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Eugene Paul Wigner
1902–1995
A. S. Wightman
Eugene Wigner died in Princeton, NJ, on Janu- as a physicist, assistant to the physicist Richard
ary 1, 1995, at the age of 92. He had been one Becker.
of the last survivors of the generation that wit- During the next decade and a half Wigner
nessed the creation of quantum mechanics and continued his study of the theory of chemical re-
participated in the exciting initial years of its de- actions but used the then new quantum me-
velopment. He spent most of six active decades chanics. He did related work with Victor Weis-
on the faculty of Princeton University. Although skopf on the theory of line breadth in atomic
he was best known for his physical and mathe- spectra as well as a study of nuclear reaction
matical analyses of symmetry in quantum me- rates with Gregory Breit. However, the main
chanics, he also made important contributions focus of his effort was in the application of
to solid-state physics, physical chemistry, nuclear group theory to the study of the symmetry prop-
engineering, and epistemology. In his later years, erties of stationary states of atoms, molecules,
he found himself in the unusual position of atomic nuclei, and crystals.
being highly esteemed by physicists, mathe- It was also during this period that Wigner
maticians, chemists, engineers, and philoso- made a transition from Germany to the United
phers. States. From 1930 to 1933 Wigner and von Neu-
Eugene Wigner was born Jenö Pál Wigner in mann had a common arrangement: they spent
Budapest, Hungary, on November 17, 1902. Since one term each year at their jobs in Berlin and one
he was a somewhat sickly child, his parents at Princeton University. In the spring of 1933 the
arranged for his early education to occur at National Socialists came to power in Germany,
home. However, later on he spent four years at and the Berlin positions of von Neumann and
the famous Lutheran gymnasium (high school) Wigner vanished. Von Neumann then joined the
of Budapest, where he had the good fortune to faculty of the new Institute for Advanced Study.
have as friend and classmate (one class behind Wigner spent three years full-time in Princeton
him) Jancsi (=Johann=John) von Neumann. and then went to Wisconsin for two years. In the
Wigner was attracted by mathematics and fall of 1938 he was back in Princeton in an en-
physics, but, following his father’s wish that he dowed professorship, just in time to hear the
study something that could be useful in the news of the discovery of nuclear fission, a phe-
leather tannery where his father was a foreman, nomenon whose consequences dominated the
Wigner got a degree in chemical engineering next decade of his life.
from the Technische Hochschule in Berlin. His Wigner and his friend Leo Szilard foresaw as
thesis (1925), written under the supervision of clearly as anyone the disastrous consequences
Michael Polanyi, was on the theory of chemical of the Third Reich’s acquiring nuclear weapons
reactions. Wigner’s acumen so impressed Polanyi before the Allies. They persuaded Albert Einstein
that he recommended him for his first position to write a letter alerting President Roosevelt.
The result was the Manhattan Project, a large-
A. S. Wightman is professor emeritus of mathematical scale effort to separate U 235 from U 238 in ura-
physics at Princeton University, Princeton, NJ. nium ore and to create nuclear reactors to pro-
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duce plutonium as well as to design bombs However, he found the case of n ≥ 4 electrons
which used these products as explosives. Wigner too complicated to do by hand. On the advice of
was heavily involved in design studies for the nu- von Neumann, he studied the pre-World War I
clear reactors. Most of the work took place in the papers of Frobenius, Schur, and Burnside on the
mathematics department of the University of representation theory of finite groups, as well
Chicago, code-named the Metallurgical Labora- as the later papers of Weyl and of Schur on con-
tory. It was here that Wigner acquired his repu- tinuous groups. The latter enabled him to enlarge
tation as a formidable nuts-and-bolts engineer. his study to the consideration of the action on
When his design work for the plutonium pro- eigenfunctions, of rotations R of the coordi-
duction reactors was done, Wigner turned to nates of n electrons ~ x1 , ..., ~
xn :
the design of power reactors. This continued
~ xn → R~
x1 , ...~ x1 , ...R~
xn .
after the war; he spent 1946–47 as director of
research at Oak Ridge on leave from Princeton. He recognized that if the Hamiltonian commutes
Wigner and his coworker Alvin Weinberg col- with the action on wave functions of permuta-
lected their knowledge and tions of coordinates or with
experience in the definitive the action on wave func-
treatise, The Physical The- tions of rotations of coor-
ory of Neutron Chain Re- dinates, then the linear
actors (1958). subspace spanned by the
He returned to academic Although he was eigenfunctions of a fixed
life in 1947 but, over the
next three decades, often
best known for his eigenvalue is left invariant
by these actions and, in the
served as a consultant to physical and subspace, yields a unitary
the federal government in representation of the per-
nuclear matters. Both his mathematical mutation and the rotation
scientific achievements and group. Such a representa-
his service were richly rec- analyses of tion is a direct sum of ir-
ognized with prizes and
awards, of which only a few
symmetry in reducibles, so the dimen-
sion of the linear space
will be mentioned: Medal quantum spanned by the eigenfunc-
for Merit, 1946; Fermi tions must be a sum with
Award, 1958; Max Planck mechanics, he also possible multiplicities of
Medal, 1961; Nobel Prize
for Physics, 1963; National
made important the dimensions of the irre-
ducible representations of
Medal of Science, 1969. contributions to the groups. This is the ele-
After his retirement in mentary group theoretical
1971, he remained active solid-state physics, explanation for the ubiq-
until the late 1980s. uitous appearance in
In the remainder of this physical chemistry, atomic physics of degen-
obituary I have singled out
for discussion a few points
nuclear erate multiplets of multi-
plicity 2j + 1 where j is a
from Wigner’s work on the engineering, and positive integer or half-odd
application of group the- integer. Later on, in the
ory to the study of sym- epistemology. context of nuclear physics,
metry in quantum me- this argument led to a the-
chanics. More details and ory of super-multiplets in
more complete coverage which the group SU(2) is
are to be found in Volume I replaced by the group
of Wigner’s Collected Works in the magistral an- SU(4).
notations of Brian Judd, Part II “Applied Group Up to this point, Wigner had treated the un-
Theory 1926–1935”, and George W. Mackey, Part physical case of spinless electrons. However, in
III “Mathematical Papers”. the very same issue of the Zeitschrift für Physik
Wigner began his study of symmetry in quan- in which he had published these considerations,
tum mechanics with the problem of classifying there appeared Pauli’s paper on nonrelativistic
the behavior of eigenfunctions of the Schrödinger electrons with spin 1/2. Within a few months von
Hamiltonians for atoms under permutations of Neumann and Wigner published the first of three
the electrons. Inspired by a paper of Heisenberg papers generalizing everything that Wigner had
on helium (two electrons), he first treated the done to the case of spin 1/2 Pauli electrons.
case of three electrons, without recourse to the These papers were not easy to read, and it seems
representation theory of the permutation group. plausible that they, together with Hermann
770 NOTICES OF THE AMS VOLUME 42, NUMBER 7
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Weyl’s classic book Gruppentheorie und Quan-
tenmechanik (1928) were the origin of the use
by physicists of the phrase “die Gruppenpest”
to describe this approach to spectroscopy. (The
German word is often translated “group pest”,
but the alternative “group plague” is probably
better, if you take into account some of the pas-
sionate animadversions on group theory by
physicists in those days.) In any case, it is in-
structive to compare Weyl’s book with Wigner’s
Gruppentheorie und Ihre Anwendung auf die
Quantenmechanik der Atomspektren (1931).
Both books have introductory chapters on
linear transformations, groups, and quantum
mechanics. Here Weyl puts more emphasis on
vector spaces; Wigner on calculations with ma-
trices. Wigner confines his attention to the per-
mutation group and rotation group or its cov-
ering group SU(2) . However, he goes into much
more detail on the representation theory of these
groups. For example, he derives the Wigner
Eckart formula for the matrix elements of ten-
sor operators. This permits him to derive the in-
tensity relations for spectral lines that follow
from rotation invariance. He also gives a quan-
titative and general analysis of the splitting of
spectral lines in the presence of external sym-
metry-breaking interactions. Weyl, on the other leaving a vector of mass 2 = m2 fixed. When
hand, discusses the Lorentz group, its covering m2 > 0 , the little group is isomorphic to SO(3)
group SL(2, C), and the relation of their finite- or SU(2) and so is determined by a positive in-
dimensional representations to quantum field teger or half-odd integer, the spin. For m2 = 0 ,
theory. Physicists interested in spectroscopy the little group is isomorphic to the euclidean
naturally preferred Wigner’s book to Weyl’s, but, group of a two-dimensional plane or to the two-
of late, there has also been mathematical inter- sheeted covering of such a group; the physically
est in the kind of detailed formulae for the interesting irreducible representations are de-
Clebsch-Gordon coefficients that Wigner’s termined by a helicity which is an integer or
book contains. half-odd integer.
The work of Wigner best known among math- Wigner came to the problem of the determi-
ematicians is undoubtedly his construction of a nation of the unitary ray representations of the
class of irreducible unitary representations of the inhomogeneous Lorentz group by adopting a
inhomogeneous Lorentz group. This group is space-time point of view in a discussion of sym-
not compact, and all its irreducible unitary rep- metry in quantum mechanics. By an analogue of
resentations except the trivial one are infinite di- the argument he had presented in his book for
mensional. The representation theory of such the case of symmetry in space at a fixed time,
groups was still unknown territory when Wigner he showed that a quantum mechanical theory in-
published his fundamental paper in 1938. Of variant under inhomogeneous Lorentz trans-
course, later, as a result of the work of Gelfand, formations has an associated unitary ray repre-
Naimark, Bargmann, and others on such groups sentation of the inhomogeneous Lorentz group.
as SL(2, C) and SL(2, R) , this theory became It is a remarkable fact that the law of evolution
highly developed. Wigner limited his consider- of states in the most general quantum mechan-
ations to those irreducible representations in ical theory can be characterized by a measure
which the spectrum of the representation of the class and multiplicity function on the masses
translation subgroup lies in or on the future spins and helicities.
cone: In his later years Wigner devoted most of his
scientific effort to sharpening what he saw as the
(k0 )2 − (k1 )2 − (k2 )2 − (k3 )2 ≥ 0.
paradoxes in the standard interpretations of the
The irreducibles turned out to be characterized quantum theory of measurement. He became
by the squared mass (= the left-hand side of the convinced that an essential extension of
inequality) and the representation of the so- physical theory to include consciousness
called little group, the group of transformations was necessary.
JULY 1995 NOTICES OF THE AMS 771
