') (x'+ct'). (2)
has appeared before in the literature, and its It is in the further derivation of the equations
usefulness has been noted. I However, explicit connecting x and t with x' and t' that we make a
applications seem to be lacking in the elementary simplification, From Eq, (2) it follows that a
literature' Therefore, we have presented a number space-time event with x=ct in S must have
of simple applications in which the use of the x' = ±ct' in S', The minus sign can be eliminated
diagonal form of the special Lorentz transforma- by imagining a light pulse originating from the
tion is most direct. These include the linear common space-time origin and traveling along the
Doppler effect, aberration, and the relativistic positive x axis, The x' as well as the x coordinate
addition of any number of parallel velocities, of the pulse must be positive because the origin
We also apply the diagonal form of the trans- of S' moves at a velocity v which is less than e
formation to Maxwell's equations in free space, with respect to S, Thus, if the coordinates of a
in order to arrive at an analogous form of the space-time event satisfy in S the equation
transformation of the electromagnetic field. The x- ct = 0, then they satisfy in S' the equation
diagonal form of the Lorentz transformation is x' -et' =0, Similarly, if x+el=O, then x' +el' =0.
useful in applications involving light propagation It now follows from the presumed linearity of the
and Maxwell's equations because the combina- transformation, and Eq, (2), that
tions of x and t appearing in the transformation
vanish on the light cone, Such linear combinations x-ct=A (v) (x' -ct')
of x and t are known as null coordinates,3 x+ct=A-I(v) (x'+ct') , (3)
218

Reprinted from Am. J. Phys. 38, 218 (1970).

LOREN1Z TRANSFORMATIONS 459

219 S PEe I A L R E L A T I V I T Y AND D I AGO N A L T RAN S FOR MAT ION S

where A (v) is independent of the coordinates, p

but can depend on the relative velocity v.
The quantity A (v) is determined by dividing
the first of Eqs. (3) by the second, and noting
that the origin of S' has the coordinates (x=vt, t)
in S and (x' =0, t'l in S'. Thus,
(vt-et)/(vt+et) =A2(V) (-et'/et ' ),
whence
A(v) = (c-v)1/2/(e+v)1/2. ( 4)
o
FIG. 1. Light ray relative to S.
The positive square root is chosen because if an
event on the x axis satisfies the condition x> ct,
then also x'> ct, since a light signal originating linear Doppler effect
at the common space--time origin could not have v' =A (v) v. (7)
reached the event in question relative to either
frame.' B. Aberration
We have thus arrived at the special Lorentz Consider a light ray in the X--1J plane, as
transformation in the form
shown in Fig. 1. Suppose that a pulse on the
x-ct=A (v) (X'_ct ' ) light ray passes through the coincident origins of
8 and 8 ' , represented by 0 in the figure, at
x+ct=A-l(V) (X'+ct ' )
t=t' =0. At an earlier time t= -r/e, that pulse was
y=y', Z=z'. (5) at point P with coordinates x and y in S, as shown
in Fig. 1. Using the trigonometric identity
We call this the diagonal form of the special
Lorentz transformation because the matrix trans- tan(a/2) =csca-cota,
forming the variables ~=x+ct, 1/=x-ct, y, z into we obtain the equation
e=x'+ct', 1/ 1 =X'-ct', V', Zl is a diagonal matrix.
The conventional form of the special Lorentz tan(a/2) =- (x+ct) /y.
transformation is readily obtained from Eq. (5) Similarly,
using the identities
tan(a' 12) = (x' +et / ) Iy',
A +A-l= 2(I_V2/C 2)-1/2
where a ' is the angle that the light ray makes
A - A -1= - 2(1-v'/c')-I1'v/c. (6) with the x' axis in S', and x', t' are the coordinates
We now turn to some applications in which the in S' of the pulse which has coordinates x, t in S.
direct use of Eqs_ (5) is very convenient. Application of Eqs_ (5) to the last two Eqs.
yields the aberration formula7 :
II. ELEMENTARY APPLICATIONS tan(a'/2) =A (v) tan (aI2) . (8)
A. Linear Doppler Effect
C. Relativistic Addition of Several Parallel Velocities
Consider a plane light wave traveling in the
positive x direction, represented by Consider three inertial frames 8, S', and S" in
standard configuration. Let VI be the velocity of 8'
Ao sin21rv(t-x/e). relative to S, and V2 be the velocity of S" relative
to S'. Let u=dx/dt be the velocity of a particle
For such a wave the phase is invariant6 : moving in the x direction relative to 8, and
v(t-x/c) =v/(t'-x'/e). u" = dx" I dt" be the corresponding velocity rela-
tive to 8". The problem is to relate u" to u_
Comparison with Eqs. (5) immediately yields the The coordinates of the particle relative to 8
460 CHAPTER IX

I,. PARKER AND G. M. SCHMIEG 220

and S' satisfy A. Transformation Properties of the Fields

x+ct=A-l(Vl) (x'+ct'), Consider the homogeneous Maxwell equations
in vacuum1':
x-ct=A (VI) (x'-ct'),
v xE+c-1(aB/at) =0 (12a)
while relative to S' and S" they satisfy
v·B=O. (12b)
x'+ct'=A-l(V2) (x"+et") ,
Derivatives in x and t can be replaced by deriva-
x'-ct' =A (V2) (x" _et").
tives in ~ and '1/ by using the identities
Eliminating x' ±ct' yields the relation between the
a/ax= (a/a~) + (0/0'1/),
coordinates of Sand S",·
c-1(a/at) = (a/a~) - (0/07/).
x+ct=A-l(Vl)A-l(V,) (x"+ct"),
This procedure is advantageous because of the
x-ct=A(Vl)A(v,) (x"-ct"). (9) simple transformation properties
Forming the differentials of Eqs. (9) and dividing a/a~=A (a;a~'),
yields at once the relation a/a'f/=A-1(a/ar/) , (13)
A(u) =A(Vl)A(v,)A(u"). (10) which follow from Eqs. (5).
The y and z components of Eq. (12a) then have
This is a form of the relativistic velocity addition the explicit forms
formula. 9 The generalization when any number of
intermediate inertial frames are involved is ob-
vious. From Eq. (10) it is easy to see that when
a;a E.- ~a (E.-B.) - a;;a (E.+B.) =0
u"· v, the.\ also u = c, and when u" <c, then also
a a a
.. < c. That is not so easy to deduce, even when - ay E.+ ~ (E.+B.) + a;; (E.-B.) =0. (14)
only two inertial frames are involved, if the more
conventional form of the velocity addition By subtracting Eq. (12b) from the x component
formula is used. W of Eq. (12a), one obtains
III. MAXWELL'S EQUATIONS a a a
ay (E.-B.)- a; (E.+B.)-2a;;B.=O. (15)
The simple transformation properties of the
variables Substituting Eqs. (13) and a/ay=a/ay', a/az=
~=x+cl, a/az' into Eq. (14), we get the equations
'I/=x-ct (11) a a a
,E.- ,A(E.-B.l - -;A-1(E.+B.) =0
under special Lorentz transformation make them az a~ a'l/
very useful in dealing with Maxwell's equations. a a a
Using the variables ~ and '1/, we obtain the trans- - -;E.+ ,A(E.+B.l+ ,A-1(E.-B.) =0.
formation equations of the Maxwell fields under ay a~ a'f/
special Lorentz transformation directly in a form (16)
analogous to Eq. (5). The derivation is certainly Clearly, the requirement of form invariance, i.e.,
no more difficult, and probably is somewhat that Eqs. (16) have the same form as Eqs. (14)
simpler, than the corresponding derivations which in the primed system, is satisfied if
also do not use tensors, but use the variables
x and t. II We also show that the most general E.' =E. (17a)
plane wave solution of Maxwell's equations in E.'-B.'=A(E.-B.) (17b)
free space follows immediately from the first-order E.'+B.'=A-1(E.+B.) (17c)
Maxwell equations written in terms of ~ and '1/,
without introduction of the second-order wave E.'+B,'=A(E.+B.) (17d)
equation. E.'-B.'=A-1(E.-B,). (17e)
LORENTZ TRANSFORMATIONS 461

221 S PEe I A L n E L A T I V I T Y AND D I AGO N A L T RAN S FOR MAT ION S

One then readily finds, with the aid of Eqs. One can easily show that Eqs. (17) are
(17b) and (17d), that Eq. (Vi) is form invariant equivalent to the more conventional form of the
if transformation equations by using Eqs. (G).
However, as with Eqs. (;"i), the transformation
Bx'=Bx. (l7f)
Eqs. (17) are often useful in their original form.
Equations (17a)-(17f) are analogous to Eqs. Ui) For example, multiplication of Eq. (l7b) by
in that they only involve multiplication by the Eq. (17c) , and Eq. (l7d) by Eq. (17e), followed
factor A" by addition yields the invariance of E'-B'.
The above considerations arrive at Eqs. (17) Similarly, multiplication of Eq. (17c) by Eq.
in a natural and simple marmer, and are therefore (17d), and Eq. (17b) by Eq. (17e) yields, upon
suitable for use in an introductory course. In subtraction, the invari~1nce of E· B. The invariance
fact, analogous derivations of the more conven- of these quantities can immediately be generalized
tional form of the transformation equations are to arbitrary Lorentz transformations because the
given in the texts cited in Ref. 11. However, it quantities are clearly invariant under three-dimen-
should be noted that the considerations given sional rotations.
show only that form-in variance is satisfied if the
B. Plane Wave Solutions
correct transformation equations of the Maxwell
fields hold. Those considerations alone are not It is worth noting how simply one can extract
sufficient to arrive uniquely at the transformation the general plane wave solution from Maxwell's
equations, since six field quantities arc involved, equations written in terms of i; and~. We seek solu-
while less than six equations are used. Further- tions of Maxwell's equations which are inde-
more, since the equations used were all homo- pendent of y and z. It is then evident in the usual
geneous, the transformation equations arrived at way that the x component of Eq. (12a) together
could obviously all be mUltiplied by a common with Eq. (12b) implies thatBx is a constant, which
factor dependent on v. In the paper cited in Ref. we set equal to zero. Similarly one finds that Ex
11, Einstein avoids the above difficulties by is a constant, which we put equal to zero. It is easy
considering six of the Maxwell equations in the to show that the remaining four components of
absence of charges, thereby deriving uniquely the Maxwell's equations are equivalent to
transformation of the fields, to within a common
(a/a~) (E,-B y) =0 (a/a~) (E,+By) =0
factor", (v). He then shows by simple considera-
tions that "'(v) = 1. (a/ao (Ey+B,) =0 (a/a.,,) (Ey-B,) =0. (18)
We can overcome the uniqueness difficulty in
The solutions, to within additive constants,
our derivation by noting that Eqs. (12a) and which we put equal to zero, are
(12b) become the remaining Maxwell equations
in the absence of charges, if one makes the sub- E,-By=2jl(~) E,+By=2g 1 W
stitutions E-+B and B-+-E. That substitution in
Eqs. (14) and (15) then gives three new equa- E y+B,=2j,(.,,) Ey-B,= 2g2(~). (19)
tions, which are form-invariant if Eqs. (17) with Hence
E->B and B-+-E hold. However, the full set of
Eqs. (17) are simply interchanged among them- E y=j2(.,,)+02W E,=jl(~)+glW
selves by that substitution. Hence, if Eqs. (17) (20)
Bu= -!t(.,,)+g,W B,=j2(~) -g2(~).
hold, then the six independent equations given by
Eqs. (14), (Li), and the equations obtained Equations (20) involve four arbitrary differen-
from them under the substitution E-+B and tiable functions, and represent the most general
B-+-E are form-invariant. Consequently, form plane wave solution of Maxwell's equations (the
invariance of Maxwell's equations in the absence x axis has been chosen to be along the direction of
of charges uniquely determines Eqs. (17) to the wave motion, without significant loss of
within a common factor", (v). We refer the reader generality). The solution clearly consists of a
to Einstein's paper cited in Ref. 11, for the superposition of a wave traveling at velocity c in
proof that y;( v) = 1. the positive x direction (the." dependence), with a
462 CHAP1ERIX

L. PARKER AND G. M. SCHMIEG 222

wave traveling at velocity c in the negative x connection with the linear Doppler effect [see the deriva-
direction (the ~ dependence). By comparing tion of Eq. (7)]. Bondi emphasizes the reflection of radar
signals in his applications, whereas Ollr viewpoint is
Eqs. (19) with Eqs. (17) and Eqs. (5), it is generally quite different. Also, our considerations with
evident that the plane wave solutions in one regard to Maxwell's equations are unrelat~d to the k-cal-
inertial frame transform into similar plane wave culus. For the k-calculus, see H. Bondi, Relativity and
solutions under special Lorentz transformations. Comman Sense (Doubleday & Co., Inc., Garden City,
N. Y., 1964).
IV. CONCLUSIONS • The use of null coordinates in treating radiation is well
known. See, for example, R. Penrose in Relativity, Groups
We feel that the diagonal form of the special and Topowgy, C. and B. DeWitt, Eds. (Gordon and
Breach Science Publ., Inc., New York, 1964), p. 565.
Lorentz transformation, and the corresponding • See, for example Ref. 1, or A. P. French, Speciat
transformation of the electromagnetic field, can Relativity (W. W. Norton & Company, Inc., New York,
be advantageously employed when covering 1968); R. Resnick, Introduction to Special Relativity
certain material in a relativity course. A derivation (John Wiley & Sons, Inc., New York, 1968); W. G. V.
which arrives directly at Eqs. (5) is evidently Rosser, An Introduction to the Theory of Relativity (Butter-
wortl", Scientific Publications Ltd., London, 1964), as well
somewhat simpler than the more conventional as many others.
derivations of the Lorentz transformation. Equa- • Or because A (v) must clearly reduce to unity as v
tions (5) can be applied directly to the Doppler approaches zero.
effect, aberration, and the addition of any number • Invariance of the phase follows from a consideration of
of parallel velocities. 14 the counting of wave crests. See for example, W. G. V.
Rosser, Ref. 4, p. 154, or C. Mpller, Theory of Relativity
By expressing the vacuum form of Maxwell's (Oxford University Press, London, 1952), p. 7.
equations in terms of the variables x+ct and x- ct , The derivation of Eq. (8) was suggested by a problem
we arrived at the transformation Eqs. (17). in Ref. 1, p. 54.
Equations (17) are clearly analogous to Eqs. (5) • The fact that Eqs. (9) have the form of a special
in that they involve only multiplication by factors Lorentz transformation illustrates the group property of
the special Lorentz transformations.
of A. We also pointed out that the plane wave
• The general form of Eq. (10) has been given in a
solutions follow naturally from thc above form of group theoretic context by P. Malvaux, Compt. Rend.
Maxwell's equations. 236,1009 (1952).
In group theoretic language, the variables 10 D. Bohm, Special Relativity (W. A. Benjamin, Inc.,

x-ct, x+ct, y, and z form the basis of the diagonal New York, 1965), p. 68; N. D. Mermin, Space and Time in
Special Relativity (McGraw-Hill Book Co., New York,
representation of the special Lorentz transforma-
19G8), p. 132.
tion, which is given by Eqs. (5). Similarly, the 11 A. Einstein, Ann. Physik 17, 891 (1905); translation in
linear combinations of the fields appearing in Principle of Relativity (Dover Publications, Inc., New
Eqs. (17) form the components of an antisym- York, 1923), p. 37; R. Resllick, Ref. 4, pp. 178-181;
metric tensor which transforms like the direct W. G. V. Rosser, Introductory Relativity (Plenum Press,
Inc., New York, 1967), pp. 224-227.
product of two basis vectors of the diagonal
"We use Heaviside-Lorentz units. To change the
representation of the special Lorentz transforma- formulas in this paper into rationalized MKS units,
tion. To go more deeply into such matters would simply replace B by cB. (That procedure works for the
take us beyond the scope of this article. formulas in this paper, but not in general.)
13 We are not aware of any reference where it is pointed

out that the transformation of the electromagnetic field can

1 W. Rindler, Special Relativity (Wiley-Interscience, be written in the form of Eqs. (17).
Inc., New York, 1966), p. 22. " For a further application of Eqs. (.5) to accelerating
'However, H. Bondi's k-calculus is related to the frames of reference, see R. T. Jones, Amer. J. Phys. 28, 109
diagonal form of the Lorentz transformation because of its (1960).
LORENlZ TRANSFORMATIONS 463

Reprinted from AMERICAN JOURNAL OF PHYSICS, Vol. 38, No. 11, 129S-1302, November 1970
Printed in U. S. A.

A Useful Form of the Minkowski Diagram

LEONARD PARKER AND GLENN M. SCHMIEG
Department of Physica, University of Wiseonsin, Milwaukee, Wiseonsin 59S01
(Received 2 Apri11970; revision received 22 May 1970)

We give a diagrammatic representation of the diagonal form of the special Lorentz trans-
formation. The null coordinates z:!::ct are plotted along a single set of orthogonal axes. Special
Lorentz transformations are then represented only by a change of scale along those orthogonal
axes. This diagram, which we call a null coordinate diagram, and the Minkowski diagram are
closely connected. To demonstrate the use of the null coordinate diagram, we apply it to the
linear Doppler effect, time dilation, and Lorentz contraction.

In a previous paper,' we considered a form of as a complement to the other diagrammatic repre-

the special Lorentz transformation, the so-called sentations of the special Lorentz transformation, 3
diagonal form, which is especially convenient in each representation being most convenient in
certain applications. Our methods were mainly particular applications.
algebraic in nature. In this paper, we will show
that the diagonal form of the Lorentz transfor- I. THE NUI.L COORDINATE DIAGRAM
mation can be conveniently represented on a
A. Explanation of the Diagram
diagram, and we will apply this geometrical repre-
sentation to several elementary probleIllS. This Let S' and S be two inertial frames in standard
representation is equivalent to the Minkowski configuration, with S' moving at velocity v in the
diagram. However, our version of the diagram positive x direction relative to S. The special
has the favorable feature that only one set of Lorentz transformation relating the coordinates
orthogonal axes are used to represent any number of S and S' can be written in the so-called diagonal
of frames related by special Lorentz transforma- form':
tion. A special Lorentz transformation is repre- e=A(vH,
sented simply by a change of scale of the orthog-
r,' = A-'(v)q,
onal axes. Because our version of the Minkowski
diagram uses null coordinates x±ct, we will call y'=y, z'=z, (1)
it a null coordinate diagram to distinguish it from where
the customary form of Minkowski diagram.2 ~=x+ct, ~'=x'+ct',
To demonstrate its use, we apply the null co-
7/=x-ct, 7/'=x'-ct', (2)
ordinate diagram to the linear Doppler shift, the
time dilation, and the length contraction. We and
feel that the null coordinate diagram can serve A(v) = (C-V)I/2/(C+V) 1/2, (3)

Reprinted from Am. J. Phys. 38, 1298 (1970).

464 CHAPTER IX

1299 USEFUL FORM OF MINKOWSKI DIAGRAM

The coordinates ~ and 1] are known as null co-

ordinates.
In a null coordinate diagram, we draw the ~ and
1] axes as shown in Fig. 1. By convention, the .t
unprimed coordinates ~ and 1] are always plotted
with equal scales along the two axes. The coordi-
nates ~ and 'I of a space-time point are obtained
by orthogonal projection onto the axes. The
e
primed coordinates and 1]' are plotted along the -----=-r-----'lf-.......!.r'----.... "t ,'I.'
same axes, respectively, as ~ and 1]. However, in
accordance with the transformation given in
e
Eq. (1), one unit of is equal to A-'(v) units
of ~, and one unit of 1]' is equal to A (v) units of1].
For positive v, since A(v) <1, it follows that the
e
units of will be dilated with respect to those of FIG. 2. Equivalence of null coordinate and Minkowski
~, and the units of 1]' will be contracted with diagrams. Relation of coordinate scales is shown, as well
respect to those of 'I. The e and 'I' coordinates as the x' and ct' coordinates of a point P.
of a space-time point are also obtained by orthog-
onal projection. A given event corresponds to a the various position and time axes on a null co-
single point on the diagram, regardless of which ordinate diagram and determine how the posi-
coordinate system we refer to. tion and time coordinates of a space-time point
In practice, as will become evident in the ap- can be obtained using those axes rather than the
plications, it is sufficient to know that the primed null coordinate axes. The x axis is the locus of
and unprimed coordinates are related by Eq. (1), points with t = 0 or ~ = 1]. Similarly, the x' axis
so that it is not actually necessary to plot the e
is the line = 'I'. The et axis is the locus of points
various scales along the axes. If more than two with x=O or ~= -'I. Similarly, the et' axis is the
inertial frames in standard configuration are in- e
locus of points with = -1]'. The direction of the
volved, the additional null coordinates are also axes can be determined by noting that as x or et
plotted along the single set of orthogonal axes, increase, Ii also increases. The various position
with scales which depend on the relation of the and time axes are plotted in Fig. 2 for the case
additional null coordinates to ~ and 'I. Note that when v>O. From Eq. (2), it follows that the lines
only a single orthogonal proj ection is necessary to of constant x' are parallel to the et' axis, and the
read the null coordinates of a given event in any lines of constant et' are parallel to the x' axis.
Lorentz frame. Hence, to find the x' and et' coordinates of a point,
such as the point P shown in Fig. 2, one projects
B. Equivalence with the Minkowski Diagram
lines parallel to the et' and x' axes, respectively.
To illustrate that the null coordinate diagram Thus, our diagram is equivalent to a Minkowski
is equivalent to the l\Hnkowski diagram, we plot diagram rotated through 45°.' The null coordinate
diagram also provides a convenient method of
determining the scales on the x' and et' axes as
shown in Fig. 2. The examples given below will
demonstrate the use of null coordinates diagrams
and will show that they are very convenient in
some cases.

II. APPLICATIONS

---+-------\.1( A. Linear Doppler Effect

Suppose that a source at the origin of the inertial
FIG. 1. Null coordinate diagram. Equal scales for I; and ~. frame S emits pulses periodically in the +x di-
U neq"al scales for I;' and .'. rection with a proper period T. Figure 3 shows the
LORENTZ TRANSFORMATIONS 465

L. PAR K ERA N D G. M. S C H M lEG 1300

.t' For any light pulse moving in the +x direction

we have .:l~=2c.:lt=2.:lx, while for any light
ct pulse moving in the -x direction we have .:l1/=
L, 2c.:lt=2.:lx, as is easily obtained from Eq. (2).
The analogous equations hold in the primed
coordinates. Thus, the distances and times of
travel of light pulses are proportional to the
lengths of their world lines, as measured in the
___ ..l...-_~~ ____ "I,\' appropriate null coordinate units.
Therefore, we can see directly from Fig. 4 that

FIG. 3. Doppler effect. The first pulse L, is emitted (and

(9)
received) at t ~ t' ~ o. and
.:ltBC' = A -I (v) .:lIBC. (10)
world lines LI and L2 of two such pulses, 5 as well The factors of A and A -I appear because of the
as the time axes of Sand S', which are the world ditlerent scales for the primed and unprimed null
lines, respectively, of the source at the origin coordinates, in accordance with Eq. (1). In the
of S and an observer at the origin of S'. Because proper system of the rod, the time intervals .:ltAB
x' =0 for both the events A and 0, it follows that 6 and .:ltBa are clearly equal, and will be denoted
(4) by .:ll. The period r' of the clock with respect to
S' is therefO! e
Similarly, because x = 0 for both the events B
and 0, we have r' = .:lIAB'+.:ltBC' =![A (v) +A-'(V) Jr, (11)
.:l1/BO = Cr. (5) where r = 2.:l1 is the proper period of the clock.
By projection onto the 1/, 1/' axis, we see that Using Eq. (3), the expression for r' becomes
(6) (12)

Thus, making use of Eqs. (1) and (6), we have which is the familiar form of the time dilation
equation. The ease with which Eqs. (9) and (10)
.:l1/AO' =A-l (v) .:l'1AO =A-I(V).:l'1BO. (7)
were obtained from Fig. 4 illustrates the utility
This gives the linear Doppler effect: of null coordinate diagrams in problems involving
r'=A-'(v)r light pulses.
or
C. Length Contraction
(8)
Suppose that a rod of proper length I is at rest
where v and v' are the frequencies relative to S
along the x axis of S, with its center at x=O. If a
and S', respectively. light flash is emitted at the center of the rod, it
B. Time Dilation
To illustrate further the usefulness of the null
coordinate diagram, consider a clock7 consisting
of two mirrors at the ends of a rod, with a light
pulse bouncing back and forth between the :;""'_--1 8
mirrors. The period of the clock is the round trip
time of the light pulse. Figure 4 shows the world
line of the light pulse in one such round trip. The _ _ _ _ _ _ _+A"-_ _ '1''1'
rod is at rest in the inertial system S. The light
starts from the mirror at space-time point A, is
reflected from the other mirror at space-time
point B, and returns to the original mirror at FIG. 4. Round trip of light pulse bouncing between two
space-time point C. mirrors.
466 CHAPTER IX

1301 USEFUL FORM OF MINKOWSKI DIAGRAM

will reach the ends of the rod simultaneously rela-

tive to S, and the pulses in the +x and -x direc-
tions will each travel a distance 1/2. The world
lines Lt and i.PJ of the ends of the rod and the world
lines of the light pulses (emitted at t = 0) are
shown in Fig. 5.
Let a different light flash be emitted from
another source on the rod, such that the flash
will reach the ends of the rod simultaneously
relative to S'. Then, relative to S', the initial
position of the source must be halfway between
the final positions of the ends of the rod at the
time when the flash reaches them." Therefore,
relative to S', the wave front of the flash travels
FIG. 6. World lines L. and L, of ends of rod, the x axis of S,
to reach the ends of the rod a distance l' /2, where
find the x' axis of S'.
l' is the length of the rod. The world lines AD
and DE of the flash which reaches the ends of the Substituting (16) and (17) into (15), and using
rod simultaneously with respect to S' are also (13) and (14), we obtain
shown in Fig. 5. 9
We have A-'(v)/'-I=I-A(v)I'
(13) or
and I=KA (vHA-1(V) Jl'. (18)
(14) Hence,
(19)
Since the world line Lt makes a 45° angle with
the -I) axis, it follows that The length contraction can also be obtained
directly without the use of light rays. Figure C
AEcF=AI)EF shows the world lines of the same rod at rest with
or respect to S with its center at x=O. As in the
AEAD - AEAB = AI)BC - AI)DE. (15) previous derivation, because the world line L,
But
AEAD=A-1(V)AEAD
, (16)
makes a 45° angle with the -I) direction, it
follows that the perimeters of the two dotted
,and figures with vertices at A and B, respectively,
AI)DE =A (v) AI)DE'. (17) are equal. Thus,
AEoA + AI)OA = t1EOB+ t1I)OB
f,.~'
or
AEoA+AI)OA =A-1(V)AEoB'+A (v) AI)OB'. (20)
From Eq. (2), we have
AEoA + t1I)OA = I
and
t1Eon' + AI)OB' = I'.
But since A lies on the x axis and B lies on the
x' axis, we must have
(21)
and
AEon' = Al)on' = 1'/2. (22)

FIG. 5. World lines L. and L, of ends of rod. World lines Then Eq. (20) yields the length contraction
AB and Be of flash emitted from center of rod. World l=~[A-'(V) +A (v) Jl'.
lines AD and DE of flash emitted from position such that
events A and E are simultaneous in S'. The time dilation can be derived analogously.
LORENTZ TRANSFORMATIONS 467

L. PAR K ERA N D G. M. S C H M lEG 1302

The above examples should be sufficient to vertically and horizontally, or rotated through a 45·
familiarize the reader with the use of the null angle, is purely a matter of taste.
o The world lines of light pulses are always parallel to the
coordinate form of the Minkowski diagram. t or ~ axis.
o We use the convention that A~PQ denotes the positive
1 L. Parker and G. M. Schmieg, Amer. J. Phys. 38, 218 change in ~ between events P and Q. Similarly, in our
(1970). notation A followed by any variable always denotes the
• Although diagrams involving null coordinates are not positive increment in that variable. An alternate procedure
new, we have not previously seen null coordinate diagrams would be to use coordinates ct±x in Eq. (2). If this were
used to represent special Lorentz transformations, nor done, many examples could be easily discussed in the first
have we seen them applied to elementary problems in quadrant of the t, ~ plane.
special relativity. 7 Often called a "Feynman clock."

• The Minkowski, Brehme, and Loedel diagrams are • Since the rod is moving relative to S', the source at the
described in A. Shadowitz, Special Relativity (Saunders, time of emission must be closer to the leading end of the rod.
Philadelphia, Pa., 1968). o For convenience, in Fig. 5 both flashes are drawn so
• The question of whether to draw the t and ~ axes that they reach the end of the rod at x =1/2 simultaneously.
468 CHAPTER IX

PHYSICAL REVIEW A VOLUME 33, NUMBER 6 JUNE 1986

SU(2) and SUO,I) interferometers

Bernard Yurke, Samuel L. McCall, and John R. Klauder

AT&T Bell Laboratories, Murray Hill, New Jersey 07974
(Received 30 October 1985)
A Lie-group-theoretical approach to the analysis of interferometers is presented. Conventional in-
terferometers such as the Mach-Zehnder and Fabry-Perot can be characterized by SU(Z), We intro-
duce a class of interferometers characterized by SUU,Il, These interferometers employ active ele-
ments such as four-wave mixers or degenerate-parametric amplifiers in their construction. Both the
SU(2) and SU(l, 1) interferometers can in principle achieve a phase sensitivity 6.¢ approaching 1/ N,
where N is the total number of quanta entering the interferometer, provided that the light entering
the input ports is prepared in a suitable quantum state. SUO,l) interferometers can achieve this sen-
sitivity with fewer optical elements.

I. INTRODUCTION light pumping the active devices. SU(J,1I interferometers

can achieve a phase sensitivity of 1/N with fewer optical
In a conventional interferometer such as the Mach- elements than the SU(2) interferometers and hence present
Zehnder) depicted in Fig. I light is fed into one of the in- a more practical way of doing sensitive interferometry,
put ports. The light beam is split into two beams which once sufficiently low-noise parametric amplifiers or four-
propagate along different paths and suffer a phase shift wave mixers become available.
relative to each other of <fl. The light beams are combined
and interfere with each other at a second beam splitter.
The relative phase shift <fl can be determined by measuring II. SU(2) CHARACTERIZATION
the position of the interference fringes in the output OF PASSIVE LOSSLESS DEVICES
beams. Such an interferometer can achieve a phase sensi- WITH TWO INPUT AND TWO OUTPUT PORTS
tivity
In this section the connection between a linear lossless
(1.1) passive device having two input ports and two output
ports and the group SU(2) is presented. Since SU(2) is
where N is the total number of photons that have passed equivalent to the rotation group in three dimensions, this
through the interferometer during the measurement time. will allow one to visualize the operations of beam splitters
Caves' has pointed out that by feeding suitably construct- and phase shifte" as rotations in 3-space. This insight
ed squeezed states into both input ports of the interfero- will be exploited in the next section to discuss the perfor-
meter the phase sensitivity can approach mance of the Mach-Zehnder interferometer.
Let a) and a, denote the annihilation operators for two
I (1.2) light beams which may be, for example, the two light
t:.<fl=/i'
beams entering a beam splitter or the two light beams
Bondurant and Shapir0 3.' and Ni' have also investigated leaving a beam splitter. These operators and their Hermi-
the use of squeezed states in increasing interferometer sen- tian conjugates satisfy the boson commutation relations:
sitivity. [a;,aj]=[a/,aJ]=O,
The interferometers considered by Caves' and Bon- (2.11
durant and Shapiro 3,4 were primarily passive lossless de- [a;,a]J=Il;j,
vices with two input ports and two output ports. We will
where i and j take on the values 1 and 2. One can intro-
show that the group SU(2) naturally characterizes such in-
duce the Hermitian operators
terferometers and present group-theoretical arguments in-
dicating the ultimate sensitivity that can be achieved by Jx=T(ara,+a!a)) ,
such devices. We will then introduce a class of active
lossless interferometers characterized by the group itt
JY =-'2(a)a,-a2a )) , (2.2)
SUO,II. In these devices the interference arises not from
recombining light beams via beam splitters, but from the Jz=T(ara)-a!a,) ,
phase-sensitive response of active elements such as
degenerate-parametric amplifiers and four-wave mixers. and
In contrast to SU(2) interferometers, SUO,II interferome- (2.3)
ters can achieve a phase sensitivity of 1/ N with only vac-
uum fluctuations entering the input ports and coherent The operators (2.2) satisfy the commutation relations for

33 4033 © 1986 The American Physical Society

Reprinted from Phys. Rev. A 33,4033 (1986).

LORENTZ TRANSFORMATIONS 469

4034 BERNARD YURKE, S. L. McCALL, AND JOHN R. KLAUDER 33

the Lie algebra of SU(2):

(2.1 I)
[Jx,Jy]=iJz ,
[Jy,J,]=iJx , (2.4)
One can alternatively work in a Schriidinger picture where
the operators JX' Jy, and Jz remain unchanged but the
[Jz> Jx ]=iJy . state vector, after interacting with the beam splitter, be-
comes
The Casimir invariant for this group, using (2.2) and (2.3),
can be put into the form (2.12)

where I in) is the state vector for the light before it has
(2.5)
interacted with the beam splitter. Throughout this paper
we will hop back and forth between the Heisenberg pic-
in fact, N itself commutes with all the operators of (2.2), ture where J is rotated while the state vector remains
Why one should want to characterize a lossless passive fixed and the Schriidinger picture where J remains fixed
device with two input ports and two output ports with the depending on which picture is most convenient for the
operators (2.2) and (2.3) will now be explained. Let a I in discussion at hand.
and a2in denote the annihilation operators for the light Another realizable scattering matrix for a beam splitter
entering the two input ports and similarly let a lout and is
a20ut denote the annihilation operators for two light
beams leaving the two output ports. The scattering ma- cos fi -sin fi
trix for the device will have the form 2 2
U= (2.13)
sin fi cos fi
(2.6) 2 2
At radio frequencies devices with the scattering matrix
Since the creation and annihilation operators for the two Eq. (2.8) and Eq. (2.13) would be distinguished, respec-
input beams and the two output beams must satisfy (2. I) tively, as 90' and 180' couplers. The scattering matrix Eq.
the matrix (2.13) transforms J according to

(2.7) o sin/31 Jx
I 0 Jy (2.14)

must be unitary. Such a transformation will in general

o cos/3 Jz
transform JX' Jy, and Jz among themselves.
This transformation represents a rotation of J about the y
How J = (JX' Jy,Jz) transforms under U will now be
axis by an angle /3. This transformation can be written as
determined for some common optical elements.

1
Consider a beam splitter with the scattering matrix

a
cos- -lsm~
.. a [
Jx J =e,f3J, 1
. [Jx Je -,f3J,
. . (2.15)
2 Z out Z

U= .. a a
(2.8)
-lsmT COST Hence in the Schriidinger picture where J remains fixed
the state vector for the light after interacting with the
beam splitter is
This transformation will transform J according to
lout)=e- if3J, lin). (2.16)
Jx
Jy =
[I0 cosa
0 0
-sma
J Jx
Jy (2.9)
How J transforms under a phase shift or change in op-
tical path length is now determined. Let light beams I
Jz out 0 sina cosa Iz and 2 incur a phase shift 1'1 and 1'2. respectively. The uni-
tary matrix associated with this process is
That is, the abstract angular momentum vectors are rotat-
ed about the x axis by an angle a. This transformation e'O'Y 1
can be expressed in the form U= [ (2.17)

[~Z
J =/oJ,
out
[~ Ie
Z
-ial, , (2.10) Under this transformation J transforms as

Jx COS(1'2-1'I) -sin(1'2-1'I) 0 Jx
Jy sin(l'2-1'I) COS(1'2-1't) 0 Jy (2.18)
where the angular momentum operators on the right-hand
side are evaluated for the input beams a I in and a 2in' The Jz out 0 0 Jz
equivalence of (2.9) and (2.10) can be checked using the
operator identity This represents a rotation about the z axis by the angle
470 CHAPTER IX

SU(2) AND SU(1,I) INTERFEROMETERS 4035

r2-rl corresponding to the relative phase shift between depicted in Fig. 1. It consists of two 50-50 beam splitters
the two light beams. This transformation can be ex- Sl and S2. The relative phase shift <P=<P2 - <Pl is mea-
pressed as sured by observing the interference fringes in the light
leaving S2. Here, as depicted in Fig. I, the case will be
Jx Jx considered where the photodetector is placed in each of
=/(Y2- YI IJ, -I(Yz-rI IJ,
J, J, e . (2.19) the two output beams 0 t OUI and 02 oul' By counting the
number of photoelectrons generated by each detector, D t
J, oul J, and 9 2, separately, onf measures the operators
N1=010,,010uI and N2=02 0uI02 0uI' From Eqs. (2.2) and
Hence in the Schriidinger picture this represents a
(2.3) one sees that this is equivalent to measuring both
transformation of the incoming state vector \ in) accord-
N out and Jzouto
ing to
A geometrical picture of the operation of the inter-
\ out) =e - i lr,-r liJ, \ in) . (2.20) ferometer will now be developed. For definiteness the
beam splitters S 1 and S2 will be chosen to have scattering
It is worth noting that under the full transformation Eq. matrices of the form (2 .8). For a 50-50 beam
(2.17) the incoming state transforms as splitter a must take on the value rr/2 or - rr/2. For the
\ out)=edYI+y,IN/2e -dr,-rlIJ'\in) but since N com- beam splitter SI we take a= + rr/2, for the beam splitter
mutes with J the operator e' IYt+r,IN/l gives rise to phase S2 we take a= -rr/2. Let \ in) denote the state vector
factors which do not contribute to the expectation values for the light in the two light beams entering the inter-
or moments of number-conserving operators such as J ferometer. From Eq. (2.12) the state \ 1ft) of the light
and N. In fact, it is the insensitivity of photodetectors
(photon counters) to the extra phase ei lYI+r,INIl that
allows one to fully characterize an interferometer by the ,
........
SU(2) transformations described above. It has now been
~~ . _. {
shown that the transformations the beam splitters and
phase shifters perform on the two incoming light beams \ /
can be visualized as rotations of the vector J. Further,
\ .'
"
since the operator (0101 or 0 lOll characterizing the num-
ber of photons counted by a photodetector placed in one
of the light beams can be expressed in terms of the opera-
tor Nand J" interferometry can be visualized as the pro-
cess of measuring rotations of J. The operators giving
rise to the mode transformations of Eqs. (2.12), (2.16), and
(2.20) have also been recently discussed by Schumaker in
Ref. 6 where they are referred to as two-mode mixing
operators.

III. THE MACH-ZEHNDER INTERFEROMETER

The formalism of the last section will now be applied to

the Mach-Zehnder' interferometer. This interferometer is

ICI
I.'
FIG. 2. A rotation-group picture of the performance of a
Mach-Zehnder interferometer. When light enters only one input
port of the interferometer the input state has the form
ij, m ) = Ij,j) in a fictitious (Jx.J"J,) space and can be
represented by a cone centered along the z axis with height j (a).
The first beam splitter performs a -1T/2 rotation about the x
axis. The cone now lies along the y axis Ibl. The phase shifts
accumulated by the two light beams in the interferometer corre-
spond to a rotation -,p about the z axis Ic). The second beam
splitter performs a 1T 12 rotation about the x axis (dl. Since J, is
proportional to the difference in the number of photons counted
by the two pholodetectors in the interferometer output beam the
FIG. 1. A Mach-Zehnder interferometer. Light entering one interferometer can resolve states whose overalJ rotation is suffi-
of the two input ports O lin or alin is split into two beams by ciently far from the z axis so that on average the Jz measured
beam splitter S1. The two light beams bl and b, accumulate a will differ from j by one. In order for this to be the case the
phase shift ,pI and ,p" respectively, before entering beam splitter cone must be rotated by approximately the width of its base
S2. The photons leaving the interferometer are counted by which is VJ. Hence the minimum detectable <P is of order
detectors D I and D2. IN}.
 LORENTZ TRANSFORMATIONS 471

4036 BERNARD YURKE, S. L. McCALL, AND JOHN R. KLAUDER 33

upon leaving the S I is As shown in Fig. 2(d), the net result of this sequence of
rotations is a rotation of the initial state vector about the
11/1) =e -ii_/2IJ, I in) (3.\)
y axis by an amount "'.
which amounts to a rotation of the state vector about the As pointed out earlier, by placing photodetectors in
x axis by an amount -7T/2. This is depicted in Figs. 2(a) both of the output beams one can measure both N (the to-
and 2(b) where for definiteness I in) was chosen to be the tal number of photons passing through the interferometer)
state I j,m = j), that is, J lies on the circle surrounding and lz (the difference in the number of photons arriving
the base Llf the cone in Fig. 2(a). With a -7T /2 rotation at each detector divided by 2), Because N commutes with
about the x axis this cone now lies along the y axis. J it by itself gives one no useful information about "'. It
Upon reaching the input ports of S2 one light beam CI does, however, give one useful information about I in), in
has undergone a phase shift of "'I while the other C2 has particular the total number of photoelectrons n counted
undergone a phase shift "'2' Thus, from Eq. 12.19), upon after the light has passed through the interferometer tells
arriving at S2 the light is in the state I vi): one that I in) was in an eigenstate of N:

11/1') =e -il. 2- . 111, 11/1) Nlin)=nlin) . (3.4)

From this, using 12.5) one concludes

(3.2)
Hence, as depicted in Fig. 2(c), the phase shift rotates the jlj+l)=1- [1-+1] (3.5)
state vector about the axis by an amount
-cP=-lcP'-"'I). or
The second beam splitter rotates the state vector I 1/1')
about the x axis by an amount 7T /2. The state vector . n
lout) for the light leaving the interferometer is thus J=2 ' (3.6)

lout) =e"_12I1, 11/1') that is, I in) was an eigenstate of 12

The other variable measured, 1" allows one to infer
13.3) what the value of cP was. In particular,

(3.7)

One can show From (3.6) and (3.11) one concludes that the incoming
light beam was in an eigenstate of Ij ,m )
!in)= Ij=n!2,m=n/2). (3.12)
= -lsincP)l, +lcoscPiJz. (3.8) Hence the incoming light is in the eigenstate that was de-
Hence picted in Fig. 2.
Intuitively the smallest cP that can be measured is one
(Jz ) = (out i lz lout) where the cones of Fig. 2(d) do not appreciably overlap.
The distance from the apex of one of the cones to a point
=-sincP(inIJ, iin)+coscP(inIJ,iin) 13.9)
on the circle of the cone's base is the square root of the
and eigenvalue of J2 or Vj (j + I ). The distance from the
apex of one of the cones to the center of its base is the
(J,2 ) = (out IJ,2 lout)
eigenvalue of Jz or J. Hence the radius of the base of one
= sin'cP (in IJ;: in) of the cones is [j Ij + 1) - /]1/2 = vJ. The minimum
detectable cP is thus of order cPm,"~j-I/2, and since from
-sincPcoscP(in iJ']z+J,}, I in) Eq. 13.12) j = n /2,
+cos'dJ(inIJ; in). 13.10) 4>mm-:::::=.n -1/2. (3.13)
To proceed further one needs additional information on Hence the sensitivity of an interferometer operated in the
I in). Let us suppose the interferometer is operated in the mode where light enters only one of the two input ports
usual manner where light enters the interferometer only has a sensitivity that goes as the square root of the num-
along one of the input beam paths, say a l' Then from the ber of photons passing through the interferometer.
total number of photons n counted by D I and D2 one Equation (3.13) is now made more rigorous by a direct
knows that there were n photons in the incoming light calculation from Eq. (3.9) and Eq. (3.10). For the state
beam. Hence I in) is an eigenstate of lz: 13.12) the mean value of Jz is
- n
lz in)=~ !in). (3.11) lZ=2coscP . (3.14)
472 CHAPTER IX

33 SU(2) AND SUO,\) INTERFEROMETERS 4037

The mean-square fluctuation (4Jz )2 about this value is A Fabry-Perot interferometer is depicted in Fig. 3(a). It
consists of semitransparent mirrors Ml and M2. This in-
(4Jz)'=.7}-J; terferometer measures the phase shift rf> suffered by light
as it propagates from one mirror to the other. This device
=1"sin 2rf> • (3.15)
has two input ports a I in and a, in' and two output ports
al out and a'out. Although alin and a'out and a'in and
The mean-square noise in rf> is thus
a lout are collinear they can be separated with optical cir-

(4rf»
2

=r- =-.
(4Jz )'
aJz J
arf>
1
n
(3.16)
culators as shown in Fig. 3(b). In this manner one can
place photodetectors in both beams a lout and a, out
without obstructing the light injected into a I in or a, in'
Hence one is allowed to measure N and Jz for the two
output beams.
Hence the rms fluctuation of rf> due to photon noise goes
An analysis of the Fabry-Perot interferometer is now
asn- l12 ,
carried out. The mirrors M I and M2 will be taken to
(3.17) have scattering matrices of the form Eq. (2.8). In particu-
lar for the mirror M 1 we take
in agreement with the intuitive argument based on Fig. 2.
We will refer to (3.17) as the "standard noise limit" for an bl=COS(+flJalin+isin(+flJb, ,
interferometer. (4.1)
Note that no assumption was made about the quantum a,out=+isin(+flJalin+COS(+{3)b, ,
statistics of the source of light entering the interferometer.
The total number of photons n entering the interferometer and for the mirror M2
completely characterizes the ultimate sensitivity that can
be achieved with an interferometer in which light is fed a lout =cos( +(3)cl -i sin( +(3)a'in ,
(4.2)
into only one input port. If instead of using photodetec-
tors in both output ports and measuring Nand Jz one
c, = -i sin( +mci +cos( +(3)a'in .
chooses to use only one photodetector or to measure only In writing (4. I) and (4.2) it has been assumed that both
J" then one is throwing away information. In this case mirrors have the same transmission coefficient
knowledge about the photon statistics of the source be-
comes important. For this situation the performance of (4.3)
the interferometer will generally degrade although for
some particular values of ci>1 -</>, the n 1/2 phase sensitivi-
e
The phase shift sustained by the light as it propagates
between the two mirrors is given by
ty can still be achieved.
As will be pointed out in Sec. V, the sensitivity of an in- Cj =ei(Jb\ ,
terferometer can be dramatically improved if photons are (4.4)
allowed to enter both input ports provided the photons are b 2 = ei9c 2 .
prepared in the right quantum state.
Equations (4.1)-(4.4) can be solved to obtain a I out,a'out
in terms of a 1 '" and alin' One finds
IV. THE FABRY-PEROT INTERFEROMETER

In the last section a geometrical picture of the operation

of the Mach-Zehnder interferometer was presented in
terms of rotations of the operators (J.. Jy , Jz) defined by
Eq. (2.2). Photodetectors placed in the output beam of the
interferometer measure the operator Jz. A relative phase
shift between two optical beams produces a rotation of J
101
about the z axis, see Eq. (2.19). A measurement of J"
however, is only sensitive to rotations in a plane contain-
ing the z axis. The function of the two 50-50 beam spli-
tters is thus to convert a rotation about the z axis into a
"Olin OIOU'/
rotation in a plane containing the z axis. For the particu-
lar set of beam splitters chosen in the last section this cor-
O=[-~]=U
responds to a net rotation in the x-z plane as depicted in
Fig. 2.
A", C C a~
The Fabry-Perot interferometer,' by employing semi-
( bl
transparent mirrors, also converts a rotation about the x FIG. 3. A Fabry-Perot interferometer. The device (a) con-
axis into a rotation lying in a plane containing the z axis. sists of two semitransparent mirrors M I and M2. The light
One can also take advantage of the multiple passes of the propagating between M 1 and M2 suffers a one~way phase shift
light between the two mirrors to enhance the sensitivity of e. In (b) circulators C have been placed behind the mirrors 10
the interferometer, but at the expense of the physically separate the incoming and outgoing light beams. It is
interferometer's bandwidth. Here expressions are ob- evident that the Fabry.Perot interferometer has two input ports
tained for the phase sensitivity of a Fabry-Perot. and two output ports.
LORENTZ TRANSFORMATIONS 473

4038 BERNARD YURKE, S. L. McCALL, AND JOHN R. KLAUDER 33

or in terms of the transmission coefficient T for the mir-

(4.51 rors
4(1-- TlI/2
where (4.151
T
cos'( + f3le ,8
Hence the smallest rms fluctuations in flB achieved by a
Jl = l-sin2( +f3)e 2i8 Fabry-Perot is
(4.61
i sin( +f3)(e'i8_ 1) M. =~= Tfl1> (4.16)
V=-
l-sin2( +f3)e"8
rnm I<i.i.1 4(1- TlI/2 .

de Imax
The scattering matrix of Eq. (4.51 is unitary. Using Eq.
(2.21 one can determine how J transforms under this uni- For mirrors with a small transmission coefficient T, and
tary transformation. One finds using (4.91
T
(4.17)
o sin</> 1 Jx flO min ---- 4n 1/2 .
I 0 Jy (4.7)
Hence, as with the Mach-Zehnder, the sensivity of the in-
o cos</> J,
terferometer scales as n -1/2 where n is the total number
where of photons entering the interferometer. As with the
derivation of (3.17), Eq. (4.17) is based on the assumption
cos</> = 1Jl i2- 'I y 'I 2 , that light enters only one port of the interferometer.
(4.8) In the next section it is shown that the sensitivity of an
sin</> =Jl'v+ y' Jl . interferometer can be greatly enhanced if light, prepared
So the Fabry-Perot interferometer, for the mirrors chosen, in a suitable quantum state, is allowed to enter both ports
performs a rotation of J about the y axis. Hence, follow- of the interferometer. Although the arguments will be ap-
ing the same line of reasoning as in the last section, if plied to the Mach-Zehnder, with the tools developed in
light enters the Fabry-Perot in only one input port the ul- this section, they can be applied to the Fabry-Perot inter-
timate phase sensitivity fl</> is given by ferometer as well.

fl</>=n -II'. (4.9)

In order to determine what this implies for the ultimate V. SURPASSING THE STANDARD NOISE LIMIT
phase sensitivity MJ one needs to evaluate 1d</>Ide I. In the last two sections it was shown that an inter-
With Eq. (4.6), Eq. (4.8) becomes
ferometer can be regarded as a device which performs ro-
cos 4 ( +f3)-4sin'( +f3)sin'e tations on the operators (Jxo Jy '/,) defined by Eq. (2.2).
cos</>= --~-----:----:--­ (4.10) Photodetectors placed in the output beam of the inter-
cos4 ( +f3) +4 sin 2( + f3)sin 2e ferometer measure the operator Jz. Hence the overall ro-
tation must lie in a plane containing the z axis. For the
. 4cos 2( +f3lsin( +msinB choice of beam splitters used in the Mach-Zehnder of Sec.
SIO</I= + 1 1 (4.111 III and the mirrors used in the Fabry-Perot of Sec. IV this
cos4 ( ,f3l+4sin2( ,f3lsin 2e rotation was in the x-z plane. [See Eqs. (3.8) and (4.7)].
For the Mach-Zehnder the sequence of rotations per-
and thus </I is given by formed is depicted in Fig. 2. A cone was used to
represent a Jz eigenstate. Based on how such an object
4cOS2( +f3)sin( +13)sinB 1 transforms under the rotations performed by the inter-
ferometer a minimum detectable phase shift of order
<h=arctan l cos4( ,f3l-4sin2(
[
,f3)sin2e
1 (4.12)
n -1/2 was derived. As will be shown, a J, eigenstate is
not the optimum eigenstate for interferometry. In partic-
Differentiating this equation with respect to e one obtains ular, by forming a linear superposition of J, eigenstates
near m =0, one might imagine constructing a squashed
<i.i. _ 4cos 2( +f3lsin( +f3lcose
(4.13) cone or "fan-shaped" state lying in the x-y plane as de-
de - cos4 ( +131 +4 sin 2( +f3lsin2e picted in Fig. 4(a). Such a state constructed from a super-
position of Jz eigenstates near m = I would have an ex-
The Fabry-Perot is most sensitive for those angles B for tent along the z axis of order unity. Figure 4 indicates
which 1d1>lde 1 is maximized. From Eq. (4.131 one sees how such a geometrical object would transform under the
that the sensitivity is greatest when 1case 1 = I and rotations performed by the interferometer. Since the ex-
sin4> =0. So tent of the state along the z axis is - I and the distance
41 sin( +1311 from the origin to the edge of the cone is - j, Fig. 4(d)
:IdB
<i.i.1 rna> cos 2( + (3)
(4.14) would indicate that the minimal detectable </> is 1>min - 1!j.
Or, from Eq. (3.6),
474 CHAPTER IX

SU(2) AND SUO ,I) INTERFEROMETERS 4039

(,v,)2=+[j(j+1)-1] •
(5.7)
(,v,)2= t U (}+ I)- t ]·

Equations (5.5) and (5.7) indicate that the state is oriented

along the x axis and is very broad along the x and y axes.
This state could thus be represented by a geometrical ob-
ject similar to that depicted in Fig. 4(a). One can a1so
show that

la) (5.8)

and consequently
(inIJ,J,+J,J, l in)=tU(}+1ljl l2. (5.9)

The rms fluctuations in q, for this state will now be

determined. Substituting Eqs. (5.3) and (5.5) into Eq.
0.9), Jz • half the mean differenced photocurrent. is given
by

(5.10)

Ie) Substituting Eqs. (5.6) and (5.9) into Eq. 0.10) one has
I"
J;= tU(} + 1) - t ]sin2q, + tcos2q, . 15.11)
FIG. 4. The perfonnance of a Mach-Zehnder interferometer
in which an input state, of length j when depicted in the The mean-square fluctuation in Jz is then
(J~,J"Jt) space, is a flattened cone whose width along the z axis
is of order unity. The sequence of rotations perfonned by the (,vz)2= +U() + I )-I]sin 2q,+ +cos 2q, . (5.12)
interferometer is the same as that of Fig. 2. In contrast to the
state depicted in Fig. 2, an overall rotation '" - II j can be
The mean-square fluctuation in q, is given by
resolved with the state depicted here. 2 (,v,)2
(5.13)
(~q,)= [~r
(5.])
or
Hence, by choosing the appropriate incoming state I in),
an interferometer's sensitivity can be greatly improved I~q,)'= U(j+I) - ljsin2f +cos'f . 15.14)
over the n -112 sensitivity of Eq. (3.17) or Eq. (4.9). lUI} + I)f12cosq,+sinq,I'
The ~bcve discussion is now made rigorous by explicitly
exhibiting a state with the properties described above. This quantity has its minimum va1ue when sinq, =0. then
Consider the state 1~.I.)2 _ _I_
'I' mon- }I} +1)
15.15)

or in terms of the number of photons passing through the

interferometer, since} =n 12,
From tbis equation one can immediately show
2 4
(in IJ,k I in) = t . (5.3) (~q,)min= n (n +2) . (5.16)

Hence this state lies close to the x-y plane and has a Hence when the state Eq. (5.2) is fed into the input ports
mean-square height of order unity, of an interferometer a minimum rms fluctuation ~<Pmin in
(,v,)2=+ . (5.4) the phase of order n - I can be achieved:

It is alsO straightforward to show that (5.17)

(in IJ,in) = t[j(j + 1)]112 , This maximum sensitivity is however achieved only at
{inIJ, lin)=O,
(5.5)
particular values of q, satisfying sint,6 =0. For other values
of <P the sensitivity of the interferometer is degraded.
and Since q,=<PI-<P" q,1 may be tracked as a function of time
with the precision Eq. (5.17) by controlling <P2 with a feed-
(in IJi lin) =tU(j + 1)- t l. back loop which maintains <PI -q,2 at zero. The error sig-
(5.6)
{in IJ; I in)=tU(}+ 1)- t 1. nal for this loop is the differenced photodetector current
2Jz . The use of feedback loops with be further discussed
So tbe mean-square uncertainties in J, and J, are in Sec. VII.
LORENTZ TRANSFORMAnONS 475

4040 BERNARD YURKE, S. L. McCALL, AND JOHN R. KLAUDER 33

A state i in) which allows an interferometer to achieve where J, is given in Eq. (2.2). In fact, the operator J,
phase uncertainty of order n ~ I has now been presented. commutes with all the K i .
How one prepares light in such a state, or a state similar There has been a considerable amount of theoretical
to it, is the topic of the next section. Here we simply work, beginning with Yuen and Shapiro,1O on four-wave
point out some properties of the state I in) of Eq. (5.2). It mixers as possible sources of squeezed states. The reader
is a superposition of the states I},O) and I}, 1 ). For the is directed to Reid and Walls II and references therein for
state I},O), N has the eigenvalue n =2} and J, has the work that has been done on four-wave mixers. For the
eigenvalue m =0. Equations (2.2) and (2.3) allow one to purposes of this paper, a four-wave mixer will be regarded
recognize this state as one in which exactly j photons as a device with two input ports alin,02m and two output
enter each of the two input ports of the interferometer. ports Glou"G2o", which performs the mode transforma-
For the state I}, 1 ), N has the eigenvalue n = 2} and J, tion of the form 12.1l
has the eigenvalue m = 1. This state can be recognized as
one in which exactly} + 1 photons enter the input port
(6.7)
G I '" while exactly} -1 photons enter the input port G, ,"'

VI. THE TWO-MODE FOUR-WAVE MIXER Both backward degenerate four-wave mixing in which
two counter propagating pump beams pass through the
In the last section it was shown that the sensitivity of nonlinear medium, and forward four-wave mixing, in
an interferometer could be greatly improved provided one which the pump beam propagates in only one direction
could prepare the light delivered to the input ports of the through the nonlinear medium, perform mode transforma-
interferometer in a state which consists of a superposition tions l4 of the form (6.71. Since the incoming and outgoing
of two states, one in which exactly} photons enter each of creation and annihilation operators must satisfy (2.1), the
the two input ports in the interferometer and a state in following restrictions are placed on the Sij:
which} + 1 photons enter one port while} - 1 photons
enter the other port. In this section it is shown that states i SIII'-ISI, ;'=1,
similar to this can be generated with two-mode four-wave (6.8)
mixers. For the analysis of such a device it will be con-
venient to introduce a set of operators whose commuta- SIIS~I =S12S~2 .
tion relations are those for the generators of the group
From these relationships one cal' show
SU(1,II.
In particular we introduce the Hermitian operators ISIII'=I S22!',
16.9)
K x =+(a1ai+aj a2) , ISI,I'= IS,II'·
itt (6.1) The phases of the Sij are controlled by the pump phase.
KY=-2IaIG2-GIG,I,
How the operators 16.1) transform under the scattering
matrix (6.7)
K,=+laial+a,Gil.

[~:: ~:: I
The commutation relations for these operators,
(6.10)
S=
[K"Kyl = -iK, ,
[Ky,K,l=iK x , (6.2) will now be determined for some particular examples. A
possible realization of S is
[K"Kxl=iKy,
can be recognized as those belonging to the group'"
SUIl ,Ii. It is also useful to introduce the raising and
_I
cosh(+f3I e~issinhl+f3I I'
16.11)
S-s I I
lowering operators e'sinhl T f3I cosh(T/3)

K+=Kx+iKy=aiai,
16.3) where 8 is controlled by the phase of the pump light rela-
K _ =K x -iKy =a1 Q 2 tive to some master clock and /3 is related to the reflectivi-
ty R of the four-wave mixer (when it is used as a phase-
which satisfy the commutation relations' conjugating mirror) via sinh'( +/3I=R.
[K~,K+l~2K, , When the pump phase is set such that 8=1T/2 Eq.
16.4) 16.11) becomes
[K"Ktl=±K±
-isinh~+/3) J
The Casimir invariant K2 is

16.51
S= I coshl +/3)

i sinhl +/3) cosh( T/3)

16.12)

which upon the substitution of Eq. 16.1) becomes

Under this transformation, the vector K=(K"Ky,K,)
K'=J,IJ,+I) , (6.6) transforms as
476 CHAPTER IX

SU(2) AND SUO,\) INTERFEROMETERS 4041

~ [~ =
o o
cosh{3 sinh{3
I K.
Ky (6.13)
K.
Ky
K.
=e -;(~I+~2)Kz Ky /(I/JI+t/lzlKz .
(6.22)
sinh{3 cosh{3 K
out in Kz out Kz
•
Z

which represents a Lorentz boost along the y axis, where z In the Schriidinger picture the state vector is transformed
transforms as time. This transformation can be expressed as
in the form
(6.23)

(6.14) The transformation (6.11) can be factorized into the

form

Equivalently, in the Schriidinger picture where K remains cosh( +{3) e -i6sinh( +m ]

fixed the state vector transforms as S(/»S({3)S( -/»= [
e
i6· h( I (.I)
SIn TfJ cosh( +m '
(6.15)
(6.24)
When the pump phase is set at /) =0, the scattering matrix where

I
(6.11) becomes

+
_ [COSh( {3) sinh( (3) + S(/»= [e~i5 ~), (6.25)
S- . (6.16)

+mI
I I '
smh( ,(3) cosh( ,{3)

Under this transformation K transforms as

S({3)= [ .
+
COSh( {3) sinh(
I I ' (6.26)
smh( T {3) cosh( T {3)
K. [COSh{3 0 Sinh{3] K.
Ky
Kz out
= 0 I 0
sinh{3 0 cosh{3
Ky
Kz
(6.17)
e i6
S(-/»= [ 0
0)I . (6.27)

This transformation has the form of a Lorentz boost From (6.20) the transformation S( -/» can be recognized
along the x axis and can be expressed in the form as a rotation about the z axis by an angle -/). S({3)
represents a Lorentz boost along the x axis, and S(/»
K. K. represents a rotation about the z axis by the angle /). The
-iPK ifJK y product of transformations Eq. (6.24) thus represent a
Ky =e ' Ky e . (6.18)
Lorentz transformation along a direction making an angle
Kz out
Kz Ii with respect to the x axis. Hence, in the SchrOdinger
picture, after the incoming light I in) has passed through
In the Schriidinger picture the state vector is transformed
a four-wave mixer, it will be in the state
as
(6.28)
(6.19)
It has now been demonstrated that a four-wave mixer
The operators performing the transformations of Eqs.
performs Lorentz transformations on the vector K, the
(6.15) and (6.19) are two-mode squeeze operators. 6.,z.IJ
direction of the Lorentz boost being determined by the
At this point it will be useful to determine how K
pump phase which is at the experimenter's control. Since
transforms when the two input light beams sustain phase
Jz commutes with K, it remains unchanged under the
shifts. Letting a'in undergo a phase shift of q" and aZin
transformations performed by the four-wave mixer.
undergo a phase shift of q,z, then
From Eq. (2.2) one sees that this invariant is equal to half

[eoi~t
the difference in the number of photons entering the input
(6.20) port of the four-wave mixer. This invariant has been not-
S=
ed by Graham's and Reid and Walls.'6
Let us now consider the case when no light enters the
Under this transformation, K transforms as input ports of the four-wave mixer. The state delivered to
the output is then given by Eq. (6.28) where Iin) is the
COS(q" +q,2) sin(q,,+q,z) 0 K. vacuum state I0).
-sin(q,,+q,2) cos(q" +q,2) 0 Ky (6.21) The probability amplitude that n, photons will appear
in the output beam a, out and nz photons in the beam
o o K z out a20ut is
which can be recognized as a rotation about the z axis by (n"nzl out) = (n"n21 e -i6K'eiPK'ei5K, 10), (6.29)
an angle q,= -(q" +q,2)' This transformation may be ex-
pressed as where the state In "nz) is
LORENTZ TRANSFORMATIONS 477

4042 BERNARD YURKE, S. L. McCALL, AND JOHN R. KLAUDER 33

t nIt
> (a lout)
n~
(a 2 out) ~ n, photons and the beam a, out n 2 photons is thus
In"n, = ~, 10). (6.30)
P(n, ,n,) =On,.n,sech'l +(3)[tanh'( +(3)]n, . (6.411
From (6.11 one sees that K, can be put in the fonn
From this equation one sees that P(n"n,) is zero if
K,=+IN,+N,+lJ, (6.31) n I ¥en,. For the vacuum state one has
(6.42)
where Nj=aioutQloul and .""'2=a;out Q 2ouI are the number
operators for output beams I and 2, respectively. With Since J, is an invariant for the four-wave mixing process,
this equation it is readily apparent that when there are n I photons in beam I there must be n,
(6.32) photons in the second beam as well, that is, the photons
are emitted in correlated pairs. These photons are in fact
and more highly correlated than allowed classically.""·
From (6.411 the mean (n) and the mean-square (n 2)
(6.33) number of photons emitted by the four-wave mixer can be
So Eq. (6.291 simplifies to computed:
00

( n I, n 2 lout) =e -ilo/21<n I +n,' (n, ,n, I e'PKy 10) . (n)= ~ (n,+n,)P(n"n2)

(6.34)

In order to simplify things further we make use of the =2 sinh 2( +(3) , (6.43)
identity'
(n')= i (n,+n,)'P(nl>n,)
exp(TK + -T*L )=exp I[~tanh IT' ]K+ I 1!]n Z

Xexp[ -2llncosh T I K,l

The mean-square fluctuation in n is rather large:

(Ll.n )'=4 sinh'( +(3)cosh 2( +(3) (6.45)

so
(6.35) (Ll.n)' 2 ,
(,;)2 =coth (-;:(3)? I . (6.46)
Hence, noting (6.11 and (6.3), e iPK, can be put into the
fonn Hence the light emitted is super-Poissonian but ap-
proaches Poissonian as (3 becomes large.
e iPKy =exp[i tanh( +(3)K + lexp[ -2Incosh( +(3)K,l We are now in a position to argue that a four-wave
mixer can generate states similar to the one described in
X exp[i tanh( +(3)K -1 . (6.36) the last section, Eq. (5.2). The case when no photons
enter the input port of the four-wave mixer has already
From 16.3) K_ =a,a" hence been discussed. The eigenvalue m of lz is O. Hence if a
exp[itanhl+(3)K_1IO)= 10). (6.37) total of n photons were measured coming out of the four-
wave mixer one can infer that the light leaving the four-
Making use of Eq. (6.31) one has wave mixer was in the state Ij,O) where, from Eq. (2.5),
j=n /2.
exp[ -2Incosh( +(3)K,11 0) =sech( +(3) I 0). (6.38)
If instead the state
Finally, using (6.3) t'
12,0)=~ 10) (6.47)
exp[i tanh( +(3)K + 11 0) = i
11=0
[i tanh( +(3)]" I n,n) ,
is fed into the input ports, that is, if two photons are
(6.39) forced to enter the input port a, in of the four-wave mixer,
then the eigenvalue m for J, is I. If one thus measures a
where I n,n) is defined by (6.30). total of n photons leaving the four-wave mixer one can
Collecting the results, Eq. (6.34), (6.37), (6.38), and infer that the light leaving the four-wave mixer is in the
(6.39), one has state ij, I ) where again j = n /2.
If now the light entering the input port is a superposi-
( n],nZ Iout) =un1,n/
< -i1ol'lIn,+n,'
tion of a vacuum state and the 12,0) state of Eq. (6.47),
say,
Xsech( +(3)[i tanh( +(3)(' . (6.40)
(6.48)
The probability P(nl>n2) that the beam alOll! will contain
478 CHAP'rERIX

SU(2) AND SUII,!) INTERFEROMETERS 4043

then upon measuring n photons leaving the four-wave (7.1)

mixer all one can infer is that the state leaving the four-
mixer is in a superposition of the states !j,O} and Ij,l }. and
Hence a four-wave mixer can generate states of the form (7.2)
(5.2) provided the state (6.48) is fed into its input.
From a practical point of view the state Eq. (6.48) may that is, I a} is a coherent state for a I and a vacuum state
be hard to generate. It would be more practical to attenu- for a2.
ate laser light until on average there is one photon per unit Since Jz is an invariant under the transformation (6.7)
coherence time propagating along the beam. The input performed by the four-wave mixer, its expectation values
state generated by feeding this coherent state into the in- can be computed at the input port,
put port a I in of the four-wave mixer will have a strong
Jz =+(ata ,-aia2) . (7.3)
overlap with states I nl,n2} only when nl is small and n2
is o. Consequently, (Jz) and AJz for the light fed into One can readily show
the four-wave mixer will still be of order unity. Such
light when passed through the four-wave mixer should (Jz}=+laI 2 (7.4)
still produce "fan-shaped" states that will allow an inter-
and
ferometer to reach a phase sensitivity Acf> of order 1/ n.
Interferometry, using the light coming from a four-wave IMz )2=tl a l'· (7.5)
mixer fed with coherent states, will be discussed in detail
in the next section. For I a I 2 of order unity such a state will lie near the x- y
plane and have a spread along the z axis of order unity.
In order to compute the expectation values of Jx and Jy
VII. ACHIEVING A PHASE SENSITIVITY OF 1/N at the output of the four-wave mixer it is necessary to ex-
The device to be considered in this section is depicted in press Jx and Jy in terms of the operators al and a2.
Fig. 5. It consists of a Mach-Zehnder interferometer Again we choose the scattering matrix for the four-wave
whose input ports are fed by the output beams b I and b, mixer to be given by Eq. (6.11),
of a four-wave mixer. The four-wave mixing medium is
pumped with a laser. Part of the laser light is split off of
(7.6)
the main beam phase shifted, attenuated and then fed into
the input port a I of the four-wave mixer. The other input
port a, is terminated with a cold blackbody absorber so Using this transformation, the output Jx and Jy expressed
that no light enters the four-wave mixer from this port. in terms of the input creation and annihilation operators
The pump light's phase is controlled with the phase are
shifter /).
Letting a, and a, denote the creation operators for the Jx = tcosllsinh{3(a ia i +alal +aiai +a,a2)
light beams fed into the input ports of the four-wave i . /) . h{3( t t t t )
-"4sm SIn U1Q)-QIQ1+a2U2-Q2Q2
mixer, the state vector for this light I a} is defined by
++cosh{3(ata2+aial)' 17.7)
; . t t t t
Jy = - "4cos/)smh{3lala I -alai -a,a2 +a2a2)

i
- t sin/)sinh{3(a ia +alal -aiai -a,a,)
itt
- Z"cosh{3(a la, -a 2a I) . (7.8)

The expectation values of J x and Jy for the state Ia}

can now be readily evaluated. Writing
a=lale- iB (7.9)

one has
(aIJ.la)=+ laI 2sinh{3cosI20-/),
(7.10)
FIG. 5. A method by which the state depicted in Fig. 4 can (a IJy I a) = + Ia 1 2sinh{3sin(20-/) .
be generated and fed into an interferometer. The state is gen-
erated via a degenerate four-wave mixer IFWM) pumped via a The mean-square fluctuation in Jx and Jy is independent
laser. A small fraction of the pump light is split off of the of cf> and /):
pump beam, phase shifted by 9, attenuated by A and then fed
into one of the FWM inputs, a,. The input port a, is terminat- (Mx )2=(My )2
ed with a cold blackbody absorber B. The two output ports b,
and b, of the four-wave mixer are fed into the input ports of the =¥(Sinh2{3++)+tsinh'{3. (7.11)
Mach-Zehnder interferometer. 8 is a phase shifter for the pump
light before it enters FWMI. One also has
LORENTZ TRANSFORMATIONS 479

4044 BERNARD YURKE, S. L. McCALL, AND JOHN R. KLAUDER 33

(a IJ.Jz +JzJ. Ia) (IltP)'= ( Ia 1'+ I)' (7.22)

~ Ia I'[(N + 1)'_( Ia 1'_1)')
= 2 (I a 1'+ I )sinhtlcos(20-1l). (7.12)
This expression can be optimized for Ia I holding N
Since from Eq. (3.8) J zou" measured at the output of the fixed. One finds that for large N (IltP)' is smallest when
interferometer, is Ia I' is close to I, hence
(7.13) IltP".)~ . (7.23)
N
one can now evaluate
This equation implies that the interferometer of Fig. 5
(Jz OUI) = - +Ia I'sintP sinh{3 cost 211 -0 >+ +Ia I'costP can achieve a phase sensitivity approaching liN, and that
photons are most economically used by the interferometer
(7.14) when the coherent state Ia) fed into the input port a I
and has its intensity reduced to I a 1'",,1, that is, on average
only one photon per unit coherence time of the four-wave
(t..!zout )'=~sin'.l.sinh'{3 mixer enters the input port a I'
2 II'
The sensitivity IltP of Eq. (7.23) with the particular nu-
merical coefficient 2 cannot be achieved in practice. The
- ¥ s i ntP cos,p sinh{3 cost 20 -0) reason for this is now indicated. Equation (7.23) holds
only for tP < liN. Hence a practical interferometer em-
+~sin'tPsinh'{3+~. (7.15) ployed to ~easure tPl must incorporate a feedback loop
8 4 which adjusts tP, to follow <PI such that tP=<P,-tP, =0.
However, for angles tP outside the narrow range
The mean·square phase uncertainty IltP can be evaluated
:s
I tP I liN the uncertainty in tP, defined by Eq. (7.16), be-
via
comes
(IlJzou,)'
(7.16) IltP= [4 Ia I2 +1 jl12 tantP (7.24)
1a(~;uI) I' 21 a 14
which for small tP and Ia I' = I becomes
This quantity is minimized with respect to the pump
phase 0 when 211-0=0, that is, cos(2tP-0)= I. This IltP",,(f)I/'tP . (7.25)
quantity is also near its minimum value when tP=O.
That is, the uncertainty in tP is greater than tP itself.
When tP is set to zero Eq. (7.16) reduces to
A feedback loop presented with a measurement' of tP
whose uncertainty is greater than tP will generally not be
(7.17)
Ia I 'sinh'{3 . able to adjust tP, properly to drive tP to zero. The uncer-
tainty Il<p can be decreased by increasing I a I or by
The parameter (3 will now be expressed in terms of the averaging several 4 successive measurements of tP. In the
mean number of photons (N) leaving the four-wave next section the problem of locking an interferometer to
mixer and i a I· This will allow us to optimize IltP hold- tP=O will be discussed in more detail and the maximum
ing (N) fixed. From Eq. (6.31) the number operator N uncertainty in Il<p that a feedback loop can tolerate will be
can be expressed in terms of K" determined.
N=2Kz - I . (7.18)
Upon leaving the interferometer the state I a) has been VIII. TRACKING THE PHASE
transformed, according to Eq. (6.28), into the state 11/1),
In the last section an interferometer capable of achiev-
11/I)=e-ibK'eiIiKYeibK'la) . ing a phase sensitivity IltP approaching 1/N, where N is
(7.19)
the number of photons passing through the interferometer
From Eq. (6.18) and Eq. (6.22) one has per unit measurement time, was discussed. This sensitivi-
ty is only achieved, however, for a small range of angles
e -i6Kze -ifJKYe i6K zKze -i6Kze jPKYe i6Kz
within IIN of tP =0. The interferometer can be made to
follow the phase tPl with a sensitivity approaching 1/N
= (costP)( sinh{3)K. + (sintP )(sinhtl>Ky + (coshf/)K z .
provided a feedback loop is employed to adjust a controll-
(7.20) able phase shifter <p, such that tP=tP,-tPl is maintained at
zero. The operation of the interferometer of the last sec-
The mean number of photons leaving the four-wave mixer tion with a feedback loop will now be discussed in more
can then easily be shown to be detail.
N =( Ia 1'+ I)cosh{3-1 . (7.21) The parameters 11, 0, I a I , and f/ are under the control
of the experimenter. It will be assumed that 20-0=0
This equation is easily solved for sinh'{3. Equation (7.17) and that Ia I and tl are known. Then the quantum statis-
then becomes tics of the light entering the interferometer is well charac-
480 CHAPTER IX

SU(2) AND SU(1,1) INTERFEROMETERS 4045

terized and in particular one knows the numbers (J,) and .p1(O). It will be determined how rapidly .p2 approached
(Jx), which according to Eq. (7.4) and (7.10) are .p1(O) given the feedhack algorithm (S.8). Equation (8.S)
iteratively substituted into itself yields
(Jz)=+laI 2 ,
II-I n-l n-I
(S.l)
.p( n)= l: (1 -AA k ).p(o)+A l: BK 11 (1 -AAm) ,
k=O k=O m=k+1
The differenced photocurrent is measured at the output of (8.9)
the interferometer, that is, the photodetectors measure
2Jz . A sequence of measurements will generate a string where the product is defined in the usual way, except that
of numbers, each of which is an eigenvalue of 2Jz. One is "-I
free to process these numbers and in particular one can 11 F(m)=I. (8.10)
subtract (Jz) from them and divide them by -2(J.).
Then the sequence of numbers Idl>d2, ... 1are eigenstates The mean value (.p(n» is, using Eq. (S.5),
of the operator D
(.p(n»=(1-A)"(.p(O» . (8.11)
(sin.plJx (cos.p lJ, - (Jz )
D=--- (8.2)
(J. ) (Jx) It is apparent that the mean value of .p(n) will converge to
zero only if 11- A 1< I. Hence the feedback parameter is
For simplicity it will be assumed that .p is small so that restricted to the range
the approximations sin.p"",.p and cos.p"", I can be made.
Then one can write (S.12)

(S.3) The mean-square value of .p( n) can be obtained by

squaring (8.9) and then taking the expectation value. One
where obtains

"-I
(8.4) +A2(B2) l: W_AA)2)k
k=O

-A2(AB+BA ).p(O)
Since
(A)=l, "-I
(8.5) xl: ((l-AA)2)"-I-k(l_AA)k.
(B)=O k=O
(8.13)
it is immediately evident that The sums can be evaluated to yield
(S.6) ([.p(n)f) = « I-AA )2)"[.p(O)f
that is, the sequence of numbers Id I>d 2 , ••• 1 are esti-
+A2(B2) 1-({\_AA2»"
mates of.p.
1- ((l-AA )2)
The phase shifter .p2 of Fig. 5 will be taken to be con-
trollable. A feedback algorithm that will track .p 1 main- -A2(AB+BA) ((l-AA)2)"_(I-AA)"
taining .p=.p2-.p1 at zero will now be described. Let .p2(i) ((l-AA )2) - (I-AA)
be the setting of .p2 during the ith measurement. The
(8.14)
measurement provides the estimate of .pU)=.p2(i)-.pI(i),
d" which is an eigenvalue of The expectation values « 1- AA )2) and (1- AA ) can be
written, keeping in mind Eq. (8.5), as
D,=.p(j)A,-B, . (8.7)
(I-AA)=I-A,
The feedback loop then adjusts .p2 to the new setting (8.15)
.p2(i + 1)=.p2(i)-Ad, ,
Substituting these expressions into (8.14) one finally has
or in operator form
([.p(n)]2)
.p2(j +1)=.p2(i)-AAM2(i)-.pM)]+AB, , (S.S)
=[( I_A)2+A2(~A )2]"[.p(O)f
where A is a feedback parameter.
It is now assumed that the successive measurements are +A2(B2) 1-[(1 _A)2+A2(~A )2]"
performed on a time scale equal to the characteristic 1-[( I_A)2+A2(~A )2]
coherence time of the four-wave mixer so that the ith
-A2(AB+BA) [( I_A)2+A2(~A )2]"_( I-A)"
operators A, and B, are independent of the j operators Aj
[(I_A)2+A2(~A )']-(I-A)
and Bj • Then Eq. (8.8) can readily be iteratively substitut-
ed into itself. For convenience .pIU) will be held fixed to (S.l6)
LORENTZ TRANSFORMATIONS 481

4046 BERNARD YURKE, S. L. McCALL, AND JOHN R. KLAUDER

From this equation one sees that in order for ([~(n)]') approximately 1.44 measurements in order for ~, to adjust
to converge one must, in addition to (8.12), have itself to the new ~l. Hence, on the average, the total num-
[(1-A)'+A'(..1A)'J"<1. This expression yields the re- ber of photons NT used to detect this displacement is of
striction the order N T = 1.44N and ~~ in terms of the total num-
ber of photons used is
(8.17)
As a particular example, consider the case when A= I, ~~"",,: . (8.27)
NT
then
Hence, by increasing the number of photons fed into the
([~(n)]2) =(~A )2"[~(O)]2+ (B2) I +(~A ),. four-wave mixer from I to 1a 1'",,2+ v'S ",,4. 24, the in-
I-(~A)' terferometer can be operated stably in a feedback mode
and ~, can detect changes in ~l as small as that given by
-(AB+BA)(ilA)2.-1. (B.18)
(8.27).
Consider now the case where, instead of choosing 1a I'
The mean-square value of'" converges to large enough so that the feedback loop would be stable
with the feedback parameter A set to unity, one chose
("").-~
'f' mm- I-(~A)' .
(B.19) I ai' = I as was done in Sec. VII in order to optimize the
sensitivity (7.22) with N fixed. In this case
As ~ is driven to zero, the characteristic number of
measurements il that must be made to reduce ~' to 1/e of (~A)'=+ ,
its original value is
(B,) _ _ I- (8.28)
_ I - sinh'/3 '
(8.20)
n = - In(~A )2 . 2
(AB+BA) = sinh/3 .
We now substitute the results of Sec. VII into these ex-
pressions. One has Since (~A)'> I it is apparent from (8.18) that the feed-
back loop cannot be operated stably with the feedback pa-
21 a 1'(sinh'/3+ +)+ +sinh'/3
+
(~A )'= - - - - - : - - - : ; - - - (8.21) rameter set to unity. In fact, from Eq. (8.17) it follows
Ia 14sinh'/3 that A must be less than if the feedback loop is to be
operated stably. In order to make the e-folding time for
(8.22) ~' as short as possible we choose the value of A which
minimizes
and
[(l-A)'+A'(~A )']
(AB+BA)=_--:2~­ (8.23) of Eq. (8.16), that is,
Ia l'sinh/3
In the large-/3limit Eq. (8.21) reduces to A= _ _I__ =l (B.29)
I+(~A)' 7·
21 1'+ I
(~A)2= aT. (8.24) If ~I(O) is held fixed (~') settles to a steady-state value
I a 14
( "") . = A'(B') (830)
In order that the sensitivity (~~)' not be degraded too 'f' mm 1_[(l_A)2+A2(~A )'] .
much from its minimum value (il~)2=(B2) [see Eq.
(8.19)], let us choose (~A)2 = +. Then the characteristic The characteristic number of measurements which must
number of measurements necessary to reduce ~' to 1/e of be made to reduce ~2 to 1/ e of its initial value is
its original value is il = 1.44. From (8.24) one sees that _ I
(B.31)
lal' has the value lal'=2+v'S. Using (7.21), Eq. n=-ln[(l_A)'+A'(~A)'] .
(8.19) becomes, for large N,
("") . ~2( la 1'+1)'
For A= t and (ilA )'= +one has, upon using Eq. (7.21),
'f' mm la I'N' ( ",2) .
"P mln-
_.!.:..!!
N2 (8.32)
12.9
(8.25) and
"" N' .
il=3.0. (8.33)
Suppose ~ I is stationary so that ~, has settled down and ~
fluctuates with the mean-square value of (8.25), i.e., Again consider the case where ~l has remained constant
for a long time so that the rms fluctuations in ~ have set-
(8.26) tled down to the value determined by (8.32), il~",,1.07lN.
Suppose now that ~ I is displaced instantaneously to a new
Then if a small disturbance should come along to displace value a distance ~~ from its old value. It takes character-
~l by an amount ~~ from its quiescent value it will take istically three measurements for ~, to adjust itself to the
482 CHAPTER IX

33 SU(2) AND SU(1, 1) INTERFEROMETERS 4047

new 4>,. Hence, on the average, the total number of pho-

tons NT used to detect the change in 4>, is NT = 3R. The
sensitivitr of the interferometer, operated with 1a 12= I
and A= 7' is thus expressed in terms of the total number
of photons needed to observe the change as

A.I.=E (8.34)
'I' NT'
a number that is somewhat better than Eq. (8.27).
FIG. 6. An SU(1,1) interferometer. The beam splitters of a
In this section it has been shown that by using suitable conventional interferometer have been replaced by the four·
feedback loops the interferometer of Sec. VII can track wave mixers FWMI and FWM2. The light pumping FWM2 is
changes in 4>, in a stable manner and can achieve a phase phase shifted from the light pumping FWMI by the angle.p.
sensitivity of order I! N. Hence the two problems en-
countered in Sec. VII, namely the fact that the inter-
ferometer achieves its optimum sensitivity only for a

I
small range of phases, 4> < I!N, and that the fluctuations particular let FWMI have the scattering matrix
in Jz ou" the interferometer's output, are greater than
(lzou,) for 1a 12 set at its optimum value, can be over- cosh( t{3) +i sinh( t{3)
come be operating the interferometer with 1a 12 slightly S( -{3)= f . ' I , (9.1)
- I smh( ,{3) cosh( ,(3)
degraded or by choosing the response of the feedback loop
to be such that it averages enough successive measure- As can be seen from Eq. (6.13), K transforms as a
ments of 4> that a useful error signal can be generated. Lorentz boost L ( - (3,y) along the - y axis under this
In the literature2 - 4 a number of schemes for achieving scattering matrix:
interferometer sensitivities of II N have been described.

[~ -S~nh/3j
All of these schemes employ standard interferometers into
which light from degenerate-parametric amplifiers or L(-{3,y)= C:h/3 (9.2)
four-wave mixers is injected. In the next section we will o - sinh/3 cosh{3
describe a novel set of interferometers which dispense
with beam splitters and use the SUIl,1) boosts to convert The scattering matrix for FWM2 is
phase shifts into light amplitude changes rather than the
SU (2) rotations employed by a conventional interferome- cosh( t{3) -i sinh( t/3) 1
S({3)= fi sinh( t{3)
(9.3)
ter. cosh( t{3)

IX. AN SU(1,O MACH-ZEHNDER INTERFEROMETER This scattering matrix transforms K as a Lorentz boost
L ((3,y) along the + y axis. The transformation per-
In S-c. III it was shown how the operation of a Ma~h­ formed by the phase shifters 4>, and 4>2 is, from Eq. (6.20),
Zehnder mterferometer could be viewed in terms of rota-
tions of the vector I under the rotation group SU(2).
In this picture relative phase shifts between two light S(4))=
eoi¢'
beams correspond to rotations about the z axis while pho-
f (9.4)

todetectors are sensitive to rotations in a plane containing

the z axis. The function of the beam splitters was to con- Under this scattering matrix K transforms as a rotation
vert a rotation about the z axis into one perpendicular to R(4),z) about the z axis by an angle 4>=-(4),+4>2)'
the z axis.
COS4> - sin4> 0]
In this section an interferometer whose operation can be R(4),z)= [ sine/! cose/! 0 . (9.5)
viewed in terms of transformations of the vector K Eq.
(6.1), under the Lorentz group SUIl,\) is considered. o 0 I
From Eq (6.21) one sees that the common mode phase The overall scattering matrix for the device of Fig. 6 is
shift of two light beams corresponds to a rotation of K
about the z axis. But from Eq. (6.1) one sees that photo- S=S({3)S(4»S( -(3) , (9.6)
detectors placed in the two light beams will be sensitive and the overall transformation perfonned on K is
only to transformations perpendicular to the z axis.
Again, a device is required which will convert rotations
K ou,=L(/3,y)R(4>,z)L(-/3,y)Kin · (9.7)
about the z axis into transformations perpendicular to this
axis. The four-wave mixers described in Sec. VI can carry
out such transformations. These transformations consist It will be useful to reexpress this transformation as fol-
of Lorentz boosts. lows:
As a specific example, consider the device of Fig. 6.
The phase shifter", in the pump beam is. adjusted such L(/3,y)R (4),z)L (-/3,y)=R (O,z)L(yx)R(II,z) , (9.8)
that four-wave mixer FWM2 performs the inverse of the
transformation performed by four-wave mixer FWMI. In where L ( y ,x) denotes a Lorentz boost along the x axis,
LORENTZ TRANSFORMATIONS 483

4048 BERNARD YURKE, S. L. McCALL, AND JOHN R. KLAUDER 33

COSh Y o sinhy j
L (y, x) = [ 0 o . (9.9)
sinhy o coshy
Equation (9.8) holds when 8 and yare chosen such that

sin8= (l-cos¢>)cosh/3 (9.10)

[sin'cI> + ( I -cos¢ )'cosh'/3JI !2

cos8= sin¢>
[sin'¢ + (I -coscl> )'cosh'/3J 11l
(9.11)
'.
.., Ib,

coshy = ( l -cos¢)Cosh'/3+coscl> , (9.12)

sinhy = sinh/3[sin'¢ + 1I -cos¢ )'cosh'/3J I12 . (9.13)

Hence the transformation performed by the device of Fig.

6 on K can be regarded as a rotation 8 about the z axis,
followed by a Lorentz boost along the x axis, followed by
e
a second rotation about the z axis.
Let us now consider the operation of this device when
no light enters the input ports, that is when 1in) is the
vacuum state 10). The vacuum state is both an eigen- /
state of K, and),: Ie, " .0'
K,IO )=+ IO), (9.14)
FIG. 7. A geometrical view of the performance of an SU(J, II
),10) = 0. (9.15) interferometer. (a) The input state consisting of the vacuum
state is depicted in the (Kx,Ky.Kzl space. where Kx and Ky are
Consequently, from Eq. (6.6) the invariant K' is zero, that regarded as space coordinates and Kz as a time coordinate, as a
is, we can think of K for the vacuum state as lying on the circle on the light cone. (b) The first four-wave mixer performs
light cone. In the spirit of Fig. 2, the vacuum state is de- a Lorentz boost along the positive y axis. (c) The phase shifts

+.
picted in Fig. 7(a) as a cone whose base intersects the z accumulated by the light beams propagating in the interferome-
axis at The Lorentz boost L ( - /3,y) is equivalent, in ter result in a rotation in the xy plane. (d) The second four-
the Schriidinger picture, to a boost of the state vector in wave mixer performs a Lorentz transformation along the nega·
the opposite direction. The Lorentz boost performed by tive y axis. The total number of photons leaving the interferom·
the first four-wave mixer is depicted in Fig. 7(b)' The etef is a linear function of K z .
mean value of K, in terms of the mean number of pho-
tons (N) emitted by the four-wave mixer is, from Eq.
(6.31), explicit calculation . From Eq. (9.6) and Eqs. (6.15) and
(K,)=+((N)+I) . (6.23) the incoming state vector 1in) is transformed as

The phase shifts cI> 1 and ¢, encountered by the two light (9.17)
beams leaving the four-wave mixer then rotate the state but from 19.8) this is equivalent to the transformation
vector about the z axis by an angle -¢ = ¢I +¢,. This is
depicted in Fig. 7(c). A second Lorentz boost with the (9.18)
same rapidity, but in the opposite direction, is then per-
formed. If ¢ = 0the final state will be a vacuum state and The operator Nd for the total number of photons detected
no photons will be detected by the photodetectors in the by the photodetectors placed in the output beam is from
Eq. (6.31)
output beams. If ¢ is nonzero the state of the light
delivered to the photodetectors will be a Lorentz-boosted Nd=2K,-I. (9.19)
vacuum, the rapidity parameter being determined by Eq.
(9.12) or (9.13). Hence in order- to evaluate (Nd ) and ll.Nd one needs to
In Fig. 7(b) the projected ellipse lying in the x- y plane evaluate (out 1K, l out) and (out IK; lout). From Eq.
has a width of +
and the distance from the origin to its (9.18) one has
center is (K,)";' +(
(N) + I), Hence Fig. 7(c) suggests
(out 1K z lout) = (in 1e i6K'e -iYK'K,eiyK'e -i6K, 1in) .
that the minimum detectable phase ¢min is of the order
(9.20)
I
(9.16)
¢min= (N) +1 ' Since I in) is the vacuum state, one has
that is, this detector can achieve a phase sensitivity ap- e -i6K, 1in) =e - i612 1in) . (9.21)
proaching 1/ N.
That this is the case will now be demonstrated with an Equation (9.20) thus simplifies to
484 CHAPTER IX

SU(2) AND SU(1,1) INTERFEROMETERS 4049

(out IK. lout) = (0 Ie -yK,K./ YK, 10) . (9.22) The quantity (11t/!)2 is minimized when t/!=O, then
From Eqs. (6.18) and (6.17),
2 I
(dt/!)min=-'-2- . (9.30)
e -iyK, K./ yK, = (sinhy )K. + (coshy )K. , (9.23) smh ,B

but Expressed in terms of the mean number of photons (N)

(9.24) emitted by the first four-wave mixer, sin2,B
= (N)( (N) -2), so
so
(out! K. lout) = +cosh,B . (9.25) (9.31)
(N)(N)-2)
From Eq. (9.19), the mean number of photons (Nd )
detected by the photodetectors is Hence it has now. .been shown that the SUO, I) inter-
ferometer depicted in Fig. 6can achieve a phase sensitivi-
(9.26) ty approaching 1/N. This sensitivity is achieved when no
light is fed into the input ports. A comparison with Fig.
In a similar manner one can show
5 shows that SUO,I) interferometers achieving a sensitivi-
(9.27) ty of 1/N require fewer optical elements than an SU(2)
interferometer achieving the same sensitivity. Further, at
The dependence of y on t/! and {3 is given by Eqs. (9.12) t/! =0 no light is delivered to the photodetectors, that is,
and (9.13). Hence Eqs. (9.26) and (9.27) can be rewritten the pairs of pump photons converted into pairs of four-
as wave-mixer output photons are absorbed by the second
(Nd)=(1-coS¢)sinh 2,B , four-wave mixer and converted back into pump photons.
(9.28) Hence an SUO, I) interferometer can be very economical
(Md )2=[sin 2t/!+(I-cost/!)2cosh2,Blsinh2,B . with photons. It will absorb pump power only when t/! is
nonzero. It is also worth pointing out that the beams I
The mean-square fluctuation in t/! due to the photon
and 2 need not be at the same frequency, as long as they
statistics is thus
are placed symmetrically about the pump frequency, "'0,

r
that is, with beam 1 at the frequency "'0+ d", and beam 2
at the frequency "'0- d",; the scattering matrix 6•12. 13 for
[a(~d) the four-wave mixer will still have the form (6.11).
By using techniques similar to those used in deriving
Eq. (6.41) one can show that the probability P(N) of
sin 24>+( l-coS¢)2cosh2,B detecting a total of N photons leaving the output ports of
(9.29)
sin 2t/! sinh2p the interferometer is

o ifNisodd
NI2
P(N)= 2 [ (I-cost/!)sinh 2p (9.32)
]
if N is even.
(l-cost/!)sinh 2,B+2 (l-cost/!)sinh 2,B+2

Let Ni denote the number of photons counted during the The sum has the form
ith measurement in a sequence of measurements. One is
free to take the square root of each of these numbers.
Hence it is meaningful to talk about the average and rms
value of v'N. The motivation for investigating the statis-
tics of v'N stems from the fact that (N) is an even func- which can be approximated by the integral
tion of t/!. Now
(v'N)= l: v'NP(N) (9.33) (9.35)
Neven

can be put into the form Hence

(v'N) 23/2 ( v'N ) _ v'2rr
( 1- cost/! )sinh2,B + 2 (I-cost/!)sinh 2,B+2
X i VI [ (l-cost/!)sinh 2p ]k (9.34) X [In [( l-cos4»sinh 2p+2]1-312 (9.36)
k=O (l-cost/!)sinh 2,B+2 (I_cos,p)sinh2P
LORENTZ TRANSFORMATIONS 485

4050 BERNARD YURKE, S. L. McCALL, AND JOHN R. KLAUDER 33

Since this approximation holds reasonably well for Hence for N > 4 it has been shown that the uncertainty in
the inferred oorm of the phase 4' is to a good approxima-
(l-cosp)sinh2p~ 4 (9.37) tion
the logarithm can be approximated via In(l+x)~ and al4'1 =0.52214'1· (9.44)
one has
It is also instruEtive to ask what the probability P( I4'J :
(v'N)"" ~1r (l-co~)ll2sinhp. (9.38) (l-a)I4>1 < 14>1 dl+a)I4>I) is that a measured 14>1
will lie in the range

If
Approximating (1- co~) as 4>2/2 one finally has
(I-a)I4>1 < 14'1 <O+a)I4>1 .
(v'N )=t f1214> Isinhp . (9.39) From (9.40) this is equivalent to determining the probabil-
ity PIN: Nl <N <N 2) that N lies in the range
Hence the norm of the phase 4' inferred from a measure- Nl <N <N2 where
mentof Nis
N =~(I-a)24>2sinh2p
1 8 '
(9.45)
N2=f(l+a)24>2sinh2p.
(9.40)
One can show rigorously
From (9.39) one has

( 14'1 )= 14> I ' (9.41)

that is, the mean value of 14' I inferred from a measure-
ment of N is equal to the norm of the actual phase setting
4> of the interferometer. Now

(9.42) [
(I-co~)sinh 2 P ]N212] (9.46)
- (l-co~)sinh2f:l+2 .
so
(9.43) So from Eq. (9.45)

P( I ~l (I-a) 14>1 < 14'1 dl+a) 14>1)= II .

(l-cosp)slnh
2
P
j ("'/16111-al'~',jnh2p
(l-co~)slnh2f:l+2

_ [ (l-cosp)sinh 2p jl"./1611l+aI2;"jnh2P j. (9.47)

( l-cos.p)sinh2p+2

Approximating I-cos,p by 4>2/2, this expression can be P(I4'I:(l-a)I<I>1 < 14'1 dl+a)I<I>I)
put into the form
""e -1".12111-aI2-e -1 ..I2IIl+al' (9.50)

I[ +, then
P(lhO-a)I4>1 < 14'1 dl+a)I4>I)
As an example, let a =
= x :4 r16111-a12x - [ x :4 r16111+a12X j, p<lh 1<1>1/2< If I dl<l>I/2)
""e - ../8_ e -9.-/8=0.646 . (9.51)
(9.48)
where x = <l>2sinh2fJ. Hence from a single measurement of 14' lone has 65%
Now confidence that I <I> I /2 < If I d I <I> I /2.
The interferometer described here suffers from draw-

'x_co
[x+4 j =e-
lim _x_
x
4 • (9.49)
backs similar to those of the SU(2) interferometer of Sec.
VII. Maximum sensitivity occurs at 4>=0 and the sensi-
tivity rapidly degrades as 4> is adjusted away from zero. It
This limiting value is not a bad approximation for was shown in Sec. VIII that such drawbacks can be over-
[x/Ix +4)]' even for x as low as 10, the level at which come with feedback. However, implementing a feedback
on average five photons are counted in the interferometer algorithm for the SUO,I) interferometer described here is
output beams. Hence complicated by the fact that (N) is an even function of 4>
486 CHAPTER IX

33 SUIZ) AND SUIl,1) INTERFEROMETERS 4051

and hence the sign of the error signal cannot be deter- [Lx>Lyl=-iL, ,
mined from the ~umber of photons counted by the photo-
'detector during a single measurement. [LyLzl=iLx' (10.3)
The sign of the error signal can be generated by chang- (L"Lxl=iLy.
ing (dithering) </>, between successive measurements and
constructing the derivative signal (N; + I -N;)IIl</>,. Again, it is useful to introduce the raising and lowering
Alternatively, one could impleIl]ent the feedback algo- operators
rithm which will now be described. Make repeated mea-
L+ =Lx+iLy=fata t ,
surements of Ii: until I</> I is determined to some 110.4)
predetermined precision: Ill</> I =a I</> I where a is a con- L_ =Lx -iLy = Taa
stant. Then move </>2 according to
which satisfy the commutation relations
cb,(new)=</>,Iold)+ Iii. (9.52)
[L_,L + 1=2L, ,
i
Make repeated measurements of I I at this new setting (10.5)
so that a new i </> I can be inferred with the precision [L"L±l=±L± .
Ill</> I =a I</> I· If I</> I inferred for the new setting of </>, The Casimir invariant
is less than I</> i inferred for the old setting one assumes
that one has moved in the right direction. If, on the other L'=L;-L;-L; , 110.6)
hand, the inferred value of I</> I for the new setting of </>,
when expressed in terms of the operators a and at,
is greater than the inferred I</> I for the old setting one as-
reduces to the number
sumes that one has moved </>2 in the wrong direction and
cb, is then readjusted so that (10.7)
(9.53)
It is useful to determine how L=(Lx>Ly,L,)
The process is then repeated. transforms under specific cases of Eq. (10.1). Under the
If I</> I is determined to sufficient precision this algo- mode transformation
rithm will move one closer to <I> = 0 most of the time. On aout=cosh(+(3)Qln+sinh(+f3)a~l , 110.8)
the occasions when this algorithm moves one in the wrong
direction it generally does not move <1>, very far in the L transforms as a boost along the x axis:
wrong direction and the lost ground is regained during the
next few iterations of the feedback procedure. Further, Lx ] [COSh{3 0 Sinh{3] Lx
since a single measurement already determines I cb I with Ly = 0 1 0 Ly 110.9)
a precision Il</> ",,0. 51</> I at a 65% confidence level, one
Lz out sinh{3 0 cosh{3 Lz
does not have to make very many repeated measurements
of I '" I in order for the feedback algorithm to work. Under the mode transformation

x. SINGLE-MODE SUO,I) INTERFEROMETERS a out =cosh( T{3)ain - i sinh( T{3)a;n (10.10)

In this section interferometers based on devices having L transforms as a boost along the y axis:
the scattering matrix
Lx
j Lx
will be described. Such a single-mode device can be re-
(10.1) Ly
L, out
= [~ o 0
cosh{3 sinh{3
sinh{3 cosh{3
Ly
L,
110.11)

garded as a limiting case of the four-wave mixers of Sec.

VI in which the two input and the two output beams are A phase shift
made collinear and are sufficiently close in frequency that aout =e -i f ain (10.12)
they cannot be resolved during the coherence time of the
device. Both four-wave mixers and parametric amplifiers transforms K as a rotation about the z axis by the amount
configured properly!7,!8 are capable of performing the 2</>:
mode transformation Eq. (10.1). Connected with Eq.
(10.1) it is convenient to introduce the operators Lx [COS2</> -sin2</> 0] Ly
Ly = sin2</> cos2</> 0 Ly 110.13)
Lx=+(atat+aa) ,
L, out 0 0 1 Lz
L =_i.(atat_aa) 110.2)
y 4 ' It can be shown that K transforms under Eq. (10.1) as a
Lz=+(ata+aa t ) . rotation of K about the z axis by the angle -0/2 fol-
lowed by a boost {3 along the x axis followed by a rotation
These operators behave as generators 6 •9 of the group about the z axis by the angle 012. Equivalently, in the
SUII,I) satisfying commutation relations identical with Schriidinger picture the state vector transforms according
Eq. (6.2): to
LORENTZ TRANSFORMATIONS 487

4052 BERNARD YURKE, S. L. McCALL, ANO JOHN R. KLAUOER

lout) =e -;(612)L,//JL'e'1612)L, Iin) , (10.14) photons will be counted leaving the two-port four-mixer
of Sec. VI. From Eq. (6.41)
where e'/JL, is a single-mode squeeze operator l9 and O, n odd
e '(612)L, has been called a single-mode rotation opera- { (10.25)
PT(n)= sech 2( t(3)[tanh 2( t(3)f, n even.
tor. 6• 12.13
More generally one could consider a device which With the identity
transforms a state vector according to
lout) =e'<;L'e'/JL'e'9L, I in) . (10.15) k~J: 1[2~n~:)l=22n (10.26)
The probability distribution for the number n of photons
in the output beam will now be determined for the case one can show
when the input consists of vacuum fluctuations. A more
general case, when the input consists of coherent states,
PT(n)= ! P(nl)P(n2)' (10.27)
nl,n2
has been treated by Yuen. 20 A photodetector in the out- n l +n2=n
put beam measures N =a t a, which can, from Eq. (10.2),
be written in the form This equation implies that the statistics of the total num-
ber of photons leaving the two-mode four-wave mixer is
N=2Lz -t· (10.16) the same as the statistics of the total number of photons
coming from two independent single-mode devices. This
The amplitude that n photons will be counted in the out-
observation will allow a simplification of the discussion of
put beam is (n lout), hence the probability P(n) that n
feedback loops for the interferometer discussed in this sec-
photons will be counted is
tion, since the results of Sec. IX can be made to apply by
P(n)= I (n lout) 12 . (10.17) pairwise averaging successive measurements made with a
single-mode device.
Now for an n-photon state In) one has The interferometer to be considered in this section is

Lzln)= [f+± lin). (10.18)

depicted in Fig. 8 where for the sake of definiteness,
degenerate-parametric amplifiers OPA I and OPA2 are
used to perform the Lorentz boost. OPAl will be taken
Hence one has to perform the boost e'/JL, on the incoming state vector.
The phase shifter is taken to perform a phase shift e -,<;L,
(10.19) on the light beam. The last degenerate-parametric ampli-
and fier is taken to perform the boost
-i6Lz -ifJL;r i6L z
e -'';L, In) =e -'<;(n12+1/4) In) . (10.20) e e e ,

The probability distribution (10.17) thus reduces to where Ii is proportional to the phase of the pump light
entering OPA2. Letting I in) denote the state vector for
P(n)= I (n Ie'/JL, 10) 12 . (10.21) the incoming light, the state vector lout) for light leav-
ing the interferometer is
Hence P(n) is independent of the phase angles <P and 0,
i.e., P(n) depends only on the magnitude of the boost. lout) =e -'6L'e -'/JL'e'6L'e -'<;L'e'/JL, I in). (10.28)
Again, using (6.35) one has
The behavior of this device when I in) is the vacuum
e '/J L, =exp(i tanh( t(3)L + Jexp[ -2Incosh( t(3)L z J state will now be considered. Figures analogous to Fig. 7
can be drawn to illustrate the behavior of the interferome-
(10.22) ter. However, in this case the Casimir invariant Eq. (10.6)
Hence by using the techniques used to arrive at Eq. (6.41)
bas the numerical value - +.-.
Hence L lies on a space-
like hyperboloid instead of the light cone of Fig. 7. The
one can show vacuum state is an eigenstate of L,

0, n odd

P(n)=
;n [i j cosh( t(3) ,
n even (10.23)

PUMP
where

[: l= (n-~)!m!
FIG. 8. A single-mode SU(J,I) interferometer. The device
(10.24) employs two degenerate-parametric amplifiers OPAl and
OPA2. The output of the device is sensitive to the difference
It is useful to compare this probability distribution with between the phases '" and Ii accumulated by the signal and pump
distribution PT ( n) for the probability that a total of n beam, respectively.
488 CHAPTER IX

sum AND SUO,I) INTERFEROMETERS 4053

110.29) ter. Since the number operator N for the total number of
and hence could be represented as a circle drawn around photons counted at the output of the detector is linear in
the hyperboloid at a height along the z axis of +. L" E~. (10.16), one would like to determine
We now determine the mean and variance in the num- (outlLz lout). One can readily show from Eq. (l0.28)
ber of photons counted at the output of the interferome- that

(out IL: lout) = (in I e -i/iL,/i¢-bIL,/IiL'L:e -i/iL'e -'i¢-bIL'ei/iL, I in) . (10.30)

Using the techniques of Sec. IX this can be further reduced to

(out IL: lout) = (in I eiBL'e -iYLyeiBL, L:e -iBL,/YLYe -iBL, I in) = (in I e -iyL yL:/yLy I in) , (10.31)

where in analogy with Eqs. 19.12) and 19.13), taining 4>-0=0 by using the error-correcting signal to
adjust the phase shifter Ii in the pump beam delivered to
coshy=[ l-cosl4>-0)]cosh 2/3+cosl4>-0) ,
(10.32) DPA2. As was mentioned earlier the statistics of the total
number of photons counted in two successive measure-
sinhy=sinhtJ! sin'I4>-o)+ [1-cos(4)-0)j2cosh'/3jll' . ments of 4> are the same as for the total number of pho-
tons leaving the interferometer of Sec. IX. Hence the
It is straightforward then to show that
feedback algorithms discussed in Sec. IX will also work
(N) "" (out IN lout) =+coshy , 110.33) for the single-mode device discussed here.

and XI. CONCLUSION

{N')"" (out IN'I out) A geometric or Lie-group-theoretical approach to the
= +sinh'y++(coshy-l)' (10.34) analysis of interferometers was presented. Such an ap-
proach facilitates identifying the input states which op-
or, using Eq. 110.32) timize the interferometer's sensitivity. It was shown that
ordinary interferometers are characterized by the group
(N)=+!I-cos(4>-o)]lcosh'/3-I) , SU(2) which is equivalent to the group of rotations in
110.35)
three dimensions. With suitable input states such an in-
IAN)2= +sinh'tJ! sin'I4>-o)+ [1-cosl4> -o)]'cosh'/3) terferometer can achieve a phase sensitivity 114> approach-
ing 1 IN where N is the total number of photons passing
The mean-square fluctuation in the readings for 4>, through the phase-shifting element 4> of the interferome-
given by ter. Although this sensitivity can only be achieved for 4>
within 1! N of 4>=0, it was shown that by employing a
110.36) feedback loop the interferometer could track phase as a
function of time with a precision of 1 IN.
A class of interferometers in which four-wave mixers
serve as active analogs of beam splitters was also present-
is readily evaluated and has a minimum given by ed. Such interferometers are characterized by the group
SUI!,I) and have the virtue of being able to achieve a
( A4>min) ' = --.1- , - . (10.37) phase sensitivity approaching liN with only vacuum
2 smh /3
fluctuations entering the input port. SU(I,I) interferome-
The mean number of photons (N[) in the light beam ters can consequently achieve the liN phase sensitivity
passing through the phase shifter 4> is much more readily and in fact have a simpler construc-
(N[) = (0 I e -i/iL'Nei/iL, 10) tion than sum interferometers. The output of a four-
wave mixer depends on the relative phase between the
= +lcosh/3-1) . 110.38) pump and the incoming signal. It is this phase sensitivity
which the SU(I,I) interferometers employ. In fact, it was
Solving this equation for sinh'/3 one finally has shown that degenerate-parametric amplifiers which are
also phase-sensitive devices can be used to construct
IA'" . )2_ 1 (10.39) SU(I,I) interferometers.
'l'mm - S{N[)(N[)-I)

Hence it has been shown that the device of Fig. 8 can ACKNOWLEDGMENT
indeed achieve a phase sensitivity approaching lin. This
minimum sensitivity is achieved when 4>-15=0. Hence We would like to thank R. E. Slusher for stimulating
by implementing a feedback loop one can track 4> main- discussions on the work presented here.
LORENTZ TRANSFORMATIONS 489

4054 BERNARD YURKE, S. L. McCALL, AND JOHN R. KLAUDER 33

1M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, IOH. P. Yuen and J. H. Shapiro, Opt. Lett. 4,334 (1979).
1975). 11M. D. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1985).
'C. M. Caves, Phys. Rev. D 23, 1693 119811. 12c. M. Caves and B. L. Schumaker, Phys. Rev. A 31, 3068
3R. S. Bondurant and J. H. Shapiro, Phys. Rev. D 30, 2548 (1985).
(1984). 13B. L. Schumaker and C. M. Caves, Phys. Rev. A 31, 3093
4R. S. Bondurant, Ph.D. thesis, Massachusetts Institute of Tech- (1985),
nology, 1983. 14P. Kumar and J. H. Shapiro, Phys. Rev. A 30, 1568 (1984).
lWei-Tou Ni (unpublished). lsR. Graham, Phys. Rev. Lett. 52, 117 (1984).
6B. L. Schumaker, Phys. Rep. (to be published). 16M. D. Reid and D. F. Walls, Phys. Rev. Lett. 53, 955 (1984).
'w. H. Steel, Interferometry, 2nd ed. (Cambridge University, I'B. Yurke, Phys. Rev. A 32, 300 (1985).
Cambridge, 1983). 18B. Yurke, Phys. Rev. A 29, 408 (1984).
8K. W6dkiewicz and J. H. Eberly, J. Opt. Soc. Am. B 2, 458 19J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979).
(1985). 2oH. P. Yuen, Phys. Rev. A 13,2226 (1976),
9D. R. Truax, Phys. Rev. D 31, 1988 (1985).
490 CHAPTER IX

Thomas precession, Wigner rotations and gauge

transformations

D Hant, Y S Kim* and D Son§

t National Aeronautics and Space Administration, Goddard Space Flight Center (Code
636), Greenbelt, MD 20771, USA
*USA
Department of Physics and Astronomy, University of Maryland, College Park, MD 20742,
.
§ Department of Physics, Kyungpook National University, Taegu 635, South Korea

Received 10 March 1-987, in final form 1 July 1987

Abstract. The exact Lorentz kinematics of the Thomas precession is discussed in terms of
Wigner's 0(3 )-like little group which describes rotations in the Lorentz frame in which the
particle is at rest. A Lorentz-covariant form for the Thomas factor is derived. It is shown
that this factor is a Lorentz-boosted rotation matrix, which becomes a gauge transformation
in the infinite-momentum or zero-mass limit.

1. Introduction

To most physicists, the Thomas precession is known as an isolated event of the ! factor
in the spin-orbit coupling in the hydrogen atom [1-6]. The purpose of this paper is
to point out that the Thomas rotation plays a very important role in studying the
internal spacetime symmetries of massive and massless particles. Einstein's E = me 2 ,
which means E = (p2+ m 2)1/2, unifies the momentum-energy relations for massive and
massless particles, as is illustrated in table 1. We are interested in the question of
whether the internal spacetime symmetries can be unified in a similar manner.
The Thoma~ precession is caued by the extra rotation the particle in a circular orbit
feels in its own rest frame [1,3, 5, 7]. In this paper, we point out first that Wigner's
little group [8] is the natural language for the Thomas effect and perform the exact
calculation of the precession angle. We shall then examine its implications.

Table 1. Significance of Wigner's little groups. Einstein's formula, commonly known as

E = me 2 , unifies the energy-momentum relation for massive and massless particles. Like-
wise, Wigner's little grollps unify the internal spacetime symmetries of massive and massless
particles. The Thomas rotation occupies an important place in this figure as an element
of the 0(3 )-like little group for a massive particle.

Massive, slow Between Massless, fast

Energy, momentum E = p2/2m Einstein's E = (m 2+ p2)'/2 E=p

Spin, gauge, helicity S3' S"S2 Wigner's little group S3, Gauge transformations

0264-9381/87/061777+07$02.50 © 1987 lOP Publishing Ltd 1777

Reprinted from Class. Quantum Grav. 4,1777 (1987).
LORENn TRANSFORMATIONS 491

1778 D Han, Y S Kim and D Son

The little group is the maximal subgroup of the Lorentz group which leaves the
4-momentum of a given particle invariant [8]. The little groups for massive and massless
particles are locally isomorphic to 0(3) and E(2) respectively. The 0(3)-like little
group for a massive particle becomes the three-dimensional rotation group in the
Lorentz frame in which the particle is at rest, which is therefore the Thomas frame.
In the case of massless particles, the rotational degree of freedom of E(2) is
associated with the helicity [8,9] and the translational degrees of freedom correspond
to the gauge degrees of freedom [10-12]. It has been established that the O(3)-like
little group becomes the E(2)-like little group in the limit of infinite momentum and/ or
vanishing mass [13-16]. This means that both the 0(3)- and E(2)-like little groups
are two different manifestations of a single little group, just as E = p2/2m and E = cP
are two different limits of E = [( CP)2 + (mc 2f]\/2. We shall show in this paper that the
Thomas effect stands between these two limiting cases.
In § 2, the Wigner rotation is constructed for the kinematics of the Thomas pre-
cession. In § 3, the Thomas effect is shown to be an element of the 0(3)-like little
group. In § 4, it is shown that the Thomas factor becomes a gauge transformation in
the large-momentum or zero-mass limit.

2. Wigner rotations

Let us consider a system of three Lorentz boosts, as is described in figure 1. We start

with a massive particle at rest whose 4-momentum is
Pa=(O,O,m) (1)
in the 4-vector convention: XIL = (x, Z, t), where we omit the y component which is
not affected by the transformations discussed in this paper. B\ boosts the above
4-momentum along the Z axis with the boost parameter 11:
Pb = B\Pa = m(O, sinh 71, cosh 71)· (2)
The 3 x 3 matrix for B\ is

B\ =(~ co~ 11 sin~ 11). (3)

o sinh 71 cosh'T/

a 8, b

Figure I. Closed Lorentz boosts. Initially, a massive particle is at rest with 4-momentum
p•. The first boost B, brings p. to Pb' The second boost B2 transforms Pb to Pc. The
third boost B3 brings Pc back to p•. The net effect is a rotation around the axis perpendicular
to the plane containing these three transformations. We may assume for convenience that
Ph is along the z axis, and Pc in the zx plane. The rotation is then made around the y axis.
492 CHAPTER IX

Thomas effect and gauge transformations 1779

The second boost B2 transforms Ph into Pc whose momentum has the same magnitude
as that of Ph but makes an angle 0 with the direction of Ph:
(4)
From the triangular geometry of figure 1, the direction of the boost for B2 becomes
( 0 + 7T) /2 and its boost parameter is
A =2tanh- I ([sin(0/2)] tanh 1)). (5)
The transformation matrix is
1 + (sin 0/2)( cosh A -1) ! sin 0(1- cosh A) (cos 0/2) sinh A)
B2 = ( ! sin 0(1- cosh A) 1 + (sin 0/2)2(cosh A -1) -(sin 0/2) sinh A . (6)
(cos 0/2) sinh A -(sin 0/2) sinh A cosh A

The momentum of Pc is in the zx plane. We can also obtain Pc by rotating Ph around

the y axis by 0 as is illustrated in figure 1. The third boost B3 brings Pc back to the
state of zero momentum:
Pa = B3 Pc = (0, 0, m). (7)
B3 takes the form:
1+(Sin 0)2(cosh 1)-1) ! sin 20( cosh 1)-1) -(sin 0) sinh 1))
B3 = ( hin 20(cosh 1)-1) 1+ (cos Of(cosh 1)-1) -(cos 0) sinh 1) . (8)
-(sin 0) sinh 1) -( cos 0) sinh 1) cosh 1)

This means that the above three successive boosts will leave the 4-momentum
Pa = (0, 0, m) invariant. Since all the boosts are made in the zx plane, the net effect
will be a rotation around the y axis, which does not change Pa:
B3B2BI =W or B3B2BI W- I =I (9)
where I is the identity matrix. W is a one-parameter matrix representing a rotation
around the y axis in the Lorentz frame in which the particle is at rest and its form is
COS a
W= (
-s~n a (10)

with
. -I ( (sin O)(sinh 1)/2)2 )
a =2sm . . (11)
[(cosh 1))2 - (smh 1) )\sin 0/2)2]1/2
W is definitely a rotation matrix of Wigner's O(3)-like little group which leaves Pa
invariant. Since Wigner was the first to introduce the concept of this little group in
terms of rotations in the Lorentz frame in which the particle is at rest [8], it is quite
appropriate to call W(ll') the 'Wigner rotation' [17].

3. Thomas effect

The Thomas effect is caused by two successive boosts with the same boost parameter
in different directions [1-7]. The transformation needed for this case is (B3 )-I(B 1)-1
LORENTZ TRANSFORMATIONS 493

1780 D Han, Y S Kim and D Son

which brings Pb to Pa and then transforms Pa into Pc. However, this is not B 2 , but
requires an additional matrix T:
(B 3 )-I(B 1)-I=B2 T (12)
where T is the Thomas factor. According to (9)
(B 3)-I(B1)-1 = B2BI W-'(B1)-'. (13)
If we compare (13) and (14):
T=B,W-I(B,) 1 (14)
While W is the rotation matrix whose form is given in (10), T is not a rotation matrix
but is a Lorentz-boosted rotation matrix. Its form is
' cos a -(sin a) cosh 1/ (sin a) sinh 1/ )
(
T = (sin a) cosh 1/ 2
(cos a) cosh 1/ - sinh 1/2 (cosh 1/)[ cosh 1/ - (sinh 1/) cos a] .
(sin a) sinh 1/ (sinh 1/)( cosh 1/)( cos a -1) cosh 1 1/ - (cos a) sinhl1/
(15)
Furthermore, T can be written as
T = (B1 )-I(B 3 )-I(B 1) -I. (16)
This matrix leaves the 4-momentum Pb invariant and is therefore an element of the
O(3)-like little group for Pb.
The right-hand side of (12) is different from the decomposition given in the existing
literature [2-6, 18-20]. (B 3 )-'(Bd- ' can also be written as a boost preceded by a
rotation:
(17)
As is illustrated in figure 2, R represents a rotation in the Lorentz frame in which the
particle is not at rest and the boost B' is quite different from B2 . The rotation angle
for R is
, _I ( (sin O)[sinh( 1//2)f )
(18)
a =2tan [cosh(1// 2 )]2-[sinh(1// 2 )]2cosO.

a~----~----~----~~b

Figure 2. Lorentz boosts which are not closed. Two successive boosts (B 3 )-'(B,)-' result
in boost B' preceded by rotation R performed in the Lorentz frame where the particle is
not at rest. It is quite clear that B' is not B2 •
494 CHAPTER IX

Thomas effect and gauge transformations 1781

This angle has been reported in the literature [18-20] but is conceptually quite different
from a of equation (11). Figures 1 and 2 will illustrate this difference. a and a' are
numerically different.
The Thomas precession of a charged particle in an electromagnetic field and its
connection with the Wigner rotation has been thoroughly discussed in a recent review
article by Chakrabarti [21]. Our main interest in this paper is the Thomas effect in
the large-momentum or zero-mass limit.

4. Large-momentum or zero-mass limit

Let us go back to the T matrix of (15). In the low-energy (small-1) limit, T becomes
a rotation matrix and the rotation angle a becomes equal to a' in the same limit. The
above discussion therefore does not alter the existing treatment of the Thomas pre-
cession in the small-1) limit. If 1) is not small, (15) has a new meaning. In the limit
of large 1), T takes the form:
-u

T=(~ ~ u 0
(19)

u 0

with

For a finite value of u, a vanishes as 1) becomes very large. We have restored the y
coordinate in order to compare this form with those given in the literature [8, 10, 11].
The T matrix is now applicable to the 4-vector (x, y, z, t).
The above expression for the T matrix represents an element of the E(2)-like little
group which leaves invariant the 4-momentum of a massless particle moving in the z
direction [8]. Furthermore, when applied to the 4-potential of a plane electromagnetic
wave, it performs a gauge transformation [10-12]. Therefore, the Lorentz-boosted
rotation given in (19) becomes a gauge transformation. Indeed, the Thomas rotation
stands between slow particles with spin degrees of freedom and massless particles with
the helicity and gauge degrees of freedom, as is illustrated in table 1.
Using techniques different from the Thomas kinematics given in this paper, we
have reported in our earlier publications [15,16] that Lorentz-boosted rotations become
gauge transformations in the limit of large momentum and/ or small mass. What is
new in the present paper is that the Thomas effect is one concrete physical example
of the Lorentz-boosted rotation which stands between massive and massless particles.
Let us illustrate what we did above using the representation of SL(2, c) for spino!
particles. The group of Lorentz transformations is generated by three rotation
generators Sj and three boost generators K j • They satisfy the Lie algebra:

[Sj, Sj] = ieijkSk [Sj, Kj ] = ieijkKk

(20)
[K j , KJ = -iejjkSk'
Since these commutation relations remain invariant when K j are replaced by - K j , we
have to consider both signs of the boost generators.
LORENTZ TRANSFORMATIONS 495

1782 D Han, Y S Kim and D Son

For spin-! particles, the generators of rotations take the form

(21)
and the rotation matrix W of (10) is translated into

W = (cos al2 2).

-sin a I (22)
sin al2 cos al2
Since there are two different sets of boost generators:
(23)

there are two different boost matrices for B 1 :

BI
(±) _
-
(eX P(±1] 12) (24)
1
Consequently, there are two different forms of T:
T(±) = B\±) W-I(B\±»-I. (25)
In the limit of large 1]

(26)

This reflects the fact that, in the SL(2, c) regime, there are two different representations
of the E(2)-like little group for a given direction [22]. The 4 x 4 gauge transformation
matrix of (19) can be constructed from the direct product of the above 2 x 2 matrices
[16, 22].

Acknowledgment

We would like to thank Professor Eugene P Wigner for very illuminating discussions
on his little groups and table 1.

References

[I] Thomas L H 1927 Phil. Mag. 3 I

[2] Moller C 1952 The Theory of Relativity (Oxford: Oxford University Press)
[3] Corben H C and Stehle P 1960 Classical Mechanics (New York: Wiley) 2nd edn
[4] Taylor E F and Wheeler J A 1966 Spacetime Physics (San Francisco: Freeman)
[5] Jackson J D 1975 Classical Electrodynamics (New York: Wiley) 2nd edn
[6] Goldstein H 1980 Classical Mechanics (Reading, MA: Addison-Wesley) 2nd edn
[7] Hestenes D 1974 J. Math. Phys. 15 1768
[8] Wigner E P 1939 Ann. Math. 40 149
[9] Wigner E P 1957 Rev. Mod. Phys. 29 255
[10] Weinberg S 1964 Phys. Rev. 13S BI049
VI] Kupersztych J 1976 Nuovo Cimento B 31 I
[12] Han D and Kim Y S 1981 Am. J. Phys. 49 348
[13] Inonu E and Wigner E P 1953 Proc. Natl. Acad. Sci. USA 39 510
[14] Talman J D 1968 Special Functions, A Group Theoretical Approach Based on Lectures by E P Wigner
(New York: Benjamin)
[15] Han D, Kim Y S and Son D 1983 Phys. Lett. BIB 327
496 CHAPTER IX

Thomas effect and gauge transformations 1783

[16] Han D, Kim Y S and Son D 1986 1. Math. Phys. 27 2228

[17] Chakrabarti A 1964 J. Math. Phys.5 1747
[18] Hestenes D 1966 Space-time Algebra (New York: Gordon and Breach)
[19] Ben-Menahim A 1985 Am. J. Phys. 53 62
[20] Salingaros N 1986 J. Math. Phys.27 157
[21] Chakrabarti A 1987 Preprint Centre de Physique Theorique de l'Ecole Polytechnique, Palaiseau,
A.767.0187
[22] Kim Y Sand Noz M E 1986 Theory and Applications of the Poincare Group (Dordrecht: Reidel)
LORENTZ TRANSFORMATIONS 497

PHYSICAL REVIEW A VOLUME 37, NUMBER 3 FEBRUARY I, 1988

Linear canonical transformations of coherent and squeezed states in the Wigner phase space

D. Han
/Vational Aeronautics and Space Administration, Goddard Space Fligh"t Center (Code 636), Greenbelt, Maryland 20771

Y. S. Kim
Department of Physics and Astronomy, Unh'ersity of lvlary/and, College Park, Maryland 20742

Marilyn E. Noz
Department of Radiology, New York University. New York, l\/cw York ]0016
(Received 8 September 19871
It i, ~h()wn that classical linear canonical transformations are possible in the Wigner phase
. . pace. Coherent and squeezed states arc shown to be linear canonical transforms of the ground-
state harmonic oscillator. It is therefore possihlc to evaluate the Wigner functions for coherent
and squeezed states from that for the harmonic oscillator. Since the group of linear canonical
transformations has a subgroup whose algebraic property is the same as that of the (2 + 0-
dimensional Lorentz group, it may be possible to test certain properties of [he Lorentz group using
optical devices. A possible experiment to measure the Wigner rotation angle is discussed.

I. INTRODUCTION The basic advantage of the Wigner function is that

these operators commute with each other in phase space.
Coherent and squeezed states now form the basic In this paper we study coherent and squeezed states in
language for quantum optics.' -.1 They preserve the the Wigner phase space. We shall show that these states
minimum-uncertainty product in the phase space con- are canonically transformed states of the ground-state
sisting of phase and intensity. The Wigner phase space, harmonic oscillator. A subset of these transformations
which was initially formulated in 1932,),4 is also becom- form a group whose algebraic properties are identical to
ing the standard scientific language in many branches of that of the (2 + I I-dimensional Lorentz group. It may
physics, including quantum optics. '.6 It is therefore of therefore be possible to design an optical experiment to
interest to formulate the coherent and squeezed states test the properties of the Lorentz group.
within the framework of the Wigner phase-space repre- In Sec. II we briefly review the linear canonical trans-
sentatIon. formations in classical mechanics. In Sec. III we discuss
The Wigner distribution function for the coherent the canonical transformations of the Wigner distribution
states has been discussed in the literature 7 The Wigner function in phase space. The canonical transformation
function for the squeezed states has also been studied re- of the Wigner function is much simpler than the conven-
cently by Schleich and Wheeler for the deformation tional Weyl transformation applicable to the
along the "x" or "p" axis caused by real or purely imagi- Schri)dinger picture. In Sec. IV the Wigner phase-space
nary parameters.) However, the deformation in phase formalism is discussed in detail for the harmonic oscilla-
space of squeezed states with complex parameters has tors.
not been systematically studied. In Sec. V we discuss coherent and squeezed states in
In this paper we shall study the squeezed states with terms of canonical transformations in phase space. It is
complex parameters. It will be shown that for a com- possible from this formalism to determine the Wigner
plex value of the squeeze parameter, the deformation is function for the squeezed state with a complex parame-
along the direction of the phase angle of the squeeze pa- ter. It is noted in Sec. VI that the algebra of squeezed
rameter. We shall achieve this purpose not by perform- and coherent states is the same as that for the (2 + 1)-
ing a direct calculation but by studying transformation dimensional Lorentz group. This enables us to discuss a
properties in phase space. possible experiment to measure the Wigner rotation an-
Classical mechanics can be effectively formulated in gle using optical devices.
terms of the Poisson brackets and canonical transforma-
tions. 8 Although the Poisson brackets become II. LINEAR CANONICAL TRANSFORMATIONS
Heisenberg'S uncertainty relations in quantum mechan- IN CLASSICAL MECHANICS
ics, it is cumbersome to use canonical transformations in
quantum mechanics because the translation operators in The group of linear canonical transformations consists
phase space, which are x and p, do not commute with of translations, rotations, and squeezes in phase
each other. 9,10 space 8 • to, 1t These operations preserve the area element

37 807 © 1988 The American Physical Society

Reprinted from Phys. Rev. A 37, 807 (1988).

498 CHAPTER IX

808 D. HAN, Y. S. KIM, AND MARILYN E. NOZ

in phase space. We present in this section a short for- (8)

malism which will be useful for studying coherent and
squeezed states in quantum optics. The squeeze matrix can be written as
In order to define the word "squeeze" in phase space,
Sx =e -illB I (9)
let us consider a circle around the origin in the coordi-
nate system of x and p. If we elongate the x axis by where
multiplying it by a real number greater than I and con-
tract the p axis by dividing it by the same real number, o
the circle becomes an ellipse. The area of the ellipse -i/2 (10)
remains the same as that of the circle. This is precisely o
an act of squeeze. If we combine this operation with ro-
tation around the origin, the squeezing can be done in In addition, if we introduce the matrix B 2 defined as
every possible direction in phase space.
The coordinate transformation representing transla-
0 i/2 0 J
tions, B 2 = [i/2 0 0 , (II)
x'=x+u, p'=p+v , (I) o 0 I
can be written as which generates tbe squeeze along the direction which
makes 45' with the x axis, then the matrices L, B I' and
(2)
B 2 satisfy the following commutation relations:

The matrix performing the rotation around the origin by

(J/2takes the form (12)

(J (J This set of commutation relations is identical to that for

cos"2 -sin"2 0 the generators of the (2 + I)-dimensional Lorentz
group.12 The group generated by the above three opera-
(J (J
R ((J)= sin "2 0 (3) tors is known also as the symplectic group Sp(2),13.14 and
cos "2
its connection with the Lorentz group has been exten-
0 0 sively discussed in the Iiterature. ls
If we take into account the translation operators, the
The matrix which squeezes along the x axis is commutation relations become

S.(7/)=
e V/2 0 0]
[ 0 e- v/2 0 . (4)
[B I ,Nd=(i/2)NI' [B I ,N2]=(-i/2)N 2 ,

o 0 I [B 2,Nd=(i/2)N2, [B 2,N2]=(i/2)NI'
(13)
[N I ,L]=(i/2)N2' [N I ,L]=(-i/2)N I ,
The elongation along the x axis is necessarily the con-
traction along the p axis. [N I ,N2 ]=0.
Since a canonical transformation followed by another
one is a canonical transformation, the most general form These commutators, together with those of Eq. (12),
of the transformation matrix is a product of the above form the set of closed commutation relations (or Lie
three forms of matrices. We can simplify these algebra) of the group of canonical transformations. This
mathematics by using the generators of the transforma- group is the inhomogeneous symplectic group in the
tion matrices. If we use T(u,v) for the translation ma- two-dimensional space or ISp(2).1I
trix given in Eq. (2), it can be written as
The translations form an Abelian subgroup generated
(5)
by NI and N 2 • Since their commutation relations with
all the generators result in N I' N 2' or 0, the translation
subgroup is an invariant subgroup. The translations and
the rotation form the two-dimensional Euclidean group
oOil
generated by N I ,N2 , and L, which have closed commu-
0 , N 2 = [0
0 00 01
i . (6)
tation relations. This group also has been extensively
o0 0 0 0 discussed recently in connection with the internal space-
time symmetries of massless particles. 16. 17
The rotation matrix is generated by
Indeed, it is of interest to see how the representations

~ j,
-i/2 of the Lorentz group can be useful in optical sciences. It
o (7) is also of interest to see how the experimental resources
o in optical science can be helpful in understanding some
of the "abstract" mathematical identities in group
and theory.
LORENTZ TRANSFORMATIONS 499

LINEAR CANONICAL TRANSFORMATIONS OF COHERENT AND ... 809

Ill. LINEAR CANONICAL TRANSFORMATIONS -

N1=-i ax '
a N,=x
IN THE WIGNER PHASE SPACE

[=±l[a~r-x'l
If ",Ix) is a solution of the Schriidinger equation, the
Wigner distribution function in phase space is defined as (18)

Wlx,p )=(1 /7T) f "'-Ix +y )tblx -y)e';PYdy. (14)

This is a function of x and p which are c numbers. This BI=-i[fla~' B'=±[X'+[a~n

function is real but is not necessarily positive everywhere
in phase space. The properties of this function have These operators satisfy the commutation relations given
been extensively discussed in the literature.4~ 7 in Eqs. (12) and (13), except the last one. The operators
When we make linear canonical transformations of NI and N, do not commute with each other, and
this function in phase space, the infinitesimal generators (19)
are
Therefore, it appears that the operators applicable to the
N .
1 =-1
a N,=-i~
ax' ap Schriidinger wave function do not satisfy the same set of
commutation relations as that for classical phase
space. 9• ID
L=-t.[ xap
a -Paxa J (\5)
Let us consider the translation along the x axis fol-
lowed by the translation along the p axis, and the opera-
tion in the opposite order. From the Baker-Campbell-
Hausdorff formula for two operators,"10
(e ~;uNI )(e,;,N')=(e;")(e ~;"N2)(e ~;uNI) • (20)

These operators satisfy the commutation relations given The interchange of the above two translations results in
in Eqs. (12) and (13). We can therefore derive the alge- a mUltiplication of the wave function by a constant fac-
braic relations involving the above differential forms us- tor of unit modulus.
ing the matrix representation discussed in Sec. II. However, this factor disappears when the Wigner
The rotation of the translation operators takes the function W is constructed according to the definition of
form Eq. (14). Therefore, the translation along the x direction
and the translation along the p direction commute with

RWIN1R( -/1)= [cos~ 1NI-lsin~ 1N , , each other in the Wigner phase space. This means that
the commutation relation [N 1,N,1=O in the Wigner
(16) phase space and the Heisenberg relation [N I' N ,1 = - i

RI/lI\',RI /11= [Sill i INI + [cos~ IN, are perfectly consistent with each other. The basic ad-
vantage of the Wigner phase-space representation is that
its canonical transformation property is the same as that
Under the same rotation, the squeeze generators become of classical mechanics.
We now have three sets of operators. The first set
RW)B1R I -/l)=(cos/l)B 1+ (sin/l)B, ' consists of the three·by-three matrices in Eqs. (6), (7),
(17) (10), and (III, and this set is for classical mechanics.
R 1!JIB,R I -!J)= -lsin!J)B 1+lcos8)B, The differential operators in two-dimensional phase
space form the second set, and they are for the Wigner
Likewise, we can derive all the algebraic relations using function. The third set consists of the differential opera-
matrix algebra. The important point is that the group of tors of Eq. (18) applicable to the Schriidinger wave func-
canonical transformations in the Wigner phase space is tion. The first and second sets are the same. While both
identical to that for classical mechanics. the second set of double-variable operators and the third
Next, let us consider the above transformations in set of single-variable operators are extensively used in
terms of operators applicable to the Schriidinger wave the literature,II,14,18 it is interesting to see that the con-
function. From the expression of Eq. (14) it is quite nection between these two sets can be established
clear that the operation e ~h>x on the wave function leads through the Wigner function.
to a translation along the P axis by u. The operation of The transformations discussed in this section consti-
exp[ -u(a/ax)1 on the wave function leads to a transla- tute the basic language for coherent and squeezed states
tion of the above distribution function along the x axis in quantum optics. The relevance of the translation in
by u. phase space to coherent states has been noted before,l
Likewise, the operation in the Wigner phase space of The word squeeze comes from quantum optics. It has
ix(a/ap) and ip(a/ax) become x' /2 and f(a/ax )', re- been also noted that its mathematics is like that of
spectively. Thus, the transformations in phase space can (2 + II-dimensional Lorentz transformations. As was
be generated from the operators applicable to the wave emphasized in the literature, I', 17 combining translations
function. The generators applicable to the wave func- with Lorentz transformations is not a trivial problem.
tion are We shall discuss the problem in Secs. V and VI.
500 CHAPTER IX

810 D. HAN, Y. S. KIM, AND MARILYN E. NOZ

IV. HARMONIC OSCILLATORS

The one-dimensional harmonic oscillator occupies a

unique place in the physics of phase space. For the
Hamiltonian of the form
(21)

the Wigner function is a function only of (p2+x2), and

is thus invariant under rotations around the origin in
phase space. The Wigner function for the ground-state
harmonic oscillator is" to This transformation is illustrated in Fig. I. As we shall
see in Sec. Y, the translated and deformed Wigner func-
(22) tions will he useful for studying coherent and squeezed
states, respectively.
This function is localized within the circular region In the meantime, let us observe other useful properties
whose boundary is defined by the equation of the harmonic oscillator. We noted above that, in or-
der to study the harmonic oscillator, we can start with a
x 2+p2=1. (23) circle in phase space. How does this rotational invari-
Therefore, the study of the Wigner function for the har- ance manifest itself in the Schrodinger picture? The
generator of rotations is

f-
monic oscillator is the same as the study of a circle on

L=± [[ :x
the two-dimensional plane. The canonical transforma-
tion consists of rotations, translations, and area-
preserving elliptic deformations of this circle. These
x2 j=+(-m. (29)

transformations are straightforward. If the wave function is a solution of the time-

Under the translation by r along the x axis, the above independent Schriidinger equation with the above Ham-
circle becomes iltonian, the application of the rotation operator
exp( -iOL) will only generate a constant factor of unit
(x_r)2+ p 2=1 (24) modulus. This is the reason why the Wigner function
for the above Hamiltonian system is invariant under ro-
This circle is centered around the point (r,O). We can tations in phase space.
rotate the above circle around the origin. Then the re- In order to study rotations more carefully in the
sulting Wigner function is Schr6dinger picture, let us use a and a t, defined in this

r
case as

R (O)T(r,O)W(x,p)= ;exp { [ [x -r cos%

a =(1/11'2) [x+ a: 1,
[x- a: j.
(30)

a t =(I/1I'2)

(25)
p
where T(r,O) and R(8) are the translation and rotation
operators. Because the circle of Eq. (23) is invariant un-
der rotations around the point where x =r and p =0, the
above Wigner function is the same as the translated
Wigner function,

T [r cos%,r sin% 1W(x,p )=R(O)T(r,O)W(x,p). (26)

re7j/2
Let us next elongate the translated circle of Eq. (24)
along the x direction. The circle will be deformed into
e -V(x _r')2+e vp2= 1 , (27) FIG. 1. Coherent and squeezed states in the Wigner phase
space. The circle centered around the origin describes the
where
ground-state harmonic oscillator. The circle around (r,O) is for
r'=re Tl !2 . the coherent state. This coherent state can be squeezed to el-
lipse along the x axis, with a real value of the squeeze parame-
If we rotate this ellipse, the resulting Wigner function ter. When the squeeze parameter becomes complex then the
will be ellipse is rotated around the origin in tbe Wigner pbase space.
LORENTZ TRANSFORMATIONS 501

LINEAR CANONICAL TRANSFORMATIONS OF COHERENT AND ... 811

These operators serve two distinct purposes in physics. We can obtain this state by applying the translation
They are step-up and step-down operators for the one- operator to the ground state,
dimensional harmonic oscillator in nonrelativistic quan-
tum mechanics. (38)
On the other hand, in quantum-field theory, they serve
as the annihilation and creation operators. We are here where
interested in the creation and annihilation of photons.
Then, what is the physics of the phase space spanned by T(a)=exp(aa'-a*a) .
x and p variables? Indeed, the concept of creation and
The translation operator in the phase space depends on
annihilation comes from the commutation relation
two real parameters. In the above case, the parameter a
(31) is a complex number containing two real parameters.
It is possible to evaluate the Wigner function from the
This form of uncertainty relation states also that the above expression to obtain the form given in Eq. (25),7,9
area element in phase space cannot be smaller than with
Planck's constant. The area element in the Cartesian
coordinate system is (~x)( ~p). It is also possible to
write the area element in the polar coordinate system. If
this area is described in the polar-coordinate system, the It is also possible to obtain the Wigner function starting
uncertainty relation is the relation between phase and in- from a real value of a by rotation. From the rotation
tensity.20 This is the uncertainty relation we are discuss- properties of the a and a t operators given in Sec. IV, the
ing in this paper. We are particularly interested in the rotation of this operator becomes
minimum-uncertainty states.
In both Eq. (25) and Eq. (28) the rotation plays the (40)
essential role. Let us see how the operators a and a' can
be rotated. For two operators A and B, we note the re- with
lation 21
e ABe - A=B +[ A,B ]+H A,[A,Bll This means that we can make a complex starting from a
real number r by rotation.
+HA,[A,[A,Blll+" . (32)
The squeezed state ! s,a) is defined to be , ,3,S,IS,22
and
!s,a)=S(s)!a)=S(s)T(a)!O) , (41)
[L,a]=-ta, [L,a']=ta'. (33)
where
Since R((J)=e- iOL ,
R(lllaRI !lIcit iO"la, S(s)=exp [fa'a'-f aa 1 (42)
(34)
R(IJla RI·-OJ=(e,",2)a'.
Here again the parameter S is complex and contains two
In terms of the a and a' operators, the generators of real numbers for specifying the direction and the
canonical transformations take the form strength of the squeeze.
If 5 is real, it is possible to evaluate the Wigner func-
N,=(-;/v2)(a-a'l, N 2 =(\/Vl)(a+a t J, tion by direct evaluation of the integral. If, on the other
L =t(aa' +a 'a J, (35) hand,s is complex, the present authors were not able to
manage the calculation, We can, however, overcome
B,=t(aa-ata\ B,=t(aa+ata'). this difficulty by using the method of canonical transfor-
mation developed in this paper, We can make S com-
We can rotate these operators using Eq. (34). In particu- plex starting from a real value of '1/ by rotating the above
lar, the rotations given in Eq. (17) can now be written as squeeze operator using the rotation properties of the a
R(IJ)aaR(-IJ)=e-iOaa, and a t operators.
(36) Let us start from a real value of S for which the evalu-
ation is possible. S For the real value '1/, the squeeze
operator becomes
These relations will be useful in evaluating the Wigner
function for the squeezed state.

V. COHERENT STATES AND SQUEEZED STATES

S('I/)=exp [-t [x! 11 ' (43)

This operator makes the scale change of x to (e -,I2)x.

In terms of the a and a t operators, the coherent state It is therefore possible to visualize the deformation of
is defined as the circle into an ellipse in the phase space, Let us next
!a)=[exp(-!a!'/2)] i (an/n!)(at)n!O). (37)
rotate this ellipse. From Eqs. (36) and (42),
n=O (44)
502 CHAPTER IX

812 D. HAN, Y. S. KIM, AND MARILYN E. NOZ

where TABLE I. How to evaluate the Wigner function for

coherent and squeezed states.
;=(e ~i6)'1/ .
Coherent Squeezed
The operator S(;), when applied to the wave function, states states
leads to the Wigner function which is elongated along
the 012 direction in the phase space. It is indeed possi- Direct computation Possible Not known
ble to evaluate the Wigner function for the squeezed Canonical transformation Possible Possible
state with a complex value of; simply by rotating the el-
lipse elongated along the x direction.
Table I describes how we can determine the Wigner This effect exhibits itself in the Thomas effect in atomic
functions for coherent and squeezed states. Figure I il- physics."
lustrates how the above calculation can be carried out. Since the mathematics of squeeze is the same as that
The translated circle in phase space describes the of Lorentz boost, we can discuss the possibility of
coherent state. This circle can be elongated along the x measuring the effect of the Wigner rotation in optical ex-
direction. The resulting ellipse is for the squeezed state periments. In order to illustrate how the Wigner rota-
with a real parameter. This ellipse can be rotated. This tion comes into this subject, let us start with a circle of
rotated ellipse corresponds to the squeezed state with a unit radius centered around the origin in the Cartesian-
complex parameter. coordinate system with the coordinate variables x and p,
whose equation is given in Eq. (23). If we squeeze this
VI. POSSIBLE MEASUREMENT circle by elongating along the x axis, the squeeze matrix
OF THE WIGNER ROTATION applicable. to the vector (x,p) is

We have noted in Sec. II that the transformation

group contains the subgroup Sp(2) which is locally iso-
morphic to the (2 + l}-dimensional Lorentz group. It
S(O,A)= l
e"12 0
0 e~"12
j (45)

may therefore be possible to design experiments in optics This will deform the circle into the ellipse
to test the mathematical identities in the Lorentz group. (e~")x2+(e")p2=1 . (46)
The Wigner rotation is a case in point. Two successive
applications of Lorentz boosts in different directions is If we squeeze the circle centered around the origin
not a Lorentz boost, but is a boost preceded by a rota- along the 012 direction with the deformation parameter
tion which is commonly called the Wigner rotationy~26 '1/, the squeeze matrix is

S((},A)=
cosh + +]
+ [sinh cosO
(47)

[sinh+ ]sino A-
cosh "2 [.smh"2A] cosO

and the circle is deformed into the ellipse

r
tana= (sinO)[sinhA+(tanh'1/)(coshA-1 )cosO]

e~' [x cos~ +p sin~ r +e' [x sin~ -p cos~ =I .

(sinhA)cosO+(tanh'1/)[ I +(coshA-1 )(cOSO)2]

This is an ellipse elongated along the al2 direction with

(48)
the parameter S.
In order to understand the squeeze mechanism The above calculation gives an indication that two
thoroughly, we should know how to squeeze an ellipse. successive squeezes become one squeeze. This is not
We can achieve this goal by studying two successive true. The product of the matrices S((},A)S(O,'1/) does
squeezing properties. Let us therefore consider the not result in S(a,;). Instead, it becomesI6.17,2J~25
squeeze S((},A) of the circle centered around the origin
preceded by S(O,'1/)' This will result in another ellipse, (50)

e~'[xcos-}+psin-} r+e'[xsin-}-pcos-} f=l. where

(49) (sinO)[tanh(A/2)][ tanh( '1//2)]

where I + [tanh( Al2)][ tanh( '1/ /2)]( cosO)

The right-hand side of the above equation is a squeeze

and preceded by a rotation, which may be called the Wigner
LORENTZ TRANSFORMATIONS 503

37 LINEAR CANONICAL TRANSFORMAnONS OF COHERENT AND ... 813

rotationY-26 Although Eq. (491 does not show the p

effect of this rotation which leaves the initial circle cen-
tered around the origin invariant, we need the derivation
of Eq. (491 in order to determine a, g, and eventually ¢.
The study of coherent states representations requires
transformations of a circle not centered around the ori-
gin. If we squeeze this circle by applying S(O, '1 I, the
circle is transformed into the ellipse given in Eq. (271. If
we squeeze this ellipse by applying S(8,J...I, the net effect
is the squeeze S(a,gl preceded by the Wigner rotation
R (¢ I. If we apply this rotation to the circle of Eq. (241,

(51)

FIG. 2. Two repeated squeezes resulting in one squeeze pre-

The effect of this rotation is illustrated in Fig. 2.
ceded by one rotation. The circle around (r,OI in Fig. I is ro-
Next, if we apply the squeeze S(a,gl to the above cir-
tated around the origin by "'/2 and is then elongated along the
cle, the resulting ellipse is

r
a!2 direction.

e- S [(x-alcosT+(y-bISinT

+e s [(X -a IsinT -(y -b )cOST r = I (52)

where

The effect of this squeeze is also illustrated in Fig. 2.

The Wigner rotation angle ¢ can now be determined
from a,b, which can be measured. In terms of these pa-
rameters,

'~i ~ I~ ++ i';'hf Hj-b-+.hf I"~

a [coshf - [sinhf ]cosa [sinhf ]sina
(53)

The parameters g and a can be measured or determined work indicates that some of optical experiments may
from Eq. (491. The angle ¢ determined from the above serve as analog computers for the (2 + I }-dimensional
expression can be compared with the angle calculated Lorentz group.
from '1, J..., and a according to the expression given in Eq.
(50).
Indeed, if the parameters of the coherent and squeezed
VII. CONCLUDING REMARKS
states can be determined experimentally, the Wigner ro-
tation can be measured in optical laboratories. The
question is then whether this experiment can be carried It is quite clear from this paper that the coherent and
out with the techniques available at the present time. squeezed states can be described by circles and ellipses in
While the analysis presented in this section is based on the Wigner phase space. One circle or ellipse can be
single-mode squeezed states, the squeezed states that transformed into another by area-preserving transforma-
have been generated to date are two-mode states. 3.\8 tions. The group governing these transformations is the
Hence, in order to be directly applicable to experiment, inhomogeneous symplectic group ISp(21.
the present work has to be extended to the two-mode We studied the generators of these transformations
case, unless the single-mode squeezed state can be gen- both for phase space and for the Schrodinger representa-
erated in the near future. In the meantime, the present tion. It has been shown that the connection between
504 CHAPTER IX

814 D. HAN, Y. S. KIM, AND MARILYN E. NOZ 37

these two sets of operators can be established through be possible to measure the Wigner rotation angle in opti-
the Wigner function. cal laboratories.
We also studied in detail rotations in the Wigner
phase space and their counterparts in the Schriidinger ACKNOWLEDGMENTS
representation. It is now possible to evaluate the Wigner
function for a squeezed state with a complex parameter. We are grateful to Professor Eugene P. Wigner for
The correspondence (local isomorphism) between Sp(2) very helpful discussions on the subject of canonical
and the 12 + I)-dimensional Lorentz group allows us to transformations in quantum mechanics and for main-
study quantum optics using the established language of taining his interest in the present work. We would like
the Lorentz group. At the same time it allows us to to thank Mr. Seng-Tiong Ho and Dr. Yanhua Shi for ex-
look into possible experiments in optical science to study plaining to us the experimental techniques available at
some of mathematical formulas in group theory. It may the present time.

I), R, Klauder, Ann. Phys. IN.Y.) II, 123 11960); R. J. Space, Col/ege Park, Maryland, 1986, edited by y, S. Kim
Glauber, Phys. Rev. Lett, 10, 84 11963); F, T. Arechi, ibid. and W, W. Zachary ISpringer-Verlag, Heidelberg, 1987),
IS, 9\2 11965); E. Goldin, Waves and Photons IWiley, New 7p. Carruthers and F. Zachariasen, Rev. Mod. Phys. 55, 245
York, 1982); J. R. Klauder and B. S. Skagerstam, Coherent (19831.
States (World Scientific, Singapore, 1985). BH. Goldstein, Classical Mechanics, 2nd ed. (Addison·Wesley,
'D, Stoler, Phys. Rev, D 1, 3217 II 970l; H. p, Yuen, Phys. Reading, MA, 1980),
Rev, A 13, 2226 11976), 9H. Weyl, The Theory of Groups and Quantum Mechanics, 2nd
3For some of the recent papers on the squeezed state, see C. M. ed. (Dover, New York, 1950),
Caves, Phys. Rev. D 23, 1693 1198]); D. F. Walls, Nature 10K. B. Wolf, Kinam 6, 141 (19861.
ILondon) 306, 141 (1983); R, S. Bondourant and J. H. llA. Perelomov, Generalized Coherent States (Springer-Verlag,
Shapiro, Phys, Rev. D 30, 2584 II 984); M. D. Reid and D. F. Heidelberg, 19861.
Walls, Phys. Rev. A 31, 1622 (1985); B. Yurke, ibid. 32, 300 12E, p, Wigner, Ann. Math. 40, 149119391; V. Bargmann, ibid.
(1985); 32, 311 119851; R, E. Slusher, L. W. Hollberg, B. 48, 568 119471; V, Bargmann and E, p, Wigner, Proc, Nat.
Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev, Lett. 55, Acad, Sci. U,S,A, 34, 211 119481.
2409119851; C. M. Caves and B. L. Schumaker, Phys, Rev. 11H. Weyl, Classical Groups, 2nd ed, (Princeton University,
A 31, 3068 119851; B. L. Schumaker and C, M, Caves, ibid. Princeton, N.J., 1946).
31, 3093 119851; M. J, Collet and D, F, Walls, ibid, 32, 2887 !4V. Guillemin and S. Sternberg, Symplectic Techniques in
11985); J, R. Klauder, S. L. McCall, and B. Yurke, ibid. 33, Physics ICambridge University Press, Cambridge, England,
3204 119861; S, T, Ho, P. Kumar, J, H, Shapiro, ibid. 34, 293 1984),
119861; 36, 3982 119871; Z. Y. Ou, C. K. Hong, and L. Man- lly. S, Kim and M, E, Noz, Am, J, Phys. 51, 368 119831.
del, ibid, 36, 192 119871. 16 0. Han, O. Son, and Y. S. Kim, Phys. Rev. 0 26, 3717
4E. p, Wigner, Phys. Rev. 40, 749 11932), For review articles 11982),
on this subject, see E. P. Wigner, in Perspective in Quantum l7y. S. Kim and M. E. Noz, Theory and Applications of the
Theory, edited by W. Yourgrau and A. van def Merwe (MIT Poincare Group (Reidel, Dordrecht, 1986l.
Press, Cambridge, MA, 1971); M. Hillery, R, F, O'Connell, 18B. Yurke, S. McCall, and J. R. Klauder, Phys. Rev, A 33,
M. O. Scully, and E, P. Wigner, Phys. Rep,· 106, 121 119841; 4033 (19861.
N. L. Balazs and B, K. Jennings, ibid. 104C, 347 (1984), 19S. Sh10mo, J, Phys, A 16, 3463 (1983).
5W. Schleich and J. A. Wheeler, in Proceedings of the First In· 'oW. Heit1er, Quantum Theory of Radiation, 3rd ed, (Oxford
ternational Conference all the Physics of Phase Space, College University Press, London, 1954),
Park, Maryland 1986, edited by y, S. Kim and W. W. Za- 21W, Miller, Symmetry Groups and Their Applications
chary ISpringer-Verlag, Heidelberg, 19871; Nature ILondon) (Academic, New York, 1972); D. R, Truax, Phys, Rev, D 31,
326,57411987). 1988119851.
6W. H. Louisell, Quantum Statistical Properties of Radiation "This definition is equivalent to !g,a) = T(a')S(~) 10), with a
IWiley, :-lew York, 19731; B. V, K. Vijaya Kumar and C. W. suitably adjusted value of a, since the translation subgroup is
Carroll, Opt. Eng, 23, 732 11984); R, L. Easton, A. J, Tick- an invariant subgroup which enables us to write and
nor, and H. H. Barrett, ibid. 23, 738 (19841; R. Procida and ST=STS- 1S with ST- 1 =T', which is another element in
H. W. Lee, Opt. Commun, 49, 201 11984); N. S_ Subotic and the translation subgroup.
B. E. A. Saleh, ibid, 52, 259 119841; A. Conner and Y. Li, 2lV, I. Ritus, Zh, Eksp, Teor, Fiz, 40, 352 11961) [Sov. Phys,-
App!. Opt. 24, 3825 (1985); S. W, McDonald and A, N. JETP 13, 240 (19611],
Kaufman, Phys, Rev, A 32, 1708 119851; O. T. Serima, ), '4A. Chakrabarti, J. Math. Phys, 5,174711964),
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Nuc!' Instrum, A 246, 71 119861; H. Szu, in Proceedings of 1777 11987).
the First International Conference on the Physics of Phase 26A, Ben-Menahim, Am. ). Phys. 53, 62 11985),
 Of Related Interest

Theory and
Applications of the
Poincare Group
by

Y. S. Kim
Department of Physics and Astronomy, University of Maryland, U.S.A.

and

Mari Iyn E. Noz

Department of Radiology, New York University, U.S.A.

Special relativity and quantum mechanics are likely to remain the two
most important languages in physics for many years to come. The
underlying language for both disciplines is group theory. Eugene P.
Wigner's 1939 paper on the Poincare group laid the foundation for
unifying the concepts and algorithms of quantum mechanics and
special relativity. This book systematically presents physical examples
which can best be explained in terms of Wigner's representation
theory. The examples include the relativistic quark model, hadronic
mass spectra, the Lorentz-Dirac deformation of hadrons, the form
factors of nucleons, Feynman's parton picture and the proton structure
function, the kinematical origin of the gauge degrees of freedom for
massless particles, the polarization of neutrinos as a consequence of
the gauge invariance, and massless particles as the (small-
mass/large-momentum) limits of massive particles.
This book is intended mainly as a teaching tool directed toward those
who desire a deeper understanding of group theory in terms of
examples applicable to the physical world and/or of the physical world
in terms of the symmetry properties which can best be formulated in
terms of group theory. Each chapter contains problems and solutions,
and this makes it potentially useful as a textbook. Graduate students
and researchers interested in space-time symmetries of relativistic
particles will find the book of interest.

ISBN 90-277-2141-6 FTP 17