Twistors and Particles (Lecture notes in physics)

Lecture Notes in Physics Edited by .I. Ehlers, M~Jnchen, K. Hepp, Z0rich R. Kippenhahn, M~Jnchen, H. A. Weidenmeller, He...

Author: L. P. Hughston

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Lecture Notes in Physics Edited by .I. Ehlers, M~Jnchen, K. Hepp, Z0rich R. Kippenhahn, M~Jnchen, H. A. Weidenmeller, Heidelberg and J. Zittartz, KSIn Managing Editor: W. BeiglbSck, Heidelberg

97

L. R Hughston

Twistors and Particles

Springer-Verlag Berlin Heidelberg New York 1979

Author Lane Palmer Hughston The Mathematical Institute University of Oxford Oxford England

ISBN 3-540-09244-? ISBN 0-38?-09244-?

Springer-Verlag Berlin Heidelberg New York Springer-Verlag New York Heidelberg Berlin

Library of Congress Cataloging in Publication Data. Hughston, L P 1951Twistors and particles. (Lecture notes in physics ; 97) Bibliography: p. Includes index. 1. Particles (Nuclear physics) 2. Twistor theory. I. Title. I1.Series. QC793.3.F5H83 539.7'21 ?9-13891 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/3140-543210

PREFACE The momentum of the mind is all toward abstraction.

-

Wallace Stevens, Opus Posthumous

Within the framework of twistor theory the structure of spacetime is relegated, in contrast to the position which it has held since the beginning of the twentieth century, to a status of secondary character.

Whereas in the past spacetime has al-

ways served as the background against which phenomena are to be interpreted--and indeed, according to Einstein's theory of gravitation,

spacetime serves moreover as a

basic dynamical entity itself--the new view which the twistor theorists are advocating takes twistor space, with the many rich and variegated aspects of its complex analytic structure, as the primary descriptive device and dynamical construction in terms of which phenomena are to be understood. The difficulties inherent in a spacetime description have long been appreciated by many authors.

Julian Schwinger,

for example, in his preface to Selected Papers

on Quantum Electrodynamics summarizes the situation aptly when he remarks that "... The localization of charge with indefinite precision requires for its realization a coupling with the electromagnetic field that can obtain arbitrarily large magnitudes.

The resulting appearance of divergences,

deny the basic measurement hypothesis. be formulated consistently

and contradictions,

serves to

We conclude that a convergent theory cannot

within the framework of present space-time concepts.

To

limit the magnitude of interactions while retaining the customary coordinate description is contradictory, measurements."

since no mechanism is provided for precisely localized

With a similar attitude towards this question Einstein, at the

end of The Meaning of Relativity,

concludes that "One can give good reasons why

reality cannot at all be represented by a continuous field.

From the quantum

phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers

(quantum numbers).

This does not seem to be in accordance with a continuum theory, and must lead to an attempt to find a purely algebraic theory for the description of reality."

Of

IV

course when he refers to a continuum Einstein means spacetime, taken with its usual real differentiable structure.

In twistor theory, however, the continuum

which arises is that of the complex number system, and those aspects of the geometry of twistor space which are of interest to physics stem more specifically from its complex analytic structure, rather than its real differentiable structure.

The

general characterization of the structures which can arise in the case of complex analytic manifolds has been the subject of intense investigation by mathematicians, especially with the advent of the powerful techniques of sheaf cohomology theory. One of the precepts of twistor theory is that here, within a suitably formulated sheaf cohomological framework, we have the proper basis for a "purely algebraic" description that is compatible both With the ideas of relativity and with the principles of quantum mechanics. This view has met with a reasonable degree of success, and it has been possible, using methods of algebraic geometry and complex analytic geometry, for twistor theorists to assemble the outlines of a new approach to elementary particle physics.

The subject is still in its infancy and in a rapid state of development,

and thus many of its results are only of a preliminary character and are both subject to and deserving of considerable modification and improvement.

In spite of

their tentative nature, it seemed appropriate nonetheless to prepare an account of some of these matters for a wider audience, with the hope that it might stimulate or otherwise prove a useful aid in further and more extensive research into the subject.

With this purpose in mind the following study is presented.

Although a fair amount of background material is covered in Chapters 2 and 3, the reader previously uninitiated into the mysteries of twistor theory may find it necessary to consult some additional references. formalism see Pirani and Penrose.

(1965), Penrose

For the two-component spinor

(1968a), and the forthcoming book by Rindler

For further reading in basic twistor theory see Penrose

Penrose and MacCallum (1972), and Penrose

(1975a).

(1967),

Although a specialized know-

ledge of elementary particle physics is not necessary, at the outset, for reading this volume, it is assumed nonetheless that the reader is familiar with basic

quantum mechanics, and is acquainted already, to some extent, with the quark model. The author is indebted to many of his

colleagues for their help in the

preparation and development of this material, particularly to R. Penrose who originated many of the ideas discussed here, and who has acted as a constant source of illumination and inspiration.

G.A.Jo Sparling has contributed extensively to

this work, and the author wishes to thank him for many helpful discussions. would also like to thank many of my

I

colleagues at Oxford and elsewhere, including

D.M. Blasius, M. Eastwood, M.L. Ginsberg, A. Hodges, S.A. Huggett, T.R. Hurd, R. Jozsa, E.T. Newman, A. Popovich, Z. Perj~s, I. Robinson, M. Sheppard, L. Smarr, P. Sommers, K.P. Tod, Tsou S.T., M. Walker, R.S. Ward, and N.M.J. Woodhouse, for useful conversations and suggestions related to the work described herein.

The author is grate-

ful to B.S. DeWitt, C.M. DeWitt, R. Matzner, L. Shepley, H.J. Smith, and the late Alfred Schild, as well as other

colleagues at the University of Texas at Austin, for

their hospitality shown during the author's 1974 visit, when some of the ideas preliminary to the material described here were worked out.

The author has profited

much from his regular visits, supported by the Clark Foundation, to the University of Texas at Dallas, and he would like to thank I. Ozsvath, and J.R. Robinson for their hospitality.

W. Rindler, I. Robinson,

Likewise the author has benefited from

his visits to the Astronomy Department at the University of Virginia, and gratitude is expressed to W. Saslaw, and other

colleagues there, for their hospitality.

I

am grateful to J. Ehlers, M.L. Ginsberg, C.J. Isham, R. Penrose, G.A.J. Sparling, and N.M.J. Woodhouse for reading earlier drafts of the manuscript and contributing many corrections and helpful suggestions for improvement. This work was supported by a Rhodes Scholarship at Oxford during the years 1972-1975. 1972-73.

This work was also supported by the Westinghouse Corporation during More recently the work described herein has been supported by a grant

from the Science Research Council, and by a Junior Research Fellowship at Wolfson College, Oxford.

I am very grateful to Valerie Censabella, who typed the manuscript

and who has been most helpful at all stages in the preparation of this material. This volume is dedicated to my mother and my father.

TABLE OF CONTENTS Page

Preface

.................................................................

i.

Introductory

2.

Aspects §2.3 §2.2 §2.3 §2.4 §2.5 §2.6 Notes

3.

4.

5.

6.

7.

Quantization:

5 7 8 |0 ]] 13 15

Symmetries 16 ]7 |8 20 24 28

zero Rest Mass F i e l d s

Ouantization:

Massive

29 29 31 34 35 42

Fields

O p e r a t o r s for M o m e n t u m and A n g u l a r M o m e n t u m .................... C o n t o u r Integral F o r m u l a e for M a s s i v e Fields ................... The M a s s O p e r a t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Spin O p e r a t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Internal U(n) C a s i m i r O p e r a t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 46 47 48 51

Baryons

The Quark Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The T h r e e - T w i s t o r Model for L o w - L y i n g Baryons .................. E l e c t r i c Charge, Hypercharge, Baryon Number, a n d Isospin ....... Mass and Spin for T h r e e - T w i s t o r Systems ........................ The SU(3) C a s i m i r O p e r a t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The A b s e n c e of C o l o r D e g r e e s of F r e e d o m ........................ ................................................................

Mesons, §7.1 §7.2 §7.3

Internal

W h a t is T w i s t o r Q u a n t i z a t i o n ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The H e l i c i t y O p e r a t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P o s i t i v e H e l i c i t y Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N e g a t i v e H e l i c i t y Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The P o s i t i v e F r e q u e n c y C o n d i t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................

The L o w - L y i n g §6.1 §6.2 §6.3 §6.4 §6.5 §6.6 Notes

and their

M o m e n t u m and A n g u l a r M o m e n t u m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The K i n e m a t i c a l T w i s t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The D e c o m p o s i t i o n of M a s s i v e systems into M a s s l e s s S u b s y s t e m s °. Internal S y m m e t r i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The C e n t e r of M a s s T w i s t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................

Twistor §5.1 §5.2 §5.3 §5.4 §5.5

Systems

]

of T w i s t o r space

C l a s s i c a l Systems of Zero R e s t M a s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . The A c t i o n of the P o i n c a r ~ Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Group SU(2,2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The T w i s t o r E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q - P l a n e s a n d b-Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P r o j e c t i v e T w i s t o r Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................

Twistor §4.1 §4.2 §4.3 §4.4 §4.5 Notes

.................................................

of the G e o m e t r y

Massive §3.1 §3.2 §3.3 §3.4 §3.5 Notes

Remarks

I[[

Resonances,

56 62 62 64 66 68 70

and B o u n d States

The L o w - L y i n g M e s o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The ~ - ~ P r o b l e m ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M e s o n s as Q u a r k - A n t i q u a r k Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72 76

77

V~J~

Page §7.4 §7.5 §7.6 §7.7 Notes 8.

Orbital Angular Momentum ....................................... E x c i t e d Meson States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baryon Resonances .............................................. The D e u t e r o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................

Leptons

and W e a k

Interactions

§8.1 P r o p e r t i e s of Leptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §8.2 Space R e f l e c t i o n Symmetry V i o l a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . §8.3 Leptons as T w o - T w i s t o r Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §8.4 Models for Sequential Leptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.

Sheaves

Cochains, Cocycles, and C o b o u n d a r i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . L i o u v i l l e ' s Theorem, the L a u r e n t Expansion, a n d the C o h o m o l o g y of pl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §9.3 The C o h o m o l o g y of pn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §9.4 The L o n g E x a c t C o h o m o l o g y Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §9.5 The Koszul C o m p l e x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §9.6 Line Bundles and Chern Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §9.7 Varieties, Syzygies, a n d Ideal S h e a v e s ......................... Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications Physics §10.1 §10.2 §10.3 §10.4 §10.5 §10.6 Notes

of C o m p l e x M a n i f o l d

Techniques

to E l e m e n t a r y

108 111 114 ]15 i|7 119 121 ]25

Particle

The Kerr T h e o r e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zero Rest Mass Fields as E l e m e n t s of Sheaf C o h o m o l o g y Groups .. Spin-Bundle Sequences ......................................... Remarks on the G e o m e t r y of n - T w i s t o r Systems .................. M a s s i v e Fields R e v i s i t e d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T o w a r d s the C o h o m o l o g y of n - T w i s t o r S y s t e m s ................... ...............................................................

]26 129 132 136 140 141 ]45

...............................................................

147

....................................................................

151

References Index

94 99 102 105 107

and C o h o m o l o g y

§9.1 §9.2

i0.

79 84 87 92 93

CHAPTER 1 INTRODUCTORY REMARKS

Progress in any aspect is a movement through changes in terminology.

-

Wallace Stevens, Opus Posthumous

This study will touch on a variety of topics concerning twistor theory and elementary particle physics. but none exhaustively.

A few of these topics will be treated in some detail,

The purpose of this work is to describe how it is possible,

using twistor methods, to gain some understanding of the microscopic structural degrees of freedom responsible for the properties of elementary particles. In a very general sense the methodology of twistor theory consists simply of the application of techniques of complex analytic geometry to problems in physics. Inherent in the twistor program are many changes in terminology, whereby a number of the familiar concepts of physics are reexpressed in the language of algebraic geometry and analytic geometry.

"The physicist always prefers

to sacrifice the less

perfect concepts of physics rather than the simpler, more perfect,

and more lasting

concepts of geometry, which form the solidest foundation of all his theories",

said

Mach, and there is certainly a good deal of reason in his remark: but the twistor philosophy goes one step further, and insists that within geometry itself one can discover all the laws of physics. The organization of this volume is as follows. space from the standpoint of classical physics.

Chapters 2 and 3 view twistor

Algebraic geometry is to complex

analytic geometry as classical physics is to quantum physics--and in Chapters 2 and 3 twistor space is explored with various tools of algebraic geometry. information in Chapter 2 is standard background material,

Most of the

and is summarized here

for the reader previously unacquainted with twistor theory.

Twistors are first

defined in terms of classical systems of zero rest mass--that is to say, classical special relativistic systems defined by a null momentum and an angular momentum

which is related to the momentum in such a way that twisters transform in a natural way under the action of the group SU(2,2), and, in particular, the P o i n c a r ~ group. In §2.4 it is shown that twisters can be characterized in terms of the solutions of a certain differential equation called the "twister equation".

In §§2.5 and 2.6

twisters are described in terms of the geometry of complex projective 3-space p3 . Complex projective lines in p3 correspond to points in complex Minkowski space; using this correspondence (the "Klein representation") various aspects of the geometry of spacetime are expressed in twister terms, and vice-versa. In Chapter 3 it is shown how massive systems can be built up out of two or more twisters.

The momentum and the angular momentum are described in terms of a single

two-index symmetric "kinematical twister".

Theorem 3.3.1 shows how any massive sys-

tem can be decomposed into two or more twister constituents.

Thus, massive systems

(at the classical level) can always be regarded as being "made up" out of twisters. Twisters are, in a certain sense, the elementary constituents of matter.

For a

given momentum and angular momentum there are internal degrees of freedom which yet remain, mixing the various twister constituents.

Theorems 3.4.2 and 3.4.14 show the

relevant groups which leave the momentum and angular momentum of an n-twister system invariant.

These groups are called the "n-twister internal symmetry groups", and,

for each value of n, contain U(n) as a subgroup.

It is proposed that these internal

degrees of freedom are in some sense responsible for the phenomenological unitary groups which arise naturally in elementary particle classification schemes (e.g., SU(3)).

In §3.5 a center of mass twister is introduced for n-twister systems.

This

construction plays a useful role in a number of problems. In Chapter 4 the rules of twister quantization are introduced for systems com~ posed of a single twister.

It is shown how solutions of the zero rest mass equations

can be obtained in terms of holomorphic functions defined over suitable domains of twister space.

Both positive and negative helicity fields are discussed, and the

differences in the relevant contour integral formulae for evaluating the fields, in the two cases, are noted.

The positive frequency condition is discussed in §4.5, and

the whole procedure is illustrated with the example of an elementary state. In Chapter 5 massive fields are desexibed in terms of holomorphic functions of

two or more twisters.

It is proposed that observables correspond to holomorphic

differential operators with polynomial coefficients.

Explicit expressions are pre-

sented for the operators corresponding to momentum, angular momentum, mass, and spin.

In §5.5 the operators corresponding to "internal" observables are discussed,

and are described explicitly in the cases of one, two, and three twisters. In Chapter 6 the scheme is applied to the low-lying baryons--that is to say, the N(949) octet and the A(1232) decimet.

After a brief review of the quark model

(described in a language suitable for our purposes) it is demonstrated how the lowlying baryons can be represented in terms of certain types of holomorphic functions of three twisters.

Baryons are not regarded as bound states of quarks.

No color

degrees of freedom are introduced. In Chapter 7 the methods of Chapter 6 are extended so as to apply to more general systems.

Mesons are introduced as quark-antiquark bound states, described in

terms of holomorphic functions of six twisters.

The charge

conjugation

ber plays a crucial role in the representation of these states.

quantum num-

Orbital angular mo-

mentum is described in twister terms, and it is shown how orbital excitations of the quark-antiquark system lead to meson resonances. as excitations of a quark-diquark bound state.

Baryon resonances are represented The deuteron is briefly discussed,

from a twister point of view, in the last section of Chapter 7.

In Chapter 8, after

a review of the properties of leptons and of parity violation in weak interactions, a model for sequential leptons is built up in twister terms.

Chapters 9 and i0 are

concerned with further mathematical developments in the theory.

In Chapter 9 the

methods of sheaf cohomology are introduced, and these are applied to various problems in Chapter i0, the aim being to sharpen up much of the material of the previous chapters, and to open up the doors to more extensive developments. The tentative nature of any general inferences that can now be put forward in connection with the twister particle program, or, for that matter, twister theory in general, should undoubtedly be apparent to anybody working in this subject.

One

need merely consider the vast range of phenomena which so far have resisted any formulation in twister terms whatsoever.

Nonetheless, significant conclusions are be-

ing drawn along certain lines, and are receiving continually increasing support.

In particular, theory

the central

role of the t w i s t o r p r o g r a m

seems to me n o w firmly established,

now why particle framework

physics

of twistor

as a whole

theory.

in c o n n e c t i o n

with Einstein's

and there does not s e e m to be any reason

should not be amenable

to t r e a t m e n t w i t h i n

the

CHAPTER 2 ASPECTS OF THE GEOMETRY OF TWISTOR SPACE 2.1

Classical Systems

of Zero Rest Mass.

There are various ways of b u i l d i n g up the framework of t w i s t o r theory, and it m u s t be s a i d that it is not e x a c t l y clear w h e r e to begin.

For the p u r p o s e s of in-

v e s t i g a t i o n s into e l e m e n t a r y p a r t i c l e p h y s i c s a convenient,

if not totally adequate,

place to start is w i t h the o b s e r v a t i o n that a p o i n t Z ~ in twistor space

(5 = 0,1,2,3)

can be r e p r e s e n t e d n a t u r a l l y in terms of p h y s i c a l quantities as a classical system of

zero

rest mass.

Such a system is c h a r a c t e r i z e d by its total m o m e n t u m pa , w h i c h is null and future-pointing,

and its angular m o m e n t u m M ab

(= -M ba) w i t h respect to a p a r t i c u l a r

choice of origin in spacetime. T o g e t h e r these q u a n t i t i e s m u s t satisfy a r e l a t i o n to the effect that if we form the s p i n - v e c t o r

S

(2.1 .i)

1 a = ~

b cd Sabcd P M

then the p r o p o r t i o n a l i t y S a = sP holds for some value of the number s. a tude of s is the spin of the system, and s itself is c a l l e d the helicity.

The magniPositive

h e l i c i t y systems are called right-handed, and negative h e l i c i t y systems are called left-handed. There is a certain algebraic c h a r a c t e r i z a t i o n of the m o m e n t u m and angular m o m e n tum that ensures that together they constitute a zero rest mass ated ZRM) 2.1.2

system: Proposition.

A p a i r {pa , Mab} represents a ZRM system if and only if

there exists a p a i r of spinors

(2.1.3)

(henceforth a b b r e v i -

P

(wA

, ZA,)

a

- A A'

such that

= T

and

(2.1.4)

M ab = i ~ (A-B) ~ ~A'B'

where ~A is the complex conjugate of A '

- i~(A' B ' ) A B -A'

, and ~

is the c o m p l e x conjugate of

A

Proof (1).

The existence

fied is precisely

the condition

The spin-relation is the dual of M ab.

AB

AI

such that equation

(2.1.3)

is satis-

that pa should be null and future-pointing.

S a = sP a can be written

,MabPb = sP a where *M ab

M := ~1 ~ abcd "cd

If we write

(2.1.5)

where ~

of a spinor ~

is a symmetric

.-A'B' ~ AB *M ab = -ipABt A'B' + l~

,

spinor, then the spin-relation,

using equation

(2.1.3),

reads

(2.1.6)

_i AB~B A'

Contracting

i~A'B'

-A -A A' ~B,~ = sT

this relation with ~A yields ~AB~A~ B = 0, which implies

for some choice of A , Finally,

+

the factor of -i being included

using the fact that equation

for later convenience.

implies

M ab = BABe A'B' + p-A'B' C AB

(2.1.7)

we deduce equation

(2.1.5)

-i~ AB = ( A ~ B )

,

(2.1.4). [ ]

The spinor pair

(wA , ~A, ) completely

determines

the ZRM system,

and defines a

point Z ~ in twistor space according to the scheme

(2.1.8)

(Z 0 , Z 1 , Z 2 , Z 3) =

(~00 , ~0i , 4 0, , ~i,)

Note, on the other hand, that a ZRM system determines to an overall phase factor,

its associated

twistor only up

since the m o m e n t u m and the angular m o m e n t u m are invariant

under the transformation

( A , ~A,)

(2.1.9)

It is interesting

-----+ el@( A , ZA.)

to observe that the helicity of a ZRM system can be expressed

directly in twistor terms.

For this purpose it is useful to define the complex con-

jugate twistor Z~ by the spinor pair

_n I (~A , ~ ).

A short calculation

that the inner product defined by

(2.1.10)

A-A I Z~Zd = ~ ZA + ZA 'W

establishes

7

is p r e c i s e l y twice the h e l i c i t y of the system, i.e. we have Z~Z

= 2s.

One m i g h t be inclined i n i t i a l l y to think that the f r e e d o m e x p r e s s e d in

(2.1.9)

is of an i r r e l e v a n t nature, and arises p e r h a p s on account of some slight i n a d e q u a c y in the r e p r e s e n t a t i o n that has b e e n chosen for twistors in terms of systems of zero rest mass.

N o t h i n g c o u l d b e further from the truth, however.

One of the remarkable

things about twistors is that they do, in fact, c a r r y more i n f o r m a t i o n in them than just m o m e n t u m and angular momentum.

This fact takes on great significance,

as we

shall see, w h e n q u a n t u m m e c h a n i c s is b r o u g h t into the picture.

2.2

The A c t i o n of the P o i n c a r ~ Group. It is of considerable interest to k n o w how the action of the P o i n c a r ~ group is

expressed in twistor terms.

Since our u l t i m a t e goal is to express various field

quantities in terms of twistors, and since these field quantities m u s t themselves be subject to a p a r t i c u l a r b e h a v i o r under the action of the P o i n c a r ~ group, it is of s i g n i f i c a n c e to study the action of the P o i n c a r ~ group on twistors first. Under the spacetime t r a n s l a t i o n x a - - ÷

x a + r a the angular m o m e n t u m M ab trans-

forms a c c o r d i n g to the rule

(2.2.1)

M ab - - +

M ab + 2r[ap b]

It is not d i f f i c u l t to c h e c k that for a ZRM system the t r a n s f o r m a t i o n on Z ~ that induces

(2.2.1) is A

(2.2.2)

to

AA'

A

---~ W

+ ir

HA,

'

ZA' - - ÷

ZA'

This t r a n s f o r m a t i o n can therefore be r e g a r d e d as d e f i n i n g the action of a spacetime t r a n s l a t i o n on Z ~. The action of a r e s t r i c t e d Lorentz t r a n s f o r m a t i o n on a ZRM system is specified by

(2.2.3)

Pa - - +

For a r e s t r i c t e d Lorentz t r a n s f o r m a t i o n A

(2.2.4)

--÷

AabPb

Mab b a

A c. d M a I~ cd

has the form

A b : ~ B[ B' a A A'

where ~A- is an element of the group SL(2,C), i.e. subject to the relation (2.2.5)

~ C~ D A B £CD = e A B

The action on Z ~ which induces

(2.2.3) is easily verified to be:

(2.2.6)

wA

_~AB B '

'

"~

- B' ~A' ~B'

~A' ~ - ~

By following a Lorentz transformation with a translation, we can realize the complete action of the restricted Poincare group on a twistor.

This can be conven-

iently expressed in the form

(2.2.7)

Z~ - - ÷

where the transformation matrix P ~

P~Z ~

F

is given by

-~A B (2.2.8)

P~

8

ir

~A'

= --

0

B

l

~A'

'

with the usual laws of matrix multiplication applying in the contraction of Pe~ with the spinor parts of Z ~.

(2.2.9)

wA

That is to say, we have

> _~A B wB + irAA '[A' B' ZB'

for the spinor parts of equation

2.3

(2.2.7).

The Group SU(2,2). The complex conjugate twistor Z

Z

- B' ZB' ZA' - - + ZA'

--+

undergoes the complex conjugate transformation

P ~.~ when Z ~ undergoes transformation

(2.2.7).

Since the helicity s is

Poincare invariant, the requirement that the inner product Z@Z that P ~ P y ~

= (~ , where 6~ i s t h e t w i s t o r

Kronecker delta,

be preserved implies

given in spinor parts

by: -E B (2.3.1)

~

= 0

g A'

B'

The set of all matrices U~$ satisfying U ~ $ U d = 65d forms the group U(2,2).

This

Such transformation

can be seen as follows.

matrices preserve

the norm ZeZ

,

which is given explicitly by

(2.3.2)

A-A' 01-0' -i' = ~ ZA + ZA 'w = ~ Z0 + ~ Z1 + ~0 'W + Zl '~

ZdZ

If new variables

(2.3.3)

W

0

are introduced

=

(w+y)

W

1

according to the scheme

=

where w, x, y, and z are complex,

which shows that the helicity The group U(2,2) formations

The group SU(2,2) addition

to satisfying

Hermitian

(x-z)

form of signature

{++--}.

group of complex linear trans-

form of that signature.

is the subgroup of U(2,2) = ~

~i' =

,

Hermitian

the multiplicative

a quadratic

uYpya

(w-y)

+x~- y~- ~

is a quadratic

is by definition

which preserve

~0' =

then

z ~-z : ~ ~z

(2.3.4)

g~Y~ ,

(X+Z)

consisting of matrices which,

, also preserve

in

the twistor epsilon tensor

i.e.:

(2.3.5)

U~ U@ Uy U6@ g ~ @

Condition

= ~By6

(2.3.5) amounts to the same thing as requiring

that U~@ have unit deter-

minant. SU(2,2)

is of special

phic with the 15-parameter restricted P o i n c a r ~ g r o u p ship between "infinity

importance conformal

to

physics

group of compactified

is a subgroup of SU(2,2).

Minkowski

A description

space (2) .

The

of the relation-

the two groups can be facilitated with the introduction

of the so-called

twistors", given by

(2.3.6)

IaB

=

I

0

which,

inasmuch as it is locally isomor-

according

to a scheme to be elaborated

of the null cone at infinity.

R

=

g

in Section 2.6, represent

the vertex

10

The infinity twistors are skew-symmetric,

are complex conjugates of one-another,

and satisfy the tollowing relations:

(2.3.7)

i~Si ~y = 0

Poincare transformations they preserve

2.4

,

I~

= ~1s ~BY@ 176

are SU(2,2)

I ~ = 1 s Sy@iy6

,

transformations

which have the property that

the infinity twistors.

The Twistor Equation. Another way in which twistor space arises naturally is as the solution set of

the differential

equation

(2.4.1)

= 0

which, accordingly,

2.4.2

is sometimes called the twistor equation.

Preposition.

The general solution of equation

(2.4.3) where ~

~A(x ) = A A

,

_ ix

(2.4.1)

is

AA v ~A ~

and ~A' are constant.

Proof.

Equation

(2.4.1)

(2.4.4)

can be written in the form

vB'B~C = ~ i BC~B'~DVD %

Taking a derivative,

we have

(2.4.5)

~A'AvB'B~C = ~E l BC~AA'v VD~B' D

,

which, using vA'(A~ c) = 0 , implies

(2.4.6)

B(CvA)A'

B'~D VD = 0

B'~D showing that VD is a constant spinor, which will be denoted 2iz B' , the factor of 2i being for convenience. integration then gives The pair

Substituting

(2.4.3), with A

this result back into equation

(2.4.4),

appearing as a constant of integration. []

(mA , ZA,) defines the twistor Z ~ , and ~A(x) is called the associated

spinor field (3) of the twistor Z ~.

It can be checked that the natural action of the

11

P o i n c a r e group on ~A(x) agrees w i t h the action on Z ~ d e f i n e d in Section 2.2.

2.5

~ - P l a n e s and 8-Planes. The l o c a t i o n of a twistor Z6% in complex M i n k o w s k i space can be defined as the

region for w h i c h the a s s o c i a t e d spinor field ~A(x) vanishes.

From

(2.4.3) this is

e v i d e n t l y the c o n d i t i o n that

A . ~-A I [0 = ix ZA'

(2.5.1)

Since e q u a t i o n

(2.5.1) is linear in x

An

!

, and represents a p a i r of conditions that

these c o o r d i n a t e s must satisfy, the solution for fixed A

and ZA' must be a 2-plane.

AA' M o r e o v e r it should be obvious that if x 0 represents any p a r t i c u l a r p o i n t satisfying

AA ' ~AA ' (2.5.1), then the general p o i n t s a t i s f y i n g this r e l a t i o n is x 0 +

where the spinor 1A is arbitrary.

So the location of the twistor Z6% is the 2-plane

c o n s i s t i n g of all the e n d p o i n t p o s i t i o n s of a complex vector IA A' springing from AA' the p o i n t x 0

Each such complex vector is null.

each such vector is o r t h o g o n a l to any other.

Moreover,

since ~

A'

. is fixed,

Thus Z d corresponds to a null 2-plane

in Minkowski space. A point W

in dual twistor space is r e p r e s e n t e d by a spinor p a i r

(OA

6%

Associated with W

T

A'

'

6%

) "

is a solution of the "primed" twistor equation

(2.5.2)

vA(A'D B') = 0

given by

(2.5.3)

0

By a n a l o g y w i t h P r o p o s i t i o n

A'

= T

A'

. A'A + ix UA

(2.4.2) it is not difficult to see that equation

gives the general solution of

(2.5.2).

The locus of the dual twistor W

6%

(2.5.3)

is given

by

(2.5.4)

T

the region w h e r e D solution to

A'

vanishes.

A'

= -ix

A'A

~A

l

A'A In this case if x 0 r e p r e s e n t s any p a r t i c u l a r

(2.5.4) then the general solution is given b y x 0

A'A + I A ' A

.

12

It is of interest to note that in complex Minkowski space there are two distinct systems of null 2-planes.

The so-called Q-planes are those null 2-planes which

correspond to twistors of valence

[~], i.e. the Zd-type twistors.

those null 2-planes which correspond to twistors of valence

The ~-planes are

[I], i.e. the Wd-type

twistors. Any two distinct Q-planes have a unique intersection point in complex Minkowski

space.

intersection

If the corresponding

point one must solve simultaneously A

(2.5.5)

. AA'

C01 = i x

the algebraic equations, AA'

A

rflA ,

co2 = i x

Assuming that ZIA' is not proportional equations

twistors are denoted Z 1 and Z ~2 then for an

to Z2A'

~2A'

, the unique solution to these

is given by the formula AA'

(2.5.6)

ix

A A' A A' A' = (~i~2 -~2~i )/(~IA,~2 )

,

as can readily be checked. In manifestly

twistorial terms the solution for the intersection point can be

represented by the skew product of the two twistors.

x~B :

(2.5.7)

where a normalization

d ~ 7~ B

In particular,

if we put

~ ~I

(ZIZ2-~ZI)/(ZIZ 2 d~ )

factor has been included so as to ensure that Xd~Id~ = 2 ,

then one finds 1 dAB -~ XdX e (2.5.8)

. A ] ix B'

X d@ L-ixA 'B

for the spinor parts of X d@ , after a short calculation. is represented

in twistor terms by a simple skew-symmetric

SA'B'J Thus a spacetime point x twistor x ~

that "simple" here means that X[~@X ~]~ = O) satisfying the normalization

(recall condition

X~@I @ = 2 , where Id~ is the infinity twistor defined in (2.3.6). The dual description of the same spacetime point is formed by taking Xd@ = 1 ~d@y6XT~

.

The complex conjugate

spacetime point ~a is described dually by the

AA'

13

complex

conjugate

twistor

is that the dual t w i s t o r

x~

=

xaB

x~

•

The c o n d i t i o n

should be equal

that a s p a c e t i m e

to the c o m p l e x

point

conjugate

should be real

twister,

i.e.

.

If X ~

and Y ~

represent,

according

to the d e s c r i p t i o n

given above,

the space-

a time p o i n t s

x a and y

, respectively,

(2.5.9)

-X~Y

~ =

then the q u a n t i t y

(xa-ya) (Xa-Ya) a

is the n o r m of the spacetime a

a

a

v -iw

where v

and w

are real,

That r e g i o n of c o m p l e x future-pointing

Minkowski

awkwardness)

the future - tube, and p a s t - p o i n t i n g

Twistor

An ~ - p l a n e

does not d e t e r m i n e

from e q u a t i o n

class of twistors

space

is a P o i n c a r ~

(CM)

(notwithstanding

Projective

twistor,

if x

=

then

p a r t of x a , w h i c h

w i l l be called

w h i c h w a is timelike

evident

In particular,

~i xaB~a 6 = W a W a

is the n o r m of the i m a g i n a r y

2.6

of the two points.

a

(2.5.10)

ological

separation

invariant

quantity.

for w h i c h w a is timelike some a p p a r e n t l y

and will be d e n o t e d will be d e n o t e d

and

unavoidable

CM +.

termin-

The r e g i o n

for

CM-.

Space.

(2.5.1),

a twistor

uniquely,

only up to an o v e r a l l

all of w h i c h are p r o p o r t i o n a l

and by p r o j e c t i v e

twistor

space

but rather,

scale

factor.

as should be An e q u i v a l e n c e

to each other is called

a projective

(PT) we m e a n the set of all such e q u i v a l e n c e

classes. It is clear Nevertheless, be d i v i d e d n o r m Z~Z

that p r o j e c t i v e l y

projectively

the sign of the n o r m still makes

is p o s i t i v e ,

in p r o j e c t i v e

the h o m o g e n e o u s homogeneous

a well-defined sense,

into three parts d e n o t e d PT +, PN, and PT- a c c o r d i n g zero,

or

twistor

coordinates

coordinates

norm.

and thus PT can

as to w h e t h e r

the

negative.

O f t e n we w i l l use the twistor class

a t w i s t o r does not have

space.

coordinates

Z~ to denote

the a s s o c i a t e d

equivalence

In that case we refer to the c o m p o n e n t s

for the c o r r e s p o n d i n g

has the m a r v e l o u s

point

catalytic

in PT.

effect

of Z a as

The systematic

of s i m p l i f y i n g

use of

m u c h of the

14

calculational work that crops up in algebraic geometry. A point W

in dual projective twistor space corresponds to a plane in PT.

plane consists of all those twistors Z d that satisfy W Z d = 0.

The

Note that the equa-

tion for the plane is completely scale invariant. The skew product Z[~'~]I m2 between a pair of projective twistors Z 1 and Z 2 corresponds to the complex projective line

(pl) which joints them.

correspond to points in complex Minkowski space.

~[~]

then we can normalize ml m2

'

A' A' If Z1 is not proportional to z2

so as to obtain the convenient representation of space-

time points given by equations A' tional to Z2

Thus, lines in PT

(2.5.7) and (2.5.8).

A' (If z1 is, in fact, propor-

then the skew product .[d.b] represents a point at infinity.)

~i ~2

The representation of lines in PT

(i.e. points in CM) by simple skew-symmetric

twistors - these being the "Plucker coordinates" for the lines - allows us to derive a number of interesting results concerning the geometry of PT, several of which will be mentioned here: 2.6.1

Proposition.

The intersection in PT of the line X ~8 and the plane W

is

represented by the twistor W X ~ Proof. plane W~ .

One must show that the twistor W X ~ Clearly the latter holds, since

lies both on the line X ~

(W X~8)W~ = 0.

on the line X d~ if and only if Z[~X 7@] = 0.

Now a twistor Z ~ lies

It follows therefore,

tic p-relations Xd[SX Y@] = 0 (i.e. the simplicity conditions) that W X ~

lies on X ~

2.6.2

and the

from the quadra-

by contraction with W

. []

Proposition.

The line X d~ lies entirely within PT + (that is to say, 0) if and only if the inequality

Z[~X ~Y] = 0 implies ZdZ

(2.6.3)

(W X ~ ) ( W Y ~ )

> 0

holds for every choice of a plane W Proof. plane

W

By Proposition

and the

line

X~

.

(2.6.1) above, W X ~ Clearly,

the

line

represents the intersection of the X~

lies

entirely

within

PT + i f

only if its intersection with any plane lies in PT +, which is precisely what asserts.[]

and

(2.6.3)

,

15

2.6.4

Proposition.

Lines w h i c h lie e n t i r e l y w i t h i n PT +

p r o j e c t i v e twistor space) c o r r e s p o n d to points in C M + Proof. Z~ =

The twistor Z ~ lies on the line X ~

AA v (ix ~A'

, ZA,)

for some choice of ZA'

(the "top half" of

(the f u t u r e - tube).

if and only if it is of the form a a Writing x a = v-iw , a short calcu-

"

lation e s t a b l i s h e s that A ~ w-

Z~Z a = 2w

(2.6.5)

~A~A,

N o w since ~AZA , is null and future-pointing,

it follows that Z ~ Z

> 0 for all

%A'

if and only if w a is timelike and future-pointing. [] --+ + lines lying in PT = PT U P N

Similarly,

future tube C M .

Lines in PT

c o r r e s p o n d to points in the "closed"

c o r r e s p o n d to p o i n t s in CM .

Lines w h i c h intersect

all three of PT +, PN, and PT- c o r r e s p o n d to p o i n t s for w h i c h w a is spacelike. M i n k o w s k i space points c o r r e s p o n d to lines in the h y p e r s u r f a c e PN.

Real

For further dis-

cussion see Penrose 1967, section VI.

C h a p t e r 2, Notes. i.

We require here various spinor identities, • -iSabcd

A

AB

= £

=

CACSBDCA,B, SC, D ,

,

A

WB AB

BA =

-S

See, for example, Pirani tions of spinor algebra.

SABSCDSA,C ,6B, D ,

A'

~B = w £AB

S

including the following:

,

Z

A'B'

= S

A'

ZB'

~.[ABsC]D ,

S

'

ZB' = Z

A'B' =

0

,

C

(1965), section 3, and P e n r o s e

SA'B'

B'A' =

-£

(1968a) for standard exposi-

P r o p o s i t i o n 2.1.2 appears in Penrose and M a c C a l l u m

(1972),

section 1.3.

2.

For treatments of the global geometry of c o m p a c t i f i e d Minkowski space, see Penrose

(1963), Penrose

~1965b) and, especially, P e n r o s e

lined treatment is o u t l i n e d in Penrose and Ellis

3.

(1973).

See Penrose

(1967), section V.

(1968a).

(1965a).

A somewhat more stream-

Also see the account given in H a w k i n g

CHAPTER 3 MASSIVE SYSTEMS AND THEIR 3 .i

INTERNAL SYMMETRIES

M o m e n t u m and Angular Momentum. A massive system, like a m a s s l e s s system, is characterized by its total momen-

tum and its total angular momentum.

Unlike the case for a m a s s l e s s system, however,

for a massive system it is not required that pa and M ab be related to one another d i r e c t l y in any special way.

All that is required is that the m o m e n t u m be timelike

and future-pointing, and that the angular m o m e n t u m behave a p p r o p r i a t e l y under translations. The angular m o m e n t u m can be expressed in the form

(3.1.1)

where ~

Mab = ~ AB ~ A'B' + ~A'B' AB

AB

is a symmetric spinor.

Under a change of origin in complex Minkowski space

the angular m o m e n t u m is taken to transform as follows:

AB

(3.1.2)

where q

AB ---~ ~

AA'

the old.

~(A B)B' - ~ B,q

is d e f i n e d to be the p o s i t i o n vector of the new origin w i t h respect to The c o m p l e x center of mass(1) of the system is the set of all points in C M

about w h i c h the angular m o m e n t u m vanishes. x

AA'

It is, accordingly,

given b y those points

w h i c h satisfy

(3.1.3)

AB

The general solution to equation

(3.1.4)

x~A'

where m is the mass,

= p(A xB)B ' B'

(3.1.3) is

= 2m

-2 AB A' D

PB

+ IP

AA'

and I is an arbitrary complex number.

The following result is

illustrative of the significance of the complex center of mass:

3.1.5

Proposition.

The spin-vector of a massive system is a measure of the

system's d i s p l a c e m e n t transverse to the m o m e n t u m into the complex.

17

The spin-vector

Proof.

. AB A'B' + -±~

is defined,

Writing

*M ab =

i~A'B'8 AB one obtains

(3.1.6)

S AA

Combining

as usual, by S a = *Mabp b.

(3.1.4) and

' = -l~. ABPBA' + ±~.-A'B'pA B'

(3.1.6) it follows directly that

(3.1.7)

x a = V a + i(<pa + m-2sa)

where v a = ~(xla + ~) a

and < = -i}(l - ~).

,

Thus m-2S a is the transverse displacement

of the center of mass into the complex. ~ The complex center of mass will be discussed where an expression

3.2

The Kinematical

at greater

length in Section

in explicit twistor terms will be derived for it.

Twistor.

The momentum and angular momentum of any system, massive or massless, expressed

in terms of a certain type of symmetric

system's kinematical

twistor.

(3.2.1)

A~

where P

AA'

sufficient

is the momentum, condition

and ~

AB

is the angular momentum

A~I~y

is the complex conjugate

(3.2.3)

A ~

twistor of valence

can be

[~] called the

It is defined as follows:

for a symmetric

(3.2.2)

where A ~

3.5 ,

=

twistor A ~

= A~yI ~

of A ~

spinor.

to be of the form

A necessary and (3.2.1) is that

,

, given in the spinor parts by

IO, pA

PA

B'

1

2i~A'B' B

For a massive that A ~ I ~ system,

system we require that pa be timelike and future pointing.

W W ~ must be greater than zero for any choice of W~

on the other hand, we require

that PAA'

= ~A~A ,

and ~AB

This means

For a massless IW(AZB) = . -

for some

18

choice of Z ~ =

( A , ZA,) .

In fact, a kinematical

twistor describes

a massless

system if and only if there exists a twistor Z ~ such that

(3.2.4)

A~

= 2Z(~I ~)Y~'y

which is equivalent to conditions

3.3

The Decomposition

'

(2.1.3) and

(2.1.4).

of Massive Systems into Massless Subsystems.

A very curious fact about massive

systems is that they ean always be regarded as

being composed out of a set of two or more massless systems are the "twistor constituents"

3.3.1

Theorem.

subsystems.

These massless

sub-

of the associated massive system.

For any integer n > i, given a massive system with m o m e n t u m pa AB

and angular m o m e n t u m

one can find a set of n ZP~ systems Z~l (i = 1 .... ,n) such

that

(3.3.2)

pa = ~ pa l i

where P91 and ~

described

~AB '

AB = ZP i . 1

'

label the momenta and angular momenta of the various

ZRM systems

by Z . . l

Proof. decomposed

First,

it will

be d e m o n s t r a t e d

into a pair of ZRM systems.

Let ~a be any unit spacelike vector

P1a = (pa + m~a)/2

(3.3.3)

It follows

how a m a s s i v e s y s t e m {pa , BAB} c a n be

immediately

(~a ~a = -i) orthogonal

,

P2a =

to pa.

Put:

(pa - m~a)/2

that both of the momenta P9 (with i = i, 2) are null, and i

thus that a -1A A' P1 = ~ ~i

(3.3.4)

A' A' for some choice of ~i and ~2 A' ~i

")

Now ~AB

Expanding

(3.3.5)

'

a -2A A' P2 = ~ ~2

(We have written -iA for the complex conjugate of

' being symmetric,

must be of the form ~(A~B)

for some A

-2B ~B in the spinor basis generated by -IB and ~ , one obtains

~B = @i IB- + @2~2B

and ~B .

~9

for some choice of 81 and 82 .

And then we have:

uAB = 81 ~(A~B)I + 82 ~(A~B)2

(3.3.6)

Thus equations

(3.3.2) hold

with pa given as in '

(3.3.7)

(3.3.4)

and with

i

U? = 8ie(AgB)i

Our two ZRM systems are accordingly

•

~? = 82~(AgB)2 (81 ~A , ZIA,)

then given by Z 1 =

and Z 2 =

A (82~

, Z2A,)To decompose

a massive

mutually orthogonal

vectors

system into three twistors,

take a pair ~a , D a of unit

lying in the 3-space orthogonal

to the m o m e n t u m and form

the three null momenta p.a g i v e n by(2) :

1

a =

PI

a = P2

(3.3.8)

(pa + m

a

~

{3m

+ ~-~

(pa + m a ~ 2"~ -

a

)/3

Da)/3

a

P3 =

(p2 _ m ~ a ) / 3

of which pa is obviously for an appropriate

a -IA A' Put P1 = ~ Z1

the sum.

triplet of spinors ZA'i

Writing,

can clearly put ~B = [e.~Bi for some choice of @.. • 1 1 l (3.3.9)

~B

we see that equations

= @i

(3.3.2)

(A~B)I

a -2A A' ' P2 = ~ 72

a -3A A' , and P3 = Z Z3

as before,

AB = ( A ~ B )

one

Hence, defining

, etc.

are satisfied,

as desired.

The three twistors

Z~ are 1

then defined by

(3.3.10)

Z ~i =

(0i ~ A

,

WiA ')

This method can by iteration be extended to decompose any number of twistors, subsystems,

one now recombines

system of the original null subsystems, subsystems.

as follows:

system.

Having split the massive

two of the null subsystems

system into

system into three null

to form a massive

sub-

Then that massive subsystem can be split into three

giving us a splitting of the original

This process

a massive

system n e w into four null

can be repeated over and over until the desired number of

20

null subsystems is a c h i e v e d . ~ ] Suppose that a massive system A c~ has b e e n decomposed into a collection of massless subsystems denoted A~.$ where i = i, ..., n. 1

On account of the linearity of

the kinematical twistor in m o m e n t u m and angular m o m e n t u m it follows that Ad$ = ~A.~ . i i

Each massless subsystem is described by a twistor Z? for some value of the m

index i.

The c o r r e s p o n d i n g complex conjugate twistors will be denoted Z i , raising

the index i.

It is useful to treat these indices according to the usual rules of

tensor algebra, adopting the summation convention,

and so forth.

k i n e m a t i c a l twistor of the complete massive system one has

(3)

Then for the

:

where now the contributions from all the various ZRM subsystems are a u t o m a t i c a l l y summed over.

3.4

Internal Symmetries. Expression

(3.3.11) can be regarded as the natural starting p o i n t for the de-

velopment of the twistor a p p r o a c h to elementary p a r t i c l e physics.

It shows h o w the

m o m e n t u m and angular m o m e n t u m of a massive system can be built up out of a set of twistor constituents.

A c c o r d i n g to T h e o r e m

(3.3.1) there will exist, for any k i n e -

matical twistor A ~$, a set of twistor constituents Z ~ such that A d~ is given by exi pression

(3.3.11).

An important p o i n t to notice is that a massive system A ~ unique set of twistor constituents.

does not determine a

Linear t r a n s f o r m a t i o n s of the form

Zi

~j=8 + -~8~J + R~i~ j ~ija$

{i

÷ ~$i~j ~ijz~ ej $ + ~ j

(3.4.1)

can be made such that--when RBi and twistor A ~B , as given in equation

3.4.2

Theorem.

13 are suitably r e s t r i c t e d - - t h e k i n e m a t i c a l (3.3.11), is left invariant.

AI m7 and z~ that p r e s e r v e the Linear t r a n s f o r m a t i o n s acting on z. l A

A'-i m o m e n t u m z. ~ are of the form: i A

21 A' --~ i

(3.4.3) where

j A' i ]

~,

j -k

U~I i s u n i t a r y Proof.

-iA

~.~.

7

-1-" " u_~ jA 3

--÷

@k). ]

( i . e . UiU j =

It should be evident that if the m o m e n t u m is to be preserved A -iA' must pick up no terms involving ~. or ~ 1

when transformed,

linear transformation

satisfying

this condition,

n l then zi

The most general

whilst maintaining

the conjugacy

A' ~iA relations between 7. and is given by l A' l .

(3.4.4)

-iA

Under this transformation

.~A' ~j'B

zAi:jB

-ijA B '

-ijB

the m o m e n t u m transforms

A'-iA --÷

+

(RA'J -ikA B' C' i B'Sc ' )Tj 7 k +

In order for the m o m e n t u m to be preserved, ing on the right of

(3.4,5) must survive,

that only the first term survives, A'-iA 1 =

7.

(3.4.6)

A for all values of 77 1 lowing degenerate

'

as follows:

(RA' j -Ai A' -ijA B'~kC i B'-~C + SikcSB' )Tj

~,1

(3.4.5)

_A'j B' i 0 . --

i.e. the domain for which

Thus in order to characterize a massless field as p o s i t i v e frequency it

--+ suffices to restrict ones attention to PT

By requiring f(Z ~) to have suitable

---+ analyticity p r o p e r t i e s in PT one can ensure that the related field is w e l l - d e f i n e d throughout ~ +

, and thus of positive frequency.

This brings us back to the problem, m e n t i o n e d in Section 4.3, of the domain on w h i c h f(Z ~) is to be defined, ought to exhibit.

and the question of what sort of singularities f(Z ~)

Part of the reason why this p r o b l e m is so d i f f i c u l t is due to

the fact that the q u e s t i o n is not p o s e d quite correctly.

For the sort of objects

that are b e i n g dealt w i t h here are not really functions at a l l - - a t least, in the standard sense--but rather, are elements of the sheaf eohomology group

(4.5.1)

HI ( ~ + , 0 (-2s-2))

w h e r e 0(-2s-2) -2s-2. of

is the sheaf of germs of h o l o m o r p h i c functions, h o m o g e n e o u s of degree

A twistor function f(Z ~) p r o v i d e s

(4.5.1).

,

a "representative cocycle"

for an element

It is p o s s i b l e to have several distinct twistor functions, all defined

over d i s t i n c t domains, all of w h i c h yet are r e p r e s e n t a t i v e s for the same element of the cohomology group

(4.5.1)Jthis

is why, in the older twistor literature,

for a

specified ZR~ field the domain of the c o r r e s p o n d i n g twistor function seems a bit "shifty ~'.

And it was only in 1976 that the m a t t e r b e g a n to clear up, and it emerged

that positive frequency analytic m a s s l e s s fields of h e l i c i t y s corresponded to

37

elements of (4.5.1), thereby specifying precisely the relationship between such fields and the complex analytic geometry of twistor space. point see Chapter i0 here, and also Penrose Techniques in Theoretical Physics (Eds.:

For discussion on this

(1977), and the book Complex Manifold

D. Lerner and P. Sommers; Pitman, 1979).

We shall return to matters of sheaves and cohomology later.

Let us now con-

sider more explicitly the sort of condition that must be imposed

(and that can be

refined and spelled out more explicitly within a sheaf theoretic framework) on a twistor function in order to ensure that the field it generates has positive frequency.

It is necessary first to build up some apparatus useful in evaluating

contour integrals.

The following result--well known, in a slightly disguised form,

from elementary complex analysis--is of fundamental utility: 4.5.2

Lemma.

Let ~A' and ~A' be a pair of fixed spinors: then the contour

integral formula

(4.5.3)

2Ti(~A,~A')-I = /

(~A' ~A,~ B' TB,)-IA~

is Valid, where the contour surrounds the pole ~B'~B, = 0 once in the positive sense (or, equivalently, Proof. zero, z

A'

surrounds the pole d

A'

TA, = 0 once in the negative sense).

Since the differential form to be integrated in homogeneous of degree

can be scaled such that in a suitable basis its components are given by

(4.5.4)

Z

A'

= (~ , i)

,

and the associated differential form AT (defined in equation 4.3.6) is given by

(4.5.5)

AT = -dl

Writing, in the same basis, ~A' =

.

(a,b) and ~A' = (f,g), it is straightforward

to see that

(4.5.6)

~

A'

TA,~ B T B ,)-lAw = i ( a l

+ b)-l(fl + g)-idl .

Thus, taking the contour to surround the pole at i = -g/f , elementary calculus of residues shows that the result of the integral is

38

A i -i

2~i(ag - bf) -I = 2~i(~A,~

(4.5.7)

)

[]

Armed with this lemma, we can examine the twistor function

(4.5.8)

f(Z ~) = (P Z~Q~Z~) -I

and see what sort of field it gives rise to.

,

It is worth noticing that if P

is

any fixed dual twistor with spinor parts given by

(4.5.9)

P~ = (PA ' P

then the following formula is valid

pxpZd

(4.5.10)

where p

A'

(x) = ix

A'A

PA + P

A'

AI )

(cf. equation 3.5.17):

= P A' ~A'

'

is the solution of the primed twistor equation

Section 2.5) associated with the dual twistor P

(cf.

Accordingly, we have the

identity

px(p Z~Q~Z~ ) -i = (pA' ~A 'qB' ~B,) -i

4.5.11) A i

where p

,

A i

and q

are solutions of the primed twistor equation.

Inserting this

identity into the contour integral formula

(4.5.12)

~(x) = / p x f ( Z @ ) a ~

,

we can apply Lemma 4.5.2 in order to evaluate the field @(x), obtaining: A i -1

(4.5.13)

@(x) = 2~i(PA,q

)

It is straightforward to verify--using the primed twistor equation--that @(x) satisfies the wave equation.

But what conditions must be imposed in order to ensure

that @(x) is of positive frequency? the correct answer.

A geometrical argument can be employed to give

Note that the twistor function f(Z ~) given in (4.5.8)

singular on the plane P

and on the plane Q 4 the total number of internal unitary group

built in accordance with the nested sequence 5.5.16

is 4n-6.

For certain purposes it is useful to consider schemes alternative to the "standard" arrangement just described.

Examples of various alternative schemes arise, as we

shall see, in Chapters 7 and 8, in the context of studying both hadrons and leptons. states.

Alternative schemes must be employed also in the analysis of many-particle

CHAPTER 6 THE LOW-LYING BARYONS

6.1

The Quark Model. Most of the known elementary particles fall into two broad classes: leptons and

hadrons.

The only particles which fall outside of these classes are the photon, the

hypothetical graviton, and the hypothetical "weak" bosons Z-particle).

(the W-particles, and the

In view of increasing evidence favoring a unification of the weak and

electromagnetic interactions, it is not unfair to say that the weak bosons and the photon should "morally" be thrown in with the leptons.

Then--ignoring the graviton

(whose presence in the world is a bit of a nuisance for particle physicists)--there are two broad classes of particles, i.e. leptons and hadrons, and each class can be subdivided into two subclasses: bosons and fermions. fermions have half-integral spin.

Bosons have integral spin and

Bosonic hadrons are called

mionic hadrons are called "baryons".

"mesons", and fer-

In this chapter we shall be concerned with the

twistor representation of those baryons that lie on the lower end of the mass spectrum. Hadrons can be regarded as--at least in some sense--being built up out of certain fundamental units called "quarks". Each baryon, for example, is composed of three quarks.

There are several different kinds of quarks--at least three, probably more--

and so by choosing various combinations of three quarks, various kinds of baryons are manufactured.

Many people like to think of quarks very literally, and regard

baryons as being composite particles.

In what follows we shall adopt a more con-

servative (and more reasonable) approach, and regard quarks purely from an abstract v i e w p o i n t p a c c o r d i n g to our view, the quark model merely provides a convenient descriptive language for many of the observed group-theoretical aspects of hadron phenomenology, and the remarks which follow should be interpreted in just that

(1) sense The "old" hadrons--in particular, those hadrons known before 1974--can be described in terms of three distinct types of quarks; for many of the post-1974 hadrons, it appears that at least four types of quarks are required.

For the moment,

57

let us consider just the first three types of quarks. guished by varying assignments of quantum numbers.

These quarks are distin-

In addition to these three

"light" quarks, we also have the three corresponding antiquarks.

Each quark state

has spin 1/2: the quarks have unprimed spinor indices, and the antiquarks have primed spinor indices.

The three quark states will be denoted

(6.1.1)

~ i A = (uA

dA

s A)

and the three antiquarks will be denoted A A' ~i' = (u , d A'

(6.1.2)

A' , s )

,

where the symbols u, d, and s stand for "up", "down", and "strange" to some accounts,

"sideways").

The use of the symbol i A

(or, according

for the quark triplet,

A' and the symbol ~. for the antiquark triplet, is for the sake of a bit of stylistic l augury. In hadron dynamics there are two quantum numbers which are always strictly conserved--the electric charge and the baryon number; and if a state composed of several quarks is represented by a product of quark states, then the total electric charge and baryon number are obtained by summing the values for the various constituent states.

In strong interactions and electromagnetic interactions

weak interactions)

(but not

another quantum number is conserved, which is analogous to elec-

tric charge in certain ways, called the hypercharge.

Each quark state is assigned a

particular electric charge, hypercharge, and baryon number--and,

for various reasons

which can be justified in many ways, these quantum numbers take on peculiar fractional values:

uA

quark state:

dA

sA

u A'

d A'

s A'

charge

2/3

-1/3

-1/3

-2/3

1/3

1/3

hypercharge

1/3

1/3

-2/3

-1/3

-1/3

2/3

baryon number:

1/3

1/3

1/3

-1/3

-1/3

-1/3

6.1.3

Quark quantum number assignments.

58

It is c o n v e n i e n t

A

to r e g a r d the u

states of an "isospinor"

quark state n

n aA =

( u A , d A)

n A' a =

(uA'

quark and the d A quark as b e i n g aA

according

(6.1.4)

a,b,..,

space indices).

are called

A product

said to be in a d e f i n i t e w i t h respect

of various

indices;

indices

(not to be c o n f u s e d w i t h M i n k o w s k i

quark isospinors isospin

the total

isospin

of a state.

sA counts

and symmetric the number

as an isoscalar,

eigenvalue

types of w e l l - e s t a b l i s h e d

by their h y p e r c h a r g e

is

(Y) and total isospin;

and

is denoted

A .......................

Y=

1

I = 3/2

N

.......................

Y = 1

I = 1/2

.......................

Y=

0

I = 1

A .......................

Y = 0

I = 0

Z .......................

Y=

.......................

Y = -2

For m o s t of these types there are m a n y examples includes

the p r o t o n

similar

and the neutron,

to the p r o t o n

v a l u e s of the basic q u a n t u m numbers Since b a r y o n s structing baryons of three quarks:

-i

these types

known:

I = 1/2 I=

0

type N

("nucleon"),

as w e l l as m a n y e x c i t e d

and n e u t r o n

states,

for exor

i n a s m u c h as they have the same

B, Y, and I.

have B = I, and quarks have B = 1/3, the simplest w a y of conis out of three quarks.

Here we list all p o s s i b l e

I.

baryon

as follows:

(6.1.5)

"resonances",

isospinors

is then o n e - h a l f

The isospin

in nature six d i s t i n c t

distinguished

and a n t i q u a r k

if it is t r a c e f r e e

The strange q u a r k

to the isospin

There are o b s e r v e d

are labelled

isospin

indices present.

does not contribute

states,

, d A' )

state of total

to its isospin

of free isospin

ample,

to the scheme

(a = i, 2)

The indices

(B=I)

two d i s t i n c t

combinations

59

dA d B d C

u

A

d

B

d

C

u

uA d B

d A d B sC

d

A

s

B

s

These

combinations

s

B

Combinations

are evidently

u

u

A

obtained

u

C

C

s

6.1.6

s

A uB dC

s

A

s

B

s

A

u

B

s

A

u

B

u

C

C

C

C

of Three Ouarks

by considering

the

"supermultiplet"

configuration

(6.1.7)

A i Bj Ck

and allowing

the i n d i c e s

symmetrizing

over

to the

j, a n d k to r a n g e

the s p i n o r

t a i n a set of t e n s p i n according

i,

3/2 s t a t e s .

following

A-

indices

A°

Z-

over the values

i,

in e a c h of the c o m b i n a t i o n s The q u a n t u m

numbers

of t h e s e

2, a n d in

3.

By

(6.1.6), w e o b -

states

are g i v e n

scheme:

A+

Z°

A ++

...................

Z + ........................

Y = i, I = 3/2

Y = 0, I = i

(6.1.8) ..........................

~The s u p e r s c r i p t these

ten

states

spin

.............................

on e a c h s t a t e 3/2 s t a t e s ,

denotes

the e l e c t r i c

it is a l s o p o s s i b l e

f r o m the c o m b i n a t i o n s

listed

in

(6.1.6),

Y = -i,

I = 1/2

Y

I

charge

=

-2,

value.

to f o r m e i g h t the quantum

=

0

In a d d i t i o n

independent numbers

to

s p i n 1/2

of w h i c h

a r e as

follows:

N°

(6.1.9)

~-

N+

zo Ao ~o

These

eight

binations

states

comprise

corresponding

........................

Y = i, I = 1/2

Z+

Y = 0,

.....................

........................

the basic

"baryon octet",

to t h e m are g i v e n

explicitly

I Z : i, I A : 0

Y = I, I = 1/2

and the various below:

.

s p i n 1/2 c o m -

8O

dAdBU

B

N°

...........................

B

+

UAdBU

...........................

N

dAdBSB

...........................

E-

u (AdB) sB ...........................

(6.1.10)

(proton)

o

B

+

UAUBS

...........................

SAuBdB

........................... i °

SASBdB

...........................

E

Z-

B SASBU

(neutron)

_o ...........................

Note that there are two linearly independent ways of ~educing the spinor comb i n a t i o n uAdBs C down to a spin 1/2 state: one of these gives the I = 1 state E and the other gives the I = 0 state A ° . and

The isospin m u l t i p l e t content of

o

,

(6.1.9)

(6.1.I0) can be recorded more explicitly as follows:

a

cB

b

n A nB n

8bc

.........................

N

(isodoublet)

(a b) B n (AnB) s ............................. E

[isotriplet)

(6.1.11) a nB

B s

a

nB n

where 8

ab

sA

bB

..............................

sA 8 a b

Z

......................... A

(isodoublet)

(isosinglet)

is the a n t i s y m m e t r i c isospin "epsilon" tensor.

Suntming up, we see that from the s u p e r m u l t i p l e t c o n f i g u r a t i o n

(6.1.7) we obtain

a spin 1/2 octet, and a spin 3/2 d e c i m e t - - a n d it is indeed truly remarkable fact is perhaps the true essence of the role of SU(3) in physics)

(this

that the lowest

lying b a r y o n states group themselves n a t u r a l l y into a spin 1/2 octet and a spin 3/2 decimet. in Table 6.I.

Some of the basic p h y s i c a l p r o p e r t i e s of these states are summarized It is important to notice that all of the octet m e m b e r s are

stable to strong decays, and are unstable

(with the exception of the proton, w h i c h

is completely stable) only to w e a k / e l e c t r o m a g n e t i c decay.

A m o n g s t the decimet m e m -

61

Table 6.I The Low-Lying Baryons

Particle

Mass

(MeV)

Mean Life or Full Width

Decay Modes

stable

P

938.2796(27)

n

939.5731(27)

A

115.60(05)

2.632(20)x10

E+

1189.37 (06)

.802(5)xi0 -I0 sec

918 (14) sec -10 sec

pe-~

64.2 35.8 8.07 (28)xi0 -4 1.57 (35)xi0 -4 .85 (14) xl0 -3

pzO

51.6 48.4 i. 24 (18)x10 -3 .93 (i0) xl0 -3 2.02 (47)xi0 -5

pY n~+y Ae+y

Z

~O

1192.47(08)

5.8(1.3)x10 -20 sec

i19~.35(06)

1.483(15)xi0 -10 sec

1314.9(6)

2.90(i0)xi0 -I0 sec

100%

p~n~ o pe-~ p~-~ p~-y

nz +

2°

Fraction

Ay Ae+e nzne-~ n~-~ Ae-~ nw-y A~ o

Ay

~100% 5.45xi0-3 ~100% 1.08(04)xi0 -3 .45(04)xi0 -3 .60(06)xi0 -4 4.6(6)x10 -4 ~100% O.5±O.5%

E

1321.32(13)

1.654(21)xi0 -I0 sec

ATie-~ A~-v

£(1232)

1230-1234

110-120 MeV

Nw

~99.4

Z (1385)

[+]1382.3(4) [011382.0±2.5 [-]1387.5(6)

AT Z~

88±2 12±2

H (1530)

[011531.8(3) [-]1535.0(6)

Ew

100%

~o~-

100%

(16'72)

1672.2(4)

35(2) MeV ~35 MeV 40(2) MeV 9.1(5) MeV 10.1±1.9 MeV +0 4 -10 1.1_oi3XlO sec

E -~o AK-

~100% 0.69(18)xi0 -3 (3.5±3.5)xi0 -4

62

bers, only the ~

particle

(whose existence was predicted by Gell-Mann on the

basis of SU(3) theory, and subsequently confirmed with great drama) is stable against strong decay,

and the remaining states are observed as resonances in various

strong interactions.

Aside from the basic octet and decimet members listed in

Table 6.I many additional baryon states have been observed.

These also seem to

clump together naturally into octet and decimet configurations

(as well as some

possible A singlet states), as will be discussed in Chapter 7.

6.2

The Three-Twistor Model for Low-Lying Baryons. The transition from the quark model to twistor theory is achieved by inter-

preting the "quark configuration structure" of a hadron state as the "spinor coefficient structure" which appears in the contour integral formula twistor functions to field multiplets.

(5.2.2) relating

According to this view the operator

^ iA

A' is interpreted as a quark triplet, and we interpret ~. as an antiquark triplet 1 (although this latter interpretation must--as we shall see--be suitably qualified). Thus the fields constituting the low-lying baryon supermultiplet are given by the following contour integral formula:

(6.2.1)

-ABC =

If f (Z) is in a definite quantum state, then a particular member of the supermultiplet

i j k will be picked out uniquely as being non-vanishing. ~ABC

For example,

if

f(Z) should be a proton state, then--selecting the correct spinor coefficient structure using

(6.2.2)

(6.1.10)--the only non-vanishing field will be given as follows:

~A =

xUAdB uB f(Z)A~

,

where UA ' dA ' and SA are the three components of the operator ~

6.3

.

Electric Charge, Hypercharge, Baryon Number, and Isospin. What remains to be shown is how to construct in explicit twistor terms the

various hadronic observables which we require f(Z) to be put into an eigenstate of. For our twistor triplet Z~ l

(i = 1,2,3) we shall write

63

(6.3.1)

Z ~, = (U e , D ~ , S e) l

and analogously

the three operators

î Z e

(= -~/~Z~)

will be labelled according to the

scheme

•

(6.3.2)

~

A

: (~e ' ~e ' s )

with:

(6.3.3)

U ~ = -~/~U~

The three twistor \

operators

(6.3.4)

G = _~e~

are the "total occupation

De = -$/~D~

+ 2

d=

numbers"

S ~ = -~/~S e

- DC~~)0~ + 2

"s = _SC~e ~ + 2

for the three types of q u a r k s - - e . g . ,

the number of u-quarks minus the number of u-antiquarks. charge,

u measures

The baryon number,

hyper-

and electric charge are then given by :

(6.3.5)

i (~ + ~ + ~) { : S

(6.3.6)

~=ii

(G+~_2s)

(6.3.7)

6 = ~i

(2~-~

The three generators formations

=-s+

-

for infinitesimal

s) =Au -

isospin transformations--i.e.,

8U(2) trans-

applied to U ~ and D e --are as follows:

i (u~)e De~e) ~i : - i +

% = }(oe
_

(s + ~-)

>_

(s +

1/2 states

in fact,

; one o b s e r v e s

singlet

serves

an o c t e t of S = 2 mesons;

none of the states

an e x t e n d e d

scheme

w i t h those

states.

to enable

it to a c c o u n t

states--in

it admits

the various

6.6

The A b s e n c e

of Color Degrees

One p o i n t w h i c h needs concerns

the role of

quark model, Now,

one assumes

regardless

quirements

SU(3)

of Fermi

statistics

One o b s e r v e s one observes

scheme);

are c o m p a t i b l e

one ob-

state)

an

with a descrip-

our f r a m e w o r k m u s t be g e n e r a l In the next c h a p t e r w e shall

for a greater

w h i c h exhibits

a

v a r i e t y of h a d r o n i c

states m e n t i o n e d

above w h i c h are

are considered.

of Freedom.

in h a d r o n i c

that b a r y o n s

of the nature

listed above.

of six twistors)

to be s t r e s s e d here,

"color"

of h a d r o n s - -

hadronic multi-

(as p a r t of a m i x e d

Accordingly,

(based of functions

ruled out w h e n o n l y three twistors

description

in a t h r e e - t w i s t o r

flexibility sufficient particular,

for the descrip-

b a r y o n d e c i m e t s w i t h S > 3/2;

just m e n t i o n e d

for these additional

adequate

in nature v a r i o u s

and one o b s e r v e s

tion b a s e d e n t i r e l y on three twistors.

outline

observe

(which are not a l l o w e d

ized so as to a c c o u n t

although

for a general

are incompatible

baryon

singlet:

scheme,

w i l l not suffice

plets w h o s e q u a n t u m n u m b e r s

S = 1 meson

O, l, or 2

w h i c h has not b e e n m e n t i o n e d

structure.

are e f f e c t i v e l y

of the forces b i n d i n g

bound

In the s t a n d a r d "naive" states

the quarks

d e m a n d that the quarks

have

yet,

of three quarks.

together,

in a d d i t i o n

the re-

to their

69

flavor degrees of freedom (viz.: up, down, and strange ...) also three their disposal.

colorz' at

Thus, according to the color hypothesis there are nine distinct

kinds of light quarks--each labelled by a color and a flavor.

Within a baryon the

quarks are put into a color singlet state, i.e., into a state which is totally antisymmetric with respect to its color SU(3) indices.

With this assignment baryons

then automatically possess the correct flavor and spin symmetries (i.e., totally symmetric with respect to clumped flavor SU(3) and spin indices). pothesis can then be taken one step further.

The color hy-

An octet of color SU(3) bosons

(called "colored gluons") is introduced, and it is hypothesized that the subhadronic quark binding forces are due to the exchange of virtual gluons--the resulting theory goes by the name of "quantum chromodynamics".

There is a certain

amount of evidence in favor of QCD, but this evidence rests on such a plethora of assumptions that--to the critical eye--it is not very convincing.

The theory's

chief merit is its elegance and its aesthetic simplicity.

Within twistor theory baryons are not assumed to be in any literal sense built up out of bound states of quarks, and consequently the color hypothesis is unnecessary.

Of course, if there are no color degrees of freedom, then there are no

colored g l u o n s - - a n d thus it is not obvious at all how one might begin to formulate a theory of strong interactions in twistor terms. might be followed towards this end. diagrams

(2)

There are several routes that

One approach would be to study twistor

, or appropriate generalizations thereof, and try to build up reasonable

expressions for hadronic scattering implitudes. evitably anticipate links with Regge theory.

In this connection one would in-

Another route to take, perhaps of a

more speculative character, would involve looking at deformations (3) of the complex analytic structure of the space of three twistors (or, as it may turn out, suitably related higher dimensional spaces).

Although it is not at all evident

how one would go about describing strong interaction phenomena in terms of such deformations, the utility of such an approach has been demonstrated admirably in a variety of non-linear problems

(Penrose 1976; Ward 1977a and 1977b; Atiyah and

Ward 1977; Atiyah, Hitchin, and Singer 1977; Hartshorne 1978; etc.) and it is not

70

unreasonable to p r o p o s e that hadronic interactions m i g h t be amenable to treatment by means of this sort.

Finally,

any a p p r o a c h to strong interaction physics re-

quires a d e t a i l e d knowledge of the "internal" geometry of hadrons.

In Chapter i0

some of the material n e c e s s a r y towards this end is presented; but clearly, tion to this, knowledge of a m u c h more specific character is needed.

in addi-

It is w o r t h

noting that an o p e r a t o r analogue of the center of mass twistor described in Section 3.5 can be c o n s t r u c t e d

(Hughston and Sheppard, 1979), and in a d d i t i o n to

the center of mass operator for the t h r e e - t w i s t o r system as a whole, in the case of hadrons, we also have three "partial" center of mass operators c o n s t r u c t e d from the three two-twistor subsystems.

It is not u n l i k e l y that these operators should

play a significant p a r t in u n d e r s t a n d i n g various aspects of the structure of hadrons. In particular,

the role of the center of mass operator in d e t e r m i n i n g the p r o p e r -

ties of the m a g n e t i c moments of hadrons n o w seems to be firmly established.

Chapter 6, Notes

i.

For standard discussions of the quark model and SU(3) see, for example,

Gell-Mann and N e ' e m a n

2.

(1964), Dalitz

(1969), and F e y n m a n

Twistor diagrams were introduced in Penrose and M a c C a l l u m

discussed at length in Penrose see, for example, Sparling Ryman

(1966), Feld

(1975), and Huggett

(1975a, pp. 330-369).

(1974), S p a r l i n g (1976).

(1972).

(1972), and are

For additional d i s c u s s i o n

(1975), Hodges

(1975), Harris

(1975),

A number of articles on twistor diagrams have

been w r i t t e n by A.P. Hodges for Twistor Newsletter, and in the same reference one can find an article by S.A. Huggett and M.L. Ginsberg d i s c u s s i n g the cohomological i n t e r p r e t a t i o n of certain classes of twistor diagrams.

In P o p o v i c h

(1978) one finds

a good summary of m a n y of the heuristic aspects of the analysis Of twistor diagrams for hadronic,

leptonic, and semileptonic processes.

A l t h o u g h we shall not be en-

tering into a d i s c u s s i o n of the m a t t e r here, it is perhaps w o r t h noting that there exist a number of interesting formal c o r r e s p o n d e n c e s between twistor diagrams and d u a l i t y diagrams.

A useful reference on dual theory is Jacob

(1974).

Basic

71

references to duality diagrams include Harari and Matsuoka et al

(1969).

(1969), Rosner

(1969), Neville

(1969),

Higher order duality diagrams, which also fit into the

twistor framework [where "quark loops" correspond to "helicity flux loops" in twistor diagrams], are discussed in Kikkawa et al (1969).

There is something

very curious and combinatorial about the theory of duality diagrams, suggestive of some of the principles involved in spin-network theory

[Penrose 1971a and 1971b;

also see the Twistor Newsletter articles on spin-networks by S.A. Huggett and J.P. Moussouris], and more investigation in this area is certainly called for.

3.

Standard references for the theory of deformations of complex analytic struc-

tures include Kodaira and Spencer (1958), K0daira and Spencer (1960), and Morrow and Kodaira

(1971).

It is first suggested in Penrose

(1968b) that gravitation is

in some sense due to a shift in the complex analytic structure of twistor space.

CHAPTER 7 MESONS, RESONANCES, AND BOUND STATES

7.1

The Low-Lying Mesons. Among the observed low-lying meson states two nonets stand out as particularly

striking.

These include a spin zero nonet of negative

one nonet of negative

intrinsic parity.

intrinsic parity,

and a spin

At the level of the naive quark model,

these nonets can be represented by quark-antiquark

pairs.

Since quarks and anti-

quarks are both spin 1/2, pairs of such states can be either spin 0 or spin i, assuming no orbital angular momentum

(cf. Section 7.4) between the quarks.

Table 7.I one finds a list of the relevant with the hypothesized

quark structure

states comprising

these nonets,

In together

for each case:

Table 7.I The Quark Structure of the Low-Lying Mesons

7T+

uA~A

Q+

u (A~B)

To

UAUA _ dA~A

QO

u (A-B)u - d (A~B)

~-

dAuA

p-

d (AuB)

+

Au sA

K +*

u (AsB)

K°

dAsA

K °*

d (AsB)

K

s K

-

~'

Each

A

s

A-

K-*

s uA

s

(A-B)

u

u A-u A + dA~A - 2sAs a

0J

u (A-B) u + d (A~B)

uAuA + dad A + sAsA

~

s (AsB)

nonet has a pair of I = 0 members.

For the spin 0

~' , and for the spin i- nonet we have w and 4posed of an SU(3) octet and an SU(3)

singlet.

nonet we have D and

This is because each nonet is comThe octet has an £ = 0 state, and

73

the singlet has I = 0; this makes that the o b s e r v e d

I = 0 states

No one knows w h a t the p r e c i s e mixing:

therefore,

are m i x t u r e s principles

the quark

are to some e x t e n t ad hoc.

for two I = 0 states altogether.

structures

It is p o s s i b l e

of p u r e octet and singlet

components.

are w h i c h g o v e r n the p h e n o m e n o n for these

states

as listed

of

in Table

7.I

If we write

= cos0{8}

+ sin@{l}

(7.1.1) N' = -sin@{8}

+ cose{l}

and

¢

= cos@{8}

+ sine{l}

(7.1.2) = -sin@{8}

and assume

the G e l l - M a n n m a s s

the f o ~ l o w i n g values

+ cos@{l}

formula

for the m i x i n g

(cf., however,

Section

5.3)

then we obtain

angles:

@lin

(7.1.3)

equad

-24±1 °

-ii±i °

38±1 °

40±1 °

i where

eli n is the angle

the m e s o n masses,

and 0

is b e i n g v e r y elusive properties modes

quad

if G e l l - M a n n

is the result

the P a r t i c l e

the reader

Data Group.

is a s s u m e d

to be linear

in the q u a d r a t i c

In Table

are summarized.

for w h i c h a d e f i n i t e

information

formula

obtained

about the whole matter.

of the l o w - l y i n g m e s o n s

are listed

further

obtained

The d a t a here comes

Nature

7.II a n u m b e r of the basic

In m o s t cases

lower bound on the fraction

should c o n s u l t

case.

in

only those d e c a y is known.

the m o s t r e c e n t tables f r o m the 1978

lists.

For

c o m p i l e d by

74

Table

Particle

Mass

7.II: Properties

of the Low-Lying Mesons

Mean Life or FullWidth

Decay Modes

Fraction

~+~ e+~ ~+~y e4-O%o e+Vy

~100% 1.267 (23) xlO -4 1.24 (25)xi0 -4 1.02(07)x10 -8 2.15 (50)xlO -8

+ %

%o

139.5669(12)

2. 6030 (23) xl0 -8 sec

134.9626(39)

0.828xi0 -16 sec

yy ye+e e+e-e+e -

98.85(O5) 1.15(05) 3.32X10 -5

548.8(6)

0.85 (12)keV

YY %oyy 3%0 %+%-%0

38.0 3.1 29.9 23.6(6) 4.89(13) 0.50 (12) 0.i (i) 2.2(8)xi0 -5

%+%-y e+e-y e+e-~+% D+H+ K-

493.668(18)

1.2371(26) xl0 -8 sec

Z+y %+%0 %+%+% -

%+%+~-y e%)~T Oy e+e-%+ ~+~e+e e+~ e+e -

63.50(16) 21.16 (15) 5.59(03) 1.73(05) 3.20(09) 4.82 (05) 5.8 (3.5)xi0 -3 1.8 (+2.4) (-0.6)x10 -5 3.90 (15) x10-5 0.9 (4)x10 -5 1.54 (09)xi0 -5 1.62 (47) xl0 -5 2.75 (16)x10 -4 1.0(4)x10 -4 3.7(14)xi0 -4 2.6 (5)x10 -7 ii (3)xi0-7 2 (+2) (-l)xl0 -7

%0%0 ~+~--y

68.61(24) 31.39(24) 1.85(i0)xi0 -3

%+%0%0

~+~ ~o e+~)T O ~+~y e+~%o% o e+~%+z U+~ ~+%e+~ e+~ y

9T+'rroy

K°

t ~ o

K° S

497.67(13)

0.8923(22)xi0 -I0 sec

75

Table 7.If Mean Life or Full Width

Mass

Particle

(Continued)

o KL

5. 183 (40) xl0 -8

note:

Decay Modes

see

Z°Z°T° ~±~±~ T±eiv Ze~y Z+ZTOzO ~+~-y

Ko o = s-KL

0.5349(22)xi0 I0 h see -I [ Ko i and K°decay 50% into

YY Z+U-

Fraction

21.5(7) 12.39(18) 27.0(5)

38.8(5) 1.3(8) 0.203 0. 094 (18) 6 (2)x10 -5 4.9 (5)x10 -4 9.1 (I. 8) xl0 -9

o and 50% into K °L

KS

~' (958)

957.6(3)

r)T~ pOy ~y

< 1 MeV

YY p (770)

776

155(3) MeV Ty e+e~+~ -

w(783)

782.6(3)

10.1(3)

MeV

]i+~[-70 ~i+~iT°y

e+enY ~ (i020)

K

(892)

1019.6(2)

892.2 (4)

4.1 (2) MeV

49.5 (i. 5) MeV

K+K -

66.2 (1.7) 29.8(1.7) 2.1(4] 2.0(3) ~100% .024(7) .0043(5) .0067(12) (seen) 89.9(6) 1.3(3) 8.8(5) .0076(17) (seen) 48.6(1.2) 35.1(1.2)

n7 T°y e+e U+~ -

14.7 1.6(2) 0.14(5) .031(1) .025(3)

Kn Ky

-i00 .15(7)

76

7.2

The w-~ Problem. Several methods have been suggested for describing the low-lying mesons in

twistor terms.

Evidently, one requires a scheme of considerable generality,

since,

in addition to the low-lying states, there are many many other mesons as well. One method which has been proposed is to treat mesons as holomorphic functions of îA three twistors and to consider the spinor coefficient structure ~ ~jA' spinor coefficient structure produces a multiplet of states @ ~ ,

This

by means of the

contour integral formula

(7.2.1)

~iA /Q AAi jA,(X) = x ~ ~A,jf(Z~ )a~

By taking the divergence of @iA jA' we obtain a set of spin 0 mesons, and by taking the divergence-free part of @ ~ , Unfortunately,

we obtain a set of spin 1 mesons.

this procedure leads to two grave difficulties.

problem is concerned with the spin 0- mesons. negative intrinsic parity.

These mesons are supposed to exhibit

Now in the naive quark model there is no problem, be-

cause quarks have P = i, and antiquarks have P = -I. S-state

The first

(i.e., no orbital angular momentum)

ly have negative intrinsic parity.

Therefore,

if they are in an

then the combined pair will automatical-

If one considers the spin 0- SU(3) singlet

state produced in (7.2.1), then it will be observed that what is actually being produced is the derivative of the field ~ defined by

(7.2.2)

~(x) =/~xf(Z~)A~

In other words, we have the formula

(7.2.3)

~

,(x) = iVA,~(x)

which follows at once as a consequence of

,

(7.2.1),

(7.2.2), and

(5.3.2).

Since

@(x) exhibits no quark structure whatsoever in its associated contour integral formula, it is very difficult to make a case for its being of negative intrinsic parity.

The second problem is concerned with the spin i- mesons.

even worse!

According to formula

Here matters are

(7.2.3), a spin one singlet state simply does not

77

exist within a three,twistor

framework (1) .

(6.5.3), and, in particular,

formulae

account

7.3

This result corroborates

(6.5.6).

Thus, as matters

Theorem

stand we cannot

for both ~ and 4-

Mesons as Quark-Antiquark

Systems.

So, back to the drawing board.

In Section 6.6 we discussed

the fact that

baryons need not be treated in any sense as bound states of quarks--at far as the low-lying baryons are concerned. rather different,

picture of the low-lying mesons

can be built up by following the quark model as closely as possible. in the previous

For a single quark state,

inso-

With mesons the state of affairs is

and indeed quite a reasonable

the defects mentioned

least,

In particular,

section can be eliminated.

if such states exists,

the relevant

contour integral

formula is

(7.3.1)

and for antiquarks

qAi(x)

the relevant

= /Qx~Aif(z)az

qi

a function of six twistors

~

(x) =

Now in order to characterize

(i = 1,2,3)

,

formula is:

A'

(7.3.2)

,

A' xZi f(z)a~

a bound state of a quark and an antiquark we require

f(Z.,Z.) 11 2 l

, where three of the twistors refer to a quark

and the other three refer to an antiquark. To simplify the notation

in what follows,

~ =

( A ,~A,i)

Z~ = 2i

(sA ,SA,i)

let us write

(7.3.3)

for the spinor parts of Z~ and Z~ , and write iI 21 {

~Ai = - ~ / ~ A i

(7.3.4) ~Ai = - ~ / ~ A i

for the associated

spinor operators.

Then for a quark-antiquark

system we could

78

take the spinor coefficient

structure

^~i ~ ~A'j

(7.3.5)

On the other hand, we might equally well take

(7.3.6)

~A,j~ Ai

Which of the two do we take?

Or do we, perhaps,

take some linear combination?

What a dilemma. BuG wait!

There is one quantum number which we have not yet taken into ac-

count: namely,

the charge conjugation

the operator which changes particles whilst at the same time preserving for eigenstates

number.

Charge conjugation

into antiparticles,

handedness.

and vice-versa,

variety of particles;

processes

for example,

Now, let us define the following I

the photon,

the Q-meson,

the ~-meson,

operators

(2)

-IÂ'Âi =

Selection rules in

allow one to empirically determine

~-meson all have C = -i; and the pion,

but

Evidently we have ~2 = i; thus

of C we must have C = ±i for the eigenvalue.

strong and electromagnetic

is defined to be

the w-meson,

C for a and the

and n'-meson all have C = 1.

:

i '

-IÂ'Âi =

O(0)

is surjective note

of degree zero defined on some

small open set U then we can pick a point ~A' lying outside of U and write fA, (~A ,) = ~A,f/~B,~B'

, which is non-singular

degree -i in ~A' ' and satisfies This is a very elementary interest.

~

A'

throughout

fA' = f ' as required.

exact sequence--but

U and homogeneous

of

[]

nevertheless

a sequence of great

In fact, when it is jazzed up just a bit it provides

(as we shall see) a

very direct route for establishing zero rest mass fields.

the connection between twistor cohomology and

A sequence of slightly greater generality

is given by;

118

n+l A' (9.5.3)

0

÷ O(-n-2)

~A,~B,...~C ,

) OA'B'...C' (-i)

÷ OB'...C' (0)

n+l

Now if we construct

(9.5.4)

the associated

"'" - - ÷

HI(pI, o(-n-2))

Thus, providing phism

(9.2.8)

were a global

very directly.

function of ZA' homogeneous

where k is a constant. ~ ZA'

-----+ HI(pI,OA,B,...C,(-I))

0.

But then we would have f = k/~ A

vanishes,

cycle for an element

we proceed

as follows~

of HI(pI,o(-I)). -i ~A,fij

(9.5.5)

unless

Suppose

Multiplying

Equation

(9.5.6)

implies

(9.5.7)

Substituting

(9.5.7)

-i p[ifj]~A , , showing

in

(9.5.5),

would be

f~A,Z A' = k,

which blows up at To see that

f . is a representative z]

co-

by ZA' we get

Transvecting

with z A' we get

A I

= Qj~A 'Z

that fjA'

fjA'

theorem

A'

0 = p[ifj]A ,

A I

fjA 'Z

note that if f(zA,)

f simply vanishes.

for some 0f. since we know HI(pI,o(0) ) is trivial. ]A, , A' A' p[ifj]A,~ = 0 , showing that fjA,~ is global~

(9.5.6)

(0)) - - ÷

-i , then f~A,Z

,A'

is

we obtain the isomor-

vanishes,

Whence by Liouville's

; thus we have a contradiction

HI(pI,o(-i))

of degree

the result

> 0 .

for O(-1),

To see that H0(pI,o(-I))

of degree

sequence,

(-i)) ----+ H°(pI,OB,...C,

we know that H 1 and H 0 both vanish

global and homogeneous

~A'

n

long exact cohomology

HO(pI'OA'B'...C'

> 0

'

(~A' constant)

must be of the form:

-i Pj~A' + fj~A'

we get the desired

result,

that fij is indeed cohomologically

namely:

trivial.

-i ~A,fij

=

119

A result quite analogous to P r o p o s i t i o n 9.5.1 holds for pn . h o m o g e n e o u s c o o r d i n a t e s on

pn

, as before, b y ~a'

"

Let us denote

A n d w e shall w r i t e Oa,...b,(r)

for the sheaf of germs of s k e w - s y m m e t r i c tensor v a l u e d h o l o m o r p h i c functions, twisted by r. 9.5.8

9.5.9)

Then we have:

Proposition.

0

The sequence

> O(-r-l)

> Oa,(-r)

> Oa,b,(-r+l)

... - - ÷

> ---

Oa,b,...c,(0)

--÷

0

is exact, w h e r e in each case the sheaf maps are given b y m u l t i p l i c a t i o n b y Za' and skew-symmetrizing

in an appropriate fashion.

The p r o o f is quite analogous to that of P r o p o s i t i o n 9.5.1. also, to construct an

analog

of sequence

(9.5.3).

Then b y a p p l y i n g the exact co-

h o m o l o g y sequence one can obtain directly the isomorphisms Sequence

It is not difficult,

(9.3.1) and

(9.3.2).

(9.5.9) is a special example of w h a t is k n o w n as the Koszul complex.

w i l l be d i s c u s s e d in Section 10.6,

As

it plays a special role in the analysis of the

c o h o m o l o g y of functions of several twistors.

9.6

Line Bundles and Chern Classes. It is easy to see that the set of all n o w h e r e - v a n i s h i n g h o l o m o r p h i c functions on

a region U forms a group under m u l t i p l i c a t i o n . 0".

The c o r r e s p o n d i n g sheaf is denoted

S i n c e locally any n o w h e r e - v a n i s h i n g h o l o m o r p h i c function g can be e x p r e s s e d in

the form exp(f) exact,

= g , where f is a h o l o m o r p h i c function, the following sequence is

where Z denotes the integers:

(9.6.1)

0

W h e r e the map Z - - ÷

> Z

~ O ----+ 0* - - ÷

-

,

O is simply m u l t i p l i c a t i o n by 2zi

The c o h o m o l o g y sequence a s s o c i a t e d w i t h

(9.6.2)

0

-- -----+ HI(M,O)

-

-

÷ H 1 (M,O)

(9.6.1) contains the segment:

-- ) H2(M,Z)

-----+ ---

The group H I ( M , O *) is called the group of h o l o m o r p h i c line bundles over M, and each

120

element of HI(M,O *) is called a line bundle. a representative

A line bundle is specified by giving

cocycle ~ij , which must satisfy the cocycle condition

(9.6.3)

Pijk~ij~jk~k i = 1

,

where Pijk denotes restriction to the triple intersection (9.6.3)

region U ijk .

Note that

is satisfied trivially if we put

(9.6.4)

~ij = Pij~i/~j

'

where ~i is a collection of nowhere-vanishing Thus the coboundary

holomorphic

freedom available in the specification

functions defined over U iof a line bundle is

given by

(9.6.5)

-i ~ Pij~i~ij~j

~ij

The element of H2(M,Z)

to which a line-bundle

Chern class of the line bundle.

~ij is mapped in (9.6.2)

From the exactness of (9.6.2)

is called the

it should be evident

that line bundles with vanishing Chern class are precisely those which can be obtained by "exponentiating"

elements of HI(M,O);

i.e., a line bundle ~ij has vanish-

ing Chern class if and only if it can be expressed in the form

(9.6.6)

~ij = exp(fij)

'

with f.. satisfying the additive cocycle relation 13 (9.6.7)

p[ifjk] = 0

.

The notion of line bundle is a special case of the notion of a vector bundle (5) over a space M.

A holomorphic

vector bundle is defined to be an element of the

group HI(M,O~), where O~ is the sheaf of holomorphic tions.

An element of HI(M,O~)

non-singular

is specified by a collection

over U.. satisfying 13

(9.6.8)

~a ~b ~c Pijk~i]b~jkc%kid

a = ~d

(in Uij k)

matrix valued func-

[aijb of such functions

121

9.7

Varieties,

Syzygies,

A projective lection f

r

projective

and Ideal Sheaves.

algebraic variety is defined to be the common zero set of a col-

of homogeneous polynomials n-space.

in the homogeneous

If there is but a single homogeneous

variety V defined by f = 0 is called a hypersurface degree of the polynomial called hyperplanes,

f.

Hypersurfaces

quadrics,

pIxpI as a quadric hypersurface

polynomial

Z a of complex f, then the

of degree q , where q is the

of degree q = i, 2, 3, 4,

cubics, quartics,

As a simple example of an algebraic

coordinates

... are

etc., respectively.

variety one can consider the embedding of

in p3.

Suppose we write ~ 'l

for the four homogeneous

coordinates

of p3 .

(9.7.1)

A ! BI .. ~i ~'3 eA'B'~ 13 = 0

(A' = I, 2; i = i, 2)

Then the quadratic

equation

has the solution Al AI z. = Z I .

(9.7.2)

l

The variables homogeneous

~

A !

1

and I. (which are determined by 9.7.2 only up to scale) 1

coordinates

for pIxpI .

As a somewhat more complicated as an algebraic ~iA'

variety in p5

(i = i, 2, 3; A' = I, 2).

example,

let us consider the embedding of pIxp2

For homogeneous

coordinates

on p5 let us write

Then pIxp2 is given by the locus AIB

(9.7.3)

serve as

~iA'~B'63

l

= 0

,

for which the solution is

(9.7.4)

~iA' = ~A'li

with ~A' and li acting as homogeneous

'

coordinates

should be noted that the three equations

(9.7.3)

since we have the relations

(9.7.5)

fij~kA,£ ijk = 0

,

for pl and p2 , respectively. are not completely

independent,

It

122

which are satisfied automatically,

where fij is defined by A'B'

(9.7.6)

fij = ~iA'~jB 'e

Associated with any projective

algebraic variety V is an ideal sheaf IV , de-

fined to be the sheaf of germs of holomorphic

functions which vanish on V.

The

ideal sheaf can be described by an exact sequence PV 0 ----+ Iv ----+ 0 ----+ 0 V ----+ 0

(9.7.7)

,

where PV is the restriction map down to the variety, of holomorphic

functions defined on the variety.

serve that, locally,

any holomo~phic

and O v is the sheaf of germs

In the case of plxpic p3

function which vanishes when restricted down

from its domain in p3 to the intersection

of that domain with the quadratic pIxpI

A' B' ij A' A' must be of the form ~i ~j £ £A,B,f(~i ) , where f(~i ) is homogeneous -2.

Thus, in this case (9.7.7)

(9.7.8)

0 -----+ 0(-2)

In the case of p I x 2 c p5 (9.7.8)

can be written more explicitly

-----+ Op3 -----+ Oplxp I - - +

the syzygy

is a long exact sequence.

(9.7.5)

Therefore

0

plays a role, and the analog of sequence

This is because any function on a region of p5

O13(-2)

where hA'(~)

(9.7.11)

function

(twisted by -2).

~ Oplxp2 ----+ 0

----+Op5

However, we can substitute . . . . ijk A' g z3 ~ g 13 + ~ ~A,k h (~)

(9.7.10)

invariant.

and g 13 an arbitrary holomorphic

, with fij

it follows that the sequence

(9.7.9)

is exact.

of degree

in the form

which vanishes when restricted down to plxp2 must be of the form giJfij as defined in (9.7.6)

we ob-

is an arbitrary function homogeneous Thus, on account of (9.7.5)

0 --÷

oA'(-3)

--÷

,

of degree -3, and leave gl]f.. ~3

we obtain a long exact sequence

oiJ(-2)

----+ Op5 ----+ Oplxp 2 - - ÷

0

(6)

:

123

Consequently,

since the first three of the sheaves

the cohomology

of pIxp2 can be related to various

(using the long exact cohomology

sequence).

may not have the pleasure of being products (9.7.11)

are defined on p5 •

cohomology groups defined on p5

Of course in this case we can compute

the cohomology of pIxp2 directly by other means;

analogs of

in (9.7.11)

but for other varieties

of projective

spaces)

(which~

we can construct

and reduce the p r o b l e m of computing the cohomology of V to an

elementary p r o b l e m in linear algebra. For example,

suppose one is interested

cubic surface A 8yZ~ZSZ Y = 0 in p3 .

(9.7.12)

0

where OV(n)

.

.) Op3(n~3) . . .

in the cohomology of a non-singular

In this case the relevant

~ ÷ Op3(n)

---+ Ov(n)

is the sheaf of germs of holomorphic

twisted by n, and the map

~

is multiplication

~

exact sequence

0

functions on the cubic surface,

by the function A ~ Z~Z~Z Y .

a short calculation

is

the cohomology of

(9.7.12),

(9.7.13)

H0(V,Ov(n))

= Coker(~*)

,

~*:H0(p3,O(n))

H2(V,Ov(n))

= Ker(~*)

,

~*:H3(p3,O(n-3))

Taking

gives

--+ H0(p3,0(n-3))

,

and

(9.7.14)

After a little thought one will recognize twistors

form A(d~yP~...6)

one finds the dual space to

(9.7.15)

(9.7.13),

P~SY6"''eA

~y

this being the space of symmetric

=

o

by A ~

Thus, the four independent

(9.7.14)

twistors

.

to parameterize

components

In

:

let us consider the so-called

In this case it is most convenient

p3 .

as the space of symmetric dual

, where P~...£ is of valence n-3.

of valence n which are annihilated

AS another example,

3.

H3(p3,0(n))

P ~y~...e of valence n, modulo the space of symmetric valence n dual twis-

tots of the special

p~y6...£

(9.7.13)

~+

"twisted cubic" curve in p3 .

p3 by symmetric

spinors of valence

of ~ABC act as homogeneous

The twisted cubic curve is defined to be the locus

coordinates

for

124

c ~ABC~E F = 0

(9.7.16)

C Writing ~ABEF = ~ABC~EF it is straightforward

(9.7.17)

to verify the property

~ABEF = - ~EFAB

Thinking of the symmetric is a skew-symmetric components.

Thus

index pairs AB and EF as index clumps, we see that ~ABEF

three-by-three (9.7.16)

matrix,

represents

and accordingly

the intersection

has three essential

of three quadrics.

This

gives us the exact sequence O ABEF p3 (n-2)

(9.7.18)

where OT(n)

~ABEF •

+ Op3(n)

.....

is the sheaf of germs of holomorphic

twisted by n.

In order to continue

the elementary

spinor identity

(9.7.19)

(9.7.18)

i OT(n )

functions

,

on the twisted cubic,

to form a long exact sequence,

we need

~ABEF ~ABE = 0

This gives us the sequence

(9.7.20)

0 ----+ oA(n-3)

~--~-÷ oABEF(n-2)

-----+ O(n)

~÷

OT(n)

-

~ 0

,

where the sheaf map ~ is specified by

(9.7.21)

OA

~ o(A~ B)EF - o(E~ F)AB

in order to ensure that the image has the correct symmetries. quence

(9.7.20),

Given the exact se~

it will be left to the reader to work through the details of sorting

out the associated

long exact cohomology

sequence t this being intricate but not dif-

ficult. This concludes

our brief introduction

to sheaves and cohomology.

material mentioned here is useful in one way or another theory,

with twistor

although not all that has been said will be used in the next chapter.

further material Gunning

in connection

All of the

the reader is referred to Serre

(1966), Chern

(1967), Morrow and Kodaira

(1956), Gunning and Rossi (1971), Godement

(1973),

For

(1965),

125

Shafarevich

(1977), H a r t s h o r n e

(1977), and n u m e r o u s other references.

It should be

stressed that hhere are m a n y intimate i n t e r c o n n e c t i o n s b e t w e e n q u a n t u m m e c h a n i c s and the theory of algebraic v a r i e t i e s - - i t is r e a s o n a b l e to speculate,

in fact, that all

the d i s c r e t e degrees of f r e e d o m that m a n i f e s t themselves in q u a n t u m m e c h a n i c s can be u n d e r s t o o d u l t i m a t e l y in terms of the c o h o m o l o g y of algebraic varieties.

For the

various continuous degrees of freedom that appear in q u a n t u m mechanics, however, it w o u l d appear that more general categories of complex m a n i f o l d s manifolds)

(i.e., n o n - a l g e b r a i c

must be investigated.

Chapter 9, Notes

i.

For a d e s c r i p t i o n of the limiting p r o c e d u r e involved here see Gunning,

p. 30..

Also,

1966,

see pp. 44-47 in the same reference for a d i s c u s s i o n of "Leray's

theorem" w h i c h gives a set of conditions sufficient to ensure that a covering U. is 1 general enough to calculate the cohomology of a space M.

2.

S t r i c t l y speaking in order to e s t a b l i s h this result we need to k n o w that a

c o v e r i n g of pl by two open sets suffices to compute its cohomology.

3.

C r o s s - s e c t i o n s of the sheaf O(n)

are often r e f e r r e d to as "twisted functions";

and O(n) itself is called the ~'sheaf of germs of h o l o m o r p h i c functions, twisted by n".

4.

For further d i s c u s s i o n of the long exact c o h o m o l o g y sequence,

Gunning,

5.

see, for example,

1966, pp. 32-34.

H o l o m o r p h i e line bundles and h o l o m o r p h i c vector b u n d l e s - - b u i l t over suitable

regions of p r o j e c t i v e twistor space--can be used to d e s c r i b e self-dual solutions of Maxwell's equations and the Yang-Mills equations (1977a and 1977b), A t i y a h and W a r d various details of the procedure.

6.

Note that for sequence

In the case of sequence

(without sources).

(1977), H a r t s h o r n e

(1978), and Ward

Also see B u r n e t t - S t u a r t

V

(1979) for

(1978) and Moore

(1978).

(9.7.8) we have an i s o m o r p h i s m b e t w e e n 0(-2) and IV .

(9.7.11) we have the following isomorphism: I

See Ward

.. . A = O±3(-2)/Image[O

I

(-3)]

CHAPTER i0 APPLICATIONS OF COMPLEX MANIFOLD TECHNIQUES TO ELEMENTARY PARTICLE PHYSICS

10.1

The Kerr Theorem. Standing before us we see two alternative pictures of reality.

hand there is spaoetime, and on the other there is twistor space.

On the one Einstein has

taught us that gravitation is itself but an aspect of the structure of spaeetime. According to the view of twistor theory, gravitation is to be reinterpreted in terms of the complex analytic geometry of twistor space.

Elementary particle states are

to be interpreted similarly--in fact, according to Penrose we are to think ultimately of actually in some sense incorporating elementary particle states directly "into" the complex analytic structure of twistor space. At the moment only a few examples of this procedure are known in sufficiently explicit detail to make comment worthwhile~however,

there is no reason to suppose

that these techniques cannot be generalized to accontmodate

a reasonable spectrum

of particles, and to treat certain features of their interactions as well. At the mention of interactions what springs to mind immediately is the question of how the various non-linearities of field theory are to be realized in complex analytic terms.

The Kerr theorem provides a striking illustration of the fact that

certain non-linear partial differential equations arising in connection with properties of fields on Minkowski space can be reinterpreted in a very straightforward way in terms of the complex analytic geometry of twistor space.

The Kerr

theorem has its origin in certain special classes of Maxwell's equations, called null electromagnetic

fields.

A null electromagnetic field is a solution of

Maxwell's equations for which both of the invariants FabFab and *FabFab vanish. Equivalently,

if the electromagnetic field spinor @A'B' is introduced according to

the familiar scheme

(i0.i.i)

Fab = ~A,B,EAB + ~ABEA,B ,

,

then Fab is null if and only if @A'B' is of the form

]27

(10.i.2)

@A'B'

for some choice of

@ and ~A'

"

According

if a spinor field ZA' satisfies

(10.1.3)

Z

: @ZA'ZB '

to a remarkable

the geodesic

A' B' Z VAA,ZB,

'

theorem of Robinson

shearfree condition

= 0 ,

then there will always exist a choice of @ such that @A'B' AA'

satisfies Maxwell's

equations

V

satisfies Maxwell's

equations

and is of the form

(10.1.3).

(10.1.3)

shearfree

Equation null rays.

congruences

function

time p o i n t x

then ZA' satisfies

in terms of complex analytic

of some degree n.

then the surface

surfaces

shearfree

in p3.

If f (Z~) should happen to be a homo-

f(Z ~) = 0 [which will henceforth

be denoted

surface; but more generally we simply have an analytic

Let us denote by X a complex projective AA'

(10.1.2),

if a spinor field @A'B'

surface in p3 is defined by the vanishing of a holomorphic

f (Z~) , homogeneous

S] is an algebraic

~A'B' = 0; and conversely,

asserts that ~A' is tangent to a family of geodesic

can be characterized

geneous polynomial,

, as defined in (10.i.2),

According to the theorem of Kerr (1) , such geodesic

A complex analytic

(1959),

line in p3 corresponding

Now if Z ~ is an intersection

surface.

to a space-

point of X and S, then Z ~ must be of

the form

(10.1.4)

AA ! ZA'

Z~ =

(ix

f(ix

AA I ZA'

' ~A ')

'

and must satisfy

(10.1.5)

' ZA ') = 0

If we vary the line X, then ZA' must be correspondingly remain satisfied. proportionality,

(10.1.6)

if

In this way we obtain a field of spinors ZA,(X),

(10.1.5)

is to

determined

up to

satisfying

f[ix

AA w ~A,(X)

, ~A,(X)]

In general the field ~A,(X) will possess possible

adjusted

= 0

several distinct

"branches",

for a given line X to intersect S in more than one place.

since it is

128

10.1.7 holomorphic Proof.

Theorem.

If a spinor field ZA,(X)

satisfies

function f(Z~), then it satisfies equation Since f(Z ~) is homogeneous

(10.1.8)

z c~ ~Z ~~f=

(10.i.6)

for some

(10.1.3).

of degree n we have

nf

whence on the surface S we have

(10.1.9)

Z o~

Then, restricting

~f ~-Y

=

~3f

6oA

+17 A'

to the intersection BB'

which implies the existence of a

(10.l.ll)

ix

BB '

Df ~'EB'

scalar

~f

=

o .

of S with X, we obtain

~f ~B' ~ B + ~B'

ix

(i0.i.i0)

~~----~f A'

~f

~

0

,

such that B'

B'

Storing this bit of information, (10.1.6)

let us return to equation

must remain valid if we vary x

(10.1.12)

must vanish. of (10.1.12)

(10.1.13)

Transvecting

VAA,f[ix

BB'

~B,(X)

With a straightforward

AA'

(i0.1.6).

Since

, it follows that the derivative

, ~B,(X)]

application

of the chain rule, the vanishing

implies

i~A, ~ f ~A

+ (VAA,~B,) [ixBB' ~ f ~B

(i0.i.13) with A '

and using

+ d~B'~f] = 0

(i0.i.ii),

the desired result

(10.i.3)

follows immediately°[] As an example of the Kerr theorem at work, let us consider again a cubic surface in p3 , given by the equation

(10.1.14)

Now associated with A ~

A ~ Z~Z~Z Y = 0

is a solution ~A'B'C'(x)

of the equation

129

(io.1.15)

where

~ v ( A ' ~ B'c'D')

~A'B'C'

is defined

(10.1.16)

: 0

,

by

PxA ~TZ~Z~Z Y

:

~A'B'C'

ZA,ZB,ZC,

,

!

with px Z ~ =

(ixAA ZA'

, ZA,) , as usual.

Thus,

if we put

AA' (10.1.17)

Z~ =

in equation

(10.1.14),

~A'B

solutions,

where ZA'

for a general

' ~A'

cubic

congruence.

surface,

'

ZA' z ~A'

~A'B'C'

these principal

spinors

the three branches the cubic surface geometry

to verify

exactly

27 zero-points

exercise (2) "

'

condition,

shearfree

according

all three of and thus define

rays associated

to a classical

in p3 has exactly

~A'B'C'

of this configuration

of Sheaf Cohomology

27 lines lying of

vanishes).

would undoubtedly

(10.1.15) A de~ make

Groups.

zero rest mass

We are interested

with

result of pro-

to the fact that a solution

of analyzing

sheaf cohomology.

(10.1.15)

(i.e., points where

Zero Rest Mass Fields as Elements Now we come to the question

of

shearfree

of geodesic

this result corresponds

for a highly amusing

of twistor

%A' ~ YA'

i

a cttbic surface

of the geometry

spective

w

Incidentally,

tailed investigation

10.2

has three distinct

spinors of ~ A ' B ' C '

the geodesic

due to Schl~fli,

(10.1.18)

= ~(i ~B yC )

of the congruence

spacetime

must possess

i

'

that as a consequence

satisfy

(10.1.14).

Equation

and these are given by

, and YA' are the principal

It is not difficult

on it--in

ZA' (X)ZB' (X)%c' (x) = 0

%A' ~ ~A'

(10.1.20)

jective

i

Ic

for our shearfree

(10.1.19)

TA, (x) , ~A,(X)]

then we obtain

(i0.I. 18)

as the formula

[ix

fields

from the per~

here in the cohomology

130

group HI(M,O(n)), where M is a region of p3 of h o l o m o r p h i c functions twisted by n.

, and where O(n)

is the sheaf of germs

The sort of region M in w h i c h we are in-

terested is one that is swept out by a set of p r o j e c t i v e lines in p3 corresponding to a set of points in complex Minkowski space. example,

For p o s i t i v e frequency fields, for

the region of complex M i n k o w s k i space of interest is CM + , and the cor-

r e s p o n d i n g region in p3 is PT + .

10.2.1

Proposition.

Each element of Hl(M,O(n))

i mass field of h e l i c i t y s = - ~ n - i space to w h i c h M is related.

corresponds to a zero rest

d e f i n e d over the region of complex Minkowski

D i s t i n c t elements c o r r e s p o n d to d i s t i n c t zero rest

mass fields. Proof.

First let us consider the case s > 0, i.e., n < -2.

In order to

simplify the d i s c u s s i o n we shall examine the case s = 1/2 explicitly, and the reader should have no d i f f i c u l t y in filling in the details r e q u i r e d for higher helicities. Suppose that f.. is a r e p r e s e n t a t i v e coeycle for an element of HI(M,O(-3)). i] 1 If we restrict f.. down to the complex p r o j e c t i v e line P c o r r e s p o n d i n g to a spacei3 x AA' AA' time p o i n t x , then Oxfij can be regarded as a function of x and ZA' " For fixed x AA' then Pxfij is a r e p r e s e n t a t i v e cocycle of HI(p 1 ,0(-3)). apply the analysis of Section 9.2. must be c o h o m o l o g i c a l l y trivial,

, the result

so we get

-3 -i ZA,ZB,Pxfij = p[igj]A,B,

(10.2.2)

-i for some 0-cochain giA'B' (10.2.2) with Z

If we m u l t i p l y Pxfij by ZA,ZB,

Now we can

A'

we get

, w h i c h is a function of x

AA'

and ZA'

"

Transvecting

(cf. e q u a t i o n 9.2.9): 0

(10.2.3)

p[i@j]A , = 0

,

0 where @jA' is d e f i n e d b y 0 (10.2.4)

-i

B'

@jA' = gjA'B '~

0 Since @jA' is global and h o m o g e n e o u s of degree zero in ~A'

, it m u s t be constant

131 AA

in ZA'

I

, and thus a function of x

(IQ.2.5)

alone, with

@jA' = Pj@A '(x)

To prove that @A,(X)

satisfies the zero rest mass equations we must note that

since f.. is a collection of twistor functions 13

it must satisfy

V A' ~A' A Pxfij = 0

(10.2.6)

Consequently,

if we transvect

AA

(10.2.7)

p[i V

(10.2.2)

with V A'A we get:

I

pxgj]A,B,

: 0

which says that ?AA'pxgjA,B , is global. of degree -i in HA,

,

However,

since vAA'QxgjA,B , is homogeneous

, the only way it can be global is for it to vanish:

(10.2.8)

vAA'pxgjA,B , = 0

Transvecting

(10.2.8) with %B' and using

(10.2.4)

and

(10.2.5), the desired result

?AA'@A , = 0 follows immediately. Next, we must prove that @A' is independent of the coboundary in the specification

of f.. . l]

To see this, observe that the transformation

-3 fij - - ÷

(10.2.9)

freedom available

-3 -3 fij + P[igj]

must be accompanied by the substitution -I g jA'B' ---+ gjA'B' + gjZA'~B '

(i0.2.10)

in equation

(10.2.2).

However,

under this substitution. must be cohomologically

a glance at (10.2.4)

Conversely, trivial.

we wish to see that if @A' vanishes,

From

(10.2.4)

then gjA'B' m u s t be of the form gj~A,~B, and ZA'

(10.2.11)

"

From

(10.2.2)

,

then fij

it follows that if ~A' vanishes,

for some 0-cochain gj dependent upon x

it then follows that

-3 -3 Pxfij = p[igj]

shows that @jA' is invariant

AA'

132

but we are not done yet, since it remains to be shown that gj is indeed a 0-cochain on twistor space--thus XAA , and ZA'

*

A' with VA,Z A

far, we have merely established

The situation

is ir~mediately remedied,

that gj is dependent however,

if we hit

on (10.2.11)

, thereby obtaining

~A w-3 gj]

(10.2.12)

p[iZA ,

which implies,

since ZA'

That concludes

= 0

,

gj is homogeneous

the proof for s = 1/2.

If fij is homogeneous

of degree

of degree

-2, that ZA'

gj vanishes.

Now let us consider the case s = - 1/2.

-i, then Pxfij is cohomologically

trivial,

i.e., we

have: -i -i Pxfij = p[igj]

(10.2.13)

for some 0-cochain

-i gj

,

If we operate on

(10.2.13)

A with ZA,VA

then we obtain

vA ,-i P[iZA ' A gj] = 0

(10.2.14)

which implies,

A'-I since ZAV A gj is homogeneous

(i0.2.15)

~A'~A'g~

where @A(X)

is a function of x

of degree zero, that

= Pj~A(X)

alone.

,

Transvecting

equation

(10.2.15) with

~B' VB'A we get B'A ZB 'V @A = 0

(10.2.16)

on account of the identity ~(A,VB,)A = 0 . values of ZA,

, we get the field equation V

And since A'A

(10.2.16) must hold for all

@A = 0 , as desired.

forward to check that @A is independent of the coboundary Moreover, trivial.

10.3

one can verify that if @A vanishes,

It is straight-

freedom available

to fij

"

then fij itself must be cohomologically

[]

Spin-Bundle

Sequences.

The results of Section 10.2 can be obtained of view through the consideration

from a somewhat more refined point

of various exact sequences of sheaves.

It should

133

be p o i n t e d tion,

out that quite a b i t of lore has a l r e a d y b e e n d e v e l o p e d

and it w o u l d be i m p o s s i b l e

clusive

of all the w o r k

us assume

away from i n f i n i t y

thus that the r e g i o n of twistor I d~ .

functions

sheaf F(1) bundle.

is called

isomorphic

(10.3.1)

0

is exact, of x

AA '

to O(n),

by n.

of germs

3 1 is P -PI ' where

of degree

cross

f(x,z)

'

the sheaf of germs

n in

sections

satisfying

ZA'

the

"

of the spin

A' TA.V A f(x,z)

.

~ F(n)

-

VA ' ~A' A -

O(n)

of degree

n+l in

ZA'

"

holomorphic

Sequence

(10.3.1)

functions

can be

----+ F(n)

A' ~A,VA --FA(n+I)

B'A ~B ,V ~ F(n+2)

\ /

----+ 0

A (n+l)

/\ 0

where the a u x i l l i a r y

sheaf A(n+l)

(10.3.3)

A(n+l)

0

defined

by

= Image(~A,V~' ) = K e r n e l ( T B , V

has been i n t r o d u c e d

in order to facilitate

p a i r of short exact

sequences:

(10.3.4)

0 ----+ O(n)

(10.3.5)

0

= 0

~ FA(n+I)

as follows :

0 --÷

The

and thus the sequence

is the sheaf of germs of s p i n o r ~ v a l u e d

, homogeneous

and

Let us denote b y F(n)

, homogeneous

of h o l o m o r p h i c

consisting

~ O(n)

where FA(n+l)

and ZA'

completed

(10.3.2)

of F(n)

twisted

and %A'

the sheaf of germs

The s u b s h e a f

is n a t u r a l l y

of x

space,

AA'

space,

let

ZA' = 0 , we can take ZA

we shall d e n o t e by O(n)

on twistor

sheaf of germs of functions

Minkowski

space w i t h w h i c h we are c o n c e r n e d

As before,

in-

To s i m p l i f y matters,

in c o m p l e x

Since pl is given by the e q u a t i o n I

n o w to be n o n - v a n i s h i n g . of h o l o m o r p h i c

here w h i c h is in any sense

that has been done in this vein.

that we are w o r k i n g

pl is the line I

to give an a c c o u n t

in this c o n n e c -

--+

> A(n+l)

F(n)

B'A

the d i s i n t e g r a t i o n

--

~ FA(n+I)

) A(n+l)

--÷

-----+ F(n+2)

)

of

(10.3.2)

0

> 0

into a

134

Now let us consider,

as an example, the helicity 1/2 case n = -3.

cohomology sequence associated with

(10.3.6)

H0(M,A(-2))

(10.3.4) contains the segment

----+ HI(M,O(-3))

~÷

HI(M,F(-3))

and associated with

(10.3.5) we have the following segments:

(10.3.7)

0 --+

(i0.3.8)

H 0(M,F(-I))

Now H0(M,FA(-2))

H0(M,A(-2))

--÷

which have negative twist in ~A' (i0.3.7)

--÷

,

H I(M,F A(-2))

both consist of global functions of x ; consequently

we then obtain that H 0(M,A(-2))

the map from HI(M,A(-2))

----+ HI(M,A(-2))

~ H0(M,FA(~2))

H I(M,A(-2))

and H0(M,F(-I))

The long exact

they must both vanish.

vanishes,

to HI(M,FA(-2))

AA'

and ZA'

From

and from (i0.3.8) we deduce that

is injective.

Gathering these facts to-

gether we deduce that the sequence

(10.3.9)

is exact.

0

~ HI(M,O(-3))

The group HI(M,F(-3)

--÷

HI(M,F(-3))

~÷

HI(M,FA(-2))

is the set of all primed spinor-valued

fA' (x) defined over the region of spacetime corresponding is the set of all unprimed spinor-valued

functions

functions

to M, whereas HI(M,FA(-2))

fA(x) on the same region.

not difficult to verify that the induced map between H I(M,F(-3))

It is

and H I(M,FA(-2))

is given by

(10.3.10)

fA '(x) - - +

for a typical element.

Sequence

A fA '(x)

(10.3.9)

'

asserts that HI(M,O(-3))

kernel of this map, and a glance at (10.3.10)

is precisely the

shows that the kernel consists pre-

cisely of ZRM fields of the appropriate helicity. Thus we have established Proposition point of view.

10.2.1 again, but from a slightly different

In fact, a somewhat stronger result has been obtained,

namely an

isomorphism between the twister cohomelogy group of interest and the relevant set of ZRM fields.

It is perhaps instructive

to arrive at this result from yet another

angle, using an exact sequence which codifies more directly the procedure

outlined

135

in Proposition

10.2.1.

germs satisfying example,

Let us denote by ~A' the subsheaf of FA, consisting of

the zero rest mass equations:

the following

(10.3.11)

To calculate

0

sequence

÷ 0(-3)

--

the eohomology

vAA'~A,(X,Z)

= 0 .

Then,

for

is exact:

ZA'ZB'

%B'

÷ ~A,B,(-I)

of the @ sheaves,

>

@A,(0)

~>

0

one can use the following

sequences:

VA 'A (10.3.12)

0 ----+ ~A' (n) -----+ FA, (n)

+ FA(n)

(10.3.13)

0 --÷

VA

~

0

B' ~A'B' (n) - - ÷

Now the long exact cohomology

FA,B, (n)

sequence

vAN' ~ FAA , (n) ....

associated

with

> F(n)

(10.3.11)

-----+ 0

contains

the

segment

(i0.3.14)

Using

H0(M,~A,B , (-i)) - - +

(i0.3.13)

H0(M,~A, (0))

one can establish

-

) HI(M,O(-3))

--÷

HI(M,~A,B , (-i)) .

that both H 0 and H 1 vanish for }A,B, (-1), whence

we have the isomorphism

0 --÷

(10.3.15)

And finally,

an elementary

sists of the relevant For positive

(10.3.16)

H0(M,~A, (0)) - - +

calculation

helicity

(s > 0) sequence

0 ----+ 0(-2s-2)

The map 8 is contraction For negative helicity,

(i0.3.17)

~

The map ~ is

(10.3.11)

by ZA,...ZB,ZC,

0 .

shows that H0(M,}A, (0)) con-

with

--÷

can be generalized

as follows:

) ~A,...B, (0) - - +

0

,

, the total number of Z's being

C' of

(10.3.16)

O(-2s-2)

consists of polynomials

~A". ~B

(10.3.12)

}A,...B,C, (-i) ~

the analogue

0 ----+ P(-2s-2)

The sheaf P(-2s-2)

using

--+

set of zero rest mass fields.

where the map ~ is multiplication 2s+l.

HI(M,O(-3))

(-2s-3 occurrences

of

in ~

is the following

(3 ) @A...B A

~A),

sequence:

~ ) @A...BC ---~ 0

and ZA' homogeneous

of degree -2s-2.

A'

and ~ is the operator ZA,Vc

136

By using

(10.3.16)

and

(10.3.17),

tween twistor

cohomology

incidentally,

are the o r i g i n a l

the p r o b l e m of r e l a t i n g quence

(10.3.2)

and Z R M fields, sequences

p o i n t of v i e w has been r e f l e c t e d Jr..

As another a l t e r n a t i v e

the single

suggested

etc.,

T 0

) 0(-2s-2)

w h i c h can be used to derive

A'

the r e l e v a n t

of p o t e n t i a l s

in the various

net p a r t i c u l a r l y

Remarks

serious

~

twistor

the appearance

and

R. Penrose, (10.3.17)

certain

÷ O(~2s)

cohomology

inas-

and R.O. Wells,

one can consider

Our a t t i t u d e

of n - T w i s t o r

categories

of fields,

formulae,

treating

structure

and then i n t r o d u c i n g

In this w a y we arrive

duced at an earlier

systems

arise on a c c o u n t of

but these c o m p l i c a t i o n s

are

at the c o n c l u s i o n

are m o r e

stage

"primitive"

necessary

sheaf

cohomological

complex

methods.

spaces w i t h r e l a t i v e l y

more and m o r e that certain

than others,

in the w h o l e p r o c e s s

for the t r e a t m e n t

from a somewhat more a b s t r a c t

t h e m as h i g h e r - d i m e n s i o n a l

at first,

eigenvalues)

for all

Systems.

using twistor

will be to r e g a r d n - t w i s t o r

p o i n t of view,

relationships

(4) .

on the G e o m e t r y

of more general

-----+ 0

complications

N o w we shall b e g i n to set up some of the m a c h i n e r y

initial

respects,

A'

(10.3.2),

associated

The se-

are easy to w o r k w i t h d i r e c t l y - - t h i s

) % , (-2s-2)

w i t h sequence

piece.

groups (3] .

at least in certain

(10.3.16)

helicities--as

little

in order to solve

sequence

(i0.3.18)

10.4

by R. P e n r o s e

in w o r k by M. Eastwood,

to sequences

be-

These two sequences,

sheaf c o h o m o l o g y

advantageous,

sheaves F(n),

again the c o n n e c t i o n

for all helicities.

ZRM fields and twistor

is somewhat m o r e

m u c h as the spin-bundle

one can e s t a b l i s h

structure operators

piece by (and their

since they can be intro-

and r e q u i r e

less s t r u c t u r e

in their

definition.

Let S a denote

C m+l

(m > i) r e g a r d e d

as a complex

vector

space,

and denote by

aI Sa

, S

, and Sa, the dual

conjugate twistor" means

space,

the complex

space to S a , respectively. as a p o i n t

a pair

( a

in the space

(Sa

conjugate

One can introduce , Sa,) .

, ~a, ) w i t h w a s s a and ~a,SSa,

space,

and the dual c o m p l e x

the idea of a "generalized

Thus by a g e n e r a l i z e d .

twistor

one

137 v

The dual twistor

space to

[henceforth

portunity

(Sa

S a ' ) is the space

'

(Sa

we shall d r o p the a d j e c t i v e

for confusion]

then its inner p r o d u c t

a'

,S a ).

If

"generalized"

(~a

,T

) is a dual

w h e n there

w i t h the twistor

is no op-

(wa , ~a,)

is de-

al fined to be w a u a + ~

,T a -a v (~a ,w ), and the n o r m of

The complex ( a

which,

(wa ,~ ,) is the dual twistor a

r

wa~a + Za,~

using a s t a n d a r d

of

'Za' ) is d e f i n e d by the inner p r o d u c t -a

(10.4.1)

conjugate

argument

,

[cf. S e c t i o n

2.3],

can be shown to have signature

space is p2m+l

(S a ,S ,) i s c l e a r l y C2m+2 , and t h e a s s o c i a t e d p r o j e c t i v e a 02m+2 The space is d i v i d e d into three regions , d e n o t e d C +2m+2 , _

2m+2 CO

, according

(m+l , m+l).

, and C

negative; P

The space

2m+2

the three

as to w h e t h e r

corresponding

regions

the n o r m

(10.4.1)

is positive,

zero,

or

o f p2m+l a r e d e n o t e d p2m+l+ , PO2m+l , a n d

2m+l

In the case of m = i, we recover of signature

(2,2).

Spaces

standard

of n twistors

twistor

space,

fit into the p i c t u r e

w i t h its usual n o r m for odd v a l u e s of m,

w i t h the i d e n t i f i c a t i o n s

a

(10.4.2)

w

A

= ~i

'

Za' = 7iA'

The spaces w i t h m even do not admit of an obvious it m u s t be a d m i t t e d

spacetime

that they do fit a l m o s t u n c o m f o r t a b l y

interpretation,

naturally

although

into the general

scheme. The G r a s s m a n n i a n case m = i

of p r o j e c t i v e

[where the 1-planes

g a r d e d as c o m p l e x i f i e d pretation sional

are,

compactified

is a v a i l a b l e ,

"hyperspace"--to

and for the

m-planes

in p2m+l has d i m e n s i o n

of course, Minkowski

lines] space.

"finite" p o i n t s

borrow a convenient

(m+l)

the G r a s s m a n n i a n For general

2

In the

can be re-

m a similar

of the a s s o c i a t e d

term from the literature

inter-

(m+l)21dimen of science

aa l fiction--we

can introduce

of p r o j e c t i v e realized

j-planes

as a q u a d r i c

the space G(m,2m+l) pr, w h e r e

a set of v a r i a b l e s

in pk is often d e n o t e d hypersurface

can be r e a l i z e d

r is given by the formula

in ~DS.

x

as coordinates. G(j,k).

The space

In the general

as an a l g e b r a i c

The G r a s s m a n n i a n G(I,3)

can be

case of interest

variety

of d i m e n s i o n

here,

(m+l) 2 in

138

(10.4.3)

r =

(2m+2)!/[(m+l)!] 2

this being the d i m e n s i o n of the space of skew-symmetric tensors of v a l e n c e m+l in c2m+2 in the case m=l there is a "preferred" line pl in p3 c o r r e s p o n d i n g to the I v e r t e x of scri in spacetime. w h i c h meet p3 I " avoid pl I "

The "finite" points of spacetime c o r r e s p o n d to lines in p3 w h i c h

Likewise, there is a p r e f e r r e d m - p l a n e pm in p2m+l I

of the hyperspace G(m,2m+l) avoid pm I "

The remaining points of scri c o r r e s p o n d to lines in p3

correspond, b y definition,

"Infinity" in G(m,2m+l)

in p2m+l w h i c h intersect PI m

"

"

The "finite" points

to m-planes in p2m+l w h i c h

consists of points w h i c h correspond to m - p l a n e s

The points of G(m,2m+l)

can be classified by a number

d w h i c h is the dimension of the intersection of the c o r r e s p o n d i n g m-plane w i t h P For finite points we put d = -i.

m I

In the case of standard twistor space, given b y

m=l, there are just three classes of a s s o c i a t e d spacetime points: (d = -i), n o n - v e r t e x points on scri

finite points

(d = 0), and the vertex of scri

(d = 1).

In the

general case d has the range d = -l,0,...,m; the case d = m c o r r e s p o n d i n g to the m-

PIm i t s e l f .

plane

A g e n e r a l i z e d twistor x

as'

( a '%a' ) lies on the m - p l a n e c o r r e s p o n d i n g to the p o i n t

if and only if the relation

(10.4.4)

holds.

a

ixaa '

We can characterize g e n e r a l i z e d twistors in terms of solutions of the

equation

(10.4.5)

(m+l)Vaa,~ b = @~Vca,~ c

In fact, we have the following result

(cf. Hughston,

1979), w h i c h is analogous to

P r o p o s i t i o n 2.4.2: 10.4.6

(10.4.7)

Theorem.

The general solution of equation

~a = wa _ i x a a ' a I

w h e r e w a and z a I are constant.

i

(10.4.5) is given by

139

Proof.

Differentiating

(10.4.8)

which,

one has

(m+l)Vbb,Vaa , ~c = ~aVbb,Vda,~ c d

exchanging

aa' and bb'

(10.4.9)

(m+l)V

Since the V's commute, equal;

(10.4.5)

, can be written as

~ c c d aa,Vbb,~ = 6bVaa,Vdb,~

the left hand sides of equations

(10.4.8)

and

(10.4.9)

are

therefore:

~d 6~Va a {d ~Vbb,Vda , = ,Vdb ,

(10.4.10)

Transvecting

equation

(10.4.11)

(10.4.10)

with @b gives c

Vab,Vda,~ d :

On the other hand,

if equation

(10.4.10)

(m+l)Vbb,Vda,~ d = Vba,Vdb,~ d which,

(10.4.12)

(m+l)Vaa,Vdb,~ d

is transvected

substituting

instead with 6 a one obtains c

b with a , reads:

(m+l)Vab,Vda,~ d = Vaa,Vdb,~ d

For m > 0 , equations

(10.4.11)

(10.4.13)

and

(10.4.12)

Vaa,Vdb,~ d = 0

showing that Vdb,~ d is constant;

from

together

imply

,

(10.4.5)

we can therefore

infer that ~a is

aa'

a

linear in x

Inserting the most general

the desired result

(10.4.7)

( a ,~a,).

From

and

finite points are concerned,

into

(i0.4.5),

(Wa ,~a,) is defined to be the set of points in the

corresponding (10.4.4)

for ~

follows after a short calculation (S) . []

The locus of a twistor space G(m,2m+l)

linear expression

to the pencil of m-planes (10.4.7)

in p2m+l through the point

it follows that the locus,

is the region where the associated

insofar as

spinor field ~a(x)

vanishes. When m is odd, and the n-twistor (10.4.4)

reads

relations

(10.4.2)

are assumed,

equation

140

(10.4.14)

A

. AA'j i = lXi ~jA'

where we have made the identification

(i0.4.15)

x

for the coordinates

aa'

AA'j = x. l

of finite points in G(m,2m+l).

Complex Minkowski

space is

embedded naturally as a linear subspace in this space, given by

(10.4.16)

xAA'j AA'~j i = x l

We shall take the attitude that fields on spacetime tions, down to spaeetime,

of "hyperspace"

fields.

time should be thought of as the restriction

can be regarded as restric-

Thus, a field @(x

Px of some hyperspace

AA'

) on space-

field ~(x'AA'J)I

according to the following scheme:

(10.4.17)

AA'j) ~(x m~' ) = Px'~(xi

By requiring that the hyperspace

: }(x~'~!)l

exhibit suitable properties, field ~(x AA'j) i

tions can be imposed on the spacetime field ~(x) projected

10.5

condi-

from it.

Massive Fields Revisited. The contour formulae introduced

reinterpreted

for massive fields in Section 5.2 can be

in an interesting way in the light of the remarks made in Section

10.4. In Section 5.2 we were concerned with contour integral formulae of the form

(10.5.1)

A...j°...i. ( x ) = / p x @A'.

~Aj "" .~A,if(Z~)~

Using the notation of Section 10.4, formula

(10.5.2)

(i0.5.1) can be rewritten as:

@a...a , ... (x) = S P x @ a. "'~a '" "'f(wa 'ga ')A~ '

where, in addition to (10.4.2), we use the notation ~ a = -~/$w a , along with (10.5.3)

0xf(~ a ,~a,) =

pxf(W~

,~A,i )

AA' =

f(ix

~A'

'ZA')

141

Now it turns out that formula

(10.5.2)

evaluate the twistor function

f(~0a ,Za,) on hyperspace

coefficients,

can be evaluated

as will be described below).

down to spacetime.

is denoted ~.

ation is determined ~-type.

(using a suitable

set of z-

, and restrict

the result

This gives us the field @a...a'... (x) .

Suppose the number of T-coefficients of ~-coefficients

First we

Then, we take a number of derivatives

i = ~/~ x ~ ' J using the operator VA,Aj

of the hyperfield,

in two steps.

(10.5.2)

The coefficient

is denoted

structure

^ If 7-z is positive,

as follows.

If Z-~ is positive,

in

z, and the number

for the hyperspace

^ we use z-z coefficients

on the other hand, we use ~-~ coefficients

evaluof the

of the ~-

type. The number of derivatives spacetime

we take before restricting

is simply the absolute value

the hyperfield

down to

I~-~I

It should be observed that when we examine hyperfields, fields whose indices are all of the same type.

Let us denote

we need only consider by Px the restriction

aa' down to the hyperspace

point x

Then the following three contour integral

formulas are of interest to us:

(10.5.4)

~a. ..b(£) = /Px~a.. .~bf(~ ~ a

(10.5.5)

~(x) =/px__f(

(10.5.6)

%a,...b,(X)

By taking derivatives,

(10.5.7)

which form

10.6

,~a,)A~

a 'Za')AT

=/Px~a,...zb,f(~a_

,Za,)A%

and using the identity

iV

aa

,Ox

A = Qx~a~a '

is valid for functions of w a and

a'

' we

can

recover general

fields of the

(10.5.2).

Towards the Cohomology of n-Twistor The fields defined in formulae

rather curious

feature;

Systems.

(i0.5.4),

(i0.5.5),

they satisfy the following

and

(10.5.6)

exhibit a

set of field equations:

142

(10.6.1)

Va, [a~b]...c = 0

(10.6.2)

Va,[aVb]b,~ = 0

(10.6.3)

Va[a,~b,]...c, = 0

,

,

These relations generalize the ZRM equations in a natural way, and in the case m = 1 they reduce to the Z_KM equations. have a second order equation

Note that in the case of a scalar field ~ we

(analogous to the wave equation), w h e r e a s in the

other case we have first order equations. wood, equations

(10.6.1) and

However, as was p o i n t e d out by M. East-

(10.6.3) imply the second order equations

(10.6.4)

Vp,[pVq]q,~a... b = 0

(10.6.5)

Vp, [p?q]q'~a'...b' = 0

,

,

as one m i g h t expect by analogy w i t h the case m = I. We can classify the ~-fields w i t h a h a l f - i n t e g e r r.

I ~ e n r is p o s i t i v e we have

a field w i t h 2r p r i m e d indices, and w h e n r is negative we have a field w i t h -2r u n p r i m e d indices.

W h e n r = 0 we have a scalar field.

For lack of better terminolo~

gy, we shall refer to r as the "hyperhelicity". It turns out, rather remarkably, that solutions of the free field equations (10.6.1), groups.

(10.6.2), and

(10.6.3) can be d e s c r i b e d as e l e m e n t s of certain c c h o m o l o g y

The relevant groups can be d e s c r i b e d as follows.

Let M be a region of

p2m+l swept out by a set of m - p l a n e s c o r r e s p o n d i n g to the points in a region M of hyperspace.

Recall that if we are d e a l i n g w i t h n - t w i s t o r systems, then m = 2n-i

Denote by O(q)

.

the sheaf of germs of h o l o m o r p h i c functions on M, twisted b y q.

Then we have: 10.6.6

(10.6.7)

Proposition.

Eler~ents of the g r o u p

Hm(~i,O(-2r-m-1))

correspond to solutions of the h y p e r s p a c e free field equations and

(10.6.3)

for h y p e r h e l i c i t y r, over the domain M.

(10.6.1),

(10.6.2),

143

Proof.

We shall establish

the result explicitly

for the case m = 2, with

r = 1/2.

Unfortunately,

this is not one of the cases which readily admits of a

spacetime

interpretation

(for which m must be odd); but this is not a serious draw-

back,

since the more general cases can be inferred directly

will be outlined here.

from the method that

Our proof will follow rather closely the material

in

Section 9.3. We are concerned with the group H2(M5,0(-4)), swept out by a set of 2-planes.

where M 5 is a region of p5

Let fijk be a representative

cocycle.

Restricting

aa' down to the 2-plane x

we have -4

(10.6.8)

-2

PxSa,Zb,fijk

since H 2 is trivial

= p[ifjk]a,b , (x,z)

for twist greater than -3, on p2

.

Skewing with Zc' we get

-2 p[ifjk]a , [b,Zc,] = 0

(10.6.9)

from which we deduce the existence

-i of an fka'b'c'

such that

-2 -i fjka'[b'~c '] = P[jfk]a'b'c'

(10.6.10)

Skewing with Zd' we get -i P[jfk]a' [b'c'Zd ']

(10.6.11)

=

0

whence we obtain: -i (10.6.12) 0 Now since f alone;

0

fka' [b'c'Zd '] = Pkfa'b'c'd '

a'b'c'd'

is global and has twist zero,

it must be a function of x

thus we obtain our field 0

(10.6.13)

~a'

To show that ~a' satisfies

(10.6.14)

(xaa ')

= fa'b'c'd

the field equation

Vb[b,}a, ] = 0

'~

b'c'd'

144

is a somewhat more insidious

operation.

We proceed as follows:

Let D be any operator which annihilates

the expression

Px~a,~b,fijk

.

In

our case D is defined by

(10.6.15)

D: Px~a,~b,fijk

which vanishes; of D.

From

~÷

Ve[e,~a,]~b,Pxfij k

but the results which follow are independent

of the specific choice

(10.6.8) we obtain -2 P[iDf'k] a'b']

(10.6.16)

= 0

whence we have -2 Dfjka'b'

(10.6.17) -2 for some fka'b'

"

-2 = P[jfk]a'b'

Applying Zc' and skewing, we get -2 -2 Dfjka' [b'~e '] = P[jfk]a' [b'~c ']

(10.6.18)

Now suppose we take

(10.6.10),

and hit it with D.

-2 Dfjka,[b,~c,]

(10.6.19)

Since the right hand sides of

(10.6.18)

Then we obtain

-i = p[jDfk]a,b,c~

and

(10.6.19)

are equal, we have

-2 -3_ P[jfk]a' [b'~c '] = p[jDfk]a'b'e'

(10.6.20)

which asserts that the cochain -2 (10.6.21)

is global.

-i

fka' [b'~c '] - Dfka'b'c'

Since the twist is negative,

whence we deduce,

skewing ~d'

it follows that

' that -i Dfka, [b,e,Zd, ] = 0

(10.6.22)

(10.6.21) must vanish,

from which we get D~ a w = 0 , using

(10.6.12)

and

,

(10.6.13).

However,

it is not

difficult to verify that if we trace the action of D through the various above then

(10.6.15)

implies

formulae

145

(10.6.23)

D:

~a'

showing that Oa, s a t i s f i e s

+ Ve [e'~a']

the field

equations,

a s d e s i r e d (6) , [ ]

It w o u l d be nice to sharpen up P r o p o s i t i o n 10.6.6 a bit, so as to specify p r e c i s e l y for what sort of domains an actual i s o m o r p h i s m is o b t a i n e d b e t w e e n H m and the relevant set of hyperfields. p4n-i + The

A r e a s o n a b l e candidate for M is the space

(or p o s s i b l y its closure), w h i c h in twistor terms is the space Z 0 l

(2n-l)-planes lying e n t i r e l y w i t h i n p4n-i include, as a subset, a four+

d i m e n s i o n a l family of planes c o r r e s p o n d i n g to the future tube CM + in complex Minkowski space.

The "Minkowskian"

twistors w h i c h are of the form Z~'3 = x

AA'

(2n-l)-planes are o b t a i n e d by looking at nAA' (ix ZA'•3 , ~A,j).

we obtain a (2n-l)-plane b y v a r y i n g ZA'i "

o n l y if Z.Z

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