Vol.

30 (1999) ACTA PHYSICA POLONICA B No 10

LECTURE IN HONOUR OF LEOPOLD INFELD

(EXTENDED OUTLINE ONLY)
SPINORS IN GENERAL RELATIVITY

Sir Roger Penrose

Mathemati al Institute
2429 St Giles, Oxford OX1 3LB, U.K.

(Re eived June 29, 1999)

This arti le is an extended outline of the le ture delivered at the Infeld

Centenial Meeting. In the le ture a review was given of the development of
the theory of spinors and related obje ts in spe ial and general relativity,
with some emphasis on the twistor theory and its impli ations. The le ture
was not intended as a detailed a ount of the subje t, but it rather was
a series of omments on the relevan e of various spinor-type obje ts and
their relation to some features of spa e-time stru ture. The present arti le
is also a guide, with its author's personal preferen es, to the extensive
bibliography of the subje t.

PACS numbers: 04.20.Gz

It is a great honour for me to have this opportunity to pay my respe ts

to Leopold Infeld. His seminal work showing how spinor al ulus may be
applied in general urved spa e-times has been extremely inuential, and it
has profoundly ae ted my own resear hes, these having been very greatly
on erned with the relationship between spinor theory and Einstein's general
relativity.

1. Preliminaries: at-spa e spinors

Spinors were rst found by Élie Cartan (1913; f. also Cartan 1966).
His spinor spa es onstitute the representation spa es of 2-valued represen-
tations of orthogonal groups. He did not use the name spinor at that
time, this term having been apparently introdu ed later by Ehrenfest (as I
was re ently informed by Andrzej Trautman), following Dira 's (1928) re-
dis overy of spinors and dis overy of their appli ation to the spin of the

 Presented at The Infeld Centennial Meeting, Warsaw, Poland, June 2223, 1998.

(2979)
2980 R. Penrose

relativisti ele tron (and also Pauli's earlier work, in 1927, on erning the
non-relativisti ele tron's spin). However many spinor-related ideas were
known mu h earlier, parti ularly in relation to the 2-valued (spin) rep-
resentations of parti ular rotation groups. (There are quaternions, found
by William Rowan Hamilton in 1837 and their expli it representation of
rotation matri es given by the CayleyKlein parameters, illustrating the
lo al isomorphism SU(2) ! O(3), and its omplexi ation SL(2; C ) !
O(3, C ), and there is also the lo al isomorphism SU(4) ! O(6) and its
omplexi ation SL(4; C ) ! O(6, C ), known to Sophus Lie in about 1872.
For information on the history of these matters, see Crowe 1967 and van der
Waerden 1985.) The general dis ussion of spin representations of rotation
groups stems from the algebras of William Kingdon Cliord (1878), these
arising as the ommon generalization of quaternions and Grassman algebras
( f. Grassman 1844, des ribing work done in 1825). An elegant and thor-
ough dis ussion of spinors for general orthogonal groups is to be found in
Brauer and Weyl (1935), based on Cliord algebras. (See also the appendix
to Penrose and Rindler 1986; Budini h and Trautman 1988.) Chevalley
(1954) showed how to develop a theory of spinors appli able to elds of
nite hara teristi (in luding the awkward ase of hara teristi 2).

The appli ations of spinors to physi s stemmed initially from the work
of Pauli and Dira as ited above, where the spinors were brought in spe if-
i ally for the treatment of parti les of spin 1 2. = Whereas Pauli spinors
are the 2- omponent spinors for the (non-relativisti ) rotation group O(3),
Dira spinors are 4- omponent entities, being spinors for the Lorentz group
O(1,3). These 4-spinors split down into pairs of 2- omponent entities, some-
times referred to half-spinors (or Weyl spinors), whi h were onsidered
by Weyl (1929) (and earlier, apparently, by Dira himself, f. Dira 1982)
to have relevan e to a relativisti (ree tion-non-invariant) wave equation.
This equation is now often used to des ribe the neutrino. It is a general
feature of spinors for an even-dimensional spa e that the spinors indeed
break down into su h half-spinors  or redu ed spinors  in this way.
(This is redu tion under the narrowing down of the rotation group to its
n
non-ree tive subgroup.) For a 2 -dimensional spa e, the redu ed spinors
are 2
n 1 -dimensional, n
so the unredu ed spinors are 2 -dimensional; for a
(2 n + 1)-dimensional spa e, the spinors are 2 n
-dimensional.

In the ase of a 4-dimensional spa e-time, subje t to lo al Lorentz group

symmetry, the redu ed spinors are 2- omponent entities, usually referred to
as 2-spinors. Ea h of the two 2-dimensional spa es of redu ed spinors is a
omplex spa e; these two spa es go into ea h other under spa e-ree tion,
time-ree tion, or omplex onjugation. A ording to the 2-spinor al ulus
introdu ed by van der Waerden (1929), as notationally slightly modied in
 Le ture in Honour of Leopold Infeld : : : 2981

Penrose (1960) and Penrose and Rindler (1984), the elements of these two
spin-spa es are labelled by kernel symbols having indi es whi h are, respe -
tively, apital Roman letters without primes ( A; B; C; : : : ; A0 ; B0 ; : : : ; A1 ; : : :)
and Roman letters with primes ( A0; B 0 ; C 0; : : : ; A00 ; B00 ; : : : ; A01 ;0 : :0 :). There
are invariant skew-symmetri al quantities " , " 0 0, " , "
AB A B
AB A B , that an
be used for raising and lowering indi es ( are having to be exer ised to keep
the signs onsistent). The tensor al ulus may be thought of as being em-
bedded in the 2-spinor al ulus where an unprimed and a primed spinor
index taken together ounts as a tensor index.
In this van der Waerden (1929) 2-spinor formalism, the Dira equation
for the ele tron be omes a pair of oupled linear dierential equations re-
lating an unprimed spinor A to a primed spinor A0 . This formalism was
subsequently used by Laporte and Uhlenbe k (1931) to represent, among
other things, the Maxwell equations for the ele tromagneti eld in an ele-
gant way. Then Dira (1936) showed how to write down eld equations for
higher spin, generally, in a very neat way and this was followed up signi-
antly by Fierz (1938). (In later work, Rarita, S hwinger, Dun, Kemmer,
and others formulated higher-spin equations, in spe ial ases, but not us-
ing su h a powerful general formalism; see Corson 1953 for a omprehensive
a ount of all this.)
Lorentzian 2-spinors have a very neat geometri al des ription in terms of
the null one. A non-zero 2-spinor determines a future-pointing null ve tor
and a null half-plane ontaining the dire tion of this ve tor. These are the
agpole and ag plane, respe tively, of this null ag interpretation of the
spinor; see Payne (1952), Penrose and Rindler (1984). This geometri al de-
s ription determines the 2-spinor ompletely up to a sign. (To interpret the
sign, a non-lo al des ription is needed; see Penrose and Rindler 1984. Ro-
tation of a spinor through 2  hanges its sign, whereas rotation through 4 
restores it to its original value.) The null dire tion of the agpole des ribes
the 2-spinor ompletely up to proportionality. A symmetri al n-valent spinor
(an n-index quantity  , with  = ( ) des ribes an entity of
spin n=2. It has a anoni al de omposition, a ording to whi h it an be rep-
AB:::L AB:::L AB:::L

resented (uniquely up to s alings and ordering) as a symmetrized produ t of

single-index 2-spinors  = ( : : :  ) and, therefore, up to propor-
n null dire tions.
AB:::L A B L

tionality, it orresponds uniquely to a symmetri al set of

This des ription gives the Majorana (1932) geometri al des ription of the
general spin state of a spin n=2 parti le (see Penrose 1989).
2. Spinors for urved spa e-time
The rst treatment of spinors in urved spa e-time was that of Infeld and
van der Waerden (1933), using the van der Waerden 2-spinor formalism. The
2982 R. Penrose

translation between spinors and tensors is then a hieved in lo al omponent

form by use of the Infeldvan der Waerden symbols (in the terminology of
Penrose and Rindler 1984). In this paper, the ingenious suggestion was
made that the spinor phase should be the gauge quantity that generates
ele tromagnetism. However, this idea has not stood the test of time (at
least not in its original form) be ause it appears to imply a dire t relation
between the spin and the harge of a parti le. (The spin/ harge value for the
neutron would appear to be in oni t, the neutron having been dis overed
in 1932, at about the same time as this paper was written.)

Although the initial use of the Infeld-van der Waerden formalism was for
the des ription of parti les with spin within the urved spa e-time framework
of Einstein's general relativity, this formalism an also be used to study
Einstein's theory itself. See Veblen and von Neumann (1936), Bergmann
(1957), Witten (1959), Penrose (1960), Penrose and Rindler (1984, 1986).
For example, the Weyl onformal tensor orresponds to a totally symmetri
spinor ABC D whi h resembles a spin 2 massless eld, in the sense (referred
to above) of Dira Fierz. The anoni al de omposition applied to this spinor
gives a neat lassi ation s heme for va uum spa e-times ( f. Penrose 1960,
Penrose and Rindler 1986).

When all tensor as well as spinor omponents are referred to a hoi e

of spin-frame (a basis in the lo al spin spa e at ea h spa e-time point),
then the formalism of spin- oe ients is obtained ( f. Jordan, Ehlers, and
Sa hs 1961, Newman and Penrose 1962). This turns out to have a great
al ulational utility; see, for example, Chandrasekhar (1983). In ertain
situations, it turns out that be ause of the geometry of a problem, it may
be onvenient to spe ify merely a pair of null dire tions at ea h point (the
agpole dire tions of the two spinor basis elements) rather than an entire
spin-frame. Then the more streamlined ompa ted spin- oe ient formalism
an be very onvenient to use (see Newman and Penrose 1966, Gero h, Held,
and Penrose 1973, Penrose and Rindler 1984).

Most of the foregoing remarks, in this se tion, have been on erned with
spinors only lo ally in urved spa e-time. In fa t there are important global
restri tions and ambiguities, for spinor elds to make global sense on a spa e-
time manifold. These have to do with what are alled StieelWhitney
lasses on the manifold. The essential issue is the fa t that the sign of a
spinor does not have a lo al geometri al interpretation (something that was
tou hed upon in the previous se tion), so the onsisten y of this sign globally
depends upon global properties of the spa e-time. For results of relevan e
to this issue, see Milnor (1963), Li hnerowi z (1968), Gero h (1968, 1970),
Penrose and Rindler (1984). (1984).
 Le ture in Honour of Leopold Infeld : : : 2983

3. Twistor theory
It is possible to regard spinors as being, in some sense, more primi-
tive than tensors. One may take the view that the light- one stru ture
and metri s aling are determined by the spinor stru ture of the spa e-time
( f. Bergmann 1957, Penrose and Rindler 1984). Perhaps, even the par-
ti ular dimensionality and signature of the spa e-time metri may, in some
sense, be regarded as derived on epts. However, so long as the spa e-time
manifold itself must be given beforehand, it is hard to hold to su h a view in
a serious way. The manifold's dimension is determined by its topology, and
the denition of tensors requires only its dierentiable stru ture. However,
the theory of twistors (Penrose 1967) provides the possibility of a more rad-
i al view. A ording to this s heme of ideas, the spa e-time manifold itself,
not just its light one stru ture, is indeed regarded as derived from a more
primitive spinor-type spa e.

This is not the pla e to give a detailed a ount of twistor theory. (Su h
a ounts an be found in Huggett and Tod 1985, Penrose and Rindler 1986,
Ward and Wells 1989.) Only some brief omments of relevan e will be
given here. M , whi h is to be a Minkowski
For at spa e-time 4-spa e,
# of M and the
the translation between the onformal ompa ti ation M
twistor spa e T is very dire t. The spa e T is a 4- omplex-dimensional
ve tor spa e with a Hermitian quadrati form  of signature (+ + ).
The parts of T on whi h  > 0,  < 0, and  = 0 are denoted by
T +, T , and N , respe tively. The elements of N are alled null twistors.
# is interpreted as a 2-dimensional linear subspa e x
Any point x 2 M
of M , where x  N . It is often onvenient to think in terms of the
proje tive twistor spa e P T , whi h is the omplex proje tive 3-spa e whose
points are the 1-dimensional linear subspa es of T . Thus, there is 5-real-
dimensional subspa e P N , of P T , onsisting of the proje tive null twistors.
The proje tive version P x, of x  N , is a proje tive straight line in P N . It
turns out that a proje tive null twistor represents a light ray (null geodesi )
in M
# and the generators of the light one of a point x 2 M # are thereby
represented, in P T , by the points of the line P x. Two points x and y of
M # are null separated i the orresponding lines P x and P y interse t.
This gives the basi geometry of the orresponden e between the ( om-
pa tied) spa e-time M # and the (proje tive) twistor spa e P T . In fa t,
this geometry is really a manifestation of the stru ture of Sophus Lie's lo-
al isomorphism SL(4; C ) ! O(6; C ), in its real form SU(2; 2) ! O(2; 4).
The relevan e of O(2; 4) here is that it is lo ally isomorphi to the 15-
parameter onformal group of ( ompa tied) Minkowski 4-spa e M . The
#
group SU(2; 2) a ts on the spin spa e T and preserves the (2; 2)-Hermitian-
quadrati form  . The lo al isomorphism SU(2; 2) ! O(2; 4) expresses
2984 R. Penrose

the fa t that twistors (elements of T ) are in fa t redu ed spinors for the

pseudo-orthogonal group O(2,4) that des ribes the onformal symmetries
of ompa tied spa e-time M #. The other spa e of redu ed spinors (the
analogue of Lorentzian primed spinors) turns out to be the dual spa e
T  of T .
The underlying philosophy of twistor theory is that the omplex spa e
T is to be regarded as being, in some sense, more primitive than the spa e-
time itself. (The main motivations for this twistor view ome from a desire
to bring together the basi , but in ompatible, prin iples of quantum me-
hani s and Einstein's general relativity without trying to impose one upon
the other. The omplex-number stru ture of quantum me hani s and quan-
tum non-lo ality nd manifestations in spa e-time geometry via the twistor
orresponden e.) Over the years, it has been found that many of the basi
physi al notions, parti ularly those whi h have to do with massless parti les
and elds, indeed have elegant interpretations in twistor-spa e terms; also,
twistor theory has had many appli ations within areas of pure mathemat-
i s. (See Hughston 1975, Huggett and Tod 1985, Penrose and Rindler 1986,
Ward and Wells 1989, Bailey and Baston 1990, Mason and Woodhouse 1996,
Huggett, Mason, Tod, Tsou, and Woodhouse 1998 for details).

Yet, that ru ial physi al eld, whi h any attempt at unifying quantum
me hani s with spa e-time stru ture must profoundly ome to terms with,
is Einstein's general relativity. How does twistor theory fare in this respe t?
At rst sight (in fa t a rst sight whi h lasted for some 10 years or more),
it had seemed that twistor theory is really just a s heme of things that
applies only to ( onformally) at spa e-time. However, it eventually turned
out (Penrose 1976) that by deforming (portions of ) twistor spa e it be omes
possible to en ode the Einstein va uum equations in the ase of anti-self-dual
Weyl urvature into the stru ture of the deformed twistor spa e. Anti-self-
dual Weyl urvature orresponds, in the weak-eld limit, to the left-handed
graviton eld, when we are thinking of the omplex graviton wave-fun tion
as a weak-eld deformation of omplexied Minkowski spa e CM . In fa t,
twistor theory is a profoundly hiral theory, and it has the urious feature
that the left-handed and right-handed omponents of a massless quantum
eld are treated on a quite dierent footing. In the ase of the Einstein
gravitational eld, it has turned out (using the standard twistor onventions)
that the left-handed part of the graviton nds a twistor interpretation far
more readily than does the right-handed part.

What is the present status of the problem of en oding the right-handed

graviton also into the framework of twistor theory, over twenty years after
the su essful left-handed onstru tion? For a number of years re ently
( f. Penrose 1992), I had been pinning my hopes on the striking fa t that
=
massless elds of heli ity 3 2 seem to provide an intermediary between the
 Le ture in Honour of Leopold Infeld : : : 2985

Einstein va uum equations and twistor theory. The onsisten y ondition

=
for heli ity 3 2 elds is pre isely the va uum equations, whereas, in M , the
spa e of harges for these elds is a tually twistor spa e T . In the meantime,
however, another idea has emerged (Penrose 1999) whi h does indeed en ode
the outstanding right-handed part of the gravitational eld. It remains to
be seen whether this all links together (perhaps via the agen y of elds
=
of heli ity 3 2) to provide a full twistor des ription of Einstein's general
relativity. If it does, then a new approa h to the uni ation of quantum
theory with spa e-time stru ture ould well be provided.

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