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Mim-HILQU‘An_lntr~01Iucti0n to Spmm ts"andGeonwtijv vyitlzApplications ml’I1_nsz¢:w I/UBenn Rl"lF1ckvr “I- ‘bvI I'~_0 ..'I‘OO +._,1‘-i '4 ‘ Q1"-s' -I r-' _ 0 I'.U1 & -Q-1-\-'1'”-:‘_Qg ‘f~- n 0- I J: ~S ' ._ 0"i ."-" ‘E.’ -. 4-. H '"‘""\_| ml i 1|-Q‘; I .l_pl’ ’_ ‘ii -.5 ;L.. V.’-J " ‘ -1 -Q 1'5‘-'b-_t:1.-2}’"=:- "-1'_“_.'-1':IL...H-‘MJ"*_in "5I --Q’ I.!I~,,- -_.. r .1“??-‘’!‘::‘-.‘n‘-3"";_.:'?'_a;_'-0'?-C7'1'-':;l.°.*""'13."'-‘-1-? ‘-1"\_ . -' I ‘:|‘;£-1-II’'-_2-+"inf"\'t~""""'jg‘;r:4....'_.q.1:‘$11| tak»;-'1‘'-_-*~=..-:7;-Ias-.1‘,‘r':_..:.I4-II 1?'L'QIl-D-I’ 0”.“H'P\~ l. 'M-*1 -I\- 4'v .I... . ‘I "'00 i ‘K1. "'_‘ '1.~%Ia-4"""% -‘:1.-‘4*"% ' I» I "0 Qfl kn Introduction to Spinors and Geometry with Applications in Physics I M Benn Faculty of Science, University College of The Northern Territory, Australia R W Tucker Department of Physics. University of Lancaster, UK Adam Hilger, Bristol and New York \nIntroduction toSpinors andGeometry with Applications inPhysics IMBenn Faculty ofScience, University College ofTheNorthern Territory, Australia RWTucker Department ofPhysics. University ofLancastert UK Adam Hilger, Bristol andNew York C) IOP Publishing Ltd 1987 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. British Library Cataloguing in Publication Data Berm, I. M. An introduction to spinors and geometry with applications in physics. 1. Spinor analysis I. Title II. Tucker, R. W. 512'.57 QA433 ISBN 0-85274-169-3 ISBN 0-85274-261-4 (pbk) Library of Congress Cataloging -in-Publication Data Benn. I. M. (Ian M.) An introduction to spinors and geometry with applications in physics Bibliography: p. Includes index. 1. Spinor analysis. 2. Geometry, Differential I. Tucker, R. W. (Robin W.) II. Title. QC20.7.S65B46 1988 515'.63 87-21117 ISBN 0-85274-169-3 ISBN 0-85274-261-4 (pbk) Consultant Editor: Professor R F Streater King's College, London First published 1987 Paperback edition 1989 Published under the Adam Hilger imprint by IOP Publishing Ltd Techno House, Redcliffe Way, Bristol BS1 6NX, England 335 East 45th Street, New York, NY 10017-3483, USA Typeset by KEYTEC, Bridport, Dorset Printed and bound in Great Britain by Butler & Tanner Ltd, Frome and London ©IOPPublishing Ltd1987 Allrights reserved. Nopart ofthispublication may bereproduced, stored inaretrieval system ortransmitted inanyform orbyanymeans, electronic, mechanical, photocopying, recording orotherwise, without theprior permission ofthepublisher. British Library Cataloguing inPublication Data Benn, I.M. Anintroduction tospinors andgeometry with applications inphysics. 1.Spinor analysis I.Title II.Tucker, R.W. 512’.57 QA433 ISBN 0-85274-169-3 ISBN 0-85274-261-4 (pbk) Library ofCongress Catal0ging-in-Publication Data Benn, I.M.(Ian M.) Anintroduction tospinors andgeometry with applications inphysics Bibliography: p. Includes index. 1.Spinor analysis. 2.Geometry, Differential I.Tucker, R.W.(Robin W.) II.Title. QC20.7.S65B46 1988 515’.63 87-21117 ISBN0-85274-169-3 ISBN0-85274-261-4 (pbk) Consultant Editor: Professor RFStreater King’s College, London First published 1987 Paperback edition I989 Published under theAdam Hilger imprint byIOPPublishing Ltd Techno House, Redclifi“e Way, Bristol BS16NX, England 335East 45th Street, New York, NY10017-3483, USA Typeset byKEYTEC, Bridport, Dorset Printed andbound inGreat Britain by Butler &Tanner Ltd, Frome andLondon To Shian To Daniel and Edmund ToShian ToDaniel andEdmund Contents Preface 1 Tensor Algebra ix 1 1.1 The tensor algebra 2 1.2 The exterior algebra of antisymmetric tensors 4 1.3 The exterior algebra as a quotient of the tensor algebra 10 1.4 The Hodge map 13 1.5 The mixed tensor algebra 16 Bibliography 20 2 Clifford Algebras and Spinors 21 2.1 The Clifford algebra 23 2.2 The structure of the real Clifford algebras 28 2.3 The even subalgebra 39 2.4 The Clifford group 42 2.5 Spinors 54 2.6 Spin-invariant inner products 62 2.7 The complexified Clifford algebras 80 2.8 The confusion of tongues 85 Bibliography 105 3 Pure Spinors and Triality 106 3.1 Pure spinors 106 3.2 Triality 117 Bibliography 122 4 Manifolds 123 4.1 Topological manifolds 124 4.2 Derivatives of functions IFI"' 127 Contents Preface 1Tensor Algebra 1.1 Thetensor algebra 1.2 Theexterior algebra ofantisymmetric tensors 1.3 Theexterior algebra asaquotient ofthetensor algebra TheHodge map Themixed tensor algebra Bibliography1.4 1.5 2Clifford Algebras andSpinors 2.1 TheClifford algebra 2.2 Thestructure oftherealClifford algebras 2.3 Theeven subalgebra 2.4 TheClifford group 2.5 Spinors 2.6 Spin-invariant inner products 2.7 Thecomplexified Clifford algebras 2.8 Theconfusion oftongues Bibliography 3Pure Spinors andTriality 3.1 Pure spinors 3.2 Triality Bibliography 4Manifolds 4.1 Topological manifolds 4.2 Derivatives offunctions ]B"'—>IR"ix v-dC-l>t\)I-* 13 16 20 21 23 28 39 42 54 62 80 85 105 106 106 117 122 123 124 127 CONTENTS Vii 4.3 Differentiable manifolds 129 4.4 Parametrised curves 134 4.5 Tangent vectors 136 4.6 Vector fields 141 4.7 The tangent bundle 143 4.8 Differential 1-forms 146 4.9 Tensor fields 150 4.10 Exterior derivatives 154 4.11 One-parameter diffeomorphisms and integral curves 156 4.12 Lie derivatives 161 4.13 Integration on manifolds 167 4.14 Metric tensor fields 171 Bibliography 174 5 Applications in Physics 176 5.1 Galilean spacetimes 176 5.2 Maxwell's equations and Minkowski spacetime 178 5.3 Observer curves 183 5.4 Electromagnetism 188 Bibliography 197 6 Connections 199 6.1 Linear connections 200 6.2 Examples and Newtonian force 204 6.3 Covariant differentiation of tensors 206 6.4 Curvature and torsion tensors of V 208 6.5 Bianchi identities 212 6.6 Metric-compatible connections 214 6.7 The covariant exterior derivative 216 6.8 The curvature scalar and Einstein tensor 219 6.9 The pseudo-Riemannian connection 221 6.10 Sectional curvature 223 6.11 The conformal tensor 225 6.12 Some curvature relations in low dimensions 227 6.13 Killing's equation 229 Bibliography 231 7 Gravitation 232 7.1 Lorentzian connections 232 7.2 Fermi-Walker transport 234 7.3 The Einstein field equations 234 7.4 Conservation laws 237 7.5 Some matter fields 239 7.6 The Reissner-Nordstrôm solution 243 7.7 Gravitation with torsion 249 Bibliography 250 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14CONTENTS Differentiable manifolds Parametrised curves Tangent vectors Vector fields Thetangent bundle Differential 1-forms Tensor fields Exterior derivatives One-parameter diffeomorphisms andintegral curves Liederivatives Integration onmanifolds Metric tensor fields Bibliography Applications inPhysics 5.1 5.2 5.3 5.4Galilean spacetimes Maxwell’s equations andMinkowski spacetime Observer curves Electromagnetism Bibliography Connections 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13Linear connections Examples andNewtonian force Covariant differentiation oftensors Curvature andtorsion tensors ofV Bianchi identities Metric-compatible connections Thecovariant exterior derivative Thecurvature scalar andEinstein tensor Thepseudo-Riemannian connection Sectional curvature Theconformal tensor Some curvature relations inlowdimensions Killing’s equation Bibliography Gravitation 7.1 7.2 7.3 7.4 7.5 7.6 7.7Lorentzian connections Fermi—Walker transport TheEinstein field equations Conservation laws Some matter fields TheReissner—Nordstrom solution Gravitation with torsion Bibliography viii CONTENTS 8 Clifford Calculus on Manifolds 251 8.1 Covariant differentiation of Clifford products 252 8.2 The operator 0 254 8.3 The Kahler equation 256 8.4 The Duffin—Kemmer—Petiau equations 260 Bibliography 260 9 Spinor Fields 261 9.1 Spinor bundles 261 9.2 Inner products on spinor fields 264 9.3 Covariant differentiation of spinor fields 267 9.4 Lie derivatives of spinor fields 271 9.5 Representing spinor fields with differential forms 275 Bibliography 277 10 Spinor Field Equations 278 10.1 The Dirac operator 278 10.2 Covariances of the Dirac equation and conserved 280 currents 10.3 The Dirac equation in spacetime 282 10.4 The stress tensor 290 10.5 Tensor spinors 294 10.6 The Lichnerowicz theorem 299 10.7 Killing spinors 300 10.8 Parallel spinors 303 Appendix A: Algebra 307 Bibliography 341 Appendix B: Vector Calculus on 1R3 342 References 349 Index 351 viii CONTENTS SClifford Calculus onManifolds 8.1 Covariant differentiation ofClifford products 8.2 Theoperator gal 8.3 TheKahler equation 8.4 TheDuffin—Kemmer—Petiau equations Bibliography 9Spinor Fields 9.1 Spinor bundles 9.2 Inner products onspinor fields 9.3 Covariant differentiation ofspinor fields 9.4 Liederivatives ofspinor fields 9.5 Representing spinor fields with differential forms Bibliography 10Spinor Field Equations 10.1 TheDirac operator 10.2 Covariances oftheDirac equation andconserved currents 10.3 TheDirac equation inspacetime 10.4 Thestress tensor 10.5 Tensor spinors 10.6 TheLichnerowicz theorem 10.7 Killing spinors 10.8 Parallel spinors Appendix A:Algebra Bibliography Appendix B:Vector Calculus on1R3 References Index Preface A student of theoretical physics who wishes to follow recent trends in current research is liable to be confronted with a bewildering amalgam of ideas from physics and mathematics. In particular, much of the terminology permeating developments in the theories of matter and gravitation is borrowed from classical differential geometry. In many of these theories spinors play a prominent role. A further notable develop- ment is the introduction of spaces with 'exotic' topologies and geo- metries in formulating the basic laws of Nature. Consequently the student finds it necessary to possess a broad knowledge of mathematical techniques that encompasses such generalities as well as the computa- tional skills necessary to use this information. In this book we have attempted to provide a concise but self- contained introduction to the basic properties of differential geometry and spinors accommodating some of the needs mentioned above. We feel that physicists learn most rapidly by seeing new concepts spelled out in some detail. We have attempted a blend of mathematics and theoretical physics which we hope will assist in the assimilation of new ideas and give readers a feeling that they are closer to the 'nuts and bolts' of the subject material. In writing any introduction to a subject as broad as this we have had to face the problem of what prerequisites we expect our readers to possess. Fundamental to any appreciation of tensor methods is a firm familiarity with linear algebra. Thus our book begins with algebraic notions. We have tried to encapsulate the neces- sary concepts used in Chapters 1 and 2 into Appendix A. This should provide a reservoir of compact information for those who may find some foreign vocabulary in these early chapters. Our emphasis here is on real vector spaces and their complexifications. We feel that this approach makes closest contact with what most physicists actually use when working with the complexified Clifford algebra of spacetime. We intro- duce a spinor as an element carrying an irreducible representation of Preface Astudent oftheoretical physics who wishes tofollow recent trends in current research isliable tobeconfronted with abewildering amalgam ofideas from physics and mathematics. Inparticular, much ofthe terminology permeating developments inthetheories ofmatter and gravitation isborrowed from classical differential geometry. Inmany of these theories spinors play aprominent role. Afurther notable develop- ment istheintroduction ofspaces with ‘exotic’ topologies and geo- metries informulating thebasic laws ofNature. Consequently the student finds itnecessary topossess abroad knowledge ofmathematical techniques that encompasses such generalities aswell asthecomputa- tional skills necessary tousethisinformation. Inthis book wehave attempted toprovide aconcise but self- contained introduction tothebasic properties ofdifferential geometry andspinors accommodating some oftheneeds mentioned above. We feelthatphysicists learn most rapidly byseeing newconcepts spelled out insome detail. Wehave attempted ablend ofmathematics and theoretical physics which wehope willassist intheassimilation ofnew ideas andgive readers afeeling that they arecloser tothe‘nuts and bolts’ ofthesubject material. Inwriting anyintroduction toasubject as broad asthiswehave hadtoface theproblem ofwhat prerequisites we expect our readers topossess. Fundamental toany appreciation of tensor methods isafirm familiarity with linear algebra. Thus ourbook begins with algebraic notions. Wehave tried toencapsulate theneces- sary concepts used inChapters 1and2into Appendix A.This should provide areservoir ofcompact information forthose whomayfindsome foreign vocabulary inthese early chapters. Our emphasis here isonreal vector spaces and their complexifications. Wefeel that this approach makes closest contact with what most physicists actually use when working with thecomplexified Clifford algebra ofspacetime. Weintro- duce aspinor asanelement carrying anirreducible representation of X PREFACE some Clifford algebra. This emphasis on the Clifford algebras rather than the spin groups is slightly different from that commonly adopted by most working physicists. However, the spin groups are most easily defined as sitting in the Clifford algebra, and thus we may induce representations of these groups from those of the algebras. No doubt some readers will be surprised at the classical tone that dominates our description of spinors. We offer little apology. As a mathematical entity the notion of a spinor requires no quantum theoretical overtones. Although we would have liked to develop further the basic role played by spinors in quantum field theory we feel that their role in physical models need not intrude into their basic relation to geometry. More- over, a proper appreciation of this relation is essential in relativistic quantum field theory. The introduction to differential manifolds (Chapter 4) is fairly elementary and presupposes only a basic knowledge of the calculus of many variables. We have interrupted its development with a chapter on physical applications before formally introducing the idea of a linear connection. This chapter illustrates the importance of Lorentzian geometry in relativistic physics, and is motivated by a discussion of electromagnetism. Chapter 7 is devoted to the field theory of gravitation and its sources in which many of the mathematical tools introduced earlier are put to use. The two main themes of Clifford algebras and differentiable manifolds are drawn together in the final chapters on Clifford forms and spinor fields. Here readers will find physical applica- tions involving spinors on manifolds and are introduced to some recent developments that relate geometrical properties of a space to the existence of spinor fields with particular properties. Earnest readers are invited to test their expertise by working out some of the illustrative examples that have been inserted at strategic points in the text. In the course of writing this book we have benefited from dialogues with many colleagues. In particular, we wish to thank Graeme Segal, R Al-Saad, J Brooke, C T J Dodson, E Kahler, K McCrimmond, and D Towers for helpful comments on various aspects of our enterprise. We are also grateful for correspondence with A Crumeyrolle, K McKenzie and D Plyman on aspects of Clifford algebras. The production of our manuscript was greatly assisted with the aid of TEXnical facilities generously provided by A B Clegg and P M Lee. We also thank G Hughes for all the time and effort he spent teaching us to drive the Vax-editor and its peripherals. Finally, we are happy to acknowledge the support provided by the University of Lancaster Research Fund. I M Benn R W Tucker x PREFACE some Clifford algebra. This emphasis ontheClifford algebras rather than thespin groups isslightly different from that commonly adopted by most working physicists. However, thespin groups aremost easily defined assitting intheClifford algebra, and thus wemay induce representations ofthese groups from those ofthealgebras. Nodoubt some readers willbesurprised attheclassical tone that dominates our description ofspinors. Weoffer little apology. Asamathematical entity thenotion ofaspinor requires noquantum theoretical overtones. Although wewould have liked todevelop further thebasic role played byspinors inquantum field theory wefeel that their role inphysical models need notintrude into their basic relation togeometry. More- over, aproper appreciation ofthis relation isessential inrelativistic quantum field theory. The introduction todifferential manifolds (Chapter 4)isfairly elementary andpresupposes only abasic knowledge ofthecalculus of many variables. Wehave interrupted itsdevelopment with achapter on physical applications before formally introducing theidea ofalinear connection. This chapter illustrates the importance ofLorentzian geometry inrelativistic physics, and ismotivated byadiscussion of electromagnetism. Chapter 7isdevoted tothefield theory ofgravitation and itssources inwhich many ofthemathematical tools introduced earlier areputtouse. The two main themes ofClifford algebras and differentiable manifolds aredrawn together inthefinal chapters on Clifford forms andspinor fields. Here readers willfindphysical applica- tions involving spinors onmanifolds andareintroduced tosome recent developments that relate geometrical properties ofaspace tothe existence ofspinor fields with particular properties. Earnest readers are invited totesttheir expertise byworking outsome oftheillustrative examples thathave been inserted atstrategic points inthetext. Inthecourse ofwriting thisbook wehave benefited from dialogues with many colleagues. Inparticular, wewish tothank Graeme Sega], R Al-Saad, JBrooke, CTJDodson, EKahler, KMcCrimmond, andD Towers forhelpful comments onvarious aspects ofourenterprise. We arealso grateful forcorrespondence with ACrumeyrolle, KMcKenzie andDPlyman onaspects ofClifford algebras. Theproduction ofour manuscript was greatly assisted with theaidofTEXnical facilities generously provided byABClegg and PMLee. Wealso thank G Hughes forallthetime and effort hespent teaching ustodrive the Vax-editor and itsperipherals. Finally, wearehappy toacknowledge thesupport provided bytheUniversity ofLancaster Research Fund. IMBenn RWTucker 1 Tensor Algebra This first chapter will provide a foundation for the two initially separate directions the book will take; algebra and geometry. In Appendix A we have gathered together a number of ideas relating to the study of vector spaces and algebras. These notions will be used freely within the first two chapters. The reader who is initially confronted with foreign vocabulary or new concepts should consult this Appendix for definitions where a concise development of rudimentary ideas is also to be found. The first section of this chapter introduces the tensor algebra of an arbitrary vector space. In Chapter 2 this will be the starting point for our construction of the Clifford algebra, which will be defined as a quotient of the tensor algebra. In Chapter 4 and subsequent chapters when beginning geometry we will be interested in the tangent space (and the cotangent space) of a manifold. We will then be able to apply the material of this chapter immediately to that vector space. In fact it will be the cotangent space that is taken for the arbitrary vector space V. Anticipating this we have (identifying the second dual space of V with itself) written elements of V as acting on V*, rather than the other way around. Particularly important on manifolds are the totally antisymmetric tensor fields; the differential forms. In §1.2 we introduce the exterior forms on an arbitrary vector space. These will also play a prominent role in our treatment of the Clifford algebra. To facilitate a comparison with the Clifford algebra we re-introduce the exterior algebra in §1.3 as a quotient of the tensor algebra. Only in §1.4 does a metric enter. (Our meaning of a metric is given in Appendix A.) This allows us to introduce the Hodge map which is a key ingredient of the calculus of differential forms on (pseudo-) Riemannian manifolds. We have delayed introducing the mixed tensor algebra until §1.5. Here contact is made with the classical definition of a tensor in terms of Tensor Algebra Thisfirstchapter willprovide afoundation forthetwoinitially separate directions thebook willtake; algebra andgeometry. InAppendix Awe have gathered together anumber ofideas relating tothestudy ofvector spaces and algebras. These notions willbeused freely within thefirst two chapters. The reader who isinitially confronted with foreign vocabulary ornewconcepts should consult thisAppendix fordefinitions where aconcise development ofrudimentary ideas isalsotobefound. The first section ofthischapter introduces thetensor algebra ofan arbitrary vector space. InChapter 2thiswillbethestarting point for ourconstruction oftheClifford algebra, which willbedefined asa quotient ofthetensor algebra. InChapter 4andsubsequent chapters when beginning geometry wewill beinterested inthetangent space (and thecotangent space) ofamanifold. Wewillthen beabletoapply thematerial ofthischapter immediately tothat vector space. Infactit willbethecotangent space that istaken forthearbitrary vector space V.Anticipating this wehave (identifying thesecond dual space ofV withitself) written elements ofVasacting onV*,rather than theother wayaround. Particularly important onmanifolds are thetotally antisymmetric tensor fields; thedifferential forms. In§1.2 weintroduce theexterior forms onanarbitrary vector space. These will also play aprominent roleinourtreatment oftheClifford algebra. Tofacilitate acomparison withtheClifford algebra were-introduce theexterior algebra in§1.3as Hquotient ofthetensor algebra. Only in§1.4does ametric enter. (Our meaning ofametric isgiven in Appendix A.)This allows ustointroduce theHodge map which isakey ingredient ofthecalculus ofdifferential forms on(pseudo-) Riemannian manifolds. Wehave delayed introducing themixed tensor algebra until §1.5. Here contact ismade with theclassical definition ofatensor interms of 2 TENSOR ALGEBRA transformation properties of components. Index conventions will be established that allow the traditional 'raising and lowering' of indices. 1.1 The Tensor Algebra If V is any vector space over some field F then the set of F-valued linear maps on V forms a vector space; the dual space, V. lf, as we now assume, V is finite dimensional then there is a natural way to regard elements of V as linear maps on V*. That is, if x E V and X E V* such that X acts on x to produce the scalar X(x) then we can equivalently think of this as defining an action of x on X, x(X) = X(x). In the following it will be convenient to adopt this seemingly perverse view of regarding V as the space of linear mappings on V*. Just as the F-valued linear maps on V* form a vector space so do the multilinear maps on ordered sets of elements from V*. The F-valued multilinear maps on V* x V* x ...x V* (r times) are called tensors of degree r. The notion of multilinearity is an obvious extension of the notion of a linear map; for any fixed choice of r — 1 elements of V* the map is linear in the remaining variable. Multilinearity ensures that a tensor of degree r is completely specified by its action on all ordered sets of basis vectors for V*, thus if V (and hence V*) is n-dimensional then the tensors of degreet r form an nr-dimensional vector space, T r(V). We may associate a set of r elements from V with a tensor of degree r. For Xi E V, i = 1, .. r and Xi E V*, i = 1, r we define (X1Y_.\9X20 X2, . Xr) = XI(X1)X2(X2) . . . X r(Xr). In particular, if fe') is a basis for V then the set of all n r elements {e''Oe1,0 . . . Oe'r}, where the indices take all values from 1 to n, forms a basis for T r(V). The vector space T r(V) is called the tensor product of V r times T r(V) = VOV OV OrV. More generally, the tensor product defines a mapping between tensors of different degrees : T r(V) x T s (V) T r „(V) (1.1.1) a, b a0b where t Formerly called rank. 2 TENSOR ALGEBRA transformation properties ofcomponents. Index conventions will be established thatallow thetraditional ‘raising andlowering’ ofindices. 1.1TheTensor Algebra IfVisanyvector space over some field Fthen thesetofF-valued linear maps onVforms avector space; thedual space, V*.If,aswe now assume, Visfinite dimensional then there isanatural way to regard elements ofVaslinear maps onV*.That is,ifxeVandXeV* such that Xacts onxtoproduce thescalar X(x) then wecan equivalently think ofthisasdefining anaction ofxonX,x(X) =X(x). Inthefollowing itwillbeconvenient toadopt thisseemingly perverse view ofregarding Vasthespace oflinear mappings onV*.Just asthe F-valued linear maps onV*form avector space sodothemultilinear maps onordered sets ofelements from V*.The F-valued multilinear maps onV*><V*X...><V*(rtimes) arecalled tensors ofdegree r. The notion ofmultilinearity isanobvious extension ofthenotion ofa linear map; foranyfixed choice ofr—1elements ofV*themap is linear intheremaining variable. Multilinearity ensures that atensor of degree riscompletely specified byitsaction onallordered setsofbasis vectors forV*,thus ifV(and hence V*)isn-dimensional then the tensors ofdegree’r rform ann’-dimensional vector space, T,(V). Wemay associate asetofrelements from Vwith atensor ofdegree r.Forx'eV,i= 1,...,randX,-eV*,i= 1,...,rwedefine (x1®x2® ...®x')(X1,X2, ...,X,) =x‘(X1)xZ(X2) ...x'(X,). Inparticular, if{e'} isabasis forVthen thesetofalln’elements {e"®e'1® ...®e'?}, where theindices take allvalues from 1ton, forms abasis forT,(V).The vector space T,(V) iscalled thetensor product ofVrtimes T,(V)=v®v...®vE ®'v. More generally, thetensor product defines amapping between tensors ofdifferent degrees ®:T,(V) ><T,(V) i> T,,,,(V) (1.1.1) a,b»——>a®b where TFormerly called rank. THE TENSOR ALGEBRA 3 (a0b)(X 1, X2, . . , Xr + 1, . Xr + = a(Xi Xr)b(X, +1, X,,). We may take the (external) direct sum of the vector spaces Tr(V) for all r to form an infinite-dimensional vector space. The direct sum of such a vector space with a one-dimensional space spanned by an identity element forms an associative (but not commutative) algebra under the tensor product, the tensor algebra T(V). The subspace spanned by the identity is written as To(V), and since this is just another copy of the field F with an identical rule for multiplication on tensors we shall not distinguish between these two spaces. The tensor algebra is generated by V and the identity element; any element can be written as a sum of tensor products of elements from V and the identity. Those tensors that are simply a product of vectors from V are called decomposable. By construction we have the direct sum vector space decomposition T(V) = E oTp(V). p=o The tensor product is such that the tensor algebra is a Z-graded algebra; elements in T(V) that are sums of products of p elements from V being homogeneous of degree p. The zero element (which is homogeneous for every degree) is the only term that is homogeneous for negative degree. The grading naturally gives rise to an involutary automorphism ij defined on homogeneous elements byt = (_odega a. This is certainly an automorphism since if a and b are homogeneous n(a0b) = (_odegaobaob = odega + degben‘b V.9 (since the algebra is graded) = (_odega( i)degb a0b and so ri(a0b) (1.1.3) To say that ri is involutary means that 712 = 1, which indeed follows from (1.1.2). The homomorphism Z—' Z2 induces a coarser Z 2- gradation in T(V). The Z2-homogeneous subspaces consist of the sum of all Z-homogeneous subspaces of even (odd) degree. Thus the Z2-homogeneous subspaces are eigenspaces for the automorphism ij with eigenvalues plus (minus) one. Elements of these spaces will be called even or odd, respectively. t The notation ce is also employed. (1.1.2) THETENSOR ALGEBRA 3 (a®b)(X1,X2,...,X,,X,+1,...,X,+,) = t1(X1. ..,Xr)b(Xr+1, ...,Xr+3). Wemay take the(external) direct sum ofthevector spaces T,(V) for allrtoform aninfinite-dimensional vector space. The direct sum of suchavector space withaone-dimensional space spanned byanidentity element forms anassociative (but notcommutative) algebra under the tensor product, thetensor algebra T(V).The subspace spanned bythe identity iswritten asT0(V), andsince thisisjust another copy ofthe field Fwith anidentical rule formultiplication ontensors weshall not distinguish between these twospaces. Thetensor algebra isgenerated by Vandtheidentity element; anyelement canbewritten asasumof tensor products ofelements from Vandtheidentity. Those tensors that aresimply aproduct ofvectors from Varecalled decomposable. By construction wehave thedirect sum vector space decomposition r(v)=Zoe)r,,(v).,,= Thetensor product issuch thatthetensor algebra isaZ-graded algebra; elements inT(V) thataresums ofproducts ofpelements from Vbeing homogeneous ofdegree p.Thezero element (which ishomogeneous for every degree) istheonly term that ishomogeneous fornegative degree. The grading naturally gives rise toaninvolutary automorphism 17 defined onhomogeneous elements byr 11a=(—1)d‘g"a. (1.1.2) This iscertainly anautomorphism since ifaandbarehomogeneous 11(a®b) =(—l)d°g"®"a®b =(—1)d°g” ”d‘g"a®b (since thealgebra isgraded) :(_1)dega(_1)degba®b andso 17(a®b) =17a®r]b. (1.1.3) Tosaythat 17isinvolutary means that 172=1,which indeed follows from (1.1.2). The homomorphism Z—> Z2induces acoarser Z2- gradation inT(V). The Z2-homogeneous subspaces consist ofthesum OfallZ-homogeneous subspaces ofeven (odd) degree. Thus the Z2-homogeneous subspaces areeigenspaces fortheautomorphism 17 with eigenvalues plus (minus) one. Elements ofthese spaces will be called even orodd, respectively. fThenotation a"isalsoemployed. 4 TENSOR ALGEBRA The tensor algebra is isomorphic to its opposite algebrat and admits an involutary anti-automorphism, or simply an involution, defined on homogeneous elements by (xiox2 oxpy = ox2ox1 (1.1.4) It is straightforward to see that this really is an anti-automorphism, namely (a0b)',' = b®at such that 2 = 1. If X is in V* then the interior derivative with respect to X is denoted ix. It is defined to be a linear transformation that is an anti-derivation with respect to the automorphism n, that is ix(a0b) = ixa0b + tia0i xb. (1.1.5) If x E V then ixx = X(x), whilst for A in the subspace spanned by the identity ixA 0, and so the interior derivative is a homogeneous linear mapping on T(V) (with respect to the Z-gradation) of degree —1. These properties completely characterise the interior derivative. Since i x is an anti-derivative with respect to the involution n, with ixn = —nix, it follows that ixi y + i yix is a derivation on T(V). For x E V or the subspace spanned by the identity (ixi y i yix)X = 0, and since T(V) is generated by this space (ixi y i yix)C1 = 0 for all a c T(V). (1.1.6) In particular ixix = O. 1.2 The Exterior Algebra of Antisymmetric Tensors A tensor is a multilinear mapping on an ordered set of vectors, the ordering being in general important. Many important tensors have symmetries, however, the result of the evaluation on a set of vectors being invariant under the interchange of certain pairs of vectors. To formalise this we introduce the interchange permutation 7rjk , which rearranges the set of numbers {1, 2, . . p) such that vik(i)= i if i j or k, /kW = k and ulk(k) = j. Then a degree-p tensor T is symmetric (antisymmetric) in the j,k entries if T(X,(1),X,),(2), . . X„(p)) = +(—)T(X I,X2, . . Xp).: A tensor that is symmetric (antisymmetric) under all such inter- changes is called totally symmetric (totally antisymmetric). The totally antisymmetric tensors are particularly important. The subspace of totally 1 See Appendix A. 4 TENSOR ALGEBRA The tensor algebra isisomorphic toitsopposite algebrai andadmits aninvolutary anti-automorphism, orsimply aninvolution, 5defined on homogeneous elements by (x1®x2 ...®xP)§ =xP® ...®x2®x1. (1.1.4) Itisstraightforward toseethat this really isananti-automorphism, namely (a®b)5 =b5®a5 such thatE2=1. IfXisinV*then theinterior derivative with respect toXisdenoted ix.Itisdefined tobealinear transformation that isananti-derivation with respect totheautomorphism 17,thatis iX(a®b) =iXa®b +r7a®iXb. (1.1.5) IfxeVthen ixxEX(x), whilst for/Iinthesubspace spanned bythe identity ix/1E0,andsotheinterior derivative isahomogeneous linear mapping onT(V) (with respect totheZ-gradation) ofdegree —1.These properties completely characterise theinterior derivative. Since iXisan anti-derivative with respect totheinvolution 1],with ixn=—1]iX, it follows that lxly +lylX isaderivation onT(V). For xeVorthe subspace spanned bytheidentity (lxly +iyiX)x =0,andsince T(V) is generated bythisspace (ixiy +iyiX)a =0forallaeT(V). (1.1.6) Inparticular ixix =0. 1.2TheExterior Algebra ofAntisymmetric Tensors Atensor isamultilinear mapping onanordered setofvectors, the ordering being ingeneral important. Many important tensors have symmetries, however, theresult oftheevaluation onasetofvectors being invariant under theinterchange ofcertain pairs ofvectors. To formalise this weintroduce theinterchange permutation 17,-k, which rearranges thesetofnumbers {1,2,...,p}such that1r,~,<(i) =iifi#1" ork,11,-,<(j) =kand11,-k(k) =j.Then adegree-p tensor Tissymmetric (antisymmetric) inthej,kentries if T(Xfl/k(l),Xfl]k(2), ...,Xn,k(p)) = +(_)T(X1,X2, ...,Xp).:f Atensor that issymmetric (antisymmetric) under allsuch inter- changes iscalled totally symmetric (totally antisymmetric). The totally antisymmetric tensors areparticularly important. Thesubspace oftotally ‘rSeeAppendix A. THE EXTERIOR ALGEBRA OF ANTISYMMETRIC TENSORS 5 antisymmetric tensors in T(V) is denoted by A(V), the elements of this space being called exterior p-forms, or simply p-forms. The total antisymmetry ensures that a p-form is determined by its evaluation on all distinct combinations of p vectors from a basis for V*. So if V is n-dimensional and ( ) denotes the number of distinct combinations of p objects chosen from n then dim Ap = (ç). In particular, the only p-forms for p>n are zero, and dim A„ = 1. In analogy with the case of the tensor algebra it will be convenient to identify the field F with a space Ao(V). Given an arbitrary element a c T(V), we define a new tensor sag a c T p(V) by sag a(X I,X2, . . ., X p) = —1Ee(a)a(X am,X,,(2), . . X„, p)) V X, E V* (1.2.1) Pi a where the sum is over all permutations a, e(a) being +1 if this permutation is even (i.e. an even number of pair interchanges rear- ranges the elements 1, 2, . . p into the order a(1), a(2), . . a(p)) or —1 if the permutation is odd (an odd number of such interchanges). From the definition of saga we see that it is totally antisymmetric and that sag(sa 3 - = saga. Hence sin- is a projection operator, sag:Tp(V)--Ap(V). Although CD: T,(V) x T,(V)—> T s+,(V), the map 0 will not map As(V) x A,(V) into As+,(V). Thus we devise a new composition map in terms of 0 and sin- that does have this property. It is called the exterior productt and is denoted by a A placed between the elements of A(V) and Ar(V) A :As(V) X Ar(V)-* As+,(V) a, b - > a Ab = sag(a0b). (1.2.2) t The reader is cautioned that there are other conventions for the definition of the exterior product. Other conventions involve a numerical factor which depends on the degrees of a and b. The reader should convince himself that such numerical factors cannot be arbitrarily inserted with impunity! (Why not?) The convention we have adopted is convenient for regarding the exterior algebra as a quotient of the tensor algebra modulo the kernel of si23" , as we shall do in the next section. THEEXTERIOR ALGEBRA orANTISYMMETRIC TENSORS 5 antisymmetric tensors inT,,(V) isdenoted byA,,(V), theelements of thisspace being called exterior p-forms, orsimply p-forms. The total antisymmetry ensures that ap-form isdetermined byitsevaluation on alldistinct combinations ofpvectors from abasis forV*.SoifVis n-dimensional and ("IP denotes thenumber ofdistinct combinations ofpobjects chosen from n then dimA,, = Inparticular, theonly p-forms forp>n arezero, anddimA,, =1.In analogy with thecase ofthetensor algebra itwill beconvenient to identify thefield Fwith aspace A0(V). Given anarbitrary element aeTp(V), wedefine anew tensor dig aeT,,(V) by a§£aa(X,,X,, ...,X,,) 1=F2s(0)a(X,,(,,,X,,(2), ...,X,,(,,,) VX,-eV* (1.2.1) where the sum isover allpermutations 0,5(0) being +1ifthis permutation iseven (i.e. aneven number ofpair interchanges rear- ranges theelements 1,2,...,pintotheorder 0(1), 0(2), ...,0(p)) or —1ifthepermutation isodd(anoddnumber ofsuch interchanges). From thedefinition ofs4.5£‘J aweseethatitistotally antisymmetric and that s4.§£9(d§£‘Ja) =d.§£‘Ja. Hence s4.5£°J isaprojection operator, fl§£9:T,,(V)-—>A,,(V). Although ®:T,(V) XT,(V)—+ T,,.,(V), themap ®willnotmap AS(V) XA,(V) into A,,.,(V). Thus wedevise anew composition mapinterms of®andslid thatdoes have thisproperty. Itiscalled theexterior producti andisdenoted byaAplaced between theelements ofA_,(V) andA,(V) /\3A1-(V) XA1(V) ‘T’ As+I(V) t1,bii-> (ZAb=fl§£(J(t1®b). (1.22) ’rThe reader iscautioned that there areother conventions forthedefinition of theexterior product. Other conventions involve anumerical factor which depends onthedegrees ofaandb.The reader should convince himself that such numerical factors cannot bearbitrarily inserted with impunity! (Why not?) Theconvention wehave adopted isconvenient forregarding theexterior algebra asaquotient ofthetensor algebra modulo thekernel of514.529", asweshall doin thenext section. 6 TENSOR ALGEBRA It follows from this definition that (a A b)(XI, X2, . . . Xs, Xs+i, . . Xs+t) E (s+t)! E(a)(a0b)(X 0(,), . . . ., 1 E(s+t)! E(s, t)a(X ii, . . . X j,)b(X ki, . . . where the sum is over all partitions of (1, 2, . . s+t) into GI, j2, . js) and (k1, k2, . . lc), and E(s, t) is the sign of the permutation (1, 2, .. s+t)i--> (ji, j2, js, ki, k2, kt). The exterior product has a well defined symmetry under the interchange of factors such as a and b above. To see this we introduce a permutation v, (1, 2, .. s, s+1, . . s+t) r, (t+1, t+s, 1, 2, . t). We write any permutation a as a = rv, giving E(a) = E(v)E(r). Inserting this in the above gives (a A b)(Xi, X2, . X„ X,+1, . . X s+f) E (s+t)! e(a)a(X G(1), . . .7 X „(s))b(X 0",), . . X,(5+0) E(v) E (s+t)! E(T)a(,G(,+i), . . . 7 X r(r+s))6(X,(1), . . .7 = E(v)(b A a)(X 1, .. .7 X,„). A trivial combinatorial calculation gives E(v) = (-1)", and so we have for any s-form a and t-form b a A b = ( -1)stb A a. (1.2.3) The exterior algebra A(V) is formed by the direct vector space sum of all the spaces of p-forms A(V) = E A( V) p=0 with multiplication given by the exterior product. The exterior product is defined on non-homogeneous elements by extending .94Y3 - to be distributive over addition, ensuring that the exterior product is. Unlike the tensor algebra this algebra is finite dimensional: we have dim A( V) — E n — 2. p=0 P 6 TENSOR ALGEBRA Itfollows from thisdefinition that (a,\b)(X1,XZ, ...X,,X,+1, ...,X,+,) = 2s<o><a®b)<X....,. .....X.....,> 1=GT)! Ego, t)a(X]-‘, ....,X]-I)b(X,(l, ....,xkl) where thesum isover allpartitions of(1,2,...,s+t) into(j1,j2, .., j,)and(kl, k2,...,k,),ands(s,t)isthesignofthepermutation (1,2,...,s+t)+—>(j,,j2, ...,j,,k1,k2, ...,k,). Theexterior product hasawell defined symmetry under theinterchange offactors such asaand babove. Tosee this weintroduce a permutation v, v (1,2,...,s,3+1, ...,s+t)»——>(z+1, ...,t+s,1,2,...,r). Wewrite anypermutation 0as0=rv,giving 5(0) =e(v)e(r). Inserting thisintheabove gives (a,\b)(X1, X2,...,XS,X,.+,, ...,X,+,) 1 =E,‘ 25(‘7)a(Xv(1), ---iX0(S))b(X0(S+l)’ ~--»X(7(S+l)) e(v)=(Tm2-§<r>a<X..,..,. ....X......>b(X.<.,. --..X...» =s(v)(b Aa)(X1, ...,XS+[)~ Atrivial combinatorial calculation gives e(v)=(—1)~", andsowehave foranys—form aandt-form b aAb=(—1)"b Aa. (1.2.3) The exterior algebra A(V) isformed bythedirect vector space sum of allthespaces ofp-forms /\(v)=Zea/\,(v) p=0 with multiplication given bytheexterior product. The exterior product isdefined onnon-homogeneous elements byextending $24589 tobe distributive over addition, ensuring that theexterior product is.Unlike thetensor algebra thisalgebra isfinite dimensional: wehave dimA(V) =2". (1.2.4) THE EXTERIOR ALGEBRA OF ANTISYMMETRIC TENSORS 7 The exterior algebra is, in fact, associative. This will be seen to follow from the observation that if .54Y5-ni = 0 then saY(a0m) = a1Y.5(m(Da)= 0, V a e T(V), as will now be established. Let Ck be the group of all permutations of k objects. Then the subgroup of C5+, that only permutes the first s objects is obviously isomorphic to C„, and we shall identify it as such. Let H be a set that contains one and only one element from each left coset of Cs+, relative to C,. So for each oc C5+, u = In-, tE C, and h EH, with e(a) = E(T)E(h), and then .36115(m0a)(X i, Xs,) 1 (s+t)! E e(h) E E(r)(m0a)(X 0(l), heH rEC, For some fixed h let Xh(,) = Y, then X0(0= Xhr(I)= Y r(i), and thus E E(r)(m0a)(X am, . Xa(s+0) T E = E E(T-)(m(Da)(Y T(l), Yr(s+t)) T E = E E(T)m(Yro),   , Yroa(Y„-+i,   E Cs = sin-m(Yi,   , Y0a(Y5+1,   ,Y,Fi). So indeed slYY(m0a)= 0 if sti2Fim = O. Similarly, it follows that saYff(a0m) = O. Since, as we have remarked, .9119- is a projection operator, if we set (1 — siYFI)(a0b)= m then a(Db = s4.V1(a0b)+ m, with = O. From the definition of the exterior product we have (a A b) A C = ,9a7/ (.9CYFf (a0b) Oc) = (a0b0c — m0c) since 0 is associative = .942,5 (a0b0c) from the above result, which may be used once more to give (a A b) A C = .942Y (aOstYg (b0c)) = a A (b A C). The exterior algebra inherits a Z-gradation from the tensor algebra. The zero element is the only homogeneous element of degree greater than n in the exterior algebra. Since .91Y,5 is a homogeneous mapping of degree zero on the tensor algebra it follows that n is also an automorphism of the exterior algebra, that is THE EXTERIOR ALGEBRA OFANTISYMMETRIC TENSORS 7 The exterior algebra is,infact, associative. This willbeseen tofollow from the observation that ifsiiiffim =0then s4§£9'(a®m) = s4§£€(m®a) =0,VaeT(V), aswillnow beestablished. Let Ckbethegroup ofallpermutations ofkobjects. Then the subgroup ofCH, that only permutes thefirst sobjects isobviously isomorphic toC,,andweshall identify itassuch. LetHbeasetthat contains oneandonly oneelement from each leftcoset ofCH, relative toCK. Sofor each oeCH, 0=hr, reC, and heH, with e(0) =e(r)e(h), andthen s4§£.°T(m®a)(X,, ...,X,1,) _ 1_—(;;)—!g:He(h)T;CSe(r)(m®a)(X,,(1), ...,X,,(,+,,). Forsome fixed hletX,,(,-) =Y,,then X0“) =X,,,(,-) =Y,(,-), andthus 2c£(I)(m®a)(X0(1)v "*'‘vXo(s+l)) =Ze(r)(m®a)(Y,(,), ......,r,,,,,,) reC_¢ :2 £(I)m(Yr(l)v '‘'~Yr(s))a(Ys+1v ---vY:+z)' rec‘; =s4§£.°Tm(Y1,..., Y,)a(Y,+1, ...,Y,+,). Soindeed .<24§£9'(m®a) =0 if.<24§£§m =0.Similarly, itfollows that .<24§£9'(a®m) =0. Since, aswehave remarked, &4§£?T isaprojection operator, ifweset (1—.<24§£.°T)(a®b) =m then a®b =&4§£§(a®b) +m, with siiiffim =0.From thedefinition oftheexterior product wehave (aAb)Ac=s4§£?I ($245135 (a®b) ®c) =stiff? (a®b®c —m®c) since ®isassociative =sill’? (a®b®c) from theabove result, which may beused once more togive (aAb)Ac=s4§£€ (a®s4§£€7 (b®c)) =aA(bAc). The exterior algebra inherits aZ-gradation from thetensor algebra. The zero element istheonly homogeneous element ofdegree greater than nintheexterior algebra. Since s4§£€ isahomogeneous mapping ofdegree zero onthetensor algebra itfollows that r)isalso an automorphism oftheexterior algebra, thatis 8 TENSOR ALGEBRA l(a A b) = ?la A rib . (1.2.5) Similarly exterior forms are called even or odd according to their Z2-gradation in the tensor algebra. The involution commutes with .942.7 and so it is also an involution of A(V). Taking the definition of ,99.15- and rearranging the permutations gives the following simple expression for acting on a p-form to, = (-1)[PI 21(o. (1.2.6) where [ ] denotes the integer part. The interior derivative i x has already been defined on tensors, and so it is defined the same way on exterior forms. In fact this is where it will mainly be utilised. We need to show that the result of ix on an exterior form is another exterior form, of one lower degree, and that the anti-derivation property (1.1.5) goes over to the exterior algebra with 0 replaced by A . It will be sufficient to consider decomposable tensors. If T = x1C) x2C) .. . OxP then T = xi(X 1)x20 . . . C)xP — x 2(X i)xl C) x3 . . . C)xP + x3(X i)xl C) x20 x4 . . . Ox" + . . . + (-1)P -1xP (X 1).x1CD . . . xP-1 that is x,T)(X 2,  . X p) = E E(v)T(X„ (1), . . x) (1.2.7) where y is any of the p permutations such that (1, 2, . . r, . . . , p) —> (2, 3, . . r-1, 1, r, . . . , p). Substituting s42.7 T into (1.2.7) gives x19425- T)(X 2, .   X,,) = ps42er T(X 1, . . ., X,,). (1.2.8) From the definition we have Xp) 2, E(r)(x20 OxP)(x,(2), ., X) x2 (X1) (p-1)! E(r)(x 10x3 . . . OxP)(X ,(2),    X v(p)) (-1)P-I xP (X ,) + . . . + 2, s(r)(x 10 . .  . . . , X r(,,)) where Te Cp-1 8 TENsoR ALGEBRA r)(aAb)=naAnb. (1.2.5) Similarly exterior forms arecalled even orodd according totheir Z2-gradation inthetensor algebra. The involution Ecommutes with stiff? andsoitisalsoaninvolution ofA(V). Taking thedefinition of sflifig and rearranging the permutations gives the following simple expression forEacting onap-form w, (05=(—1)|-"’2]w. (1.2.6) where []denotes theinteger part. Theinterior derivative iXhasalready been defined ontensors, andso itisdefined thesame way onexterior forms. Infactthisiswhere itwill mainly beutilised. Weneed toshow thattheresult ofiXonanexterior form isanother exterior form, ofonelower degree, and that the anti-derivation property (1.1.5) goes over totheexterior algebra with ® replaced byA.Itwillbesufficient toconsider decomposable tensors. If T=x1®x2®...®x/’ then iX1T =x'(X1)x2® ...®x” —x2(X1)x'®x3 ...®x/’ +x3(Xl)x1®x2®x4 ...®x/’ +...+(-1)/"1x"(X1)x'® ...®x"_' thatis (iX1T)(X2, ...,X,,)=Ze(v)T(X,(1), ...,x,,,,,) (1.21) where visanyoftheppermutations such that (1,2,...,r,...,p)i>(2,3,...,r—1,1,r,...,p). Substituting s2Q.§E?J'T into(1.2.7) gives (iX]d.§E9T)(X2, ...,Xp)=p&Q.§EETT(X,, ...,Xp). (1.2.8) From thedefinition wehave (.fl.§E9TiX|T)(X2, ...,Xp) : 2E(T)(X2® ...®Xp)(Xt(2), ..., 2— 2s(r)(x1®x3® ...®x/’)(X,(2), ...,x,(,,,) +...+% ;s(r)(x1®...®x""‘)(X,(2), ..., X,U,)) where reCp_1 THE EXTERIOR ALGEBRA OF ANTISYMMETRIC TENSORS 9 1 E e(cr)(x1® . . . oxo(x0„), . . X,(p)) (p-1)! for a E Cp (AYFTT)(X i, . . X p) (p-1)! and thus (s12,9-ix,T)(X2, . XI)) = (ps4,Yff TXXI,  P! . Xp). (1.2.9) So (2.8) and (2.9) give ixs42.7 = .94Z5i x. Thus ix:Ap --> Ap_i, and ix(a A b)= ix4Z5-(a0b) = .94aFf(i xa0b + gaOi xb) and hence ix(a A b) = ixa A b + qa A ib. (1.2.10) If a) e Ap(V) then slYFIN = w and sin-ixco = ixco, so (1.2.9) re- duces to (ix,w)(X2, . . X p) = pw(X i, Xp). (1.2.11) Just as the space formed by V together with the identity generates T(V) under the product 0, it generates A(V) with the product A . Thus any element of A(V) can be written as a sum of decomposable forms, these being the ones consisting of products of elements from V. If {e'} is any basis for the n-dimensional V then the (pn) p-forms e A '2 A   - A ei° for i1 < i2 <. . .<ip (p 1) form a basis for Ap(V). It is often convenient to label such p-forms by an ordered multi-index, / = (ii, i2, ip) with < i2 < . <ip with each index i, varying from 1 to n. So if w is an arbitrary p-form w = E A ei2 A    A ei° ii<i2<  <ip = where col= w ,, E F are the components of w in this basis. Care must be exercised when using the summation convention (see Appendix A) with ordered multi-indices. Since this convention operates with unconstrained summations one may equivalently write 1 w = 2 i A ei2 A    A eiP p! 1 p it being understood that the components are totally antisymmetric in the indices. If {fi} is a new basis for V related to {ei) by fi = Mijej, {Mid E Gl(n, F)t, then we can induce a corresponding change in the t The group of n x n invertible matrices with elements from F, see Appen- dix A. THE EXTERIOR ALGEBRA orANTISYMMETRIC TENSORS 9 =—1i 2s(0)(x1® ®xP)(X X ) for06C (p—1)1 ... 0“), ..., 0(1)) P ' U = (fl$gT)(X1,..., Xp) andthus (s4.§E9'iX,T)(X2, ...,Xp) =(ps4.§E9T)(X1,...,Xp). (1.2.9) So(2.8) and (2.9) give ixsflifg =sflifgix. Thus iX:Ap -—>Ap_1, and iX(aAb)=iXs4.§£°..T(a®b) =s4.SE°..T(iXa®b +17a®iXb) andhence IX(a/\ :Ixa Ab+Tia/\IXb. Ifwe/\,,(V) then sfl.§£°..Tw= wand oQ.SE9iXw= ixw, so(1.2.9) re- duces to (iX]w)(X2, ...,Xp) =pw(X1,...Xp). (1.2.11) Justasthespace formed byVtogether with theidentity generates T(V) under theproduct ®,itgenerates A(V) with theproduct A.Thus any element ofA(V)canbewritten asasumofdecomposable forms, these being theones consisting ofproducts ofelements from V.If{ei} isany basis forthen-dimensional Vthen the(Q)p-forms e“AellA...Ae‘Pfor il<i2<...<ip (p21)form abasis forAp(V). Itisoften convenient tolabel suchp-forms byanordered multi-index, I=(i1,i2,...,ip)Withi1 ‘<1-2<...<ip with each index iIvarying from 1ton.Soifwisanarbitrary p-form w=2 wi,i2...i,, ei‘/\e"’/\ ---Aei’ i,<i2<. ..<i,, =200,6’ where 01,5 0),‘,-Z ,PeFarethecomponents ofwinthisbasis. Care must beexercised when using thesummation convention (see Appendix A)with ordered multi-indices. Since thisconvention operates with unconstrained summations onemayequivalently write _L i1 iz i01- pjwi,i2...ipe /\eA~~-A9" itbeing understood thatthecomponents aretotally antisymmetric inthe indices. If{fl} isanew basis for Vrelated to{ei} byfl=M’,-ell, {Mil-} eGl(n, F)i, then wecaninduce acorresponding change inthe TThe group ofnXninvertible matrices with elements from F,seeAppen- dixA. 10 TENSOR ALGEBRA components of a p-form. Since the space of n-forms is one-dimensional the n-forms formed by the products of the two bases must be related by a multiple of F. In fact it follows from the antisymmetry that fInf2A- A.r = detMel A e2 A    A en (1.2.12) where detM is the determinant of the matrix {M'1} that relates the bases. Any n-form 52 can be used to classify frames {X,} for V*. These frames fall into two classes according to the sign of Q(X 1, X2, . Xn). Frames in different classes are said to be of opposite orientation. The Gl(n, F) related frames {e"} and (t) are of the same orientation if and only if det M is positive. This is consistent since the determinant of a product of two matrices is positive if the determinant of each factor is positive. 1.3 The Exterior Algebra as a Quotient of the Tensor Algebra We have introduced the exterior algebra as the set of totally anti- symmetric tensors with the product A constructed out of 0 and This algebra is isomorphic to a quotient of the tensor algebra; indeed the definition in terms of the quotient offers certain advantages. In the next chapter we will define the Clifford algebra as a quotient of the tensor algebra, and it is useful to see the exterior algebra introduced in a parallel way. We will use bold-face type to denote the quotient algebra and its product, the use of the same symbols anticipating its isomorphism with the exterior algebra of antisymmetric tensors already defined. Let I be the ideal in T(V) consisting of sums of terms of the form aOx0x0b where x E V and a, b are arbitrary elements of T(V). Then we define the exterior algebra A(V) by A(V) = T(V)II. (1.3.1) Elements in A(V) are equivalence classes of elements in T(V), where the equivalence relation is defined by a — b if a = b + c for some c E I. The equivalence class that contains a is denoted [a]. The vector space structure of A(V) is defined by [a] + Â[b] = [a + Ab] a, be T(V), E F (1.3.2) and the multiplication which is denoted by A is given by [a] A [b] = [a0b]. (1.3.3) 10 TENSOR ALGEBRA components ofap-form. Since thespace ofn-forms isone-dimensional then-forms formed bytheproducts ofthetwobases must berelated by amultiple ofF.Infactitfollows from theantisymmetry that f‘Af1A...Af"=detMe1Ae2A...Ae" (1.2.12) where detM isthedeterminant ofthematrix {Mi}-} that relates the bases. Any n-form Qcanbeused toclassify frames {X2} forV*.These frames fallintotwoclasses according tothesignofQ(X1, X2,...,X,,). Frames indifferent classes aresaid tobeofopposite orientation. The Gl(n, F)related frames {e"}and{fl}areofthesame orientation ifand only ifdetM ispositive. This isconsistent since thedeterminant ofa product oftwomatrices ispositive ifthedeterminant ofeach factor is positive. 1.3TheExterior Algebra asaQuotient oftheTensor Algebra We have introduced theexterior algebra asthesetoftotally anti- symmetric tensors with theproduct Aconstructed outof®and$559. This algebra isisomorphic toaquotient ofthetensor algebra; indeed thedefinition interms ofthequotient offers certain advantages. Inthe next chapter wewilldefine theClifford algebra asaquotient ofthe tensor algebra, anditisuseful toseetheexterior algebra introduced in aparallel way. Wewill use bold-face type todenote thequotient algebra anditsproduct, theuseofthesame symbols anticipating its isomorphism withtheexterior algebra ofantisymmetric tensors already defined, LetIbetheideal inT(V)consisting ofsums ofterms oftheform a®x®x®b where x6Vanda,barearbitrary elements ofT(V).Then wedefine theexterior algebra A(V)by A(V)=T(V)/I. (1.3.1) Elements inA(V) areequivalence classes ofelements inT(V), where theequivalence relation isdefined bya~bifa=b+cforsome ceI. Theequivalence class thatcontains aisdenoted [a].Thevector space structure ofA(V) isdefined by [a]+/l[b] =[a+Ab] a,beT(V), A6F (1.3.2) andthemultiplication which isdenoted byAisgiven by [a]A[b] =[a®b]. (1.3.3) EXTERIOR ALGEBRA AS A QUOTIENT OF THE TENSOR ALGEBRA 11 The ideal I is a Z-gradedt subspace of T(V) and so A(V) inherits a natural Z-gradation given by deg [a] = deg a. The automorphism n and the involution preserve the ideal I and they thus extend in an obvious way to A( V) by n[a] = [ im] [a] = [al. (1.3.4) Similarly interior multiplication preserves I and so we may define ix [a] = [ixa]. (1.3.5) If x, y E V then 2xC)y = (x0y — y0x) + (x + y)0(x + y) — x0x — y0y hence x0y = x A y + 1{(x + y)O(x + y) — x0x — y0y). (1.3.6) The A denotes the antisymmetrised tensor product as defined in (1.2.2). The term in brackets is in I and so x0y—x Ay. That is, [x] A [Y] = [x®Y] = [ X A Y]. More generally, it follows that the ideal I is just the kernel of and so [a] = [staff a]. We have already seen, in proving that A is associative, that this kernel is an ideal. To see that it is in fact I we will prove that X 0 0) — X A 0) for x E V, 0) E A( V). (1.3.7) The recursive application of this result gives x10x20. . . Ox" — siY9 - (x10x20. . .Ox") = X 1 A X2 A . . . AX. We will prove (1.3.7) by induction on the degree of co. It is certainly true when co is a 1-form; we assume it is true for co of degree less than p. It is sufficient to consider the case of co decomposable. The definition of A involves the permutation of the arguments in the evaluation, but this is obviously equivalent to permuting the factors in the product. Thus from the definition of Awe have 1 Y° AY' ... AY y (p+1)! eavc(0)0y,(1)0 . . . '', \ where a permutes the set (0, 1, 2, . . ., p). We will characterise each permutation according to the first number in the reordered set. With one interchange we swap the elements 0 and r, and with r — 1 further t Grading is discussed in Appendix A. EXTERIOR ALGEBRA ASAQUOTIENT OFTHETENSOR ALGEBRA 11 The ideal IisaZ-graded? subspace ofT(V) and soA(V) inherits a natural Z-gradation given bydeg[a]=dega.The automorphism 17and theinvolution .7;preserve theideal Iandthey thus extend inanobvious waytoA(V) by vial=[rial[(115=[(15]. (1.3.4) Similarly interior multiplication preserves Iandsowemay define ix[a]=[ixa]. (1.3.5) Ifx,yeVthen 2x®y =(x®y —y®x) +(x+y)®(x +y)—x®x —y®y hence x®y =xAy +%{(x +y)®(x +y)—x®x —y®y}. (1.3.6) TheAdenotes theantisymmetrised tensor product asdefined in(1.2.2). The term inbrackets isinIand sox®y~x Ay. That is, [X]Alyl=[X®y] =[X/\YI- More generally, itfollows thattheideal Iisjustthekernel of.9133? , and so[a]=[s£.§E‘Ja]. We have already seen, inproving thatA is associative, thatthiskernel isanideal. ToseethatitisinfactIwewill prove that x®w~ xAw forxe V,weA(V). (1.3.7) Therecursive application ofthisresult gives x1®x2®. ..®x/’ ~s1§£‘J(x1®x2®.. .®x/’) =x1Ax2A. ..Ax/’. Wewillprove (1.3.7) byinduction onthedegree ofw.Itiscertainly true when toisa1-form; weassume itistrue fortoofdegree lessthan p.Itissufficient toconsider thecase ofwdecomposable. Thedefinition ofAinvolves thepermutation ofthearguments intheevaluation, but this isobviously equivalent topermuting thefactors intheproduct. Thus from thedefinition ofAwehave 1 O O Oy°Ay‘ Ay” =—(1F1Y;s(0)y(°)®y (‘l®...®y (Pl where 0permutes theset(0,1,2,...,p).Wewillcharacterise each permutation according tothefirst number inthereordered set.With oneinterchange weswap theelements 0andr,andwith r—1further TGrading isdiscussed inAppendix A. 12 TENSOR ALGEBRA interchanges bring the 0 to the second position. So if v, is the permutation such that vr (0, 1, r, . . p) (r, 0, 1, . . p) where F denotes that r is missing from this sequence, then E(vr) = (-1)'. We can now write any permutation a as a = TrVr for some r, where Tr permutes the set with r removed, then Y°AY1  AY 1 +1)! E.(T„,)yooy,„(1)0 03,0p) (p 1 0 rE(Tr)yr0)0 . (p+i)! oyi-,(P) 1 (p+1) \Y°0(Y1 A    A YE) (_i)ryr0(y0 A A - 7- Y A   A YP)) r=1 Substituting x for y° gives (_ 1 +1) x Oy 12 x A y12 p (p  'P (-1)ry0(x A yl  r r=1 )) where y'2 . p yl A y2 A A yP, and again the hat means that a term is missing. Now yr0(x A yl ... 7- yrOx0y1 .  P since (1.3.7) is assumed true for (p-1)-forms -—x0yrOy i since x0y + yOx — 0 -—x®(Yr Y I  7- P) from (1.3.7) again, -(_ orxoy 12 p where the sign comes from moving yr through r-1 terms. So xAy1 A . . . AY x0y12  P. Thus if (1.3.7) holds for co of degree less than p it is also true when co is a p-form. This completes the proof. Thus every equivalence class of A( V) is represented by an element of A(V), and the product of the classes under A is the class of the product of the representatives under A' Thus A(V) is indeed isomorphic to A(V). In practice it is more convenient to work with representatives, the antisymmetric tensors, rather than with their equivalence classes. 12 TENsoR ALGEBRA interchanges bring the0tothesecond position. Soifv,isthe permutation such that V!‘ (0,l,...,r,...,p)i>(r,0,l,..., ?,...,p) where ?denotes that rismissing from this sequence, then s(v,) =(—1)’. Wecannow write anypermutation 0as0=r,v, for some r,where T,permutes thesetwith rremoved, then Y0/\y1---AYP 1 to — 23e(rO)y°®y (1)® ...®y (*0) 1 P+H 2(_1)ryr ® £(.[r)yr,(0)® ___®y1,(r-i)®y1,(r+i) 'r=l r, ...®y"9’l I 0 P=m(y ®(y‘/\ '--Ay) P +2(_1)')"®()’0/\ ~~-/\y' A---/\)’P))- r=1 Substituting xfory°gives 1 P -)12... = l2...p _1rr l...r...pMy P(p+1)(x®y +gl1( )y®(x/\y ) where y'2 PEy‘Ay2A ... AyP, and again thehatmeans that a term ismissing. A Now y’®(xAy‘ f P)~y’®x®y‘ 'Psince (1.3.7) is assumed truefor(p—1)-forms ~—x®y’®y‘"~”-"P sincex®y+y®x~0 ~—x®(y’ Ay‘1"1f~~P) from (1.3.7) again, ~(_1)rx®ylZ...p where thesigncomes from moving y’through r—1terms. SoxAy‘ A... AyP ~x®y‘2 P.Thus if(1.3.7) holds forwof degree lessthan pitisalsotrue when wisap-form. This completes the proof. Thus every equivalence class ofA(V) isrepresented byanelement of A(V), andtheproduct oftheclasses under Aistheclass oftheproduct oftherepresentatives under A.Thus A(V) isindeed isomorphic to A(V). Inpractice itismore convenient towork with representatives, theantisymmetric tensors, rather than with their equivalence classes. THE HODGE MAP 13 1.4 The Hodge Map When the vector space V has a (non-degenerate) metric g then the Hodge dual, or * map, may be defined on exterior forms. Since (n) = n n — p we have dim Ap(V) = dim An_p(V), and thus these two vector spaces are isomorphic. We may use the metric g to set up a standard isomorphism between these spaces: the Hodge map, denoted by *. (Although one can define a Hodge map for a non-symmetric non- degenerate metric, we shall take g to be symmetric as well as non- degenerate.) If V has a metric then one can use a g-orthonormal frame {e} to construct a standard n-form = e A e2 A . .. A en. (1.4.1) Since the determinant of the matrix relating orthonormal frames is plus or minus one, depending on the relative orientations, we see from (1.2.12) that there are two possibilities for co, differing by a sign. The members of a g-orthonormal frame for V are sometimes called n-beins in the physics literature, generalising the familiar triad of orthonormal vectors in Euclidean three space. Some authors, however, associate this term with the r2 elements {M il E Gl(n, F) that relate an orthonormal frame to an arbitrary one {t}, e' = N'ifj. If the components of g in the frame {ei} are where = 0 if i j and for each value of i, rill = ±1, and the components in the frame {f} are en, then 1711 = Nt k skl(f). Hence det (0) --= det(e-n)(det N)2. The components of the metric on the dual space form the inverse matrices, gee' = 6k, and ggiD = bk,. (For further details see Appen- dix A.) So if t = det(r hi) = ±1 then, since det(m -1) = (det m)' for all matrices m, det(e) = t(det N)2. But co = (det N)f' A f2 A . . . At" so if we write the sign of det N as det N 11N = Idet NI THEHODGE MAP 13 1.4TheI-lodge Map When thevector space Vhasa(non-degenerate) metric gthen the Hodge dual, or*map, may bedefined onexterior forms. Since (;1)=(.':..) wehave dimAp(V) =dim/\,,_7,(V), andthus these twovector spaces areisomorphic. Wemay usethemetric gtosetupastandard isomorphism between these spaces: theHodge map, denoted by*. (Although one can define aHodge map foranon-symmetric non- degenerate metric, weshall take gtobesymmetric aswell asnon- degenerate.) IfVhasametric then one canuseag-orthonormal frame {el} to construct astandard n-form co, (0:81/(C2/\ ... Ae". Since thedeterminant ofthematrix relating orthonormal frames isplus orminus one, depending ontherelative orientations, weseefrom (1.2.12) that there aretwo possibilities forco,differing byasign. The members ofag-orthonormal frame forVaresometimes called n-beins inthephysics literature, generalising thefamiliar triad oforthonormal vectors inEuclidean three space. Some authors, however, associate this term with ther2elements {N‘7} eGl(n, F)that relate anorthonormal frame toanarbitrary one{f’}, ei = Ifthecomponents ofgintheframe {el} are17"/,where 17"/'=0ifi#=j andforeach value ofi,17”=il,andthecomponents intheframe {fl} areg’/97, then Ill!=N’./<Nj18k!m- Hence det(r7'7) =det(g’7(0)(detN)2. The components ofthemetric onthedual space form theinverse matrices, TI,-,-17"" =51‘andgfflg/"‘<0 =6f‘.(For further details seeAppen- dixA.)Soift=det(17,-7) =i1then, since det(m") =(detm)“‘ forall matrices m, det(g§{7) =t(det N)2. Butco=(detN)f‘ AfzA...Af", soifwewrite thesignofdetN as _detN “N7|detN| 14 TENSOR ALGEBRA then = ph,,{tdet(gP)} /I 2.» A f2 A At' (1.4.2) If the frames fel and {f } are related by a Gl(n, F) transformation that preserves the orientation, then uN = 1. A metric on V naturally gives rise to a metric on A(V). We start by defining a metric gp on the space of p-forms, A(V), for any p > 1. Since gp is defined to be bilinear it is sufficient to specify its action on decomposable p-forms. If A = cr1 A a'2 A   A (VP and B = f31 A132 A pi' then gp(A, B) = det{g(cri, 00}. (1.4.3) It is convenient to define g o to simply multiply the two 0-forms. Having defined a metric on the homogeneous subspaces we define a metric G on A(V) by requiring it to be diagonal in the homogeneous subspaces. That is, if (1), 111 c A(V) with, for example, O p denoting the projection of 10 into the subspace of degree p, then Go), qo = E gp(cDp, gip). (1.4.4) p=0 As we have remarked the spaces of p-forms and (n — p)-forms are of the same dimension, and we are now in a position to establish a standard isomorphism between them. The Hodge map, *, is a linear map from the space of p-forms to the space of (n — p)-forms: *: A( V) An _p( a *a where *a is given implicitly by b A *a = gp(b, a)co V b c Ap(V). (1.4.5) The standard n-form w is defined as in (1.4.1). The definition may be completed by defining the map on a 0-form, *1 = w. This is called the volume n-form. Linearity extends the definition to inhomogeneous elements of the exterior algebra. Thus the definition of the Hodge map depends not only on the metric but on a choice of orientation. The non-degeneracy of g (and hence of gp) ensures that such a definition does indeed determine the * map. It immediately follows from the symmetry of g (and hence of gp) that a A *b = b A *a Va, b E Ap(V). (1.4.6) A useful calculus can be set up relating the * map to the interior product. We may use the metric g to establish an isomorphism (denoted by a tilde) between V and V*. If x E V then the metric dual, is in V*; 14 TENsoR ALGEBRA then w=/1~{td@I(gl,”)}"2f‘ Afz/\---Af"- (1-4-Z) Iftheframes {e‘} and {f}arerelated byaGl(n, F)transformation thatpreserves theorientation, thenux=1. Ametric onVnaturally gives risetoametric onA(V). Westart by defining ametric gponthespace ofp-forms, A,,(V), foranyp>1. Since gpisdefined tobebilinear itissufficient tospecify itsaction on decomposable p-forms. IfA=a‘Aa2A...Aal’and B=B‘A,/52A ...ABP then gp(A, B)=det{g(o/L, B/')}. (1.4.3) Itisconvenient todefine gotosimply multiply thetwo 0-forms. Having defined ametric onthehomogeneous subspaces wedefine a metric GonA(V) byrequiring ittobediagonal inthehomogeneous subspaces. That is,if<1),II-‘eA(V) with, forexample, GDPdenoting the projection ofCDintothesubspace ofdegree p,then G(CD,\I1) =fig-,(<1>,, WP): (1.44) P=0 Aswehave remarked thespaces ofp-forms and(n—p)-forms areof thesame dimension, and wearenow inaposition toestablish a standard isomorphism between them. The Hodge map, *,isalinear map from thespace ofp-forms tothespace of(n—p)-forms: *i M An—p(V) ai——> *a where *aisgiven implicitly by bA*a=gp(b, a)w VbeAp(V). (1.4.5) The standard n-form wisdefined asin(1.4.1). The definition may be completed bydefining themap ona0-form, *1=w.This iscalled the volume n-form. Linearity extends thedefinition toinhomogeneous elements oftheexterior algebra. Thus thedefinition oftheHodge map depends notonly onthemetric butonachoice oforientation. The non-degeneracy ofg(and hence ofgp)ensures that such adefinition does indeed determine the*map. Itimmediately follows from the symmetry ofg(and hence ofgp)that aA*b =bA*a Va, be./\p(V). (1.4.6) Auseful calculus canbesetuprelating the*map totheinterior product. Wemay usethemetric gtoestablish anisomorphism (denoted byatilde) between VandV*.IfxeVthen themetric dual, 2,isinV*; THE HODGE MAP given by y(i) = g(x, y) V y E V. It then follows from the definition of * that xE V, OE A(V). 15 (1.4.7) This formula can be applied recursively to a decomposable p-form to produce *(xIA X2 A    AX) = 17P17P    iT i *1- (1.4.8) It is convenient to display the action of * on exterior products of basis vectors. Suppose that {e'} and {X,} are dual bases, with er(X) = We will often use the shorthand x, ' i1. The metric dual, of ea is gabxb X° and we write i = j". Equation (1.4.8) takes the following simple form for the product of p basis vectors *(el A e2 A    A eP) = iPiP-1   di*l. From this it can be seen that the dual of a product of p orthonormal 1-forms is the product of their complement in the basis. Duals of the orthonormal basis forms can be expressed in terms of the Levi—Civita antisymmetric &symbol. This is defined such that 0 if = +1 (-1) if (j1, i2, . . i„) is an even (odd) permutation of the standard sequence (1, 2, 3, . . n). (1.4.9) With the summation convention the volume n-form can be written in the orthonormal frame {e'} as 1 *1 — E "    A (1.4.10) n 1112 If the components of the metric in this orthonormal frame are rig we have *(e11 A A    A e''') = 1 (n—p)! jr_I e1P-■ . . . where ' =_ . . . niPiP pipni in. It is sometimes necessary to rearrange expressions such as ea A *(eb, A eb2 A    A ebP) THE HODGE MAP 15 given by yo?)=g(X»y) VyEV- Itthen follows from thedefinition of*that *(¢>Ax)= i;*<I> xeV,<1>eA(V). (1.4.7) This formula canbeapplied recursively toadecomposable p-form to produce *(Xl/\X2/\ /\Xp)=l}'pl}'r--l...l}11*I. Itisconvenient todisplay theaction of*onexterior products ofbasis vectors. Suppose that{e'}and{X,-} aredual bases, with e'(X,~) =6",-. Wewilloften usetheshorthand IX! El/-. Themetric dual, E”,ofe”isg"”Xb EX”andwewrite ix”Ei”. Equation (1.4.8) takes thefollowing simple form fortheproduct ofp basis vectors *(el/\€2/\ ...A6”) :lplp_l ...il*1. From thisitcanbeseen thatthedual ofaproduct ofporthonormal 1-forms istheproduct oftheir complement inthebasis. Duals ofthe orthonormal basis forms canbeexpressed interms oftheLevi—Civita antisymmetric s-symbol. Thisisdefined suchthat 0iflj=lk. 5,7,-:_ ,2= +1('_1)If(l-1, i2,...,i,,)isaneven (odd) permutation ofthestandard sequence (1,2,3,...,n). (1.4.9) With thesummation convention thevolume n-form canbewritten in theorthonormal frame {ei}as I . . *1: ‘,3 87'|,':__ 7'"6" /\6!:/\ .../\€"'. Ifthecomponents ofthemetric inthisorthonormal frame are17”we have *(@"A6”/\ ---/\@'”) =_i,9""""”1,,-,...i/”“ /\---/\@"'01-11)- where -‘3'4"_’''''t’1,., ...1,:7li‘j'7li1f2--- 77”,” 51,/2...j,1,., ...1,~ Itissometimes necessary torearrange expressions such as e”A*(e”' Ae”1A ...Aebr). 16 TENSOR ALGEBRA This may be accomplished by using (1.4.8), for example ea A *(eb ec) , ea A ic*eb = ic(ea A *eb) gca*eb _gabic*i gca*eb (since ea A *eb = gab*i) = _gab*ec gca*eb. and similarly ea A *(eb A ec A ed) = gab*(ec A ed) gac*(eb A ed) gad*(eb A ec). 1.5 The Mixed Tensor Algebra Just as the tensor product rV is the space of multilinear mappings on V* x V* x . . . x V* (r times), the tensor product of V* with itself, 0 rV*, is the space of multilinear mappings on Vx Vx...x V (r times). More generally we have the vector space of multilinear mappings on V* x V* x . . . x V* X Vx Vx...x V, r times s times the space 0170sV*. This space is called the space of mixed tensors of covariant degree r and contravariant degree s, T rs (V) . (The assignment of the terms covariant and contravariant is a matter of convention. The way we have indexed our bases accords with the classical component conventions.) Tensors in T rs (V) will be referred to as being of type (r, s). It will be seen that we have defined tensors to be multilinear maps on sets of vectors ordered such that those from V* occur first; that is, our space of tensors is formed by tensor products of V with itself followed by products with V*. One might envisage a more general definition that formed the tensor product of the spaces V and V* in no definite order. However, such tensor product spaces are naturally isomorphic to the canonically ordered product. For example, the ordered pairs V x V* are certainly distinct from V* x V, the bilinear mappings on these spaces being V*0 V and VO V* respectively. How- ever, we may define a map op by cp:V*OV VC) V * T ço T where (cpT)(X, co) = T(o), X) V X E V*, co E V. 16 TENSOR ALGEBRA This may beaccomplished byusing (1.4.8), forexample eaA,,(ez=Aec)=eaAicxeb =_l-@7811 A*eb) +gcaxeb =—g"”i‘*1 +g‘”*e” (since e”A*e"=g""*1) =_g@i1,,e@ +gra*eb_ andsimilarly e"A1-(ehAesAed)=3"“-(esAed)-graterAed)+3"“-(ehAer). 1.5TheMixed Tensor Algebra Justasthetensor product ®’V isthespace ofmultilinear mappings on V*XV*X...XV* (rtimes), thetensor product ofV*with itself, ®’V*, isthespace ofmultilinear mappings onVXVX...XV(r times). More generally wehave thevector space ofmultilinear mappings on V*XV*X...XV* >< VXVX...xV, _J MM rtimes stimes thespace ®’V®‘V*. This space iscalled thespace ofmixed tensors of covariant degree randcontravariant degree s,T,i(V). (The assignment oftheterms covariant andcontravariant isamatter ofconvention. The waywehave indexed ourbases accords with theclassical component conventions.) Tensors inT,‘(V) willbereferred toasbeing oftype (r,s).Itwillbeseen thatwehave defined tensors tobemultilinear maps onsetsofvectors ordered such thatthose from V*occur first; that is,ourspace oftensors isformed bytensor products ofVwith itself followed byproducts with V*.One might envisage amore general definition thatformed thetensor product ofthespaces VandV*inno definite order. However, such tensor product spaces arenaturally isomorphic tothecanonically ordered product. For example, the ordered pairs VXV*arecertainly distinct from V*XV,thebilinear mappings onthese spaces being V*®V andV®V*respectively. How- ever, wemaydefine amapqrby <p:V*®V 1—-> V®V* Tii> <pT where (qJT)(X, w)=T(w, X) VXe V*,weV. THE MIXED TENSOR ALGEBRA 17 It is easy to see that cp defines an isomorphism between V*0 V and VO V*. Further it is natural (or canonical), depending on no choice of bases for these spaces. Similarly, any tensor product containing V r times and V* s times is naturally isomorphic to the canonically ordered rVOs V*. We shall not distinguish between these naturally isomorphic spaces, and shall always form tensor products with the factors from V collected at the left. Thus we adopt the convention that tensors will be evaluated on a set ordered with elements from V* occurring first. If (e) is a basis for V, with {X,} a dual basis for V*, such that ei(Xj)= 6'1, then a basis for T rs (V) is provided by the n(r+s) elements {eliOei20 Oel'OX hOX;20 . . . OX I). If T is any element of T rs (V) then T = e',Oe120 . . . OelrOX 1i0 OX1., where the summation convention is employed. If {e'i} is a different basis for V, with dual basis {X'}, then if e' = Mijej and X'; = NiiX; it follows from e'i(X' ;)= 6' that MIkNik = (5' j. So if the transformation coefficients are arranged into matrices M and N, the transpose of N is the inverse of M. If the components of T in the basis labelled with a prime are then T71 = T(X, X:2, e'h) = Mh Mh N N Pr T q' qi    q, ti    1, Pi This is the classical expression for the change in the components of a tensor induced by a change of basis. The contravariant components, placed as superscripts, transform contragradiently to the covariant components, placed as subscripts. We may classify the symmetry of a mixed tensor according to the behaviour under permutations of the vectors from V, and those from V*: of course it makes no sense to talk of a symmetry that mixes these spaces. Since dual bases transform contragradiently we can define a contrac- tion map that reduces both the contravariant and the covariant degrees by one: : T(V) --> T;:1 (V) T 1—> Cl,T THE MIXED TENSOR ALGEBRA 17 Itiseasy toseethat cpdefines anisomorphism between V*®V and V®V*. Further itisnatural (orcanonical), depending onnochoice of bases forthese spaces. Similarly, any tensor product containing Vr times and V*stimes isnaturally isomorphic tothecanonically ordered ®’V®’V*. Weshall notdistinguish between these naturally isomorphic spaces, andshall always form tensor products with thefactors from V collected attheleft. Thus weadopt theconvention that tensors willbe evaluated onasetordered with elements from V*occurring first. If{e‘}isabasis forV,with {X,-} adual basis forV*,such that e'(X7~) =6'7,then abasis forT,‘(V) isprovided bythen(’*‘) elements {e"'®e"2® ...®e"®X7-,®X7-2® ...®X7-5}. IfTisanyelement ofT,‘(V) then T=T,-'7‘,-",1_‘_‘_",_,'i* e"®e’1® ...®e"'®X7-l® ...®X7-I where thesummation convention isemployed. If{e"} isadifferent basis forV,with dual basis {X’,-}, then ife"=Mi]-e/I andX’,-=N,-IX; itfollows from e”'(X’7-) =6‘,that MikNjk =éij. Soifthetransformation coefficients arearranged intomatrices Mand N,thetranspose ofNistheinverse ofM.Ifthecomponents ofTin thebasis labelled with aprime are Tr_/'1..._j, ll |r then =T(X;-,.X;,.....X:-,.e'1-. ....ea) =1' ; . .Pr-1----1 Mlql ...M’,1N,]P* ...N,’ T,,,‘___,,§. This istheclassical expression forthechange inthecomponents ofa tensor induced byachange ofbasis. The contravariant components, placed assuperscripts, transform contragradiently tothe covariant components, placed assubscripts. Wemay classify thesymmetry ofamixed tensor according tothe behaviour under permutations ofthevectors from V,and those from V*:ofcourse itmakes nosense totalkofasymmetry that mixes these spaces. Since dual bases transform contragradiently wecandefine acontrac- tion map that reduces both thecontravariant andthecovariant degrees byone: CliTill’) E’ Tlii (V) Tii> CQT 18 TENSOR ALGEBRA /th entry CikT ( „ . . . , ; , . . . ,) = 7' („ . . X, , . . . , ; , , . . . , e', , . . ) kth entry (1.5.1) where {e'} is dual to (X,}. Since the dual frames transform contragradiently the linearity of T ensures that the definition of Ck is basis independent. If, in some basis, T has the components T is then the components of Ci T are Tit  j_lflhll.   ik-Im kt where the 'dummy' index m is summed over. For the special case of T E T(V) the contraction CI maps T to the field F. In this case the contraction map is sometimes called the trace of T, Tr T. When V has a metric there is a canonical isomorphism -, between V and V*. Similarly we can use a metric on V to define a mapping between tensors of different contravariant and covariant degrees. For example, given a tensor TE T(V) we can define an SE T(V) as follows: S(X,, . . . el, . es, es+i) = T(X,, . . ., X — k-1, Xk, . . X,_1; e', . . ei+1, . . e+1). In a similar way we could associate with T a tensor in T(V) or more generally a tensor in T(V) with p+q=r+s. We give an example. Given TE T(V) we define S c T(V) by S(W, Y, co) = T(W , , i 7) VW, Ye V* , we V. (1.5.2) If {e'} and {x,} are dual bases for V and V* respectively such that T = l'e1C)eiC)Xk S = Se'OefOX k then writing W, Y and co in this basis gives WI PO) kS =1P1 71 CO kgqkgmTfq. Since this must hold for all W, Y and co gqkgpiTfq. (1.5.3) Such expressions can be simplified by adopting a convention for raising and lowering indices with the components of the metric, similar to the case for vectors. However, such a procedure would be ambiguous 18 TENsoR ALGEBRA lthentry _ Cl(T(99'-119"9):T(99"-9Xfs' 7;19""9e'1?"'7) Mia kthentry (1.5.1) where {e’}isdualto{X,-}. Since thedual frames transform contragradiently thelinearity ofT ensures thatthedefinition ofCLisbasis independent. If,insome basis, Thasthecomponents ...]Tf,1]...l,S then thecomponents ofCLTare ---l[—lm.l-l_+l --~.ls_ 11...ik_1mt;,,,1...i, where the‘dummy’ index missummed over. Forthespecial case of TeT}(V) thecontraction C}maps Ttothefield F.Inthiscase the contraction map issometimes called thetrace ofT,TrT. When Vhasametric there isacanonical isomorphism “,between V and V*. Similarly wecan useametric onVtodefine amapping between tensors ofdifferent contravariant and covariant degrees. For example, given atensor TeT§(V) wecandefine anSeTfi}(V) as follows: S(X1, ...X,_1; e1,...,e’,e‘*1) =T(X1, ...,Xk_1, El,Xx,...,X,_1; e‘,...,e/'1. e/+1, ...,e‘+‘). Inasimilar way wecould associate with Tatensor inT§§}(V) or more generally atensor inT';(V) with p+q=r+s.We give an example. Given TeT2(V)wedefine SeT2(V)by s(w,Y,w)=r(w,tn,Y‘) vw,YeV*,wev.(15.2) If{e"}and{x,} aredual bases forVandV*respectively such that T=Tl‘,-e"®e/®Xk S=Si‘,-e"®e/®X,, then writing W,Yandtointhisbasis gives W‘Ylo1,(Sf7 =W"Y/'w,.g‘?"g7,7-Tfq. Since thismust hold forallW,Yandto Sf}=g‘l"g7,,~T{-’q. (1.5.3) Such expressions can besimplified byadopting aconvention for raising andlowering indices with thecomponents ofthemetric, similar tothecase forvectors. However, such aprocedure would beambiguous THE MIXED TENSOR ALGEBRA 19 with the tensor components arranged in the way we have them: it not being clear, for example, where the upper index should be lowered to. To enable a raising and lowering convention to be employed, from now on we will order the upper indices relative to the lower ones. We can always specify a tensor with the indices in a canonical order; the lower indices occurring first. Components can then be raised and lowered with the components of the metric, maintaining the ordering. Thus one obtains an array of components not in canonical order, some super- scripts occurring before subscripts. If we return to the example we were considering, only this time stagger the components in the canonical order, T = TelOefOX k S = Se'® ei®Xk then the relationship (1.5.2) between S and T relates the components by Suk = gqkg piT,,IP This can now be compactly written as = "1. (1.5.4) There are a couple of points relating to this index convention that are worth emphasising. The first is that a raising and lowering convention need not be adopted at all: in which case there is no need to order the upper indices relative to the lower ones. No inconsistencies would arise, only relationships between tensors such as (1.5.2) would have the untidy component form of (1.5.3). The second point concerns the ordering of the basis. We have decided to work always with tensors formed with products from V to the left. Nevertheless relationships such as (1.5.4) involve components that are not indexed in the canonical order. As we earlier remarked one could work with the larger class of tensors in which the factors from V and V* occur in no definite order. In this case one might adopt the convention that the basis is attached in the order in which the components occur; an element from V going with a subscript for example. Such a tensor would, however, as we have pointed out, be naturally isomorphic to a tensor with the same components but with a canonically ordered basis. Thus the adopted ordering of the basis is in no real sense a restriction, and in particular we have the freedom to employ the raising and lowering conventions that introduce the non- canonically ordered components. Sometimes we may speak, for example, of a degree two tensor being symmetric and trace free. Such imprecise statements should be under- stood to mean that T is a symmetric tensor in T(V), and that SE T;(17) is traceless, where S(X, w) = T(X, VX€V*, we V. Equivalently, T(Xl, X,) = 0, where X' = gYX,. THE MIXED TENSOR ALGEBRA 19 with thetensor components arranged intheway wehave them: itnot being clear, forexample, where theupper index should belowered to. Toenable araising andlowering convention tobeemployed, from now onwewillorder theupper indices relative tothelower ones. Wecan always specify atensor with theindices inacanonical order; thelower indices occurring first. Components canthen beraised andlowered with thecomponents ofthemetric, maintaining theordering. Thus one obtains anarray ofcomponents notincanonical order, some super- scripts occurring before subscripts. Ifwereturn totheexample wewere considering, only this time stagger thecomponents inthecanonical order, T=T,7"e‘®el®Xk S=S,»/‘e'® el®Xk then therelationship (1.5.2) between SandTrelates thecomponents by 511"=3""8p/Tn” This cannow becompactly written as S,-7"=T,-"7. (1.5.4) There areacouple ofpoints relating tothisindex convention that are worth emphasising. The first isthat araising andlowering convention need notbeadopted atall:inwhich case there isnoneed toorder the upper indices relative tothelower ones. Noinconsistencies would arise, only relationships between tensors such as(1.5.2) would have theuntidy component form of(1.5.3). The second point concerns theordering of thebasis. Wehave decided towork always with tensors formed with products from Vtotheleft.Nevertheless relationships such as(1.5.4) involve components that arenotindexed inthecanonical order. Aswe earlier remarked one could work with thelarger class oftensors in which thefactors from VandV*occur innodefinite order. Inthiscase onemight adopt theconvention thatthebasis isattached intheorder in which thecomponents occur; anelement from Vgoing with asubscript forexample. Such atensor would, however, aswehave pointed out, be naturally isomorphic toatensor with thesame components butwith a canonically ordered basis. Thus theadopted ordering ofthebasis isin noreal sense arestriction, and inparticular wehave thefreedom to employ theraising and lowering conventions that introduce thenon- canonically ordered components. Sometimes wemay speak, forexample, ofadegree twotensor being symmetric andtrace free. Such imprecise statements should beunder- stood tomean that Tisasymmetric tensor inT§(V),andthat SeT}(V) istraceless, where S(X,o))= T(X, 67) VXeV*,we V. Equivalently, T(X‘, X,-)=O,where X’=g’/X,-. 20 TENSOR ALGEBRA Bibliography Abrahams R, Marsden J E and Ratiu T 1983 Manifolds, Tensor Analysis and Applications (New York: Addison-Wesley) Dodson C T J and Poston T 1977 Tensor Geometry (London: Pitman) Greub W 1978 Multilinear Algebra 2nd edn (Heidelberg: Springer) Schutz B F 1980 Geometrical Methods of Mathematical Physics (Cambridge: Cambridge University Press) 20 TENsoR ALGEBRA Bibliography Abrahams R,Marsden JEandRatiu T1983 Manifolds, Tensor Analysis and Applications (New York: Addison-Wesley) Dodson CTJandPoston T1977 Tensor Geometry (London: Pitman) Greub W1978 Multilinear Algebra 2ndedn(Heidelberg: Springer) Schutz BF1980 Geometrical Methods ofMathematical Physics (Cambridge: Cambridge University Press) 2 Clifford Algebras and Spinors In this chapter we present an account of Clifford algebras and spinors. Taken with Appendix A it is fairly self-contained. Whereas in some places we have explicitly referred to Appendix A we have often tacitly assumed knowledge of something that is to be found there. Thus a reader confronted with concepts or terminology that are unfamiliar should consult Appendix A where (we hope) further details may be found. The Clifford algebra is constructed so as to facilitate a study of orthogonal transformations. It leads to a systematic way of introducing the spin groups (the covering groups of the orthogonal groups and various subgroups) for arbitrary dimensions and signature. The irreduc- ible representations of the Clifford algebra give rise to irreducible representations of the spin groups: spinors. If the real vector space V with bilinear form g is an orthogonal space then we wish to imbed V and a copy of the real numbers as vector subspaces in the real associative algebra C(V, g) in such a way that x2 = g(x, x), VX E V. The square of x denotes its product with itself in this algebra, and the right-hand side is a real number which lies in the vector subspace of the algebra spanned by the identity. If S is any invertible element of the algebra and x' = SxS -I then obviously x'2 = g(x, x). So if x' is in V we have an orthogonal transformation. Those elements S such that x' is in V form a group, the Clifford group. Obviously elements of the Clifford group which differ by a multiple of the centre will produce the same orthogonal transformation, so that the mapping from the Clifford group to the orthogonal group is many-to-one. By suitably normalising elements of the Clifford group we obtain a subgroup such that the mapping into the orthogonal group is two-to-one, and we have a double covering of the orthogonal group. Being able to write an orthogonal transformation in terms of simultaneous multiplication from both sides by an element of the Clifford group we are led to consider those Clifford Algebras andSpinors Inthischapter wepresent anaccount ofClifford algebras andspinors. Taken with Appendix Aitisfairly self-contained. Whereas insome places wehave explicitly referred toAppendix Awehave often tacitly assumed knowledge ofsomething that istobefound there. Thus a reader confronted with concepts orterminology that areunfamiliar should consult Appendix Awhere (we hope) further details may be found. The Clifford algebra isconstructed soastofacilitate astudy of orthogonal transformations. Itleads toasystematic way ofintroducing thespin groups (the covering groups oftheorthogonal groups and various subgroups) forarbitrary dimensions andsignature. The irreduc- iblerepresentations oftheClifford algebra give risetoirreducible representations ofthespin groups: spinors. Ifthereal vector space V with bilinear form gisanorthogonal space then wewish toimbed V andacopy ofthereal numbers asvector subspaces inthereal associative algebra C(V, g)insuch away that x2=g(x, x),VxeV. Thesquare ofxdenotes itsproduct with itself inthisalgebra, andthe right-hand sideisarealnumber which liesinthevector subspace ofthe algebra spanned bytheidentity. IfSisanyinvertible element ofthe algebra andx’=SxS" then obviously x’2=g(x, x).Soifx’isinV wehave anorthogonal transformation. Those elements Ssuch thatx’is inVform agroup, theClifford group. Obviously elements ofthe Clifford group which differ byamultiple ofthecentre willproduce the same orthogonal transformation, sothat themapping from theClifford group totheorthogonal group ismany-to-one. Bysuitably normalising elements oftheClifford group weobtain asubgroup such that the mapping intotheorthogonal group istwo-to-one, andwehave adouble covering oftheorthogonal group. Being able towrite anorthogonal transformation interms ofsimultaneous multiplication from both sides byanelement oftheClifford group weareledtoconsider those 22 CLIFFORD ALGEBRAS AND SPINORS transformations obtained by multiplying from one side only; the spin transformations. The Clifford algebra can be constructed as a quotient of the tensor algebra. This is in close parallel with §1.3, where we considered the exterior algebra as a quotient of the tensor algebra. Rather than regarding elements of the Clifford algebra as equivalence classes in the tensor algebra it is more convenient to work with representatives of these classes. We show how we can choose these representatives to be the exterior forms, the Clifford product being given in terms of the exterior and interior products. In §2.2 we determine the structure of the real Clifford algebras. These algebras are Z2-gradedt, and we give the structure of the even subalgebra in §2.3. In §2.4 we introduce the Clifford group and show the relation of it and its subgroups to the orthogonal group and its subgroups. After examining the irreducible representations of the Clifford algebra and group, spinors, we move on to spin-invariant products. At this point some readers will probably feel the furthest removed from what they feel they want to know, and from relevance to physics. However, such readers should be assured that this section will enable them to determine all the spin-invariant products in whichever dimension is currently in fashion, and, for example, whether the charge conjugation matrix (defined in either of two ways) is symmetric or antisymmetric. The reader with a trusting disposition may be content to learn how to interpret the tables that summarise the results. In §2.7 we consider the complexified Clifford algebras. Anyone familiar with the y-matrices, which are usually assumed to be complex, may wonder why we have postponed the complex case for so long. However, although the y-matrices are usually assumed to be complex, conjugate—linear operations, such as the Dirac adjoint, are considered as well as complex—linear ones. Thus an underlying real structure is singled out and so one way or another we need the results of the real case. The account we have given is logically complete at the end of §2.7. It makes no reference, however, to such things as Dirac spinors and charge conjugation with which most physicists are familiar. Whilst not being intended as a dictionary, §2.8 makes contact with the y-matrices and physics vocabulary. We also mention the 'two-component spinor formal- ism' for Lorentzian spinors. Having outlined what we shall do, it is in order to state what is omitted. There are two main restrictions we have imposed: we only consider algebras over the real or complex field and we assume the bilinear form is non-degenerate. The important topic of pure spinors has been given a chapter of its own. t Grading is discussed in Appendix A. 22 CLIFFORD ALGEBRAS AND SPINORS transformations obtained bymultiplying from one side only; thespin transformations. The Clifford algebra canbeconstructed asaquotient ofthetensor algebra. This isinclose parallel with §1.3, where weconsidered the exterior algebra asaquotient ofthe tensor algebra. Rather than regarding elements oftheClifford algebra asequivalence classes inthe tensor algebra itismore convenient towork with representatives of these classes. Weshow how wecanchoose these representatives tobe theexterior forms, theClifford product being given interms ofthe exterior andinterior products. In§2.2 wedetermine thestructure ofthe realClifford algebras. These algebras areZ2-gradedi, andwegive the structure oftheeven subalgebra in§2.3. In§2.4 weintroduce the Clifford group and show therelation ofitand itssubgroups tothe orthogonal group and itssubgroups. After examining theirreducible representations oftheClifford algebra andgroup, spinors, wemove on tospin-invariant products. Atthispoint some readers willprobably feel thefurthest removed from what they feelthey want toknow, andfrom relevance tophysics. However, such readers should beassured that this section willenable them todetermine allthespin-invariant products in whichever dimension iscurrently infashion, and, forexample, whether thecharge conjugation matrix (defined ineither oftwo ways) is symmetric orantisymmetric. The reader with atrusting disposition may becontent tolearn how tointerpret thetables that summarise the results. In§2.7 weconsider thecomplexified Clifford algebras. Anyone familiar with they-matrices, which areusually assumed tobecomplex, may wonder why wehave postponed thecomplex case forsolong. However, although the7/-matrices areusually assumed tobecomplex, conjugate—linear operations, such astheDirac adjoint, areconsidered as well ascomplex—linear ones. Thus anunderlying realstructure issingled outandsoonewayoranother weneed theresults oftherealcase. The account wehave given islogically complete attheendof§2.7. Itmakes noreference, however, tosuch things asDirac spinors and charge conjugation with which most physicists arefamiliar. Whilst notbeing intended asadictionary, §2.8 makes contact with the7/-matrices and physics vocabulary. Wealsomention the‘two-component spinor formal- ism’forLorentzian spinors. Having outlined what weshall do,itisinorder tostate what is omitted. There aretwo main restrictions wehave imposed: weonly consider algebras over thereal orcomplex field and weassume the bilinear form isnon-degenerate. The important topic ofpure spinors has been given achapter ofitsown. 1‘Grading isdiscussed inAppendix A. THE CLIFFORD ALGEBRA 23 2.1 The Clifford Algebra We assume now that the vector space V has an F-valued non- degenerate symmetric bilinear form, or metric, g. Let J be the ideal of T(V) consisting of sums of terms of the form a0{x0x — g(x, x)}0b, a, b E T(V), x c V. Then the Clifford algebra associated with V is C(V, g) defined by C(V, g) = T(V)IJ. (2.1.1) The product will be denoted v , satisfying [a] y [b] = [a® b]. The ideal J is not a Z-graded subspace and so C(V, g) does not inherit a Z-gradation. However, x0x — g(x, x) is homogeneous with respect to the induced Z2-gradation of T(V) making J a Z2-graded subspace. Thus C(V, g) inherits a Z2-gradation. The ideal J is preserved by n, and ix and so all of these naturally induce operations (denoted by the same symbol) in C(V, g). If x, y E V then x0y = xAy + g(x, y) + {(x + y)0(x + y) — g(x + y, x + y) — x0x + g(x, x) — yOy + g(y, y)). The term in brackets is in J and so x0y X Ay + g(x, y). (2.1.2) More generally for co a p-form and x E V we have x0co X A co + iîco. (2.1.3) Here 5( E V* is the metric dual of x, defined by Y(y) = g(x,y), V y E V. For co a 1-form (2.1.3) reduces to (2.1.2). We may prove its general validity by induction. This will be closely analogous to the proof of (1.3.7). Suppose that (2.1.3) is true for co of degree less than or equal to p — 1, then it will be true for all p-forms if it holds for co the product of p orthogonal 1-forms. As we showed in the proof of (1.3.7) it follows from the definition of the exterior product that if x, y', i = 1, . p are in V then xnYin    AY' where y -1' p = y 1 A 3,2 A . . A yr-1 A y r+1 A . . A p y Since (2.1.3) is assumed true for co of degree p —1 or less r=1 = 1/(p+1) (x0y 1-- P yroor Ay , ... 0) (2.1.4) yr0(x A Y1 P) Yr ®(X°Y 1 P)- Use of (2.1.2) gives THE CLIFFORD ALGEBRA 23 2.1TheClifford Algebra We assume now that the vector space Vhas anF-valued non- degenerate symmetric bilinear form, ormetric, g.LetJbetheideal of T(V) consisting ofsums ofterms oftheform a®{x®x —g(x, x)}®b, a,beT(V), xeV.Then theClifford algebra associated with VisC(V, g)defined by C(V, g)=T(V)/J. (2.1.1) The product will bedenoted V,satisfying [a]v[b] =[a®b]. The ideal JisnotaZ-graded subspace andsoC(V, g)does notinherit a Z-gradation. However, x®x —g(x, x)ishomogeneous with respect to theinduced Z2-gradation ofT(V) making JaZ2-graded subspace. Thus C(V, g)inherits aZ2-gradation. The ideal Jispreserved by17,Eandix andsoallofthese naturally induce operations (denoted bythesame symbol) inC(V, g).Ifx,yeVthen X®y=X/\)’+s'(X.y)+%{(X+y)®(X +y)—s'(X+yrX+y) —X®X+s'(x.X)—y®y+s'(y.y)}- Theterm inbrackets isinJandso x®y ~xAy +g(x, y). (2.1.2) More generally forwap-form andxeVwehave x®w~xAw +iiw. (2.1.3) Here YeV*isthemetric dual ofx,defined by3?(y) =g(x,y), Vye V.Forwa1-form (2.1.3) reduces to(2.1.2). Wemay prove its general validity byinduction. This willbeclosely analogous totheproof of(1.3.7). Suppose that(2.1.3) istrue forwofdegree lessthan orequal top—1,then itwillbetrue forallp-forms ifitholds forwtheproduct ofporthogonal 1-forms. Asweshowed intheproof of(1.3.7) itfollows from thedefinition oftheexterior product that ifx,y’,i=1,...,p areinVthen X/xyl/\ AY” P =1/(p+1)(x®y1"-P +Z(—1)’y'®(xAy‘""" i~'~P)) (2.1.4) rl where y1;~~~P =y‘Ay2A Ay"1Ay'*1A ... AyP. Since (2.1.3) isassumed true forwofdegree p—1 orless yr®(x/\y1.. ?...p)~yr®(x®y1...?...p _i;yl...T...p)_ Useof(2.1.2) gives 24 CLIFFORD ALGEBRAS AND SPINORS YrO(XAY 1- î- ) 2g(yr, x)yl   X0YrOY1 P Yr Oiiyi P Since the yi are assumed orthogonal we may use (2.1.3) for co a (p-1) or (p-2)-form to show that yrO(x A y1'   - -P) 2g(yr, x)y 1   P XO(Yr Yi * ** P)— Yr Ai7Y1— We may pull the interior derivative to the front of the last term and use yrAyi... . p i)r-1 yl P to produce yr0(x A Y1  p) g(yr, , x)yl P (-1)r X0y1 P (-1)ri1yi P SO (-1)"Yro(x A yi    P) r=1 E(_orgur,x)y1p px0y1 ° P r=1 pxOyl P - (1+p)i5y1 P . Returning to (2.1.4) shows that if (2.1.3) is true for co a q-form with q p — 1 then it is true for co a p-form. Thus (2.1.2) shows that indeed (2.1.3) holds for all p-forms. Repeated use of (2.1.3) shows that an arbitrary tensor product is equivalent to a sum of exterior forms, for example x10x20x3 x10{x2 A X3 + g(x2, x3)} xi A x2 A x3 ± x2)x3 x3)x2 ex2, x3)xl. In principle we could write down an explicit formula for the relation between the class of a homogeneous tensor and classes of exterior forms. However, it is generally sufficient to know that (2.1.3) deter- mines such a relation and for practical purposes we shall be content with (2.1.3) and the following other special case. If co is an arbitrary p-form and x a 1-form then u.P0x x A nco — iW. (2.1.5) For co a 1-form this is certainly true since it reduces to (2.1.1). Again we prove its general validity by induction. Suppose that (2.1.5) holds for co of degree less than or equal to p, then (Y A (0)0x --- (yOco — i9w)Ox by (2.1.3) y0(x A WO i177(0) - X A nii(0 by (2.1.5). 24 CLIFFORD ALGEBRAS AND SPINORS yr®(xAyl...?...p) ,___2g(yr,x)yl...7...p _x®yr®yl...?...p Since they‘areassumed orthogonal wemay use(2.1.3) forwa(p—1) or(p—2)-form toshow that yr®(xAyl...'i...p) ,___2g(yr,x)y1...?...p _x®(yr/\yl...'F...p)_yr/\i_;yl...?...p. Wemaypulltheinterior derivative tothefront ofthelastterm anduse yr/\y1... r...p =(_1)r—1yl...p toproduce yr®(xAy1...?...p) ,___g(yr’ x)y1... ?...p _,|_(__1)rx®yl...p _(_1)rijyl...p SO P 2(—1)’y’®(XAy‘"""’"""”) r=1 I7 ~2(_1)rg(-yr, x)y1... 9...p _,|_px®y1...p _piiyl...p ~PX®y""” —(1+p)ii-y‘"""”-w,_. Returning to(2.1.4) shows that if(2.1.3) istrue forwaq-form with qEp—1then itistrue forwap-form. Thus (2.1.2) shows thatindeed (2.1.3) holds forallp-forms. Repeated useof(2.1.3) shows thatan arbitrary tensor product isequivalent toasumofexterior forms, for example x‘®x2®x3 ~x‘®{x2Ax3 +g(x2, x3)} ~x‘Ax2Ax3 +g(x‘, x2)x3 —g(x‘, x3)x2 +g(x2, x3)x‘. Inprinciple wecould write down anexplicit formula fortherelation between theclass ofahomogeneous tensor and classes ofexterior forms. However, itisgenerally sufficient toknow that(2.1.3) deter- mines sucharelation andforpractical purposes weshall becontent with (2.1.3) andthefollowing other special case. Ifwisanarbitrary p-form andxa1-form then w®x ~xA170)—i,217w. (2.1.5) Forwa1-form thisiscertainly truesince itreduces to(2.1.1). Again we prove itsgeneral validity byinduction. Suppose that(2.1.5) holds forw ofdegree lessthan orequal top,then (yAw)®x ~(y®w —iy-w)®x by(2.1.3) ~y®(xA 170)—i,,17o)) —xA17i,2w +i,217iy-w by(2.1.5). THE CLIFFORD ALGEBRA 25 A second application of (2.1.3) gives CY A WYDX — Y A (X A no) — iinco) + iyxwo — x A iinw — YON) + x A iM0 — iiimo where we have used nix = —ixn. Dropping the terms that cancel and a little rearranging gives (YAW)®x --. X A n(Y A (0) — iin(Y A (1)) and so if (2.1.5) holds for all co of degree less than or equal to p it also holds for all (p+1)-forms. This completes the inductive proof of the general validity of (2.1.5). We have shown that the classes of a basis for the space of all exterior forms provide a basis for C(V, g). There is thus a natural way of introducing a product, y , on the space of exterior forms that turns this vector space into an algebra, C(V, g) say, where C(V, g) = C(V, g). If a and co are exterior forms then the exterior form a v co is defined by [œ] y [co] = [61' v a]. (2.1.6) Since [a] v [a] = [aOco] the equivalence in (2.1.3) gives for x a 1-form X v co =-- x A co + i„t-co. (2.1.7) Similarly (2.1.5) gives wvx=xn nw — ii 71w- (2.1.8) As we noted earlier, the associativity of the product together with (2.1.7) completely determines y on arbitrary forms. Thus the vector space of exterior forms together with the antisymmetrised tensor product A is an exterior algebra, whereas the product y turns the same vector space into a Clifford algebra. The products are related as in (2.1.7). By quotienting the tensor algebra in a particular way we have been led to an algebra C(V, g) which satisfies the familiar relations xvy + yvx = 2g(x, y) V x, y E V. (2.1.9) It is because of this relation that the Clifford algebra is adapted to the study of orthogonal transformations of V. We would like to know if there are any other associative algebras, apart from the one we have constructed, whose product satisfies the relation (2.1.9). Suppose that C'(V, g) is an associative algebra with product A and that cp is a linear mapping of V into a subspace of C'(V, g), V', which generates the algebra, and that (p(x)4)(y) + cp(y)Acp(x) = 2g(x, y) Vx, y E V. (2.1.10) THE CLIFFORD ALGEBRA 25 Asecond application of(2.1.3) gives (yAw)®X ~yA(XAnw-i,~r1w)+iy-Xnw —XAi,-aw —iyimw +xAi,-1700 —iii,-r7co where wehave used r7ix=—ix17.Dropping theterms that cancel anda little rearranging gives (yAw)®X ~XAr1(yAw)-it-110 Aw) andsoif(2.1.5) holds forallcoofdegree lessthan orequal topitalso holds forall(p+1)-forms. This completes theinductive proof ofthe general validity of(2.1.5). Wehave shown that theclasses ofabasis forthespace ofallexterior forms provide abasis forC(V, g).There isthus anatural way of introducing aproduct, V,onthespace ofexterior forms that turns this vector space intoanalgebra, C(V, g)say, where C(V, g)=C(V, g).If aandcoareexterior forms then theexterior form aVcoisdefined by [oz]V =[ozV (2.1.6) Since [or]V[co]=[a®co] theequivalence in(2.1.3) gives forxa1-form xVco= xAco+ i,2co. (2.1.7) Similarly (2.1.5) gives coVx =xAr7co— i,2r7co. (2.1.8) Aswenoted earlier, theassociativity oftheproduct together with (2.1.7) completely determines Vonarbitrary forms. Thus thevector space ofexterior forms together with the antisymmetrised tensor product Aisanexterior algebra, whereas theproduct Vturns thesame vector space into aClifford algebra. The products arerelated asin (2.1.7). Byquotienting thetensor algebra inaparticular way wehave been ledtoanalgebra C(V, g)which satisfies thefamiliar relations xVy +yVx =2g(x, y) Vx, yeV. (2.1.9) Itisbecause ofthisrelation thattheClifford algebra isadapted tothe study oforthogonal transformations ofV.Wewould like toknow if there areanyother associative algebras, apart from theone wehave constructed, whose product satisfies therelation (2.1.9). Suppose that C’(V, g)isanassociative algebra with product Aandthat qaisalinear mapping ofVintoasubspace ofC’(V, g),V’,which generates the algebra, andthat <P(X)A<r(y) +<P(y)A<r(X) =2g(X.y) VX,y6V-(2-1-10) 26 CLIFFORD ALGEBRAS AND SPINORS The right-hand side is understood to contain the identity in C'(V, g). The mapping (4) can be extended to a homomorphism (I) from T(V) to C'(V, g): 0:T(V)---> C' (V , g) 43(x0Y) = T(x)A 019(Y). (2.1.11) Since V' generates C'(V, g), (ID[T(V)] = C'(V, g). It follows from (2.1.10) and (2.1.11) that (13{x0x — g(x, x)} = 0, and so OM = 0 where J is the ideal used to construct C(V, g). Thus if v is the mapping of T(V) onto C(V, g) defined by rra = [a] then (13 = von., where ip is some homomorphism from C(V, g) to C'(V, g). So the dimension of C'(V, g) certainly cannot be greater than that of C(V, g), and if the dimensions are the same then the algebras are isomorphic. Since the kernel of tp is an ideal of C(V, g) if the dimension of C'(V, g) is less than that of C(V, g) it must be a (non-trivial) quotient of that algebra. So the only possibility of a C'(V, g) which is not isomorphic to C(V, g) arises if C(V, g) is not simple. Conversely, it readily follows that if C(V, g) is not simple then any quotient satisfies the conditions assumed for C'(V, g). Sometimes any algebra like C'(V, g) is called a Clifford algebra, the algebra C(V, g) being termed the universal Clifford algebra. From now on, unless indicated otherwise, by Clifford algebra we shall mean the algebra of the vector space of exterior forms with the product given in (2.1.7), and shall reserve the notation C(V, g) for this algebra. We shall also henceforth omit the symbol y, it being understood that juxtapositioning of exterior forms denotes this product. Although the Clifford algebra is not a Z-graded algebra the vector space of exterior forms is a Z-graded vector space and it will be convenient to use the decomposition into Z-homogeneous subspaces: n QV , g) = E Wp (C( V, g)) (2.1.12) p=0 where n is the dimension of V and the projection operators Yp project out the homogeneous subspaces of p-forms. If A and B are homogeneous of degree p and q respectively then their Clifford product will not in general be homogeneous; rather AB = 9'1,÷1(AB) + W p+q_2(4B) + . . . + 9' ip_qi(AB). (2.1.13) This follows directly from (2.1.7) and (2.1.8). If q) and ip are arbitrary elements of the algebra then 9'0090 = E 990(99p1P,) (2.1.14) P 26 CLIFFoRD ALGEBRAS ANDSPINORS The right-hand side isunderstood tocontain theidentity inC’(V, g). The mapping <79canbeextended toahomomorphism <1)from T(V)to C’(V, g)I <I>:T(V) -—> C’(V, g) ‘P(X®y) =<P(X)A<P(y)- (2-1-11) Since V’generates C’(V, g),<I>[T(V)] =C’(V, g).Itfollows from (2.1.10) and (2.1.11) that <I>{x®x—g(x, x)}=0, and so<I>(J) =0 where Jistheideal used toconstruct C(V, g).Thus if11isthemapping ofT(V) onto C(V, g)defined by7T(1=[a]then <I>=1/1°11, where 1/1is some homomorphism from C(V, g)toC’(V, g).Sothedimension of C’(V, g)certainly cannot begreater than that ofC(V, g),andifthe dimensions arethesame then thealgebras areisomorphic. Since the kernel of1/;isanideal ofC(V, g)ifthedimension ofC’(V, g)isless than thatofC(V, g)itmust bea(non-trivial) quotient ofthat algebra. Sotheonly possibility ofaC’(V, g)which isnotisomorphic toC(V, g) arises ifC(V, g)isnotsimple. Conversely, itreadily follows thatif C(V, g)isnotsimple then anyquotient satisfies theconditions assumed forC’(V, g).Sometimes anyalgebra likeC’(V, g)iscalled aClifford algebra, the algebra C(V, g)being termed the universal Clifford algebra. From now on,unless indicated otherwise, byClifford algebra weshall mean thealgebra ofthevector space ofexterior forms withtheproduct given in(2.1.7), andshall reserve thenotation C(V, g)forthisalgebra. Weshall also henceforth omit thesymbol V,itbeing understood that juxtapositioning ofexterior forms denotes this product. Although the Clifford algebra isnotaZ-graded algebra thevector space ofexterior forms isaZ-graded vector space anditwillbeconvenient tousethe decomposition intoZ-homogeneous subspaces: C(V, g)=Eilofifp (C(V, g)) (2.1.12) where nisthedimension ofVandtheprojection operators SP7,project out the homogeneous subspaces ofp-forms. IfAand Bare homogeneous ofdegree pandqrespectively then their Clifford product willnotingeneral behomogeneous; rather AB=&r,,,(/1B) +&r,,,,_,(AB) +...+sr|,,_,,(AB). (21.13) This follows directly from (2.1.7) and(2.1.8). Ifqaand1/1arearbitrary elements ofthealgebra then erotw)=2H11(<1>,.w,.) (2-1.14) THE CLIFFORD ALGEBRA 27 where cpp gpcp and (2.1.13) has been used. If gp denotes the metric on p-forms induced from g, as introduced in the previous chapter, then we may introduce a metric on inhomogeneous forms, G, by defining G(q), ip) = Egp(opp, ipp) (2.1.15) that is, G is diagonal in the homogeneous subspaces. This metric on forms can be related to Clifford multiplication G(T,V) = go(V/P). (2.1.16) From (2.1.14) the right-hand side is seen to be diagonal in the homogeneous components of 92 and tp and so to verify (2.1.16) all we need to check is that gp(cpp, tPp) = o(q)Pp). Since both sides are linear in cpp and ipp it suffices to consider the case of cpp and ipp products of orthonormal 1-forms. If (pp = ala2 . . . aP and ipp = b' b2 . bP then from (2.1.7) go(9106) = i    ied; (b 1b2    bP)- If the fal and WI are subsets of an orthonormal basis then the right-hand side is zero unless these sets are the same up to a relabelling. Since (a1a2 ap) = eal ai)g(az, a2) 8,(ap, ap) = gp(a1a2 ap, a1a2 ap) we have verified (2.1.16). One trivial result that is important for calculations is Yo(VP) = 9'o(1P99) (2.1.17) as wo(opiP) = E wo((PoPp) = E(-1)EP' 21gp(cpp, 16) where [p/21 denotes the integer part of p12, and the result follows from the symmetry of gp. It will sometimes be useful to expand an arbitrary element of the Clifford algebra in a G-orthonormal basis. If {ea} is a g-orthonormal basis then {eA) is a G-orthonormal basis where the multi-index A takes on all naturally ordered sequences of distinct indices. We use the notation el2 p „ el A e2 A A ep = ele2 ep. If g(ea, eb) = —ab q and qab denotes the inverse matrix then we set = nabeb, giving e A an obvious meaning. Then W0(e4e8) =6 AB where THE CLIFFORD ALGEBRA 27 where (ppESfptp and(2.1.13) hasbeen used. Ifg,,denotes themetric onp-forms induced from g,asintroduced intheprevious chapter, then wemay introduce ametric oninhomogeneous forms, G,bydefining G<<1>.w> =2s,.(<P,., w.) (2-1-15> that is,Gisdiagonal inthehomogeneous subspaces. This metric on forms canberelated toClifford multiplication G019»w)=9’@(<P§w)- (2-1-16) From (2.1.14) the right-hand side isseen tobediagonal inthe homogeneous components oftpand 1/Iandsotoverify (2.1.16) allwe need tocheck isthat g,,(tpp, I/1,)=SF0(tp§I[1,,). Since both sides are linear in(ppand I117,itsuffices toconsider thecase of(ppand I117, products of orthonormal 1-forms. If q2,,=a1a2...a/’ and 171,,=blbz ...bl’then from (2.1.7) 9j0((P§’I/1p)=lg; ...15‘ (blbz ... Ifthe{a'} and {bi} aresubsets ofanorthonormal basis then the right-hand sideiszero unless these setsarethesame uptoarelabelling. Since ix;...i;,~| (a1a2.. .aP)= g(a1, a1)g(a2, a2)...g(aP, al’) =g,,(a1a2...a/’,a1a2...aP) wehave verified (2.1.16). One trivial result thatisimportant forcalculations is 5f0(‘P1ll) =5f0(1l"P) (2-1-17) as %P0(<1>w) =25/’o(<P,,1l1,.) =2(-1)“”2lg,.(<P,., 111,.) where [p/2] denotes theinteger part ofp/2, andtheresult follows from thesymmetry ofgp. Itwillsometimes beuseful toexpand anarbitrary element ofthe Clifford algebra inaG-orthonormal basis. If{e“} isag-orthonormal basis then {e"‘} isaG-orthonormal basis where themulti-index Atakes Onallnaturally ordered sequences ofdistinct indices. We use the notation e12"--P Ee‘Ae2A Ael’ =e‘e2...eP. Ifg(e“, e")=r7""and r7,,,,denotes theinverse matrix then weset 6,,=r7,,,,e", giving exanobvious meaning. Then SF0(e§4eB) =6,.,B where 28 CLIFFORD ALGEBRAS AND SPINORS 6,1 B denotes the Krônecker function that takes the value zero, unless the sequences A and B are the same in which case its value is one. If a is any element of the Clifford algebra then we can expand in this basis a = 9'0(aeAe A  (2.1.18) A The Hodge dual of a form may also be related to Clifford multi- plication. The definition of the Hodge dual, (1.4.5), of ipp, *Ipp, is given by Tp A *1Pp = gp(Cpp, 4p)*1 for all p-forms (Pp. Setting z *1(2.1.16) enables this to be rewritten as C19,9 A *Vp = 920(CPEpIP)Z = Y0(CPp4)Z. It immediately follows from (2.1.7) that 9'((Pp*IPp) = (Pp A *IP„ and from (2.1.13) that Yo(ePpli/Dz = Sn((Pp/14). Thus 99(99p"Pp) = giving 4,111 = (2.1.19) Exercise 2.1 If {ea}, {Xb} are any dual bases, ea(Xb)= sg, and a, /3 are any exterior forms, derive the relations . œvfl= 2, (111x 19„,   i)i%pa) A (i--?1   i-;i9,13) p =0 p (-1)[1'12] cvAP= 2, n1 (i X°,   i X uple a) V (i' s;L1   i -- p= 0 1-  2.2 The Structure of the Real Clifford Algebras In this section we take the field F to be the real numbers E. We shall determine the structure of C(V, g) for all real symmetric non- degenerate g. If g has a signature with p plus and q minus signs, then the structure of the Clifford algebra can only depend on p and q. We shall anticipate this by setting C(V, g) = C p, 1(E). One thing we know about the Clifford algebras is their dimension. Since we have identified the underlying vector space with the space of exterior forms the dimension of Cp. q (E) is 2n where p + q = n. Given a basis for V we can repeatedly use (2.1.7) to construct a multiplication table for the Clifford algebras, and in this sense we know its structure completely. What we would like to do is to relate the Clifford algebra to other 'standard' algebras. In particular we have already seen that if Cp,q(E) is not simple then we can construct a smaller algebra that satisfies the relation (2.1.9). Some low-dimensional examples will clarify how (2.1.7) is used in practice. It will also transpire that we can relate any Clifford algebra to a number of low-dimensional Clifford algebras. 28 CLIFFORD ALGEBRAS AND SPINORS 6,,” denotes theKronecker function that takes thevalue zero, unless thesequences AandBarethesame inwhich case itsvalue isone. Ifa isanyelement oftheClifford algebra then wecanexpand inthisbasis .1=§]9>,(a@,,§)@/1. (12.1.18) The Hodge dual ofaform may also berelated toClifford multi- plication. Thedefinition oftheHodge dual, (1.4.5), of(pp,*1pp, isgiven by(ppA*1/1p=gp((pp, 1pp)*1 forallp-forms (pp.Setting zE*1(2.1.16) enables thistoberewritten as(ppA*1pp=9’(,((pf,1/:p)z =9’0((pp1p§)z. It immediately follows from (2.1.7) thatEl’,,((pp*1pp) =(ppA*1ppandfrom (2-1-13) that9’0(<i>,,w5)z =$A(<i>pw§)- Thus9’1(<i>,,*w,,) =EP1(<z>,,w;'§1) giving *1p=(7152. (2.1.19) Exercise 2.1 If{e“}, {Xp} areany dual bases, e"(X2,) =62,and or,/3areany exterior forms, derive therelations "-1Ir/11 . . . .0zV/3 =EL)j(17P1xal ...1x” a)A(1?,, ...i~;,p/3) p=0 P "-1IF/11. . . .0zA/3 =2LL7—(1xa7 ...1x“P17Pa)V(1~e;,, ...1;p/3). '5O"P 2.2TheStructure oftheReal Clifford Algebras Inthissection wetake thefield Ftobethereal numbers IR.Weshall determine the structure ofC(V, g)forallreal symmetric non- degenerate g.Ifghasasignature with pplus andqminus signs, then thestructure oftheClifford algebra canonly depend onpandq.We shall anticipate thisbysetting C(V, g)ECp_q(lR). One thing weknow about theClifford algebras istheir dimension. Since wehave identified theunderlying vector space with thespace of exterior forms thedimension ofCp_,,(IR)is2"where p+q=n.Given abasis forVwecanrepeatedly use(2.1.7) toconstruct amultiplication table fortheClifford algebras, andinthissense weknow itsstructure completely. What wewould liketodoistorelate theClifford algebra to other ‘standard’ algebras. Inparticular wehave already seen that if Cpv,,(lR) isnotsimple then wecanconstruct asmaller algebra that satisfies therelation (2.1.9). Some low-dimensional examples willclarify how (2.1.7) isused inpractice. Itwillalso transpire that wecanrelate anyClifford algebra toanumber oflow-dimensional Clifford algebras. THE STRUCTURE OF THE REAL CLIFFORD ALGEBRAS 29 We will denote an orthonormal basis for V by {e', PI for i = 1, . . p, j = 1, ..., q where g(e', e') = —g(P, P) =1. It will be convenient to set z =e1A e2 A   eP A.fl A    AP' The two-dimensional algebra Co, 1(E) has as basis {1, f) where f2 = —1. It is thus isomorphic to the algebra of complex numbers, C0 OR) = CORY (2.2.1) A basis for C1, 0(R) is {1, e), and this algebra might not be so immediately recognisable. If P1 = 1(1 + e) and P2 =1-(1 — e) then (P1, P2) is obviously a new basis. The multiplication table is given in table 2.1. Thus PI and P2 each span mutually orthogonal one- dimensional subalgebras, each of which is isomorphic to the field R, so that C 1, 0(R) =E$R. (2.2.2) Table 2.1 PI P2 PI Pi 0 P2 0 P2 Rather than simply determine the structure of C1, 1(1R) we shall take this opportunity to demonstrate some general features of associative algebras. A basis is {1, e, f, z) where z = e Af = ef since e and f are orthogonal. The multiplication table is readily completed (see table 2.2). (For example, ez = eef = f since e is of unit norm.) Table 2.2 1 1 1 -z -1 -f —e 1 It is straightforward to see that the identity spans the centre. An immediate consequence of this is that C 1, 1(R) is not reducible. More generally, all Cp,I(R) have an identity. If the algebra were reducible TI-IE STRUCTURE OFTHEREAL CLIFFORD ALGEBRAS 29 Wewilldenote anorthonormal basis forVby{e‘,fl}fori=1,..., p,j=1,...,qwhere g(e‘, e‘)=—g(f/', fl)=1.Itwillbeconvenient IOSetZ=€1/(e2/\ ...€p/\flV/\ Afq. The two-dimensional algebra C0.1(R) has asbasis {1,f}where f2=—1.Itisthus isomorphic tothealgebra ofcomplex numbers, CO_1(lR) EC(lR). (2.2.1) Abasis forCLO(lR) is{1,e},andthisalgebra might notbeso immediately recognisable. IfP1=§(1+e)and P2=2(1 —e)then {P1, P2} isobviously anew basis. The multiplication table isgiven in table 2.1. Thus P,and P2each span mutually orthogonal one- dimensional subalgebras, each ofwhich isisomorphic tothefield IR,so that c,_,,(1R)=non. (2.2.2) Table 2.1 P, P2 P, P, 0 P2 0 P2 Rather than simply determine thestructure ofC1_,(1R) weshall take thisopportunity todemonstrate some general features ofassociative algebras. Abasis is{1,e,f,z}where zEeAf= efsince eandfare orthogonal. Themultiplication table isreadily completed (seetable 2.2). (For example, ez=eef=fsince eisofunitnorm.) Table 2.2 1 e f z N\v,t\i- N\qt\>—~\v,N>—~f\t'\1—iN\vs >—~t\\v,N Itisstraightforward toseethat theidentity spans thecentre. An immediate consequence ofthisisthat C,’1(lR) isnotreducible. More generally, allCp_q(]R) have anidentity. Ifthealgebra were reducible 30 CLIFFORD ALGEBRAS AND SPINORS then the identity would be the sum of the identities in the component algebras. The identities of the component algebras must all lie in the centre, so if an algebra with a unit element is reducible then the identity can be written as a sum of pairwise orthogonal central idempotents. Conversely if the centre of an algebra contains a set of mutually orthogonal idempotents then the algebra is reducible. Thus either C 1,1(1R) has a radical or it is simple. The multiplication table enables the two-dimensional Clifford algebras we have already encountered to be recognised as subalgebras. Both {1, el and {1, z) span subalgebras isomorphic to IRCTI, whereas the algebra spanned by {1, f) is isomor- phic to C(IR). We can use the pair of orthogonal idempotents in one of the IFICIIFI subalgebras to write C 'AIR) as a sum of two left ideals. For example, if P1 = (1 + z), P2 = (1. - Z) then C 1,1(IR) = C 11(E)P1 ± C 1,1(1R)P2. Since fPi= ePi and zPi = P1 a basis for the left ideal C11(I11)P1 is {/31, eP 1 }. Similarly a basis for C 1,1(IR)P2 is {P2, eP2}. It Is instructive to look at the multiplication table for the algebra in this basis (see table 2.3). Table 2.3 PI eP, P2 eP2 P1 131 0 0 eP2 eP, eP, 0 0 P2 P2 0 eP, P2 0 eP2 0 P1 eP2 0 The left ideals C 11(1R)P1 and C i j(IR)P2 are both minimal; they contain no smaller left ideals. So P 1 and P2 are primitivet idempotents, for if P 1 = P + Q where P and Q are orthogonal idempotents then C 1,1 (R)P 1 = C 1,1(IR)P + C 1,1(I11)Q. The sum must be a direct vector space sum. For suppose that bP = cQ for some b and c. Then since P is idempotent bP = bPP, but bPP = cQP = 0 since Q and P are orthogonal. Thus b = c = 0. So if P1 were not primitive C ij(IR)Pi would be a sum of two smaller left ideals. Could C "(R) contain any two-sided ideals? Suppose I is a two-sided ideal and that a E I. We can write a = al + a2 where a i € Ci j(IR) P1, a2 € C i JOR)P2. Now C 1 J(R)a 1 is a left ideal which is contained in the left ideal C 11(11)P1 since a1 is. But this left ideal is minimal and so C11(F)a1 = C i,i P I - Thus if a1 * 0 there is a b such that bell = P1 and so ba = P1+ ba2 t The notion of 'primitive idempotents' is discussed in (A11)—(A19) of App- endix A. 30 CLIFFORD ALGEBRAS ANDSPINORS then theidentity would bethesum oftheidentities inthecomponent algebras. The identities ofthecomponent algebras must alllieinthe centre, soifanalgebra with aunitelement isreducible then theidentity canbewritten asasum ofpairwise orthogonal central idempotents. Conversely ifthecentre ofanalgebra contains asetofmutually orthogonal idempotents then thealgebra isreducible. Thus either C{_x(lR) hasaradical oritissimple. The multiplication table enables the two-dimensional Clifford algebras wehave already encountered tobe recognised assubalgebras. Both {1,e}and{1,2}span subalgebras isomorphic toIREBIR, whereas thealgebra spanned by{1,f}isisomor- phic toC(18). Wecanusethepairoforthogonal idempotents inoneof theIRG-)lR subalgebras towrite Cx_x(lR) asasum oftwoleftideals. For example, ifPx= §(1+ 2),P2=§(1— z)then C1_,(lR)= CL,(1R)Px + C1_,(lR)P2. Since fPx=eP1and2P,=P1abasis fortheleftideal C1‘,(lR)P1 is{P2, eP,}. Similarly abasis forCx_x(lR)P2 is{P2, eP2}. It isinstructive tolook atthemultiplication table forthealgebra inthis basis (seetable 2.3). Table 2.3 P1 eP1 P2 eP2 P1 P1 0 0 CPZ eP| eP1 0 0 P2 P2 0 CP1 P2 0 CPZ 0 P] 6P2 0 The left ideals C1x(lR)P, and CH(lB)P2 areboth minimal; they contain nosmaller leftideals. SoP,andP2areprimitivef idempotents, forifPl=P+Qwhere Pand Qareorthogonal idempotents then CU(lR)P{ =Cu(lR)P +C,,1(lR)Q. The sum must beadirect vector space sum. Forsuppose that bP=cQforsome bandc.Then since P isidempotent bP= bPP, but bPP= cQP=0 since Qand Pare orthogonal. Thus b=c=0. SoifP,were notprimitive Cx‘1(lR)Px would beasum oftwo smaller leftideals. Could CH(lR) contain any two-sided ideals? Suppose Iisatwo-sided ideal andthat aeI.Wecan write a=ax+a2 where a,eCx_x(lB) P2, a2eCu(lR)P2. Now C{_x(lR)a, isaleftideal which iscontained intheleftideal Cx_x(lR)Px since a,is.But thisleftideal isminimal and soC,_2(lR)a1= CHPI. Thus ifaxE0there isabsuch that ba,=P,andsoba=P,+ba2 TThe notion of‘primitive idempotents’ isdiscussed in(A11)-(A19) ofApp- endix A. fi f2 1 f' f2 THE STRUCTURE OF THE REAL CLIFFORD ALGEBRAS 31 and baPi = Pl, which shows that P1 must be in I since a is. Similarly, there exists a c such that caPi= ePi, which must be in I. But from the multiplication table we see that right multiplying P1 and ePi by eP2 generates the remainder of the basis for the whole algebra. The situation is the same if we assume that a2 O. Thus the only ideals are the zero ideal and the algebra itself which is thus simple. Wedderburn's structure theorem, together with Frobenius's theorem on real division algebras, shows that the only simple four-dimensional associative algebras over the reals are the total matrix algebra At2(111) and the quaternions, H(E). The quaternion algebra is a division algebra whose only idempotent is the identity and so we must have C (R) AAR). (2.2.3) Of course we could have obtained this result directly, for if feu), i, j =1, 2 is an ordinary matrix basis for .42(I1:1) then a set of generators is {e, fl where e =- 12 + - a 21, f= e2 - e21. These generators anticom- mute and satisfy e2 = —f2 = 1. A basis for C0201:1) is {1, f1, f2, z) and the multiplication table is given in table 2.4. This may be recognised as the multiplication table of the standard basis for the quaternion algebra by relabelling f1 = f2 = j, z = k: CO32(1E1) -= H(1E1). (2.2.4) Table 2.4 1 fl f2 fi f2 -1 z _f2 -z -1 f, f2 _fl -1 CO33(R) is generated by an orthonormal basis for V, {A f2, f3} Since z = fif2f3 it will commute with these generators, and hence must lie in the centre. Furthermore, z2 = 1 and so P1 = 1(1 + z), P2 = 1(1 - Z) are a pair of orthogonal idempotents in the centre. Thus CO33(IFI) is reducible, CO33(E) = C0.3(E)PICCO33(E)P2. A basis for CO33(E) is {1, fi, f2, f3, fif2, f2f3, f3fi, z) and since zpi = PI, flf2pi = _f3p1, f2f3pi = flp i, f3flpi = -f2 P, a basis for CO3(1R)P1 is {P1, flP 1, f2P 1, f3/31). The resulting multiplication table is given in table 2.5. The identity in this algebra is P1. Again we have the quaternion algebra with a standard basis {P1, PP', f2P1, —f3Pi). The mapping 77 is an automorphism of CO33(E), but maps one THE STRUCTURE OFTHEREAL CLIFFORD ALGEBRAS 31 andbaP1 =P1,which shows that P1must beinIsince ais.Similarly, there exists acsuch that caP1 =eP1, which must beinI.Butfrom the multiplication table weseethat right multiplying P1and eP1 byeP2 generates theremainder ofthebasis forthewhole algebra. The situation isthesame ifweassume that a2=#0.Thus theonly ideals arethezero ideal and thealgebra itself which isthus simple. Wedderburn’s structure theorem, together with Frobenius’s theorem on real division algebras, shows that theonly simple four-dimensional associative algebras over thereals arethetotal matrix algebra A/t2(lR) andthequaternions, H(lB). Thequaternion algebra isadivision algebra whose only idempotent istheidentity andsowemust have C1_1(lR) =A/t2(lR). (2.2.3) Ofcourse wecould have obtained thisresult directly, forif{ep}, i, j=1,2isanordinary matrix basis forA/t2(lR) then asetofgenerators is {e,f}where e=e12 +621, f=e12—e21. These generators anticom- mute andsatisfy e2=—f2=1. Abasis forC11‘2(lR) is{1,fl,fl,2}and themultiplication table is given intable 2.4.This may berecognised asthemultiplication table of thestandard basis forthequaternion algebra byrelabelling fl=i, fl=j,z=k: C11‘2(1R) =H(1R). (2.24) Table 2.4 1 fl 1" Z 1 1 fl f2 2 fl fl _1 Z _fZ fl F -2 -1 1"2 Z fl -fl -1 s‘C11v1(lR) isgeperated byanorthonormal basis forV,{fl, fl,f3}. ince2=fffitwillcommute withthese generators, andhence must lie in the centre. Furthermore, 22=1and so P1= §(1+ 2),P2=§(1— 2)areapair oforthogonal idempotents inthe centre. Thus C0‘3(lR) isreducible, C0v3(lR) =C11_1(lFl)P1€)C11_3(lR)P2. A basis forC11v3(lR) is{1,fl,fl,f3,flfz, fzfl, flfl, 2}and since ZP1= P11flf2P1= -f3P1» f2flP1= —flP1» flflP1= -f2P1 =1basis forC11‘3(lR)P1 is{P1, flP1, flP1, f3P1}. The resulting multiplication table isgiven intable 2.5. The identity inthisalgebra isP1.Again we have thequaternion algebra with astandard basis {P1, flP1, flP1, —flP1}. The mapping 17isanautomorphism ofC11‘1(lR), butmaps one 32 CLIFFORD ALGEBRAS AND SPINORS component algebra into the other since rz = —z. It thus establishes an isomorphism between these component algebras and so CO33(IFI) =- H(IR)01 -/(lF1). (2.2.5) Table 2.5 pi PPI f3Pi PI Pi ppi f3Pi flPi PPi —f3Pi pp, f2pi f3Pi f3Pi PPi _f2pi —Pi It is unlikely that we will recognise the sixteen-dimensional algebra CO34(IF1) by writing out the multiplication table. An orthonormal basis for V {f', f2, f3, f4} generates the algebra. These generators mutually anticommute and square to minus one. If we can find a new set of generators that splits into two mutually commuting subsets then these subsets will generate mutually commuting subalgebras. If the product of the dimensions of these subalgebras is the dimension of C O34(I1i) then we can express that algebra as the tensor product of these subalgebras. Such a set is provided by {f, z, f 2f3, f3f4}. The first two elements certainly commute with the last two but we need to verify that they do indeed generate the algebra. We do this by checking that we can recover the original generators by forming sums of products of this new set. In fact, fizf2f3 = fa, f1zf3f4 = r2 f and so fizf3f4f2f3 = —f3 and, indeed, we have a new set of generators. The generators {P, zl mutually anticommute satisfying z 2 = (fl )2 = 1. They therefore gener- ate an algebra isomorphic to C j(11:1), that is .4 2(11:1). The anticommuting pair {f2f3;f3f4} both square to minus one, and so they generate the quaternion algebra. (In the standard basis we may choose {i, j) as generators.) Both At 2(IR) and H(1F1) are four dimensional and so we have CO34(E) H(R) ®AtAIR). (2.2.6) Of course, in a similar way, we could have quickly identified the structure of the algebras previously considered. It has been anticipated that a knowledge of some low-dimensional Clifford algebras will enable the structure of an arbitrary Clifford algebra to be determined. In fact, given that we know the structure of C1,1(E), C1,0(I1:1) and C 04(1R) for q = 1, 2, 3, 4 the following determine the structure of all the real Clifford algebras: Cp+i,q(Fi) = Cq+Lp(11) (2.2.7) 32 CLIFFORD ALGEBRAS AND SPINORS component algebra into theother since 172=—z. Itthus establishes an isomorphism between these component algebras andso c,,,(IR) =H(lR)®H(lR). (225) Table 2.5 P1 flP1 f2P1 flP1 P1 P1 flP1 f2P1 flP1 flP1 flpi -P1 “f2P1 .f2P1 f2P1 fZP1 f2P1 _P1 _flP1flP1 f’P1 —r1P1 f1P1 —P1 Itisunlikely that wewill recognise thesixteen-dimensional algebra C0_4(lR) bywriting outthemultiplication table. Anorthonormal basis for V{fl,f2,f3,fl} generates thealgebra. These generators mutually anticommute and square tominus one. Ifwecanfind anew setof generators that splits into two mutually commuting subsets then these subsets willgenerate mutually commuting subalgebras. Iftheproduct of thedimensions ofthese subalgebras isthedimension ofC0,4(lR) then we canexpress that algebra asthetensor product ofthese subalgebras. Such asetisprovided by{fl, z,f2f3, f3f‘l}. The first two elements certainly commute with thelasttwobutweneed toverify that they do indeed generate thealgebra. We dothis bychecking that wecan recover theoriginal generators byforming sums ofproducts ofthisnew set.Infact, flzf2f3 =f‘l, flzf3f‘l =f2 and soflzf3f‘lf2f3 =—f3 and, indeed, wehave anew setofgenerators. The generators {fl, z} mutually anticommute satisfying 22=——(fl)2 =1.They therefore gener- ateanalgebra isomorphic toC121(lR), thatisA/t2(lR). Theanticommuting pair {f2f3;f3f‘l} both square tominus one, and sothey generate the quaternion algebra. (Inthestandard basis wemay choose {i,j}as generators.) Both A/t2(lR) andH(lR) arefour dimensional andsowe have C0,4(lR) =H(lR) ®A/t2(lR). (2.2.6) Ofcourse, inasimilar way, wecould have quickly identified the structure ofthealgebras previously considered. Ithasbeen anticipated that aknowledge ofsome low-dimensional Clifford algebras will enable thestructure ofanarbitrary Clifford algebra tobedetermined. Infact, given that weknow thestructure of C121(lR), C1,0(lR) andC0p(lR) forq=1,2,3,4thefollowing determine thestructure ofalltherealClifford algebras: Cp+1_q(lR) ECp+1_p(lR) (2.2.7) THE STRUCTURE OF THE REAL CLIFFORD ALGEBRAS 33 cp,i,q+I(E) c c 1,1(E) (2.2.8) Cp, 0_4(1R) -= Cp,q(111)0C0.4(R). (2.2.9) Before demonstrating the truth of the above assertion we have to prove these relations. This will be done by choosing suitable generators. A set of generators for Cp+i,q(Fi) is provided by an orthonormal basis for V, {el, PI for i =1, . . p+1, j = 1, . . q. Alternatively, we could generate the algebra with {eP+1, eP+lei, eP+ 1fil, i =1, . p, j =1, q. This follows since we can easily recover the original generators from products of this set. The new generators are mutually anticommut- ing and for i =1, . . p, (eP+ ie )2 = ep+ieiep+tei =1 _(ep+1)2(ei)2 = _1; similarly (eP+1P)2 = 1. So we have a set of mutual- ly anticommuting generators, q + 1 of which square to plus one and p of which square to minus one and so (2.2.7) indeed holds. Cp.",q+I(IFI) is generated by {eP+1, et, fq+1, PI for i =1, . . p, j =1, . . q. A new set of generators are {eP+1, fq-f-1, eP+ifq+ifi) with i =1, . . p, j =1, . . q. (Although the notation assumes p 1 and q 1 the argument obviously goes through with p -= 0 or q = O.) We have only to verify that the original generators are recovered by products of the new set to be sure that they are indeed generators. The first pair of mutually anticommuting generators com- mute with the second mutually anticommuting pair. For i =1, p (ep+ifq+lei)2 epi-ifq+leiep+ifq+tei = (ep+1)2r-i eifq+iei = _(ep+1)2(fq+1)2(ei)2 = (et)2 = 1. Similarly (eP-1-Ifqi-lf))2 = _1. Thus the second pair of the set generate q(IFI), whereas the first pair obviously generate C 1,101=1). The product of the dimensions of these mutually commuting subalgebras is indeed the dimension of Cp." q+1(111) and we have proved (2.2.8). The proof of (2.2.9) proceeds in the same spirit. An orthonormal basis for V provides a set of mutually anticommuting generators for Cp, q+4(1F1). We partition the generators into two subsets, and form new generators out of the first subset and the elements of the second subset multiplied by the product of all the elements in the first set. If the first set is of even dimension, we will then have two mutually commuting subsets of generators. That is, we replace the generators {et, fifq+1, fq+2, fq+3, fq+4} i =1, . . p; j =1, q with the set ifi,fq+19fq+2,fq+3,fq+4} i =1, . . p; j = 1, . . q where î = fq-"fq+2P+3P+4. Then 27'1+1 = -f4+12, for example, and the last four generators commute with the first p + q. Since = THE STRUCTURE OFTHEREAL CLIFFORD ALGEBRAS 33 C,,+1,.,+1(1R) =Cp.q(B)®C1,1(B) (2-2-3) c,,_,,,(1P.) =cp,,(1R)®c,,,,(1n). (2.29) Before demonstrating thetruth oftheabove assertion wehave toprove these relations. This willbedone bychoosing suitable generators. Aset ofgenerators forCp+1_q(lP1) isprovided byanorthonormal basis forV, {(31,fl}fori=1, ..., p+l, j=1, ..., q.Alternatively, wecould generate thealgebra with {eP*l, eP*lel, eP*lfl}, i=1, ..., p,j=1, ...,q.This follows since wecaneasily recover theoriginal generators from products ofthisset.The new generators aremutually anticommut- ing and for i=1, ..., p, (eP*lel)2 = eP*lelePlle‘ = —(ePll)2(el)2 =-1;similarly (eP*lfl)2 =1.Sowehave asetofmutual- lyanticommuting generators, q+1ofwhich square toplus oneandp ofwhich square tominus oneandso(2.2.7) indeed holds. Cp+1_,,+1(lP1) isgenerated by{eP*l, e‘,f"*l, fl}fori=1, ..., p, j=1, ..., q.Anew setofgenerators are{eP*l, f‘1*l, eP*lf‘1"lel, eP*lf‘1llfl} with i=1, ..., p,j=1,..., q.(Although thenotation assumes pE1 and qE1 theargument obviously goes through with p=0orq=0.)Wehave only toverify thattheoriginal generators are recovered byproducts ofthenew settobesure that they areindeed generators. The first pair ofmutually anticommuting generators com- mute with thesecond mutually anticommuting pair. Fori=1,...,p (ep+lfq+1e[)2 :ep+1fq+1eiep+lfq+1ei =(ep+1)2fq+leifq+1ei :_(ep+1)2Urq+1)2(ei)2 =(ei)2 =1_ Similarly (eP*lf‘1*lfl)2 =-1.Thus thesecond pairofthesetgenerate Cp_q(IR), whereas thefirstpairobviously generate C1_1(lR). The product ofthedimensions ofthese mutually commuting subalgebras isindeed thedimension ofCp+1_q+1(lR) andwehave proved (2.2.8). The proof of(2.2.9) proceeds inthesame spirit. Anorthonormal basis forVprovides asetofmutually anticommuting generators for Cp_,,+,,(lPr). Wepartition thegenerators into twosubsets, andform new generators outofthefirst subset andtheelements ofthesecond subset multiplied bytheproduct ofalltheelements inthefirst set.Ifthefirst setisofeven dimension, wewillthen have twomutually commuting subsets ofgenerators. That is,wereplace thegenerators leivfjvfqflv .fq+2r.fq+31 .fq+4} :11 ~''1pi=1r '''1q with theset {2el,2fl,f‘1"l,f‘1"2,f‘1"3,f‘1"“} i=1,...,p;j =1,...,q where 2=f‘1*lf‘1*2f‘1*2f‘1*“. Then 2f‘1"l =—f‘l*l2, forexample, andthe lastfour generators commute with thefirst p+q.Since Eel=elf, 34 CLIFFORD ALGEBRAS AND SPINORS 1f) = fil for i = 1, . . p, j = 1, . . q and 12 = 1 we have C„ ,(11) and CO34(1F1) as mutually commuting subalgebras. The dimensions Of the algebras are such that we have proved (2.2.9). Of course we could equally well have shown that C p+4, q(1F1) Cp, (11)0C 4, 0(E). The low-dimensional examples and periodicity relations we have given have been judiciously chosen to enable the structure of an arbitrary Clifford algebra to be determined. We show first how the structure of Cp,q(11) can be determined assuming q > p. Repeated use of (2.2.8) gives Cp, =- Co, q_p(11)0C i(Fi)0 . . . p terms If we set q —p = 4A + m with m. < 4 then use of (2.2.9) shows that Cp, AR) = C o, (1E1) Co, 4(1E)0 .. . Co, 4(11)0C 1, 101:00 . . . A terms p terms Since we know the structure of all the C0 m(Ili) for m. < 4, we have expressed C as a tensor product of factors of known structure. Now we do the same thing assuming that p < q; by (2.2.8) Cp, (AIR) Cp-q, 0(I11)0C1 , I(E)0  OC 1(11). q terms Now we use (2.2.7) for the first time: Cp, q(11=1) Ci, p_q_1(11:1)01i, i(E) . . . C i, 1(11). q terms If p — q -= 1 or 2 then there is nothing left to do, and in the former case we will need our knowledge of the structure of C l, 0(IF1). If not then one more application of (2.2.8) gives Cp, q(lF1) -= Co, p_q_2(1F1)0C i(IR)0 . . . OC i(IR). q+1 terms If we set p — q — 2 = 4œ + )3, with 0 < 4 then (2.2.9) produces Cp, q(F3) Co, (lF1)0C O3 4(11)0 . . . 0C 0, 4(1)0C 1, i(IF1)10 a terms q+1 terms Again we have expressed the algebra in terms of products of algebras whose structures are known. So what are the possibilities for Cp, q(IR)? Since C1, (R) .4t 2(R) and At„,(E)att n(IFI) .ht,„ n(R), repeated tensor 34 CLIFFORD ALGEBRAS AND SPINORS 2fl=fli fori=1,...,p,j=1,..., qand 22=1wehaveCp7q(]R) andC0_4(lR) asmutually commuting subalgebras. The dimensions ofthe algebras aresuch that wehave proved (2.2.9). Ofcourse wecould equally well have shown thatCp+4, ,,(lR) ECp,,,(lR)®C4‘ 0(lR). Thelow-dimensional examples andperiodicity relations wehave given have been judiciously chosen toenable thestructure ofanarbitrary Clifford algebra tobedetermined. Weshow first how thestructure of Cp_q(]R) canbedetermined assuming q>p.Repeated useof(2.2.8) gives cp_,(I=i) =C0Y,,_p(lR)®C1_1(lR)® ...®c1,1(I=i). pterms Ifwesetq—p=4/I+14with 11<4then useof(2.2.9) shows that Cp,q(lR) 2C0.1((lR) ®C70.-r(lR)® ---C0,4(ll:l)®_C1,1(lR)® ---®C1,1(ll:l) /Iterms pterms Since weknow thestructure ofalltheC0_p(lR) for11<4,wehave expressed Cp_q(lR) asatensor product offactors ofknown structure. Now wedothesame thing assuming thatp<q;by(2.2.8) Cpv,,(lR) ECp_,,,0(lFl)®C1_1(lR)® ...®C1_1(lR). qterms Now weuse(2.2.7) forthefirsttime: Cp.q(lB) ZC1.p-q-1(lR)®C1.1(lR) ~--C1.1(lR)- qterms Ifp—q=1or2then there isnothing lefttodo,andintheformer case wewillneed ourknowledge ofthestructure ofC1_11(lR). Ifnotthen onemore application of(2.2.8) gives c,...,<1R>=c@.._.,_t<1R>®c.,.<1R>® ...®C1.1(lR)' q+1terms Ifwesetp—q—2=4a+B,with B<4then (2.2.9) produces c....,<1R> =c@,1(r1>®,c@..(r1>®... ®c@..<B>®c.. .<1R>®...®C1.1(B) ctterms q+1terms Again wehave expressed thealgebra interms ofproducts ofalgebras whose structures areknown. Sowhat arethepossibilities forCp_q(lR)? Since C1_1(lR) EA/t2(lR) andA/t,,,(lR)®JI/t,,(lP1) EA/t,,,,,(lP1), repeated tensor THE STRUCTURE OF THE REAL CLIFFORD ALGEBRAS 35 products of C1, 1(R) are isomorphic to a total matrix algebra. We have seen that C o, 4(R) H(1F1)0.M 2(1R), and since 1/(1R)OH(R) At(B) the tensor product of C0 ,4(R) an even number of times is isomorphic to a total matrix algebra, whereas an odd number of products produces the product of the quaternions and a total matrix algebra. So any Clifford algebra is either isomorphic to a total matrix algebra or isomorphic to the tensor product of C 0,0(R), 13 < 4, with either a total matrix algebra, or the tensor product of the quaternions and a total matrix algebra. In the former case equations (2.2.1), (2.2.4) and (2.2.5) show that Cp,I(Fi) = .9411)®-M.,(E) (2.2.10) where si C, H or HSH, and r2dim.s4 = 2P+q. Since C(IR)OH(R) --- C(111)0.#2(1B), and as we have already noted 1/(l11)0H(111) .M. 4(IR) the second case would lead to (2.2.10) with al = C, R or R + Ill. So any real Clifford algebra can be expressed as in (2.2.10) with .94 = R, C, H, ROE or HIGH. Since we know the dimension of the real Clifford algebras their structure is characterised by the algebra .s4. The possibili- ties for al show that the real Clifford algebras are either simple or semi-simple, in the latter case being the direct sum of two isomorphic simple components. Obviously the values of p and q determine .99 , in fact from (2.2.8) it can be seen that .94 is determined by p — q. Two applications of (2.2.9) give Cp, q+8(11l) Cp, q+4(1F)®CO3 4(E) = C p, q(R)0CO3 4(R)000. 4(R) Cp, q(11)0H(R)01t 2(1F1)0H(F1)0.4 2(1F1) (by (2.2.6)) thus Cp, 0.8(R) p — q mod 8. The low-dimensional algebras given in equations (2.2.1) to (2.2.6) provide examples of p — q mod 8 being 7, 1, 0, 6,5 and 4. So all that is missing is p — q mod8 equal to 2 and 3. From (2.2.7) we have C2,0(R) = C1 I(E) = 42(R) and C3, 0(R) CI, 2(R), and so by (2.2.8), C3,0(R) C1 l(F)OCO, i(E) hi2(R)0C(11). We now have the struc- ture of all the Clifford algebras, namely C p, q(R) .siakt where .9sl is given in table 2.6. Some of this table is easy to understand and remember. If p + q is even, then C is central simple, whereas for P + q odd the centre is spanned by the identity and z. If z2 = —1 then the centre must be C, and this will be the case if p — q mod 8 is 3 or 7. If z2 -= 1 then the centre is isomorphic to IFi3OR and the algebra is reducible. It can be checked that z 2 = 1 for p — q mod 8 equal to 1 or 5. The involution will induce an involution on the components of one of the reducible algebras if and only if z;.` = z. The only reducible Clifford algebras occur when V has odd dimension and in that case Z'1 = —z and so either or ij induce an involution on the simple components. At16(E)0Cp, q(E). So in fact si is determined by THE STRUCTURE OFTHEREAL CLIFFORD ALGEBRAS 35 products ofC1_1(lR) areisomorphic toatotal matrix algebra. Wehave seen that C111.,(lB) EH(lB)®M2(lB), and since H(lR)®H(lB) EM.,(lR) thetensor product ofC11_.,(lR) aneven number oftimes isisomorphic to atotal matrix algebra, whereas anoddnumber ofproducts produces the product ofthequaternions andatotal matrix algebra. SoanyClifford algebra iseither isomorphic toatotal matrix algebra orisomorphic to thetensor product ofC017,(lR), B<4,with either atotal matrix algebra, orthetensor product ofthequaternions andatotal matrix algebra. In theformer case equations (2.2.1), (2.2.4) and(2.2.5) show that c,,,(I=1) =s4(1R)®A/l,(1R) (2.2.10) where s4EC,HorHCBH, andr2dims4 =2””. Since C(lR)®H(lR) E C(lR)®M2(lR), andaswehave already noted H(lB)®H(lR) EA/l4(lR) the second case would lead to(2.2.10) with sd=C,IRorIR+IR.Soany realClifford algebra canbeexpressed asin(2.2.10) with s4=IR,C,H, IRCBIR orHCBH. Since weknow thedimension ofthereal Clifford algebras their structure ischaracterised bythealgebra sd.The possibili- tiesfors4show that thereal Clifford algebras areeither simple or semi-simple, inthelatter case being thedirect sum oftwo isomorphic simple components. Obviously thevalues ofpand qdetermine sd,in factfrom (2.2.8) itcanbeseen that s4isdetermined byp—q.Two applications of(2.2.9) give Cp,q+s(lR) ZCp,q+4(lB)®C0,4(lR) =Cp.q(lR)®C0,4(lR)®C0.4(lR) ECp‘,,(lR)®H(lR)®Jl1t2(lR)®H(lB)®M2(lR) (by(2.2.6)) thus Cp_,,+8(lR) EM16(lR)®Cp,,(lB). Soinfact s4isdetermined by p-qmod8.Thelow-dimensional algebras given inequations (2.2.1) to (2.2.6) provide examples ofp -qmod8 being 7,1,0,6,5and4.Soall thatismissing isp—qmod8 equal to2and3.From (2.2.7) wehave C2.0(lR) 2C1.1(lR) =Jl/l2(lR) andC3.0(lR) 2C1.2(lR)» and5°by(2-2-8), C3_11(lR) EC1_1(lR)®C0_ 1(lR) EA/t2(lR)®C(lR). Wenow have thestruc- ture ofalltheClifford algebras, namely Cp_,,(lR) Es4®Jt/t where sdis given intable 2.6. Some ofthis table iseasy tounderstand and remember. Ifp+qiseven, then Cp_p(lR) iscentral simple, whereas for P+qoddthecentre isspanned bytheidentity andz.If22=-1then thecentre must beC,andthiswillbethecase ifp—qmod8 is3or7. If22=1 then thecentre isisomorphic tolR(~9lR and thealgebra is reducible. Itcanbechecked that 22=1forp—qmod8 equal to1or 5.The involution Ewillinduce aninvolution onthecomponents ofone ofthereducible algebras ifand only if25=z.The only reducible Clifford algebras occur when Vhasodd dimension and inthat case Z5”=-25 and soeither §or§17induce aninvolution onthesimple Components. 36 CLIFFORD ALGEBRAS AND SPINORS Table 2.6 p — q mod8 0 , 2 3 , 7 4 , 6 H 1 Fi Fi 5 H C.) H Of paramount physical importance is the algebra C3, 01). From table 2.6 we see that C3, i(R) = .44,4(1R) and so the algebra admits an ordinary matrix basis {e,j} with i, j =1, ..., 4. It is instructive to construct such a basis. This construction provides a concrete example of Wedderburn's structure theorem for simple algebras. The identity is of rank four and first we seek a set of four pairwise orthogonal primitive idempotents. We seek an a and b which commute and square to one, for then taking all sign choices the set {1(1 ± a)1(1 ± b)) consists of pairwise orthogonal idempotents. For example, if (ea), a = 0, 1, 2, 3 is an orthonormal coframe with (e°)2 = —1 we choose a = set , b = e°2 and PI = 1(1 + e')(1 + 6,02) P2 = 1(1 ± e')(1 — 02) (2.2.11) P3 = — e')(1 + e°2) P4 = 14(1 — e1)(1 — e°2) where e°2 = e0 A e2. These four primitives are all similar, for example e3 Pi(e3)-1 = P3 e° P = P4 (2.2.12) e°3P1(e°3)-1 = P2. Thus, e° 3Pi C P2C3_1(R)Pi, e3Pi C P3C3,1(1R)Pi and e°P i C P4C3,1(1R)P1 and we set = = e°31', e3, = e3Pi e41 = e°Pi (2.2.13) 36 CLIFFORD ALGEBRAS ANDSPINORS Table 2.6 p—qmod8 .91 <§U)© O\\II\) 3165 U1r—* 3125Q90)3125 Ofparamount physical importance isthealgebra C2,1(lR). From table 2.6weseethat C3_1(lR)EA/t4(lR) and sothealgebra admits an ordinary matrix basis {e,~,-} with i,j=1, ..., 4.Itisinstructive to construct such abasis. This construction provides aconcrete example of Wedderburn’s structure theorem forsimple algebras. Theidentity isof rank fourandfirstweseek asetoffourpairwise orthogonal primitive idempotents. Weseek anaandbwhich commute andsquare toone, forthen taking allsign choices theset{2(1 i-a)2(1 1-b)}consists of pairwise orthogonal idempotents. Forexample, if{e“}, a=0,1,2,3is anorthonormal coframe with(e°)2 =-1wechoose a= =ell2and setW B‘ P1= P2= P2= P4=%(1 .l(1 %(1 %(1+ +el)(1 el)(1 el)(1 el)(1+ +602) 602) 602) 602)(22.11) where ell2=ellAe2.These fourprimitives areallsimilar, forexample e3P1(@3)—l =P3 e°P1(ell)'l =P4 (2.2.12) 603Pl(603)—1 =P2_ Thus, ell3P1 CP2C3,1(lR)P1, elP1 CP2C3_1(lR)P1 and e°P1 C P4C2_1(lR)P1 andweset 911=P1 921 : €03P1 (2.2.13) 931=e3P1 941: COP1. THE STRUCTURE OF THE REAL CLIFFORD ALGEBRAS 37 If the {e11} for j = 1, . ., 4 are given by = e 12 = (e°3) 1P2 e 13 = (e3) 1P3 e14 = (0)-1P4 (2.2.14) then e11 c P1C3, 1(}1)/); and eveil = Pp If now ei; = e 11 then the e do indeed form an ordinary matrix basis. The resulting e are tabulated in table 2.7. Table 2.7 e„ P1 e°3P2 e3P3 e°3P1 P2 e°P3 —e3P4 e3P1 —e°P2 P3 e°3P4 e°P1 —e3P2 e°3P3 P4 Any element of C3. i(IR) can be expanded in this basis. In particular, the orthonormal 1-forms can be written as = E ya e (2.2.15) where the arrays of components form a real representation (or Majo- rana representation) of the familiar Dirac 7-matrices. In principle we can determine these components from the formula = ebeaeik but it is here easier to proceed by inspection. From (2.2.11) = 131 + P2 — P3 — P4 so if the components 4 are arranged as a matrix: (70 = 1 0 0 , 0 1 0 0 0 0 —1 0 0 ) 0 0 —1 and again from (2.2.11) e02 pl p3 — p2 — p4 THE STRUCTURE OFTHE REAL CLIFFORD ALGEBRAS Ifthe{e17-} forj =1,...,4aregiven by 911=Pr e=(e°3)‘lP1’312 (2.214) 913=(9)_P3 914=(e0)_1P4 ll1CIl C P1C3' and =Pl. HOW =E,-181; ll1Cl‘l e,-,-doindeed form anordinary matrix basis. The resulting e,»,~are tabulated intable 2.7. Table 2.7 91'/-> l P1 e°lP2 e3P3 -e°P., e°3P1 P2 e°P; —elP., e3P1 —e°P2 P3 e°3P., e°P1 —e3P2 e°lP3 P4 Any element ofC3,1(lR) canbeexpanded inthisbasis. Inparticular, theorthonormal 1-forms canbewritten as 8“ : Y“,-,6,-,~ 1.} where thearrays ofcomponents form areal representation (orMajo- rana representation) ofthefamiliar Dirac y-matrices. Inprinciple we candetermine these components from theformula 1'3": 29/<r@“9jk k butitishere easier toproceed byinspection. From (2.2.11) €l=P1+P2—P3—P4 soifthecomponents y,l,-arearranged asamatrix: ©©©P—‘ ©©>—*© ©>—‘©© P-l©©©(Yij) : andagain from (2.2.11) €02=P1+P3—P2—P4 38 CLIFFORD ALGEBRAS AND SPINORS SO e2 = —e°Pi — e°P3 + e°P2 + e0P4 e — e 23 — e32 e 14 similarly e° = e 2 p _ e2p3 e2p2 e2p4 = e2e02p1 _ e2e02p3 e2e02p2 _ e2eo2p4 = e°Pi + e°P3+ e°P2 + e°P4 = e41 Thus we have (72,) (7?) = , = e23 — e32 — e o o o —1 0 0 —1 0 0 —1 0 0 —1 0 0 0 o o o —1 0010 0 —1 0 0 , 1 0 0 0 Writing e3 = e3 (P + P2 + P3 + P4) gives ,3 — e31 — e42 + e13 e24 and hence (70 = 0 0 0 —1 0010 1000 0 0 —1 0 Since the algebra C3 , 1(F1) is central simple, the transposition can be related to the involution by an inner automorphism, namely aT = C'a.=C Va E C3.1(E) (2.2.16) where C can be chosen such that C4 = ±C. The choice of a C in (2.2.16) is determined up to a multiple of the centre, and so we have no choice in the symmetry of C under For the basis given in table 2.7 we may take C = e2 e3 , and have C = —C. Since el, e2 and e3 commute with C their components will form symmetric matrices (as we have already seen). The components of C are related to the charge conjuga- tion matrix: exactly how will be seen in §2.8. In the above example of C3, I(R) the Clifford algebra was isomorphic 38 CLIFFORD ALGEBRAS AND SPINORS so e2=—ellP1 —ellP3 +ellP2 +ellP4 =—941 _923_932_914 similarly ell=—e2P1 —e2P3 +e2P2 +e2P4 :_62602P] _62602P3 _62602P2 _62602734 =ellP1 +e°P2 +e°P2 +ellP,, =941+ 923_932E914- Thus wehave J >-‘COO ©*—*©© ©©>—~© ©©®1—\(rt)= _ >-‘OOO @>—‘©@ ©©>—‘© OOOI-‘(291)=_ Writing e3=e2(P1 +P2+P3+P4)gives 93=931—942‘l’913—924 andhence ©*—'©© *—‘©©© ©®®>—- ©©>—‘©(ri,-)= _ Since thealgebra C2’1(lR) iscentral simple, thetransposition canbe related totheinvolution §byaninner automorphism, namely al=C'la5C VaeC3_1(lR) (2.2.16) where Ccan bechosen such that C5=iC. The choice ofaCin (2.2.16) isdetermined uptoamultiple ofthecentre, andsowehave no choice inthesymmetry ofCunder 5.Forthebasis given intable 2.7we may take C=ele2e3, andhave C5=-C. Since el,e2ande3commute with Ctheir components will form symmetric matrices (aswehave already seen). The components ofCarerelated tothecharge conjuga- tionmatrix: exactly how willbeseen in§2.8. Intheabove example ofC3_1(lR) theClifford algebra wasisomorphic THE STRUCTURE OF THE REAL CLIFFORD ALGEBRAS 39 to a total matrix algebra, generated by a real set of Dirac y-matrices. As a less familiar example we now consider C4.0(11) = Hati 2(F1). (Although many physicists will be used to working with y-matrices that satisfy the anticommutation relations with a positive-definite metric such matrices are always complex, generating the complexified Clifford algebra. This complexified algebra will be discussed in §2.7.) As usual z denotes the volume 4-form with here z2 = 1. Thus a pair of orthogonal primitive idempotents is given by 131= ;(1+ z), P2 = — z). Since P2 = elPiel we may choose a basis for /RAH) as follows: eq 131 elP2 elPi P2 {p1, e23pi, e34p1, e24p1} is a basis for PIC4, 0(IR)P1. This is a canonical basis for the quaternion algebra. Replacing P1 with P2 gives a basis for P2C4, 0(IR)P,. Thus, a quaternion subalgebra of C4. 0(1R) that commutes with all the e, is spanned by {1, e23, e34, e24). 2.3 The Even Subalgebra The Z2-gradation of the Clifford algebra ensures that elements of even degree form a subalgebra, C p+.q(IF1). That is, a E C4(1R) if and only if TN= a. Since V generates the Clifford algebra the 2-forms must generate the even subalgebra. However, a basis for 2-forms provides a set of generators with redundant elements, that is, a subset will generate the even subalgebra. If {e`, El for i =1, ..., p+1, j = 1, . q are an orthonormal basis with (e1)2 = —(P)2 = 1 then a set of generators, with no redundant members, for C7,,i.q(E) is {eP+1e1, eP+1P} for i =1, p, j =1, ..., q. Since, for example, el9+leteP+lei = - —e'el we see that products of this set produce a basis for 2-forms and so the set generates Cp++,4(11). It is not hard to see that there are no redundant generators. These generators are mutually anticommuting with (ep+let)2 = _1 and (eP+1P)2 =1 thus C q. p(1E1). (2.3.1) So if Cp. q(IFI) .910,4,. and C4(E) A0A,.. the algebra -A is obtained by relabelling table 2.6. Since dim Cp.q(IFI) = dim Cp.1(1R) it follows that r'2dim A = 2n-1 (see table 2.8). Whereas more than one value of p — q mod8 can give rise to the same .94 or A no combination of sti and A is repeated in table 2.8. An important example of the even subalgebra is provided by C1(11:1). From table 2.8 we see that this algebra is isomorphic to the algebra of THE sTRuCruRE OFTHEREAL CLIFFORD ALGEBRAS 39 toatotal matrix algebra, generated byarealsetofDirac ‘y-matrices. As aless familiar example we now consider C4_11(lR) EH®A/t2(lR). (Although many physicists willbeused toworking with ‘y-matrices that satisfy theanticommutation relations with apositive-definite metric such matrices are always complex, generating the complexified Clifford algebra. This complexified algebra willbediscussed in§2.7.) Asusual 2 denotes thevolume 4-form with here 22=1.Thus apair oforthogonal primitive idempotents isgiven byP1E§(1+2),P2E§(1—2).Since P2=elP1el wemay choose abasis forA/t2(lR) asfollows: eij —> ‘L P1 €1P2 elP1 P2 {P1, e23P1, e3‘lP1, e24P1} isabasis forP1C4_11(lR)P1. This isa canonical basis forthequaternion algebra. Replacing P1with P2gives a basis forP2C4_O(lR)P2. Thus, aquaternion subalgebra ofC4,11(lR) that commutes with allthee1,»isspanned by{1,e23,e34,e24}. 2.3TheEven Subalgebra The Z2-gradation oftheClifford algebra ensures that elements ofeven degree form asubalgebra, C,f_,,(lR). That is,aeC,j_,,(lR) ifand only if 17a=a.Since Vgenerates the Clifford algebra the 2-forms must generate theeven subalgebra. However, abasis for2-forms provides a setofgenerators with redundant elements, thatis,asubset willgenerate theeven subalgebra. If{el,fl}fori=1,..., p+1, j=1,..., qare anorthonormal basis with (el)2 =-(fl)2 =1then asetofgenerators, with noredundant members, forC,f,1_,(lR) is{eP*lel, eP*lfl} fori=1, ..., p,j=1,...,q.Since, forexample, e1’*leleP*le/'= —elel wesee that products ofthis setproduce abasis for2-forms and sotheset generates Cp,1v,,(lR). Itisnothard toseethat there arenoredundant generators. These generators are mutually anticommuting with (eP*lel)2 =-1and(eP*lf/)2 =1thus C7:+1.q(lR) 2 SoifCp_,,(lR) Es4®A/t, and C,f_,(lR) E9B®A1t,- the algebra 95is obtained byrelabelling table 2.6. Since dimC,f_,(lR) =§dim Cp_,,(lR) it follows that r’2dim% =2"'l (see table 2.8). Whereas more than one value ofp-qmod8 cangive risetothesame .24or93nocombination Of.24and97$isrepeated intable 2.8. Animportant example oftheeven subalgebra isprovided byC§_1(lR). From table 2.8weseethat thisalgebra isisomorphic tothealgebra of 40 CLIFFORD ALGEBRAS AND SPINORS Table 2.8 p—q mod 8 .54 .013 0 E FICAR 1 IRSIR IR 2 fi C 3 C H 4 H HOH 5 HOH H 6 H c 7 C E complex matrices of order two. The centre of the algebra, which is isomorphic to C, is spanned by {1, z} where, as usual, z = ele2e3e0. The involution leaves z invariant and so induces an involution on C1(E) which is similar to transposition. That is, if 4,13), œ, fi = 1, 2 is an ordinary matrix basis and the involution over C, t, is defined by = cpc, then there is a c E Cl1(1R) such that at = Va e CMIR) (2.3.2) with c = ±c. The element c is determined up to a multiple of the centre and so we can have only one of these signs. In fact it must be the minus sign since elements are invariant under if and only if they are in the centre, so c must be a 2-form. Thus, although similar, t and cannot be equivalent since c = —c. Equation (2.3.2) may be naturally extended to define t on the whole of C3, 1(E). If j is any odd regular element of C31(IF1) then the involution I, defined by al = jaj -1 Va EC i(IF1) (2.3.3) will induce an involution in CMFI). Since z/ -= —z this involution must be similar to Hermitian conjugation in CMFI). That is, if is the involution over IR in C1(IF1) defined by c o' = t13, then there is a b E CiAll) such that al = b 1atb V a EC1(11) (2.3.4) where ht = ±b. Since b is only determined up to an element of the centre, which is C, we can have either sign. This equation is naturally extended to define on C3,1(11). Equations (2.3.2) to (2.3.4) show that transposition and Hermitian conjugation in CMIR) differ by an inner automorphism of C3,1(111). This automorphism is not an inner auto- morphism of C1(11:1). We have = vazt)-1 (2.3.5) 40 CLIFFORD ALGEBRAS AND SPINORS Table 2.8 p—q mod8 at 913 \lO'\LJ\-J>bJI\J>—'OE ficnmgmfifiIR lR(-DIR IRC-DIR HC-DH complex matrices oforder two. Thecentre ofthealgebra, which is isomorphic toC,isspanned by{1,2}where, asusual, 2Eele2e2ell. The involution lg‘leaves zinvariant and soinduces aninvolution on C§'_1(lB) which issimilar totransposition. That is,if{sap}, a/,/3=1,2is anordinary matrix basis and theinvolution over C,t,isdefined by sap‘=app.thenthere isaceC§_1(lR) suchthat 2'=C_ld§C VaeC§_1(lR) (23.2) with c5=ic.Theelement cisdetermined uptoamultiple ofthe centre andsowecanhave onlyoneofthese signs. Infactitmust bethe minus signsince elements areinvariant under Eifandonly ifthey arein thecentre, socmust bea2-form. Thus, although similar, tand5 cannot beequivalent since cg=—c.Equation (2.3.2) maybenaturally extended todefine tonthewhole ofC2_1(lR). Ifjisanyoddregular element ofC3_1(lB) then theinvolution Q defined by allEja5j"l VaeC3_1(lR) (2.3.3) willinduce aninvolution inC§_1(lR). Since 24‘E-2thisinvolution must besimilar toHermitian conjugation inC§_1(lR). That is,iflisthe involution over IRinC§_1(lR) defined bysapl=£71,,then there isa beC§_1(lR) such that a5Eb'lalb VaeC§_1(lR) (2.3.4) where bl=ib.Since bisonly determined uptoanelement ofthe centre, which isC,wecanhave either sign. This equation isnaturally extended todefine ionC2_1(lB). Equations (2.3.2) to(2.3.4) show that transposition andHermitian conjugation inC§'_1(lR) differ byaninner automorphism ofC2_1(lB). This automorphism isnotaninner auto- morphism ofC§'_1(lR). Wehave alEva'v"l (2.3.5) THE EVEN SUBALGEBRA 41 where y = bjc. The inner automorphism of C3, i(IR), a—> vav-1, in- duces the involutary outer automorphism # on C1(11), where # complex conjugates the matrix components in the basis {to). It in fact follows that we can find a unit-norm 1-form x such that a# = xax Va E C1(IR) and xc = ex (2.3.6) for an appropriate choice of c in (2.3.2). For we know that a# = for some odd y, and since #2 = 1, V2 lies in the centre of C1(F1). Suppose that y = y + wz for the 1-forms y and w. Then u2 = y2 ± w2 (yw wy)z =y2 ± 2 W ± 2(y A W)Z. The first two terms are 0-forms, whilst the last is a 2-form, and so for w 0 we must have y = Aw, A E Fi. Thus y = (A — z)w, and since A — z is in the centre of the even subalgebra a# = waw-1 for all even a. Now y2 0 and so W2 * 0, so we have a# = xax-1 where x = wl(lw21)1/2, giving x2 = ±1. Since Eo# = co, x must commute with the matrix basis, giving = 0. Thus the to must lie in the even subalgebra of the orthogon- al complement to x, whereas CI.1(1F1) .4 2(1F1), C3t0(IFI) = H and so we must have x2 = 1. We can choose the c of (2.3.2) to lie in the subalgebra C 1(1R) and then xc = cx. We give an explicit example. A basis for C1(IF1) is {1, e", e02, e", e12, en, e31, z), where we use the previously introduced notation. In exactly the same way as we constructed a matrix basis for C3, I(IF1), we can construct the matrix basis given in table 2.9 for C1(FI) where P; = 1(1 + e°2) and — e°2). This matrix basis spans the even subalgebra associated with the vector space spanned by {e°, e2, e3). We may choose the c of equation (2.3.2) to be e23. The 1-form e1 commutes with the matrix basis and squares to one, and we may choose it to be the x of equation (2.3.6). This element can be used together with the primitives in the even subalgebra to form primitives in the full algebra. For example, if P1 = (1 + x)Pt, P2 = 1(1 + x).13, P3 = 1(1 - x)Pi1 and P4 = 1(1 X)P2 -1- then we have a set of pairwise orthogonal primitives of Cj, 1(F1). These are the primitives used to construct the matrix basis given in table 2.7. Notice that the involution that corresponded to transposition in that matrix basis induces Hermitian conjugation in the basis for the even subalgebra given here. Table 2.9 e03.13,' e°3Pi' THEEvEN SUBALGEBRA 41 where oEbjc.The inner automorphism ofC3_1(lR), a—>vav‘l, in- duces the involutary outer automorphism #onC§_1(lR), where # complex conjugates thematrix components inthebasis {sap}. Itinfact follows thatwecanfindaunit-norm 1-form xsuch that allExax VaeC§1(lR) andxcEcx (2.3.6) foranappropriate choice ofcin(2.3.2). Forweknow that ailEvav'l forsome oddv,andsince #2E1,v2liesinthecentre ofC§'_1(lR). Suppose that vEy+wz for the 1-forms yand w. Then 02Ey2+w2+(yw—wy)z Ey2+w2+2(yAw)z. The first two terms are0-forms, whilst thelastisa2-form, andsoforwE0wemust have yEAw,heIR.Thus vE(A—2)w, andsince A—2isinthecentre oftheeven subalgebra allEwaw‘l foralleven a.Now v2E0andso w2E0,sowehave all=xax'l where xEw/(|w2|)l’2, giving x2Ei1. Since £afl# Esap, xmust commute with thematrix basis, giving i,,e,,,pE0.Thus thesapmust lieintheeven subalgebra oftheorthogon- alcomplement tox,whereas C{1(lR) EA/l.2(lR), C§_0(lR) EHandsowe must have x2E1.Wecanchoose thecof(2.3.2) tolieinthe subalgebra C2'_1(lR) andthenxcEcx.Wegiveanexplicit example. Abasis forC§,1(lR) is{1,elll, e°2, e°3, el2, e23, e3l, 2},where We usethepreviously introduced notation. Inexactly thesame wayaswe constructed amatrix basis forC3,1(lR), wecanconstruct thematrix basis given intable 2.9forC§_1(lR) where Pf’E§(1+e°2) and P2E 2(l—e°2). This matrix basis spans theeven subalgebra associated with thevector space spanned by{e°,e2,e3}. Wemay choose thecof equation (2.3.2) tobee23.The 1-form elcommutes with thematrix basis andsquares toone, andwemaychoose ittobethexofequation (2.3.6). This element canbeused together with theprimitives inthe even subalgebra toform primitives inthefullalgebra. Forexample, if P1E2(1+ x)P1*, P2E2(1+ x)P2', P3E2(1— x)P1* and P4= 2(1— x)P2* then wehave asetofpairwise orthogonal primitives of C3_1(lR). These aretheprimitives used toconstruct thematrix basis given intable 2.7. Notice that theinvolution that corresponded to transposition inthat matrix basis induces Hermitian conjugation inthe basis fortheeven subalgebra given here. Table 2.9 $mg__) l P; e°3P§air: P; 42 CLIFFORD ALGEBRAS AND SP1NORS 2.4 The Clifford Group Those regular (that is, invertible) elements, s, such that SXS -1 E V VXEV (2.4.1) form the Clifford group, F. It is straightforward to see that they do indeed form a group. The vector representation of F, x, maps F into the group of automorphisms of the Clifford algebra: F Aut Cp, q(11:1) s x(s) where x(s)x = sxs -1. (2.4.2) Since 2g(x(s)x, x(s)y) = sxs -'sys-1 + sys-'sxs-1 = 2g(x, y) x clearly maps the Clifford group into the orthogonal group. If n is the dimension of V then the range of x depends on n. If n is even then Ar) = 0(p, q) (2.4.3a) whereas for n odd X(F) = SO(p, q). (2.4.3b) Let a be any orthogonal transformation on V. Then since V generates the Clifford algebra, a extends uniquely to an automorphism of the algebra, that is, we define cl(x ix2 . . . xp) = crx iax2 . . . axe. If n is even then the Clifford algebra is central simple and all automorphisms are inner, so in this case x(F) = 0(p, q). If n is odd then the centre is spanned by {1, z), the identity and the volume n-form. Clearly, any orthogonal automorphism that does not leave the volume n-form invariant cannot be inner. However, any automorphism that does leave the centre invariant is inner. For if Cp, q(E1) is simple all automorphisms over the centre are inner. If Cp, q(IFI) is not simple then it is the sum of two central simple components Cp. q(Fi) = Cp, q(11)PiCiCp, q(F0P2 where {P1, P2) are orthogonal idempotents that span the centre. If a is an orthogonal automorphism that leaves the centre invariant then it induces an automorphism on the simple components, and this must be an inner automorphism of the component algebras. That is, for any a, a(aP,)= S,(aP,),S7 1 where SS ïl = P„ the identity in Cp, q(E)P„ i = 1, 2. If S = S1+ S2 then S-1 = + S2-1, for SS-' = SiS-, + S2S-71 since S1S2-1 = S2S-, = 0 and P1 + P2 = 1. Now ua = cr(aP,) + a(aP 2) = S1aP1S1-1 + S2aP2S2-1 = (S 1 + S2)(aP1 + aP2)(S1 + S2)-1 = SaS-1. 42 CLIFFORD ALGEBRAS ANDSPINORS 2.4TheClifford Group Those regular (that is,invertible) elements, s,such that sxs'le V VxeV (2.4.1) form theClifford group, F.Itisstraightforward toseethat they do indeed form agroup. The vector representation ofT,)5,maps Finto the group ofautomorphisms oftheClifford algebra: )5:I'—-—> AutCp_,,(lR) s1-—> )5(s) where )5(s)x Esxs"l. (2.4.2) Since Zg(X(S)X, x(S)y) =Sxflsyfl +Syflsxfl =280.y) )5clearly maps theClifford group intotheorthogonal group. Ifnisthe dimension ofVthentherange of)5depends onn.Ifniseven then 2(F)=O(p,<1) (Z-4-3(1) whereas fornodd x(r)=SO(p, q). (2.4.3b) Let0beanyorthogonal transformation onV.Then since Vgenerates theClifford algebra, 0extends uniquely toanautomorphism ofthe algebra, thatis,wedefine 0(x1x2 ...xp)E0x10x2 ...oxp. Ifnis even then theClifford algebra iscentral simple andallautomorphisms areinner, sointhiscase)5(F)EO(p, q).Ifnisoddthen thecentre is spanned by{1,z},theidentity and thevolume n-form. Clearly, any orthogonal automorphism that does notleave thevolume n-form invariant cannot beinner. However, anyautomorphism that does leave thecentre invariant isinner. ForifCp_,(lR) issimple allautomorphisms overthecentre areinner. IfCp,,,(lR) isnotsimple then itisthesumof twocentral simple components Cp_,,(lR) ECpv,,(lR)P1®Cp, ,,(lR)P2 where {P1, P2}areorthogonal idempotents thatspan thecentre. If0is anorthogonal automorphism that leaves thecentre invariant then it induces anautomorphism onthesimple components, andthismust be aninner automorphism ofthecomponent algebras. That is,foranya, 0(aP,-) ES,~(aP,-)S,Tl where S,-S,-‘l EP,-,theidentity inCp_,,(lR)P,~, iE1, 2.IfSES1+S2then S‘lESf‘+S§l, forSS'l ES1S1'l +S2S§l since S1S2l ES2S1'l E0andP1+P2E1.Now 0aE0(aP1) +0(aP2) ES1aP1Sfl +S2aP2S2l =(S1 ‘l’S2)(l1Pl ‘l’aP2)(S1+ S2)_l :SaS_l. THE CLIFFORD GROUP 43 We have shown that for n odd any orthogonal automorphism that leaves the volume n-form invariant is inner, that is, x(F) = SO(p, q). Obviously the Clifford algebra does not transform irreducibly under the vector representation of F, the Z-homogeneous subspaces being preserved. In fact these spaces of p-forms carry irreducible representa- tions. It will be convenient to be able to express any element of the Clifford group in a standard form. To do this we firstly show how any element of the orthogonal group can be written in a standard form, as the product of reflections. Let y be a non-null (non-isotropic) vector with g(y, y) = a, a O. Then the reflection of x in the plane orthogonal to y is given by Syx = x — 2a -lg(x, y)y Vx€ V. (2.4.4) If we write g(x, y) x — y + r a where r is orthogonal to y then g(x, y) S x = r Y Y a so Sy indeed corresponds to the usual notion of a reflection. It is readily verified that reflections are orthogonal transformations, for g(Syx, Syx) = g(x, x) + 4a -2g(x, y)2g(y, y) — 4a-lg(x, y)g(x, y) = g(x, x). The following theorem has already been anticipated. Any orthogonal transformation of a finite-dimensional vector space with non-degenerate bilinear form is expressible as the product of a finite number of reflections. (2.4.5) The truth of this statement will be proved by induction on the dimension of the vector space V. Note firstly that any two vectors of the same non-zero length can be related by at most two reflections. For if g(x, x) = g(y, y) 0 and x — y is not null then 2g(x, x — y) Sx_yx = x (x y) g(x — y, x — y) 2[g(x, x) — g(x, y)] = x (x y) [g(x, x) + g(y, y) — 2g(x, y)] = x — (x — y) if g(x, x) = g(y, y) = Y. THE CLIFFORD GROUP 43 Wehave shown thatfornoddanyorthogonal automorphism thatleaves thevolume n-form invariant isinner, thatis,)5(T) ESO(p, q). Obviously theClifford algebra does nottransform irreducibly under thevector representation ofF,theZ-homogeneous subspaces being preserved. Infactthese spaces ofp-forms carry irreducible representa- trons. Itwillbeconvenient tobeable toexpress anyelement oftheClifford group inastandard form. Todothiswefirstly show how anyelement of theorthogonal group canbewritten inastandard form, astheproduct ofreflections. Letybeanon-null (non-isotropic) vector with g(y, y)E a,aE0.Then thereflection ofxintheplane orthogonal toyisgiven by SyxEx—2a'lg(x, y)y Vxe V. (2.4.4) Ifwewrite FM,“ where risorthogonal toythen s(x,y)SyxEr—T y soS,indeed corresponds totheusual notion ofareflection. Itisreadily verified thatreflections areorthogonal transformations, for g($,X- Syx)=g(X.X)+40‘2g(x. y)2g(y, y)—4@‘lg(X. y)g(x. y) Eg(x,x). Thefollowing theorem hasalready been anticipated. Anyorthogonal transformation ofafinite-dimensional vector space with non-degenerate bilinear form isexpressible asthe product ofafinite number ofreflections. (2.4.5) The truth ofthis statement will beproved byinduction onthe dimension ofthevector space V.Note firstly thatanytwovectors ofthe same non-zero length canberelated byatmost tworeflections. Forif g(x, x)Eg(y, y)E0andx—yisnotnullthen 2s'(X.X—y)S,,_. E —-i—i(x—y)lxXs(X—y-X-y) 2 1 _ v =X_ [s(XX)soy)l (X_y) [s(x.x)+g(y,y)—2s(x.y)l =X—(X—y) ifs(X-X)=g(y-y) Z y_ 44 CLIFFORD ALGEBRAS AND SPINORS If x — y is null then x + y cannot be since x and y are not. Then 2g(x, x + y) Sx±yx = x (x + y) = —y g(x + y, x + y) and so SySx+yx = —Syy = y. Suppose now that (2.4.5) is true for n-dimensional orthogonal spaces and that V is of dimension n + 1. If y is any non-null vector then its conjugate space (the space of all vectors orthogonal to y) is an n-dimensional orthogonal space (since g is non-degenerate). Furthermore, since y is non-null the restriction of the non-degenerate g to its conjugate is also non-degenerate. If a is any orthogonal transformation of V then, since it has the same length as y, ay can be transformed into y by the product of at most two reflections. That is, there exists a u which is a product of reflections such that uay = y. Since ua leaves y invariant it must transform the conjugate space into itself, that is it is an orthogonal transformation on this n-dimensional orthogonal space. By hypothesis then ua = y, where y is a product of reflections and so a = u-ly which is also a product of reflections. For n = 1 relation (2.4.5) is obviously true and so we have proved its general validity. As a step towards writing an arbitrary element of the Clifford group in a standard form we observe the following. If x E V and g(x, x) 0 then x E F and x(x) = nSx. (2.4.6) It is sufficient to show that x(x)y = —S1y for y E V since V generates the algebra. We have x(x)y = xyx-1 = {2g(x, y) — yx}x-1 = —y + 2g(x, y)x -1 and since x2 = g(x, x)* 0 then 2g(x, y) x-1 = and xyx-1 = y + x — g(x, x) g(x, x) Together (2.4.5) and (2.4.6) give a canonical form for any element of the Clifford group. If s E F then s = Ax' . . . xh where A is in the centre and the x' are non-isotropic vectors in V. (2.4.7) Suppose firstly that n is odd, and so if s E F, x(s) E SO(p, q). Since det S, = —1 (as is readily seen in a basis consisting of x and vectors from its orthogonal complement) it follows that x(x) can be written as an even number of reflections. If then x(s) = S1 . . . Sxh with h even, then x(s) = x(x1 . . . X"). The kernel of the vector representation is obviously the centre and so (2.4.7) follows. If n is even then n = x(z) where z is the volume n-form and Sx = x(zx). Since zx is a product of 44 CLIFFORD ALGEBRAS AND SPINORS Ifx—yisnullthen x+ycannot besince xandyarenot.Then SW,‘=x_£Wi(x+y): _y g(x+y,X+y) and soSyS,+yx =—Syy =y.Suppose now that (2.4.5) istrue for n-dimensional orthogonal spaces andthat Visofdimension n+1.Ify isanynon-null vector then itsconjugate space (the space ofallvectors orthogonal toy)isann-dimensional orthogonal space (since gis non-degenerate). Furthermore, since yisnon-null therestriction ofthe non-degenerate gtoitsconjugate isalso non-degenerate. If0isany orthogonal transformation ofVthen, since ithasthesame length asy, 0ycanbetransformed intoybytheproduct ofatmost tworeflections. That is,there exists auwhich isaproduct ofreflections such that uoy=y.Since uoleaves yinvariant itmust transform theconjugate space into itself, that isitisanorthogonal transformation onthis n-dimensional orthogonal space. Byhypothesis then uo=v,where vis aproduct ofreflections and so0=u'1v which isalso aproduct of reflections. Forn=1relation (2.4.5) isobviously true andsowehave proved itsgeneral validity. Asastep towards writing anarbitrary element oftheClifford group inastandard form weobserve thefollowing. IfxeVandg(x, x)450then xeFandX(x) =r7S,,. (2.4.6) Itissufficient toshow that)((x)y =—S,y foryeVsince Vgenerates thealgebra. Wehave x(x)y=xyx“ ={2g(X, y)—yX}X“ =—y+2g(x,y)X" andsince x2=g(x, x)450then -12X -1=_iwl =_x g(x, X) andxyx y+g(x, X)x S,y. Together (2.4.5) and (2.4.6) give acanonical form foranyelement of theClifford group. IfseFthens=Ax‘...x"where Aisinthecentre andthe x’arenon-isotropic vectors inV. (2.4.7) Suppose firstly that nisodd, andsoifseF,)((s)eSO(p, q).Since detS,=-1(asisreadily seen inabasis consisting ofxand vectors from itsorthogonal complement) itfollows that X(x) canbewritten as aneven number ofreflections. Ifthen )((s) =S,»...S,~with heven, then )((s) =)((x‘ ...x"). The kernel ofthevector representation is obviously thecentre andso(2.4.7) follows. Ifniseven then 17=95(2) where zisthevolume n-form andS,=X(zx). Since zxisaproduct of THE CLIFFORD GROUP 45 n — 1 non-isotropic vectors it follows that for any s E r, X(s) = xh), where h need not now be even, and so (2.4.7) again follows. If n is even then the Clifford algebra is central simple and so in this case elements of the Clifford group are even or odd. If F is the subgroup of F consisting of all elements that are either even or odd, then for n odd Fi is a non-trivial subgroup. When n is odd the vector representation maps the Clifford group onto the special orthogonal group and not the whole orthogonal group. The twisted vector repre- sentation is introduced to map F± onto 0(p, q) for n odd as well as even: : F Aut Cp, q(R) s cp(s) where cp(s)x = snxs -1 for x E V. (2.4.8) Notice that (2.4.8) gives the action of p(s) on elements of V by Clifford multiplication, and since V generates the algebra the action on the whole algebra is defined: cp(Fi) = 0(p, q). (2.4.9) If x is a regular element of V then X E 1-± and for y E V cp(x)y = —(x)y = Sty. Thus (2.4.9) follows from (2.4.5). If n is even then F± = F and if sq = s then cp(s)= x(s). If sq = —s then q)(s)x = —sxs-1 = szxz 1s1 = x(sz)x. The kernel of cp is the multiplicative group of non-zero real numbers, IR*. For if sqxs-1 = x VX E V and s is written in terms of even and odd parts as s = s + s_ we have s +x = xs+ and xs_ + s_x = 0 VX E V. The condition on the odd part of s is iis_ = 0 for all x and so s _ = 0. Thus s is in the even part of the centre which is JR*. (Sometimes the Clifford group is defined differently. It is defined to be the group G consisting of all regular s such that sqxs-I E V,Vx E V. It follows that G = The even elements in the Clifford group form a subgroup F+. In this case the 'twisted' representation and the vector representation coincide and we have X(r+) = SO(13, q). (2.4.10) It follows from (2.4.7) that if n is even and s E F+ then s = Axl . . . x" where Ac R and h is even. From (2.4.6) then x(s) = (-1)hSx,    Sxh which, since h is even, is an even number of reflections. Hence in this case x(F+) = SO(p, q). If n is odd then x(F+)C SO(p, q). It again follows from (2.4.7) that if s E F then x(s) = x(x' . xh) for some x1. If h were odd then x(x . . . xh ) = x(zx . . . xh ) where z is the volume n-form which, for n odd, lies in the centre. If h is odd then zx' . Xh is even and in F+ so x(r) -= x(F) = SO(p, q). THECLIFFORD GROUP 45 n—1non-isotropic vectors itfollows thatforanyse1",X(s) =X(xl ... xl‘),where hneed notnow beeven, andso(2.4.7) again follows. Ifniseven then theClifford algebra iscentral simple andsointhis case elements oftheClifford group areeven orodd. If1":-is the subgroup of1"consisting ofallelements that areeither even orodd, then fornodd1"’isanon-trivial subgroup. When rtisoddthevector representation maps theClifford group onto thespecial orthogonal group and notthewhole orthogonal group. The twisted vector repre- sentation isintroduced tomap I“:onto O(p, q)fornoddaswellas even: <p:1"1' i> AutC,,_q(lB) sii> q2(s) where <p(s)x =s"xs" forxeV. (2.4.8) Notice that (2.4.8) gives theaction ofq2(s) onelements ofVbyClifford multiplication, and since Vgenerates thealgebra theaction onthe whole algebra isdefined: fP(F’) =0(1)»q)- (Z4-9) Ifxisaregular element ofVthen xel“ and foryeVq2(x)y = —X(x)y =Sxy. Thus (2.4.9) follows from (2.4.5). Ifniseven then 1":=Fandifs"=sthen q2(s) =)((s). Ifs”=—s then q2(s)x =—sxs_l =szxz_‘s" =X(sz)x. The kernel oftpisthe multiplicative group ofnon-zero real numbers, lB*. For ifs"xs_1 =x VxeVandsiswritten interms ofeven andoddparts ass=s++s_ wehave s+x=xs+ andxs_+s_x=0VxeV.The condition onthe oddpart ofsisiis_ =0forallxandsos_=0.Thus sisintheeven partofthecentre which islR*.(Sometimes theClifford group isdefined differently. Itisdefined tobethegroup Gconsisting ofallregular s such thats"xs'1 eV,VxeV.Itfollows that G=Pi.) The even elements intheClifford group form asubgroup I“.Inthis case the‘twisted’ representation andthevector representation coincide andwehave X(r+)=SO(p,q). (24.10) Itfollows from (2.4.7) that ifniseven andseI“then s=Ax‘...x" where /le1Bandhiseven. From (2.4.6) then X(s) =(—1)"Sxi ...S,»- which, since hiseven, isaneven number ofreflections. Hence inthis case X(1"+) =SO(p, q).Ifnisodd then )((1"*)C SO(p, q).Itagain follows from (2.4.7) thatifseFthen X(s) =X(x‘ ...x")forsome xi.If hwere odd then X(x‘ ...x")=X(zx‘ ...x")where zisthevolume n-form which, fornodd, liesinthecentre. Ifhisodd then zx‘...x" iseven andin1“soX(I‘*) =)((1") =SO(p, q). 46 CLIFFORD ALGEBRAS AND SPINORS If s E F+ and n is even then s is a product of an even number of non-singular 1-forms whereas if n is odd, s can be written as a product of non-singular (n — 1)-forms. (2.4.11) The case of n even is taken care of by (2.4.7). For n odd we can write s = . . xh with A in the centre. Since s is even if h is even then c Fi and s = ±A(xlz) . . . (xhz). By redefining x' the factor of ±A can be absorbed. If h were odd then A would be proportional to the volume form, say s = pzxl . . . xh with p E R. Once more, s = -±p(zxl) . . . (zxh) and we have proved (2.4.11). The kernel of the 'twisted' representation (and the vector representa- tion for n even) is By suitably 'normalising' elements of F± we obtain a subgroup whose image under these representations is the same as that of F±, whereas the kernel is smaller. The norm homomorphism A is a group homomorphism: A : r-± Fi* s 4s) = (2.4.12) If s is invertible then so is s 4 with (.0)-' = (s-Y . If s E F then (sxs-1)'> = sxs1 Vx c V so (s-1).xs = sxs -1 or .s.s.x = xs=s. Since V generates the algebra ss lies in the centre. If s is even or odd then ss is even, and so A does map r-± into Fi*. It is straightforward to see that A(s is 2) = i)A(s2). We denote the subgroup of F± which consists of those elements whose norm is plus or minus one by „F±; the subgroup of unit norm elements „F±. We define „F+ and +F+ similarly. The group +r- is sometimes called PIN(p, q), ,F+ called SPIN (p, q) and „F+ called SPIN+(p, q). If s Eft then S/(14S)1) 1/2 E 4 and cp{s1(1)1.(s)()'9 = q2(s) and so indeed the image of „F± under cp is 0(p, q) and the kernel consists of the multiplicative group formed by plus and minus one, which is isomorphic to Z2. Similarly x( +F+) = SO(p, q) with kernel Z2. We can introduce a slightly different norm, p: s p(s) = s (2.4.13) Obviously p(s) = ±A(s) depending on whether s is even or odd and so the only new subgroup is the group of those s with p(s) = 1, + The various subgroups of F± that have been introduced can be arranged as follows: : ---> +F± +r+ i ±r± +r-± +r+. (2.4.14) ,F+ In this last diagram (2.4.14) the appropriate mathematical symbol 46 CLIFFORD ALGEBRAS AND SPINORS Ifsel"*andniseven then sisaproduct ofaneven number ofnon-singular 1-forms whereas ifnisodd, scanbewritten asaproduct ofnon-singular (n—1)-forms. (2.4.11) The case ofneven istaken care ofby(2.4.7). Fornoddwecanwrite s=/lxl...x”with /Iinthecentre. Since siseven ifhiseven then /is1Bands=i/l(x'z) ...(x"z). Byredefining x‘thefactor ofi/Ican beabsorbed. Ifhwere oddthen /Iwould beproportional tothevolume form, says=uzx1 x"with tie1B.Once more, s=i/.i(zx‘) ... (zx") andwehave proved (2.4.1l). The kernel ofthe‘twisted’ representation (and thevector representa- tion forneven) islB*. Bysuitably ‘normalising’ elements of1":we obtain asubgroup whose image under these representations isthesame asthat ofF1“,whereas thekernel issmaller. The norm homomorphism /Iisagroup homomorphism: /1:1“: ———> lB* s+——> /l(s) =s5s. (2.4.12) Ifsisinvertible then soiss5with (s§)" =(s“)5. IfseFthen (sxs‘l)5 =sxs” Vx6Vso(s_')*5xs§ =sxs"1 orsgsx =xsgs. Since V generates thealgebra s5sliesinthecentre. Ifsiseven oroddthen sgs iseven, andso/Idoes map F‘into lB*.Itisstraightforward toseethat M5152) =/l(51)/l(52)- Wedenote thesubgroup ofFtwhich consists ofthose elements whose norm isplus orminus onebyti“; thesubgroup ofunit norm elements +1"? Wedefine :1“and+1“similarly. Thegroup :1":issometimes called PIN(p, q),11“ called SPIN (p,q)and+1“ called SPIN*(p, q). IfseF1’then s/(I/l(s)])"2 e:1“: and <p{s/(I/l(s)[)”2} =<p(s) and so indeed theimage of1,1“: under (pisO(p, q)andthekernel consists of themultiplicative group formed byPlus and minus one, which is isomorphic toZ2.Similarly )((i,l“+) =SO(p, q)with kernel Z2. Wecanintroduce aslightly different norm, MI /4:F:———> lB* sI-—> /4(s) =s§"s. (2.4.13) Obviously ti(s) =i/l(s) depending onwhether siseven oroddandso theonly newsubgroup isthegroup ofthose swith u(s) =1,*1"? The various subgroups ofI":that have been introduced can be arranged asfollows: E———> +1“: E $41 ,r*E—-—> +r*%l> ,r+. (24.14)2->:r+s Inthis last diagram (2.4.14) theappropriate mathematical symbol THE CLIFFORD GROUP 47 here for 4-- is and for ---> is Here denotes that „F+ is a normal subgroup of +F±. This is certainly the case, for if a e ,F+ and SE +1-± then (sus')' = sas -' since sq = ±s and 2(sas-1) = 1 since /1.(s) = ±1. If we look at all (four) quotients modulo ,F+ this gives all (seven) quotients obtainable from this diagram. For example, ±F±/,F+ r±/ r+ , , ' Firstly consider ,F±/,F+. If there are no odd elements of unit norm then obviously ,F± ,F+, so assume that a is odd with 4a) = 1. If s_ is any odd element in ,F± then s_ = (s _cr -l)a, where s_a-1 is even with norm plus one so that s_ u. Similarly if s + is any even element s, — 1 and so +r±i+r+ is the multiplicative group of plus and minus one, isomorphic to Z2. The argument above applies in exactly the same way to +F±/„F+ and ,F+/,F+. In the general case ,F± will contain even elements with norms plus and minus one, +y+ and _y+, and odd elements with both norms, +y and _y-. It readily follows that ,F±/,1-+ has four elements [„y+], [4+], [+y-] and [_ y-1. Each element is labelled by an ordered pair of indices which take the values plus or minus one. The multiplication rule is defined by multiplying the values of these indices pairwise, and so +Fiv,s+ Z2 X Z2. In various special cases this quotient group can have less than four elements as will be made clear in the following. The kernel of cp from 1—± to 0(p, q) is the group of plus and minus one, Z2, which is contained in all the subgroups in (2.6.14), and so the kernel of 4p restricted to these subgroups is the same. Thus, for example (p(F) F±/Z2 r± cp(+F±) +r-±/z2 „F± We have already determined the images of ,_F± and ,F+ under (p, and now turn to the unit-norm subgroups. If x is a non-singular element of V then op(x) = Sx and A(x) = g(x, x). So the image of unit-norm elements of F± under go contains an even number of reflections in planes orthogonal to negative length, `timelike', vectors. Such orthogonal transformations are said to be `orthochronous'; the subgroup of orthochronous transformations being denoted 0 I (p, q). For x E V, p(x) = —g(x, x) and so the unit 0-norm elements have images in the orthogonal group containing an even number of reflections in planes orthogonal to positive length, spacelike, vectors. Such orthogonal transformations will be called 'parity preserving' and the subgroup denoted 0+(p, q). If elements of SO(p, q) are orthochro- nous then they must also be parity preserving and so the notation SO+(p, q) is unambiguous. The following summarises the images of the various subgroups under cp: THE CLIFFORD GROUP 47 here for<i isii andfor———> isll. Here +I“"il:I“1' denotes that +1“ isanormal subgroup of:1“? This iscertainly thecase, forif ae+1“ and seti": then (sas"1)" =sas” since s"=is and /1(sas'1) =1since /1(s) =il.Ifwelook atall(four) quotients modulo +1“ thisgives all(seven) quotients obtainable from thisdiagram. For example, 1' +,,.,,,.Z +1":/+1“ Firstly consider +I‘*/+I"'. Ifthere arenooddelements ofunit norm then obviously +1“ =+1“, soassume that0isoddwith /1(0) =1.Ifs_ isanyodd element in+1“: then s_—(s_a'1)a, where sac" iseven with norm plus onesothat s_~0.Similarly ifs+isanyeven element s+~1andso+1“:/+I"" isthemultiplicative group ofplus andminus one, isomorphic toZ2.The argument above applies inexactly thesame wayto"I":/+I“* and1.1""/+1“. Inthegeneral case 11"’ willcontain even elements with norms plus andminus one, ,1)/" and _y*, andoddelements with both norms, +y' and _y'. Itreadily follows that 1:1“/.,l"" hasfour elements [+y"], [_y+], [+y'] and[_y']. Each element islabelled byanordered pair of indices which take thevalues plus orminus one. The multiplication rule isdefined bymultiplying thevalues ofthese indices pairwise, and so :1“:/+1“ =Z2XZ2.Invarious special cases thisquotient group can have lessthan four elements aswillbemade clear inthefollowing. The kernel oftpfrom I“:toO(p, q)isthegroup ofplus andminus one, Z1,which iscontained inallthesubgroups in(2.6.14), andsothe kernel oftprestricted tothese subgroups isthesame. Thus, forexample <P(F*) ~F:/Z1 ~1" <P(+F*) +1":/Z1 J‘ Wehave already determined theimages of:1“: andI1“ under tp,and now turn totheunit-norm subgroups. Ifxisanon-singular element ofVthen <p(x) =S,and/l(x) =g(x, x).Sotheimage ofunit-norm elements of1"‘under rpcontains aneven number ofreflections inplanes orthogonal tonegative length, ‘timelike’, vectors. Such orthogonal transformations aresaidtobe‘orthochronous’; the subgroup oforthochronous transformations being denoted Ol(p, q).ForxeV,ti(x) =—g(x, x)andsotheunitti-norm elements have images intheorthogonal group containing aneven number of reflections inplanes orthogonal topositive length, spacelike, vectors. Such orthogonal transformations will becalled ‘parity preserving’ and thesubgroup denoted O+(p, q).Ifelements ofSO(p, q)areorthochro- nous then they must also beparity preserving and sothenotation SO+(p, q)isunambiguous. The following summarises theimages ofthe various subgroups under rp: 48 CLIFFORD ALGEBRAS AND SPINORS +F± 0(p, q) +Ft 0 1(p, q) +r± 0+(p, q) (2.4.15) H-F+ SO(p, q) +F+ SO+(p, q). If the dimension of V is even then the image of the Clifford group under x is the same as under cp. If q is even then the volume form is of unit norm, A(z) = ,u(z) = 1. As has already been noted if s is an even element of F then x(s) = cp(s), whereas if s is odd x(s) = cp(sz). Since, for q even, X(sz) = X(s) and p(sz) = p(s) the images of the subgroups under x are the same as under cp. If, however, q is odd then X(sz) = --)1(s) and p(sz) = —j(s) and thus for s odd il(sz) = p(s) and ,u(sz) = /1.(s). So in this case x(j-±-) = o±(p, q) and x(+F±) = 01 (p, q). The groups 01(p, q) and 0+(p, q) have been identified with subgroups whose elements contain an even number of reflections in timelike and spacelike planes respectively. (A timelike (spacelike) plane is the conjugate of a timelike (spacelike) vector.) The nomenclature reflects -the fact that these groups preserve the timelike and spacelike orientations of V in a way that will now be defined. Let V be written as a direct sum of a p-dimensional positive-definite orthogonal space and a q-dimensional negative-definite conjugate space, V = POQ. If O c 0(p, q) then we define a linear mapping on P: m(a): P --> P x m(a)x = where P,a denote the projections onto the subspaces P and Q. This mapping must be one-to-one, for if m(a)x = 0 then ax E Q and since a is an orthogonal transformation x must be zero. Thus det m(a) O. If det m(a) > 0 then a will be said to preserve the spatial orientation of V. Of course for this definition to make sense it is necessary to verify that this criterion does not depend on the particular orthogonal decomposi- tion of V chosen. If x1, X2 E P then g(xi, m(a)x2) = g(xl,(axi)) = g(x i, ax2) = ax2) = x2) = g(3'(a -ixi), x2) = g(m(a -1)x1, x2). So if m(a)t denotes the adjoint map, with respect to the induced 48 CLIFFORD ALGEBRAS AND SPINORS (P 1F: Z) O(p, (I) (P +r: i) OT(p1 (P +F: Z> 0+(p, q) (2.415) (P 1-IMF i>sO(P~ (I) (P +F*—>50+(P, q)- Ifthedimension ofViseven then theimage oftheClifford group under Xisthesame asunder qa.Ifqiseven then thevolume form isof unit norm, 1(2) =/.i(z) =1.Ashasalready been noted ifsisaneven element ofFthen ;((s) =qJ(s), whereas ifsisodd;5(s) =qJ(sz). Since, forqeven, }.(sz) =}.(s) and/.i(sz) =/.i(s) theimages ofthesubgroups under Xare the same asunder qa.If,however, qisodd then }.(sz) =—}.(s) and/.i(sz) =—/.i(s) and thus forsodd }.(sz) =/.i(s) and /.i(sz) =}.(s). Sointhiscase;5(+l“¢) =O+(p, q)and)5(*I“*) =OT(p,q). The groups OT(p,q)and O+(p, q)have been identified with subgroups whose elements contain aneven number ofreflections in timelike andspacelike planes respectively. (Atimelike (spacelike) plane istheconjugate ofatimelike (spacelike) vector.) The nomenclature reflects thefactthat these groups preserve thetimelike and spacelike orientations ofVinawaythatwillnow bedefined. LetVbewritten as adirect sum ofap-dimensional positive-definite orthogonal space anda q-dimensional negative-definite conjugate space, V=P®Q. If 0eO(p, q)then wedefine alinear mapping onP: m(o) :P———> P xl-—> m(o)x =(3P(ox) where 97>,92denote theprojections onto thesubspaces Pand Q.This mapping must beone-to-one, forifm(0)x =0then oxeQandsince 0 isanorthogonal transformation xmust bezero. Thus detm(0) #=0.If detm(0) >0then 0willbesaidtopreserve thespatial orientation ofV. Ofcourse forthisdefinition tomake sense itisnecessary toverify that thiscriterion does notdepend ontheparticular orthogonal decomposi- tionofVchosen. Ifx1,x2ePthen g(X1, 171(0)/X2) =g(Xi, (3P(UX2)) =glxi» UX2) =g(UU_lX1> OX2) :g(U~lXi, X2)=g(9>(U—1X1), X2)=g(m(U_1)X1, X2)- Soifm(o)’ denotes theadjoint map, with respect totheinduced Syx = x y — x g(Y, Y) m(S,)x = x 2g(x, u) g(Y, Y) y 2g(x, u) g(Y  Y) u. for x E P 2g(x, y) THE CLIFFORD GROUP 49 positive-definite orthogonal metric on P, we have m(a)( = m(a -1). Since reflections are involutary the linear transformation associated with a reflection is symmetric. It is thus diagonalisable with determinant the product of the eigenvalues. If y is non-singular then y = u + v where u, are in P and Q respectively and There are p — 1 linearly independent vectors in P orthogonal to u and these are obviously eigenvectors of m(Sy) with eigenvalues one. A basis of eigenvectors is completed by u, with m(Sy)u = (1 2g(u, u ))4 (g(v, v) — g(u, u)) u g(Y, Y) g(Y Y) det m(S) — g(v, v) — g(u, u) y g(Y, Y) The numerator is negative-definite and so reflections in timelike planes preserve spatial orientation. Any orthogonal transformation is a product of reflections and it will preserve a spatial orientation if it contains an even number of reflections in spacelike planes. This criterion obviously does not depend on any particular orthogonal decomposition of V. In exactly the same way any orthogonal transformation induces a linear transformation on the negative-definite space Q. If the determinant is positive then the orthogonal transformation is called time-orientation preserving, or orthochronous. Such transformations contain an even number of reflections in timelike planes. The orthogonal group has (in general) four disconnected pieces containing 1, P, T and PT respectively. Here P(T) denote transforma- tions which change the spacelike (timelike) orientation whilst perserving the timelike (spacelike) orientation. The component containing the identity is a subgroup as is the sum of that component with any other component. thus (2.4.16) Dl ( : :  ISO.(p,q) Ot(p,q) (p,q) r PT I THE CLIFFORD GROUP 49 positive-definite orthogonal metric onP,wehave m(0)' =m(0"). Since reflections areinvolutary thelinear transformation associated with areflection issymmetric. Itisthus diagonalisable with determinant the product oftheeigenvalues. Ifyisnon-singular then y=u+vwhere u, vareinPandQrespectively and 2 7 2 7 Sy,=,_i<:L>,,=,_i,, fmxep g(y,y) g(y,y) m(S,,)x =x—M u.' g(y,y) There arep—1linearly independent vectors inPorthogonal tou andthese areobviously eigenvectors ofm(Sy) with eigenvalues one. A basis ofeigenvectors iscompleted byu,with thusdetmgy) 2go.4)—go.14> g(y.y) ' The numerator isnegative-definite andsoreflections intimelike planes preserve spatial orientation. Anyorthogonal transformation isaproduct ofreflections anditwillpreserve aspatial orientation ifitcontains an even number ofreflections inspacelike planes. This criterion obviously does notdepend onanyparticular orthogonal decomposition ofV.In exactly thesame way any orthogonal transformation induces alinear transformation onthenegative-definite space Q.Ifthedeterminant is positive then theorthogonal transformation iscalled time-orientation preserving, ororthochronous. Such transformations contain aneven number ofreflections intimelike planes. The orthogonal group has(ingeneral) four disconnected pieces containing 1,P,TandPTrespectively. Here P(T) denote transforma- tions which change thespacelike (timelike) orientation whilst perserving thetimelike (spacelike) orientation. The component containing the identity isasubgroup asisthesum ofthat component with anyother component. _________\ l_ L @.@+.....\- ' 0 \ 0 uunnonunan--0-nnununonnncanon -.-“...:/-.2:Q.Lfl.-/ L____l\ 0 \ \\ I \\ (2.4.16)I/ml \:\ ‘\5\\ x 00000000000 \\\ sot/2.4) \ \ \_1 i i _ 50 CLIFFORD ALGEBRAS AND SPINORS The Clifford group is in fact a Lie group, and its Lie algebra can be identified with a subspace of the Clifford algebra, the Lie bracket being the Clifford commutator. The regular representation maps the Clifford algebra into a total matrix algebra, thus the group of all regular elements, Cp*,q(F1), and hence the Clifford group and its subgroups are all subgroups of some general linear group. The general linear group is certainly a Lie group and the charts of this group induce charts on C ' p*q(Fi) and I' which give them a manifold structure. The exponential map is defined on the Clifford algebra in the obvious way a" exp a = 2, — a E Cp,q(B). (2.4.17) n=o n! Since the Clifford algebra is isomorphic to a subalgebra of a total matrix algebra where the exponential map can be defined, the limit implicit in this definition does indeed exist. Since exp (—a) = (exp a)-1 the exponential maps the Clifford algebra into the group of all invertible elements, Cp*,q(1E1). Thus the vector space of the Clifford algebra with the product of Clifford commutation can be identified with the Lie algebra of Cp*,q(F1). With this identification the vector representation of Cp*,q01:1), x, is seen to map the group into the automorphism group of the Lie algebra; this corresponds to the adjoint representation of Cp*,q(111), Ad. Similarly if we define ad : C p,q (IFI) --> End C p,q (E) a 1—> ad a (2.4.18) where (ad a)b = [a, b] with the bracket denoting a Clifford commu- tator, [a, 13]= ab —ba, then ad is the adjoint representation of the Lie algebra of Cp*4(11=1). The Clifford group is a Lie subgroup of the group of all invertible elements and its Lie algebra must be a vector subspace of the Clifford algebra. Suppose that m is in the Lie algebra of I", then exp (Am)x exp (—Am) C V V x E V, VA E Fi. (2.4.19) The standard group theory result, Ad exp (Am) = exp (ad Am), shows that this can hold for all A if and only if the Clifford commutator of m with x is in V. This can be seen directly by defining, for fixed m and x, the Clifford-algebra-valued function f(A) = exp(Am)xexp (—Am). We then have df(A)/dA = exp (Am)[m, x] exp (—Am) and more generally d"f(A)IdA" = exp (Am) (ad m)"x exp (—Am). 50 CLIFFORD ALGEBRAS AND SPINORS The Clifford group isinfactaLiegroup, anditsLiealgebra canbe identified with asubspace oftheClifford algebra, theLiebracket being theClifford commutator. The regular representation maps theClifford algebra into atotal matrix algebra, thus thegroup ofallregular elements, Cf,_q(R), andhence theClifford group anditssubgroups are allsubgroups ofsome general linear group. The general linear group is certainly aLiegroup and thecharts ofthis group induce charts on C§,,,(lB) andFwhich give them amanifold structure. The exponential mapisdefined ontheClifford algebra intheobvious way °°an expa=2W a6CM(lB). (2.4.17) n=0 ' Since theClifford algebra isisomorphic toasubalgebra ofatotal matrix algebra where theexponential map canbedefined, thelimit implicit inthisdefinition does indeed exist. Since exp(—a) =(exp a)‘1 theexponential maps theClifford algebra intothegroup ofallinvertible elements, C;§‘q(lB). Thus thevector space oftheClifford algebra with theproduct ofClifford commutation can beidentified with theLie algebra ofC;“,vq(lB). With thisidentification thevector representation of C§',,(lB), X,isseen tomap thegroup into theautomorphism group of theLie algebra; this corresponds tothe adjoint representation of C§,,,(lB), Ad.Similarly ifwedefine ad:CM(lB) -—> EndCM(lB) ai—> ada (2.4.18) where (ada)b=[a,b]with thebracket denoting aClifford commu- tator, [a,b]=ab—ba, then adistheadjoint representation oftheLie algebra ofC;,q(lB). The Clifford group isaLiesubgroup ofthegroup ofallinvertible elements anditsLiealgebra must beavector subspace oftheClifford algebra. Suppose thatmisintheLiealgebra ofF,then exp(/1m)x exp(—Am) CV VxeV,VA61B. (2.4.19) The standard group theory result, Adexp(km) =exp(adkm), shows thatthiscanhold forallAifandonlyiftheClifford commutator ofm with xisinV.This canbeseen directly bydefining, forfixed mandx, theClifford-algebra-valued function f(/1) =exp(/1m)x exp(—/1m). Wethen have df(/1)/d/1 =exp(/lm)[m, x]exp(-/lm) andmore generally d"f(/1)/d/1" =exp(Am) (adm)"x exp(—Am). THE CLIFFORD GROUP 51 By expanding f(A) in a Taylor series about A -= 0 it can easily be seen that f(A) E V VA if and only if [Fri, E V. If m is written in terms of even and odd parts, m -= m, + m_, then, from (2.3.7), for m to be in the Lie algebra of F we must have x A M_ =0 IiM+ E V VX e V. If the dimension of V is even then the odd part of m must be zero, whilst if the dimension is odd m_ can be an n-form, which is then in the centre. The even part of m has to be a sum of 0-forms and 2-forms. The exponential of an even element will be even whilst the one- parameter subgroup generated by the volume form for n odd will consist of elements that are in general neither even nor odd. Thus the Lie algebra of 1"-± consists of the ;n(n — 1) 2-forms and the identity. Since (expAm)t = exp Am we must have m = —m if m is in the Lie algebra of +F±, similarly rOn -= —m if m is in the Lie algebra of +F±. Thus the Lie algebra of these groups is the commutator algebra of the 2-forms. The exponential map sends the Lie algebra into that component of the group which is connected to the identity. This connected component is a subgroup, so products of exponentials are also connected to the identity. Conversely, every element of that component of the group which is connected to the identity can be written as a finite product of exponentials. Since 2-forms are even under n and odd under the exponential maps the Lie algebra of +1-± into ,F+, and so this must contain the component of „F± connected to the identity. We will now demonstrate that, except for one exceptional case, +F+ is a connected group. If s E ,F+ then s = crxix 2 . . x2", with a E IR* and the x' non- singular elements of V. By suitably scaling a , we can obviously arrange that x`2 = Ei = ±1. Then the norm of s is given by /1,(s) = X(a)E 1 . . . E2h , and so if s E „r+ we must have an even number of negative-norm vectors and a, = ±1. The negative-norm elements can be collected at the left-hand side, for if Ei = 1 and E'+1 = —1 then we write xixi+i (xixi+ixi)xi ix,i+i where (x')2 = xixi+Ixixixi+Ixi = —1. The overall factor of plus or minus one can be absorbed by redefining x1 and thus if s c +r+ s = a1cr2 ah where each a can be written = xy x, y E V with x2 = y2 = ±1. (2.4.21) Thus every element of ,r+ will be connected to the identity if and only if all such products of vectors are. If y = ±x then a = ±x2 and so for ,r+ to be connected —1 must be connected to +1. For an indefinite (2.4.20) THE CLIFFORD GROUP 51 Byexpanding f(/1) inaTaylor series about A=0itcaneasily beseen thatf(/1)eVV/1ifandonly if [m,x]eV. Ifmiswritten interms ofeven andodd parts, m=m++m_, then, from (2.3.7), formtobeintheLiealgebra ofFwemust have x,\m_ =0 _ (2.420) 1,;m+eV VxeV. Ifthedimension ofViseven then theodd part ofmmust bezero, whilst ifthedimension isoddm_canbeann-form, which isthen inthe centre. Theeven part ofmhastobeasum of0-forms and2-forms. The exponential ofaneven element will beeven whilst theone- parameter subgroup generated bythevolume form forrtodd will consist ofelements that areingeneral neither even norodd. Thus the Liealgebra ofF1consists ofthe§n(n —1)2-forms and theidentity. Since (exp/lm)§ =exp/1m‘? wemust have m5=—m ifmisintheLie algebra of+1“, similarly m5”=—mifmisintheLiealgebra of"T". Thus theLiealgebra ofthese groups isthecommutator algebra ofthe 2-forms. The exponential map sends theLiealgebra into that component of thegroup which isconnected totheidentity. This connected component isasubgroup, soproducts ofexponentials arealso connected tothe identity. Conversely, every element ofthat component ofthegroup which isconnected totheidentity canbewritten asafinite product of exponentials. Since 2-forms areeven under 17and odd under Ethe exponential maps theLiealgebra of+1“ into +1“, andsothismust contain thecomponent of1.1"" connected totheidentity. Wewillnow demonstrate that, except foroneexceptional case, +1” isaconnected group. Ifse+1” then s=axlxz x2", with orelB*and thexinon- singular elements ofV.Bysuitably scaling orwecanobviously arrange that xiz=s‘=i1.Then thenorm ofsisgiven by/l(s) =,1(0z)s‘ ... .92",andsoifse+1“ wemust have aneven number ofnegative-norm vectors anda=1-1.Thenegative-norm elements canbecollected at the left-hand side, for if2‘=1and 2”‘=-1 then we write xixi+l :(xixi+lxi)xi EX/ix/[+1 Where (x'i)Z =xixi+lxixixi+lxi =_1‘ The overall factor ofplus orminus onecanbeabsorbed byredefining xlandthus ifse+1“ s=0102 ...0"where each 0canbewritten 0=xy x,yeVwith xl=yz=i1. (2.4.21) Thus every element of+1” willbeconnected totheidentity ifandonly ifallsuch products ofvectors are. Ify=ixthen 0=ixz andsofor +1“ tobeconnected -1must beconnected to+1.Foranindefinite 52 CLIFFORD ALGEBRAS AND SPINORS metric in two dimensions the Lie algebra of ,F+ is spanned by the volume 2-form z, where z2 = 1. In this case exp(az)exp(fiz) = exp[(a + 13)z] V a,i6 e Fi, so if —1 were connected to +1 we would in fact be able to write it as an exponential. However, exp(Oz) = cosh 0 + z sinh 0, and so exp (0z) —1 for any O. Thus in this case ,F+ is not connected. Ruling out this exceptional case we always have a pair of orthogonal vectors a, b with a2 = b2 = ±1. So (ab)2 = —1 and since exp(rab)= —1 the identity is connected to minus one by a one-parameter subgroup. We still need to show that a general a is connected to the identity. We consider three cases. Suppose firstly that x and y are linearly independent, spanning an orthogonal plane with positive- or negative-definite metric. Then we have an orthonormal basis {x, u) where x2 = u2 = E, E = ±1. Since y2 = x2 we can write y -= cos Ox + sin Ou, and xy = E(cos 0 + sin Osxu) = Eexp(E0xu). We have already shown that —1 is connected to +1 and so xy is also connected to +1. If x and y span a non-degenerate orthogonal plane with orthonormal basis {x, ul with x2 -= —u2 = E then (xu)2 = 1. Now we must have y ----- cosh Ox + sinh 0 u and xy = E(cosh 0 + sinh 0 Exu) = Eexp (E0xu). Again the fact that —1 is connected to the identity ensures that all such products xy are. If x and y span an isotropic plane then we let (x, u) denote a basis in which u is an isotropic vector orthogonal to x. Then y = ±(x + Ou) and xy = ±s(1 + 0Exu). Since xu is nilpotent we have xy = -±eexp(0Exu) and, since —1 is connected to +1, we have demonstrated that xy is connected to the identity. We have shown that +F÷ is a connected Lie group unless V is two-dimensional with indefinite metric. Thus save for this exceptional case +F+ is a connected double covering of that component of the orthogonal group which is connected to the identity, and it follows from the topology of the orthogonal group that ,F+ is simply connected. In suitably low dimensions it is particularly easy to identify the spin groups, due to the following: If dim V 5 then if s" = ±s and s = ±s-' then s E +F±. (2.4.22) All we need to check is that if x e V then sxs-I C V for such an s. If we set x' = sxs-i then x"I= —x' and = x' if x E V and s is even or odd under both i and In five or fewer dimensions the only elements that are both odd under ti and even under are linear combinations of 52 CLIFFORD ALGEBRAS AND SPINORS metric intwo dimensions theLiealgebra of+1“ isspanned bythe volume 2-form z,where zz=1.Inthis case exp(az) exp(Bz) = exp[(a +,3)z] Va,/ielfi, soif-1were connected to+1wewould in fact be able towrite itasan exponential. However, exp(6z) =cosh0 +zsinh 0,and soexp(9z) ab-1forany 0.Thus in this case +1“ isnotconnected. Ruling outthis exceptional case we always have apairoforthogonal vectors a,bwith a2=b2=1-1.So (ab)2 =-1andsince exp(1rab) =-1theidentity isconnected tominus onebyaone-parameter subgroup. Westillneed toshow thatageneral 0isconnected totheidentity. Weconsider three cases. Suppose firstly that xand yarelinearly independent, spanning an orthogonal plane with positive- ornegative-definite metric. Then we have anorthonormal basis {x,u}where x2=uz=s,s=i1. Since yz=x2 we can write y=cosBx+sinBu, and xy= s(cos6 +sinBsxu) =sexp(s6xu). Wehave already shown that -1is connected to+1andsoxyisalsoconnected to+1. Ifxandyspan anon-degenerate orthogonal plane with orthonormal basis {x,u}with x3=-uz =2then (xu)2 =1.Now wemust have y=cosh Bx+sinh6u and xy=s(cosh6 +sinh6sxu) =sexp(s6xu). Again thefactthat -1isconnected totheidentity ensures that allsuch products xyare. Ifxandyspan anisotropic plane then welet{x,u}denote abasis in which uisanisotropic vector orthogonal tox.Then y=i(x+6u)and xyé1-s(l +Osxu). Since xuisnilpotent wehave xy=isexp(6sxu) and, since -1isconnected to+1, wehave demonstrated that xyis connected totheidentity. We have shown that +1“ isaconnected Liegroup unless Vis two-dimensional with indefinite metric. Thus save forthisexceptional case +1“ isaconnected double covering ofthat component ofthe orthogonal group which isconnected totheidentity, anditfollows from thetopology oftheorthogonal group that+1“issimply connected. Insuitably lowdimensions itisparticularly easy toidentify thespin groups, duetothefollowing: IfdimV <5then ifs"=isands5=is“ then se:1"? (2.4.22) Allweneed tocheck isthat ifxeVthen sxs" CVforsuch ans.If wesetx’=sxs“ then x’'1=-x’andx’5=x’ifxeVandsiseven or oddunder both 17and5.Infiveorfewer dimensions theonly elements thatareboth oddunder 17andeven under Earelinear combinations of THE CLIFFORD GROUP 53 1-forms and 5-forms. So if n < 5 the result follows immediately. If n 5 then the 5-form is in the centre of the algebra, so if x' = a + b where a is a 1-form and b a 5-form then x'2 = a2 + b2 + 2ab. Now a2 and b2 are both 0-forms whereas ab is a 4-form. But since x2 is a 0-form and x' = sxs-I then x'2 is a 0-form and thus ab = 0, that is either a = 0 or b = 0. However, a cannot be zero since an inner automorphism cannot take an element that is not in the centre into the centre, and so b = 0 and (2.4.22) follows. Needless to say (2.4.22) does not go through in six dimensions. For example, if {e} i = 1, . . ., 6 is an orthonormal basis for a positive- definite orthogonal space and s = (1/V1)(e12 e3456,, ) where e'2 = ele2 etc, then sg = s and s s-1. However, sels-1 = _e23456 and so Da. The results of this section will now be illustrated by considering the algebra C In this case r-± = F. An orthonormal basis for V is {e") a = 0, 1, 2, 3 where —(e°)2 = (e92 = 1. Let P = e°, T = 6'123 then Te°7-1 = —e° Pe°P -1 = e° (2.4.23) = e' Pe'P-1 = —e' i = 1, 2, 3. The norms of these elements are easily seen to be /1.(P) = —1, p(P) = 1, yl(T)= 1 and te(T) = —1. So Pe +I- whereas T E ,r. Suppose now that 5' E +F such that s7 = s 1, yl.(s ,) = —1. Then s = IPTXPT) -1 and (sIPT)q = s 'PT, A(siPT) = 1 thus sl= ai(PT) where a l E +F+. Similarly if s2 E +F such that s',1 = —s2, A(s2) = 1, then s2 = a2 T; and if S3 E r such that 53 = —S3, 453) = 1, then s3 = a3P, (32,03 E r+. We know that the six 2-forms generate +F+, the Lie bracket being a Clifford commutator. If we take a product of two spacelike 1-forms, for example el2, then (e12)2 = —1 and exp(Oe12\ ) -= cos 0 + sin 0 e'2. Such elements thus generate rotations, and the Clifford commutators are seen to give the familiar Lie algebra of the rotation group, [e12, e23] = 2e13 etc. Elements such as e01 generate 'boosts', with (e°1)2 = 1 giving exp (oeoi) = cosh 0 + sinh Oe°1. The commutator of two boosts gives a rotation, for example [e°', e°2] = 2e12. The remaining structure con- stants are determined by looking at the commutator of a boost with a rotation, for example [ew, = 2e° 2. The group +F+ can be recog- nised as a matrix group by using (2.4.22). This result shows that +1-+ is the group of unit-norm regular elements of C1(11=1). In §2.3 it was shown that the even subalgebra was isomorphic to the algebra of all complex two by two matrices, and so +F+ must be the subgroup of G1(2, C) consisting of unit-norm elements. Since we have already explicitly constructed a matrix basis for CME) it can be directly verified that the norm corresponds to the determinant. If {cm3) is the basis given in THE CLIFFORD GROUP 53 1-forms and 5-forms. Soifn<5theresult follows immediately. If n=5then the5-form isinthecentre ofthealgebra, soifx’=a+b where aisa1-form andba5-form then x’2=a2+b2+2ab. Now a2 and b2areboth O-forms whereas abisa4-form. But since x2isa O-form and x’=sxs" then x’2isaO-form and thus ab=O,that is either a=0 orb=0. However, acannot bezero since aninner automorphism cannot take anelement that isnotinthecentre into the centre, andsob=Oand(2.4.22) follows. Needless tosay(2.4.22) does notgothrough insixdimensions. For example, if{el} i=1,...,6isanorthonormal basis forapositive- definite orthogonal space ands=(1/\/2)(el2 +(23456), where e12=elez etc,then s"=sands5=s'l. However, seals“ =—e23“5(’ andsos¢I‘. The results ofthissection willnow beillustrated byconsidering the algebra C3_1(lPr). Inthiscase F1’=F.Anorthonormal basis forVis {e”} a=0,1,2, 3where —(e°)2 =(e‘)2 =1.LetP=e°,T=emthen Te°T" =—e° Pe°P" =e° (2.4.23) Te'T'1 =el P€iP_l =—€l i=1,2,3. Thenorms ofthese elements areeasily seen tobe}.(P) =-1,it(P) =1, )l(T)=1 and ;i(T)= -1. SoPe+1‘whereas Te +1".Suppose now that sle ,1‘such that s{'=sl,)l(s,) =-1. Then s,=(s,PT)(PT)'1 and(s]PT)" =s1PT, }.(s1PT) =1thus s1= a,(PT) where 0,e+1“. Similarly ifs2e,1‘such thats§=-s2, )l(s2) =1,then s2=UZT; andif s3eifsuch thats§'=-s3, }.(s3) =-1,then s3=a3P, 02,03 e+1“. Weknow that thesix2-forms generate +1“, theLiebracket being a Clifford commutator. Ifwetake aproduct oftwospacelike 1-forms, for example e12, then (e‘2)2 =-1and exp(0e‘2) =cos0 +sin0 e12. Such elements thus generate rotations, andtheClifford commutators areseen togive thefamiliar Liealgebra oftherotation group, [el2, e23]=2e13 etc. Elements such asemgenerate ‘boosts’, with (e°‘)2 =1giving eXp(0e°‘) =cosh0 +sinh0e°1. Thecommutator oftwoboosts gives a rotation, forexample [e°', e°2]=2e”. The remaining structure con- stants aredetermined bylooking atthecommutator ofaboost with a rotation, forexample [e°‘, en]=2e°2. The group .,I"* canberecog- nised asamatrix group byusing (2.4.22). This result shows that +1“ is thegroup ofunit-norm regular elements ofC§f,(lB). In§2.3 itwasshown thattheeven subalgebra was isomorphic tothealgebra ofallcomplex (W0bytwo matrices, and so+1“ must bethesubgroup ofGl(2, C) Consisting ofunit-norm elements. Since wehave already explicitly ¢0nstructed amatrix basis forC§'_,(lB) itcanbedirectly verified that the norm corresponds tothedeterminant. If{sag} isthebasis given in 54 CLIFFORD ALGEBRAS AND SPINORS table 2.9 then E E = 22, -(12> '621 ;'‘ = F 21 and y22 = EH. So if 2 S = ES11 ya, rt,13= I then ss- = s 12s21)(Eti E22) = det (s)1. Thus in this four-dimensional Lorentzian case, +F+ S1(2, C), the group of complex matrices of order two with unit determinant. The group of matrices with determinants of plus or minus one is obviously isomorphic to ,r+, ,F+ ,S1(2, C). The PIN group, ,F, is obviously a subgroup of G1(4, IR). However, since we have identified +F+ as a matrix group it is convenient to identify +r as a product of this matrix group with a discrete subgroup. We have already seen how any element of „r can be written as a product of an element of +1-+ with either 1, P, T or PT. Since P2 = T2 = —1 these elements do not form a subgroup and so it is convenient to introduce a unit-norm 1-form, x say, so that {1, x} form a subgroup, Q, isomorphic to Z2. If s is any element of +I" then it can be uniquely written as s = at, GE ,r+ and t E Q. The multiplication of two elements s1 and s, is given by s1s2 = c1t1a2t2 = 01t1a2t1-111 t2 = ai{X(ti)a7}tit2 Now x(ti) acts on +r+ as an outer automorphism and x(Q) = Q. We can equivalently write elements of „F as an ordered pair of an element of ,F+ and an element of Q with the multiplication defined by (ai, t2) = (a1X(t1)a2, t1t2)- In this form ,F is recognised as a semidirect product of ,F+ and a Z2 group of automorphisms, ,F =- ,r+oz2. We have shown that +F+ ,SI(2, C) and the gener- ator of the automorphism group, x, sends a to xax. As was discussed in §2.1 we can always choose such an x, which complex conjugates the matrix components and thus ,F ,S1(2, C)C)Z,, where the auto- morphism group is generated by complex conjugation. 2.5 Spinors From the irreducible representations of the Clifford algebra and its even subalgebra we obtain irreducible representations of the Clifford group: the spinor representations. It should be noted that minor variations exist in the literature as to the precise nomenclature for these representa- tions. The regular representation maps the Clifford algebra into its endo- morphism algebra; that is, into the algebra of linear transformations on 54 CLIFFORD ALGEBRAS AND SPINORS tabl€ Ihfin £115 = £22, £125 = “£12, £215: —£21and £22: Z£11. SO Z S=2 Sal; fa‘; 0.13:1 then $35=(511522 -Sl2S2l)(£ll +322)=d(‘3((5)1- Thus inthis four-dimensional Lorentzian case, +1“ =Sl(2, C), the group ofcomplex matrices oforder twowith unitdeterminant. The group ofmatrices with determinants ofplus orminus one isobviously isomorphic to11“, 11“=1Sl(2, C).The PIN group, :1‘,isobviously asubgroup ofGl(4, IR).However, since wehave identified :1“ asa matrix group itisconvenient toidentify :1“asaproduct ofthismatrix group with adiscrete subgroup. Wehave already seen how anyelement of:1“canbewritten asaproduct ofanelement of+1“ with either 1, P,TorPT. Since P2=T2=-1these elements donot form a subgroup andsoitisconvenient tointroduce aunit-norm 1-form, xsay, sothat {1,x}form asubgroup, Q,isomorphic toZ2.Ifsisany element of:1"then itcanbeuniquely written ass=at,oeil“ and teQ.Themultiplication oftwoelements s1ands2isgiven by 5152 =01t102(2 =0ltlU2tl~ltlt2 =Ul{X(tl)U2}tl(2- Now )((r,) acts on11“ asanouter automorphism and)((Q) =Q.We canequivalently write elements of1.1“asanordered pair ofanelement of:1“ and anelement ofQwith the multiplication defined by (0,, t1)(02, I2)=(01;((r1)02, tltz). Inthisform :1“isrecognised asa semidirect product of:1“ and aZ2group ofautomorphisms, :.I“= iI“*@Z2. Wehave shown that :1“ =iSl(2, C)andthegener- ator oftheautomorphism group, x,sends 0toxax. Aswasdiscussed in §2.1 wecanalways choose such anx,which complex conjugates the matrix components and thus iI“= iSl(2, C)@Z2, where theauto- morphism group isgenerated bycomplex conjugation. 2.5Spinors From theirreducible representations oftheClifford algebra anditseven subalgebra weobtain irreducible representations oftheClifford group: thespinor representations. Itshould benoted thatminor variations exist intheliterature astotheprecise nomenclature forthese representa- tions. The regular representation maps theClifford algebra into itsendo- morphism algebra; that is,into thealgebra oflinear transformations on SPINORS 55 the vector space structure of the Clifford algebra. This representation will not be irreducible; certain vector subspaces will be preserved under multiplication from the left, namely the left ideals. It is a truism to say that the minimal left ideals transform irreducibly under the regular representation. If the Clifford algebra is simple then the regular repre- sentation induces a faithful representation on any minimal left ideal. The mapping into the endomorphism algebra of any _minimal left ideal induced by the regular representation is called the spinor representation of the simple Clifford algebra and the minimal left ideal is called the space of spinors. The choice of a different minimal left ideal gives another equivalent representation. When the Clifford algebra is not simple it is the sum of two simple component algebras, and any minimal left ideal must lie in one of these simple components. The regular representation of a non-simple Clifford algebra induces a faithful repre- sentation on the left ideal which is the sum of two minimal left ideals, one lying in each simple component. The mapping into such an endomorphism algebra induced by the regular representation will be called the spinor representation of the non-simple Clifford algebra, and such an ideal will be termed the spinor space. The minimal left ideals will be termed semi -spinor spaces and the mapping that the regular representation induces on a minimal left ideal will be called the semi -spinor representation of the Clifford algebra. The kernel of such a representation is obviously the simple component algebra that does not contain the semi-spinor space. Thus the spinor representation of a non-simple Clifford algebra is reducible, being the sum of two inequi- valent semi-spinor representations. The spinor representation of the Clifford algebra induces a representation of any subset by restricting to left multiplication on the ideal by elements of that set. In particular it induces a representation of the Clifford group. Irreducible representations of the Clifford algebra induce irreducible representations of the Clifford group. (2.5.1) That is, the spinor representation of a simple Clifford algebra, or the semi-spinor representation of a non-simple one, induces an irreducible representation of the Clifford group. This will also be called the spinor or semi-spinor representation. The proof of the statement follows immediately from the observation that non-singular vectors generate the Clifford group and the Clifford algebra. In fact the Clifford group could be replaced with the subgroup ,r, and the statement would obviously still be true. If an irreducible representation of the Clifford algebra induces a reducible representation of the even subalgebra then that induced representation is the sum of two irreducible ones. For suppose that I is a minimal left ideal of the Clifford algebra that splits into invariant SPINoRs 55 thevector space structure oftheClifford algebra. This representation willnotbeirreducible; certain vector subspaces willbepreserved under multiplication from theleft, namely theleftideals. Itisatruism tosay that theminimal leftideals transform irreducibly under theregular representation. IftheClifford algebra issimple then theregular repre- sentation induces afaithful representation onany minimal leftideal. The mapping into theendomorphism algebra ofany_minimal leftideal induced bytheregular representation iscalled thespinor representation ofthesimple Clifford algebra andtheminimal leftideal iscalled the space ofspinors. The choice ofadifferent minimal leftideal gives another equivalent representation. When theClifford algebra isnot simple itisthesum oftwosimple component algebras, andanyminimal leftideal must lieinone ofthese simple components. The regular representation ofanon-simple Clifford algebra induces afaithful repre- sentation ontheleftideal which isthesum oftwominimal leftideals, one lying ineach simple component. The mapping into such an endomorphism algebra induced bytheregular representation will be called thespinor representation ofthenon-simple Clifford algebra, and such anideal willbetermed thespinor space. The minimal leftideals will betermed semi-spinor spaces and themapping that theregular representation induces onaminimal left ideal will becalled the semi-spinor representation oftheClifford algebra. The kernel ofsuch a representation isobviously thesimple component algebra that does not contain thesemi-spinor space. Thus thespinor representation ofa non-simple Clifford algebra isreducible, being thesum oftwoinequi- valent semi-spinor representations. The spinor representation ofthe Clifford algebra induces arepresentation ofanysubset byrestricting to leftmultiplication ontheideal byelements ofthat set.Inparticular it induces arepresentation oftheClifford group. Irreducible representations oftheClifford algebra induce irreducible representations oftheClifford group. (2.5.1) That is,thespinor representation ofasimple Clifford algebra, orthe Semi-spinor representation ofanon-simple one, induces anirreducible representation oftheClifford group. Thiswillalsobecalled thespinor Orsemi-spinor representation. The proof ofthe statement follows immediately from theobservation that non-singular vectors generate the Clifford group andtheClifford algebra. InfacttheClifford group could bereplaced with thesubgroup if: andthestatement would obviously stillbetrue. Ifanirreducible representation oftheClifford algebra induces a reducible representation ofthe even subalgebra then that induced representation isthesum oftwoirreducible ones. Forsuppose thatIisa minimal left ideal oftheClifford algebra that splits into invariant 56 CLIFFORD ALGEBRAS AND SPINORS subspaces under left multiplication by the even subalgebra. Let W be such an invariant subspace of smallest dimension. Then if x is any odd regular element let xW = X, giving dim X = dim W. If S = W + X, where the sum is not necessarily direct, then S is preserved under multiplication by the Clifford algebra. For Cp.p(IFI) = C .p(11=1) + C-;.,(1F1)x SO Cp.p(IF1)W = C ptg(F)W + xC7,, q(F)W C W + xW and C p,q(IR)xW = Cp4(11i)W. Since S C I and I is a minimal left ideal we must have S = I. If wnx= Y then Cp+. q(B)YC Y since W and hence X are preserved under left multiplication by C But W is an invariant subspace of minimal dimension and so either Y = 0, and I is the sum of two invariant subspaces, or Y= W= X=I and I transforms irreducibly. Having shown that irreducible representations of the Clifford algebra induce a representation of the even subalgebra that is either irreducible or the sum of two irreducible representations, we would like to know in which cases each possibility occurs. Suppose firstly that the even subalgebra is reducible; this can only occur in even dimensions in which case the Clifford algebra is simple. Then the spinor representation of the Clifford_algebra induces a faithful representation of the even subalgebra, that is, the kernel is zero. This must therefore be a reducible representa- tion of the reducible subalgebra, being the sum of the two inequivalent irreducible representations whose kernels are the different simple ideals. The irreducible representations of a non-simple even subalgebra will again be called semi-spinor representations of that algebra. Suppose now that the Clifford algebra is reducible; this can only occur in odd dimensions in which case the even subalgebra is simple. In this case the semi-spinor representations induce irreducible representations of the even subalgebra. For let I be a minimal left ideal (the semi-spinor space) and z denote the volume form. Then if, for example, the kernel of the semi-spinor representation is the simple ideal Cp4(11)(1 + z) the semi-spinor space is an eigenspace of the volume form, zq9 = V cp€ I. Since z is odd and regular we have Cp. q(11:1) = Cp+401i) + Cp+.q(1E1)z and Cp. = Cp+.q(IFI)I. So I can have no in- variant subspaces under multiplication by Cp+.q(Fi) since it is a minimal left ideal of Cp, q OR). The irreducible representations of the Clifford algebra can induce a reducible representation on the even subalgebra even when that algebra is simple. The general criterion is given by the following. 56 CLIFFORD ALGEBRAS AND SPINORS subspaces under leftmultiplication bytheeven subalgebra. Let Wbe such aninvariant subspace ofsmallest dimension. Then ifxisanyodd regular element letxW=X,giving dimX =dimW.IfS=W+X, where thesum isnotnecessarily direct, then Sispreserved under multiplication bytheClifford algebra. For C,,_,,(lR) =C,‘,‘_,,(lR) +C,f_,,(lR)x so C,,_,,(lR)W =C;_,,(lR)W +xC;'_,,(lR)W CW+xW and C,,yq(lR)xW =CM(lR) W. Since SCIand Iisaminimal left ideal wemust have S=I.If Wt) X= Ythen C;,,(lR)YCY since Wand hence Xarepreserved under leftmultiplication byCf,,,(lR). ButWisaninvariant subspace of minimal dimension and soeither Y=0,and Iisthesum oftwo invariant subspaces, orY=W=X=IandItransforms irreducibly. Having shown that irreducible representations oftheClifford algebra induce arepresentation oftheeven subalgebra thatiseither irreducible orthesum oftwoirreducible representations, wewould liketoknow in which cases each possibility occurs. Suppose firstly that the even subalgebra isreducible; thiscanonly occur ineven dimensions inwhich case theClifford algebra issimple. Then thespinor representation ofthe Cliffordalgebra induces afaithful representation oftheeven subalgebra, thatis,thekernel iszero. This must therefore beareducible representa- tionofthereducible subalgebra, being thesum ofthetwoinequivalent irreducible representations whose kernels arethedifferent simple ideals. The irreducible representations ofanon-simple even subalgebra will again becalled semi-spinor representations ofthatalgebra. Suppose now that theClifford algebra isreducible; thiscanonly occur inodddimensions inwhich case theeven subalgebra issimple. Inthis case thesemi-spinor representations induce irreducible representations oftheeven subalgebra. ForletIbeaminimal leftideal (thesemi-spinor space) andzdenote thevolume form. Then if,forexample, thekernel ofthesemi-spinor representation isthesimple ideal C,,_,,(lR)(1 +z)the semi-spinor space isan eigenspace of the volume form, zqv=-(pVqae I.Since zisodd and regular wehave Cp_,,(lR) = C;_q(lR) +C,f,q(lR)z and Cp_,,(lR)I =C,f_,,(lR)I. SoIcan have noin- variant subspaces under multiplication byC,f_,,(lR) since itisaminimal leftideal ofCp_q(lR). The irreducible representations oftheClifford algebra caninduce a reducible representation ontheeven subalgebra even when that algebra issimple. Thegeneral criterion isgiven bythefollowing. SPINORS 57 Irreducible representations of the Clifford algebra induce reducible representations of the even subalgebra if and only if primitives in the subalgebra are primitive in the full algebra. (2.5.2) What we need to show is that the minimal left ideals of the full algebra have twice the dimension of the minimal left ideals of the even subalgebra if and only if primitives in the subalgebra are primitive in the full algebra. Let P+ be a primitive idempotent of C;.,(11:1). Then Cp,q(11)P+ is a left ideal and Cp,q(IFI)P+ = C p+,q(F)P+ + xCp+.q(R)P+ for any odd regular x. Since P+ is primitive in Cp+,q(IFI) then Cp+,q(B)P+ is a minimal left ideal of the even subalgebra and so the dimension of Cp q(Fi)P+ is twice that of the minimal left ideals of C .;q(11). So the minimal left ideals of the full algebra are twice the dimension of those of the subalgebra if and only if Cp,q(111)P+ is a minimal left ideal, that is, if and only if P+ is primitive in Cm(IFI). If a minimal left ideal of the full algebra is projected out by a primitive of the subalgebra then the Cp+.,(IF1)-irreducible subspaces are obviously the even and odd subspaces. Just as the irreducible representations of the Clifford algebra gave representations of the Clifford group the irreducible representations of the even subalgebra induce representations of the even Clifford group and in particular: Irreducible representations of the even subalgebra induce irreducible representations of ,F+. (2.5.3) Again this follows from the fact that ,F+ generates Cp+,q(11). First we note that the Clifford algebra is generated by non-singular vectors of the same norm. For if {el, fl with i = 1, . . p, j =1, . . q is an orthonormal basis, and if p * 0, then a new basis of unit-norm vectors is {e1, V2et + PI. Thus Cptg(IFI) is generated by products of unit-norm vectors, and such products are in ,F+. The relationship between the irreducible representations of the Clif- ford algebra and its even subalgebra is summarised in table 2.10. The structure of Cp,q(Fi) is determined by p — q mod 8 where p + q = n. The eight different cases have been grouped in pairs. For the first pair the semi-spinor representation of the full algebra induces an irreducible representation of the subalgebra; whereas for the second pair the Clifford spinor representation induces an irreducible even Clifford spinor representation. For the third pair of algebras the spinor repre- sentation splits into a pair of equivalent spinor representations of the subalgebra, whereas in the final case the spinor representation is the sum of two inequivalent semi-spinor representations of the subalgebra. In table 2.10 we give the dimensions of the irreducible representations of the Clifford algebra and its even subalgebra. We have used C — S/S to denote that the irreducible representation of the Clifford algebra is a SPINORS 57 Irreducible representations ofthe Clifford algebra induce reducible representations oftheeven subalgebra ifandonly if primitives inthesubalgebra areprimitive inthefullalgebra. (2.5.2) What weneed toshow isthat theminimal leftideals ofthefullalgebra have twice the dimension ofthe minimal left ideals ofthe even subalgebra ifandonly ifprimitives inthesubalgebra areprimitive inthe full algebra. Let P*beaprimitive idempotent ofC;_,,(lR). Then Cp_,,(lR)P" isaleftideal and Cp_,,(lR)P* =C,f_,,(lR)P* +xC;,q(B)P* foranyoddregular x.Since P*isprimitive inC,f_,(lR) then C§_,(lR)P+ isaminimal leftideal oftheeven subalgebra andsothedimension of Cp_,,(lR)P* istwice that oftheminimal leftideals ofC;,,,(lB). Sothe minimal leftideals ofthefullalgebra aretwice thedimension ofthose ofthesubalgebra ifandonly ifCp_,,(lR)P* isaminimal leftideal, that is,ifandonly ifP+isprimitive inCM(R). Ifaminimal leftideal ofthe fullalgebra isprojected outbyaprimitive ofthesubalgebra then the C,',*,,(lR)-irreducible subspaces areobviously theeven andoddsubspaces. Just astheirreducible representations oftheClifford algebra gave representations oftheClifford group theirreducible representations of theeven subalgebra induce representations oftheeven Clifford group andinparticular: Irreducible representations oftheeven subalgebra induce irreducible representations of+1“. (2.5.3) Again thisfollows from thefact that +1“ generates C,',*,q(lR). First we note thattheClifford algebra isgenerated bynon-singular vectors ofthe same norm. For if{e’,fl}with i=1, ..., p,j=1, ..., qisan orthonormal basis, andifp¢0,then anew basis ofunit-norm vectors is{e',\/2e‘ +fl}. Thus C;,q(lB) isgenerated byproducts ofunit-norm vectors, andsuch products arein+1“. The relationship between theirreducible representations oftheClif- ford algebra anditseven subalgebra issummarised intable 2.10. The structure ofCp_,,(lR) isdetermined byp—qmod8 where p+q=n. The eight different cases have been grouped inpairs. Forthefirst pair thesemi-spinor representation ofthefullalgebra induces anirreducible representation ofthe subalgebra; whereas forthesecond pair the Clifford spinor representation induces anirreducible even Clifford spinor representation. Forthethird pair ofalgebras thespinor repre- sentation splits into apair ofequivalent spinor representations ofthe subalgebra, whereas inthefinal case thespinor representation isthe sum oftwoinequivalent semi-spinor representations ofthesubalgebra. Intable 2.10 wegive thedimensions oftheirreducible representations oftheClifford algebra anditseven subalgebra. Wehave used C-S/S todenote thattheirreducible representation oftheClifford algebra isa Table 2.10 Dimensions of the irreducible representations of the Clifford algebra C and its even subalgebra C. S denotes a spinor representation and S/S a semi-spinor representation. (p —q) mod8 Dimension 5 1 3 2 6 , 7 4 0 2(2["/2]) C — S/S C — S C — S C — S C+ — S C+ — S 2['/21 C — S/S C — S C+ — S C+ — S/S C — S C+ — S C+ — S 1(21"/2)) C+ — S/S Table 2.10 Dimensions oftheirreducible representations oftheClifford algebra Cand itseven subalgebra Ct. S denotes aspinor representation andS/Sasemi-spinor representation. (p—q)mod8 Dimension 5 1 3 2 6,7 4 0 2(2i~/11) c-s/s c-s c-s c-s c+-s c+-s21»/11 c-s/s c-s c+-s c+-s/s c-sc+-s c+-s;(2i»/11) 0+-s/s SPINORS 59 semi-spinor representation, C+ — S to denote the induced spinor repre- sentation of the even subalgebra, and similarly for the other two cases. The integer part of n12 is denoted by [n/2]. This table is an immediate consequence of table 2.8. Since we are concerned with algebras over the real field the spinor spaces are IR-linear vector spaces, the dimensions of which are given in table 2.10. As well as obviously being left Cp,q(IR) modules the spinor spaces are also right A-modules where si is the algebra given in table 2.8. Whereas, in general, right multiplication will not preserve a left ideal it will be preserved under right multiplication by elements of sti. When the Clifford algebra is simple al is a division algebra, whereas when the Clifford algebra is not simple si = aioa where g is a division algebra. In this case the semi-spinor spaces are right g-modules. It is an immediate consequence of associativity that left multiplication induces a 21-linear transformation on the minimal left ideals. Similarly the irre- ducible representations of the even subalgebra may be regarded as a-linear transformations where a is one of the real division algebras Fi, C or H. The dimensions of the spinor and the semi-spinor spaces regarded as g-linear spaces can be found from tables 2.8 and 2.10 since d(dim2) = dime where a is a d-dimensional Fl-algebra. For those Clifford algebras whose centre is C the spinor space may be regarded as a C-linear space by using the complex structure of right (or left) multiplication by the volume form z. For example, we may define multiplication by the imaginary unit by itp = tpz, where ip lies in a minimal left ideal. Alternatively, we could define itp —viz. Although we have already noted that all irreducible representations of a simple algebra are equivalent, when representing a simple JR -algebra on a C-linear space the question of equivalence needs treating carefully. If p and p' are representations of any simple JR -algebra .94, where 54(JR) = C(IFI)GA/1.„,(E), on JR-linear spaces V and V' then there is an JR-linear transformation S from V' to V such that p'(a) = p(a)S for all a of at If, however, V and V' are regarded as complex vector spaces by defining iv = p(z)v Vv E V (where z generates the centre) then there is a C-linear transformation S such that p'(a) = p(a)S Va if and only if iv' = p'(z)v'. Thus by defining p(z)v = iv and p'(z)v' = —iv' we get two complex-inequivalent representations of a simple JR -algebra. This is easily understood in terms of the complexified algebra. Regarded as a complex vector space V carries an irreducible representation of the complexified algebra sic &IOC. The representation p extends by C-linearity to sic, p(ia)v = ip(a)v. Since COC CC, .94 c is reduci- ble and its irreducible representations have as kernel one of the simple ideals, and the irreducible representations are equivalent if and only if the kernels are the same. If p(z)v = iv then p(1 + iz) = 0 and the kernel of p is projected by the central idempotent ;-(1 + iz). If, Si>rN0Rs 59 semi-spinor representation, C*—Stodenote theinduced spinor repre- sentation oftheeven subalgebra, andsimilarly fortheother twocases. The integer part ofn/2isdenoted by[n/2]. This table isanimmediate consequence oftable 2.8. Since weareconcerned with algebras over thereal field thespinor spaces arelPt-linear vector spaces, thedimensions ofwhich aregiven in table 2.10. Aswell asobviously being leftC,,_q(lPt) modules thespinor spaces arealso right sfl-modules where sflisthealgebra given in table 2.8. Whereas, ingeneral, right multiplication willnotpreserve a leftideal itwillbepreserved under right multiplication byelements of .94.When theClifford algebra issimple atisadivision algebra, whereas when theClifford algebra isnotsimple at=E.‘D®(.;.‘1> where 91>isadivision algebra. Inthiscase thesemi-spinor spaces areright (£1)-modules. Itisan immediate consequence ofassociativity that leftmultiplication induces a E31)-linear transformation ontheminimal leftideals. Similarly theirre- ducible representations oftheeven subalgebra may beregarded as E31)-linear transformations where 91>isoneoftherealdivision algebras lR, CorH.The dimensions ofthespinor and thesemi-spinor spaces regarded as91>-linear spaces canbefound from tables 2.8and2.10 since a'(dim<1.) =dimj, where 91>isaa’-dimensional IR-algebra. Forthose Clifford algebras whose centre isCthespinor space may be regarded asaC-linear space byusing thecomplex structure ofright (or left) multiplication bythevolume form z.Forexample, wemay define multiplication bytheimaginary unit byiip= 1,02, where tpliesina minimal leftideal. Alternatively, wecould define iip=-1112. Although wehave already noted that allirreducible representations ofasimple algebra areequivalent, when representing asimple lPt-algebra ona C-linear space thequestion ofequivalence needs treating carefully. Ifp and p’are representations ofany simple lPt-algebra at,where &4(lB) =C(lB)®Jl/l,,,(lR), onlPt-linear spaces Vand V’then there isan IR-linear transformation Sfrom V’toVsuch thatp'(a) =S"p(a)S for allaofsfl.If,however, VandV’areregarded ascomplex vector spaces bydefining iv=p(z)v VveV(where zgenerates thecentre) then there isaC-linear transformation Ssuch that p’(a) =S'1p(a)SVa ifand onlyifiv’=p’(z)v’. Thus bydefining p(z)v =ivandp’(z)v’ =—iv’we gettwo complex-inequivalent representations ofasimple lPt-algebra. This iseasily understood interms ofthecomplexified algebra. Regarded asacomplex vector space Vcarries anirreducible representation ofthe complexified algebra sic=&4®C. The representation pextends by C-linearity tosic, p(ia)v =ip(a)v. Since C®C =CCBC, aicisreduci- bleanditsirreducible representations have askernel oneofthesimple ideals, andtheirreducible representations areequivalent ifandonly if thekernels arethesame. Ifp(z)v =iv then p(1+iz)=0 and the kernel ofpisprojected bythe central idempotent §(1+iz). If, 60 CLIFFORD ALGEBRAS AND SPINORS however, p'(z)v' = —iv' then 1(1 — iz) is in the kernel, thus p' and p are inequivalent representations of sic. When the spinor space is a right H-modulet then it can be regarded as a complex vector space by choosing as complex structure any complex subalgebra of the quaternions. If q E H such that q2 = —1 then we may define multiplication by complex numbers on spinors by iv = zpq Again, regarded as complex vector spaces, these minimal left ideals carry irreducible representations of the complexified algebra by extend- ing the spinor representation by C-linearity. Since HOC CaR2 the complexified algebra is simple and hence all irreducible representations are equivalent. Thus in this case all irreducible representations of the simple E-algebra on complex vector spaces are complex-equivalent. We shall return to a discussion of the complexified Clifford algebras later. The first example we give is of CO32(1F1) H(E). Here the spinor space is the algebra itself. If {P, f2} is an orthonormal basis then {1, fi, f2, f1,2 = Z} is a standard basis for the quaternions. We may choose as complex structure right multiplication by z and define ia = az for a E CO32(11). Then {1, f1} is a basis for the corresponding complex vector space. If p denotes the spinor representation then with respect to this basis and choice of complex structure we have the matrices of the transformations, p(a), as follows: p(1) = ( 1 0 ) p(f l) = 01) 0 1 0 Pp(f2) = ( —i 0 p(z) = 01 Had we instead chosen ia = —az then we would have the complex conjugate matrices. These give a complex-equivalent representation; we have p(a)* = p(fl a(fi)-1). Regarded as a complex vector space, H carries an irreducible repre- sentation of the complexified algebra HOC. If P, = iz) then P- are primitive idempotents in HOC and (H®C)P± are minimal left ideals such that u±z = Tiu± for all u± E (HOC)P ±. Since P_= (f 1)-1P,f1 then right multiplication by P is a C-linear trans- formation between the two left ideals which obviously commutes with left multiplication and hence establishes the equivalence of these com- plex representations. The even subalgebra is isomorphic to C(E) and the spinor representa- tion of CO32(I1:1) induces a reducible representation of C2(1F1), the even and odd quaternions transforming irreducibly. These irreducible repre- t The notion of an `H-module' is to be found at the end of Appendix A where the quaternion algebra. H, is also introduced. 60 CLIFFORD ALGEBRAS AND SPINORS however, p’(z)v’ =-iv’ then §(1—iz)isinthekernel, thus p’andp areinequivalent representations ofsic. When thespinor space isaright H-modulei then itcanberegarded asacomplex vector space bychoosing ascomplex structure anycomplex subalgebra ofthequaternions. IfqeHsuch that qz=-1then wemay define multiplication bycomplex numbers onspinors byiip=ipq. Again, regarded ascomplex vector spaces, these minimal leftideals carry irreducible representations ofthecomplexified algebra byextend- ingthespinor representation byC-linearity. Since H®C =C®Jtt2 the complexified algebra issimple andhence allirreducible representations areequivalent. Thus inthiscase allirreducible representations ofthe simple IR-algebra oncomplex vector spaces arecomplex-equivalent. We shall return toadiscussion ofthecomplexified Clifford algebras later. The first example wegive isofC0_2(lB) =H(lB). Here thespinor space isthealgebra itself. If{f1,fl}isanorthonormal basis then {1. fl,f2,flfz =2}isastandard basis forthequaternions. Wemay choose ascomplex structure right multiplication byzand define ia=azfor aeC0,2(lB). Then {1,fl}isabasis forthecorresponding complex vector space. Ifpdenotes thespinor representation then with respect tothis basis and choice ofcomplex structure wehave thematrices ofthe transformations, p(a), asfollows: p<1>=(§ ‘,’)pt/1>=(‘f 11,) p(f1>=(_‘§ 5)4(1)=(5I1)- Had weinstead chosen ia=—az then wewould have thecomplex conjugate matrices. These give acomplex-equivalent representation; wehavep<4>*=pt/lat/1):’). Regarded asacomplex vector space, Hcarries anirreducible repre- sentation ofthecomplexified algebra H®C. IfPi=§(l1"iz)then P: areprimitive idempotents inH®C and (H®C)Pi areminimal left ideals such that uiz =Tia’ for all uie(H®C)Pi. Since P_=(f‘)"P+f‘ then right multiplication byflisaC-linear trans- formation between thetwoleftideals which obviously commutes with leftmultiplication andhence establishes theequivalence ofthese com- plex representations. Theeven subalgebra isisomorphic toC(18) andthespinor representa- tion ofC0_2(lB) induces areducible representation ofC5fZ(lB), theeven andodd quaternions transforming irreducibly. These irreducible repre- TThe notion ofan‘H-module’ istobefound attheendofAppendix Awhere thequaternion algebra, H,isalsointroduced. SPINORS 61 sentations of the simple algebra are equivalent: right multiplication by any odd quaternion interchanges the even and odd subspaces and commutes with left multiplication. However, right multiplication by z induces a complex structure on the even and odd subspaces that enables them to be regarded as complex one-dimensional vector spaces. These are complex-inequivalent, right multiplying by any odd element not being C-linear. We next consider C3,1(1R) itt. 4(1F1). Here the four-dimensional spinor representation induces an irreducible representation of the even sub- algebra C ME) C(1R)0.4 2(R). We may choose as spinor space the minimal left ideal whose basis is the first column in table 2.7. By defining up = zip for all spinors tp the spinor space may be regarded as a complex vector space with left multiplication by the even subalgebra a C-linear transformation. A basis for this complex vector space is (P1, e°P1). With this basis and choice of complex structure the matrices of these transformations for a basis for the even subalgebra are as follows: po) = 1 0) p(e12) = ( 01 -01 ) p(e23) = —oi) P(e31) = oi) The matrix representations of the generators of the rotation group will be recognised as the Pauli matrices (up to conventional factors of i). Defining hp = -np gives the complex conjugate representation which is complex-inequivalent. In this section we have naturally represented the Clifford algebra, and hence the Clifford group, on its left ideals. We can also represent the algebra on its right ideals. Associating each element of the algebra with the linear transformation obtained by multiplying with that element from the right gives a mapping into the endomorphism algebra, namely R: Cp,q(FI) End Cp,q(11:1) a 1---> R(a), R(a)b = ba. Since {R(a)R(b)lc R(a){R(b)c} = cba = R(ba)c this correspond- ence is not an algebraic isomorphism. Given an involution of the Clifford algebra we can use this correspondence to define a representa- tion "0": p(z) = ip(1) p(eo3) = ip(e12) p(eol) = ip(e23) p(e°2) = ip(e31). SPrN0Rs 61 sentations ofthesimple algebra areequivalent: right multiplication by any odd quaternion interchanges the even and odd subspaces and commutes with leftmultiplication. However, right multiplication byz induces acomplex structure ontheeven andoddsubspaces thatenables them toberegarded ascomplex one-dimensional vector spaces. These arecomplex-inequivalent, right multiplying byany odd element not being C-linear. Wenext consider C3_1(lB) =A/l4(lB). Here thefour-dimensional spinor representation induces anirreducible representation oftheeven sub- algebra C§_1(lB) =C(lR)®./I/t2(lB). Wemay choose asspinor space the minimal left ideal whose basis isthefirst column intable 2.7. By defining itp=zipforallspinors tpthespinor space may beregarded as acomplex vector space with leftmultiplication bytheeven subalgebra a C-linear transformation. Abasis forthiscomplex vector space is{P1, e°P1}. With thisbasis andchoice ofcomplex structure thematrices of these transformations forabasis fortheeven subalgebra areasfollows: PU)=ié §)po>=imn pe">=(f -5) per)-ime"> .... ...o\_/O-pe“>=(fl C) pe“>=me“> pa“)-( 0-pe”>=1me"t The matrix representations ofthegenerators oftherotation group will berecognised asthePauli matrices (uptoconventional factors of.i). Defining itp=—z1p gives thecomplex conjugate representation which is complex-inequivalent. Inthissection wehave naturally represented theClifford algebra, and hence theClifford group, onitsleftideals. Wecanalso represent the algebra onitsright ideals. Associating each element ofthealgebra with thelinear transformation obtained bymultiplying with that element from theright gives amapping intotheendomorphism algebra, namely R:C,,_,,(lFt) —-> EndC,,_q(lFt) a|——> R(a), R(a)b =ba. Since {R(a)R(b)}c ER(a){R(b)c} =cba=R(ba)c this correspond- ence isnotanalgebraic isomorphism. Given aninvolution }ofthe Clifford algebra wecanusethiscorrespondence todefine arepresenta- tion Z7: 62 CLIFFORD ALGEBRAS AND SPINORS P-: Cp. 4(1R) —> End C p. p(IFI) a 7-3(a) = R(a$). (2.5.4) Indeed we have a representation since p-(a) = R(0)R(0) = R(blas) = R(1abP)= Mat)). Obviously the minimal right ideals transform irreducibly under this representation. Just as the minimal left ideals may be regarded as right a-modules these minimal right ideals can be regarded as left 2h-modules. A minimal right ideal is naturally identified with the space of a-valued g-linear mappings on a minimal left ideal. For if tp c C p,q(11:1)P and (13 E PC p.q(Iii) with P primitive then we may write 43: OOP) = 0/P with (D(w) E PC p.q(IR)P -= 9). Obviously (13(tpq) = (13(tp)q for q E g. Similarly the Clifford algebra itself (or a simple component thereof) may be identified with the space of g-valued linear transformations on the Cartesian product of a minimal left ideal and a minimal right ideal. For if (13 E PCp.p(11:1) and 1p E Cp.q(IFI)P then for any a E Cp.q(IFI) we may write a(43, tp) = 430, giving a(q4:1), tp) = qa(cto, tp), a(13, zpq) = a(0, tp)q for q E 9. If the minimal left ideal carries the spinor representation p and the minimal right ideal carries the representation 75 then we may induce a representation r on the Clifford algebra (or a simple component) by defining r(s)(0) = [p(s)ta P-(s)40] = stp43s./. If we choose j = then s/ = s' for s E ,r+ and the representation r and the vector representation x coincide on +P. In this case the representations p and induce contragradient representations of +r-±, and since we have seen how the minimal right ideal can be identified with the dual space of the left ideal we can construct a a-valued PP-invariant product. This will be discussed in the following section. 2.6 Spin -Invariant Inner Products Having identified the elements of certain minimal left ideals as spinors we now examine spin-invariant products of two such elements. Since Clifford multiplication from the left induces a linear transformation on 62 CLIFFORD ALGEBRAS AND SPINORS prc,,_,,(1n) —>Endc,,_,,(1n) at_->5(a)=R(a§‘). (2.54) Indeed we have a representation since p'(a) )7(b) = R(a?)R(b-5‘) =R(b9a3) =R([ab]9) =)7(ab). Obviously the minimal right ideals transform irreducibly under thisrepresentation. Just asthe minimal leftideals may beregarded asright (ED-modules these minimal right ideals canberegarded asleft9D-modules. Aminimal right ideal isnaturally identified with thespace of 9D-valued ‘ED-linear mappings onaminimal leftideal. ForifweC,,_,,(lPi)P and(DePC,,_q(lB) withPprimitive thenwemaywrite (P11/Ii’ ¢’(1/1): (P1/1 with <I>(1/1) ePC,,‘,,(lB)P =‘£5.Obviously <I>(1/Jq) =<I>(1/1)q forqei/‘E. Similarly theClifford algebra itself (orasimple component thereof) may beidentified with thespace of(ED-valued linear transformations onthe Cartesian product ofaminimal leftideal andaminimal right ideal. For if_<I>ePC,,_,,(lB) and 1peC,,_,,(lB)P then forany aeC,,_,,(lB) wemay write a(<I>, 1p)=(bat/1 giving a(q<I>. 1/1)=qa(<I>. 1/1).a(<I>.wq)=a(<I>.1/1)qforqE95- Iftheminimal leftideal carries thespinor representation pandthe minimal right ideal carries therepresentation )7then wemay induce a representation 1.’ontheClifford algebra (orasimple component) by defining T(8)(1//(P) =lP(S)1/1lli>'(8)¢’l =S1/1¢’S’- Ifwechoose Q=5then sf=s'1forse+1“ andtherepresentation 1.’ and thevector representation )5coincide on+1“? Inthis case the representations pand pinduce contragradient representations of+1“, andsince wehave seen how theminimal right ideal canbeidentified with thedual space oftheleftideal wecanconstruct a‘ED-valued +1‘:-invariant product. This willbediscussed inthefollowing section. 2.6Spin-Invariant Inner Products Having identified theelements ofcertain minimal leftideals asspinors wenow examine spin-invariant products oftwo such elements. Since Clifford multiplication from theleftinduces alinear transformation on SPIN-INVARIANT PRODUCTS 63 the spinor space we may use a product on the spinor space to define an involution on the Clifford algebra by sending every element to that which induces the adjoint linear transformation. Such an involution will be termed the adjoint involution. We shall construct a product of spinors q) and tp which is the same as that of sep and stp when s E The adjoint involution of such a product will be either or Conversely, any product on the minimal left ideal with or 07 as adjoint involution will be invariant under (at least) ,r+. We shall first consider an arbitrary simple 11-algebra and show how any involution is the adjoint of some product on the minimal left ideals. These products fall into a finite number of distinct classes, and any two involutions are equivalent (as defined in Appendix A) if and only if the associated products are in the same class. The case of the direct sum of two isomorphic simple algebras is treated similarly. Returning to the Clifford algebras we shall determine into which class the products associated with and ij fall. Similarly, we can classify the products on the minimal left ideals of the even subalgebra. As a corollary in up to five dimensions we can use (2.4.22) to express ,r+ as the invariance group of some product. Let ,s4 be simple over Fi and $ be some involution. If P is any primitive idempotent then P$ = JPJ -1 for some element J with .0 = E= +1. (2.6.1) For if j leaves elements of the centre invariant and ?i" denotes transposition in a matrix basis in which P is diagonal then we are assured (by (A23) of Appendix A) of a J with J./ = ±J such that a3 = .1-10.1V a E al; in particular, 135. = P = J -1P$ P. In the same way if the centre is C with $ inducing complex conjugation the argument can be repeated with Hermitian conjugation replacing transposition. If then E .94P then .1-11754 c Psi and we define ( , ) : x p,9qp 92, ip (cp, iP) = J -101P. If a is any element of al then (9), 91P) = (919), 9)) (2.6.3) and j is the adjoint involution of this product. The minimal left ideal 4P is a right 2l-module. If q E a then (ep, vq) = (92, oq (2.6.4) and the product is a-linear in the second entry. If we define = J'q$J for q E g then j is readily seen to be an involution of a such that (2.6.2) SPIN-INVARIANT PRODUCTS 63 thespinor space wemayuseaproduct onthespinor space todefine an involution ontheClifford algebra bysending every element tothat which induces theadjoint linear transformation. Such aninvolution will betermed theadjoint involution. Weshall construct aproduct of spinors goandlpwhich isthesame asthat ofsipandsipwhen se+1“. The adjoint involution ofsuch aproduct will beeither 2;‘or£17. Conversely, anyproduct ontheminimal leftideal with 5or517as adjoint involution willbeinvariant under (atleast) +1“. Weshall first consider anarbitrary simple IR-algebra andshow how anyinvolution is theadjoint ofsome product ontheminimal leftideals. These products fallinto afinite number ofdistinct classes, andanytwoinvolutions are equivalent (asdefined inAppendix A)ifand only iftheassociated products areinthesame class. The case ofthedirect sum oftwo isomorphic simple algebras istreated similarly. Returning totheClifford algebras weshall determine intowhich class theproducts associated with 5and517fall.Similarly, wecanclassify theproducts ontheminimal left ideals oftheeven subalgebra. Asacorollary inuptofivedimensions we canuse(2.422) toexpress +1“ astheinvariance group ofsome product. Letsfbesimple over1Band}besome involution. IfPisanyprimitive idempotent then P9‘=JPJ” forsome element JwithJ9=sJ,s=i1. (2.6.1) Forif§leaves elements ofthecentre invariant and 9'denotes transposition inamatrix basis inwhich Pisdiagonal then weare assured (by(A23) ofAppendix A)ofaJwith J5“=i]such that ag=J"a¥J Vaesf;inparticular, Pg=P=J'1P5“P. Inthesame way ifthecentre isCwith Einducing complex conjugation theargument can berepeated withHermitian conjugation replacing transposition. Ifthen qvesfiPthenJ"q0-9 ePsiandwedefine (,):sfiPx sfiP—>PsfiPE€t> ¢>.1/1*~—>(qv.w)=J"<P’w- (Z-6-Z) Ifaisanyelement ofsflthen (¢>~aw)=(4%.w) (Z-6-3) and,9istheadjoint involution ofthisproduct. Theminimal leftideal &1Pisaright (Z1)-module. Ifqe§Dthen (¢>.W)=(¢>.1/»)q (Z-6-4) and theproduct is91)-linear inthesecond entry. Ifwedefine q’=J“‘q5J forqe§Dthen jisreadily seen tobeaninvolution of(Z1) suchthat 64 CLIFFORD ALGEBRAS AND SPINORS (cpq, ip) = (q), 11)) (2.6.5) The involution j will reverse the order of terms in a product; in fact (4), = J-1(40, 10$-, = J-V-1921/Prf = J -11/*P-V1$J = Ej— I ip5. and thus cp) = IVY. (2.6.6) Such a product will be called a'-symmetric or al-skew as E is plus or minus one. We may use this product to define a mapping from the minimal left ideal to its dual space. If L(s4P, a) is the space of 21-linear maps from s61P to a then we define 4-9 where rp(V) = (cp, V). (2.6.7) We shall refer to (17 as the adjoint of cp with respect to ( , ). We remarked in the previous section that L(s4P, 2) is naturally identified with a minimal right ideal; elements acting on the minimal left ideal by the algebraic product. With this identification we have = (2.6.8) Having chosen some arbitrary minimal left ideal on which to define a product we can obtain a product on any other minimal left ideal. If P and P' are primitives then the simplicity of si ensures an element S such that P' = SPS -1. Given the product of (2.6.2) we define , } :s4P' x --* -=- a', 01—* {a,, = S(aS, I3S)S -1. (2.6.9) We can write this as {a, fi) = J'-'a40 where J'1 = SJ-1S1 and which satisfies P'5 = J' PT-1. An involution on a' equivalent to the involu- tion j on a is defined by pi' = S(S-lpS)1S-1 for p E'. It then follows that {0, a) = Eta, 13)1' and {ap, 13) = 01. The product we have constructed in (2.6.2) involves not only the involution but also the element J as defined in (2.6.1). Obviously such an element cannot be unique. Suppose that P5 = J'PJ-1 with J' $ = eV'. Then f -1.1P = Pr -1,1 and so .11-1JP = P.11-1J = A say, where A e a. Since Ai = J-1,11,1 = = EE'r-lJP then = EE1A. (2.6.10) 64 CLIFFORD ALGEBRAS ANDSPINORS (<Pq.w)=q’(<P.w)- (2-6-5) Theinvolution jwillreverse theorder ofterms inaproduct; infact (<11.w)’=J“(¢.W1=J"(J"<P’w)’J =1-‘w’¢1“"J =£J_lw}(p andthus (1/1.<11)=¢=‘(<P,WY (2-6-6) Such aproduct willbecalled 92/-symmetric or92/-skew assisplusor minus one. Wemay usethisproduct todefine amapping from the minimal leftideal toitsdual space. IfL(s.¢P, 92>)isthespace of92>-linear maps from s.¢Pto9Dthen wedefine ~:sdP--> L(sdP, 92>) <P'—>47 Where W111)=(<P.I/1)(2-6.7) Weshall refer to¢astheadjoint oftpwith respect to(,).We remarked intheprevious section that L(sdP, 92>)isnaturally identified with aminimal right ideal; elements acting ontheminimal leftideal by thealgebraic product. With thisidentification wehave $=J"<p9. (2.6.8) Having chosen some arbitrary minimal leftideal onwhich todefine a product wecanobtain aproduct onanyother minimal leftideal. IfP andP’areprimitives then thesimplicity ofatensures anelement Ssuch thatP’=SPS'1.Given theproduct of(2.6.2) wedefine {,}:sdP’ ><s.¢P’-->P's4P’ -22' (1/,fl 1-> {a/, =S(a/S,/9S)S_1. (2.6.9) Wecanwrite thisas(ix,/3}=J"‘a/3/3 where J"1 =SJ'1S5 andwhich satisfies P’?=J’P’J"l.Aninvolution on92>’equivalent totheinvolu- tionjon92>isdefined bypi’=S(S'1pS)7S‘1 forpe92)’. Itthen follows that{IiLY}=¢=‘{¢Y»5}"and{(11%5}=P"{¢Y. 5}- Theproduct wehave constructed in(2.6.2) involves notonly the involution fabutalsotheelement Jasdefined in(2.6.1). Obviously such anelement cannot beunique. Suppose that P5=J’PJ"1 with J’?=s’J’. Then J"‘JP =PJ"1J and soJ"1JP =PJ"lJ =Asay, where Ae92>.Since A1=J'W‘J =J'1J5J"‘9P-‘J =ss’J"1JP then 1"=ss’A. (2.6.10) SPIN-INVARIANT PRODUCTS 65 If (cp, = J'-`cpl'ip then (cp, ip)' = J-1JJ-1(Plp and since (q9, tp) E a we have (49, IP)' = 499, 1P). (2.6.11) When a E, then j must be the identity and so we must have E = and the products are related by a real multiple. The complex numbers have two distinct involutions, the identity and complex conjugation. When j is the former then the products are related by an arbitrary complex multiple. When j is complex conjugation then j is the adjoint of a (pseudo-) Hermitian-symmetric product, determined up to a real multiple, or equivalently the Hermitian-skew product which differs from it by a multiple of the imaginary unit. The quaternions have two inequivalent involutions, conjugation and reversion. Quaternion con- jugation, denoted by a bar, is the only involution in its equivalence class. In contrast there are distinct involutions equivalent to some 'standard' representative called reversion and denoted A. Suppose that (tp, cp) = E(cp, tp)A. Then if (cp, = A(q), ip) with A = AA then (V, Tr = EA(9), 0-149), P)} ^A-1 = EA{(99, Since A = AA then (as demonstrated in Appendix A) we can set A = ptsA for some p. Thus Aq AA-1 =poci I and we see that if is the adjoint of an HA-symmetric (or skew) product then it is also the adjoint of an HI-symmetric (or skew-) product for any j equivalent to reversion. If is the adjoint of a quaternion-conjugate-symmetric product then it will also be the adjoint of the reversion-skew product obtained by multiplying this product by any vector quaternion. The conjugate-skew and reversion-symmetric products are likewise related. The above considerations show how any involution is the adjoint of some al-symmetric or ai-skew product. Certain of these products can be further labelled by a signature. First we note that these products are non-degenerate; for if (.1-10)4, = ovip€.91P then J-10 = 0 since the regular representation of a simple induces a faithful representation on any minimal left ideal. Consider now a non-degenerate al-symmetric product on a right 91-module. Then if the mapping from a into the j-symmetric quantities of a, q —> qlq, is surjective then there is an orthogonal basis of unit-norm elements. If this mapping is not surjective but any j-symmetric quantity can be written as ±qlq, then there is an orthogonal basis of elements normalised to plus or minus one. This is just an obvious generalisation of the result guaranteeing an orthonormal basis for a real symmetric product and can be proved by induction on the dimension of the module. The two different cases are seen to arise when normalising a non-zero-norm quantity. Suppose that (ip, ip) = A, then if the product is 2J-symmetric A = . If we can write A = qlq for some q E g then vq-' will have unit norm. The mapping q —> qiq is not SPIN-INVARIANT PRODUCTS 65 If(tp,1/1)’=J"‘(/291/2 then (tp,1/1)’=J"‘JJ"(p5‘1/2 andsince (tp,1/1)e9D wehave (<0,1/1)’=Mm.1/»)- (2-6-11) When 92>=IR,then jmust betheidentity andsowemust have e=e’ andtheproducts arerelated byareal multiple. The complex numbers have two distinct involutions, theidentity and complex conjugation. When jistheformer then theproducts arerelated byanarbitrary complex multiple. When jiscomplex conjugation then ,9istheadjoint ofa(pseudo-) Hermitian-symmetric product, determined uptoareal multiple, orequivalently theHermitian-skew product which differs from itbyamultiple oftheimaginary unit. The quaternions have two inequivalent involutions, conjugation and reversion. Quaternion con- jugation, denoted byabar, istheonly involution initsequivalence class. Incontrast there aredistinct involutions equivalent tosome ‘standard’ representative called reversion anddenoted A.Suppose that (1/2,(p)=e((p, 1/1)“. Then if(tp,1/1)’=)t(q2, 1/1)with A=1“then <1/».<p>'--1<¢.1/»>A=£11414.~.~>wi-1=£1114.1/»>'m-1. Since A=Mthen (asdemonstrated inAppendix A)wecansetA=/1/4“ forsome /i.Thus )tq“)t'1 =/i(/flq/1)“/1" andweseethat if,9isthe adjoint ofanHA-symmetric (orskew) product then itisalsotheadjoint ofanH/-symmetric (orskew-) product foranyjequivalent toreversion. If,9istheadjoint ofaquaternion-conjugate-symmetric product then it will also betheadjoint ofthereversion-skew product obtained by multiplying thisproduct byanyvector quaternion. The conjugate-skew andreversion-symmetric products arelikewise related. The above considerations show how anyinvolution istheadjoint of some 9D/-symmetric or9D/'-skew product. Certain ofthese products can befurther labelled byasignature. First wenote that these products are non-degenerate; forif(J"n5’)1/1 =0V1/ze sQPthen J'1n5’ =0since the regular representation ofasimple dinduces afaithful representation on any minimal leftideal. Consider now anon-degenerate 92>/-symmetric product onaright 92>-module. Then ifthemapping from 9Dinto the j-symmetric quantities of92>,q->q/iq, issurjective then there isan orthogonal basis ofunit-norm elements. Ifthismapping isnotsurjective butanyj-symmetric quantity canbewritten asiq/Sq, then there isan orthogonal basis ofelements normalised toplusorminus one. This is justanobvious generalisation oftheresult guaranteeing anorthonormal basis forareal symmetric product andcanbeproved byinduction on thedimension ofthemodule. The twodifferent cases areseen toarise when normalising anon-zero-norm quantity. Suppose that (1/;,1/2)=1, then iftheproduct is91>/4-symmetric A=A/.Ifwecanwrite it=qlqfor some qe92>then 1/zq" willhave unitnorm. Themapping q—>qiqisnot 66 CLIFFORD ALGEBRAS AND SPINORS a surjection from a to the j-symmetric quantities when g is R, C or H with j the identity, complex and quaternion conjugation respectively. Thus the R-, C*- and H --symmetric products are further characterised by their signatures (the number of positive- and negative-norm elements in an orthogonal basis). The smallest of these two numbers will be called the index (or Witt index). The complex numbers have the important property that any complex number can be written as a square. Similarly any reversion-symmetric quaternion can be written as a square of a reversion-symmetric quantity (as demonstrated in Appendix A). Thus any C-symmetric or HA-symmetric product has an orthogonal basis of unit-norm elements. An R-skew or C-skew product can be non- degenerate only if the vector space is of even dimension, 2n say. In this case there is a canonical basis {pi, q,} for i = 1, . . n with (Pr, qi) = 45y Example 2.1 Take .94 = C4,0(E) with j At the end of §2.2 we constructed a basis for this algebra. Let P be what was there called P 1, that is P = ;(1 + z). The division algebra PAP = 2 is isomorphic to the quatern- ion algebra, with standard basis (P, e23P, e34 3, 6,24p} Since 1:)- = P, in this case the involution induces an involution on a. This is quaternion conjugation since, for example, (e23P) = —e 23 P. An H--symmetric product on AP is given by (99, 1P) = OP. An H-linearly independent basis for AP is {P, ell)}. We have (P, P) = P (P, elP) = PelP = 0 (eip, eip) = p eteip = P. So this basis is in fact orthonormal, the product being of index zero. Thus far we have shown how any involution can be put into one and only one class determined by the a'-symmetric or g1-skew product (further labelled by an index where appropriate) for which it is the adjoint involution. We may choose the representatives given in table 2.11 for the classes of product. Where we have chosen, for example, a C*-symmetric product we could have chosen a C*-skew one. For the same reason we only further classify products by the index rather than the signature. These classes of product define an equivalence relation on the associated involutions. We have already termed involu- tions and Vf equivalent if there is an automorphism J) such that = ((a)1)( 1) for all a E In fact it follows that with this notion of equivalence: 66 CLIFFORD ALGEBRAS AND SPINORS asurjection from 9Dtothej-symmetric quantities when 9DisIR,CorH with jtheidentity, complex and quaternion conjugation respectively. Thus theIR-,C*-andH‘-symmetric products arefurther characterised bytheir signatures (the number ofpositive- andnegative-norm elements inanorthogonal basis). The smallest ofthese two numbers will be called the index (or Witt index). The complex numbers have the important property thatanycomplex number canbewritten asasquare. Similarly anyreversion-symmetric quaternion canbewritten asasquare ofareversion-symmetric quantity (asdemonstrated inAppendix A). Thus anyC-symmetric orH"-symmetric product hasanorthogonal basis ofunit-norm elements. AnIR-skew orC-skew product can benon- degenerate only ifthevector space isofeven dimension, 2nsay. Inthis case there isacanonical basis {p,-,q,-} for i=1,...,nwith (Pt,(Ii)=51;- Example 2.1 Take .54=C4_0(lB) with §3EE.Attheendof§2.2 weconstructed abasis forthis algebra. Let Pbewhat was there called P1, that isP= %(1+z).The division algebra P.s4P E9Disisomorphic tothequatern- ionalgebra, with standard basis {P,e23P, e34P, e24P}. Since P’?=P,in thiscase theinvolution §induces aninvolution on9D.This isquaternion conjugation since, forexample, (e23P)5 =—e23P. AnH“-symmetric product on.s4Pisgiven by (<11,1/»)=will»- AnH-linearly independent basis for.s4Pis{P,e1P}. Wehave (P,P)=P (P,@111)=Pe1P=0 (e1P, e‘P) =Pe1e1P =P. Sothisbasis isinfactorthonormal, theproduct being ofindex zero. Thus farwehave shown how anyinvolution canbeputinto oneand only one class determined bythe9D/-symmetric or9D/-skew product (further labelled byanindex where appropriate) forwhich itisthe adjoint involution. We may choose the representatives given in table 2.11 forthe classes ofproduct. Where wehave chosen, for example, aC*-symmetric product wecould have chosen aC*-skew one. For thesame reason weonly further classify products bytheindex rather than thesignature. These classes ofproduct define anequivalence relation ontheassociated involutions. Wehave already termed involu- tions §3and fifequivalent ifthere isanautomorphism Sf’such that ail‘=((a")3’)(5’H) forallae.54.Infactitfollows that with thisnotion of equivalence: SPIN-INVARIANT PRODUCTS 67 Two involutions are equivalent if and only if they are the adjoints of equivalent products. (2.6.12) Table 2.11 (1)E -symmetric, of index v (2)E -skew (only in even dimensions) (3)C -symmetric (4)C -skew (only in even dimensions) (5)C* -symmetric, of index v (6)H--symmetric, of index v (7)HA-symmetric Here products are equivalent if they are both of the same one of seven main types and, where appropriate, of the same index. If $ and X are adjoints of equivalent products then we can introduce two al- symmetric or skew products, ( ,) j and ( , )K of (where appropriate) the same signature, with $ and X their respective adjoints. Both products admit a canonical basis of the same type, and any change of basis may be effected by left multiplication by a regular element. Since both products are j-linear in the first variable, and linear in the second there must be a regular a such that (q', 7p)j = (acp, mp)K for all cp and tp in the minimal left ideal, that is J-Lcplip = ICAacp) xalp = K'cplca xalp = IC-'(axcr)(o-la)-'cpx(axa)tp. If we introduce an involution Fl defined by = (aNa)-V(axa) V a e 59 then P9- = TPT-1 with T = (0a) -11‘. We have J'cplip = T'cp 5tp, that is (CPI 0.1 = 09, T V ço, )ES/1P. But (q), wip)i = (alq9, 1p)., and (cp, = (6'34, = (aerq9, tp), giving ((al — a'7).79, qi)j = O. Since this is true for all tp E siP and the product is non-degenerate (al — ag)q) = 0 V cp€ 59P and =- cd- Va e sg. Recalling the definition of g we have = a-i(craa-')xa, that is, $ and X are equivalent. To prove the converse we suppose that $ = 9 2XJ-1 for some automorphism 9'. Then if ( x saiP Psip is a product with X as adjoint -involution we define ( , x 59.13-1 SPIN-INVARIANT PRODUCTS Two involutions areequivalent ifandonly ifthey arethe adjoints ofequivalent products. (2.6.12) Table 2.11 (1)IR-symmetric, ofindex v (2)IR-skew (only ineven dimensions) (3)C-symmetric (4)C-skew (only ineven dimensions) (5)C*-symmetric, ofindex v (6)H"—symmetric, ofindex v (7)H"-symmetric Here products areequivalent ifthey areboth ofthesame oneof seven main types and,where appropriate, ofthesame index. IfEandfit areadjoints ofequivalent products then wecanintroduce two91>’- symmetric orskew products, (,),and(,)Kof(where appropriate) the same signature, with Eandfittheir respective adjoints. Both products admit acanonical basis ofthesame type, andanychange ofbasis may beeffected byleftmultiplication byaregular element. Since both products arej-linear inthefirstvariable, andlinear inthesecond there must bearegular 0such that(tp,1/1),=(otp, 01/1)K forall<pand1/Jin theminimal leftideal, thatis J“<12’</2 =K"(<2<12)i"<2</2 =K“<12i‘<2*‘<2</2 =K'1(0w0)(0%0)"¢%(0%)¢- Ifweintroduce aninvolution 97defined by ag=(o5’(o)'1a*7((oi’(o) Vae94 then P9=TPT" with T=(o*7‘o)"K. Wehave J"<p’1/J= T'1<p91//, thatis (<12.</2),=(<12.</2)1 V22.</2E24P- But(<12,2</2)]i(a’<12. </2),and(<12.2</2)1=(2%, </2)1=(2g<12. </2)] giving ((a<9—a”)<p, 1/1),=0.Since thisistrueforall1/1e&4P andthe product is non-degenerate (a?—a~°T)<p =0Vtpes4P and a9=agVaesfl. Recalling the definition of E’T we have 11$‘=o‘1(oao")*7(o, that is,}and 17fareequivalent. Toprove the converse wesuppose that }=9365?" forsome automorphism SP.Then if (,):&4P X.941’-> P941’ isaproduct with 3fasadjoint-involution wedefine (,};saP2“ ><94122“ —>P*’*‘saP***‘ 68 CLIFFORD ALGEBRAS AND SPINORS by {a', /3) = (o, , then {cy, no} = m/3')' (noway, 13y)y-' = ([m&rox9,-icr]y, p9y-1 = Similarly if (13, a) = E(a, 16)k then {/3, al = (f3Y, aY)Y -' = fin"-1 = Eta, firkY-1 = Efcr, PP where j = Da-1 is equivalent to k. Thus we have a product on stil3 with I as an adjoint-involution of the same type as the product on AP, which has 1C as adjoint-involution. As already noted a product on any given minimal left ideal enables an equivalent product to be defined on any other minimal left ideal. Thus if j = 9W9' -' then and Jf are adjoint-involutions of equivalent products on any minimal left ideal. Although for complex matrices not all automorphisms are inner a corollary to the above is that if involutions are related by = 9W99-1 for any automorphism 9' then in fact there is an inner automorphism such that j = EXE -1. The result of (2.6.10), together with table 2.11, gives the number of inequivalent involutions for a simple 11-algebra. This is displayed in table 2.12. Table 2.12 The number of inequivalent involutions. COAL HOER,. r even 1r + 2 + 3 r+2 r odd 1(r + 1) ,;(r + 3) 1(r ± 3) Since not all Clifford algebras are simple we now consider involutions of semi-simple algebras that have two simple components. Let = 03CA where 03 and cC are simple with 03 = AP, = AQ for central idempotents P, Q with PQ = QP = 0 and P + Q = 1. If JC is an involution of A then 1;°(, Wf are central idempotents with P'Q = Q 1.13( = 0 and 1:°c + Q = 1. So A = AP7f0.9iQ/c and, since the expression of a semi-simple algebra as a sum of simple ones is unique up to ordering ((A17) of Appendix A) then either AP' = 91 and AQ' =%, or AW6 = 91 and .5IP7' = (C. In the former case N induces an involution on the simple algebras 03 and ce and may thus be classified in the manner already treated. In the second case every element of 03 is sent to (C, and this can only arise when % is isomorphic to the opposite 68 CLIFFORD ALGEBRAS AND SPINORS by{<2/,5}=(2/i’.5i’)"Pl, then (21,mfl}=(219,m”5*')”" =("1‘”‘<Y*"»l2")*"_' =([m‘”“"'<Yl”, fi")”i‘ ={m’<>/. B}- Similarly if (5,2/)=t=‘(<Y,5)"then(5.2/}=(59,<19)” =t=‘(<2/5’,,5")"i"' =t=‘{<Y.fi}""*"l =t=‘{<Y,5}’ where j=9’kSf“1 isequivalent tok.Thus wehave aproduct onflP9_' with Qasanadjoint-involution ofthesame type astheproduct onflP, which hasfifasadjoint-involution. Asalready noted aproduct onany given minimal leftideal enables anequivalent product tobedefined on anyother minimal leftideal. Thus ifQ=Sf’?]{3"1 then Qand Yifare adjoint-involutions ofequivalent products onanyminimal leftideal. Although forcomplex matrices notallautomorphisms areinner a corollary totheabove isthat ifinvolutions arerelated byQ=5f";7{3"1 foranyautomorphism 8’then infactthere isaninner automorphism Z such thatQ=Z3752". The result of(2.6.10), together with table 2.11, gives thenumber of inequivalent involutions forasimple IR-algebra. This isdisplayed in table 2.12. Table 2.12 Thenumber ofinequivalent involutions. A/t, C®J1/t, H®J1/t, reven §r+2 §r+3 §r+2 rodd §(r+1) §(r+3) §(r+3) Since notallClifford algebras aresimple wenow consider involutions ofsemi-simple algebras that have two simple components. Let vi=973C-9(6 where 9/3and (6aresimple with 973=flP, (6=flQ for central idempotents P,Qwith PQ=QP=0andP+Q=1.Iffifis aninvolution ofdthen P9‘, Q5”arecentral idempotents with Pi'(Q?( =QPPW =0andPi"+Q?‘=1. Sod=s§P?(C9flQi7( and, since theexpression ofasemi-simple algebra asasum ofsimple ones is unique uptoordering ((A17) ofAppendix A)then either flPi7( =973and s§Qi'( =(6,orflQ?( =973and.§1QPP =(6.Intheformer case Yifinduces aninvolution onthesimple algebras 973and(6andmay thus beclassified inthemanner already treated. Inthesecond case every element ofQ3is sent to(6,andthiscanonly arise when (6isisomorphic totheopposite SPIN-INVARIANT PRODUCTS 69 algebra of T JP". There is in fact only one such involution, up to equivalence. If si is the sum of two simple algebras, with X and involutions that do not preserve these simple components, then X and are equivalent. (2.6.13) If X and are as described then is an automorphism of s4 that induces automorphisms on the simple components gi and T. We introduce an automorphism 9' of si by defining V b ER 9' C = C VC E Then 5' is an automorphism of such that for b E@ b" = bX, which is in T, and so b'' b"-` = 135 c. Similarly for c E T c91 = cs' and so c"'-' = = = c and we have established the equiva- lence of and X. When si = 20.Ato2lakt where At is a total matrix algebra and the division algebra satisfies a = aoP then any involution in the class not preserving the simple components will be called a 9h-swap. (The real division algebras Fi, C and H are isomorphic to their opposite algebras.) Any such involution is the adjoint of a non-degenerate product on the left ideal formed from the direct sum of two minimal left ideals from the two different simple components. For let P be some primitive idem- potent (necessarily in one simple component). If is some 9l-swap involution then Q = P + 1 31 is a i-symmetric idempotent. We may define a', f; 1---> (a, = a'40. Such a product is non-degenerate for if alp= 0 V )3 then choosing fi to lie in one simple component shows that the component of a in the other must vanish, and hence a, = O. It immediately follows from (2.6.14) that mfi) = (m ,a, 0) V M E ,94 (aq, )3) = ql(a, )3) V q E ac,a. We can equally well introduce a product with different symmetry. If s is any regular element lying in a, ss-i = 12, then let S = s — sl . This J-skew element of acia$ has an inverse given by S-1 = s SS-' = ss-' + (s-ls)1 = 1 + 1 g8 = 1. (2.6.14) SPIN-INVARIANT PRODUCTS 69 algebra of973,(6=9B°P. There isinfactonly onesuch involution, upto equivalence. If.94isthesum oftwo simple algebras, with SCand Q involutions that donotpreserve these simple components, then fifandQareequivalent. (2.6.13) IffitandQareasdescribed then Qfifisanautomorphism of.94that induces automorphisms onthesimple components 973and(6.We introduce anautomorphism E!’of.94bydefining b§’=b’(9 Vbe9B c5'=c Vce(6. Then E!’isanautomorphism of.94such that forbe973bi’?=bf”,which isin(6,andsob5'95'_' =b7(5'_l =bi”.Similarly force(6 cw=c9and soc5'95'_1= cgirl =c”']7(_] =cf”and wehave established theequiva- lence ofQandTit. When .94=9?>®A/l€r)9?>®A/t where A/tisatotal matrix algebra andthe division algebra satisfies 92>=9?>°P then anyinvolution intheclass not preserving thesimple components will becalled a91>-swap. (The real division algebras IR,CandHareisomorphic totheir opposite algebras.) Any such involution istheadjoint ofanon-degenerate product onthe leftideal formed from thedirect sum oftwominimal leftideals from the two different simple components. ForletPbesome primitive idem- potent (necessarily inonesimple component). IfQissome 92>-swap involution then Q=P+ P9isaQ-symmetric idempotent. We may define (,);.<flQ><.<flQ_>Q.<flQ =226-322 (2)614) 2.I2*—>(2.B)=2%- Such aproduct isnon-degenerate forifaffi =0V[3then choosing [3to lieinonesimple component shows thatthecomponent oforintheother must vanish, andhence or=0.Itimmediately follows from (2.6.14) that (oz,mfl) =(m-‘oz, B)Vmesfi (,3.0/)=(01.5)’ (2/q.fi)=q’(2/.5) Vqe‘3?>@‘3?>- Wecanequally well introduce aproduct with different symmetry. Ifsis anyregular element lying in92>,ss“ =19, then letS=s—sf.This Q-skew element of9?>G-39?)’ hasaninverse given by S_1= s_'—(s'1)5 SS“ =ss_' -1-(s"s)=9 =19, +19,;=1. 70 CLIFFORD ALGEBRAS AND SPINORS We may now define x .54Q 9:609) a', /3 {a', /3} = S-1a1,6. This product satisfies {a, mj3) = {ma, /3} {0, = —S-1{a, #}1S {oeq, 13) = S -1q1S{cr, /3). We are now ready to return to Clifford algebras. Having established how a given involution is the adjoint of a non-degenerate product on the minimal left ideals the problem of finding products invariant under ,F+ reduces to finding involutions such that s/ = s Vs E +F'. Of course and are such involutions (which may or may not be equivalent), but before classifying the associated products we confirm that these are the only such involutions (up to equivalence). It is convenient to consider the cases of even and odd dimensions separately. Suppose firstly that p + q = n is even, so that C p,q(IF1) is central simple. Then if I is any involution there exists some J such that al = Jak1-1 with 0 = +J. If se +1-+ then .54 = Jskf+1 = Js+1J+1; thus s/ =- s-1 if and only if Js-1 = 5J VS E We know (from the proof of (2.5.3)) that ,F+ generates the even subalgebra, and so sl = Vs E „F+ if and only if J commutes with all elements of the even subalgebra. If J has this property then so will its even and odd parts separately. But the volume n-form Z is even and it anticommutes with all odd elements and so the odd part of J must vanish, hence J must be in the centre of the even subalgebra. This centre is spanned by {1, zl. If Z = —Z then, since 0 = ±J, either J GE giving j = or J = Az for A E IFI and for any a, al = zaz-1 = If z = z and z2 = —1 then the centre of C+p, q(Fi) is C(Fi). If J EC then, since C is algebraically closed, J = cr2 = crue some (ye C and al = aaaa 1u1 = a(a-laa)a-1 showing that j is equivalent to If z1 = z and z2 = 1 then C4(E) is reducible and the centre is spanned by the orthogonal idempotents {P+, P_} where P+= (1± z). If J is regular then J = AP + + 11P_ with A, non-zero reals. If A and /2 are both of the same sign then there is no loss of generality in assuming them positive since multiplying J by an element of the centre does not alter j. In this case we set J = a2P+ + v2P_=(aP, + vP_)(aP ++ vP_).= and j is equivalent to Similarly if A and jt are of opposite sign then, with no loss of generality, we assume J=a2P+— v2P_. If 70 CLIFFORD ALGEBRAS AND SPINORS Wemay now define {,}:s.4Q ><s.¢Q—>9z>(-B92 2/..3*=>{<Y..3}=5"<2/‘.55 This product satisfies (2./2115'}={m’<1..3} {/2.21} =—$"{v1. fl}’$ (M1.B}=5"<1’5{<2<.fl}- Wearenow ready toreturn toClifford algebras. Having established how agiven involution istheadjoint ofanon-degenerate product onthe minimal leftideals theproblem offinding products invariant under +1“ reduces tofinding involutions Qsuch that sf=s"Vse+1“. Of course 7;and 7,117aresuch involutions (which may ormay not be equivalent), butbefore classifying theassociated products weconfirm that these aretheonly such involutions (up toequivalence). Itis convenient toconsider thecases ofeven andodddimensions separately. Suppose firstly that p+q=niseven, sothat C,,_q(IR) iscentral simple. Then ifQisany involution there exists some Jsuch that a9=Ja5J" with J5=i].Ifse+1“ then s3=Js5J‘1 =Js"J‘l; thus s9=s'l ifand only ifJs'l =s"J Vse +1“. Weknow (from the proof of(2.5.3)) that +1“ generates theeven subalgebra, and so s3=s'l Vse,1“ ifandonly ifJcommutes with allelements ofthe even subalgebra. IfJhasthisproperty then sowillitseven and odd parts separately. Butthevolume n-form Ziseven anditanticommutes with allodd elements and sotheodd part ofJmust vanish, hence J must beinthecentre oftheeven subalgebra. This centre isspanned by {1,2}. IfZ5=—Zthen, since J5=i],either JeIR giving Q=7;orJ=A2 for/leIRandforanya,a9=2a'52" =a5". If25=2and22=-1then thecentre ofCZ,,,(IR) isC(IR). IfJeC then, since Cisalgebraically closed, J=02=005 forsome oeCand a}=ao5a5o5"a‘1 =a(a'1aa)5a'1 showing thatQisequivalent to5. If25=zand 22=1 then C,',f_,,(lR) isreducible and thecentre is spanned bytheorthogonal idempotents (Pr, P_} where Pt= §(1i2).IfJisregular then J=AP, +/iP_ with 1,/inon-zero reals. IfAand/4areboth ofthesame signthen there isnolossofgenerality in assuming them positive since multiplying Jbyanelement ofthecentre does notalter Q.Inthiscase weset J=o2P+ +v2P_ =(0P+ +vP_)(oP+ +vP_)5 andQisequivalent to7,‘.Similarly ifAand/4areofopposite sign then, with no loss ofgenerality, we assume J=o2P., —v2P_. If SPIN-INVARIANT PRODUCTS 71 k= aP,.+ vP_ then kze = (oP + vP_)(P + — P _)(aP + vP = a2p+ v2p = J. Thus al = kzeaq10) - zi -1 k-i = kz(k-1ak)z-1k-1 = and / is equivalent to For the case in which = z we have seen that the requirement that = s VS E +F+ only requires / to be equivalent to either or ij. If, however, p *0 then +F±- contains an odd element and requiring = s-1 Vs E +1-± uniquely determines / to be For if J = X + tzz then the requirement that J commute with an odd element forces /.4. to vanish. Similarly, if q * 0 then ij is the unique involution such that s/ = s-1 Vs E We turn now to the case in which n is odd, with {1, z) spanning the centre of Cm(11:1), and Cp+4(IFI) central simple. Since zq = —z one of the involutions and will leave the centre invariant, the other will not. It follows that if / is any involution then either al = -1 with J = ±J, or al = JaJ-1 with .P71 = +J. Consider the former case. Then requiring sl = Vs c +F+ shows, exactly as before, that J must commute with elements of the even subalgebra. Then if J_ is the odd part of J we can write J_=(.1_z -1)z and, since the odd element z commutes with everything, J_ will commute with the even subalgebra if and only if J _z-1 does. Since the even subalgebra is central simple J_ must be proportional to z. It then follows that J is in the centre of Cp.q(IFI) and / = In exactly the same way it follows that if s/ = Js/=71J-1 and s/ = Vs E +F+ then / = We may summarise as follows: if +1—± +F+ then s$ = Vs E +F± iff = if +1-± +F+ then sl = s-I Vs E +F-± iff = If sl = s-1 Vs E +F+ then / is equivalent to or 71; if n *4 mod 4 then either / = or / = rj. (2.6.15) The involutions and ij induce the same involution on the even subalgebra. This is the only such involution that inverts elements of J+. For if / is any involution of a central simple Cp+.q(11) then = Jak1-1 for some even J. Thus s$ = s-I Vs E +F+ only if / is in the centre, giving / = If C(1F1) is not central simple then al = Jak1-1 with J even if leaves elements of the centre invariant, or odd if ,n induces a non-trivial automorphism on the centre. In the SPIN-INVARIANT PRODUCTS 71 k=aP+ +vP_ then kzks =(aP+ +vP_)(P+ —P_)(aP+ +vP_) =a2P+ —v2P_ =J. Thus a3=k2k5a5(k§)"z‘1k“ =kz(k"ak)52"k“ =k(k"ak)5"k‘1 andQisequivalent toE11. Forthecase inwhich 25=2wehave seen that therequirement that s5=s"Vse+1“ only requires Qtobeequivalent toeither EorEr).If, however, pahOthen +1‘: contains anodd element and requiring sf=s"Vse +1‘: uniquely determines QtobeE.For ifJ=/1+/tz then therequirement that Jcommute with anodd element forces itto vanish. Similarly, ifq¢Othen E17istheunique involution such that s9=s'1Vse*I“‘. Weturn now tothecase inwhich nisodd, with {1,2}spanning the centre ofC,,_q(IR), andC;,,(lFl) central simple. Since 2"=-2oneofthe involutions EandEnwillleave thecentre invariant, theother willnot. It follows that ifQisanyinvolution then either a9=Ja5J‘1 with J5=i], ora3=Ja5"J" with J5"=iJ. Consider theformer case. Then requiring s5=s'lVse+1“ shows, exactly asbefore, that Jmust commute with elements oftheeven subalgebra. Then ifJ_istheoddpart ofJwecanwrite J_=(J_z'l)z and, since theodd element 2commutes with everything, J_will commute with theeven subalgebra ifandonly ifJ_z_l does. Since the even subalgebra iscentral simple J_must beproportional to2.Itthen follows thatJisinthecentre ofCp_q(lFl) andQ=E. Inexactly the same way itfollows that ifs3=Js5”J'l and s9=s_lVse+I"' then Q=En. Wemay summarise asfollows: if,1‘: (If+I‘* then s5=s_lVse +I‘i iffQ =E if*1‘: (if+1“ then s9=s"Vse *1‘: iffQ =Er). Ifs5‘=s“Vse +1“thenQisequivalent toEorEn;ifn¢4 mod4 then either Q=EorQ=Er). (2.6.15) The involutions Eand E17induce thesame involution ontheeven subalgebra. This istheonly such involution that inverts elements of +1“. For ifQisany involution ofacentral simple C;,,,(IR) then a5=Ja5J'l forsome even J.Thus sf=s'1Vse 11"" only ifQisin the centre, giving Q=E.IfC,,'_,,(lFl) isnot central simple then a3=Jail" with Jeven ifQEleaves elements ofthecentre invariant, orodd ifQEinduces anon-trivial automorphism onthecentre. Inthe 72 CLIFFORD ALGEBRAS AND SPINORS former case we have seen that only if = is sl = s -I Vs E ,F+. There can be no involution with this property in the second category, for there is no odd element that commutes with the even subalgebra since this contains z which anticommutes with all odd elements. We wish now to classify the involutions and and the involution they induce on the even subalgebra. This will be done for an arbitrary Clifford algebra using the same isomorphisms that enabled its structure to be determined. We use those isomorphisms for which and ij on the factors of a tensor product induce or on the product algebra. In this way knowing the class of and ij on the factors enables the class of the involution on the product to be determined, and and on arbitrary algebras can be classified by explicitly classifying these involu- tions for a few low-dimensional algebras. First we consider involutions of tensor products. Let .94 -= OA, where gi is a simple algebra over IR and .kt,, is the algebra of order n real matrices. Let I be an involution on sti that induces involutions g on At,, and X on 93. We shall write this as = X05-. If 211 = akt n, and X is2l-symmetric or skew then j is certainly either lc-symmetric or skew, the symmetry being determined by that of X and 5-, in a way to be determined. If Q is primitive in 91 and R is primitive in An then P = QR is primitive in ..91. If Co' = KQK-1 and R = TRT -1 then PI = JPJ-1 with J = KT. Since .// = K'fr , J is symmetric if K and T are both symmetric or both skew, and skew if K and T are of different symmetry. Thus if either X is ak-symmetric with g fl-skew, or X is a4-skew with g 2-skew, then is a '-symmetric; otherwise it is a k-skew. We now investigate the signature in the case in which is ak-symmetric. If ( , )j is the product on siP associated with I then for b„ bj E g3 and 111,, m E birnp)j = = since gi and .44.„ are mutually commuting subalgebras of . So (b,m,,bimp)j= (ma., mo)T(b„ bj)K where the products on the right-hand side are those on the subalgebras associated with the involutions indicated. Thus if, for example, ( , )T is symmetric admitting an orthonormal basis with r vectors normalised to plus one and s to minus one (of signature r, s) and ( , )K has a basis with r' normalised to plus one and s' to minus one then ( , )j has a basis of rr' + ss' positive-norm and rs' + sr' negative-norm vectors. Similarly it follows that if ( , )7- is R-skew and ( , )K is 2-skew then ( , ), admits an orthonormal basis with as many positive- as negative-norm basis vectors. We summarise the situation below. If j is an involution on .91, where ,s4 = glairt„ with j = X05 - then: Inicr$6,1b1mp= T-1K-1m;Tbixb1mg 72 CLIFFORD ALGEBRAS AND SPINORS former case wehave seen that only ifQ=Eissf=s" Vse +1“. There canbenoinvolution with thisproperty inthesecond category, forthere isnoodd element that commutes with theeven subalgebra since thiscontains 2which anticommutes with alloddelements. Wewish now toclassify theinvolutions EandE17andtheinvolution they induce ontheeven subalgebra. This willbedone foranarbitrary Clifford algebra using thesame isomorphisms that enabled itsstructure tobedetermined. Weusethose isomorphisms forwhich EandE17onthe factors ofatensor product induce EorE17ontheproduct algebra. In thisway knowing theclass ofEandE17onthefactors enables theclass oftheinvolution ontheproduct tobedetermined, and EandE17on arbitrary algebras canbeclassified byexplicitly classifying these involu- tions forafewlow-dimensional algebras. First weconsider involutions oftensor products. LetM=95®A/1,, where 95isasimple algebra over IRandA/1,,isthe algebra oforder nreal matrices. LetQbeaninvolution onMthat induces involutions 9onA/t,,and flfon95.We shall write this as Q=fI{®§. If95=9D®Jl/l,,, andflfis9D“-symmetric orskew then Qis certainly either 92>"-symmetric orskew, thesymmetry being determined bythat ofTlfand9',inaway tobedetermined. IfQisprimitive in95 and Risprimitive inA/1,, then P=QR isprimitive inM.If Q“=KQK'1 and R5=TRT‘l then P5=JPJ“ with J=KT. Since J5‘=K7(T*7, Jissymmetric ifKand Tareboth symmetric orboth skew, andskew ifKand Tareofdifferent symmetry. Thus ifeither flf is9D"-symmetric with 91'IR-skew, orfitis9D"-skew with 9'IR-skew, then Qis9D"-symmetric; otherwise itis9D"-skew. We now investigate the signature inthecase inwhich Qis9D"-symmetric. If(,))istheproduct onMPassociated with Qthen forb,-,b,»e95andma, m),eA/1,, (bima, =./_lm,,3b,-Pb}-ml; =T_lK_lm,,‘7b,-3(b]~m5 = T_lmagm);K—lb;7{bj since 95andA/t,,aremutually commuting subalgebras ofM.So (bimav bjm/3)! =(ma. m/3)T(bi» bj)K where theproducts ontheright-hand sidearethose onthesubalgebras associated with theinvolutions indicated. Thus if,forexample, (,)Tis symmetric admitting anorthonormal basis with rvectors normalised to plus oneandstominus one(ofsignature r,s)and(,)Khasabasis with r’normalised toplus one and s’tominus one then (,),hasabasis ofrr'+ss’positive-norm andrs’+sr’negative-norm vectors. Similarly itfollows thatif(,)Tis91-skew and(,)Kis9D"-skew then (,),admits anorthonormal basis with asmany positive- as negative-norm basis vectors. Wesummarise thesituation below. IfQis aninvolution onM,where M=9/3®A/1,, with Q=fI{®€ then: SPIN-INVARIANT PRODUCTS 73 (i)if either X is 2k-skew with er IA-symmetric, or X is 9}k -symmetric with 3- Fi-skew, then 5 is 2k-skew; (ii)if K is ak-skew and 9- is E-skew then 5, is 2k-symmetric, of maximal index (if any); (iii)if X is "-symmetric, with signature r, s, and .9- is Ti-symmetric, of signature r', s' then is 2k-symmetric of signature rr' + ss', rs' + sr'. (2.6.16) Having shown how to classify the involution on the tensor product of a simple E-algebra with a matrix algebra in terms of involutions on the factors, to classify the involutions on the product of two simple algebras it only remains to consider involutions on products of the division algebras. When one of these division algebras is E itself there is nothing to do. For the product of two copies of the complex numbers we have C(lR) 0 C(R)C(B) 0(IF1) identity 0 identity = identity e identity identity 0 conjugation = C-swap conjugation 0 conjugation =- conjugation e conjugation. (2.6.17) We use the obvious notation for an involution on a reducible algebra that induces involutions on the simple component algebras. The above can be verified by choosing a specific basis. If {1, il, {1, j) are standard bases for the factors and P, = (1 ± ij) then {P+, iP,} and {P_, iP_} are bases for the simple components. Similarly C(IFI) 0 H(IFI) =- C(IF1)0A1. 2(IFI) identity quaternion conjugation -= C-skew identity 0 reversion = C-symmetric complex conjugation 0 quaternion conjugation C*-symmetric, zero index complex conjugation 0 reversion = C*-symmetric, index one. (2.6.18) If {1, z) and {1, j, j, k) are standard bases for the factors then {e0} is an ordinary matrix basis where en = 1(1 + zi), e22 = 1(1 — zi), e21 = jell = e22j and e12 = —je 22 = —enj. Finally, H(B) 0 H(B) A4 4(E) conjugation 0 conjugation =E-symmetric, zero index reversion 0 reversion = IFI-symmetric, index two conjugation 0 reversion = E-skew. (2.6.19) Again this can be verified by constructing a basis. It is sufficient to note SPIN-INVARIANT PRODUCTS 73 (i)ifeither fitis92>‘-skew with 9IR-symmetric, ori7{is QED"-symmetric with 9IR-skew, then Qis92>“-skew; (ii)iffitis92>‘-skew and9isIR-skew then Qis92>"-symmetric, ofmaximal index (ifany); (iii)iffitis92>‘-symmetric, with signature r,s,and 9is IR-symmetric, ofsignature r’,s’then Qis95"-symmetric of signature rr'+ss’,rs’+sr’. (2.6.16) Having shown how toclassify theinvolution onthetensor product of asimple IR-algebra with amatrix algebra interms ofinvolutions onthe factors, toclassify theinvolutions ontheproduct oftwosimple algebras itonly remains toconsider involutions onproducts ofthedivision algebras. When oneofthese division algebras isIRitself there isnothing todo.Fortheproduct oftwocopies ofthecomplex numbers wehave C(lR) ®C(lR) =C(lR) ®(lR) identity ®identity =identity (Bidentity identity ®conjugation =C-swap conjugation ®conjugation =conjugation G3conjugation. (2.6.17) Weusetheobvious notation foraninvolution onareducible algebra that induces involutions onthesimple component algebras. The above canbeverified bychoosing aspecific basis. If{1,i},{1,j}arestandard bases forthefactors andP:=§(1iij)then {P+, iP+} and{P_, iP_} arebases forthesimple components. Similarly C(lR) ®H(lR) =C(lR)®A/t2(lR) identity ®quaternion conjugation =C-skew identity ®reversion =C-symmetric complex conjugation ®quaternion conjugation =C*-symmetric, zero index complex conjugation ®reversion =C*-symmetric, index one. (2.6.18) If{1,2}and{1,i,j,k}arestandard bases forthefactors then {e,-I-} is anordinary matrix basis where en=%(1+2i), en=§(1—2i), E21 =j811 :E22j and 912 =-j822 =—811j. Finally, H(lR) ®H(lR) =21/t4(lR) conjugation ®conjugation =IR-symmetric, zero index reversion ®reversion =IR-symmetric, index two conjugation ®reversion =IR-skew. (2.6.19) Again thiscanbeverified byconstructing abasis. Itissufficient tonote 74 CLIFFORD ALGEBRAS AND SPINORS that a primitive is given by P = 1(1+ il)(1+ jJ) where {1, i, j, k} and {1, I, J, K) are standard bases for the factors. A basis for the minimal left ideal in .41.4(F1) is {P, iP, jP, kP) and the index of the symmetric products can be explicitly evaluated. We are now in a position to classify all involutions on simple lR-algebras, or sums of two such algebras, that are obtained from involutions on the factors of a tensor product. If ,94 = 930T and = X09- then table 2.12 gives the class of the involution in terms of the classes of X and T. These classes are encoded into the types of table 2.11, with the 0 symbol denoting an involution on a direct sum of algebras inducing involutions on the components, and the types (8), (9) and (10) (see table 2.13) being a-swaps for a = Fi, H and C. The class of X determines the row of the table whilst .9- determines the column, the class of y being given in the intersection. (The symmetry of the table reflects the fact that si02/1 g30,34.) For example, (2.6.16) is encoded into the first two rows and the first two columns, whilst the diagonal block formed by the intersection of the third and fourth rows and columns is given by (2.6.16) and (2.6.17). There are various blanks in table 2.13, corresponding to those cases when the tensor product of the factors would be a reducible algebra with more than two components. To use the above to classify the involutions and 07 on arbitrary Clifford algebras we need to build up the Clifford algebras from tensor products of smaller ones such that the standard involutions on the factors induce and ij on the product. In §2.2 the structure of an arbitrary Clifford algebra was determined using the relations (2.2.7), (2.2.8) and (2.2.9) together with a knowledge of certain low-dimensional algebras. Examining the isomorphisms that established these relations shows that the involutions and 01 of the left-hand side of (2.2.7) do not induce either of the standard involutions on the factors. However, equations (2.2.8) and (2.2.9) give a relation between the standard involutions on the factors and the standard involutions on the product. As was noted in §2.2 there is another relation similar to (2.2.9), and in this case the standard involutions on the factors are related to those on the product. The relations are given below. Cp+1, q+I(IR) = C p, q(Ii) ®C1, 1(E) = (2.6.20) 07= ® Cp, q+4(11:1) Cp, q(E) Co. 4(E) = 0/ 07 (2.6.21) 74 CLIFFORD ALGEBRAS AND SPINORS that aprimitive isgiven byP=}(1+iI)(1+jJ) where {1,i,j,k}and {1,I,J,K}arestandard bases forthefactors. Abasis fortheminimal leftideal inA/l4(IR) is{P,iP,jP,kP} andtheindex ofthesymmetric products canbeexplicitly evaluated. We are now inaposition toclassify allinvolutions onsimple IR-algebras, orsums oftwo such algebras, that areobtained from involutions onthe factors ofatensor product. IfM=9’I®(6 and Q=fI{®9 then table 2.12 gives theclass oftheinvolution Qinterms of theclasses offitand 9.These classes areencoded into thetypes of table 2.11, with the(3symbol denoting aninvolution onadirect sum of algebras inducing involutions onthecomponents, andthetypes (8),(9) and(10)(seetable 2.13) being 92>-swaps for(5=R,HandC.Theclass offitdetermines therowofthetable whilst 9determines thecolumn, theclass ofQbeing given intheintersection. (The symmetry ofthe table reflects thefact that M699’. =9/3®M.) For example, (2.6.16) is encoded into thefirst two rows and thefirst two columns, whilst the diagonal block formed bytheintersection ofthethird andfourth rows andcolumns isgiven by(2.6.16) and(2.6.17). There arevarious blanks intable 2.13, corresponding tothose cases when thetensor product ofthe factors would beareducible algebra with more than two components. Tousetheabove toclassify theinvolutions Eand E17onarbitrary Clifford algebras weneed tobuild uptheClifford algebras from tensor products ofsmaller ones such that thestandard involutions onthe factors induce Eand E17ontheproduct. In§2.2 thestructure ofan arbitrary Clifford algebra was determined using therelations (2.2.7), (2.2.8) and(2.2.9) together with aknowledge ofcertain low-dimensional algebras. Examining theisomorphisms that established these relations shows that theinvolutions EandE17oftheleft-hand side of(2.2.7) do notinduce either ofthestandard involutions onthefactors. However, equations (2.2.8) and (2.2.9) give arelation between thestandard involutions onthefactors andthestandard involutions ontheproduct. Aswasnoted in§2.2 there isanother relation similar to(2.2.9), andin thiscase thestandard involutions onthefactors arerelated tothose on theproduct. Therelations aregiven below. Cp+1,q+1(lR)= Cp.q(lR) ®C1.1(B) E=E17®E (2.6.20) §n= 5@511 Cp,q+4(lR)= Cp.q(lR) ®C0.-K13) E= E®E (2.621) §n= En@511 Table 2.13 The classification of involutions on tensor products induced from involutions of the factors. (See table 2.11 for the classification scheme.) Involutions on tensor products 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 1 4 3 5 7 6 8 9 10 3 3 4 3 0 3 4 0 4 10 4 3 10 10 4 4 3 4 0 4 3 0 3 10 3 4 10 10 5 5 5 10 10 5 0 5 5 5 10 10 6 6 7 4 3 5 1 2 9 8 10 7 7 6 3 4 5 2 1 9 8 10 8 8 8 10 10 10 9 9 9 9 9 10 10 10 8 8 10 10 10 10 10 (8).-1R-swap, (9) H-swap, (10) C-swap. Table 2.13 The classification ofinvolutions ontensor products induced from involutions ofthefactors. (See table 2.11 forthe classification scheme.) Involutions ontensor products 1 2 3 4 5 6 7 5\osc\1c\u~4>m|\>>-* 6\ooo\1c\u~4>w|\>>-* 5\oooo~\:u~m4>»—-NAw 4>5(B(|)4>w 3 10 10-Pk»! wk w5(B(|)w4>wk 4 10 105 5 10 10 5('35 5 5 10 10 5oo~o|\>>—u-u-\4>\|c:~ 500\-O-—-|\>!.n4>L»JO\\l (8)EIR-swap, (9)EH-swap. (10)EC-swap. 76 CLIFFORD ALGEBRAS AND SPINORS Cp+4, q(1R) = Cp, q(E)®C4, AR) (2.6.22) ---- C) 01 Here we have used the same symbol to denote involutions on different algebras. For example, in (2.6.21) the on the left-hand side is the standard involution on Cp,q+4(IR) whereas the same symbol on the right-hand side denotes firstly the standard involution on Cp,q(11), and then that on CO34(IF1). Those low-dimensional algebras whose involutions must be classified by inspection are given in table 2.14. The products on the left ideals associated with the involutions are readily contructed for these algebras. Some of these products are further labelled by an index; we have only indicated the index where it is zero. This is justified by the following theorem. When the involution on Cp, q(E) is associated with a spinor product labelled by an index then if q *0 that index is maximal, whilst for q = 0 the index is zero. Similarly, any index associated with tj is maximal unless p = 0 in which case the index is zero. (2.6.23) Table 2.14 Involution classes of some low-dimensional Clifford algebras. C10(Ft) 1 0 1 8 C2.4111) 1(zero index) 2 C3.0(Fi) 5(zero index) 4 C4.0(FI) 6(zero index) 6 CO31(R) 3 5 C0.2(11) 7 6 C0.3(R) 9 6 0 6 C0.4(IR) 6 6(zero index) C1.1(11) 1 2 Suppose that ( , ) is a product on some left ideal with as adjoint involution, and that {E,} is an orthonormal basis with r positive-norm and s negative-norm elements. If q 0 then there is a vector x with x2 = —1. Then {xE,} is a new orthonormal basis for the left ideal and (xe„ xE,)= (xkxE„ £,)= (X 2Ei, £,)= —(6,, et). Thus this basis has r negative- and s positive-norm elements and so if the signature is well 76 CLIFFORD ALGEBRAS AND SPINORS Cp+4, '2Cp. E=E®E (2.6.22) En=En®$1- Here wehave used thesame symbol todenote involutions ondifferent algebras. For example, in(2.6.21) theEontheleft-hand side isthe standard involution onCM+4(lR) whereas thesame symbol onthe right-hand side denotes firstly thestandard involution onCM(lR), and thenthatonC0_4(lR). Those low-dimensional algebras whose involutions must beclassified byinspection aregiven intable 2.14. The products ontheleftideals associated with theinvolutions arereadily contructed forthese algebras. Some ofthese products arefurther labelled byanindex; wehave only indicated theindex where itiszero. This isjustified bythefollowing theorem. When theinvolution EonCp_q(lR) isassociated with aspinor product labelled byanindex then ifq#0 that index is maximal, whilst forq=0theindex iszero. Similarly, any index associated with E1]ismaximal unless p=0inwhich case theindex isZero. (2.623) Table 2.14 Involution classes ofsome low-dimensional Clifford algebras. 95 En Cl.0(]R) C2.0(]R) Ca.0(]R) C4.0(B) (30.1(15) C0.2(]R) C0115) C0_4(]R) Cl.l(R)1QB1 1(zer0 index) 5(zero index) 6(zero index) l—*O‘\\-O\lb-)U\O‘\-l>l\JO0 6 6(B6 6(zer0 index) 2 Suppose that (,)isaproduct onsome leftideal with Easadjoint involution, andthat {e,-} isanorthonormal basis with rpositive-norm andsnegative-norm elements. Ifq¢0then there isavector xwith x2=-1.Then {xs,~} isanew orthonormal basis fortheleftideal and (xs,, xs,-)=(xgxs,-, 5,)=(x2s,, s,-)=—(s,~, 5,).Thus this basis hasr negative- andspositive-norm elements and soifthesignature iswell SPIN-INVARIANT PRODUCTS 77 defined we must have r = s. So if q 0 the index is maximal. That it is in fact zero for q = 0 can be verified by repeated use of (2.6.22), together with table 2.14. The involution is treated in exactly the same way. We can now give the class of and ij for all Cp,q(E). First we note that this class only depends on p mod 8 and q mod 8: two applications of (2.6.21) and (2.6.22) enable this to be inferred from table 2.13. We have given the classes in table 2.15. To complete this table we use (2.6.21) and (2.6.22) with table 2.14 to complete the first row and the first column. Then the classes of and ij are simultaneously entered in the diagonals by using (2.6.20) with the multiplication of table 2.13. The third entry for a given p and q in table 2.15 gives the class of the involution that and induce on the even subalgebra. (The case of p = q = 0 is a degenerate case for which the class is 1.) Reference to the derivation of (2.3.1) gives C p (IR) = Thus the class of the involution on the subalgebra is obtained from a relabelling of the classification of 01. When p + q 5 we can use the classification of the involution on the even subalgebra to obtain ,F+ as the group of automorphisms of the associated product. This follows from (2.4.22). The automorphism group of an 1R-skew product on an n-dimensional vector space is denoted Sp(n, R), similarly Sp(n, C) is the automorphism group of a C-skew product. For a C*-symmetric product with signature r, s the automorphism group is U(r, s), whilst we use Sp(r, s, H) to denote the automorphism group of an H--symmetric product with this signature. When s = 0 then we simply write U(r) and Sp(r, H). The products associated with the 9)-swap have the general linear groups as auto- morphism groups. For, taking the product of (2.6.14), a general element of the automorphism group is S = s + s -11 with s any regular element of gam,. We have arranged these spin groups in table 2.16. From this table we have, for example, for C3 J (IR) ,F+ Sp(2, whereas at the end of §2.4 we demonstrated that ±r+ SL(2, C). These groups are isomorphic; in fact we have Sp(1,H) SU(2) Sp(2,1R) = SL(2, R) (2.6.25) Sp(2, C) SL(2, C). It can be seen that in two dimensions ,r, is isomorphic to the orthogonal group. (2.6.24) SPIN-INVARIANT PRODUCTS 77 defined wemust have r=s.Soifq=#0theindex ismaximal. That itis infact zero forq=0can beverified byrepeated useof(2.622), together with table 2.14. The involution §17istreated inexactly thesame way. Wecannow give theclass of§and§17forallCp_q(1R). First wenote thatthisclass only depends onpmod8 andqmod 8:twoapplications of (2.621) and(2.622) enable thistobeinferred from table 2.13. Wehave given theclasses intable 2.15. Tocomplete thistable weuse(2.6.21) and(2.622) with table 2.14 tocomplete thefirst rowandthefirst column. Then theclasses ofEandE17aresimultaneously entered inthe diagonals byusing (2.620) with themultiplication oftable 2.13. The third entry foragiven pand qintable 2.15 gives theclass ofthe involution that §and§17induce ontheeven subalgebra. (The case of p=q=0isadegenerate case forwhich theclass is1.)Reference to thederivation of(2.3.1) gives C;+1_.,(1B) =Cq,p(]R) 5=511- Thus theclass oftheinvolution onthesubalgebra isobtained from a relabelling oftheclassification of§17. When p+q<5wecanusetheclassification oftheinvolution §on theeven subalgebra toobtain +1“ asthegroup ofautomorphisms of theassociated product. This follows from (2.422). The automorphism group ofanIR-skew product onann-dimensional vector space is denoted Sp(n, IR),similarly Sp(n, C)istheautomorphism group ofa C-skew product. ForaC*-symmetric product with signature r,sthe automorphism group isU(r, s),whilst weuseSp(r, s,H)todenote the automorphism group ofanH“-symmetric product with thissignature. When s=0then wesimply write U(r) and Sp(r, H). The products associated with the22)-swap have thegeneral linear groups asauto- morphism groups. For, taking the product of(2.6.14), ageneral element oftheautomorphism group isS=s+s"? with sanyregular element of€D®A/l. Wehave arranged these spin groups intable 2.16. From this table wehave, forexample, forC3_1(lR) +1“=Sp(2, C) whereas attheendof§2.4 wedemonstrated that +1“=SL(2, C). These groups areisomorphic; infactwehave Sp(l,H):su(2) Sp(2,IR)=SL(2,IR) (2.625) Sp(2,c)=SL(2,c). Itcan beseen that intwo dimensions +1“ isisomorphic tothe orthogonal group.(2.624) Table 2.15 Classes of involutions of the real Clifford algebras. For each p and q the classes of and the involution they induce on the even subalgebra are given. Classes of involution on the real Clifford algebras C„,,(11) 9 0 1 2 3 4 5 6 7 0 1 3 7 9 6 4 2 8 1 5 6 6 0, 6 6 5 1 1 C) 1 1 40 1 1 5 6 6 0 6 6 5 1 1 1 0 1 1 5 6 6 0 6 6 5 1 8 2 4 6 9 7 3 1 1 8 2 4 6 9 7 3 2 1 8 2 4 6 9 7 3 2 2 0 2 2 5 7 7 10 7 7 5 5 2 2 CD 2 2 5 7 7 0 7 7 3 5 2 2 0 2 2 5 7 7 0 7 7 4 2 8 1 3 7 9 6 6 4 2 8 1 3 7 9 4 6 4 2 8 1 3 7 9 6 5 1 1 0 1 1 5 6 6 0 6 6 0 6 6 5 1 1 CD 1 1 5 6 5 6 8 6 6 5 1 1 0 1 1 5 6 9 7 3 1 8 2 4 6 6 9 7 3 1 8 2 4 Table 2.15 Classes ofinvolutions oftherealClifford algebras. Foreach pandqtheclasses of5,Er)andtheinvolution they induce ontheeven subalgebra aregiven. Classes ofinvolution ontherealClifford algebras C,,v,,(lR) Pq 0 1 2 3 4 5 6 7 0 1 3 7 9 6 4 2 8 1 5 6 6C-D6 6 5 1 1631 1631 l 5 6 6636 6 5 l 1 1631 l 5 6 6636 6 5 1 8 2 4 6 9 7 3 l 1 8 2 4 6 9 7 3 2 1 8 2 4 6 9 7 3 2 2632 2 5 7 7(-B7 7 5 S 2 2632 2 5 7 7(B7 7 3 5 2 2692 2 5 7 76.97 7 4 2 8 1 3 7 9 6 6 4 2 8 1 3 7 9 4 6 4 2 8 1 3 7 9 6 5 1 1631 1 5 6 6636 6636 6 5 l 1651 1 5 6 5 6C-B6 6 5 1 1631 1 5 6 9 7 3 l 8 2 4 6 6 9 7 3 1 8 2 4 Table 2.15 (cont.) Classes of involution on the real Clifford algebras C. q(11) 0 1 2 3 4 5 6 7 6 6 9 7 3 1 8 2 4 7 7 0 7 7 5 2 2 a 2 2 5 5 7 7 0 7 7 5 2 2 8 2 2 7 5 7 7 0 7 7 5 2 2 0 2 2 3 7 9 6 4 2 8 1 1 3 7 9 6 4 2 8 (1)1R-symmetric, of index v (3) C-symmetric (5) C*-symmetric, of index v (7) HA-symmetric (9) H-swap (2)IR-skew (only in even dimensions) (4) C-skew (only in even dimensions) (6) H--symmetric, of index v (8) IR-swap (10) C-swap Table 2.16 0 1 2 3 4 0 1 1 U(1) Sp(1 ,H) Sp(1,H) X Sp(1,H) Sp(2,H) 1 1 Fi* Sp(2,111) Sp(2,C) Sp(1,1,H) 2 U(1) Sp(2,R) Sp(2,IF1) x Sp(2,Fi) Sp(4,11) 3 Sp(1,H) Sp(2,C) Sp(4,IR) 4 Sp(1,H) x Sp(1,H) Sp(1,1,H) 5 Sp(2,H) Sp(1,H) SU(2), Sp(2,11) SI(2,11), Sp(2, C) SI(2,C) ,r+ for all C, ,(IR) with p + q 5. Table 2.15 (c0nt.) Classes ofinvolution ontherealClifford algebras Cp_,,(IR) P ‘I 1 2 4 5 6 7 6 79 7 7®7 7 77 7®7 7697 7 9 3 761 8 2 2 2C-92 5 2 5 22 2692 2692 4 2 8 6 4 24 5 2 --l\J 8 (1)lR-symmetric, ofindex v (3)C-symmetric (5)C*-symmetric. ofindex v(7)HA-symmetric (9)Hswap (2)lFl-skew (only ineven dimensions) (4)C-skew (only ineven dimensions) (6)H"-symmetric. ofindex v(8)lPi-swap (10)Cswap Table 2.16 P q O l 2 3 4 5 K11-J>bJl\)>-‘@1 1 I W u(1) Sp(l,H) Sp(1,H) ><Sp(l,H) Sp(2,H) $P(1,H) ><$i>(1.H) $i>(1.1.H) Sp(2,H)Sp(2,lB) Sp(2,C) Sp(l,l,H) U(l) Sp(2,lR) Sp(2.lB) XSp(2.IR) Sp(4,IR) Sp(l,H) Sp(2,C) Sp(4,lB) Sp(l,H) 2SU(2), Sp(2,lFl) 2Sl(2,lFl), Sp(2, C)2Sl(2,C) .1“forallC,,_q(lFl) with p+q55. 80 CLIFFORD ALGEBRAS AND SPINORS 2.7 The Complexified Clifford Algebras So far we have only considered real orthogonal spaces and their associated real Clifford algebras. Much of the discussion could, how- ever, be repeated with the real field replaced by an arbitrary field; in particular the complex field. If W is a complex vector space with h a complex valued, symmetric, non-degenerate C-bilinear form then the Clifford algebra can be constructed as in §2.1. Since h is not charac- terised by any signature the structure of the Clifford algebra can only depend on n, and it will be denoted C n(C). The structure of the algebra can be determined as in §2.2, only here the situation is even simpler. We have C „,2(C) C ,,(C) 0 C 2(C) (2.7.1) Ci(C) C C (2.7.2) C 2(C) = At2(C). (2.7.3) These are the analogues of (2.2.8), (2.2.2) and (2.2.3) and they may be proved in a similar way. They give the structure of all C n(C). If n is even then C n(C) Al2-,2(C) (2.7.4a) whereas if n is odd C n(C) = (2.7.46) The structure of the even subalgebra follows from the analogue of (2.3.1), namely If n is even then whereas if n is odd ,((c) c (C). C(C) .422-,(C)att2,2-.(c). C(C) --= (2.7.5) (2.7.6a) (2.7.66) Rather than proceed with the study of the Clifford groups and their relation to the complex orthogonal groups we shall show how the complex Clifford algebras may be related to real orthogonal spaces. If V is a real n-dimensional orthogonal space with bilinear form g then ye, the complexification of V, is an n- dimensional complex vector space. The real bilinear form g may be extended by C-linearity to a C-bilinear form on VC, gC. If g is non-degenerate then so is g c. If ye is regarded as a 2n-dimensional real vector space then V is canonically identified with an n-dimensional subspace. The complex algebra 80 CLIFFORD ALGEBRAS AND SPINORS 2.7TheComplexified Clifford Algebras Sofarwehave only considered real orthogonal spaces and their associated real Clifford algebras. Much ofthediscussion could, how- ever, berepeated with therealfield replaced byanarbitrary field; in particular thecomplex field. IfWisacomplex vector space with ha complex valued, symmetric, non-degenerate C-bilinear form then the Clifford algebra canbeconstructed asin§2.1. Since hisnotcharac- terised byanysignature thestructure oftheClifford algebra canonly depend onn,anditwillbedenoted C,,(C). The structure ofthealgebra canbedetermined asin§2.2, only here thesituation iseven simpler. Wehave c,.+2(C) =c,,(¢:) ®c2(o:) (27.1) C1(C) =CC-BC (2.7.2) C2(C) =A/l2(C). (2.7.3) These aretheanalogues of(2.2.8), (2.2.2) and(22.3) andthey may be proved inasimilar Way. They givethestructure ofallC,,(C). Ifniseven then C,,(C) =M2”/2(C) (2.7.4a) whereas ifnisodd C,,(C) =A/t2(~_i>/1(C). (2.7.4b) The structure oftheeven subalgebra follows from theanalogue of (2.3.1), namely C,f(C) ==C,,_1(C). (2.7.5) Ifniseven then C,f(C) =./l/t2n'Z—l(C)(B./l/tzn/2—l(C). (2.7.6a) whereas ifnisodd C,f(C) =Ji/l,2l'I—ll/2(C). (2.7.6b) Rather than proceed with thestudy oftheClifford groups and their relation tothecomplex orthogonal groups weshall show how the complex Clifford algebras may berelated torealorthogonal spaces. IfVisareal n-dimensional orthogonal space with bilinear form g then VC,thecomplexification ofV,isann-dimensional complex vector space. The real bilinear form gmay beextended byC-linearity toa C-bilinear form onVC,gc.Ifgisnon-degenerate then soisgc.IfVC isregarded asa2n-dimensional realvector space then Viscanonically identified with ann-dimensional subspace. The complex algebra THE COMPLEXIFIED CLIFFORD ALGEBRAS 81 C(1/c, gc) may be regarded as a 2n+1-dimensional real algebra. Thus regarded C(V, g) is a subalgebra which certainly commutes with the subalgebra generated by the identity over the complex field. So we have the following isomorphism of real algebras C(Vc, ge) C(V, g) 0 CORY (2.7.7) For C(V, g)OC(IR) we shall write C c(V, g). We may define the conjugate-linear operation of complex conjugation, *, on Vc: if z E VC then z = x + iy for x, y E V and z* = x — iy. Complex conjugation extends to an automorphism of C(V'c, gc) regarded as a real algebra (although not of course as a complex algebra). If the real subalgebra consists of all elements equal to their complex conjugates then the real subalgebra of C(V'c, gc) is of course C(V, g). It is worth stressing that for an arbitrary complex vector space there is no naturally defined operation of complex conjugation. It is here well defined because the complex vector space is obtained from the complexification of some underlying real vector space. We have already shown that the complex Clifford algebras are isomorphic to complex matrix algebras or sums of two such algebras. The operation of complex conjugation, *, as defined above will not, however, necessarily simply complex conjugate the components of these matrices. The situation is clarified below. Suppose that ,94(IR) C(F1)0JR,.(1F1) has some involutory auto- morphism, *, that induces a non-trivial automorphism on the centre. Let 91 be the real subalgebra, that is a E if and only if a = a*. Since any a E Si can be written as a sum of real and imaginary parts it follows that CCA. So we have C091 CO.Att,.. For this to be true 91 must certainly be simple, and since the only simple real algebras are iso- morphic to aaitt with a = IF1, C or H we must have either At., or HOhtr12. BY writing si = Caltr for some particular matrix sub- algebra At, we can define another involutory automorphism * that leaves elements of .A4., invariant and conjugates elements of the centre. If {e is an ordinary matrix basis for A,. then {CI) is another ordinary matrix basis, for Atr' say. It follows from the uniqueness of the Wedderburn decomposition ((A24) of Appendix A) that e*ii = me4m-1 for some m E .54. So if a = Eaqeq ,=, then a* = E a*,jme = m E jei ,m_, i.„.l i.,=, that is a* = ma# (2.7.8) THE COMPLEXIFIED CLIFFORD ALGEBRAS 81 C(VC, gc)may beregarded asa2"“-dimensional real algebra. Thus regarded C(V, g)isasubalgebra which certainly commutes with the subalgebra generated bytheidentity over thecomplex field. Sowehave thefollowing isomorphism ofrealalgebras C(VC, gc)=C(V, g)(>9C(]R). (2.7.7) For C(V, g)®C(]B) weshall write CC(V, g).We may define the conjugate-linear operation ofcomplex conjugation, *,onVC:ifzeV9 then 2=x+iyforx,yeVand 2*=x—iy.Complex conjugation extends toanautomorphism ofC(VC, gc) regarded asareal algebra (although notofcourse asacomplex algebra). Ifthereal subalgebra consists ofallelements equal totheir complex conjugates then thereal subalgebra ofC(VC, gc)isofcourse C(V, g).Itisworth stressing that foranarbitrary complex vector space there isnonaturally defined operation ofcomplex conjugation. Itishere well defined because the complex vector space isobtained from thecomplexification ofsome underlying realvector space. Wehave already shown that thecomplex Clifford algebras areisomorphic tocomplex matrix algebras orsums of twosuch algebras. The operation ofcomplex conjugation, *,asdefined above willnot, however, necessarily simply complex conjugate the components ofthese matrices. Thesituation isclarified below. Suppose that .v4(]B) =C(]R)®Jl/l,(]B) has some involutory auto- morphism, *,thatinduces anon-trivial automorphism onthecentre. Let 93betherealsubalgebra, that isae93ifandonly ifa=a*.Since any ae9.4canbewritten asasum ofrealandimaginary parts itfollows that $4=C®9B. Sowehave C®9B =C®Jl/1,. Forthistobetrue 93must certainly besimple, and since theonly simple real algebras areiso- morphic to@®A/l with EFD=IR,CorHwemust have either 9732A/t,or 973=H®M,/2. Bywriting 9.4=C®Jl/L, forsome particular matrix sub- algebra M,wecan define another involutory automorphism *that leaves elements ofM,invariant andconjugates elements ofthecentre. If{e,-I} isanordinary matrix basis forA/l,then {eff} isanother ordinary matrix basis, forA/l,’ say. Itfollows from the uniqueness ofthe Wedderburn decomposition ((A24) ofAppendix A)that e’j-,~=me,’-m'1 forsome me.91.Soif a=2aijeij1"./=1 then _ -1_ -1a*-2a*,-I-me,-I-m —m2a’j-je,-,~m i./‘=1 i./‘=1 thatis 0*=ma#m'1. (2.7.8) 82 CLIFFORD ALGEBRAS AND SPINORS Since * and * are involutory (2.7.8) gives m*m = p where p is in the centre. Now * and * induce the same automorphism on the centre, giving (m*m)* = (m*m)* = m-1(m*m)m. That is, mm* = m*m and p is in fact real. The defining property of m, (2.7.8), only determines it up to a multiple of the centre and so by a suitable scaling we can arrange either m*m = 1, or m*m = —1. (Equivalently m*m = 1 or m*m -= —1.) We summarise as follows: = 010C with * an auto- morphism that conjugates C and leaves 03 invariant, and s4 = AtrOC with * an automorphism that conjugates C and leaves At, invariant. The two automorphisms are related by a* = ma*m -1. There are two possi- bilities for 01, either 03 At, or 01 Hattri2; and two possibilities for m, either mm* = 1 or mm* —1. These possibilities are in fact related. If C003 = CC/At, with * and * automorphisms that conjugate the centre and leave 03 and At, respectively in- variant, then a* = ma* m-1 where we can choose either mm* = 1 <=> = At „ or mm* = —1<=> 9.3 = Hattr12- (2.7.9) We now consider the proof. Since there are two and only two mutually exclusive possibilities for and similarly for m, if we can prove that mm* = 1 <=> = At, then we must have mm* = —1 <4. 03 = HOAtr12. Suppose firstly that 03 = At„ Then if {b and {e 0} are ordinary matrix bases for 03 and AA,, respectively then e u -= sb us-1 for some s Est If we write a = E a ue then a* = E 4i(sbiis -1)* = E a II 1,1 = E a * -le uss* = s* s -1 s -1)- 1 . i.; That is, we may choose m = s*s-1 giving m* = m-1. Now the converse: we introduce a C-conjugate-linear transformation on si by defining ac = a* m -= ma*. Thus c preserves the columns of At,. If m* = m-1 then c is involutory and for any a E .91 we write a = (a + ac) + (a — ac). In particular, the minimal left ideals of s4 that are the columns of At, with entries in C can be decomposed into eigenspaces of c. Since the real dimension of a minimal left ideal of si is 2r these eigenspaces are r-dimensional. Let p be an element of one of these eigenspaces. Then if a E A atp is certainly in the minimal left ideal of si and since (aip)c = a*Vc = atpc it is in fact in the eigenspace. Hence these eigenspaces carry representations of @. That is, if m* = m irreducible representations of si induce reducible repre- sentations of 91. But either 01 -= At,, in which case its irreducible 82 CLIFFORD ALGEBRAS AND SPINORS Since *and#areinvolutory (2.7.8) gives m*m =pwhere pisinthe centre. Now *and #induce thesame automorphism onthecentre, giving (m*m)* =(m*m)* =m"(m*m)m. That is,mm* =m*m andp isinfactreal. The defining property ofm,(2.7.8), only determines itup toamultiple ofthecentre andsobyasuitable scaling wecanarrange either m*m =1,or m*m =—1. (Equivalently m*m =1or m*m =—1.) Wesummarise asfollows: .91=93®C with *anauto- morphism thatconjugates Candleaves 93invariant, and91=A/l,®C with #anautomorphism thatconjugates Candleaves M,invariant. The twoautomorphisms arerelated bya*=ma#m‘1. There aretwopossi- bilities for93,either 93=A/t,or93=H®M,,2; andtwopossibilities for m,either mm* =1ormm* E—1.These possibilities areinfactrelated. If.91= C®93 =C®A/1, with *and *automorphisms that conjugate thecentre and leave 93and M,respectively in- variant, then a*=ma#m" where wecan choose either mm* =1©93 =A/1,,ormm* =-1©93 =H®M,,2. (2.7.9) We now consider the proof. Since there aretwo and only two mutually exclusive possibilities for93,and similarly form,ifwecan prove that mm* =1<993=A/t, then we must have mm* =-1@973 =H®M,,2. Suppose firstly that93=A/1,.Then if{b,~]-} and {e,7}areordinary matrix bases for93and A/L,respectively then e,-j=sb,-is" forsome se91.Ifwewrite a=2aijeij I1 then a*=2a§}(sb,]-s")* =Ea;-s*b,-I-s*‘1 1,) 1,) _ - -_ -1 -1-1-Ea};-s*s ‘e,-jss* 1-s*s a#(s*s ). Iv] That is,wemay choose m=s*s"1 giving m*=m'1. Now the converse: weintroduce aC-conjugate-linear transformation on.91by defining a°=a*m =ma*. Thus °preserves thecolumns ofA/1,. If m*=m" then °isinvolutory and forany ae.91 wewrite a= §(a+a°)+§(a—a°). Inparticular, theminimal leftideals of.91that arethecolumns ofM,with entries inCcanbedecomposed into eigenspaces of°.Since therealdimension ofaminimal leftideal of.91is 2rthese eigenspaces arer-dimensional. Let1pbeanelement ofoneof these eigenspaces. Then ifae93 at/2iscertainly intheminimal left ideal of.91andsince (at/2)“ =a*1p° =aw“itisinfactintheeigenspace. Hence these eigenspaces carry representations of93.That is,if m*=m'1 then irreducible representations of.91induce reducible repre- sentations of93.But either 93=Jl/1,, inwhich case itsirreducible THE COMPLEXIFIED CLIFFORD ALGEBRAS 83 representations are r-dimensional, or 94 HOAt r/2 with 2r-dimensional irreducible representations. Thus m* = m-1 implies gi = A,. The argu- ment of the proof is summarised below. m = s*s-1 mm = 1 irreducible representations of .94 induce reducible representations of @. Hence mm* = 1 <=> = At, and so mm* = —1<=> 91 = HOAtr/2. The complexification of the real Clifford algebra associated with an even-dimensional orthogonal space is isomorphic to the algebra of complex matrices. Complex conjugation (that leaves the real Clifford algebra invariant) is equivalent to the automorphism that conjugates the components of these matrices when the real algebra is a total matrix algebra: in this case complex conjugation will simply conjugate the components in an appropriate basis. The algebra associated with the complexification of an odd-dimensional real orthogonal space is a direct sum of two matrix algebras. Complex conjugation is equivalent to the automorphism that conjugates the components of these matrices if and only if the real algebra is a sum of two total matrix algebras. When the real algebra is the sum of two simple algebras whose Wedderburn decomposition involves the quaternions then complex conjugation in- duces an automorphism on the simple components of the complexified algebra that is inequivalent to conjugating the matrix components. When the real algebra is isomorphic to the algebra of complex matrices then complex conjugation of the complexified algebra interchanges the simple components. The irreducible representations of the complex algebras will again be called spinor representations, or semi-spinor representations when the algebra is reducible, the spinor (or semi-spinor) spaces being identified with minimal left ideals. These minimal left ideals are obviously of complex dimension 21n/2] where [n/2] denotes the integer part of n/2. When n is even there is only one such representation, up to equiva- lence, whereas if n is odd there are two inequivalent semi-spinor representations. Irreducible representations of C(VC, g induce representations of C( V, g) which may or may not be reducible. The question of the reducibility of these representations has to some extent been anticipated in §2.5. It was shown there that when the division algebra occurring in the Wedderburn decomposition of the real Clifford algebra (or a simple THE COMPLEXIFIED CLIFFORD ALGEBRAS representations arer-dimensional, or93=H®Jl/t,,2 with 2r-dimensional irreducible representations. Thus m*=m‘1 implies 93=A/1,.The argu- ment oftheproof issummarised below. m=s*s“ $mm* =1 => U 93=A/t, ¢irreducible representations of$1 induce reducible representations of 93. Hence mm* =1¢i> 93=M, andsomm* =—1¢i> 93=H®Jl/t,,2. The complexification ofthereal Clifford algebra associated with an even-dimensional orthogonal space isisomorphic tothe algebra of complex matrices. Complex conjugation (that leaves thereal Clifford algebra invariant) isequivalent totheautomorphism thatconjugates the components ofthese matrices when thereal algebra isatotal matrix algebra: inthiscase complex conjugation willsimply conjugate the components inanappropriate basis. The algebra associated with the complexification ofanodd-dimensional realorthogonal space isadirect sumoftwomatrix algebras. Complex conjugation isequivalent tothe automorphism that conjugates thecomponents ofthese matrices ifand only iftherealalgebra isasum oftwototal matrix algebras. When the real algebra isthesum oftwo simple algebras whose Wedderburn decomposition involves thequaternions then complex conjugation in- duces anautomorphism onthesimple components ofthecomplexified algebra that isinequivalent toconjugating thematrix components. When therealalgebra isisomorphic tothealgebra ofcomplex matrices then complex conjugation ofthecomplexified algebra interchanges the simple components. The irreducible representations ofthecomplex algebras willagain be called spinor representations, orsemi-spinor representations when the algebra isreducible, thespinor (orsemi-spinor) spaces being identified with minimal leftideals. These minimal leftideals areobviously of complex dimension 29/2] where [n/2] denotes theinteger part ofn/2. When niseven there isonly onesuch representation, uptoequiva- lence, whereas ifnisodd there aretwo inequivalent semi-spinor representations. Irreducible representations ofC(VC, gt) induce representations of C(V, g)which may ormay notbereducible. The question ofthe reducibility ofthese representations hastosome extent been anticipated in§2.5. Itwasshown there that when thedivision algebra occurring in theWedderburn decomposition oftherealClifford algebra (orasimple 84 CLIFFORD ALGEBRAS AND SPINORS component of that algebra) was C or H the complex structure of right multiplication by the generator of a complex subalgebra enabled the spinor (or semi-spinor) space to be regarded as a complex vector space. In this case irreducible representations of the real algebra can be extended by C-linearity to representations of the complexified algebra. Thus conversely, in these cases irreducible representations of the corn- plexified algebra induce irreducible representations of the real algebra. When the real Clifford algebra is isomorphic to the algebra of real matrices, or the sum of two such algebras, then its irreducible repre- sentations are of real dimension 2 1n121; that is, half that of the real dimension of the irreducible representations of the complexified algebra. Thus, in these cases, irreducible representations of the complexified algebra induce reducible representations of the real algebra. The way in which this reduction can be performed was given in the proof of (2.7.9). The induced representations of the real even subalgebra may be treated in exactly the same way. The irreducible representations of the even subalgebra of the complexified algebra are of real dimension 2(2[(n —1)12]), whilst the dimensions of those of the real even subalgebra are given in table 2.10. We turn now to classifying involutions of the complexified algebras. The C-linear involutions and ij induce the standard involutions on the real subalgebra, and these have already been classified. Thus we may classify these involutions on the complexified algebra from a knowledge of the involutions that they induce on the factors of a tensor product. The involution induced on the factor C is of class 3, and so if we multiply the entries in table 2.15 by 3, using the multiplication of table 2.13, then we obtain the class of `" and ij on the complexified algebra, and that of the involution they induce on its even subalgebra. (The classes are given in table 2.11.) Now these involutions are of course involutions of the Clifford algebra associated with the complex vector space Vc, and so they can only depend on the dimension of V and not the signature of g. The classes depend on n mod 8, and are given in table 2.17. The involutions and oi commute with complex conjugation, and so they may be composed with it to form involutions and Ei which again induce the standard involutions on the real subalgebra. These involutions are certainly not involutions of C(1/c, g c) regarded as a complex algebra, but are real algebra involutions. The classes can be obtained by multiplying the entries in table 2.15 by five using the multiplication of table 2.13. Of course on the simple algebras these involutions can only be of class 5, whilst in the reducible case they either induce involutions of class 5 on the component algebras or interchange those components. The classes depend on p mod 2 and q mod 2, and are given in table 2.18. It follows from (2.6.23) that is the adjoint of a zero index Hermitian-symmetric product if and only if 84 CLIFFORD ALGEBRAS AND SPINORS component ofthat algebra) wasCorHthecomplex structure ofright multiplication bythegenerator ofacomplex subalgebra enabled the spinor (orsemi-spinor) space toberegarded asacomplex vector space. Inthis case irreducible representations ofthereal algebra can be extended byC-linearity torepresentations ofthecomplexified algebra. Thus conversely, inthese cases irreducible representations ofthecom- plexified algebra induce irreducible representations ofthereal algebra. When thereal Clifford algebra isisomorphic tothealgebra ofreal matrices, orthesum oftwo such algebras, then itsirreducible repre- sentations areofreal dimension 2l"’21; that is,half that ofthereal dimension oftheirreducible representations ofthecomplexified algebra. Thus, inthese cases, irreducible representations ofthecomplexified algebra induce reducible representations oftherealalgebra. The way in which thisreduction canbeperformed wasgiven intheproof of(2.7.9). Theinduced representations oftherealeven subalgebra may betreated inexactly thesame way. The irreducible representations oftheeven subalgebra ofthecomplexified algebra areofrealdimension 2(2[("*‘)’2]), whilst thedimensions ofthose oftherealeven subalgebra aregiven in table 2.10. Weturn now toclassifying involutions ofthecomplexified algebras. TheC-linear involutions §andE17induce thestandard involutions onthe real subalgebra, and these have already been classified. Thus wemay classify these involutions onthecomplexified algebra from aknowledge oftheinvolutions that they induce onthefactors ofatensor product. The involution induced onthefactor Cisofclass 3,and soifwe multiply theentries intable 2.15 by3,using themultiplication of table 2.13. then weobtain theclass of‘g’and E17onthecomplexified algebra, andthat oftheinvolution they induce onitseven subalgebra. (The classes aregiven intable 2.11.) Now these involutions areof course involutions oftheClifford algebra associated with thecomplex vector space VC, andsothey canonly depend onthedimension ofV andnotthesignature ofg.The classes depend onnmod8, and are given intable 2.17. The involutions Eand §17commute with complex conjugation, andsothey may becomposed with ittoform involutions 5*and §1;*which again induce thestandard involutions onthereal subalgebra. These involutions arecertainly notinvolutions ofC(VC, gc) regarded asacomplex algebra, butarereal algebra involutions. The classes canbeobtained bymultiplying theentries intable 2.15 byfive using themultiplication oftable 2.13. Ofcourse onthesimple algebras these involutions canonly beofclass 5,whilst inthereducible case they either induce involutions ofclass 5onthecomponent algebras or interchange those components. The classes depend onpmod2 and q mod2, andaregiven intable 2.18. Itfollows from (2.6.23) that5*isthe adjoint ofazero index Hermitian-symmetric product ifand only if THE COMPLEXIFIED CLIFFORD ALGEBRAS 85 q --= 0; otherwise any index is maximal. Similarly is the adjoint of a zero index product if and only if p = 0, otherwise maximal. Table 2.17 Classification of involutions of the complexified Clifford algebras. p + q = n on C;,(11) 0 C 1 3 0 3 10 3 2 3 4 10 3 10 4 0 4 4 4 4 4 4 CI 4 5 4 e 4 10 4 6 4 3 10 7 10 3 0 3 3 8 3 3 3 0 3 Table 2.18 The classes of and on C,(F1) C. and ij on Cp,,(E) C 0 1 0 1 5 55 5 5 0 10 5 5 10 5®5 5 5 5 10 2.8 The Confusion of Tongues The theory of spinors was developed independently by physicists and mathematicians, and this historical apartheid has continued. Of particu- lar physical interest is the case of a four-dimensional real vector space with a Lorentzian metric, and it was in this case that much of the terminology and notation used by physicists originated. More recently there has been much interest in physical theories set in a variety of different dimensions and the nomenclature and terminology has been extrapolated to these situations. Thus there is now a language, with many dialects, for discussing spinors in physics which makes little THE COMPLEXIFIED CLIFFORD ALGEBRAS q=0;otherwise anyindex ismaximal. Similarly 517*istheadjoint ofa zero index product ifandonly ifp=0,otherwise maximal. Table 2.17 Classification ofinvolutions ofthecomplexified Clifford algebras. p+q=n E Er] ‘§onC;_,,(B)®C 3@3 10 3 3 4 10 10 4(-34 4 4®4 4®4 10 4 OO\lO\LI1J>U)l\)>-'-P-P L»)-P )—lC 10 3®3 3 3 3 3633 Table 2.18 Theclasses of‘§*and‘§r]*onC,,,,,(B) ®C and‘§*onC;_,(Fl) ®C. P q O 1 0 5 10 5 5635 5®5 5 1 5635 5 10 5 5 10 2.8TheConfusion ofTongues The theory ofspinors was developed independently byphysicists and mathematicians, andthishistorical apartheid hascontinued. Ofparticu- larphysical interest isthecase ofafour-dimensional real vector space with aLorentzian metric, and itwas inthis case that much ofthe terminology and notation used byphysicists originated. More recently there hasbeen much interest inphysical theories setinavariety of different dimensions and thenomenclature and terminology hasbeen extrapolated tothese situations. Thus there isnow alanguage, with many dialects, fordiscussing spinors inphysics which makes little 86 CLIFFORD ALGEBRAS AND SPINORS contact with the expositions of the theory to be found in the mathe- matics literature. Physicist readers may at this point vehemently declare that it also makes little contact with the exposition given here. We will now try to redress this situation. The Dirac matrices, or y-matrices, are usually defined to be complex square matrices of minimal order that satisfy ya yb yb ya = 211ab (2.8.1) where q is diagonal with p entries of plus one and q of minus one. If p + q = n then the order of these marices is 21n/21 with the bracket denoting the integer part. These matrices are also usually assumed to have certain Hermiticity properties, and we shall examine this shortly. Here we note that the presence of such operations that are not C-linear is sufficient to infer that the y-matrix algebra is not to be regarded as a complex algebra. In fact from (2.7.4) we recognise that these matrices generate an algebra isomorphic to that of the complexified Clifford algebra or, in odd dimensions, a simple component of that algebra. For the case of n even we have C ,q .42.12(C). If {ea} is a basis for the real vector space that generates C p,q(1E1) then nI2 ea = E ye t, j=1 where { e u} is some ordinary matrix basis for the complexified algebra. The arrays of complex components, y, with the usual rules of matrix multiplication, will obviously satisfy (2.8.1). All matrix bases of the complexified algebra are related by an inner automorphism, the change of matrix basis giving a new set of matrix components for the {ea}; an equivalent representation of the y-matrices. The way in which a matrix basis can be constructed and the matrix components of any element found is contained in the proof of the Wedderburn structure theorem, (A23) of Appendix A. An explicit example was given at the end of §2.2. We now further restrict ourselves to the complexification of Cp,1(IF1), for p odd, and show how a 'standard' representation of the y-matrices can be given. (Although we shall have no need of such representations this will hopefully strengthen the link with the standard physics literature.) For p odd = At2u—Di2(C) 0 .4 2(p-012(C) and thus C 1 is isomorphic to a total matrix algebra with C 0(C) a subalgebra isomorphic to the direct sum of two algebras of matrices of half the order. In a suitable matrix basis, therefore, C pc., is the subalgebra of elements whose matrix components are block-diagonal; the two simple component algebras having matrix components in only the upper or lower blocks, that is 86 CLIFFORD ALGEBRAS AND SPINORS contact with theexpositions ofthetheory tobefound inthemathe- matics literature. Physicist readers may atthispoint vehemently declare that italso makes little contact with theexposition given here. Wewill now trytoredress thissituation. The Dirac matrices, ory-matrices, areusually defined tobecomplex square matrices ofminimal order thatsatisfy i'“i"’+i"’i'”=2'1"” (2-8-1) where 1)isdiagonal with pentries ofplus oneandqofminus one. If p+q=nthen theorder ofthese marices is21"”! with thebracket denoting theinteger part. These matrices arealso usually assumed to have certain Hermiticity properties, andweshall examine thisshortly. Here wenote thatthepresence ofsuch operations that arenotC—linear issufficient toinfer thatthey-matrix algebra isnottoberegarded asa complex algebra. Infactfrom (2.7.4) werecognise that these matrices generate analgebra isomorphic tothat ofthecomplexified Clifford algebra or,inodddimensions, asimple component ofthatalgebra. Forthecase ofneven wehave CE),=A/12»/2(C). If{e”} isabasis for therealvector space thatgenerates C,,_,,(lB) then n/2 ea= Yijeij I./=1 where {e,1}issome ordinary matrix basis forthecomplexified algebra. The arrays ofcomplex components, yfj,with theusual rules ofmatrix multiplication, will obviously satisfy (2.8.1). Allmatrix bases ofthe complexified algebra arerelated byaninner automorphism, thechange ofmatrix basis giving anew setofmatrix components forthe{e"}; an equivalent representation ofthey-matrices. The way inwhich amatrix basis canbeconstructed andthematrix components ofany element found iscontained intheproof ofthe Wedderburn structure theorem, (A23) ofAppendix A.An explicit example wasgiven attheendof§2.2. Wenow further restrict ourselves tothe complexification ofC,,_1(lB), forpodd, and show how a ‘standard’ representation ofthey-matrices canbegiven. (Although we shall have noneed ofsuch representations thiswillhopefully strengthen thelinkwith thestandard physics literature.) Forpodd C50 =-/I/l.2<p-1)/1(6) (BJl/12¢-I)/1(C) and thus C51 isisomorphic toatotal matrix algebra with C,,_0(C) a subalgebra isomorphic tothedirect sum oftwoalgebras ofmatrices of half the order. Inasuitable matrix basis, therefore, CED isthe subalgebra ofelements whose matrix components areblock-diagonal; thetwosimple component algebras having matrix components inonly theupper orlower blocks, thatis THE CONFUSION OF TONGUES 87 7, = ( 0) i 0 l I = 1, . . p \o E with o' and E' matrices of order From table 2.18 and the remark at the end of §2.7 it follows that the involution on C.c.() is the adjoint involution of a zero-index C*- symmetric product; that is, it is equivalent to Hermitian conjugation. Thus we can arrange a basis in which on C1 induces Hermitian conjugation on the diagonal blocks (but not, of course, on the off- diagonal blocks), and in such a basis a' and E' are Hermitian. If = . . . el) with A = 1 or i such that 2 = 1, and P + = ± then P, and P_ are the identities in the simple components of Cpc.0. Since = P, — P_ then if 2= Ayi . . . yP then = 0 —I ). But e° anticommutes with and so y° can only have off-diagonal components. Since also (e°)2 = —1 we must have 7o = _T_Oi To ) for some non-singular matrix T. Since e° anticommutes with all e` we must in fact have E' = — . If we now change basis so that the components transform ya —> S yaS -1 with S= — 1 ( I —iT ) S -1 = 1 ( I I ) V2 I iT V2 iT -1 —iT -1 J then we arrive at the following 'standard' representation of the y- matrices: yo = ) 0 —I ( u' = a' 0 ). (2.8.2) Here ai, and hence y', are Hermitian whilst y° is manifestly anti- Hermitian. The case of CF., may be treated similarly. Since is equivalent to Hermitian conjugation in C;;.„ we are lead to a 'standard' representation as above, but with the ai anti-Hermitian and the i removed from y°. (In this case e° denoting the one positive-norm vector.) To illustrate further the relation between the y-matrices and the more abstract approach to Clifford algebras that we have pursued, we examine C1 in more detail. First we shall choose a matrix basis for a simple component of C'c, in which coincides with Hermitian conjuga- tion giving the Hermitian {a'}. We then have from (2.8.2) a standard representation of the y-matrices and shall reverse the argument to THE CONFUSION OFTONGUES 87 , 0‘ 0 .y= 02,- z=1,...,p with 0'andZ‘matrices oforder 29")”. From table 2.18 andtheremark attheendof§2.7 itfollows that the involution 5*onCEOistheadjoint involution ofazero-index C*- symmetric product; that is,itisequivalent toHermitian conjugation. Thus wecanarrange abasis inwhich 5*onCE,induces Hermitian conjugation onthediagonal blocks (but not, ofcourse, ontheoff- diagonal blocks), and insuch abasis 0‘and E"areHermitian. If E:/lel el’withA= 1orisuch that E2=1, andPi=§(1i E), then P,and P_aretheidentities inthesimple components ofC50. Since E=P,—P-then ifY/=/lyl...yl’then3-(I1.’). But e°anticommutes with Eand soyocan only have off-diagonal components. Since also(e°)2 =—1wemust have y,:( 0T) _-|--10 forsome non-singular matrix T.Since e°anticommutes with alle‘we must infacthave Z‘=——T‘1aT. Ifwenowchange basis sothatthe components transform y”->Sy”S" with _L I-iT) _1_L( ' l) ST\/zjl iT ST\/2iT"1 —iT" then wearrive atthefollowing ‘standard’ representation ofthey- matrices: y°=i((l) _‘|]) yi= (2.82) Here 0',and hence y‘,areHermitian whilst yoismanifestly anti- Hermitian. The case ofCf, may betreated similarly. Since 517* is equivalent toHermitian conjugation inC8,,weareleadtoa‘standard’ representation asabove, but with the0'anti-Hermitian and thei removed from yo.(Inthis case e°denoting theone positive-norm vector.) Toillustrate further therelation between they-matrices andthemore abstract approach toClifford algebras that wehave pursued, we examine Cf,inmore detail. First weshall choose amatrix basis fora Simple component ofCg,inwhich 5*coincides with Hermitian conjuga- Ii0ngiving theHermitian {rr'}. Wethen have from (2.8.2) astandard representation ofthe y-matrices and shall reverse theargument to 88 CLIFFORD ALGEBRAS AND SPINORS construct the matrix basis in which these are the components of the feal. The reducible algebra C 0 is projected into simple components by the mutually commuting pair of central idempotents P, = ie123%. j Since e123P, = -T-iP+, giving e12P, = -Tie3P+, if we want cricr2 = ity3 then the {01 must be the components of the {e' } in CoP_. To start the construction of the matrix basis we seek a pair of mutually orthogonal idempotents that are invariant under the involution r: these will form the diagonals of a basis in which induces Hermitian conjugation. We choose en = (1 + e3)P _ e22 = 21(1 — e3)P - and since e22 = e2e lie2 we may complete the basis with e12 — e11e 2 = F2e22 en = e2 e11 = e22e 2 (2.8.3) (2.8.4) where If p — p -21 — -12. 2 e`P_ -= E at ,oto cr,t3=1 then 2 ==E 2=1 For example, al = F11e lF21 F21F1F22 = —iF11e23F71 iF21F23F22 since e123P_ = iP_ = iF11e2F21 iF21F2F22 where the e2 has been absorbed into E21 and c22. From the definition of F21 a112 = i(Fii E22) = iP- where, we recall, P_ is the identity in this simple algebra. In this way we construct the following: 0., =( O. i a2 ( 3 ( 1 ). (2.8.5) = \ —1 ) k 1 I o —1 We may use these matrices in (2.8.2) to obtain a standard representa- 88 CLIFFORD ALGEBRAS AND SPINORS construct thematrix basis inwhich these arethecomponents ofthe {e“}. The reducible algebra Cf),isprojected intosimple components by the mutually commuting pair of central idempotents Pi= §(1iiem). Since e‘23Pi =liPi, giving e12Pg =-?ie3Pi, ifwewant oloz =I03then the{rr'} must bethecomponents ofthe{e‘}inC§_0P_. Tostart theconstruction ofthematrix basis weseek apairofmutually orthogonal idempotents thatareinvariant under theinvolution 5*:these will form thediagonals ofabasis inwhich 5*induces Hermitian conjugation. Wechoose e=l(1+e3)P_“L2 3 (2.s.3) 622 :i(1— 8 andsince 922=ezenez wemay complete thebasis with e=ee2=eze I2 11 22 (28.4) 921: @2911: 9229“ where 9215* =912- If -‘.3..(\4..9.. EIP _= ,,j;€aj3 then 2 U1”); :2£}_a€l£B)(. I-1 Forexample, _ 1 1Uiiz —9119 521'1'5219 522 _-23 -23_-15119 521—15216 £22 since e”3P_ =iP_ _-2 -2—18118 £21 -l-18218 £22 where thee2hasbeen absorbed into£21and£22.From thedefinition of£21 (7112 =K511 +1‘-'22)=iP— where, werecall, P_istheidentity inthissimple algebra. Inthisway weconstruct thefollowing: _01 _01 ,_(10)o1—(_i 0) o2—(10) 0-0_1. (2.8.5) Wemay usethese matrices in(2.82) toobtain astandard representa- THE CONFUSION OF TONGUES 89 tion of the y-matrices. At this point we reverse the reasoning and construct the matrix basis corresponding to these components. From the diagonal y° we construct a pair of (non-primitive) idempotents, 0 + iy()) = 0 1 1 1 0 Putting (2.8.5) into (2.8.2) enables another pair of idempotent matrices to be constructed 0 1 i7172) = ( 1(1 + irly2) 1 o 1 0 1 Primitives are obtained from the four products of these two pairs of idempotents, for example 1(1 iy0)1(1 iy172) = If then 4 e° = E ya4e4 we have = _ ieow _ i e12) e22 = 41 — ie°)1(1 + ie 12) e33 = 1(1 + ie°)1(1 — ie 12) e44 = 1(1 + ie°)1(1 + le ' 2). (2.8.6) In exactly the same way we take products of the y-matrices to produce a matrix of zeroes except for a 1 in the i, j entry, for all i and j. As may readily be checked this leads to the conclusion that the remainder of the matrix basis must be as shown in table 2.19. In odd dimensions there are two inequivalent representations of the y-matrices: these being the matrix components of the {ea} projected into either of the simple component algebras. We now show how a standard representation can be constructed for C, where now p is even. In this case = ht2pn(C) Aty.(C) 1 0 0 THECONFUSION OFTONGUES 39 [ion ofthey-matrices. Atthis point wereverse thereasoning and construct thematrix basis corresponding tothese components. From the diagonal yoweconstruct apairof(non-primitive) idempotents, %<|+w°)= 6—i1)= . Putting (2.8.5) into(2.82) enables another pairofidempotent matrices tobeconstructed 0 1 %(l+i1"1'2)= I0 %(|—i1"1'2) = 01- 1 0 Primitives areobtained from thefour products ofthese two pairs of idempotents, forexample 1 %(|-i1'°)%(| —i1"1'2) =( 00 0 Ifthen i[\4,,ea=>_ I/aijeij ._- wehave en=§(1—ie°)%(1—ie12) en=§(l—ie°)§(1+ie'2) E33=%(l+ie°)%(1—iel2) e44=%(1+ie°)§(1+ie‘2).(2.s.6) Inexactly thesame waywetake products ofthey-matrices toproduce a matrix ofzeroes except fora1inthei,jentry, foralliandj.Asmay readily bechecked thisleads totheconclusion thattheremainder ofthe matrix basis must beasshown intable 2.19. Inodd dimensions there aretwoinequivalent representations ofthe y-matrices: these being thematrix components ofthe{e“} projected into either ofthesimple component algebras. Wenow show how a standard representation can beconstructed forC,‘f_,, where now pis even. Inthiscase C51 =Mgr/Z(C) (731/I/I.2PFl(¢:) 90 CLIFFORD ALGEBRAS AND SPINORS whereas g o = .42,i2(C). The involution is equivalent to Hermitian conjugation on Cpc,o, whereas it swaps the components of Cpc,,. We choose a matrix basis feill for Cpc,0 in which coincides with Hermitian conjugation. If P± are the central idempotents that project C, into simple components, and e = e,//3±, then the eu± form matrix bases for these component algebras. The involution t is defined on Cpc,, by the requirement that it conjugate the complex factor and satisfy the follow- ing properties on the generators of the real subalgebra: et = e', i =1, p, e°' = —e°. Thus t coincides with on C0 c and so is indeed p, Hermitian conjugation in the basis {e,1}. If z = ... ePe° then for p= 2 mod 4, P± = 1(1 ± iz), whereas for p = 0 mod 4, P „ = 1(1 ± iz). Now certainly zt = —zr, and as we have remarked swaps the simple components of Cpc,,, that is 13_,* = P_T, thus P ,* = P. It follows that, as the notation suggests, t induces Hermitian conjugation in the simple component algebras in the bases that we have constructed, {e u±). In such bases el', are represented by Hermitian matrices, whereas e°P_, is represented by an antiHermitian matrix. In fact for p = 0 mod 4 e°P.„ = Tie' . ePP,_, whereas for p = 2 mod 4, e°13, = Tel ... Table 2.19 A matrix basis for eu, e11 —e23e22 e3e33 e23eli e22 —e3e44 e3e e33 ieie —e3e22 e 23e23 e4.4 The representation-independent operator trace, Tr, projects a matrix algebra onto the subspace spanned by the identity. There is therefore a relation between the projection of the Clifford algebra onto the space of 0-forms, Wo, and the trace of the y-matrices. In even dimensions any element can be expanded in a matrix basis a = Eagev. Since the au are (complex) 0-forms rI2 )0(a) = Eay0(eq). and since products can be reversed under S o, (2.1.17), 9'0(e1) = Yo(euegen) = Wo(euelle„) = g'0(ezi)6,. 90 CLIFFORD ALGEBRAS AND SPINORS whereas C50=A/12»/:(C). The involution 5*isequivalent toHermitian conjugation onCED, whereas itswaps thecomponents ofC5,. We choose amatrix basis {e,-j} forC50inwhich 5*coincides withHermitian conjugation. IfPgarethecentral idempotents thatproject CE‘,into simple components, andef=e,1}-Pi, then thee,-ii form matrix bases forthese component algebras. The involution "lisdefined onCg,bythe requirement thatitconjugate thecomplex factor andsatisfy thefollow- ingproperties onthegenerators oftherealsubalgebra: e“=e’,i=1, ...,p,e°*=—e°. Thus Tcoincides with 5*onC50andsoisindeed Hermitian conjugation inthebasis {e,-,-}. Ifz=e1...ePe° then for p=2mod4, Pi=§(1_tiz), whereas forp=0mod4, Pi= §(1iiz).Now certainly 2*=-25‘, andaswehave remarked 5*swaps thesimple components ofCg}, thatisPi§* =P;,thus P;=Pt.It follows that, asthenotation suggests, Tinduces Hermitian conjugation inthesimple component algebras inthebases thatwehave constructed, {e,}-1}. Insuch bases e’Pi arerepresented byHermitian matrices, whereas e°P.£ isrepresented byanantiHermitian matrix. Infact for p=0mod4 e°Pi =iiel ... ePPi, whereas for p=2mod4, e°Pi =le‘...ePP,:. Table 2.19 Amatrix basis forC§,. er/—> 1 en —e13e,2 e363; —ie1e..,., e23e1, en ie1e,3 —e3e.,.. e3e11 —ie1e22 e3, —e23e..,., ieleu —e3e22 e23e23 644 The representation-independent operator trace, Tr,projects amatrix algebra onto thesubspace spanned bytheidentity. There istherefore a relation between theprojection oftheClifford algebra onto thespace of 0-forms, SP0,andthetrace ofthey-matrices. Ineven dimensions any element canbeexpanded inamatrix basis 2n/2 £1: 2311,76 ii. i.j Since thea,-,-are(complex) 0-forms 2/1/2 90(4) =_2a1)90(e1;)~ IvI andsince products canbereversed under S0,(2.1.17), 9>0(e1"/) =9)0(e1'1'e1";e;/) =EP0(eijejje1'i) =EP0(ei1')6I'}" THE CONFUSION OF TONGUES 91 The diagonals in the matrix basis are a set of pairwise orthogonal primitive idempotents, and these are all similar. So = 9'0(se1s-') for some s, thus 970(e0 = 9 70(e ii). By writing the identity as a sum of primitives 1 = e + e 22 + . . . + e„ we have S0(e11) = 1/(2' 2). Thus with r = 21/2 &0(a) = 1 that is, 2n 2 1 90(a)= 212/2 Tr a. (2.8.7) In odd dimensions we let P, denote the central idempotents. If, for example, {e } is a matrix basis for the simple algebra whose identity is P, then P+ = ell + + ea+ r = 2(n-1)/2. Since Yo(P+) = we have 970(e 4-) = 1/[2(2 ("-1)12)] giving Yo(aP,)= (11[2(20-012))) Tr(aP+). Thus 1 &0(a) = 2(2(12-1)/2) [Tr(aP.4.) + Tr(aP_)]. (2.8.8) In calculating cross sections in quantum theory one uses various trace theorems for the y-matrices. The following illustrative properties of So are equivalent to some of the most important. If {al, ..., an} is a set of 1-forms then a'0(a1a2 . . . an) = 0 for n odd, and J0(a1a2... an) = 520(a12 . . . a2a1). These follow from the more general relations ?Op = = Sp The 0-form component of a product of n 1-forms, with n even, can be related to that of products of n-2 terms, from (2.1.7) &0(a1a2 .. an) = 9'0{a1 A (a2    an) + iwt(a2    an)) = j'o{iiia2a3 . . a, — a2i-a-,a3a4 . a, + + a2 an_lidian} = g(a,, a2)990(a3a4 . . . an) — g(ai, a3)f0(a2a4 . . an) + . . . + g(ai, a12)990(a2a3  a12-1). THE CONFUSION OFTONGUES 91 The diagonals inthematrix basis areasetofpairwise orthogonal primitive idempotents, andthese areallsimilar. So 9)0(ejj) =9o(5°115_1) forsome s,thus 90(9)‘/) =9)0(e11‘)- Bywriting theidentity asasum ofprimitives 1=e,,+e22+...+e,, withr=2"’2 wehave S0(e,-,-) =1/(2”’2). Thus 1 90(9) :W 2911' thatis, 1 90(0) =w TI£1. (28.7) Inodd dimensions weletPidenote thecentral idempotents. If,for example, {e,-7*} isamatrix basis forthesimple algebra whose identity is P,then P+=e11++...+B,,+ r‘=2("_U/2. 5iI1¢¢ 9’o(P+) =3Wehave 9’o(°11+) =1/l3(2(”")’2)l giving 5/’o(¢1P+) = {1/[2(2("'1)’2)]} Tr(aP+). Thus 19’0(a) = [Tr(aP+) +Tr(aP_)]. (2.8.8) Incalculating cross sections inquantum theory oneuses various trace theorems forthey-matrices. The following illustrative properties ofS0 areequivalent tosome ofthemost important. If{ah ...,a,,}isaset of1-forms then 9’0(a1a2 ...a,,)=0fornodd, and 9’0(a1a2. ..a,,) =9’0(a,, ...azal). These follow from the more general relations 179,, =Sfpn, 59’,=SP5.The 0-form component ofaproduct ofn 1-forms, with neven, canberelated tothatofproducts ofn—2 terms, from (2.1.7) 5f0(aIa2 ~-an)=3)0{aI/\(a2-~-¢1,1)‘l'I5,(¢12---1111)} =3’0{i,;|a2a3 ...a,,—a2i,;,a3a4 ...an+...+ a2...a,,_l1,;la,,} =SW1» a2)9)0(a3a4 an)_SW1» a3)3)0(a2a4 -- a,,)+...+g(a,, a,,)3’0(a2a3 ...a,,_1). 92 CLIFFORD ALGEBRAS AND SPINORS In physics, elements of the vector space carrying an irreducible representation of the complexified Clifford algebra are termed Dirac spinors. Thus whilst in even dimensions this accords with what we have simply called a spinor of the complexified Clifford algebra, in odd dimensions a Dirac spinor is what we have called a semi-spinor. In n dimensions Dirac spinors are obviously elements of a 2W21-dimensional complex vector space which we will identify with some minimal left ideal. If n is even the different minimal left ideals all carry equivalent representations, whilst for n odd the two inequivalent representations are carried by minimal left ideals lying in different simple component algebras. Any minimal left ideal can be taken as the first column in some matrix basis. If p E Cn(C)P with P primitive then we may form a matrix basis, le u}, with en = P, giving tp = 11pe11. If = if/S-1 for some invertible S then tp' lies in the first column of the matrix basis {e} where e = Se S-'. If we write tp' = E,tp:e,, then if S-1 = ImS,,-,le'pq we have 11.); = EIS,-, lip]. Thus although a change of minimal left ideal is effected by Clifford multiplication from the right, the components in matrix bases for which the spinors form the first columns are related by matrix multiplication from the left. The Dirac adjoint spinor, 1-p, is a 'row' spinor which enables spin- invariant products to be defined. Thus 1-p is the adjoint of tp with respect to some spin-invariant product, it being an element of the dual space carrying a contragradient representation. From table 2.18 we see that unless p is odd and q is even the involution Or is the adjoint involution of a pseudo-Hermitian product. When p is odd with q even then is the adjoint involution of such a product. We consider the former case first. For some choice of matrix basis let t be the involution of Hermitian conjugation. (In odd dimensions t induces Hermitian con- jugation in the simple component algebras.) Then t is related to Or as follows, d;rri* =A aTAI Va EC pC.q (2.8.9) with AI* = A (equivalently A = A). If cp and tp are Dirac spinors, lying in the first column in the matrix basis in which t is Hermitian conjugation, then we may define a spin-invariant product (49, = A -I cer (2.8.10) This product, which having Or as its adjoint involution is invariant under +1—=, is a special case of (2.6.2). As such it takes values in the algebra of complex numbers whose identity is the primitive en. We can trivially obtain a product with values in the underlying complex field. For if (cp, 1p)En. =(q), p)e 11 then = Tr(99, 1P)Eq.. (2.8.11) 92 CLIFFORD ALGEBRAS AND SPINORS Inphysics, elements ofthevector space carrying anirreducible representation ofthecomplexified Clifford algebra aretermed Dirac spinors. Thus whilst ineven dimensions thisaccords with what wehave simply called aspinor ofthecomplexified Clifford algebra, inodd dimensions aDirac spinor iswhat wehave called asemi-spinor. Inn dimensions Dirac spinors areobviously elements ofa2["’21-dimensional complex vector space which wewill identify with some minimal left ideal. Ifniseven thedifferent minimal leftideals allcarry equivalent representations, whilst fornodd thetwo inequivalent representations arecarried byminimal leftideals lying indifferent simple component algebras. Any minimal leftideal canbetaken asthefirst column in some matrix basis. IfweC,,(C)P with Pprimitive then wemay form a matrix basis, {e,7},with ell=P,giving 1/1=Z,-1/1,~e ,-1.If1p’=1/1S“ for some invertible Sthen 1p’liesinthefirst column ofthematrix basis {e},-} where e§,~=Se,-,-S-1. Ifwewrite 1p’=2,-ipfej-,~ then ifS‘1= Z,,,,,S,j,,‘e’,,,, wehave 1p}=2}-S,-jll/J/-. Thus although achange ofminimal left ideal iseffected byClifford multiplication from theright, the components inmatrix bases forwhich thespinors form thefirstcolumns arerelated bymatrix multiplication from theleft. The Dirac adjoint spinor, 171,isa‘row’ spinor which enables spin- invariant products tobedefined. Thus 171istheadjoint of1/1with respect tosome spin-invariant product, itbeing anelement ofthedual space carrying acontragradient representation. From table 2.18 weseethat unless pisoddandqiseven theinvolution 517*istheadjoint involution ofapseudo-Hermitian product. When pisodd with qeven then 5*is theadjoint involution ofsuch aproduct. Weconsider theformer case first. For some choice ofmatrix basis let‘ibethe involution of Hermitian conjugation. (Inodd dimensions ‘iinduces Hermitian con- jugation inthesimple component algebras.) Then 1‘isrelated to517*as follows, a§”“=Aa*A " VaeCg, (2.8.9) with A5’? =A(equivalently A1=A).Ifcpand1pareDirac spinors, lying inthefirst column inthematrix basis inwhich ‘lisHermitian conjugation, thenwemaydefine aspin-invariant product (31,¢)?.,,.=A-1<p§'1‘¢. (2.8.10) This product, which having 517*asitsadjoint involution isinvariant under “Ti, isaspecial case of(2.6.2). Assuch ittakes values inthe algebra ofcomplex numbers whose identity istheprimitive e,1.Wecan trivially obtain aproduct with values intheunderlying complex field. F0rif((P1 I/’)§n* =(‘P1I/(>911 then (‘pi :Tr((pv 'P).=1r~ THE CONFUSION OF TONGUES 93 The adjoint of zp with respect to the product in (2.8.10) is the Dirac adjoint, that is = A -lie* = A-1. (2.8.12) The defining relation for A, (2.8.9), involves Hermitian conjugation which is defined in some matrix basis, le 01. If {e} is another matrix basis with e'i; = Se11S-1 then e'iit = S-11-e1iSt. Thus if St = S' then eV = efi and the involution also induces Hermitian conjugation in this basis. So in fact the involution and hence the relation (2.8.9), involves a class of bases the elements of which are related by unitary transforma- tions. Suppose that we consider that class of matrix basis for Cpc, in which e°' = —e°, eit = e', i =1, ..., p. Equation (2.8.9) is equivalent to eat = —A-leaA, and so in such a basis we may choose A -1 = ie°. This gives the familiar relation = (2.8.13) (The factor of i is absent in the case of C.q.) It is this relation (in component form) that is usually taken as the definition of the Dirac adjoint. It is the choice of A -1 = ie° that arbitrarily restricts the representations of the y-matrices to be related by unitary transforma- tions. There is no need for this restriction. The notable exception to this restrictive definition of the Dirac adjoint is the book by Jauch and Rohrlich [4]. In the above we excluded the case in which p is odd and q is even. In this case it is rather than that is the adjoint involution of a pseudo-Hermitian product. Unless p is even and q is odd in analogy with (2.8.9) we may define crr = Ba'13 -1 V a eCc9 (2.8.14) P with B* = B' = B. Instead of (2.8.12) we define = 11)*. (2.8.15) Here the Dirac adjoint is defined with respect to a +F±-invariant product. A Dirac spinor and its adjoint are used to form the so-called bilinear covariants. If cp and p are Dirac spinors then, as explained at the end of §2.5, the spinor representation gives rise to a representation r. We define r(s)(pp) = scp(s/p). When, for example, the Dirac adjoint is defined as in (2.8.12) then r(s)(94)= s(qytp)svi". Thus for s E F ± the representation r coincides with the vector representation, that is r(s)(99/) = s(97)s-1. THEcoNFusIoN OFTONGUES 93 The adjoint of1pwith respect totheproduct in(2.8.10) istheDirac adjoint, thatis (L=A":/15'" =wt/1'1. (2.8.12) The defining relation forA,(2.8.9), involves Hermitian conjugation which isdefined insome matrix basis, {e,,-}. If{ej-j} isanother matrix basis with ef-7=Se,-,-S“ then ejf=S““e,~,-Si. Thus ifSi=S" then eff=e,~,-andtheinvolution *alsoinduces Hermitian conjugation inthis basis. Soinfacttheinvolution l,andhence therelation (2.8.9), involves aclass ofbases theelements ofwhich arerelated byunitary transforma- tions. Suppose that weconsider that class ofmatrix basis forCg, in which e°*=—e°, e”=e‘,i=1,...,p.Equation (2.8.9) isequivalent toe“*=—A"e“A, and soinsuch abasis wemay choose A" =ie°. This gives thefamiliar relation (L=we“. (2.813) (The factor ofiisabsent inthecase ofCf_,,.) Itisthis relation (in component form) that isusually taken asthedefinition oftheDirac adjoint. Itisthe choice ofA‘1=ie° that arbitrarily restricts the representations ofthe1/-matrices toberelated byunitary transforma- tions. There isnoneed forthisrestriction. The notable exception tothis restrictive definition oftheDirac adjoint isthebook byJauch and Rohrlich Intheabove weexcluded thecase inwhich pisoddandqiseven. In thiscase itis5*,rather than 511*, that istheadjoint involution ofa pseudo-Hermitian product. Unless piseven and qisodd inanalogy with (2.8.9) wemay define a5'=Ba*B‘1 VaEC5,, (2.8.14) with B‘?=Bl=B.Instead of(2.8.12) wedefine J1=B“1)’-". (28.15) Here theDirac adjoint isdefined with respect toa+1“-invariant product. ADirac spinor anditsadjoint areused toform theso-called bilinear covariants. If1pand1})areDirac spinors then, asexplained attheendof §2.5, thespinor representation gives rise toarepresentation T.We define T(5)(¢P9) =W671)- When, forexample, theDirac adjoint isdefined asin(2.8.12) then 1:(s)((p(}) =s(q>1})s5"'. Thus forse*1": therepresentation 1'coincides with thevector representation, thatis I(S)(<P171) =S(<P¢)S"- 94 CLIFFORD ALGEBRAS AND SPINORS When q is odd the image of +1-± under the vector representation is the timelike-orientation-preserving subgroup of the orthogonal group; whilst for q even it is the spacelike-orientation-preserving subgroup. As pointed out in §2.4, the p-forms transform irreducibly under the vector representation of the Clifford group. Using (2.1.18) we expand op as a sum of p-forms, = E wo(pPeA )eA A = E wo(tPeA(P)eA (by (2.1.17)). A This gives the p-form components in terms of the product (2.8.10), or (2.8.11), TIV = E (TP, eA4p)wo(e ii)eA A For the particular case of C1 we have 4W0(zpip) = Tr(ii) 4991(0p) = Tr(vea ip)e a 49' 2(0p) = .1'r(Ipeab V)eba (2.8.16) (2.8.17) 4923(0p) = Tr(peazIp)e az 49)4(0p) = —Tr(pztp)z. The components of these homogeneous forms are the familiar scalar, vector, tensor, pseudo-vector and pseudo-scalar. As was noted above, the spinor representation on lp induces the representation r on these bilinears. In particular, the spinor representation of +1-'± induces the vector representation on the bilinears, the image of +1—± under the vector representation being the group of orthochronous orthogonal transformations. It is the behaviour under the parity transformation that, for example, distinguishes between the scalar and the pseudoscalar. The vector representation of the elements of the Clifford group which change time orientation cannot be induced on these bilinears from the spinor representation. The Wigner time-reversal operator on spinors is not a representation of the Clifford group, neither does it induce on these bilinears the transformations one would expect from the nomen- clature of 'vector'. It is, however, a symmetry of the Maxwell—Dirac equations, as will be discussed in §10.3. In the physics literature the action of the spinor representation of that element of the Clifford group whose vector representation gives time reversal is called the Racah time reversal on spinors. It is not a symmetry of the Maxwell—Dirac equa- tions, which accounts for its infrequent mention these days. 94 CLIFFORD ALGEBRAS AND SPINORS When qisoddtheimage of*1": under thevector representation isthe timelike-orientation-preserving subgroup oftheorthogonal group; whilst forqeven itisthe spacelike-orientation-preserving subgroup. As pointed outin§2.4, thep-forms transform irreducibly under thevector representation oftheClifford group. Using (2.1.18) weexpand (11171asa sumofp-forms, @171=29’6(<P1/7@fi)@“ =29’6(171@x§<P)@" (by(21-17))- This gives thep-form components interms oftheproduct (2.8.10), or (2.8.11), (PIP=g(1/J, eA§q7>SF0(e1l)eA- (2-8-16) Fortheparticular case ofCf,wehave 45/’0(1l"P) =TT(WP) 45/’1(1l"P) =TT(1P@"1l’)@.1 45(2(1l"P) =iTr(1P@ab1l’)@1m (2-8-17) 45/’3(w17) =Tr(17e"zw)@..z 45/’.(1/11/7)= —Tr(17zw)z- The components ofthese homogeneous forms arethefamiliar scalar, vector, tensor, pseudo-vector andpseudo-scalar. Aswas noted above, thespinor representation on111induces therepresentation ronthese bilinears. Inparticular, thespinor representation of“T1 induces the vector representation onthebilinears, theimage of‘Ti under the vector representation being thegroup oforthochronous orthogonal transformations. Itisthebehaviour under theparity transformation that, forexample, distinguishes between thescalar andthepseudoscalar. The vector representation oftheelements oftheClifford group which change time orientation cannot beinduced onthese bilinears from the spinor representation. The Wigner time-reversal operator onspinors is notarepresentation oftheClifford group, neither does itinduce on these bilinears thetransformations onewould expect from thenomen- clature of‘vector’. Itis,however, asymmetry oftheMaxwell-Dirac equations, aswill bediscussed in§10.3. Inthephysics literature the action ofthespinor representation ofthatelement oftheClifford group whose vector representation gives time reversal iscalled theRacah time reversal onspinors. Itisnotasymmetry oftheMaxwell—Dirac equa- tions, which accounts foritsinfrequent mention these days. THE CONFUSION OF TONGUES 95 The Dirac adjoint is associated with the pseudo-Hermitian product for which or is the adjoint involution. We also have the spin- invariant products for which the C-linear involutions ij and are the adjoints. From table 2.19 we see that unless n =1 mod 8 or 5 mod 8 the involution ii induces an involution on the simple components of the reducible Clifford algebras. If 5 denotes transposition in some matrix basis then, excepting the dimensions mentioned, we have a4n = CagC-1 Va E CpC.q (2.8.18) with C = C7= ±C. The symmetry of C determines the symmetry of the complex bilinear product defined by (9), 1P),7 = C -101P. (2.8.19) Here op and tp are Dirac spinors lying in the first column of the matrix basis in which 5 is the transposition. The symmetry of this product, for which ij is the adjoint involution, is given in table 2.1. The defining property of C, (2.8.18), is equivalent to =—C-leaC the matrix components of which are usually taken as the definition of the charge conjugation matrix. If if) is the adjoint of ip with respect to the product in (2.8.19) then = C-Lten = ipErC-1. (2.8.20) This adjoint spinor is often called the Majorana conjugate. Except for n = 3 mod 8 or 7 mod 8 the involution induces an involution on the simple components of the reducible Clifford algebras. We may define = Dag D-1 Va E Cc (2.8.21) with 1Y. =- Erg = ±D. This gives eg = The symmetry of the product defined by (9), TP). = D-VIP (2.8.22) is given in table 2.18. We shall also use ip to denote the adjoint with respect to this product, specifying the relevant product whenever con- fusion is likely. In §2.7 we were careful to distinguish the automorphism *, referred to as complex conjugation, from the automorphism *. Complex conjuga- tion leaves invariant the real subalgebra generated by the real orthogon- al space with signature p, q, whilst * is defined to complex conjugate the matrix components in some matrix basis. Thus the definition of * THECONFUSION OFTONGUES 95 TheDirac adjoint isassociated with thepseudo-Hermitian product for which 517* or5*,istheadjoint involution. We also have thespin- invariant products forwhich theC-linear involutions 517and5arethe adjoints. From table 2.19 weseethat unless n=1mod8 or5mod 8the involution 517induces aninvolution onthesimple components ofthe reducible Clifford algebras. If9denotes transposition insome matrix basis then, excepting thedimensions mentioned, wehave a5"=Ca‘7C'1 VaeC§_q (2.8.18) with C5”=C‘7= i-C. The symmetry ofCdetermines thesymmetry of thecomplex bilinear product defined by (<11,11),,=cwpfiw. (28.19) Here qzand1pareDirac spinors lying inthefirst column ofthematrix basis inwhich 9isthetransposition. The symmetry ofthisproduct, for which 51]istheadjoint involution, isgiven intable 2.1. The defining property ofC,(28.18), isequivalent to em=—C‘1e“C thematrix components ofwhich apeusually taken asthedefinition of thecharge conjugation matrix. If1/1istheadjoint of1/1with respect to theproduct in(2.8.19) then 1/7=c-upii =1//Jc-1. (28.20) This adjoint spinor isoften called theMajorana conjugate. Except forn=3mod8 or7mod8 theinvolution 5induces an involution onthesimple components ofthereducible Clifford algebras. Wemaydefine a5=DagD_1 VaeCg, (2.821) with D5=D9=iD. This gives e“°J=D‘1e“D. Thesymmetry oftheproduct defined by (<11,1/1);=D"<P51l1 (2-8-22) isgiven intable 2.18. Weshall alsouse1ptodenote theadjoint with respect tothisproduct, specifying therelevant product whenever con- fusion islikely. In§2.7 wewere careful todistinguish theautomorphism *,referred to ascomplex conjugation, from theautomorphism 1‘.Complex conjuga- tionleaves invariant therealsubalgebra generated bytherealorthogon- alspace with signature p,q,whilst #isdefined tocomplex conjugate thematrix components insome matrix basis. Thus thedefinition of# 96 CLIFFORD ALGEBRAS AND SPINORS depends on the choice of some matrix basis. In (2.7.9) we showed that, excepting the case in which the real subalgebra is isomorphic to the algebra of complex matrices, these two automorphisms are related by a* = metm -1 Va. When the real subalgebra is a real matrix algebra, or a sum of two such algebras, we may choose mm* -= 1. When the real subalgebra is the tensor product of a matrix algebra with the quater- nions, or a sum of two such algebras, we may choose mm* = —1. The real subalgebra is isomorphic to the algebra of complex matrices when p — q =3 or 7 mod 8. In this case complex conjugation of the complex- died algebra swaps the simple components. Save for this exceptional case we use this relation between the two automorphisms to define the charge conjugate spinor pc pc _p*m. (2.8.23) This can be rewritten in terms of the Dirac adjoint and the charge conjugation matrix. Unless n = 1 or 5 mod 8, or p is odd with q even, we may use (2.8.9) and (2.8.18) to produce (a?1`).=7/ = (A -YnCat5C-1A Since complex conjugation commutes with the involution 01 and tT = Tt = # we have a* = mem -1 with m = A -1*C (2.8.24) where we have used A 11* = A. We know that we can scale m such that mm* = ±1, which can be accomplished by choosing C suitably. With m given by (2.8.24) equation (2.8.23) becomes = Civ. (2.8.25) In exactly the same way, except for the case of n = 3 or 7 mod 8 or p even with q odd, (2.8.23) can be written as lpC= D ,FpFl (2.8.26) where now 'Fp is given by (2.8.15). The only cases in which we can use neither (2.8.25) nor (2.8.26) are for p + q = 3 or 7 mod 8 with q even, or p + q = 1 or 5 mod 8 with q odd. These cases can only occur for p — q = 3 or 7 mod 8, which is the case we excluded from the definition of the charge conjugate spinor. When the Dirac spinors carry a reducible representation of the real subalgebra, elements of the irreducible subspaces are called Majorana spinors. As was pointed out in §2.7 this ocurs when the real subalgebra is a real matrix algebra, or a sum of two such algebras, and this occurs when p —q = 0, 1, 2 mod 8, as is seen from table 2.8. In these dimensions reference to §2.7 shows how the space of Dirac spinors can be decomposed into eigenspaces of the charge conjugation operator. 96 CLIFFORD ALGEBRAS AND SPINORS depends onthechoice ofsome matrix basis. In(2.7.9) weshowed that, excepting thecase inwhich thereal subalgebra isisomorphic tothe algebra ofcomplex matrices, these two automorphisms arerelated by a*=ma#m" Va. When therealsubalgebra isarealmatrix algebra, or asum oftwosuch algebras, wemay choose mm* =1.When thereal subalgebra isthetensor product ofamatrix algebra with thequater- nions, orasum oftwosuch algebras, wemay choose mm* =—1.The realsubalgebra isisomorphic tothealgebra ofcomplex matrices when p—q=3or7mod8.Inthiscasecomplex conjugation ofthecomplex- ified algebra swaps thesimple components. Save forthisexceptional case weusethisrelation between thetwoautomorphisms todefine the charge conjugate spinor 10° 1/1°=1p*m. (2.823) This can berewritten interms oftheDirac adjoint and thecharge conjugation matrix. Unless I1=1or5mod8, orpisoddwith qeven, wemay use(2.8.9) and(2.8.18) toproduce (a§n‘)§n =(A-1)§nCat@C—1A§n_ Since complex conjugation commutes with the involution 517and T9=91=#wehave a*=ma#m'1 with m=A"*C (2.824) where wehave used A51‘ =A.Weknow that wecanscale msuch that mm* =i1,which canbeaccomplished bychoosing Csuitably. With m given by(2.8.24) equation (2.823) becomes we=cw. (28.25) Inexactly thesame way, except forthecase ofn=3or7mod8 orp even with qodd, (2.8.23) canbewritten as 1,0‘=D1115 (2.826) where now 1pisgiven by(2.8.15). The only cases inwhich wecanuse neither (2.8.25) nor(2.8.26) areforp+q=3or7mod8 with qeven, orp+q=1or5mod8 with qodd. These cases canonly occur for p—q=3or7mod 8,which isthecase weexcluded from thedefinition ofthecharge conjugate spinor. When theDirac spinors carry areducible representation ofthereal subalgebra, elements oftheirreducible subspaces arecalled Majorana spinors. Aswaspointed outin§2.7 thisocurs when therealsubalgebra isarealmatrix algebra, orasum oftwosuch algebras, andthisoccurs when p—q=0,1,2mod8, asisseen from table 2.8. Inthese dimensions reference to§2.7 shows how thespace ofDirac spinors can bedecomposed into eigenspaces ofthecharge conjugation operator. THE CONFUSION OF TONGUES 97 Thus a Majorana spinor is an eigenspinor of the charge conjugation operation = ±/Pc- (2.8.27) This can be written in terms of the Dirac and Majorana conjugates by using (2.8.25) or (2.8.26). In an even number of dimensions the irreducible representations of the complex Clifford algebra induce a reducible representation of the even subalgebra; the spinor representation splitting into two inequi- valent semi-spinor representations of the even subalgebra. The central idempotents that project the even subalgebra into simple components are P± = (1 ± where either = z or = iz ensuring 2 = 1, z denoting the volume n-form. If p is a Dirac spinor then it may be decomposed into subspaces that transform irreducibly under the even subalgebra, 11) = + - where tp+ = 13±tp. (2.8.28) The semi-spinors 1p+ are called Weyl spinors, or chiral spinors. The Weyl spinors can carry a reducible representation of the real even subalgebra. From table 2.10 this is seen to occur when p — q = 0 mod 8. In this case the real even subalgebra is the direct sum of two real matrix algebras, having the real central idempotents P± = (1 ± z). The 'Ma- jorana condition' (2.8.27), can be consistently imposed together with the 'Weyl condition', (2.8.28), to decompose a Dirac spinor into subspaces transforming irreducibly under the real even subalgebra. The resulting spinors are called Majorana—Weyl spinors. In an odd number of dimensions irreducible representation of the complexified Clifford algebra induce irreducible representations of the even subalgebra. These can induce a reducible representation of the real, even subalgebra. Obviously this is the case for p — q = 1 mod 8 where, as we have noted, Dirac spinors carry a reducible representation of the whole real subalgebra. From table 2.10 we see that for p — q = 7 mod 8 Dirac spinors carry irreducible representations of the real subalgebra and the even subalgebra. However they carry a reducible representation of the real even subalgebra. For p — q = 7 mod 8 Cp,q(B) C 0.420,-.0(R) and C7,,,(Fi) .41,2(—w,(Fi) where p + q = n. We may thus choose a matrix basis for the Clifford algebra in which the automorphism n simply complex conjugates the components. The complexified algebra qc,„ is reducible, with inter- changing the simple components. Complex conjugation, *, also swaps Tm;CONFUSION OFTONGUES 97 Thus aMajorana spinor isaneigenspinor ofthecharge conjugation operation zp=iqfi. (2.8.27) This canbewritten interms oftheDirac andMajorana conjugates by using (2.8.25) or(2.8.26). Inaneven number ofdimensions theirreducible representations of thecomplex Clifford algebra induce areducible representation ofthe even subalgebra; thespinor representation splitting into two inequi- valent semi-spinor representations oftheeven subalgebra. The central idempotents that project theeven subalgebra into simple components arePi=§(1iE),where either E=zorE=izensuring E2=1, z denoting thevolume n-form. If1/1isaDirac spinor then itmay be decomposed into subspaces that transform irreducibly under theeven subalgebra, zp=111++1p_ where zpi=Pit/1. (2.8.28) The semi-spinors wtarecalled Weyl spinors, orchiral spinors. The Weyl spinors can carry areducible representation ofthereal even subalgebra. From table 2.10 thisisseen tooccur when p—q=0mod 8. Inthiscase therealeven subalgebra isthedirect sum oftworealmatrix algebras, having therealcentral idempotents Pi=§(1iz).The ‘Ma- jorana condition’ (2.8.27), canbeconsistently imposed together with the ‘Weyl condition’, (2.8.28), todecompose aDirac spinor into subspaces transforming irreducibly under thereal even subalgebra. The resulting spinors arecalled Majorana—Weyl spinors. Inanodd number ofdimensions irreducible representation ofthe complexified Clifford algebra induce irreducible representations ofthe even subalgebra. These can induce areducible representation of thereal, even subalgebra. Obviously this isthecase forp—q=1 mod8 where, aswehave noted, Dirac spinors carry areducible representation ofthewhole realsubalgebra. From table 2.10 weseethat forp—q=7mod8 Dirac spinors carry irreducible representations of thereal subalgebra and theeven subalgebra. However they carry a reducible representation ofthereal even subalgebra. For p—q=7 mod8 Cp_q(lR) =C®M2<~-1»/z(1B) and C;'_q(lR) =A/{W-1>/z(1B) where p+q=n.Wemay thus choose amatrix basis fortheClifford algebra inwhich theautomorphism 11simply complex conjugates the components. The complexified algebra Cg,’ isreducible, with 11inter- changing thesimple components. Complex conjugation, *,also swaps 98 CLIFFORD ALGEBRAS AND SPINORS the component algebras. The automorphism Tr will certainly preserve the simple components, and in a suitable basis we see from (2.6.17) that it coincides with #, the operation that complex conjugates the matrix components. A Dirac spinor ip can be decomposed into spinors trans- forming irreducibly under the real even subalgebra = 1P + + - with Ip = letp ±Ip'r). (2.8.29) (Such spinors have attracted no special terminology in the physics literature.) Of importance in many calculations, especially those involving super- symmetric theories, is the Fierz rearrangement formula. This allows products of bilinears to be rewritten in terms of different bilinears. Many similar results can be given, we illustrate the basic result below. Let /3, i , 92 be Dirac spinors lying in some minimal left ideal projected by the primitive P. If M and N are arbitrary elements of the Clifford algebra then 5eMpipNcp = &S0(M131pNeA)e4cp (by 2.1.18) = cleA 00(A/16i-pNeA). The terms in the brackets can be reordered using (2.1.13), and since 92 = 92P c1M)31pNcp = Cre AçoS0(VNeA4113)P. Now the term in brackets is in PC", which is isomorphic to the algebra of complex numbers with P as identity. That is, PXP = AP for A a complex 0-form, giving so(pxp) = Aso(P) and So(PXP)P = S o(P)PXP SO CrM01pNcp = Cre ANeAWPS0(P). In terms of the product in (2.8.11) we have Mfi)(ip, NOP = eAcp)(tp, Ne A4113)S0(P)P whose 0 -form component is the basic Fierz formula (a, Mfi)(ip, Nip) = (a, eAcp)(1p, Ne A4113)So(P). (2.8.30) (The factor of So(P) arises from our normalisation of the e'.) The approach to spinors that we have pursued is essentially algebraic. From the Clifford algebra we can define the spin groups, and from the 98 CLIFFORD ALGEBRAS AND SPINORS thecomponent algebras. Theautomorphism 17*willcertainly preserve thesimple components, andinasuitable basis weseefrom (2.6.17) that itcoincides with #,theoperation thatcomplex conjugates thematrix components. ADirac spinor 1/1canbedecomposed intospinors trans- forming irreducibly under therealeven subalgebra 1/»=1/».+1/»_ withwt=%(w1w"')- (2-8-29) (Such spinors have attracted nospecial terminology inthephysics literature.) Ofimportance inmany calculations, especially those involving super- symmetric theories, istheFierz rearrangement formula. This allows products ofbilinears toberewritten interms ofdifferent bilinears. Many similar results canbegiven, weillustrate thebasic result below. Leta,[3,1/1,qvbeDirac spinors lying insome minimal leftideal projected bytheprimitive P.IfMandNarearbitrary elements ofthe Clifford algebra then ¢iM[31I1Nq9 =&S0(M/31I1NeA 5)e“‘<p (by2.1.18) =Zze"q9S0(M[3t71NeA5). The terms inthebrackets canbereordered using (2.1.13), and since w=¢>P &M/3t]1N<p =&e"<pS0(1I1NeA 5M[3)P. Now theterm inbrackets isinPC§,,,P, which isisomorphic tothe algebra ofcomplex numbers with Pasidentity_. That is,PXP =APfor/1 acomplex 0-form, giving S0(PXP) =,1S0(P) and S0(PXP)P =S0(P)PXP so &M/31I1N<p =¢Ye"qnI1NeA§M[3S0(P). Interms oftheproduct in(2.8.11) wehave (01,Ml3>(1P, N<P>P =(01,@"<P><1l1, N@A§Ml3>5@(P)P whose 0-form component isthebasic Fierz formula (01,Ml3><1P, MP)=(04@"<P>(1P, NeA§MB>S0(P)- (2-3-30) (The factor ofS0(P) arises from ournormalisation oftheed.) Theapproach tospinors thatwehave pursued isessentially algebraic. From theClifford algebra wecandefine thespingroups, andfrom the THE CONFUSION OF TONGUES 99 representations of the algebra we induce representations of these groups. One can, however, start from a knowledge of the covering group of the connected component of the orthogonal group and intro- duce its irreducible representations as spinors. Representations of the component of the orthogonal group connected to the identity can then be found from the tensor product of these spinor representations. For the case of four dimensions, and Lorentzian signature, such an approach has developed its own rather specialised notation and conventions. That is the Infeld—van der Waerden formalism, or 'two-component spinor formalism'. Given that the double covering of SO+(3, 1) is SL(2, C) one introduces 'two-component spinors' as carrying irreducible repre- sentations of SL(2, C). The complex conjugate representations of this group are inequivalent, and a special notation is used to distinguish them. If u is a vector carrying an SL(2, C) representation such that the components of u transform with a matrix m then, say, the components of u are labelled by a Greek superscript. If the vector y transforms with the complex conjugate matrix then the components of u are labelled by a Greek superscript with a dot above it. (It is perhaps significant that such a notation was introduced before the advent of frequent photo- copying!) The vector spaces carrying these representations both admit SL(2, C)-invariant symplectic products, and the adjoint of u, say, with respect to such a product has its components with respect to a dual basis written as subscripts. A similar situation holds for v. Thus indices are 'lowered' with the symplectic matrix, which can be taken to have plus one in the top right-hand entry. Because of the antisymmetry of this matrix a convention must be adopted as to which side the matrix is multiplied from to lower an index. The tensor product of these two representations, with themselves and each other, gives a representation of SO+ (3, 1). Thus SO+(3, 1) irreducible representations are identified with certain expressions written with two Greek indices, either with or without dots, up and down, or a mixture. Of course, starting with SL(2, C) irreducible representations only produces SO+(3, 1) repre- sentations, not 0(3, 1) representations. One can extend the representa- tions of SL(2, C) to include other transformations so that the tensor representation extends to a representation of 0(3, 1). However, such extensions are not unique and there is certainly no universal convention for complex phase factors. Without being exhaustive we shall show the relation between the 'two-component formalism' and the algebraic approach. We shall consider the Clifford algebra associated with a four- dimensional Lorentzian space. Starting with the real even subalgebra we shall construct a basis for the complexified Clifford algebra. We saw in §2.3 that C1(1F1) CO, 112 and that if {E,i3} is a matrix basis there exists a 2-form c relating transposition, t, to the involution ct = ca`c-I THECONFUSION orrouourzs 99 representations ofthe algebra weinduce representations ofthese groups. One can, however, start from aknowledge ofthecovering group oftheconnected component oftheorthogonal group andintro- duce itsirreducible representations asspinors. Representations ofthe component oftheorthogonal group connected totheidentity canthen befound from thetensor product ofthese spinor representations. For thecase offour dimensions, andLorentzian signature, such anapproach hasdeveloped itsown rather specialised notation andconventions. That istheInfeld—van derWaerden formalism, or‘two-component spinor formalism’. Given that thedouble covering ofSO*(3, 1)isSL(2, C) one introduces ‘two-component spinors’ ascarrying irreducible repre- sentations ofSL(2, C).The complex conjugate representations ofthis group areinequivalent, and aspecial notation isused todistinguish them. Ifuisavector carrying anSL(2, C)representation such thatthe components ofutransform with amatrix mthen, say,thecomponents ofuarelabelled byaGreek superscript. Ifthevector vtransforms with thecomplex conjugate matrix then thecomponents ofvarelabelled by aGreek superscript with adotabove it.(Itisperhaps significant that such anotation was introduced before theadvent offrequent photo- copying!) The vector spaces carrying these representations both admit SL(2, C)-invariant symplectic products, andtheadjoint ofu,say, with respect tosuch aproduct hasitscomponents with respect toadual basis written assubscripts. Asimilar situation holds foru.Thus indices are ‘lowered’ with thesymplectic matrix, which canbetaken tohave plus oneinthetopright-hand entry. Because oftheantisymmetry ofthis matrix aconvention must beadopted astowhich side thematrix is multiplied from tolower anindex. The tensor product ofthese two representations, with themselves andeach other, gives arepresentation ofSO*(3, 1).Thus SO"(3, 1)irreducible representations areidentified with certain expressions written with twoGreek indices, either with or without dots, upand down, oramixture. Ofcourse, starting with SL(2, C)irreducible representations only produces SO*(3, 1)repre- sentations, notO(3, 1)representations. Onecanextend therepresenta- tions ofSL(2, C)toinclude other transformations sothat thetensor representation extends toarepresentation ofO(3, 1).However, such extensions arenotunique andthere iscertainly nouniversal convention forcomplex phase factors. Without being exhaustive weshall show the relation between the ‘two-component formalism’ and the algebraic approach. We shall consider the Clifford algebra associated with afour- dimensional Lorentzian space. Starting with therealeven subalgebra we shall construct abasis forthecomplexified Clifford algebra. Wesawin §2.3thatC§_1(lR) =C®J1/£2 andthatif{sag} isamatrix basis there exists a2-form crelating transposition, I,totheinvolution E a5=ca'c"‘ 100 CLIFFORD ALGEBRAS AND SPINORS and a 1-form x that squares to one and commutes with the co and c. Thus c must have real components and, since it is certainly anti- symmetric, we can choose it such that its components form the standard symplectic matrix, that is c = E C 43E0 (2.8.31) p where the matrix of components cal3 is cal3 101 —1 )‘ 0 1 0 The even subalgebra of the complexified algebra is the direct sum of two algebras of complex order-two matrices. Cr .M.2(C) .M.2(C). If P± = 1(1 ± iz), with z the volume 4-form, then {e0P+} and {EoP_} are bases for the simple component algebras. The complexified Clifford algebra is isomorphic to the algebra of order-four complex matrices, and so we can choose a matrix basis in which the even subalgebra is block diagonal. In such a basis any odd element must have off-diagonal components. If, as usual, we identify the space of Dirac spinors with the minimal left ideal formed by the first column then the upper two components and the lower two components will transform irreducibly under the even subalgebra. These are the even and odd parts of the spinor, forming the two inequivalent Weyl spinors. In this language one refers to a Dirac spinor as a bispinor, as it carrys a reducible representation of SL(2, C). We can use the element x to form the off-diagonal elements in a matrix basis for C„ {e11}. We can schematically display the basis we have constructed as follows: toP, xt1 3 ai3_ e :( xtapP, Ea.0P _) (2.8.33) It can be checked that this is indeed an ordinary matrix basis. In this basis the diagonal blocks are related by complex conjugation, as are the off-diagonal blocks. If T denotes transposition in this basis then (E0P+)5 = But from the defining property of c, (2.3.2), eP, = = (since c is even) = Similarly (xeoP±)5 = xfp,P (2.8.32) 100 CLIFFORD ALGEBRAS AND SPINORS anda1-form xthat squares tooneandcommutes with the5,5andc. Thus cmust have real components and, since itiscertainly anti- symmetric, wecanchoose itsuchthatitscomponents form thestandard symplectic matrix, thatis C=%¢,,,,E,,,, (2.831) where thematrix ofcomponents caflis 01cat;-(10 (2.8.32) The even subalgebra ofthecomplexified algebra isthedirect sum of twoalgebras ofcomplex order-two matrices. Cgi =M-2(C) (9M2(C)- IfPi=§(1iiz), with zthe volume 4-form, then {s,,BP,} and {saBP_} arebases forthesimple component algebras. Thecomplexified Clifford algebra isisomorphic tothealgebra oforder-four complex matrices, and sowecan choose amatrix basis inwhich theeven subalgebra isblock diagonal. Insuch abasis anyoddelement must have off-diagonal components. If,asusual, weidentify thespace ofDirac spinors with theminimal leftideal formed bythefirst column then the upper two components and thelower two components will transform irreducibly under theeven subalgebra. These aretheeven andoddparts ofthespinor, forming thetwo inequivalent Weyl spinors. Inthis language one refers toaDirac spinor asabispinor, asitcarrys a reducible representation ofSL(2, C).Wecanusetheelement xtoform theoff-diagonal elements inamatrix basis forC§_,, {e,~,»}. Wecan schematically display thebasis wehave constructed asfollows: e,,BP+ xs,,,BP_)e,,.( xEaBP+ EaBP_ . (2.8.33) Itcanbechecked that thisisindeed anordinary matrix basis. Inthis basis thediagonal blocks arerelated bycomplex conjugation, asarethe off-diagonal blocks. If‘Jdenotes transposition inthisbasis then (EaBPi)5 =£}3aPi' Butfrom thedefining property ofc,(2.3.2), EBHP: =c'1c,,55cPt =c'1s,,55Pic (since ciseven) =c‘1(s,,BPi)5c. Similarly (xs,,5Pi)‘T =xsB,,P; THE CONFUSION OF TONGUES 101 and xes,PT. = xclEtcP c'x4013-,,c = c-iP+EOE,XC = C-1(XE0P 1-)C (since x commutes with c) (since x13± = and so = Va E C1. (2.8.34) If ip is a Dirac spinor we can write ip in terms of its even and odd parts as ip = u + y. If we introduce the notation EcrlP + ba, xEaiP = then u = uab y = vat, ua, va E C. This accords with the conven- tional labelling since u and y carry complex conjugate representations of the real even subalgebra, and hence +F+ which is isomorphic to SL(2, C). The first row in the matrix basis is naturally identified with the dual space of the first column. If we define EierP+ = xtraP_ = Ba then 13"13p = A -4E11P +, B'b fi = 0, Baba = Baba = 0. We may define the Majorana conjugate of ip as = v5c-1 (this is a special case of 8.21), then = 145 C-1 ± v = liaBa C-1 + V&B&C-1 We now introduce Ba = Bac -1 B a = B (2.8.37) (2.8.38) giving, by (2.8.31), Ba, = c;,!Bv and Ba = c-a7,11:3''. We can write the Majorana conjugate as uaB, + vaB a = uaBa + v aBa where, for example, ua, = ut'cial. That is, indices are lowered with the components of the symplectic matrix. If (4) is another Dirac spinor written in even and odd parts as cp = w + y then ipq) -= + y,ya)E1113+. (2.8.35) (2.8.36) THE CONFUSION OFTONGUES 101 and xs,;,,P; =xc‘1s,,fl5cP; =c"1x£f,flP;c (since xcommutes with c) =c‘1Pi.'-:,,fl5xc (since xPi=Pix) =c“1(xs,,flP¢)5c andso a5=cage" VaeC§_,. (2.8.34) If1/1isaDirac spinor wecanwrite 1/1interms ofitseven andodd parts as1/1=u+v.Ifweintroduce thenotation £a1P+ :ba(28.35) x£a1P+ :bd then u=u"‘b,,, v=v"b,~,, u“,odeC.This accords with theconven- tional labelling since uandvcarry complex conjugate representations of thereal even subalgebra, and hence +1“ which isisomorphic to SL(2, C).The first row inthematrix basis isnaturally identified with thedual space ofthefirstcolumn. Ifwedefine s,,,P =B“‘* _ (28.36) x£1,,P_ =Ba then B"bfl =<5§s11P+, B“b,-3 =0,Bab’; =6‘Zs,1P+, Bab); =O.We may define theMajorana conjugate of1/1as 17=i/flrl (2.837) (this isaspecial case of8.21), then i/7=1250" +090" =u"B“c'1 +v"‘B'5'c". Wenow introduce B,=B"‘c" _ (2.838) Bl", =BaC_1 giving, by(2.831), B,=¢;3B" andB,,.=¢;3B*. Wecanwritethe Majorana conjugate as 17=we,+MB,=ii,,B~+D6186‘ where, forexample, ua=uflcga‘. That is,indices arelowered with the components ofthesymplectic matrix. Iftpisanother Dirac spinor written ineven andoddparts asrp=w+ythen '7}J(P :(uawa +Udyd/)£11P+. 102 CLIFFORD ALGEBRAS AND SPINORS Thus this product on the Dirac spinors induces the SL(2, C)-invariant symplectic products on the Weyl spinors. So far we have relabelled the first row and the first column of our matrix basis to facilitate a correspondence with the two-component formalism. Any element of the matrix basis can be written as a product of the first column by the first row, eu =e1e11, and so we can apply this relabelling to the whole basis roP,c-1 = 13B,6 EoP_c-1 = ba,13,6 xE 0P, c = bal3p XeoP = b p. The products on the right-hand side with no dots or two dots are even, whilst the terms with mixed indices are odd. Under complex conjugation a dotted index is replaced with an undotted one, and vice versa. These terms also have simple properties under the involution for example (b,Bp).= = —c-1P+e,,j = —c-leocP+c-1 = = —b,613„. Similarly we obtain for the full set (ba.13/3) = —13013, (13B0) = (2.8.40) (1:03/3) = —1)013a. (bŒBW = If then n is any real odd form n = n0b,B fi + naMb B = —na0b 0B — n6131)0B„. So if n is a 1-form, even under nO = —n*. That is, the components can be arranged as an anti-Hermitian matrix (with different conventions the matrix of components is Hermitian). In particular the basis 1-forms, e°, can be expanded in the matrix basis as e° = G00116,13 + (2.8.39) 102 CLIFFORD ALGEBRAS AND SPINORS Thus thisproduct ontheDirac spinors induces theSL(2, C)-invariant symplectic products ontheWeyl spinors. Sofarwehave relabelled the first row and thefirst column ofour matrix basis tofacilitate a correspondence with thetwo-component formalism. Any element ofthe matrix basis canbewritten asaproduct ofthefirst column bythefirst row,e,-j=e,~1e,,-, andsowecanapply thisrelabelling tothewhole basis £afiP+C_1 :baBfi sP_c‘1 =b-B‘“'3 _1"‘'3 (28.39) .X£a’3P+C =bdBB xs,,,3P_c" =b,,B5. The products ontheright-hand side with nodots ortwo dots are even, whilst theterms with mixed indices areodd. Under complex conjugation adotted index isreplaced with anundotted one, andvice versa. These terms alsohave simple properties under theinvolution E, forexample (b,,B,;)5 =—c"P+s,.4;5 =—c"safi§cP+c*‘ =—s5aP+c" =—bfiB,,. Similarly weobtain forthefullset (M155): =_bI3Ba (b,-,BB)‘3 =—b,;B,,. (b,,.B,;)§ =—b,;B,, (b,,B,;)§ =—b;;B,,.(2.8.40) Ifthen rtisanyrealoddform n=n"‘5bdB5 +n‘*5'b,.B,g n5=—n""3b,;B,,, —n°"5'b,;B,,. Soifrtisa1-form, even under E,nap=—nB"‘*. That is,thecomponents canbearranged asananti-Hermitian matrix (with different conventions thematrix ofcomponents isHermitian). Inparticular thebasis 1-forms, e“,canbeexpanded inthematrix basis as e"=0“"’5b,,,B,; +0”‘*'3'b0,Bj;. THE CONFUSION OF TONGUES 103 The anti-Hermitian matrices, aa°13, give the correspondence between a 'vector' and a 'rank-two spinor' (or a 'valence two spinor'). Similarly a real 3-form has components that form a Hermitian matrix. If m is real and even then m = meo 3b,,B13 + mc'O'b a.130. Requiring that m be odd under is equivalent to it being a 2-form. From (2.8.40) it follows that this gives m"13 = mfia. Obviously the components of a 0-form or 4-form must form an anti-symmetric matrix, but we still need to disentangle the two. We have Wo(1),13/3) = J0(13113,) = 0(1313c-lbo,.) = o(Eigc -itaiP+) = W0(c/32E11P+) For the primitive Eli P + we have Wo(eliP+) = 1, giving W0(ba13#) = also b.E3,6) = —Jo(Eloc-lE,IP+z)z and since Pz = —i13, it follows that 9P4(bBp) =icjz. If the inverse matrix is introduced such that cc-1,A = (5/3 then, if ma'fl is anti- symmetric, net3 = /1013 for some complex A. It then immediately follows that Yo(m) = 2 Re A whilst Y4(m) = —2 Im Az. In this section we have established contact with the most usual notations and nomenclature used for spinors in physics. There is, however, yet one more impediment to multilingual fluency. For many applications in physics one works with `anticommuting' spinors. That is, whenever the order of two spinor fields is reversed a minus sign is introduced. One rationale is that the components of the spinors take values in the odd part of some exterior algebra. Certain other fields are assigned values in the even part of this algebra; bilinears in the spinors being even, for example. In practice the rationale seems unimportant as the rules are easy to understand. The consequences are, for example, that certain expressions which are antisymmetric in 'commuting' spinors become symmetric in `anticommuting' spinors. Thus although many of the results presented in this chapter are changed (for example the properties of the spin-invariant inner products) they are easily adapted to accommodate `anticommuting' spinors. THE CONFUSION OFTONGUES 103 Theanti-Hermitian matrices, 0”“, give thecorrespondence between a ‘vector’ anda‘rank-two spinor’ (ora‘valence twospinor’). Similarly a real3-form hascomponents that form aHermitian matrix. Ifmisreal andeven then m=ma/5baB)3 +ma/3‘bdBB. Requiring that mbeodd under Eisequivalent toitbeing a2-form. From (2.8.40) itfollows that this gives m“/5 =m/5“. Obviously the components ofa0-form or4-form must form ananti-symmetric matrix. butwestillneed todisentangle thetwo. Wehave Ef0(bCl’BB) =9;0(B/fiber) =5f0(B“¢"b.i) =9o(£1¢s¢‘_1£iiiP+) I5f0(C§.il£iiP+)- Fortheprimitive s11P+ wehave Sf’0(s,1P+) = giving Sf0(baB}3) =icéi also 5f4(b¢iB8) =_Sf0(£l}3C—l£a1P+Z)Z andsince P+z =—iP+ itfollows that $f’4(b,,B5) =jicgaiz. Iftheinverse matrix isintroduced such that c/5“c“,,)= 65) then, ifm"5 isanti- symmetric, m“/’ =/la“/5 forsome complex /1.Itthen immediately follows thatSf0(m) =2Re/I whilst §f’4(m) =—2Im/12. Inthis section wehave established contact with themost usual notations and nomenclature used forspinors inphysics. There is, however, yetone more impediment tomultilingual fluency. Formany applications inphysics oneworks with ‘anticommuting’ spinors. That is, whenever theorder oftwo spinor fields isreversed aminus sign is introduced. One rationale isthat thecomponents ofthespinors take values intheoddpart ofsome exterior algebra. Certain other fields are assigned values intheeven partofthisalgebra; bilinears inthespinors being even, forexample. Inpractice therationale seems unimportant as therules areeasy tounderstand. The consequences are, forexample, thatcertain expressions which areantisymmetric in‘commuting’ spinors become symmetric in‘anticommuting’ spinors. Thus although many of theresults presented inthis chapter arechanged (for example the properties ofthespin-invariant inner products) they areeasily adapted toaccommodate ‘anticommuting’ spinors. -= A/1.16(C) 16 (Dirac spinors) C(IR) .M. 8(C) C.) .4 8(C) 8 (Weyl spinors) C(lR) 4132(C) 32 (Dirac spinors) C(111) =14-16(C) 0A/116(C) 16 (Weyl spinors) sym Hermitian sym (index 0) Hermitian sym (index 8) Hermitian sym (index 0) skew Hermitian sym (index 16) Hermitian sym (index 16) swap sym sym sym swap Exercise 2.2 Abstract from tables 2.8, 2.15, 2.17 and 2.18 the information in the following tables. In part (a): C2 gives real dimension of irreducible algebra representation; C3 gives real-valued spin-invariant product associated with C4 gives real-valued spin-invariant product associated with 07. In part (b): C2 gives complex dimension of irreducible algebra representation; C3 gives complex-valued spin-invariant product associated with C4 gives complex-valued spin-invariant product associated with ,17; C5 gives complex-valued spin-invariant product associated with C6 gives complex-valued spin-invariant product associated with -ti*. (a) Clifford algebra C2 C3 C4 C4.4(1H) At16(R) 16 (Majorana spinors) sym(index 8) sym(index 8) C:4(1H) h18(E) 0 A18(11i) 8 Majorana—Weyl spinors) sym(index 4) C8,0(E) ht16(E) 16 (Majorana spinors) sym(index 0) sym(index 8) C;0(R) = Ais(IR) 0 ht(1H) 8 Majorana—Weyl spinors) sym(index 0) C08(JR) hti6(E) 16 (Majorana spinors) sym(index 8) sym(index 0) Cô.8(lF1) A/1-8(Fi) A1.9(E) 8 Majorana—Weyl spinors) sym(index 0) C9.1(R) A32(E) 32 (Majorana spinors) sym(index 16) skew C)-1(1F1) = .416(E) 0 A1.16(IF1) 16 (Majorana—Weyl spinors) swap C1.1(1F0 AA) 2 (Majorana spinors) sym(index 1) skew C B 11R 1 (Majorana—Weyl spinors) swap (b) Clifford algebra C2 C3 C4 C5 C6 Exercise 2.2Abstract from tables 2.8, 2.15, 2.17 and2.18 theinformation inthefollowing tables. Inpart (a): C2 gives realdimension ofirreducible algebra representation; C3gives real-valued spin-invariant product associated with Z5;C4gives real-valued spin-invariant product associated with E17.Inpart (b): C2gives complex dimension of irreducible algebra representation; C3gives complex-valued spin-invariant product associated with Z5;C4gives complex-valued spin-invariant product associated with $1];C5gives complex-valued spin-invariant product associated with55*,C6gives complex-valued spin-invariant product associated withf§17*. (H) CilffOrd algebra c2 c3 c4 C4_4(lR) =A/t,6(lR) 16(Majorana spinors) sym(index 8) sym(index 8) CI_.(lR) =A/t,,(lR) ®At_i(lR) 8(Majorana—Weyl spinors) sym(index 4) C,,_@(lR) =A/t,i,(lR) 16(Majorana spinors) sym(index 0) sym(index 8) C;,,(R) =Ma,(lB) (-DM,,(lR) 8(Majorana—Weyl spinors) sym(index 0) C0_,,(lB) =A/t,,-,(lR) 16(Majorana spinors) sym(index 8) sym(index 0) C§,,((lR) =./i/i,g(B) (-3A/t,((lR) 8(Majorana—Weyl spinors) sym(index O) C9_,(R) =A/t32(B) 32(Majorana spinors) sym(index 16) skew CJ_,(lR) =A/L1(,(lR) ®A/t,(,(lR) 16(Majorana—Weyl spinors) swap C,,|(R) =./I/L2(R) 2(Majorana spinors) sym(index 1) skew C,‘_,=IR@IR 1(Majorana—Weyl spinors) swap (b) Clifford algebra c2 c3 c4 cs co C,'§_@(lB) =A/L,(,(C) 16(Dirac spinors) sym sym Hermitian sym (index 0)Hermitian sym (index 8) C,‘;]’{(lB) =./“.g(C) @A/1.,((C) 8(Weyl spinors) sym Hermitian sym (index O) C§,(lB) =A/t32(C) 32(Dirac spinors) sym skew Hermitian sym (index 16)Hermitian sym (index 16) C§j(lB) =A/L,(,(C) (-3/1/L,6(C) 16(Weyl spinors) swap Swap BIBLIOGRAPHY 105 Bibliography Chevalley C 1954 The Algebraic Theory of Spinors (New York: Columbia) Coquereaux R 1982 Phys. Lett. 115B 389 Crumeyrolle A 1969 Ann. Inst. Henri Poincaré A 11 19 — 1971 Ann. Inst. Henri Poincaré A 14 309 — 1972 Ann. Inst. Henri Poincaré A 16 171 — 1974 Algebres de Clifford et spineurs Cours et Seminaires de l'Universite de Toulouse III Greub W 1978 Multilinear Algebra 2nd edn Lounesto P 1980 Ann. Inst. Henri Poincaré 33 53 Porteous I 1981 Topological Geometry 2nd edn (Cambridge: Cambridge Uni- versity Press) BIBLIOGRAPHY 105 Bibliography Chevalley C1954 TheAlgebraic Theory ofSpinors (New York: Columbia) Coquereaux R1982 Phys. Lett. 115B 389 Crumeyrolle A1969 Ann. Inst. Henri Poincare All19 -— 1971 Ann. Inst. Henri Poincare A14309 -— 1972 Ann. Inst. Henri Poincare A16171 ,— 1974 Algebres deClifford etspineurs Cours etSeminaires del’Universite de Toulouse III Greub W1978 Multilinear Algebra 2ndedn Lounesto P1980 Ann. Inst. Henri Poincare 3353 Porteous I1981 Topological Geometry 2nd edn (Cambridge: Cambridge Uni- versity Press) 3 Pure Spinors and Triality This chapter contains some further properties of Clifford algebras and spinors. They may be regarded as more advanced material and the presentation will be adapted accordingly. Some readers may prefer to defer a study of these topics until later: they are not essential pre- requisites for understanding the bulk of the material that follows, although we shall briefly make reference to certain properties of pure spinors in the last chapter. 3.1 Pure Spinors In certain cases spinors may have a rather direct geometrical interpreta- tion. As was observed by Cartan [5] certain spinors of C( V, g) may be correlated with maximal totally isotropic subspaces of V: these spinors being called pure. (An isotropic subspace of V is one on which g induces the zero bilinear form.) The account of pure spinors that we shall give follows that given in Chevalley [6]. We shall only consider the case in which V is even-dimensional. It turns out that in four (and six) dimensions all complex Weyl spinors are pure. For the physically interesting Lorentzian case this gives a correlation between Weyl spinors (or Majorana spinors) and null planes. In the positive-definite case maximal isotropic subspaces, and hence pure spinors, can be put into correspondence with complex structures. In more than six dimensions not all spinors are pure. The possibility of constraining spinors to be pure in physical theories formulated in higher dimensions has been investigated ([7], [8]). Let V be an F-linear space with dim FV = 2r, and g an F-valued F-bilinear form with maximal index. (Here F will be either R or C.) We can express V in terms of maximal (r-dimensional) totally isotropic Pure Spinors andTriality This chapter contains some further properties ofClifford algebras and spinors. They may beregarded asmore advanced material and the presentation willbeadapted accordingly. Some readers may prefer to defer astudy ofthese topics until later: they arenotessential pre- requisites forunderstanding thebulk ofthe material that follows, although weshall briefly make reference tocertain properties ofpure spinors inthelastchapter. 3.1Pure Spinors Incertain cases spinors may have arather direct geometrical interpreta- tion. Aswasobserved byCartan [5]certain spinors ofC(V, g)may be correlated with maximal totally isotropic subspaces ofV:these spinors being called pure. (An isotropic subspace ofVisone onwhich g induces thezero bilinear form.) The account ofpure spinors that we shall give follows thatgiven inChevalley [6].Weshall only consider the case inwhich Viseven-dimensional. Itturns outthat infour (and six) dimensions allcomplex Weyl spinors are pure. For the physically interesting Lorentzian case thisgives acorrelation between Weyl spinors (orMajorana spinors) andnullplanes. Inthepositive-definite case maximal isotropic subspaces, and hence pure spinors, canbeputinto correspondence with complex structures. Inmore than sixdimensions notallspinors arepure. The possibility ofconstraining spinors tobe pure inphysical theories formulated inhigher dimensions hasbeen investigated ([7], [8]). Let VbeanF-linear space with dimFV= 2r,and ganF-valued F-bilinear form with maximal index. (I-Iere Fwillbeeither IRorC.)We canexpress Vinterms ofmaximal (r-dimensional) totally isotropic PURE SPINORS 107 subspaces M and N as V = M 0 N. A Witt basis for V is formed from the isotropic bases {xl} for M and {y'} for N such that xiyj + yjxi = 6 4. (3.1.1) Since g is of maximal index the Clifford algebra is a total matrix algebra C(V, g) = A1.2,(F) (3.1.2) whilst the structure of the even subalgebra is given by C+(V, g) = At2,-,(F) At2,-.(F). (3.1.3 Let be the 2r-form with i2 = 1 so that the idempotents P, = (1± reduce C+(V, g) to simple ideals. In terms of a Witt basis for V we may choose = [x', yt][x 2, y21 . . . [xr, yr] (3.1.4) the brackets denoting Clifford commutators. Let zm be the r-form product of some basis for M. Since M is totally isotropic its Clifford algebra is just its exterior algebra A(M) and so the r-form product of a different basis will differ from zm by the deter- minant of the general linear transformation relating the bases. Given the Witt decomposition V= MC) N we can express any element of C(V, g) in terms of products of the x' and the y'. Using the relations (3.1.1) the elements of M can be positioned at the right-hand side of any terms so that we see that C(V, g)zm = C(N, g)zm = A(N)z m. Thus the left ideal C(V, g)zm has the dimension of the exterior algebra of N, 2', and is hence a minimal left ideal. We may take this minimal left ideal as the space of spinors. If p E C(V, g)Zm then ip = Bzm for B E A(N). Thus = Bnizm. We have -z-zm = [xi, y ii[x2, y2] [xr, yr]xl x2 xr = [xl, y1]x1[x27 y2]x2 [xr, y r]xr and from (3.1.1) [x', y']x1 = x'y'x' = - (1 — y'x')x' = x' SO ‘iZA4 = Zm. Thus 'z'tp = finzm and the even and odd (under n) subspaces of C(V, g)zm form the semi-spinor spaces of the even subalgebra. Just as a maximal totally isotropic subspace can be used to define a minimal left ideal it can also be used to define a minimal right ideal. Since the Clifford algebra is a total matrix algebra the intersection of a minimal left ideal with a minimal right ideal is a 1-dimensional F-linear space. (For if P and P' are primitive idempotents with P' = SPS -1 then P'C(V, g)P = SPC(V, g)P and PC(V, g)P = AP for A E F.) So if we use a maximal totally isotropic subspace M to define PURE SPINORS 107 subspaces MandNasV=M(BN.[AWitt basis forVisformed from theisotropic bases {x’}forMand{y’}forNsuch that x‘“y/P+y/ix‘)=6'7. (3.1.1) Since gisofmaximal index theClifford algebra isatotal matrix algebra C(V, g)=A/t2i(F) (3.1.2) whilst thestructure oftheeven subalgebra isgiven by C*(Vi 3)=442'-1(F) (*9Ma'-1(F)- (3-1-3 Let2bethe2r-form with 22=1sothat theidempotents Pi.= §(1iE)reduce C*(V,g)tosimple ideals. Interms ofaWitt basis for Vwemaychoose E=[x‘,y‘][x2, yz]...[x’,y'] (3.1.4) thebrackets denoting Clifford commutators. LetzMbether-form product ofsome basis forM.Since Mistotally isotropic itsClifford algebra isjustitsexterior algebra A(M) andsothe r-form product ofadifferent basis will differ from zMbythedeter- minant ofthegeneral linear transformation relating thebases. Given the Wittdecomposition V=MGBNwecanexpress anyelement ofC(V, g) interms ofproducts ofthex’andthey‘.Using therelations (3.1.1) the elements ofMcanbepositioned attheright-hand sideofanyterms so thatweseethatC(V, g)zM =C(N, g)zM =A(N)zM. Thus theleftideal C(V, g)zM hasthedimension oftheexterior algebra ofN,2',andis hence aminimal leftideal. Wemaytake thisminimal leftideal asthe space ofspinors. If1peC(V, g)zM then tp=BzM forBeA(N). Thus ftp=B"EzM. Wehave 22M=[x‘,y‘][x2, yz]...[x’,y’]x‘ x2...x’ =[X‘iY1]X1[X2, YZIXZ ~--[X5Y’]X' andfrom (3.1.1) [xl,yilxi =xiyixi :(1_yixi)xi =xi S0EZM =zM. Thus 21/1= B"zM and theeven and odd (under 17) subspaces ofC(V, g)zM form thesemi-spinor spaces oftheeven subalgebra. Just asamaximal totally isotropic subspace canbeused to define aminimal leftideal itcanalso beused todefine aminimal right ideal. Since theClifford algebra isatotal matrix algebra theintersection ofaminimal leftideal with aminimal right ideal isa1-dimensional F-linear space. (For ifPand P’are primitive idempotents with P’=SPS'1 then P’C(V, g)P=SPC(V, g)P and PC(V, g)P=APfor XeF.)Soifweuseamaximal totally isotropic subspace Mtodefine 108 PURE SPINORS AND TRIAL1TY our space of spinors any other maximal totally isotropic subspace T can be used to define a minimal right ideal and hence a one-dimensional subspace of the spinor space. If M, T are maximal totally isotropic subspaces then any element of z g)z m is a representative spinor for T (with respect to M). A spinor that represents some T is called pure. (3.1.5) An immediate consequence of this definition is the following: If T = x(s).M for s c F then a representative for T is u = szm. (3.1.6) The space of representative spinors for M is spanned by zm, so if u is a representative for M then xu = 0 V x E M. If now u is any element of C( V, g)zm then u = Bzm for some B E A(N). For any y' E N we can write B = yiB i + B2 with B1 and B2 in the exterior algebra of the subspace of N spanned by the remaining y. So xiu = B iu and x`u = 0 only if B lies in the exterior algebra of the (r — 1)-dimensional subspace of N spanned by the remaining y. Thus u is a representative for M if and only if xu = 0 V x c M. Because of (3.1.6) this can be couched more generally. A spinor u is a representative for T if and only if xu = 0 V x E T. (3.1.7) Given the totally isotropic M there is no unique N such that V=MON. If T is a maximal totally isotropic subspace with dim( T n M)= h then we can always choose a Witt basis such that {x'} is a basis for M and {x', . . xh yh+1, y is a basis for T. Starting with a basis {x', . . } for T n M the Witt basis can be completed by a Gram—Schmidt type of construction. If we adapt the Witt basis in this way to the isotropic subspaces M and T then a representative for T is u = yh+1 . . . yr. It will often be useful to have this canonical form for a pure spinor. In even dimensions all elements of the Clifford group are either even or odd. Thus, by (3.1.6), all pure spinors are either even or odd. This property of the representative spinors can be used to classify the maximal totally isotropic subspaces as either even or odd. If 7.1, T2 are maximal totally isotropic subspaces then T, and T2 are both even or odd if and only if dim( T, n T2) = r mod 2. (3.1.8) There is some S E F such that x(s).1, = M. If u1, u2 are representatives for T, and T, then zm= su2 and if u = sui then u is a representative for T = x(s).T 1. Since s is either even or odd then u and zm behave the same under 71 if and only if u, and u2 do. Moreover, T n M = x(s).(T i n T2) so it is sufficient to prove that representatives 108 PURE SPINORS AND TRIALITY ourspace ofspinors anyother maximal totally isotropic subspace Tcan beused todefine aminimal right ideal and hence aone-dimensional subspace ofthespinor space. IfM,Taremaximal totally isotropic subspaces then any element ofzTC(V, g)zM isarepresentative spinor forT(with respect toM).Aspinor that represents some Tiscalled pure. (3.1.5) Animmediate consequence ofthisdefinition isthefollowing: IfT=)((s).M forsEFthen arepresentative forTisu=szM. (3.1.6) Thespace ofrepresentative spinors forMisspanned by2M,soifuisa representative forMthen xu=0Vx eM.Ifnowuisanyelement of C(V, g)zM then u=BzM forsome BeA(N). Foranyy'eN wecan write B=y’B1 +B2with B1and B2intheexterior algebra ofthe subspace ofNspanned bytheremaining y.Sox’u=Blu andxiu=0 only ifBliesintheexterior algebra ofthe(r—1)-dimensional subspace ofNspanned bytheremaining y.Thus uisarepresentative forMif andonlyifxu=0VxEM.Because of(3.1.6) thiscanbecouched more generally. Aspinor uisarepresentative forTifandonlyifxu=0VxeT.(3.1.7) Given the totally isotropic Mthere isnounique Nsuch that V=MG9N.IfTisamaximal totally isotropic subspace with dim(T F)M)=hthen wecanalways choose aWitt basis such that {xi} isabasis forMand{x‘, ...,x",yh“, ...,y’}isabasis forT.Starting with abasis {x1, ...,xh}forTF)MtheWitt basis canbecompleted byaGram—Schmidt type ofconstruction. Ifweadapt theWitt basis in thiswaytotheisotropic subspaces Mand Tthen arepresentative forT isu=y"*' ...y’zM. Itwilloften beuseful tohave thiscanonical form forapure spinor. Ineven dimensions allelements oftheClifford group areeither even orodd. Thus, by(316), allpure spinors areeither even orodd. This property ofthe representative spinors can beused toclassify the maximal totally isotropic subspaces aseither even orodd. IfT1,T2aremaximal totally isotropic subspaces then T1and T2 are both even or odd if and only if dim(T, F)T2)=rmod2. (3.18) There issome sEFsuch that)((s). T2=M.Iful,u;arerepresentatives forT,and T;then zM= S112andifu=sulthen uisarepresentative forT=X(s).T,. Since siseither even oroddthen uand2Mbehave the same under 17ifand only ifu, and I12 do. Moreover, TF)M=)((s).(T] F)T2)soitissufficient toprove that representatives PURE SPINORS 109 for T and M are both even or odd if dim(T n M)= r mod 2. If we adapt a Witt basis to T and M then a representative u for T has the canonical form u = yh+1 yrZm where dim( T n M)= h. So u and zm are both even or odd if r — h = 0 mod 2, that is h = r mod 2. In general not all spinors will be pure; whereas we can always choose a basis of pure spinors, linear combinations of pure spinors will not in general be pure. The following gives the conditions for the sum of two pure spinors to be pure. If ul, u2 represent T1 and T2 then a necessary and sufficient condition for u1 + u2 to be pure is that dim(Ti n T2) = r or r — 2. If this is the case then non-trivial linear combinations of u1 and u2 represent all T such that T n T2= T1 n T2. (3.1.9) As in the proof of (3.1.8) it is sufficient to consider representatives for T and M. We adapt a Witt basis to these subspaces. A non-trivial linear combination of representatives for these subspaces will be pure if u is, where u = Az,vi + yh+1 . . . yrzm E F. (3.1.10) Now x'u = 0 if and only if i =1, . . ., h and = 0 if and only if Â, = 0 V j = h + 1, . . . r, so if u is pure, representing T' say, then nm= T n M. If this is the case then we can choose a Witt basis {x', y") i =1, . . r with {x1, xh yh+1, . . .3 )1'} a basis for T'. In this basis representatives for T' will take the canonical form, so if u is pure Az m yh+1 yrzm = ittyh+Ir . yrrzm (3.1.11) for some y E F. By repeatedly using (3.1.1) the Clifford products in yh+1 yrzm    yrZ m can be written in terms of exterior products, yh+1 having homogeneous (h, h+2, . . 2r—h)-form components. Similarly yh-f-i, yrizm will have homogeneous components of the same degrees. Equating h-form components in (3.1.11) gives = 1. (3.1.12) If h + 2 r then this can be used to equate (h+2)-forms in (3.1.11): x1 xh(yh+I A xh+1 yr A xr) xh(yh+1, A xh+1 . Y .ri xr) = 1 . (3.1.13) The {y"} can be written as linear combinations of the basis {x', y`). Since {x', y") is also a Witt basis we have = y' + E + Ni i = h + 1, . . , r (3.1.14) j=h+1 PURE SPINORS 109 forTand Mareboth even orodd ifdim(TF) M)=rmod 2.Ifwe adapt aWitt basis toTand Mthen arepresentative uforThasthe canonical form u=y""‘ y’zM where dim(TF) M)=h.Souand ZMareboth even oroddifr—h=Omod 2,thatish=rmod 2. Ingeneral notallspinors willbepure; whereas wecanalways choose abasis ofpure spinors, linear combinations ofpure spinors willnotin general bepure. The following gives theconditions forthesum oftwo pure spinors tobepure. Iful,u2represent T1and T2then anecessary andsufficient condition forul+u2tobepure isthatdim(T] F)T2)=ror r-2.Ifthisisthecase then non-trivial linear combinations ofu]andu2represent allTsuch that TF)T2=T1F)T2. (3.1.9) Asintheproof of(3.l.8) itissufficient toconsider representatives for TandM.Weadapt aWitt basis tothese subspaces. Anon-trivial linear combination ofrepresentatives forthese subspaces willbepure ifuis, where u=/12M +yh“ ...y'zM heF. (3.1.10) Now x’u=0ifandonly ifi=l,...,handE,’;,,+l)t]-x/u =0ifandonly if)1)=OVj=h+l,...r,soifuispure, representing T’say, then T’F)M=TF)M.Ifthisisthecase then wecanchoose aWitt basis {x',y”} i=1, ...,rwith {x‘, ...,x",y”"1', ...,y”} abasis forT’. Inthisbasis representatives forT’willtake thecanonical form, soifu ISpure MM +y"*‘ ...y'zM =/.¢y"+" ...y”zM (3.1.11) forsome iteF.Byrepeatedly using (3.1.1) theClifford products iny"*1 ...y’zM canbewritten interms ofexterior products, y""1 ...y'zM having homogeneous (h,h+2, ...,2r—h)-form components. Similarly y”"" ...y"zM will have homogeneous components ofthe same degrees. Equating h-form components in(3.1.11) gives it=1. (3.1.l2) Ifh+25*rthenthiscanbeused toequate (h+2)-forms in(3.1.11): X1...x"(y""1,(x"+‘ +...+y’,(x’) =x1...x"(y""1’,(x"*‘+ ...+ y”Ax’). (3.1.13) The {yi’} canbewritten aslinear combinations ofthebasis {x’,y‘}. Since {x',y"}isalso aWitt basis wehave yr/=yt+;lM"/x1+N' i=h+1,..,r (3.114) j=+ 110 PURE SPINORS AND TRIALITY where m9 = -m9 and 111` is a linear combination of {x . . . }. Inserting (3.1.14) in (3.1.13) gives m9 = O Vi, j = h + 1, . . r and hence {x', . . yhn' , . . y"} is just a new basis for T; that is, T' = T. So the only non-trivial case is h + 2 = r. In this case (3.1.11) is seen to be satisfied by = yr-i Axr Nr-1 yr, = yr )r-1 Nr Here 2 is seen to parametrise all T' with rnm=Tnm. Although in general, as we have stated, not all spinors are pure, in sufficiently low dimensions the above result can be used to show that all semi-spinors are pure. If r 3 then all semi-spinors are pure. (3.1.15) In general we can always choose a set of pure spinors as a basis for the spinor space. Any semi-spinor will be a linear combination of pure spinors that are all even or odd. From (3.1.8) we know that if ul, u2 are two such pure spinors representing T1 and T2 then dim( T1 n T2) = r mod 2, whereas from (3.1.9) linear combinations of u1 and u2 will be pure if dim(Ti n T,) = r or r — 2. Thus if r 3 linear combinations of any two even or odd pure spinors are pure and hence all semi-spinors are pure. Through (3.1.7) a pure spinor is related to the maximal isotropic subspace that it represents. However, given a semi-spinor this does not give a very practical way of determining whether or not it is pure. Given a spin-invariant inner product then the tensor product of a spinor with its adjoint can be identified with an element of the Clifford algebra. Necessary and sufficient conditions for a spinor to be pure can be given in terms of these tensors on the space of spinors (or `spinor bilinears'). These conditions give a practical way of determining whether any given spinor is pure or not, and, in the case in which it is, recovering the associated maximal totally isotropic subspace. Let ( , ) be an F-valued, symmetric or skew, product on spinors with `4 as adjoint involution. Let u be the spinor adjoint to u with respect to this product. If u1, u2 represent T1 and T2 then T1 n T2 0 if and only if (u1, u2) = 0. (3.1.16) Suppose firstly that there is some x in T1 n T2. Then there is some y such that xy + yx = 1 and (u1, u2) = (u1, (xy + yx)u2) = (u1, xyu2) since x E T2. Since the spinor product has as adjoint involution then (u1, xyu2) = (xui, yu,) and this is zero if x E T1. So Ti n T2 0 implies that (ul, u2) = 0. To prove the converse we let M and N be any 110 PURE SPINORS AND TRIALITY where Mil=—M/'0 and N’isalinear combination of{x‘, x"}. Inserting (3.1.14) in(3.l.13) gives M‘7=0Vi,j= h+1,..., rand hence {x‘,.x“,y"*", ...,y”}isjustanewbasis forT;thatis, T’=T.Sotheonlynon-trivial caseish+2=r.Inthiscase(3.1.11) is seentobesatisfied by yr-ll =yr—] +Axr +Nr—l yr! =yr _Axr—l +Nr_ Here Aisseen toparametrise allT’with T’F)M=Tr)M.Although ingeneral, aswehave stated, notallspinors arepure, insufficiently low dimensions theabove result canbeused toshow thatallsemi-spinors arepure. Ifrs3thenallsemi-spinors arepure. (3.1.15) Ingeneral wecanalways choose asetofpure spinors asabasis for thespinor space. Any semi-spinor willbealinear combination ofpure spinors thatarealleven orodd. From (3.1.8) weknow thatiful,u2are two such pure spinors representing T,and T2then dim(T, OT2)= rmod2.whereas from (3.1.9) linear combinations ofu,andu2willbe pure ifdim(T, OT2)=rorr—2.Thus ifrs3linear combinations of anytwoeven orodd pure spinors arepure andhence allsemi-spinors arepure. Through (3.1.7) apure spinor isrelated tothemaximal isotropic subspace that itrepresents. However, given asemi-spinor thisdoes not give avery practical wayofdetermining whether ornotitispure. Given aspin-invariant inner product then thetensor product ofaspinor with itsadjoint canbeidentified with anelement oftheClifford algebra. Necessary andsufficient conditions foraspinor tobepure canbegiven interms ofthese tensors onthespace ofspinors (or‘spinor bilinears’). These conditions giveapractical wayofdetermining whether anygiven spinor ispure ornot,and, inthecase inwhich itis,recovering the associated maximal totally isotropic subspace. Let(,)beanF-valued, symmetric orskew, product onspinors with Easadjoint involution. Let17bethespinor adjoint touwithrespect to thisproduct. Iful,u2represent T1andT2then T1F1T2¢Qifandonly if(M1,u2)=0. (3.116) Suppose firstly that there issome xinT1OT2.Then there issome y such thatxy+yx=1and ('11-'12) =("D (W+}’x)u2)=(u1>x)’u2) since xeT2.Since thespinor product has§asadjoint involution then (u1,xyu2) =(xu1,yu2) and this iszero ifxeT1.SoT1OT2¢@ implies that(u1,u2)=0.Toprove theconverse weletMandNbeany PURE SPINORS 111 two maximal isotropic subspaces such that V = M 8 N. Then a spinor basis, each element of which is pure, is given by {yizm} with I a multi-index. Now we have already shown that (zm, yizm) = 0 unless y1 = ZN, and since the spinor product is non-degenerate we must have (zm, zNzm) * O. But zNzm is a spinor representing N which was any maximal totally isotropic subspace not intersecting with M. So if u1 and u2 represent T, and T2 then (u1, u2) can only be zero if T, n T2 0. If iv = +(—)v then for u any spinor 972,._p(uti) = for all p. (3.1.17) Since i2 = 1 W2r-p( 14173) = 992r-p(U6).i Z = p(14173j) = as the adjoint spinor is defined with respect to a product with as the adjoint involution. Since is a 2r-form = (-1)r i and W2r_p(u/3) = (-1)r97p(u(2"()))'i and the result follows. If y = Bzm for B c A(N) then = Blizm and (-1)r zv = 1.)'1. So if v = ±v then 2r_ p() = + (—)9 p(u0'i If u1, u2 represent T1 and T2 with dim(Ti n T2) = h then 52p(u2ii1) = 0 if p <h or p > 2r — h, whilst 92h(u2i(i) = z Tin T2. (3.1.18) If s c F then 1p(su2s711) = 2,.(s)sYp(u2iii)s-1 so without loss of general- ity we can assume that u1 represents M with u2 representing some T with dim(TnM) = h. In an adapted Witt basis we need to consider yhn yrzmim. In the proof of (3.1.16) we showed that (zm, yizm) = 0 unless yl = zN. Now zmzNzm = ±zm so we can always normalise the spinor product such that (zm, zNzm)zm = zmzNzm. The definition of u2it1 is that u2a1v -= u2(u1, y), so zA42-my'zm = (zm, ylzm)zm. This is zero unless y' = zN and for the normalisation just mentioned (zm, zNzm)zm = zmzNzm. But zmyizm = 0 unless yl = zN and so (zmim)yizm = zmyizm for all multi-indices I and so zm2 m = zm. Hence yh+1 . . . yrzm2m = yh+1 . yrzm. The form of lowest degree in yh+1 yrzm is proportional to xl . xh, which is just the product of a basis for T n M. Since all pure spinors are semi-spinors it follows from (3.1.17) that there is no non-vanishing p-form for p > 2r — h. A semi-spinor u is pure if and only if?(uû) = 0 V p * r. (3.1.19) From (3.1.18) we see that if u is pure then certainly Wp(uil) = 0 V p r, so what we need to do is to show that this condition on a semi-spinor is sufficient for it to be pure. Any spinor u can be written as U = Bzm where B E A(N). There is some s E r such that su ---- (1+ b)z m where b E A(N) and 9'46) = O. If u is a semi-spinor then so is su and hence b must be an even element of A(N). Suppose PURE SPINORS 111 twomaximal isotropic subspaces such that V=M69N.Then aspinor basis, each element ofwhich ispure, isgiven by{y'zM} with Ia multi-index. Now wehave already shown that (zM,y'zM) =0unless y’=2N,andsince thespinor product isnon-degenerate wemust have (ZM,zNzM) 750.ButzNzM isaspinor representing Nwhich wasany maximal totally isotropic subspace notintersecting with M.Soifu,and L42represent T,andT2then (u,,u2)canonlybezeroifT,F)T2#9Q. IfEv=+(—)v then for uany spinor EI’2,_,,(u5) = +(-)(—1)'El’p(u5) Eforallp. (3.1.17) Since E2=1 er(u13)- er(aa)2:~ —er(iiimz —er(a(}?v))rZr—P _Z'—P —P _P astheadjoint spinor isdefined with respect toaproduct with Easthe adjoint invgllition. Since 2isa2r-form 25=(—1)’f andEI’2,_,,(uz3) = (—1)’EI’,,(u(z'v))§ andtheresult follows. Ifv=B2,, forBeA(N) then 20=B”zM and (—1)"iu =0". S0 if0"=iv then Sf2,_p(u5) =+(—)5I’,,(m3) 2. Ifu,,112represent T,andT2with dim(T, F)T2)=hthen Sfp(u2i1,) =0ifp<horp>2r—h,whilst El’,,(u2L7,) = 221,,” (3.1.18) IfseFthen 9’,,(su2s'i1,) =/1(s)sEI’p(u2i7,)s‘1 sowithout lossofgeneral- itywecanassume thatu,represents Mwith u2representing some T with dim( TF)M) =h.Inanadapted Witt basis weneed toconsider y"*1 y’zM2M. Intheproof of(3.1.16) weshowed that(zM,y’zM) =0 unless y'=ZN.Now zMzNzM =izM sowecanalways normalise the spinor product such that (zM, zNzM)zM =ZMZNZM. The definition of u2fi, isthat u2fi,v =u2(u,, v),sozM2My'zM =(ZM, y'zM)zM. This is zero unless y'=2N and for the normalisation just mentioned (ZM,z,.,zM)zM =zMzNzM. But zMy’zM =0unless y’=2),,and so (ZM2M)y’zM =zMy'zM forallmulti-indices Iand soZMZM =2M. Hence y""‘ ...y’zM2M =y"*‘ ...y’zM. Theform oflowest degree in y"*‘ ...y’zM isproportional tox‘...x“,which isjusttheproduct of abasis forTF)M.Since allpure spinors aresemi-spinors itfollows from (3.1.17) thatthere isnonon-vanishing p-form forp>2r—h. Asemi-spinor uispureifandonlyifEI’p(ui7) =0 Vpabr.(3.1.19) From (3.1.18) weseethat ifuispure then certainly EI’,,(ui7) =0 Vpabr,sowhat weneed todoistoshow thatthiscondition ona semi-spinor issufficient forittobepure. Anyspinor ucanbewritten as H=BZM where BeA(N). There issome sel" such that SH=(1+b)zM where be/\(N) and EI’0(b) =0.Ifuisasemi-spinor then soissuandhence bmust beaneven element ofA(N). Suppose 112 PURE SPINORS AND TRIALITY that 2(b) * 0, then exp(—Y 2(b)) E F n A(N). Now the Clifford algebra of N is just its exterior algebra and so 2[exp(—Y 2(b))b] = 1 0[exp(—J 2(b))]if2(b) + 2[exp(—W 2(b))1920(b) = 2(b) since Jo(b) = O. Thus exp(—Y,(b))su = (1 + b')z m where b' E A±(N) with Y0(6') = = 0. Suppose that the non-vanishing homogeneous component of b' of lowest degree is an h-form. In an appropriate basis we assume that exp(—J2(b))su = (1 + Ay1y2 . . yh + . .. )z m where the extra terms are of degree h or higher. Multiplying by yyr-1 . y'+' will annihilate these other terms so if xh+1 xry, exp(-922(b))su then v = (1 + Ay 13,2 yh)zm. (3.1.20) Now we come to the point of this construction. If u is any spinor and SE F then f(sus) =)1(s)s9 2(uil)s-1 andJ 2(uû) = 0 Vp * r <=>Jp(sus74) = 0 V p r. If a is any element of V then audit = aufia. By (2.1.7) and (2.1.8) autia = g(a, a)(urt) 11 — 2a A i,(uit)n and J'p(audii) = (-1)Pg(a, a)J p(urt) — 2( —1)Pa A id&p(11/7). So if p(urt) = 0 Vp * r then p(audit) = 0 Vp *rVaE V. Thus if the semi-spinor u that we started with satisfies 5' p(urt) = 0 V p r then the y we have constructed in (3.1.20) also satisfies these conditions. We will now show that this can only hold if A. = 0; that is exp(-9 22(b))su = zm and hence u is pure. Now y is the sum of two pure spinors and, as we have already noted, a pure spinor will satisfy the conditions of the theorem. So if u satisfies these conditions then ap{zm(yl yhzm) + y1... yhzmim} = 0 Vp * r. As we noted in the proof of (3.1.18) zmz—m = zm and so zm(y1 yhzm) yl yhzm2m = z oh . yl yl yhz m. We now rearrange these terms, remembering that h is even: z myh yl yl yhz m = {(xly1) (xhyh) (_1)h/2( y1x1) (yhxh ))xh-r1 xr Now xiyz = 4 + xi A)" whereas yixi = 4 — xi A y'. So if h/2 is even there will be a non-vanishing 0-form in { }, whereas if hI2 is odd there will be a non-vanishing 2-form. Thus in the first case the total expres- sion has a non-vanishing (r — h)-form, whilst in the second the 112 PURE SPINORS AND TRIALITY thatSf2(b) ah0,then exp(—SV2(b)) eFF)A(N). Now theClifford algebra ofNisjustitsexterior algebra andso 5fil¢XP(—5fz(b))bl =9"<il<'=XP(—5fz(b))l5f2(b) +9zl<'=XP(—9’z(b))l5/’o(b) =92(1)) since Sf,,(b) =0.Thus exp(—92(b))su =(1+b')zM where b’e/\*(N) with 9’,,(b’) =Sf2(b’) =0.Suppose thatthenon-vanishing homogeneous component ofb’oflowest degree isanh-form. Inanappropriate basis weassume that exp(—{-f2(b))su =(1+)ly1yZ...y” +...)zM where theextra terms areofdegree horhigher. Multiplying byy’y"1 ...yh“ willannihilate these other terms soif u=xh“ ...x’y’ ...y"*1exp(—Et’2(b))su then u=(1+)ly'y2 ...y")zM. (3.1.20) Now wecome tothepoint ofthisconstruction. Ifuisanyspinor and s6F then $fp(susT4) =l(s)s9’,,(ui2)s'1 and Efp(ui7) =0Vp=#r <:st,,(.mm) =0Vp#=r.Ifaisanyelement ofVthen aua"12 =aufia. By (2.1.7) and (2.1.8) aufla =g(a, a)(u17)" —2a,\l,;(MI1)’7 and §f’p(auW) =(—1)Pg(a, a)Sl’p(u12)— 2(—1)Pa Ai,,Sl’p(u17). SoifEl’p(u17) = 0Vp=#rthen Ef,,(aufl) =0Vp=#rVa6V.Thus ifthesemi-spinor u that westarted with satisfies Efp(ui7) =0Vp=#rthen theuwehave constructed in(3.1.20) alsosatisfies these conditions. Wewillnow show thatthiscanonlyhold ifA=0;thatisexp(—9’2(b))su =2Mandhence uispure. Now uisthesum oftwopure spinors and, aswehave already noted, apure spinor willsatisfy theconditions ofthetheorem. Soifu satisfies these conditions then 1§fp{zM(y1...y”zM)+y1...y"zMZM} =0 Vp=#r. Aswenoted intheproof of(3.1.18) ZMZT4 =ZMandso ZM(y[..._)/ zM)+y1...y"zMiM=zMy"...y1+y1...y"zM. Wenow rearrange these terms, remembering thathiseven: zMy"...y1+ y1.. .y"zM ={(x1y1)...(x"y")+(—1)"’2(y'x1)... (y"x")}x"*1. ..x’. Now x’y’=+x’Ay’ whereas y’x’=—x’Ay’. Soifh/2iseven there willbeanon-vanishing 0-form in{},whereas ifh/2isoddthere willbeanon-vanishing 2-form. Thus inthefirst case thetotal expres- sion has anon-vanishing (r—h)-form, whilst inthe second the PURE SPINORS 113 (r — h + 2)-form component is non-zero. Since h > 2 then in both cases there is a non-vanishing p-form with p < r, so -= 0 Vp r p(vD) =0Vp±rÂ= O. As we have already noted this shows that u is pure. Eight dimensions are interesting as the lowest number of dimensions in which not all semi-spinors are pure. If dim FV = 8 with F = 11=1 or C and g is of maximal index, then from tables 2.15 and 2.17 we see that ( , ) induces a symmetric product on the semi-spinors. Hence (u, e Au) = (e Au, u) = (u, e Au) and p(titi.) = 0 if [hp] is odd. If u is a semi-spinor then uti = 'z'u(1/4) = 'ha-4 uIV in eight dimensions, and so uU = (uti)q. So if u is any semi-spinor then uû = Ep=0,4,8p(") The 0-forms and 8-forms are related by (3.1.17) so in eight dimensions a semi-spinor u is pure if and only if Wo(uii) = 0, that is, (u, u) = 0. In this section we have taken the space of spinors to be a particular minimal left ideal of the Clifford algebra. This is convenient, enabling a basis of spinors to be constructed so as to facilitate the various algebraic proofs. However, it is not essential. Indeed all we really need is that the spinor space carry an irreducible representation of the Clifford algebra. Then (3.1.7) can be taken as the definition of a pure spinor, the stated results for pure spinors then following from this. Of course in general it would make no sense to talk about the behaviour of a spinor under the involution ii, but all references to 'even' and 'odd' spinors can be interpreted as referring to their behaviour under multiplication by (-1) r . For a real (pseudo-) orthogonal space whose metric has maximal index the pure spinors of the real Clifford algebra have a direct geometrical interpretation. For the remaining real Clifford algebras we cannot apply the above theory of pure spinors directly. However, if V is any real even-dimensional orthogonal space we may correlate the pure spinors of C c(V, g) with maximal totally isotropic subspaces of Vc. In certain cases these maximal totally isotropic subspaces of ye can be interpreted in terms of structures on the real vector space V. Of particular physical interest is the case in which V is a four- dimensional Lorentzian vector space (g has signature (p, q)=(3, 1)). Then if M is a maximal totally isotropic subspace of ye we have dimcM = 2. Suppose that u and o are respectively even and odd semi-spinors of C c( V, g) representing T1 and T2. Then because of (3.1.15) they are both pure. From (3.1.8) we see that dimc(Ti CI T2) must be odd (for r is here even, namely two). Hence dimc(Ti n T2) = 1. If z is the volume 4-form of V then st. = iz. So if superscript c denotes the conjugate-linear charge conjugation operation (here involu- tory) and u is even then tic is odd. The intersection of the maximal totally isotropic subspaces of VC represented by u and r.tc is one PURE SPINORS 113 (r-h+2)-form component isnon-zero. Since h>2then inboth cases there isanon-vanishing p-form with p<r,so Efp(u17)=0Vp=/=r I>3’,,(vi7)=0Vp=/=rI>)l=0. A5wehave already noted thisshows that uispure. Eight dimensions areinteresting asthelowest number ofdimensions inwhich notallsemi-spinors arepure. Ifdim;V =8with F=lRorC andgisofmaximal index, then from tables 2.15 and2.17 weseethat (,)induces asymmetric product onthe semi-spinors. Hence (11,e,,u) =(e,,11, u)=(11,e,,iu) andH’,,(11iZ) =0if[§p] isodd. If11isa semi-spinor then 1117=E11(iu) =5111725 =E1117? ineight dimensions, andso1417=(1117)". Soif11isanysemi-spinor then 1117=Ep=0,4’8H’p(u17). The0-forms and8-forms arerelated by(3.1.17) soineight dimensions a semi-spinor 11ispure ifandonly if9’0(1117) =0,thatis,(u,11)=0. lnthissection wehave taken thespace ofspinors tobeaparticular minimal leftideal oftheClifford algebra. This isconvenient, enabling a basis ofspinors tobeconstructed soastofacilitate thevarious algebraic proofs. However, itisnotessential. Indeed allwereally need isthatthe spinor space carry anirreducible representation oftheClifford algebra. Then (3.1.7) canbetaken asthedefinition ofapure spinor, thestated results forpure spinors then following from this. Ofcourse ingeneral it would make nosense totalkabout thebehaviour ofaspinor under the involution 11,butallreferences to‘even’ and ‘odd’ spinors can be interpreted asreferring totheir behaviour under multiplication by (—1)’E. For areal (pseudo-) orthogonal space whose metric has maximal index thepure spinors ofthereal Clifford algebra have adirect geometrical interpretation. Fortheremaining realClifford algebras we cannot apply theabove theory ofpure spinors directly. However, ifVis anyrealeven-dimensional orthogonal space wemaycorrelate thepure spinors ofCc(V, g)with maximal totally isotropic subspaces ofVc. In certain cases these maximal totally isotropic subspaces ofVCcanbe interpreted interms ofstructures ontherealvector space V. Ofparticular physical interest isthecase inwhich Visafour- dimensional Lorentzian vector space (ghassignature (p,q)=(3, 1)). Then ifMisamaximal totally isotropic subspace ofVCwehave dim¢;M =2.Suppose that 11and uarerespectively even andodd Semi-spinors ofCC(V, g)representing T,and T2.Then because of (3.1.15) they areboth pure. From (3.1.8) weseethat dim,;(T, F)T2) must beodd(forrishereeven, namely two). Hence dim,;(T, F)T2)= 1.Ifzisthevolume 4-form ofVthen E=iz.Soifsuperscript c denotes theconjugate-linear charge conjugation operation (here involu- tory) and11iseven then 11°isodd. Theintersection ofthemaximal totally isotropic subspaces ofVcrepresented byuand 11°isone 114 PURE SPINORS AND TRIALITY dimensional, containing n say. Thus nu = nu` = O. But nu = 0 implies that n*uc = 0, and similarly nu` = 0 implies that nu = O. So n* lies in the one-dimensional intersection of the subspaces represented by u and u e and n* = An for some A E C. Since complex conjugation is involu- tory, A must satisfy AA* = 1, that is A E U(1). There is some /I E U(1) such that A = ii2, and if x = ,in it follows that x* = x. Thus x is a real null vector such that x(u + ue) = 0. (3.1.21) This real null vector is determined up to multiplication by a real number. Suppose that u represents T which has a basis (x, w). Now x(wu e) = — wxue = 0 since xue = 0: and certainly w(wue) = 0 since w2 = O. Thus wue and u both represent T. Since the space of repre- sentative spinors for T is one dimensional there is some A E C such that wue = Au. We cannot have A = 0 since w does not lie in the subspace represented by ue. So if co =- w then coue = u. (3.1.22) The charge conjugate of this is wu = V. So amo*u = coue = u, and since co E T we have (cow* + co*co)u = u, and thus cow* + co*co = 1. (3.1.23) From the (complex) null 1-form co we can construct a unit 1-form a: a =- w + w*. (3.1.24) We have because of (3.1.22) a(u + uc) = u + uc. (3.1.25) The real unit 1-form a is determined up to the addition of an arbitrary multiple of the null 1-form x. So equivalently we have extracted from the complex semi-spinor u a real null 1-form x and a real decomposable 2-form F F x Aa (3.1.26) both determined up to a real multiple. If tp u + ue then p is a Majorana spinor and because of (3.1.21) and (3.1.25) we can equivalent- ly think of the real forms x and F as being determined by V. The theorems (3.1.18) and (3.1.19) enable the real forms x and F to be expressed in terms of u, ue and their adjoint spinors. There is a freedom to scale the spinor product ( , ) whose adjoint is by a complex number. In the Lorentzian case we can always choose a spinor basis such that charge conjugation simply conjugates the spinor components. Thus we can require that the spinor product satisfies 114 PURE SPINORS AND TRIALITY dimensional, containing nsay. Thus nu=nu“=0.Butnu=0implies thatn*u° =0,andsimilarly nu‘=0implies that n*u=0.Son*liesin theone-dimensional intersection ofthesubspaces represented byuand u°and n*=/inforsome lieC.Since complex conjugation isinvolu- tory, /1must satisfy /111*=1,that is/leU(1). There issome /.teU(1) such thatA=/.12,andifx=unitfollows that x*=x.Thus xisareal nullvector such that x(u+u‘)=0. (3.1.21) This real null vector isdetermined uptomultiplication byareal number. Suppose that urepresents Twhich hasabasis {x,w}. Now x(wu°) =—wxu° =0since xu°=0:and certainly w(wu°) =0since wz=0.Thus wu° and uboth represent T.Since thespace ofrepre- sentative spinors forTisonedimensional there issome heCsuch that wu°=/iu.Wecannot have /1=0since wdoes notlieinthesubspace represented byu“.Soifw=)l'1w then wuc =u. (3.1.22) Thecharge conjugate ofthisisw*u=u°.Soww*u =mu“=u,and since weTwehave (ww* +w*w)u =u,andthus 0101* +w*w =1. (3.1.23) From the(complex) null1-form wwecanconstruct aunit 1-form a: aEw+w*. (3.1.24) Wehave because of(3.1.22) a(u+u‘)=u+u‘. (3.1.25) The realunit 1-form aisdetermined uptotheaddition ofanarbitrary multiple ofthenull 1-form x.Soequivalently wehave extracted from thecomplex semi-spinor uarealnull1-form xandarealdecomposable 2-form F F=x,\a (3.126) both determined uptoareal multiple. If1p= u+u°then 1pisa Majorana spinor andbecause of(3.1.21) and(3.1.25) wecanequivalent- lythink oftherealforms xandFasbeing determined byip. The theorems (3.1.18) and(3.1.19) enable therealforms xandFto beexpressed interms ofu,u°andtheir adjoint spinors. There isa freedom toscale thespinor product (,)whose adjoint is‘g’byacomplex number. IntheLorentzian case wecanalways choose aspinor basis such thatcharge conjugation simply conjugates thespinor components. Thus wecanrequire thatthespinor product satisfies PURE SPINORS 115 (u1, u2)* = u2c) (3.1.27) this leaving only a real scaling freedom. Taking a spinor product which satisfied (3.1.27) we turn to (3.1.18). The intersection of the subspaces represented by u and uc is spanned by the real 1-form x. So (3.1.18) tells us that Wi(iutic) is a complex multiple of x. The factor of i is inserted to ensure that this 1-form is in fact real. For = o(iuVea)e° = (U°, ieau)e° and 571(iurcc)* = —(u, ieaue)ea = —(ie au, tc)e° = (U°, ieau)e° = i(ivac). (by (3.1.27)) (since is the adjoint) (since the product is skew) and thus wi(iutic) = (3.1.28) where x is, of course, only determined up to a real multiple. Let fx, col be a basis for T, represented by u, where co is the complex 1-form satisfying (3.1.22). Then co is determined up to the addition of a multiple of x. From (3.1.18) we know that iurc is a complex multiple of xot, say iuit = 2exp(i0)xco for an appropriately scaled x. So if G = (iutt — iuciic) then G = exp(i0)xot + exp(—i0)xof and G(co + co*) = cos 0 x + 2i sin Ox A (0 A . This G will be nothing other than the F of (3.1.26) if in fact O = O. We have 2G(co + co*) = iu[(co + co*)u] — iu°[(co + co*)u -`) = iuti° — jUCÜ since cou = 0 and cou` = u. But if a', )3, cp, tp are any spinors then (oe, (951—P)/6) = ((qt-P)c r = (iP, a')(9), = 0(q, /6) = (IP-9-9)/3)- and so (cp) = —tpc7). Thus 2G(o) + co*) = iuti° + (iurc°) and since izu = u then (utc-c)vl = —zuficz = —zu(zuc) = —izu(izu)' = and so utic = V'i(ufic) + W3(ufic). Since 3-forms change sign under we have G(o) + co*) = 9 91(iuric) = x by (3.1.28). That is, the F of (3.1.26) can be written as F = Re(iurc). (3.1.29) From (3.1.21) and (3.1.25) we see that x and F can equivalently be thought of as being associated with the Majorana spinor p = u + u° We can also express x and F in terms of tp and its adjoint. Since PURE SPINORS 115 (u,,u2)* =(u,°, u2°) (3.1.27) thisleaving only arealscaling freedom. Taking aspinor product which satisfied (3.1.27) weturnto(3.1.18). Theintersection ofthesubspaces represented byuandu‘isspanned bythereal1-form x.So(3.1.l8) tellsusthat$f’,(i1117°) isacomplex multiple ofx.Thefactor ofiis inserted toensure thatthis1-form isinfactreal. For $f’,(iui7°) =Ef’,,(iui7°e,,)e" =(u°,ie,,u)e“ and $f’,(iui7°)* =—(u, ie,,u°)e" (by(3.1.27)) =—(ie,,u, u°)e“ (since 5istheadjoint) =(u°,ie,,u)e” (since theproduct isskew) = $f’,(lLtl7°). andthus 8/*,(i1ia*) =X (3.1.28) where xis,ofcourse, onlydetermined uptoarealmultiple. Let{x,cu} beabasis forT,represented byu,where cuisthecomplex 1-form satisfying (3.1.22). Then cuisdetermined uptothe addition ofa multiple ofx.From (3.1.18) weknow thatiufiisacomplex multiple of x(u, sayiufi=2exp(i6)xw foranappropriately scaled x.SoifG= §(iu1"1 —iu°u°) then G=exp(i6)xcu +exp(—it9)xcu* andG(cu+cu*)= cos19x+2isinBxAcu,(w*. This Gwillbenothing other than theFof (3.1.26) ifinfact0=0.Wehave 2G(w +02*)=iu[(w +w*)u] —iu°[(w +w*)u°] =iui7° —iu°u since wu=0andw11°=u.Butifa/,B,tp,111areanyspinors then (tr,(<P1l’)El3) =(((P17})a’fi) =(1/1,¢1’)(<P>l3) =—(v/2'P)(<P»5) and so((12171)? =-1116). Thus 2G(w +w*)=iu17° +(iu17°)5 and since izu=uthen (uu‘°)'l =—zufi°z =—zu(2~u°) =—izu(i?I)° =—ui7°. andsou17°=$f’,(uu°) +$f3(ui7°). Since 3-forms change signunder 5we have G(w +w*)=Sf’,(iuu°) =xby(3.1.28). That is,theFof(3.1.26) canbewritten as F=Re(iiia). (3.129) From (3.1.21) and(3.1.25) weseethatxandFcanequivalently be thought ofasbeing associated with theMajorana spinor 1p=u+u“. Wecanalsoexpress xandFinterms ofipanditsadjoint. Since 116 PURE SPINORS AND TRIALITY u = izu we have p(z) = + iu`ric) + i(ufic + (uric)9 where, since (cpv--). = , the first term is odd under whilst the second is even. Comparison with (3.1.28) and (3.1.29) shows that '4W i(lXzIP)) = (3.1.30) = F. (3.1.31) If u' is related to u by u' = exp(i0)u (3.1.32) then, from (3.1.28), we see that u' determines the same null direction as u. If u' determines the 2-form F' then F' = Re(cos20iuti — sin 20 ua). and since u = izu F' = Re(cos26iuit — sin2Oziva) = cos 20F — sin 20 zF = exp(-20z)F. Since zF = — * F we see that the 2-form determined by u' is related to that determined by u by a duality rotation. We have established the relationship between a complex Lorentzian semi-spinor and the null direction x and 2-form F by using the previously established results on pure spinors. This correspondence between Weyl spinors and 'null flags' has been emphasised by Penrose and Rindler [9]. We now consider the case of V a real even dimensional orthogonal space with the metric g positive-definite. A complex structure on V is a 1-1 tensor (or linear transformation) J satisfying J2 = —1. This complex structure is compatible with g if g(a, b) = g(Ja, Jb) Va, b c V (3.1.33) that is, J is an isometry of V. We will show that any such J is in one-to-one correspondence with a maximal totally isotropic subspace of Vc. Hence the one-dimensional space of pure spinors of the complex- ified Clifford algebra is in one-to-one correspondence with a complex structure on Vt. Suppose firstly that we have such a J. Then by complex linearity J defines a tensor on ye. Define M C ye by tWe thank G Segal for pointing this out to us. 116 PURE SPINORS AND TRIALITY u=izuwehave 11(5)) =(—iu17 +iu°17°) +i(u17°+(l.ll7c)‘§) where, since ((P'l[I)§ =—1p<p, thefirstterm isoddunder 5whilst the second iseven. Comparison with (3.1.28) and(3.1.29) shows that %§Pi<wG?I>> =x <3-1.30) 292(ip(zi/1)) F. (3.1.31) Ifu’isrelated touby u’=exp(i6)u (3.1.32) then, from (3.1.28), weseethatu’determines thesame nulldirection as u.Ifu’determines the2-form F’then F’=Re(cos26iu17 —sin26u17). andsince u=izu F’=Re(cos26iu17 —sin26ziu17) =cos26F —sin26zF =exp(—26z)F. Since zF=—*Fweseethatthe2-form determined byu’isrelated to thatdetermined byubyaduality rotation. Wehave established therelationship between acomplex Lorentzian semi-spinor and the null direction xand 2-form Fbyusing the previously established results onpure spinors. This correspondence between Weyl spinors and‘null flags’ hasbeen emphasised byPenrose andRindler [9]. Wenowconsider thecaseofVarealeven dimensional orthogonal space with themetric gpositive-definite. Acomplex structure onVisa 1-1tensor (orlinear transformation) Jsatisfying J2=—1.This complex structure iscompatible with gif g(a, b)=g(Ja, Jb) Va, beV (3.1.33) thatis,Jisanisometry ofV.Wewillshow thatanysuch Jisin one-to-one correspondence with amaximal totally isotropic subspace of VC.Hence theone-dimensional space ofpure spinors ofthecomplex- ified Clifford algebra isinone-to-one correspondence with acomplex structure onVi. Suppose firstly that wehave such aJ.Then bycomplex linearity J defines atensor onVC.Define MCVCby ‘FWe thank GSegal forpointing thisouttous. PURE SPINORS 117 x e M iff Jx = ix (3.1.34) and y c M* iff y* E M. Then Vc = M 0 M*. If J satisfies (3.1.33) then g(x I, x2) = g(Jx1, Jx2), and for x X2 E M we have g(x1, x2) = O. Hence M is a maximal totally isotropic subspace of ye. Conversely now suppose that we have a maximal totally isotropic subspace M. We can define J on elements of M by (3.1.34). Requiring Jx* = (Jx)* defines J unambiguously on the whole of Vc and, by restriction, on V. Such a J certainly satisfies J2 = —1. For any a E VC we can write a = a+ + a- with a+ c M and a- E M*. Then g(a, b) = g(a+ , b -) + g(a- , b+) and it follows that if Ja+ = ia+ and Ja- = —ia - then J satisfies (3.1.33). This correspondence between pure spinors and complex structures will be used in Chapter 10. 3.2 Triality Let V be an F-linear space with an F-bilinear symmetric metric g. If S is the space of spinors of C(V, g) then we may define a spin-invariant product on S. In certain cases (for F = IR or C) there is an F-bilinear symmetric product on S, h say. We can then ask 'when is C(V, g) =. C(S, h)?'. These algebras will be isomorphic when dim FV = dim FS and the index of g is the same as that of h. If S = S+ S -, with S+ and S- semi-spinor spaces carrying inequivalent irreducible representations of C +(V, g), with h inducing a product on the semi- spinor spaces, then we can also ask the question 'when is C(V, g) --- C(S+ , h) = C(S- , h)?'. Again this will be when dimFV = dim FS± with the index of g the same as that of the metric induced by h on S. We now examine the possibility of this latter situation occurring. If dim FV = n then n must be even if C +(V, g) is to be reducible with S splitting into semi-spinor spaces. Then dim FS = 2n/2 and for dim FS to be equal to dim FV we need 2n/2 = 2n, which requires n = 8. If F = C then we see from table 2.17 that the situation we are looking for does occur in eight dimensions, with h being the spin-invariant spinor metric associated with the involution For F = F the situation depends on the signature of g. For given p and q the third entry in table 2.15 classifies the spin-invariant product associated with h say, on the irreducible representation spaces of the even subalgebra. If this entry is 1 CI 1 then the even subalgebra has two semi-spinor representations, with an JR-bilinear symmetric product on each. Such entries occur for (p, q) = (8, 0), (0, 8) or (4, 4). From (2.6.23) we see that in all these cases the index of h is the same as that of g. Actually a little care is needed in reaching this conclusion for the case of C4.4(E) We know PURE SPINORS 117 xeM iffJx=ix (3.1.34) andyeM*iffy*eM.Then V‘:=M(9M*. IfJsatisfies (3.1.33) then g(x‘,x2) =g(Jx‘,Jx2), and forx1,xZeM wehave g(x‘,x2) =0, Hence Misamaximal totally isotropic subspace ofVC.Conversely now 5uppOS6 that wehave amaximal totally isotropic subspace M.Wecan define Jonelements ofMby(3.1.34). Requiring Jx*=(Jx)* defines J unambiguously onthewhole ofV‘:and, byrestriction, onV.Such aJ certainly satisfies J2=—1.ForanyaeV‘:wecanwrite a=a*+a“ with a*eManda‘eM*.Then g(a,b)=g(a*, b‘)+g(a‘, b*)and itfollows thatifJa*=ia"andJa'=—ia‘ then Jsatisfies (3.1.33). This correspondence between pure spinors andcomplex structures will beused inChapter 10. 3.2Triality LetVbeanF-linear space with anF-bilinear symmetric metric g.IfS isthespace ofspinors ofC(V, g)then wemay define aspin-invariant product onS.Incertain cases (for F=IRorC)there isanF-bilinear symmetric product o_n S,hsay. We can then ask ‘when is C(V, g)=C(S, h)?’. These algebras willbeisomorphic when dim,aV = dim,aS and theindex ofgisthesame asthat ofh.IfS=S"69S“, with S"and S'semi-spinor spaces carrying inequivalent irreducible representations ofC"(V, g),with hinducing aproduct onthesemi- spinor spaces, then we can also ask the question ‘when is C(V, g)=C(S", h)=C(S_, h)?’. Again this will bewhen dim;V = dim,aS"with theindex ofgthesame asthatofthemetric induced byh onS1.Wenow examine thepossibility ofthislatter situation occurring. Ifdim,aV =nthen nmust beeven ifC"(V, g)istobereducible with S splitting into semi-spinor spaces. Then dimFS=2"”andfordim,aS" to beequal todim,=V weneed 2"/Z=2n,which requires n=8.IfF=C then weseefrom table 2.17 that thesituation wearelooking fordoes occur ineight dimensions, withhbeing thespin-invariant spinor metric associated with theinvolution 5ForF=IRthesituation depends onthe signature ofg.Forgiven pandqthethird entry intable 2.15 classifies thespin-invariant product associated with 5,hsay, ontheirreducible representation spaces oftheeven subalgebra. Ifthisentry is1691then theeven subalgebra has two semi-spinor representations, with an IR-bilinear symmetric product on each. Such entries occur for (P,q)=(8,0), (0,8) or(4,4).From (2.6.23) weseethat inallthese Cases theindex ofhisthesame asthat ofg.Actually alittle care is needed inreaching thisconclusion forthecase ofC.,_.,(lR). Weknow 118 PURE SPINORS AND TRIALITY that h on S has maximal index, but we could have h inducing a positive-definite product on S+ and a negative-definite one on S. However, if x E V with x2 = 1, then for y E S- there is a u E S+ such that y = xu. Then h(v, y) = h(xu, xu) = h(u, x 2u), since the adjoint involution of h is and the indices of the metrics induced by h on S+ and S- are the same. In the following V will either be a complex eight-dimensional vector space or a real eight-dimensional vector space with g having signature (8, 0), (0, 8) or (4, 4). By taking the direct sum of the vector spaces V and S we form a 24-dimensional vector space E: E = V ® S+ (i) S-. (3.2.1) If elements 0, of E are decomposed into these subspaces as 4),= x + y, then a bilinear form B is defined on E by B(01, 02) = g(x 1, x2) + h(u l, u2) + h(vi, y2) . (3.2.2) (We shall frequently decompose an element 0 as above, the symbols x, u and y being reserved for the components of 0 in the subspaces V, 5+ and S-.) We can introduce a totally symmetric (3, 0) tensor T on E in terms of the inner product h. We define T(01, 4)2, (1)3) h(ui, x2u3) + h(ul, x3y2) + h(u2, x1v3) + h(u2, x3y1) + h(u3, xiv,) + h(u 3, x2y1). (3.2.3) Each term on the right-hand side is linear in each 0„ thus T is indeed multilinear. By construction T is totally symmetric. We can use the bilinear B and trilinear T to define a bilinear map 0: 0:ExE-->E such that T(0 1, 02, 03) = B(0 1 o02, 03) .(3.2.4) The non-degeneracy of B ensures that . is indeed well defined. Its bilinearity follows from the trilinearity of T. Since T is totally symmetric 0, . 4)2 = . If 01 and 02 are both in the same subspace, either V, S+ or S-, then T(0 1, 02, 433) = 0 from (3.2.3) and hence . 02 = 0. For x E V, u E S+ and V E S- we have B(x o u, v) = T(x, u, v) = h(u, xv) = h(xu, v) = B(xu, v) and so similarly x o u = xu (3.2.5) x o y = xu. (3.2.6) If i is the adjoint of u with respect to h then B(u o v, x) = T(u, v, x) = h(xu, v) = iiv = o(fixv) = 0(xvit) = 0(xW i(v-a)) = B(x,W 118 PURE SPINORS AND TRIALITY that honShasmaximal index, butwecould have hinducing a positive-definite product onS‘and anegative-definite one onS‘. However, ifxeVwith x2=1,then forueS‘there isaueS" such that u=xu.Then h(u, u)=h(xu, xu)=h(u, xzu), since theadjoint involution ofhisE,andtheindices ofthemetrics induced byhonS" andS‘arethesame. Inthefollowing Vwilleither beacomplex eight-dimensional vector space orarealeight-dimensional vector space with ghaving signature (8,0),(0,8)or(4,4). Bytaking thedirect sum ofthevector spaces Vand Sweform a 24-dimensional vector space E: E= V®S*(-BS‘. (3.2.1) Ifelements (D,ofEare decomposed into these subspaces as (D,=x,-+u,-+u,-thenabilinear form Bisdefined onEby B((DIa (DZ) =g(xl> x2) +h(ul> L42) +h(Ul» U2) ' (We shall frequently decompose anelement <1)asabove, thesymbols x, uandubeing reserved forthecomponents of(Dinthesubspaces V,S+ andS‘.) Wecanintroduce atotally symmetric (3,0)tensor TonEinterms of theinner product h.Wedefine T(‘D1- (P2,(D3)Eh(”1, X203) +h(”i, X302) +h(”2. X103) ‘i" ll(lJ2, X301) + h(M3, X,U2) + h(M3, X2111). Each term ontheright-hand sideislinear ineach <D,~,thus Tisindeed multilinear. Byconstruction Tistotally symmetric. Wecanusethe bilinear Bandtrilinear Ttodefine abilinear mapoz oiE X E-'9 E Sl1Cl1T((I),, (D2, (D3) = O(D2, (D3) The non-degeneracy ofBensures that Oisindeed well defined. Its bilinearity follows from thetrilinearity ofT.Since Tistotally symmetric <1),0<D2=<D2<><I>,.If<1),and<D2areboth inthesame subspace, either V,S‘ orS‘, then T(<I>,, (D2,(D3)=0from (3.2.3) and hence <D,=><I>2 =0. ForxeV, ueS‘ andueS‘ wehave B(x<>u,u)=T(x, u,u)=h(u, xv)=h(xu, u)=B(xu, u) andso x=>u=xu (3.2.5) similarly x<>u=xu. (32.6) If17istheadjoint ofuwithrespect tohthen B(u=>u,x)=T(u, u,x)=h(xu, u)=x710 =5/’,,(!7xu) =EF0(xui7) =S/’,,(x€/’,(ui7)) =B(x,S/’,(ui7)) TRIALITY SO u o y = The product o is not associative, for example we have x o (x o u) = x o xu = x2u = g(x, x)u 119 (3.2.7) (3.2.8) whereas x x = O. The norm of the spinor x o u is related to the norms of x and u by h(x 1. u, x2. u) --= g(x 1, x2)h(u, u). (3.2.9) This follows from (3.2.5), (3.2.6) and the fact that the adjoint of h is The 24-dimensional vector space E forms a non-associative algebra .54 under the o product. The spinor representation of the Clifford group, p on S± , and the vector representation x on V naturally induce a reducible representation Y on E by Y(s).(x + u + y) x(s).x + p(s).0 + p(s).v. (3.2.10) Whereas g in invariant under x(s) Vs E F, h is only invariant under p(s) for s E +F and so B(c1:01, <132,) = B(Y(s).(13 1, Y(s).(13 2) Vs E F. (3.2.11) It readily follows that in addition T(01, 02, 03) = T(Y(S).01, Y(S).02, Y(S).03) Ys E j. (3.2.12) From these last two relations we can infer from (3.2.4) that Y(s).(0 1 0 432) = (Y(s).(121 1) o (Y(s).(13 2) Vs E +F (3.2.13) that is, Y(s) is in the automorphism group of the non-associative algebra ,91. Conversely it follows that if a is any automorphism of .94. that transforms V and S into themselves then a = Y(s) for some s E +F. (The starting point of the argument is that for VE S then a.lp = sip for some regular element s of the Clifford algebra.) The orthogonal space V under consideration has been carefully selected to ensure that V, S+ and S- are all isometric. The existence of an isometry that cyclicly permutes these three orthogonal spaces can be taken as being Cartan's 'principle of triality'. Such an isometric map will be constructed out of a mapping that interchanges two of these three Spaces. Let u0 E SI- be some unit-norm semi-spinor, h(u0, u0) = 1. Then a linear transformation r(u0) from V to S- is defined by TO 0) . X = X 0 u0. (3.2.14) It immediately follows from (3.2.9) that r(u0) is in fact an orthogonal transformation from V to S. The linear transformation r(u0) is uniquely extended to an automorphism of period two on V '0 S-: that TRIALITY 119 so uov=9,(vu). (3.2.7) Theproduct <>isnotassociative, forexample wehave xQ(x<>u)=x<>xu=xzu=g(x,x)u (3.2.8) whereas x0x=0.The norm ofthespinor x0uisrelated tothenorms ofxanduby h(x,0u,x2=u)=g(x,, x2)h(u, u). (3.2.9) This follows from (3.2.5), (3.2.6) andthefactthattheadjoint ofhisE. The 24-dimensional vector space Eforms anon-associative algebra 81 under theQproduct. The spinor representation oftheClifford group, ponS1, and the vector representation XonVnaturally induce areducible representation YonEby Y(s).(x +u+v)=X(s).x +p(s).u +p(s).v. (3.2.10) Whereas gininvariant under X(s) VseF,hisonly invariant under p(s) forse+1"andso B(<I>,, <I>2,) =B(Y(s).<I>,, Y(s).<I>2) Vse+1". (32.11) Itreadily follows thatinaddition T((I>,, 1112,11);)=T(Y(s).<I>,, Y(s).<I>2, Y(s).<I>3) Yse+1".(3.2.12) From these lasttworelations wecaninfer from (3.2.4) that Y(s).(<I>, =><I>2)=(Y(s).<I>,) <>(Y(s).<I>2) Vse,1"(32.13) thatis,Y(s) isintheautomorphism group ofthenon-associative algebra d.Conversely itfollows that ifoisany automorphism of81that transforms Vand Sinto themselves then o=Y(s) forsome se+1". (The starting point oftheargument isthat forweSthen o.1/1=sipfor Some regular element softheClifford algebra.) The orthogonal space Vunder consideration has been carefully selected toensure thatV,S‘andS‘areallisometric. Theexistence of anisometry thatcyclicly permutes these three orthogonal spaces canbe taken asbeing Cartan’s ‘principle oftriality’. Such anisometric map will beconstructed outofamapping that interchanges two ofthese three Spaces. LetuoeS‘besome unit-norm semi-spinor, h(u,), uo)=1.Then alinear transformation r(u0) from VtoS‘isdefined by 1.'(u,,).x =x<>uo. (32.14) Itimmediately follows from (32.9) that r(u0) isinfact anorthogonal transformation from VtoS‘.The linear transformation r(u0) is uniquely extended toanautomorphism ofperiod twoonV(+3S‘:that 120 PURE SPINORS AND TRIALITY is, if v e S- such that v = r(u0).x for some unique x then we define r(uo).v = x. Finally we define r(u0) on S+ by r(u0).0 = 2h(u, u o)uo — u. (3.2.15) That is, r(u0) acts on S+ by sending u to minus its reflection in the plane orthogonal to u0. Thus r(u0) is an orthogonal transformation of S+ , and hence of E. In addition, r(u0) leaves T invariant. Note first that since the image under r(u0) of any of the three subspaces, V, S+ or S-, lies in only one subspace we need only consider T acting on elements lying in distinct subspaces. If u = r(u0).a for some a then r(u0).vr(u0).x = axuo = 2g(a, x)u o — xauo = 2h(t(u 0).a, r(u0).x)u0 — xv since r(uo) is an isometry from V to S- and so r(u0).vr(u0).x = 2h(v, xuo)uo — xv = 2h(xv, u 0)u 0 — xv = (r(u 0).x)v. Since T(r(u0).431, r(u0).102, r(u0).(133) = T(r(u0).0 1, r(u0).v2, r(u0).x3) + . . . it follows from (3.2.3) that T(r(u0).111, r(u0).432, r(u0).03) = T(4:131, 02, (D3). (3.2.16) Whereas r(u0) is an orthogonal transformation of E that inter- changes V and S-, Y(s) is an orthogonal transformation of E that interchanges S± and S-. If xo c V is a unit vector, g(xo, xo) = 1, then X0 E +1-' and Y(x0) is of period two, Y(x0)2 = 1. Out of these two involutory transformations of E we construct an orthogonal transforma- tion of period three. The triality map F.(x0, tto) is defined by al(x0, /40) Y(xo)r(uo). (3.2.17) To see that 2-7(x0, u0) is of period three we want to show that r(u0)Y(x0)-r(u0) = Y(xo)r(uo)Y(x0). (3.2.18) For example, if x c V then T(14 0)Y (X 0)T(Ii 0) . X = I 0)Y (X 0) .(Xl 0) = 4 0) .(x0XU 0) = 2h(X 0X14 0, 14 0)14 0 - X °nip = 2h(x o uo, xo. uo)uo — xoxuo = 2g(x, x o)uo — xoxuo (by (3.2.9)) = XX0/40. On the other hand 120 PURE SPINORS AND TRIALITY is,ifveS‘such that v=r(u,,).x forsome unique xthen wedefine r(u0).v =x.Finally wedefine r(u0) onS‘by r(u,,).u =2h(u, uO)u,, —u. (32.15) That is,r(u0) acts onS‘bysending utominus itsreflection inthe plane orthogonal touo.Thus r(uO) isanorthogonal transformation of S‘,andhence ofE.Inaddition, r(u0) leaves Tinvariant. Note first thatsince theimage under r(u0) ofanyofthethree subspaces, V,S‘or S‘, liesinonly one subspace weneed only consider Tacting on elements lying indistinct subspaces. Ifv=r(u0).a forsome athen r(u0).vr(u0).x =axuo =2g(a, x)u0 —xauo =2h(r(u,,).a, r(u,,).x)u0 —xv since duo) isanisometry from VtoS‘andso r(u,,).vr(u0).x =2h(v, xu,,)u0 —xv=2h(xv, u,,)u,, —xv=(1'(u0).x)v. Since T(r(u0).<D,, r(u0).<D2, r(u,,).<D2) =T(1(u0).u,,1(u0).v2, r(u,,).x3) +... itfollows from (3.23) that T(r(u0).<D,, r(u,,).<D2, r(u0).<D3) =T(<D,, (D2,(D2). (32.16) Whereas r(u0) isanorthogonal transformation ofEthat inter- changes Vand S‘,Y(s) isanorthogonal transformation ofEthat interchanges S‘andS‘.Ifxoe Visaunit vector, g(x,), x0)=1,then x0e+1"andY(x0) isofperiod two, Y(x,,)2 =1.Outofthese two involutory transformations ofEweconstruct anorthogonal transforma- tionofperiod three. The triality map E(x0, uo)isdefined by E(x0, uo)EY(x,,)r(u0). (32.17) ToseethatE(x,,, uo)isofperiod three wewant toshow that T("0)Y(Xo)T("0) =Y(X6)I(u6)Y(X6)- (3-2-13) Forexample, ifxeVthen T("0)Y(X0)T("0)-X =T("0)Y(X0)-(xuo) =T("o)-(xoxuo) =2h(x,,xu0, u,,)u0 —xoxuo =2h(x <>uo,x0Ou,,)u,, —xoxuo =2g(x, x0)u0 —xoxuo (by(3.2.9)) =xxouo. Ontheother hand TRIALITY 121 Y(x0)r(u0)Y(x0).x = Y(x0)/(140). (xoxx 0) = Y(x0). (xoxx ou 0) = xxou 0. The validity of (3.2.18) can be similarly demonstrated on elements from the other two subspaces. Given (3.2.18) we have E(x0, /03 = (Y(xo)r(uo)Y(x0))(r(uo)Y(x0)r(u0)) = (Y(x o)r(u0)Y(x0))2 and since both Y(x0) and r(u0) are of period two 2.".(x0, u0)3 = 1. (3.2.19) Because Y(x0) and r(u0) both have these properties separately we have B(431, 432) = B(E(x o, u0).(131, E(x0, uo).432) (3.2.20) and T(431,43132, 03) = T(E(X0, U0).4 31, 72(.1CO3 L10).432, E*()CO3 /10).(133). (3.2.21) The three subspaces of E are permuted under E(x0, uo) as follows: E(xo, uo). V C S' uo).S+ C S' E(x0, u0).S- C V. (3.2.22) We have focused on a V such that C(V, g) = C(S+ , h) = , h). The map E(xo, uo) isometrically permutes these three spaces. Any isometry between two orthogonal spaces uniquely extends to an iso- morphism between their Clifford algebras. Let N be the isomorphism obtained from E(x0, u0): N:C(V, g) C(S+ , h) 1---> C(S - , h) C(V, g). (3.2.23) Because S+ 0 .5-- is the spinor space of C(V, g) the map N enables any two of the three spaces V, S± and S- to be taken as the spinor space of the Clifford algebra of the third! For example, S- 0 V can be taken as the spinor space of C(S+, h). Let o denote the Clifford product of C(S+, h). Then for x E V and tpE S N(xip) = N(x) o N(ip). That is, if u E S+ and ty E S' S- 0 V then u = N((N-1u)(N-11p)). (3.2.24) Under this multiplication by u the spaces S- and V are interchanged; these being the semi-spinor spaces of C ±(S± , h). Exercise 3.1 Show that if V is a complex vector space then C(V, g) = C(S, h) if dime V = 2, 4. In the real case what signatures can g have? (Remember that the spinor inner product could be associated with either or ri.) TRrAr.iTY 121 Y(X())T(”0)Y(X0)-X =Y(X0)T("o)~(X0XX0) =Y(X0)-(xoxxouo) =XX0"o- Thevalidity of(32.18) canbesimilarly demonstrated onelements from theother twosubspaces. Given (3.2.18) wehave 5(Xu» ”0)3 =(Y(X0)T("0)Y(X0))(T("0)Y(x0)T(”0)) =(Y(X0)T(”0)Y(x0))2 andsince both Y(x,,) and1'(u0) areofperiod two E(x0, u,,)3 =1. (32.19) Because Y(x,,) andt(u,,) both have these properties separately wehave B(<I>,, <I>2)=B(E(x,,, u0).<I>,, E(x,-,, u0).<I>2) (3.2.20) and T(<I>,, <I>2,(D3)=T(E(x,,, u,,).<I>,, E(x0, u,,).<I>2, E(x,,, u0).<I>3). (3.2.21) Thethree subspaces ofEarepermuted under E(x0, u,,)asfollows: E(x,,, u,,).V CS‘ E(x,,, u,,).S‘ CS‘ E(x,,, u0).S‘ CV.(3.2.22) Wehave focused onaVsuch that C(V, g)=C(S‘, h)=C(S‘, h). The map E(x0,u,,) isometrically permutes these three spaces. Any isometry between two orthogonal spaces uniquely extends toaniso- morphism between their Clifford algebras. LetNbetheisomorphism obtained from E(x,,, u,,): N2C(V, g)i-—-> C(S*, h)i-—-> C(S‘, h)i-—-> C(V, g).(32.23) Because S‘C-DS‘isthespinor space ofC(V, g)themap Nenables any twoofthethree spaces V,S‘andS‘tobetaken asthespinor space of theClifford algebra ofthethird! Forexample, S‘("DVcanbetaken as thespinor space ofC(S‘,h).Let<>denote theClifford product of C(S‘, h).Then forxeVandweS /)/(X111) =A/(X)<>NW)- That is,ifueS‘ and1})’eS’=S‘(9Vthen U8qr=/1/((/1/-1i1)(/1/-11)/)). (3.2.24) Under thismultiplication byuthespaces S‘andVareinterchanged; these being thesemi-spinor spaces ofC‘(S‘, h). Exercise 3.1 Show that ifVisacomplex vector space then C(V, g)=C(S, h)if dimCV =2,4.Intherealcase what signatures canghave? (Remember thatthespinor inner product could beassociated with either EorE11.) 122 PURE SPINORS AND TRIALITY Bibliography Chevalley C 1954 The Algebraic Theory of Spinors (New York: Columbia University Press) 122 PURE SPINORS ANDTRIALITY Bibliography Chevalley C1954 The Algebraic Theory ofSpinors (New York: Columbia University Press) 4 Manifolds Like many concepts in mathematics that of a manifold is based on intuitive ideas which require some sophistication to make precise. Perhaps the simplest example of a manifold is Euclidean three-space. Of necessity at this stage we must refrain from defining Euclidean space, but shall nevertheless assume that the reader has some intuitive ideas about this model description of our perceived three-dimensional world. (The term Euclidean space is not synonymous with Euclidean vector space. A Euclidean vector space is a real vector space with a positive- definite symmetric metric.) At an early age we all learnt how a Cartesian coordinate system can be introduced to put points in Eucli- dean space into correspondence with an ordered triple of real numbers, an element of 11:13. However, it is important that we distinguish Eucli- dean three space from 1R 3. Euclidean space has no preferred coordinate system. Indeed we need not of course even be restricted to Cartesian coordinates. Despite our emphasis on the distinction between Euclidean three-space and 11V it is nonetheless in 1113 that the familiar calculus of differentiation and integration is introduced. Through the introduction of a coordinate system one may then apply this calculus to Euclidean space. It is the correspondence of Euclidean space to 1R", through the introduction of a coordinate system, that generalises to provide the definition of a manifold. This is defined, in a sense that will be made precise, to be locally like Fin. Because we can define differentiation and integration on E" we can extend these notions to a manifold. Unlike Euclidean space, for an arbitrary manifold we cannot choose some origin to put all points on the manifold into a unique correspond- ence with points in Fin. For example, we could take the two-dimensional outer surface of a hollow rubber ball. Whilst any cap of the ball could be put into one-to-one correspondence with points in a plane (by cutting the section out and flattening it), we cannot do this with the whole surface. (If we simply squashed the ball then two points on the surface 4 Manifolds Like many concepts inmathematics that ofamanifold isbased on intuitive ideas which require some sophistication tomake precise. Perhaps thesimplest example ofamanifold isEuclidean three-space. Of necessity atthisstage wemust refrain from defining Euclidean space, butshall nevertheless assume thatthereader hassome intuitive ideas about thismodel description ofourperceived three-dimensional world. (The term Euclidean space isnotsynonymous with Euclidean vector space. AEuclidean vector space isarealvector space with apositive- definite symmetric metric.) Atanearly agewealllearnt how a Cartesian coordinate system canbeintroduced toputpoints inEucli- dean space intocorrespondence withanordered triple ofrealnumbers, anelement ofB3.However, itisimportant that wedistinguish Eucli- dean three space from B3.Euclidean space hasnopreferred coordinate system. Indeed weneed notofcourse even berestricted toCartesian coordinates. Despite ouremphasis onthedistinction between Euclidean three-space andB3itisnonetheless inB3thatthefamiliar calculus of differentiation andintegration isintroduced. Through theintroduction ofacoordinate system onemay then apply thiscalculus toEuclidean Space. Itisthecorrespondence ofEuclidean space toIR",through the introduction ofacoordinate system, that generalises toprovide the definition ofamanifold. This isdefined, inasense thatwillbemade precise, tobelocally likeIR".Because wecandefine differentiation and integration onIR”wecanextend these notions toamanifold. Unlike Euclidean space, foranarbitrary manifold wecannot choose Some origin toputallpoints onthemanifold intoaunique correspond- encewithpoints inIR".Forexample, wecould takethetwo-dimensional Outer surface ofahollow rubber ball. Whilst anycapoftheballcould beputintoone-to-one correspondence withpoints inaplane (bycutting thesection outandflattening it),wecannot dothiswith thewhole surface. (Ifwesimply squashed theballthen twopoints onthesurface 124 MANIFOLDS would be mapped to the same point on the plane.) The fact that the surface is locally like IF1 2 is sufficient to establish a differential calculus on the surface. This does not require a knowledge of embedding in three-space. The intuitive examples of the Euclidean plane and the two-sphere convey ideas of more structure than that of an arbitrary manifold. Although locally any manifold resembles, in some sense, En this does not imply the existence of any metric or distance function on the manifold. Rather the resemblance relates to topology, this being an abstraction of the concept of 'nearness' from that given by distance. We start by defining a topological space. By making precise the idea of being 'locally like En' we arrive at the definition of a topological manifold. After reviewing differentiation on IFIn we show how a system of coordinates on a topological manifold enables differentiation to be defined, giving a differentiable manifold. From its introduction in En the concept of a tangent vector will undergo a metamorphosis, the imago emerging in a form appropriate to the environment of an arbitrary differentiable manifold. This leads naturally to vector fields, and hence tensor fields. After introducing the computationally powerful exterior and Lie derivatives we define integration on manifolds. Similar to the case of differentiation, the definition reduces integration on manifolds to integration on En. Only at the end of the chapter do we consider metric tensor fields. We are then equipped to apply our heavy artillery to the example of Euclidean three-space. This is done in Appendix B. Actually there is still an important facet of Euclidean space that will not be discussed until the following chapter, that of parallelism. 4.1 Topological Manifolds The usual definition of continuity of a function f: U —> W where U and W are subsets of IR relies on the notion of 'nearness' of different elements of E. Such 'nearness' is measured by a proximity function d : IR x E —> E with the properties: d(x, y) = d(y, x), d(x, y) = 0 if and only if x = y, d(x, z) d(x, y) + d(y, z). (Note x, y E R.) A natural proximity function for the real line that has these properties is the absolute value or modulus map, (x, y) —> — yJ and f is said to be continuous at x E 1E1 if one can find a positive c5E E for any positive E belonging to E such that if d(x, y) < 6 then d(f(x), f(y)) < E. Thus one probes the neighbourhood of the image of f induced by a neighbour- hood about x in the domain of f. The first generalisation of this idea to arbitrary sets consists of defining a new set called the neighbourhood 124 MANIFOLDS would bemapped tothesame point ontheplane.) The fact that the surface islocally like1R2issufficient toestablish adifferential calculus onthesurface. This does notrequire aknowledge ofembedding in three-space. The intuitive examples oftheEuclidean plane and thetwo—sphere convey ideas ofmore structure than that ofanarbitrary manifold. Although locally anymanifold resembles, insome sense, 1R"thisdoes notimply theexistence ofany metric ordistance function onthe manifold. Rather theresemblance relates totopology, this being an abstraction oftheconcept of‘nearness’ from thatgiven bydistance. Westart bydefining atopological space. Bymaking precise theidea ofbeing ‘locally like lR"’wearrive atthedefinition ofatopological manifold. After reviewing differentiation on1R"weshow how asystem ofcoordinates onatopological manifold enables differentiation tobe defined, giving adifferentiable manifold. From itsintroduction in1R" theconcept ofatangent vector will undergo ametamorphosis, the imago emerging inaform appropriate tothe environment ofan arbitrary differentiable manifold. This leads naturally tovector fields, andhence tensor fields. After introducing thecomputationally powerful exterior andLiederivatives wedefine integration onmanifolds. Similar tothecase ofdifferentiation, thedefinition reduces integration on manifolds tointegration on1R". Only attheendofthechapter dowe consider metric tensor fields. Wearethen equipped toapply ourheavy artillery totheexample ofEuclidean three-space. This isdone in Appendix B.Actually there isstill animportant facet ofEuclidean space that will notbediscussed until thefollowing chapter, that of parallelism. 4.1Topological Manifolds The usual definition ofcontinuity ofafunction f:U—>Wwhere Uand Waresubsets of1Rrelies onthenotion of‘nearness’ ofdifferent elements oflFl.Such ‘nearness’ ismeasured byaproximity function d:lFl><lFl—> lFlwith theproperties: d(x, y)=d(y, x),d(x, y)=0if and only ifx=y,d(x, z)=d(x, y)+d(y, z).(Note x,yelR.) A natural proximity function forthereal linethat hasthese properties is theabsolute value ormodulus map, (x,y)—>|x—y]andfissaid tobe continuous atxe1Rifonecanfind apositive 6e1Rforanypositive 5 belonging tolRsuch thatifd(x, y)<6then d(f(x), f(y)) <e.Thus one probes theneighbourhood oftheimage offinduced byaneighbour- hood about xinthedomain off.The firstgeneralisation ofthisidea to arbitrary setsconsists ofdefining anew setcalled theneighbourhood TOPOLOGICAL MANIFOLDS 125 nbh(x, 6) C S if x E S. This is the set of elements y E S such that d(x, y) < 6, that is a set of all points that are within a 'distance' 6 from x as measured by some proximity function d. One often refers to d as a distance or metric function, although since we do not assume here that the set has any vector space structure it is logically distinct from the metric g defined earlier on vector spaces. Indeed what we have called a metric on a vector space would not in general define a distance function for a metric space. Here there is no requirement that d should be linear in either of its arguments. With this caveat in mind one refers to the pair (S, d) as a metric space. The defining properties of the proximity function d of course remind one of the properties of distances between points in Euclidean space (for example, the triangle inequality) and indeed it is worth noting that if IR" is given a vector space structure one can choose d(x, y) = [g(x — y, x —A]112 provided g is the positive- definite Euclidean metric. If one does use the Euclidean metric to define d then the set nbh(x, 6) in Euclidean IR" looks like an open ball (open because of the inequality d(x, y) < 6, V y enbh(x, 6). The triangle inequality property of d ensures that all points y E nbh(x, 6) have some neighbourhoods that are contained in nbh(x, 6). In general the proximity function on JR" need not coincide with the metric on 1R'2 regarded as a vector space. A boundary element x of a set S' contained in the set S with distance function d is an element such that nbh(x, 6), for some positive 6E TR, contains both elements in S' and elements not in S'. The set of all boundary points of S' is called the boundary of S'. In particular if S' = nbh(x, 60) C S then S' does not contain its boundary and is called an open set in (S, d). If any boundary points are not in the set then it is an open set. If all boundary points are in the set it is closed. In general it is possible to find different distance functions that determine the same class of continuous functions. A valuable genera- lisation then is to concentrate on the open sets themselves as the primitive notions and reformulate 'nearness' directly in terms of them rather than in terms of any particular proximity function. The immediate usefulness of open sets is a reformulation of the definition of a continuous function f: U —* W. f is continuous at p E U if and only if, for any neighbourhood W' containing f(p) there is a neighbourhood U' containing p whose image f(U')CW'. Such a notion of continuity relies on the open set structure of the spaces related by f and not on a particular choice of proximity function used in specifying these open sets. Consequently one attempts to bypass any mention of a proximity function and establish a more general definition of open sets on any space. The declaration of which subsets of a space are to be considered as open is called a definition of its topology provided such a family of subsets satisfy the following axioms. TOPOLOGICAL MANIFOLDS 125 nbh(x, 6)CSifxeS. This isthesetofelements yeS such that d(x, y)<6,thatisasetofallpoints that arewithin a‘distance’ 6from xasmeasured bysome proximity function d.One often refers todasa distance ormetric function, although since wedonotassume here that thesethasanyvector space structure itislogically distinct from the metric gdefined earlier onvector spaces. Indeed what wehave called a metric onavector space would notingeneral define adistance function forametric space. Here there isnorequirement that dshould belinear ineither ofitsarguments. With thiscaveat inmind onerefers tothe pair (S,d)asametric space. The defining properties oftheproximity function dofcourse remind oneoftheproperties ofdistances between points inEuclidean space (for example, thetriangle inequality) and indeed itisworth noting that ifIR"isgiven avector space structure one canchoose d(x, y)=[g(x —y,x—y)]1’2 provided gisthepositive- definite Euclidean metric. Ifone does use theEuclidean metric to define dthen thesetnbh(x, 6)inEuclidean IR"looks likeanopen ball (open because oftheinequality d(x, y)<6,Vyenbh(x, 6).The triangle inequality property ofdensures that allpoints yenbh(x, 6) have some neighbourhoods that arecontained innbh(x, 6).Ingeneral theproximity function on1R"need notcoincide with themetric on1R" regarded asavector space. Aboundary element xofasetS’contained inthesetSwith distance function disanelement such that nbh(x, 6),forsome positive 6elR, contains both elements inS’and elements notinS’.The setofall boundary points ofS’iscalled theboundary ofS’.Inparticular if S’=nbh(x, 6)CSthen S’does notcontain itsboundary andiscalled anopen setin(S,d).Ifanyboundary points arenotinthesetthen itis anopen set.Ifallboundary points areinthesetitisclosed. Ingeneral itispossible tofind different distance functions that determine thesame class ofcontinuous functions. Avaluable genera- lisation then istoconcentrate ontheopen setsthemselves asthe primitive notions and reformulate ‘nearness’ directly interms ofthem rather than interms ofanyparticular proximity function. Theimmediate usefulness ofopen sets isareformulation ofthe definition ofa continuous function f:U-> W.fiscontinuous atpeUifandonly if, foranyneighbourhood W’containing f(p)there isaneighbourhood U’ containing pwhose image f(U’) CW’. Such anotion ofcontinuity relies ontheopen setstructure ofthespaces related byfandnotona particular choice ofproximity function used inspecifying these open Sets. Consequently one attempts tobypass anymention ofaproximity function andestablish amore general definition ofopen setsonany Space. Thedeclaration ofwhich subsets ofaspace aretobeconsidered asopen iscalled adefinition ofitstopology provided such afamily of subsets satisfy thefollowing axioms. 126 MANIFOLDS (i)The whole space and the empty set belong to the family. (ii)The intersection of any finite number from the family belong to the family. (iii)The union of any number of sets from the family belong to the family. With these definitions we now refer to any open set containing a point p in a topological space as a neighbourhood Nbh(p) and the definition of continuity of a function between topological spaces is now independent of any choice of proximity function; it has been replaced by the choice of open sets. The definition of boundary points of a set and the boundary generalises simply to arbitrary topologies by replacing nbh(p, 6) by Nbh(p). A space with a topology defined on it is called a topological space. If a map between topological spaces is continuous with a continuous inverse then it is called a homeomorphism. One further property defines the topology as being Hausdorff: (iv)Disjoint neighbourhoods can be defined about distinct elements of the space. That is, one may find open sets whose intersection is the empty set. If a space has a proximity function d then we may if we wish define Nbh(p) = nbh(p, o) and the space is said to have a metric topology (which is always Hausdorff). One of the commonest metric topologies is associated with En and d(x, y) = Ix — y x, y E E'. With the above d(x, y) on lR the open sets may be visualised as all possible open hypercubes in En. It is useful to have such examples of a natural metric topology in IFIn since they can be used to induce topologies on subsets of IRn . The induced topology on a subset 3 of a topological space S is the collection of all sets formed by the intersection of 3 with all open sets of S. These are then declared to be open in 3 (they need not be open in S) and g is called a topological subspace of S. Subsets of Euclidean IFI3 provide some of the simplest visualisable models of topological spaces. Thus the sphere S2 is the subset of F13 defined by 1x1 = 1, x E 113 with a topology induced from the metric topology of IR3. It is topologically equivalent (homeomorphic) to the ellipsoid (a2x2 b2y2 c2z2 = 1, a, b, c EIR) with the topology induced from that of 11:13; that is one can establish a homeomorphism between them. Neither is homeomorphic to the 2- torus, S1 x 51 However all these examples (and indeed any two- surface) have points with neighbourhoods homeomorphic to the open disc {x1 1x1 < 1, x E IF12). Such spaces are said to be locally homeomor- phic. The fact that they need not be homeomorphic is sometimes phrased by saying that they have different global topologies. If one exploits the vector space structure of 11 3 one can project any sufficiently small region of a two-surface onto a suitable two-plane in 1113 126 MANIFOLDS (i)Thewhole space andtheempty setbelong tothefamily. (ii)The intersection ofanyfinite number from thefamily belong to thefamily. (iii)The union ofanynumber ofsetsfrom thefamily belong tothe family. With these definitions wenow refer toanyopen setcontaining apoint p inatopological space asaneighbourhood Nbh( p)andthedefinition of continuity ofafunction between topological spaces isnow independent ofanychoice ofproximity function; ithasbeen replaced bythechoice ofopen sets. The definition ofboundary points ofasetand the boundary generalises simply toarbitrary topologies byreplacing nbh(p, 6)byNbh(p). Aspace with atopology defined onitiscalled a topological space. Ifamap between topological spaces iscontinuous with acontinuous inverse then itiscalled ahomeomorphism. One further property defines. thetopology asbeing Hausdorff: (iv)Disjoint neighbourhoods canbedefined about distinct elements ofthespace. That is,onemay findopen setswhose intersection istheempty set. Ifaspace hasaproximity function dthen wemay ifwewish define Nbh(p) =nbh(p, 6)and thespace issaid tohave ametric topology (which isalways Hausdorff). One ofthecommonest metric topologies is associated with IR"and d(x, y)=Ix-yI,x,yelR". With theabove d(x, y)onlR"theopen sets may bevisualised asallpossible open hypercubes inIR”. Itisuseful tohave such examples ofanatural metric topology inIR" since they canbeused toinduce topologies onsubsets oflR". The induced topology onasubset Sofatopological space Sisthecollection ofallsetsformed bytheintersection ofSwith allopen setsofS.These arethen declared tobeopen inS(they need notbeopen inS)andSis called atopological subspace ofS.Subsets ofEuclidean IR3provide some ofthesimplest visualisable models oftopological spaces. Thus the sphere S2isthesubset ofIR3defined byIxI=1,xeIR3with atopology induced from themetric topology of1R3.Itistopologically equivalent (homeomorphic) totheellipsoid (azxz +bzyz +czzz=1,a,b,ceIR) with thetopology induced from thatofIR3;thatisonecanestablish a homeomorphism between them. Neither ishomeomorphic tothe2- torus, S‘XS‘.However allthese examples (and indeed any two- surface) have points with neighbourhoods homeomorphic totheopen disc{xIIxI<1,xe1R2}. Such spaces aresaidtobelocally homeomor- phic. The fact that they need not behomeomorphic issometimes phrased bysaying thatthey have different global topologies. Ifoneexploits thevector space structure ofB3onecanproject any sufficiently small region ofatwo-surface onto asuitable two-plane in1R3 TOPOLOGICAL MANIFOLDS 127 to obtain a neighbourhood in IR2 and a bijective map with a continuous inverse. This suggests the definition of an n-dimensional topological manifold. An n-dimensional topological manifold is a Hausdorf topolo- gical space, with a countable basis for its topology, that is locally homeomorphic to an open set of IFIn. A collection of open sets is a basis for a topology if every neighbourhood can be expressed as the union of members in the basis. The elements of a topological manifold are often referred to as points. It is clear from the examples above that one cannot in general find a homeomorphism from the whole topological space to an open set of The above definition of a topological manifold is sufficiently general that not all topological two-manifolds are subsets of 113. 4.2 Derivatives of Functions 11 Fi n Our discussion of continuity culminated in the definition of a topological manifold as being locally homeomorphic to Fin. This local correspond- ence with can be used to establish a criterion for differentiability of maps on manifolds. We first briefly review the differentiation of vector-valued functions on Elm. If f is a function from Flm to En then the derivative of f at p E Rim in the direction of V E IRm is given by Dvf(p) iim (AP ± 1110 — AP)) (4.2.1) h—■0 k h where h E IR. (Other commonly used notations for D vf(p) are df(p)V, dfp(V) and f' p V.) Whereas the discussion of the continuity of f only involved the topology of lim and En, the right-hand side of this equation manifestly uses the vector space structure of these spaces. If all the directional derivatives of f exist at p then f is said to be differen- tiable at p. In this case Df(p) is a linear transformation from IR'n to Df(p) : V F D vf(p), determining the linear part of an approximation to f in the vicinity of p. The function f sends the point p to f(p): if the point p starts to move in the direction of V then f(p) will correspon- dingly start to move in the direction D vf(p) (refer to figure 4.1). Intuitively we think of the derivative of f as sending an 'arrow' in with its tail at p and tip at p + V, to an 'arrow' in R", with f(p) as tail and f(p) + D vf(p) as tip. We may formalise this by defining the tangent space to at p, Tptim, to be the set of pairs (p, V) for all V E IFI'n. These pairs (tangent vectors) form a vector space, isomorphic to En', with the rule ToPoLooicAi. MANIFOLDS 127 toobtain aneighbourhood inlR2andabijective map with acontinuous inverse. This suggests thedefinition ofann-dimensional topological manifold. Ann-dimensional topological manifold isaHausdorf topolo- gical space, with acountable basis foritstopology, that islocally homeomorphic toanopen setoflR”.Acollection ofopen setsisabasis foratopology ifevery neighbourhood canbeexpressed astheunion of members inthebasis. Theelements ofatopological manifold areoften referred toaspoints. Itisclear from theexamples above that one cannot ingeneral find a homeomorphism from thewhole topological space toanopen setoflR". Theabove definition ofatopological manifold issufficiently general that notalltopological two-manifolds aresubsets ofIR3. 4.2Derivatives ofFunctions lR”'—>lR” Ourdiscussion ofcontinuity culminated inthedefinition ofatopological manifold asbeing locally homeomorphic tolR".This local correspond- ence with lR"canbeused toestablish acriterion fordifferentiability of maps onmanifolds. We first briefly review the differentiation of vector-valued functions onlR'". Iffisafunction from lR'"tolR"then thederivative offatpelR'"in thedirection ofVelR'"isgiven by ,2Vf(,,, =,3,( ) (42,) where helR.(Other commonly used notations forD,,f(p) aredf(p)V, df,,(V) andf’PV.) Whereas thediscussion ofthecontinuity offonly involved the topology oflR'" and lR", the right-hand side ofthis equation manifestly uses thevector space structure ofthese spaces. Ifall thedirectional derivatives offexist atpthen fissaid tobedifferen- tiable atp.Inthiscase Df(p)isalinear transformation from lR'"tolR". Df(p):VI-—>Dvf(p), determining thelinear partofanapproximation tofinthevicinity ofp.The function fsends thepoint ptof(p): ifthe point pstarts tomove inthedirection ofVthen f(p)willcorrespon- dingly start tomove inthedirection Dvf(p) (refer tofigure 4.1). Intuitively wethink ofthederivative offassending an‘arrow’ inlR'", with itstailatpandtipatp+V,toan‘arrow’ inlR",with f(p) astail and f(p) +Dvf(p) astip. We may formalise this bydefining the tangent space tolR'"atp,T,,lR'", tobethesetofpairs (p,V)forall VelR'". These pairs (tangent vectors) form avector space, isomorphic tolR'", with therule 128 MANIFOLDS 4p, V) + p(p, U) = (p, AV + pU) A, p E IR. (4.2.2) We may now define the derivative of f at p, or tangent map, f*p: f: Tf(p)Fin (P, V)1—* (AP), D vf(P))- (4.2.3) Since Df(p) is a linear transformation on Tr it follows that f" is a linear map on Tp1F1m. The tangent space of at p is just a subspace of the direct sum of lBtm with itself, and so there is a natural way of adding tangent vectors lying in different tangent spaces. This feature will not carry over to the following section where we generalise to the concept of a tangent space to a manifold. Since in general the manifold itself will have no vector space structure, there will be no natural way of adding vectors from tangent spaces associated with different points on the manifold. Figure 4.1 The tangent map of f:IFim —> R^. If {e,} and {e'i} are the natural bases for Rtm and lRz then the component functions of f, fi : i = 1, . . n, are given by AP) = Eff(p)e;. (4.2.4) 1=1 The directional derivatives of these component functions along the basis vectors for Fim are called the partial derivatives, and a special notation is customary: Deft(p) = (afilaxl)(p). (4.2.5) For any V E m n Dvf(p) = E D (p)e, = E E viD,f,(p)e; ,=, =1 by the linearity of Df(p). Thus the matrix of the linear transformation Df(p) is formed by the partial derivatives. The n x m matrix Raf/3x0(p)], with i labelling the rows, is called the Jacobian and we 128 MANIFOLDS /l(P.V)+tt(P.U)=(P./W+MU) /1.MEIR (4-Z-2) Wemay now define thederivative offatp,ortangent map, ft1,: -f*PITnmm "—-) Tf(t=)]Bn (P,V)r—->(f(p). Dvf(P))- (4-2-3) Since Df(p)isalinear transformation onIR”itfollows thatft1,isa linear map onT,,lR"'. Thetangent space oflR'"atpisjustasubspace of thedirect sumoflR"‘withitself, andsothere isanatural wayofadding tangent vectors lying indifferent tangent spaces. This feature willnot carry overtothefollowing section where wegeneralise totheconcept of atangent space toamanifold. Since ingeneral themanifold itself will have novector space structure, there willbenonatural way ofadding vectors from tangent spaces associated with different points onthe manifold. //=’i*t»&"v“/1*’ IR” IRF" lp,V) f /"V flpl+D,,flp) P f(p) V D,/f(p) Figure 4.1Thetangent map off:lR'" —>1R". If{e,} and {e}} arethenatural bases forlR’" and IR”then the component functions off,f‘1lR”'t—>IRi=1,...,n,aregiven by f(p)=§r<p>@:» (42.4) Thedirectional derivatives ofthese component functions along thebasis vectors forlR"‘arecalled thepartial derivatives, andaspecial notation iscustomary: D,.'f"(p) =(3f‘i/3x/’)(p). (42.5) ForanyVelR'" Dvf(P) =;Dvf’(P)@Z- = 'D.»,f’(P)@i bythelinearity ofDf(p). Thus thematrix ofthelinear transformation Df(p) isformed bythe partial derivatives. The n><mmatrix [(8)9]/8x")(p)], with 1'labelling therows, iscalled theJacobian andweM;M=S DERIVATIVES OF FUNCTIONS Tim-43.n 129 have m n D vf(p) = E E [(V lax0(p)11 71 e:. (4.2.6) J=1 1 The partial derivatives may be regarded as real functions of the point p and hence higher partial derivatives may be formed. A map between subsets of Em and En for which all partial derivatives up to order k exist and are continuous is said to be a Ck map. A homeomorphism that is a Ck map with a Ck inverse is called a Ck diffeomorphism. We shall be primarily concerned with C maps, which will be called smooth. Example 4.1 Let f : R2 R3,p (x 1, x2) H4 fip = ((x1)2, xlx2 + 1, x2) Taking V = (y1, v2) in (4.2.1) gives D vf(p) = (2v1x1, x1 y2x2v1, v2) = 2x1 x2 O) 0 x1 1 where the entries in the matrix are recognised as the partial derivatives of the function f. 4.3 Differentiable Manifolds With the notion of smooth maps between Rm and established we proceed now to define a differentiable manifold. A topological manifold is locally homeomorphic to IR". By setting up a system of charts that map neighbourhoods of the manifold onto neighbourhoods of En we can use the differential structure on En to define the differential structure on topological manifolds. In order to motivate the definition of a differentiable manifold let us first discuss the problem of coordinating a patch of a topological manifold by returning to the example of S2 as a subset of W. Suppose this subset is constructed from thin perspex and the boundary of a region is marked out by painting a closed curve on the perspex surface. Furthermore paint a fishnet of curves within and on this boundary so that distinct curves in the net intersect only once and each intersects the boundary image once also. Imagine a light is shone through this net of curves and examine the image shadow on any two-plane placed conve- niently to collect the shadow. If each intersection in the net of painted curves casts a unique shadow on the two-plane then the neighbourhood chosen on the sphere yields a proper coordinate patch with respect to the projection scheme. Each intersection can be uniquely labelled by labelling all the curvilinear line shadows uniquely. If a lens of suitable DERIVATIVES 0FFUNCTIONS lR’"—>lPi” 129 have Dvf(.v)=§i[(8f‘/8x")(v)lV"ei» <4-2.6) Thepartial derivatives may beregarded asrealfunctions ofthepoint pandhence higher partial derivatives may beformed. Amap between subsets oflR’"and IR”forwhich allpartial derivatives uptoorder k exist andarecontinuous issaidtobeaC"map. Ahomeomorphism that isaCkmap with aCkinverse iscalled aCkdiffeomorphism. Weshall beprimarily concerned with C°’maps, which willbecalled smooth. Example 4.1 Letf:1R2t—>lR3,p =(x1, x2)t—+f(p) =((x1)2, xlxz +1,x2). Taking V=(v1,v2)in(42.1) gives 12 Dvf(p) =(2v‘x1,x1v2 +xzvl, v2)=(v1,v2)(23 Z, where theentries inthematrix arerecognised asthepartial derivatives ofthefunction f. 4.3Differentiable Manifolds With thenotion ofsmooth maps between lR"’andlR"established we proceed now todefine adifferentiable manifold. Atopological manifold islocally homeomorphic toIR".Bysetting upasystem ofcharts that mapneighbourhoods ofthemanifold onto neighbourhoods oflR"wecan usethedifferential structure onlR"todefine thedifferential structure ontopological manifolds. Inorder tomotivate thedefinition ofadifferentiable manifold letus first discuss the problem ofcoordinating apatch ofatopological manifold byreturning totheexample ofS2asasubset of1R3.Suppose this subset isconstructed from thin perspex and theboundary ofa region ismarked outbypainting aclosed curve ontheperspex surface. Furthermore paint afishnet ofcurves within andonthisboundary so thatdistinct curves inthenetintersect only once andeach intersects the boundary image once also. Imagine alight isshone through thisnetof curves andexamine theimage shadow onanytwo-plane placed conve- niently tocollect theshadow. Ifeach intersection inthenetofpainted curves casts aunique shadow onthetwo-plane then theneighbourhood chosen onthesphere yields aproper coordinate patch with respect to theprojection scheme. Each intersection canbeuniquely labelled by labelling allthecurvilinear lineshadows uniquely. Ifalens ofsuitable 130 MANIFOLDS material is placed between the image and perspex patch one can even arrange that the shadow lines appear orthogonal with respect to the induced Euclidean metric on the two-plane. Such a projection system establishes a homeomorphism from the open set U of S2 containing the net onto the open set of 1H2 formed by the shadow. To each point p c U we assign two real coordinates p(p) = (cpl(p), cp 2(p)) E F12. The set of images labelled cpi(p) = constant (j = 1, 2) are sometimes called coor- dinate lines (or planes in general). There are many ways of establishing such an optical arrangement and equally many ways of painting lines on 52 yielding alternative coordinate systems. Thus there is no unique way of assigning coordinate labels to points in U. We choose a projection system such that cp is a homeomorphism for then and only then will a sequence of points in the topological manifold with a limiting point (in the manifold topology) map into a sequence of coordinates with a corresponding limit. To completely coordinate a topological manifold we shall in general need several overlapping patches, as the example of a sphere shows. We are then prompted to examine the relations between the different coordinates assigned to points in the region of overlap. Returning to the general case of an n-dimensional topological man- ifold M we recall that by definition each point of M has a neighbour- hood U, homeomorphic to an open set of En. If we label one such homeomorphism cpa : (la—) cpa(Ua) then the pair (U (pa) is called a coordinate chart for Va (with the chart domain Va). The image Ta(p) for pEU, assigns to the point p the n real coordinates (cpla(p), cp2a(p), .  cpna(p)). For each chart labelled by a the real-valued function : Va IR, (j = 1, . . n) is called the jth coordinate function and is projected from Pa by the j-projection map 7Ti IR" IR, Ta(P) ° Ta(P) —= Va(P) (4.3.1) for all p E Va. When we work in a prescribed chart we often drop the chart label 'a' on V, and a common notation for the set of n numbers {OP)} is {x1(P)}. One of the most important hurdles to overcome when first working with general coordinates is to resist the instinct to infer any metric or distance properties of the manifold from the use of the symbol x]. Whereas the coordinates {xl(p)} of p are elements of En, regarded as a Euclidean vector space, the metric on IR" need not define any metric or distance function on the manifold. For example, x and x2 could be the 'usual' polar coordinates 0, yo for a neighbourhood of the two-sphere. Although the Euclidean metric is used on (0(p), p(p)) to differentiate functions on the sphere this is not necessarily related to any metric on the sphere, certainly not to the standard metric. A collection of charts (U a, cpa) a = 1, 2, .. . becomes an atlas for M 130 MANiFoi.Ds material isplaced between theimage andperspex patch onecaneven arrange that theshadow lines appear orthogonal with respect tothe induced Euclidean metric onthetwo-plane. Such aprojection system establishes ahomeomorphism from theopen setUofS2containing the netonto theopen setof1R2formed bytheshadow. Toeach point peU weassign tworeal coordinates (p(p) =((p‘(p), (p2(p))elRZ. The setof images labelled (pl(p) =constant (j=1,2)aresometimes called coor- dinate lines (orplanes ingeneral). There aremany ways ofestablishing such anoptical arrangement andequally many ways ofpainting lines on S2yielding alternative coordinate systems. Thus there isnounique way ofassigning coordinate labels topoints inU.Wechoose aprojection system such that(pisahomeomorphism forthen andonlythen willa sequence ofpoints inthetopological manifold with alimiting point (in themanifold topology) map into asequence ofcoordinates with a corresponding limit. Tocompletely coordinate atopological manifold weshall ingeneral need several overlapping patches, astheexample ofasphere shows. We arethen prompted toexamine therelations between thedifferent coordinates assigned topoints intheregion ofoverlap. Returning tothegeneral case ofann-dimensional topological man- ifold Mwerecall that bydefinition each point ofMhasaneighbour- hood U,homeomorphic toanopen setoflR".Ifwelabel onesuch homeomorphism (pa:U,—>(p,,(U,,) then thepair (U2, (pa)iscalled a coordinate chart forU,(with thechart domain U2). The image (p,,(p) forpeU2assigns tothepoint pthenreal coordinates ((p,‘,(p), (p§(p), ..., (pZ(p)). For each chart labelled byathereal-valued function (p{,:U,—>1R,(j=1,...,n)iscalled thejthcoordinate function andis projected from (pabythej-projection mapof Tt’I1R" ——>1Ri<t>a(P)*—> TI’°<t>a(P) E<t>'i(P) (4-3-1) forallpeU2.When wework inaprescribed chart weoften drop the chart label ‘a’on andacommon notation forthesetofnnumbers i<P’(P)} is{x’(P)}~ One ofthemost important hurdles toovercome when firstworking with general coordinates istoresist theinstinct toinfer anymetric or distance properties ofthemanifold from theuseofthesymbol xi. Whereas thecoordinates {xl(p)} ofpareelements of1R”,regarded asa Euclidean vector space, themetric on1R"need notdefine anymetric or distance function onthemanifold. Forexample, x‘andx2could bethe ‘usual’ polar coordinates 6,(pforaneighbourhood ofthetwo-sphere. Although theEuclidean metric isused on(6(p), (p(p)) todifferentiate functions onthesphere thisisnotnecessarily related toanymetric on thesphere, certainly nottothestandard metric. Acollection ofcharts (U2, (pa)a=1,2,...becomes anatlas forM DIFFERENTIABLE MANIFOLDS 131 provided the union of all the Ua is M itself. Two charts (Un, cpa) and (Ub, cpb) such that ua n rib* Ø give rise to a homeomorphism between neighbourhoods of IFin. If U ,, n Ub =- Uab then we define (see figure 4.2) hab =(Pb ° 92.(Uab) Pb( 1b). (4.3.2) Figure 4.2 The chart maps for Ua n Ub C M. Then Tb(P) = kb° Ta(P) expresses the n coordinates q(p) of p in the `1,' chart in terms of n continuous functions leab of the coordinates cpia(p) of p in the 'a' chart, that is a coordinate transformation expresses the coordinates of p in one chart in terms of the coordinates of the same point in another overlapping chart. If as is often done we write cpta(p) and y' =cp'b(p) then x' = h (y 1 , y 2 ,  ., yn) i = 1, .. n. Similarly h;b1 is a homeomorphism from cpb(Uab) to (pa(Uab) and gives the inverse mapping between the coordinates. The maps [had between all overlapping members of the atlas are called the chart transform- ations. If all these maps are differentiable the atlas is said to be differentiable. It is this new property that turns a topological manifold into a differentiable one. Since haa is the identity map and ° hab = ha, then 11,1 = hba and so the inverse chart transformations are differentiable; hence they are diffeomorphisms on 1FIn. New charts (U, cp) can be added to the atlas [(U,„ cpa)] provided cp° cpa-I and T.° 40-' are differentiable for all a, in which case (U, cp) is compatible with the atlas. If every member of one atlas is compatible with every member of another atlas then the two atlases are compatible. A differentiable structure on a topological manifold is specified by giving a differentiable atlas from the class of all compatible differentiable atlases for M. If a topological manifold can be provided with two differentiable atlases that are incompatible then the topological manifold is said to admit two different differentiable structures. An n-dimensional C' DIFFERENTIABLE MANIFOLDS provided theunion ofalltheU,isMitself. Two charts (U,, (p,)and (U,,, (p,,)suchthatU,F)U,#=I6giverisetoahomeomorphism between neighbourhoods oflR". IfU,F)U,=U,,, then we define (see figure 4.2) hub E(Pb0(PEI: q7a(Uab) W) q7b(Uab)' ItU,,EU,l)U, ha1=\vio\va q>,lU,,l Figure 4.2Thechart maps forU,F)U,CM. Then (p,,(p) =h,,,0(p,(p) expresses thencoordinates (pI,(p) ofpin the‘b’chart interms ofncontinuous functions hf,,,ofthecoordinates q>{,(p)ofpinthe‘a’chart, thatisacoordinate transformation expresses thecoordinates ofpinonechart interms ofthecoordinates ofthe same point inanother overlapping chart. Ifasisoften done wewrite x‘=(pf,(p) andy’=(pI,(p) then x’=hf,,,(y‘, y2,...,y”)i=1,...,n. Similarly h,j,,’isahomeomorphism from (p,,(U,,,) to(p,(U,,,) andgives theinverse mapping between thecoordinates. Themaps [h,,,] between alloverlapping members oftheatlas arecalled thechart transform- ations. Ifallthese maps aredifferentiable theatlas issaid tobe differentiable. Itisthisnew property that turns atopological manifold into adifferentiable one. Since h,, isthe identity map and h,,,0h,,,=h,,then h;,,‘=h,,,andsotheinverse chart transformations aredifferentiable; hence they arediffeomorphisms on1R".New charts (U,(p)canbeadded totheatlas [(U,, (p,)] provided (p=>(p;‘ and rp,Q(p“aredifferentiable foralla,inwhich case (U,(p)iscompatible with theatlas. Ifevery member ofone atlas iscompatible with every member ofanother atlas then thetwo atlases arecompatible. A differentiable structure onatopological manifold isspecified bygiving a differentiable atlas from theclass ofallcompatible differentiable atlases forM.Ifatopological manifold canbeprovided with twodifferentiable atlases thatareincompatible then thetopological manifold issaidto admit twodifferent differentiable structures. Ann-dimensional C“ 132 MANIFOLDS manifold (or smooth manifold) is defined as an n-dimensional topologic- al manifold together with a C differentiable structure. As an example of how the topological space JR (the real line) can be assigned different C' structures consider the atlas with single chart (IR, cp) with cp:E R , x —> x. Consider another atlas for IR with chart (IR, )3) where )6 :1:11—>111, x —> x3. Then (po /3')(x) = x 113 which is not differentiable at x = O. Hence OR, (p) and (IR, 13) are not compatible and each atlas defines a different C' structure on the same underlying topological manifold. In what follows we shall always assume that our manifolds have been given a particular differentiable structure. If the manifold admits a covering by charts such that each hab is orientation preserving (that is the determinant of the Jacobian of the map (k b), is everywhere of the same sign for all a, b) then the manifold is said to admit an orientation. Every oriented differential manifold admits two orientations corresponding to the two signs of the Jacobian determinant. The ribbon with one twist (Möbius band) is an example of a two-dimensional differential manifold that is non- orientable. If it is regarded as being a subset of Euclidean three- dimensional space one notices that it is not possible to assign unambi- guously a smooth field of everywhere normal unit vectors to such a surface. Having used the differentiability of functions on 1R to establish the notion of a smooth manifold we can now similarly define differentiable maps between smooth manifolds. A map f from a smooth manifold M 1 to a smooth manifold M2 is said to be differentiable at pc Mi if, for some charts (U 1, cpi) for M, and (U2, cp2) for M2, the map cp2 of o cpi-1 is differentiable at cp,(p). Since a change of chart is a differentiable operation the differentiability of f does not depend on the chart used to represent it. A homeomorphism between smooth manifolds is a diffeo- morphism if both it and its inverse are differentiable. A map f such that P2 = .4/ 31) p2 E M2, p E MI may be represented in local coordinates by writing (P2(P2) = 992 ° i) = cP2 of 0 TT' ° (pl(pi) = f21 ° (PI(P1) where f21 - Ç2 of ° §ol If we write x(p2) - cp;(p2)= 77-J(cp2(p2)) i =1, n, for the coordin- ates of pz in (U2, go2) and yl(p q(p1) = T*Pi(P1)) j = 1,    , m, for the coordinates of p in (U1, TO, then "(P2) = y2(pi),   , ym(p (4.3.3) If we take M, to be R and write MI = M then f is usually called 132 MANIFOLDS manifold (orsmooth manifold) isdefined asann-dimensional topologic- almanifold together with aC“differentiable structure. Asanexample ofhow thetopological space lR(the realline) canbe assigned different C’structures consider theatlas with single chart (lR,(p)with (p:1Rl—>lR,x—>x.Consider another atlas forlFlwith chart (lR,B)where B:lR l—>lR,x—> x3.Then ((p<>/i"1)(x) =x1’3which isnot differentiable atx=0.Hence (lR,(p)and(lR,B)arenotcompatible and each atlas defines adifferent C°°structure onthesame underlying topological manifold. Inwhat follows weshall always assume that our manifolds have been given aparticular differentiable structure. Ifthemanifold admits acovering bycharts such that each h,,,is orientation preserving (that isthedeterminant oftheJacobian ofthe map (h,,,)» iseverywhere ofthesame sign foralla,b)then the manifold issaid toadmit anorientation. Every oriented dijferential manifold admits twoorientations corresponding tothetwosigns ofthe Jacobian determinant. The ribbon with one twist (Mobius band) isan example ofatwo-dimensional differential manifold that isnon- orientable. Ifitisregarded asbeing asubset ofEuclidean three- dimensional space onenotices that itisnotpossible toassign unambi- guously asmooth field ofeverywhere normal unit vectors tosuch a surface. Having used thedifferentiability offunctions on1R"toestablish the notion ofasmooth manifold wecannow similarly define differentiable maps between smooth manifolds. Amap ffrom asmooth manifold M, toasmooth manifold M2issaid tobedifferentiable atpeM,if,for some charts (U,, (p,)forM,and(U2, (p2)forM2, themap (p2<>f<=(p,“ isdifferentiable at(p,(p). Since achange ofchart isadifferentiable operation thedifferentiability offdoes notdepend onthechart used to represent it.Ahomeomorphism between smooth manifolds isadiffeo- morphism ifboth itanditsinverse aredifferentiable. Amap fsuch that P2=f(P1) P2EM2»P1€M1 may berepresented inlocal coordinates bywriting ‘P2(P2) =‘P2°f(P1)= ‘P2°f° <Pi_1° ‘P1(P1)= f21° ‘P1(P1) where f21E ‘P2°f° ‘Pi’- Ifwewrite x’(p2) =(p§(p2) =1r‘((p2(p2)) i=1,...,n,forthecoordin- ates°fP2in(U22vb)andy’(Pi) E<t>’i(pi) =rr’(<ri(Pi)) 1'=1,2~-.mt forthecoordinates ofp,in(U,, (p,), then X’(Pa) =f§i(y’(pi)~ y’(pi). ~--~y"’(Pi))~ (43-3) Ifwetake M2tobelFland write M,=Mthen fisusually called DIFFERENTIABLE MANIFOLDS 133 simply a function on M. If f is defined on an open set W of M f: W-+ Fi, then in a local chart (U, cp) it defines a function fq,: cp(U n E (4.3.4) by the rule fcp= fo q)-1, that is f(P) = (fT ° cP)(P) = MOP), 492(P),   ce(p)) Vp E W. We define cp* by the rule (40*.f(p) = f,. (P. (4.3.5) Writing f =f, o q = cp*fT, the map f said to be pulled back from cp(U n to u n W. This notion generalises to any diffeomorphism lp between the mani- folds M and N. For f: N IR we define lef M P (1P*f)(P) = f0P(P)) and say that the real-valued function f on N has been pulled back to the real-valued function tef on M (see figure 4.3). It follows immediately that under a composition of diffeomorphisms: (cP ° 1P)* = 1,0* ° T*. (4.3.7) Figure 4.3 The pull-back map. Suppose f is a smooth map from a manifold M to a manifold N. If dim(f*(TpM)) = r then f is said to have rank r at p E M. The tangent map fi,, is said to be injective at p if r = dim M (dim M dim N). If r = dim N then f„ is said to be surjective. The mapping f for which f" is injective for all p E M is called an immersion and M is an immersed submanifold of N. When the immersion f is injective it is referred to as an imbedding and M is an (imbedded) submanifold of N. Unless specified otherwise by submanifold we shall mean an imbedded subman- ifold. In this case coordinate systems for N exist around f(p) endowing (4.3.6) DIFFERENTIABLE MANIFOLDS 133 simply afunction onM.Iffisdefined onanopen setWofM f:W—> 1R,then inalocal chart (U,rp)itdefines afunction f,,:rp(U F)W)—>lR (4.3.4) bytherulef,,=ft»(p‘1, that is f(p)=(fa°F/>)(P) =fa(¢’(t>), t/>’(t>). --'7t/>"(t>)) VP6W- Wedefine rp*bytherule (¢*fa) =faQ¢- (4-3-5) Writing f=f,,<> rp=(p*f,,, themap f,,issaid tobepulled back from (p(UF)W)toUF)W. This notion generalises toanydiffeomorphism (/1between themani- folds MandN.Forf: N->1Rwedefine ¢*f=M—>IR P*—>(¢*f)(P) =f(¢(P)) (4-3-6) andsaythatthereal-valued function fonNhasbeen pulled back tothe real-valued function 1/1*f onM(see figure 4.3). Itfollows immediately thatunder acomposition ofdiffeomorphisms: (rt~(W=1/1*°¢*- (4.31) ti M p /v W F IR Figure 4.3Thepull-back map. Suppose fisasmooth map from amanifold Mtoamanifold N.If dim(f,.(T,,M)) =rthen fissaid tohave rank ratpeM.The tangent map f,,,,issaid tobeinjective atpifr=dimM (dimM sdimN).If r=dimNthenft,issaidtobesurjective. Themapping fforwhich fr, ISinjective forallpeMiscalled animmersion andMisanimmersed submanifold ofN.When theimmersion fisinjective itisreferred toas animbedding and Misan(imbedded) submanifold ofN.Unless specified otherwise bysubmanifold weshall mean animbedded subman- ifold. Inthiscase coordinate systems forNexist around f(p) endowing 134 MANIFOLDS f(M) with a smooth manifold structure. As an example consider the map f: IR2 where the image point traverses the figure 0 once without stopping. Then f is an injective immersion since both f and f are injective, and AS') is a one- dimensional imbedded submanifold of R2. If the map uniformly traverses the image set more than once it becomes an immersion, with f no longer injective. Similarly if the image f(SI) is the figure 8 traversed uniformly once the map is an immersion, since although again f is injective f is not. The map f: [-1, 1] ---> E, x I--> x3, is neither an immersion nor an imbedding since although f is injective the map fails to be injective at x = O. 4.4 Parametrised Curves Having defined real-valued functions on a manifold we now examine the generalisation of the directional derivative. We cannot simply apply the definition (4.2.1) since there is no vector space structure to enable points on a manifold to be added. By suitably defining curves on a manifold we can define differentiation of functions in the direction of a curve. Just as differentiation of maps between manifolds is defined by using the chart maps the derivative of a function along a curve will be defined by using a parametrisation of the curve; the derivative being defined for a real function of a real variable. A parametrised curve C on a manifold M is a map from an open interval I C E to M. If p is any point on the image of C and (U, cp) is a chart for the neighbourhood of p then C may be specified in this neighbourhood by n real-valued functions 7'cp[C(t)] —= cp' o C(t) t e I. (4.4.1) Thus denoting çoi o C by Ci we write in a local chart the representation of C xt(p) = e(t). (4.4.2) Different parametrised curves can have the same image on M. If h maps the open interval J C JR into I C R then C' : J M is said to be a reparametrisation of C: I —> M if C' = C oh (see figure 4.4). Where- as reparametrised curves have the same image, if we think of the parameter as a time, a change of parameter affects the rate at which that image evolves. 134 MANIFOLDS f(M) with asmooth manifold structure. Asanexample consider themap f:S‘—>1R2where theimage point traverses thefigure 0once without stopping. Then fisaninjective immersion since both fand f,areinjective, and f(S') isaone- dimensional imbedded submanifold of1R2. Ifthe map uniformly traverses theimage setmore than once itbecomes animmersion, with f nolonger injective. Similarly iftheimage f(S') isthefigure 8traversed uniformly once themap isanimmersion, since although again f,is injective fisnot. The map fr[-1, 1]—>lR, x1—>x3, isneither an immersion noranimbedding since although fisinjective themap f, failstobeinjective atx=0. 4.4Parametrised Curves Having defined real-valued functions onamanifold wenow examine the generalisation ofthedirectional derivative. Wecannot simply apply the definition (42.1) since there isnovector space structure toenable points onamanifold tobeadded. Bysuitably defining curves ona manifold wecandefine differentiation offunctions inthedirection ofa curve. Just asdifferentiation ofmaps between manifolds isdefined by using thechart maps thederivative ofafunction along acurve willbe defined byusing aparametrisation ofthecurve; thederivative being defined forarealfunction ofarealvariable. Aparametrised curve Conamanifold Misamap from anopen interval IClRtoM.Ifpisanypoint ontheimage ofCand(U,(p)isa chart fortheneighbourhood ofpthen Cmay bespecified inthis neighbourhood bynreal-valued functions rr‘(p[C(t)] =(pi=>C(t) reI. (4.4.1) Thus denoting tp’OCbyC‘wewrite inalocal chart therepresentation ofC x’(p) =C'(t). (4.42) Different parametrised curves can have thesame image onM.Ifh maps theopen interval JC1RintoIC1Rthen C’:Jt—>Missaidtobe areparametrisation ofC:I—>MifC’=C=>h(see figure 4.4). Where- asreparametrised curves have thesame image, ifwethink ofthe parameter asatime, achange ofparameter affects therate atwhich thatimage evolves. PARAMETRISED CURVES 135 Figure 4.4 Different parametrised curves with the same image. If f is a smooth function defined in the neighbourhood of po = C(to), with C smooth at t o, then the derivative of f along C at p o, V(f) is defined to be Vpc„(f)= —d (f o C)(to). (4.4.3) dt (The reason for adopting the notation V(f) will be clear later.) Since fo C is a map from Ito E, smooth at t o, the derivative in (4.4.3) needs no further explanation. If (U, cp) is a chart for a neighbourhood of Po = C(t o) then the chart map yo can be used to express V(f) in terms of the directional derivative of f,t,= fo yo-1. We may write fo C as the composition of maps from / to Rn and En to R: f o c = (fo (to') o (q) C)- The chain rule of differentiation then gives d — dt (f o C)(to) = (afepiax1)(92(Pon dC' (to). dt If if is the vector in R" with components dC'(t o)/dt then (4.4.4) expresses the derivative of f along C as the directional derivative of fcr vcp,,(f) = plf,p(cP(p0)). (4.4.5) Since this relation holds for all functions f we have a correspondence between the curve C, with image containing p o, and the tangent vector to En, (cp(p0), if). A curve C 1 with C I(A0) = pa will be called equivalent to C at po if V(f)= V pc„(f) for all functions f. Thus, for some choice of chart map, equivalent curves at p o correspond to the same tangent vector in Tq,(pollin. By taking all curves passing through p o we obtain a one-to-one correspondence between equivalence classes of curves and vectors in Tegpollin (see figure 4.5). (4.4.4) PARAMETRrsED CURVES 135 C I h J Figure 4.4Different parametrised curves withthesame image. Iffisasmooth function defined intheneighbourhood ofp0=C(t,,), with Csmooth atto,then thederivative offalong CatPo.V§u(f) is defined tobe V2.6")=§(toC)(t.2>- (4-4-3) (The reason foradopting thenotation Vf,,,(f) willbeclear later.) Since foCisamap from ItoIR,smooth atto,thederivative in(4.43) needs nofurther explanation. If(U, (p)isachart foraneighbourhood of pr,=C(t,,) then thechart map (pcanbeused toexpress VC,,(f) interms ofthedirectional derivative off,,=fo(p‘1. Wemay writc foCasthe composition ofmaps from 1toIR"andIR"toIR: f°C= (f°</>“)°(</><=C)- Thechain ruleofdifferentiation then gives gut0(6)=(afq:/axi)(§0(p0)) ([0)- (4.4-4) If1/isthevector inIR"with components dC’(t,,)/dt then (4.4.4) expresses thederivative offalong Casthedirectional derivative off, v.‘J..<t>=D)’f¢((i0(p0))- (4.4-5) Since thisrelation holds forallfunctions fwehave acorrespondence between thecurve C,with image containing po,andthetangent vector tolR", ((p(p,,), ‘l/). Acurve C,with C,(/10) =p,,will becalled equivalent toCatp0ifV,§,I(f) =V,§,,(f) forallfunctions f.Thus, for some choice ofchart map, equivalent curves atp0correspond tothe same tangent vector inT,,(,,,,lR". Bytaking allcurves passing through p,, weobtain aone-to-one correspondence between equivalence classes of curves andvectors inT,,,,,,,ll-'1" (seefigure 4.5). 136 MANIFOLDS 11R Figure 4.5 This diagram illustrates the relation between real func- tions on M and curves. 4.5 Tangent Vectors In view of the previous section we could define a tangent vector to the manifold M at the point Po to be an equivalence class of curves passing through Po. Such a class of curves defines a direction at the point p o and enables functions to be differentiated. Further, for any chart map we can put this class of curves into correspondence with a tangent vector in En, this having been previously defined. It is most convenient (and usual) to adopt an equivalent definition of tangent vectors, modelled on the abstraction of differentiating along a curve. A tangent vector at p will be defined to be a certain mapping from real-valued functions, defined in the neighbourhood of p 0. Such a mapping is given by any curve passing through p, namely the mapping to the derivative of the function along the curve. For this reason we used the notation V(f) to denote the derivative of f along C at p : with the definition that we shall give V c will be identified with a tangent vector, the tangent to the curve C at Po. whose action on f is given by (4.4.3). Similarly the definition of the tangent vector to Ili", based on the intuitive idea of a directed line segment, is equivalent to the more abstract definition of being a derivation into IR on functions. Given the tangent vector (p, V) E TpIR" we may take the directional derivative of the function f along V at p. In the following the reader should check that the properties we require of a tangent vector are satisfied by the derivative of a function along a curve. Later in this chapter we shall show, as is intuitively clear, that every tangent vector has a curve tangent to it. 136 MANIFOLDS f I Io rwot ‘P IR” Figure 4.5This diagram illustrates therelation between real func- tions onMandcurves.U5 4.5Tangent Vectors Inview oftheprevious section wecould define atangent vector tothe manifold Matthepoint p,,tobeanequivalence class ofcurves passing through p0.Such aclass ofcurves defines adirection atthepoint p0 andenables functions tobedifferentiated. Further, foranychart map wecanputthisclass ofcurves intocorrespondence with atangent vector inlR",thishaving been previously defined. Itismost convenient (and usual) toadopt anequivalent definition oftangent vectors, modelled on theabstraction ofdifferentiating along acurve. Atangent vector atp,, will bedefined tobeacertain mapping from real-valued functions, defined intheneighbourhood ofp0.Such amapping isgiven byany curve passing through p0,namely themapping tothederivative ofthe function along thecurve. Forthisreason weused thenotation V,§,,(f) to denote thederivative offalong Catpozwith thedefinition that we shall giveVinwillbeidentified withatangent vector, thetangent tothe curve Catp,,,whose action onfisgiven by(4.4.3). Similarly the definition ofthetangent vector tolR”,based ontheintuitive idea ofa directed linesegment, isequivalent tothemore abstract definition of being aderivation into lRonfunctions. Given thetangent vector (p,V)e T,lR" wemay take thedirectional derivative ofthefunction f along Vatp.Inthefollowing thereader should check that the properties werequire ofatangent vector aresatisfied bythederivative ofafunction along acurve. Later inthischapter weshall show, asis intuitively clear, thatevery tangent vector hasacurve tangent toit. TANGENT VECTORS 137 The notion of a tangent vector at a point p on a manifold is a local one. Therefore it is convenient to classify together all maps in the neighbourhood of some point with similar properties. So we take the set of differentiable maps defined on some neighbourhood of p E M and say that two maps in this set are equivalent if their restrictions to a common neighbourhood agree. Maps satisfying this property belong to an equiva- lence class which is denoted [f xfp] and is called a (differentiable) germ of a map from M to N at p. The collection of all such equivalence classes is called the collection of germs of C' maps at p. Clearly elements in [fm,] yield the same image for p. For example consider the germs of C maps C : IR N at t. These 'path' germs yield the images of curves in N that all pass through C(t) with the same velocity. Such curves were called equivalent in the previous section, and we expect the general notion of a tangent vector to be related to a germ [Ce,] rather than to be related to a particular curve in this class. If f: M --> N and g : N ---> P are any representatives of the germs [fmj and [g7,1,] then the composition [gN]o[fiv] is the germ obtained by composing representatives : namely g o f. Similarly we define the pull-back of germs in terms of any representitives [f ]*[g] = [rg] = [g on (4.5.1) It is convenient not to distinguish notationally between [f]* and f* since no confusion need arise in practise. We denote by 5-,(M) the set of real-valued smooth functions on the manifold M. The elements of .9-,(M) form a ring with (f + g)(p) f(p) + g(p) and (fg)(p)= f(p)g(p). By identifying the constant functions with the real numbers the ring 5;(M) may be regarded as a real vector space, and hence an algebra. A derivation into IR on [5-,(M)p] is a linear map X :[(M)p]---> IFI that obeys the Leibnitz rule X(f1f2) = X(f1)f2(P) f1(P)X(f2)- (4.5.2) Since linear combinations of derivations are derivations they form a vector space over Fi at p. If we set fi= f2=1, the identity map, then (4.5.2) implies X(1) = 0 and hence, by linearity, X annihilates any element of ri. The vector space of derivations of the above germs at p E M is defined as the tangent space TM of the smooth manifold at p. We introduced earlier the pull-back map f* associated with the diffeomorphism f: M --> N, p q = f(p). The tangent map at p associated with f is denoted f" and is defined in terms of f* by f":TpM--> T qN X 1--> f"X = Xf*. (4.5.3) Thus (see figure 4.6) f"X is a derivation on elements g EP(N)A p)] obtained by pulling back g with f* and then acting with X, that is TANGENT VECTORS 137 The notion ofatangent vector atapoint ponamanifold isalocal one. Therefore itisconvenient toclassify together allmaps inthe neighbourhood ofsome point with similar properties. Sowetake theset ofdifferentiable maps defined onsome neighbourhood ofpeMandsay thattwomaps inthissetareequivalent iftheir restrictions toacommon neighbourhood agree. Maps satisfying thisproperty belong toanequiva- lence class which isdenoted [fMP] andiscalled a(differentiable) germ of amap from MtoNatp.The collection ofallsuch equivalence classes iscalled thecollection ofgerms ofC‘maps atp.Clearly elements in if/up] yield thesame image forp. Forexample consider thegerms ofC°°maps C11B-—> Natt.These ‘path’ germs yield theimages ofcurves inNthat allpass through C(t) with thesame velocity. Such curves were called equivalent inthe previous section, andweexpect thegeneral notion ofatangent vector toberelated toagerm [CB1] rather than toberelated toaparticular curve inthisclass. Iff:M-> Nand g:N-> Pareany representatives ofthegerms [fMp] and[gm] then thecomposition [gNq] Q[fMp] isthegerm obtained bycomposing representativesznamely gof.Similarly wedefine the pull-back ofgerms interms ofanyrepresentitives [fl*[3l =[F8]=[8°fl- (4-5-1) Itisconvenient nottodistinguish notationally between [f]*andf*since noconfusion need arise inpractise. Wedenote by@(M) thesetofreal-valued smooth functions onthe manifold M.The elements of§(M) form aring with (f+g)(p) =f(p)+g(p) and (fg)(p) =f(p)g(p). Byidentifying the constant functions with thereal numbers thering §(M) may beregarded asa real vector space, and hence analgebra. Aderivation into IRon [§(M)p] isalinear map X:[@(M)p] -—>1Bthatobeys theLeibnitz rule X(flf2) =X(f1)f2(P) +fl(p)X(f2)- (4-5-2) Since linear combinations ofderivations arederivations they form a vector space over IRatp.Ifwesetfl=f2=1,theidentity map, then (4.5.2) implies X(1) =0andhence, bylinearity, Xannihilates any element ofIR.The vector space ofderivations oftheabove germs at peMisdefined asthetangent space TPM ofthesmooth manifold atp. We introduced earlier thepull-back map f“associated with the diffeomorphism f:M-> N,p»—>q=f(p). The tangent map atp associated with fisdenoted fipandisdefined interms off‘by f,,p; TPM i> TqN X|-——>f,,,PX =Xf“. (4.5.3) Thus (see figure 4.6) f,.,,X isaderivation onelements ge[@(N)f(,,)] obtained bypulling back gwith f“andthen acting with X,thatis 138 MANIFOLDS (fX)(g) = X(r(g)) = X(g f). (4.5.4) From this point on we shall also apply the definition of a tangent vector being a derivation into 1F3 on functions, to tangent vectors to Rm. We must therefore show the equivalence with the previous definition of a tangent vector being an ordered pair of elements from 11:3" 1. Let X (p, V) E T1Rm. If h is a real-valued function on Rim then we define X to map h to R by taking the directional derivative, that is X(h) = D vh(p). With this rule the tangent vector X is a derivation on functions in the neighbourhood of p. It also ensures the consistency of the definition of the tangent map given in (4.5.3) with the earlier definition (4.2.3), as will be explicitly demonstrated in a moment. IR Figure 4.6 The tangent map f„:7',111— We now construct a local basis for TM in terms of a local chart germ at p, [q] : M 11=3", that assigns the point p E M to the origin in R". As usual let xv, y = 1, . n denote the coordinate maps cpv : Um —> R. Then ep* is a map from function germs in R" to function germs in M and cp.p maps tangent vectors from TM to T0IFI". Of all the derivations on real-valued functions on IFI" we denote by X. E Tolin , the partial derivative: X,:r1(IFin) 01--> Raf/axv)(0)]. (4.5.5) Suppose a vXv = 0 for some n real numbers a v, then since (Xv(xP))(0) = (5, acting on xP gives aP = O. Thus the Xv are linearly independent and the n tangent vectors ()GI form a local basis for the n-dimensional vector space T 01F3". We may express any tangent vector X E TM in terms of 138 MANIFOLDS (f*,,X)(s') =X(f*(s')) =X(s'°f)~ (4-5-4) From thispoint onweshall alsoapply thedefinition ofatangent vector being aderivation into lBonfunctions, totangent vectors tolB"‘. We must therefore show theequivalence with theprevious definition ofa tangent vector being anordered pairofelements from lB"'. LetXE(p, V)eTplB'". Ifhisareal-valued function onlB'"then wedefine Xto map htoIRbytaking thedirectional derivative, thatis X<h>=Dvh(P)- With thisrule thetangent vector Xisaderivation onfunctions inthe neighbourhood ofp.Italsoensures theconsistency ofthedefinition of thetangent map given in(4.5.3) with theearlier definition (4.2.3), as willbeexplicitly demonstrated inamoment. IR got 9 fw, \ / XETNP IRf*,,XET,7N > N f N Figure 4.6Thetangent mapf*,,:TPM —>TqN. Wenow construct alocal basis forTPM interms ofalocal chart germ atp,[<p1,] :M—> lB", that assigns thepoint peMtotheorigin in1R". As usual letx", v=1,...,ndenote the coordinate maps tp"2UM->lB.Then <p*isamap from function germs inlB"tofunction germs inMand<p,,1,maps tangent vectors from T_,,M toTOIB". Ofall thederivations onreal-valued functions onIR"wedenote byXVeTOIB”, thepartial derivative: Xvi[9(1B")o] -—>13 [fl*—>[(@f/5X”)(9)]- (4-5-5) Suppose a"X, =0forsome nrealnumbers a”,then since (X,(x"))(0) =6Q‘,acting onx"gives a"=0.Thus theXVarelinearly independent andthentangent vectors {XV}form alocal basis forthen-dimensional vector space TUB". We may express any tangent vector XeTPM interms of TANGENT VECTORS 139 Tolin. If f E Fi(M) then by writing f = op* f q, we have = X(9)*.f(p) = (9)*pX)fg). (4.5.6) In a natural basis associated with the chart (Um, 0 q)* pX = E av(3/30, (4.5.7) where it is to be understood that the derivative acts at xv = 0, this gives X(f) =RE av(a/3xv))f q,1(x1(p), . . xn(p)). (4.5.8) v=1 Often for computations, real-valued maps f on M are specified locally in terms of their local representatives fq, = fo cp-' on Tin and the details of the chart op are suppressed. However it may be important when dealing with global properties of manifolds to remember the distinction between f and fcp since for a general manifold it is not possible to find an atlas consisting of a single chart. Just as the charts are often suppressed when discussing real-valued maps, in a similar way the representative ipofo q2-1 of a map between manifolds is often written with the charts ip and ço omitted. In the following we shall denote such a map by f. We may specify any Xe TM by giving cpX, as in (4.5.7). It is common not to distinguish cp*pX from X, identifying (aIaxy) with a tangent vector to M. Having pointed out the distinction we shall nevertheless employ this abuse of notation in the following sections. Consider the expression for the tangent map f*p where f is a representative of a germ at p from some n-dimensional manifold M to some m-dimensional manifold N. Suppose (x', . ., .0) are local chart functions that assign to pE M the origin of and (y', . . yn) are local chart functions that assign to f(p) E N the origin of IRm. Thus f may be specified in terms of the m real-valued functions (A . . fn) and we represent it by the map 1: U(Rn) Em, (x .7 xn) (yi fl(x xi% yrn fM(x We recall that {X„} = {(alax v)} is a basis for Toffin in this chart. If g is any element of [5;(1Rm)0] then (ft*o(a/axv))g = (a/axv)(:rg) = (alaxv)(g o h = E(ag/aym)(0)(afP/axv)(0) 1.4=1 or more simply L,0(3taxv) = E (3fm(0)/ax v)(3/ay 11. (4.5.9) ti=I TANGENT VECTORS 139 (p,pXe T0113". Iffe 97(M) then bywriting f=<p*fq, wehave X0)=X<¢*f¢> =(¢*,.X>f..- (4.5-6) Inanatural basis associated with thechart (UM,(p) <p,,,X=Za"(8/8x’), (4.51) v=1 where itistobeunderstood thatthederivative actsatx”=O,thisgives X(f) =[(;::]a"(€9/8x"))fq,] (x‘(p), ...,x"(p)). (4.5.8) Often forcomputations, real-valued maps fonMarespecified locally interms oftheir local representatives fq,=f0Q9"on1R"andthedetails ofthechart (paresuppressed. However itmay beimportant when dealing withglobal properties ofmanifolds toremember thedistinction between fandfq,since forageneral manifold itisnotpossible tofind anatlas consisting ofasingle chart. Just asthecharts areoften suppressed when discussing real-valued maps, inasimilar way the representative wefoQ9"ofamap between manifolds isoften written withthecharts ipand(pomitted. Inthefollowing weshall denote sucha map byf.Wemay specify anyXeTPM bygiving <p*pX, asin(4.5.7). Itiscommon nottodistinguish <p*pX from X,identifying (8/8x") with a tangent vector toM.Having pointed out thedistinction weshall nevertheless employ thisabuse ofnotation inthefollowing sections. Consider the expression forthe tangent map f,,p where fisa representative ofagerm atpfrom some n-dimensional manifold Mto some m-dimensional manifold N.Suppose (x‘, ...,x")arelocal chart functions that assign topEMtheorigin of1R"and (yl, ...,y'") are local chart functions that assign tof(p) eNtheorigin oflR"’. Thus f may bespecified interms ofthemreal-valued functions (fl, ...,f’") andwerepresent itbythemap J";U(lR")—>lam, (x1,...,x“)+—> (yl=f‘(x‘, ...,x”), ...,y"’=f’"(x‘, ...,x")). Werecall that{Xv} ={(8/€9x')} isabasis forTOR" inthischart. Ifgis anyelement of[@(1R"’)0] then (Ma/@x">>g =(8/@x"><f*g> =<8/av/><g 0i) =Ztag/@y~><0>(@f~/@x"><0> ormore simply ]’.@(a/ax") =i(8f"(O)/8x")(8/8y”). (4.5.9) u=l 140 MANIFOLDS The action of Ito on an arbitrary vector in ToTin now follows directly since 1,K0 is linear: 1,0(av(a/axv)) = a o(a/ax v) = a v(afm/ax v)(0)(8/aym). (4.5.10) The Jacobian matrix gives a representation of the linear map between TM and Tfip)N. Equation (4.5.10) expresses the chain rule of differentiation and establishes the equivalence of definitions (4.2.3) and (4.5.3) for the tangent map on TpEn. If A e T ollin is regarded as an ordered pair, A = (0, a) with a = Env=,ave, in the natural basis for IR" then A is equivalent to the derivation av(a/axv)1 0. The effect of 1„0 on this derivation is given in (4.5.10). The derivation on the right-hand side of (4.5.10) is equivalent to the ordered pair (f(0), av(afP13xv)(0)e ) where {e} is the natural basis for Rm. From (4.2.6) we recognise this as (f(0), Dpf(0)), which is the form of li3O21 given in (4.2.3). Figure 4.7 summarises the relationship between op and f and the maps that they induce. Let us next observe that if ip : u,(Fin) —> u2(Iii") = xn) (4.5.11) we may infer from the above that v,o(a/axv) = (31pP/axv)(0)(3/ax' 0). (4.5.12) The tangent vector X at p c Um, that was represented in the chart (Um, cp) by cppX = av(313x1), will have a different representation in the chart (Um, 1p0 cp), since co,px = = v*0(av(alaxv)) = av(avP13x v)(0)(a/ax'P) (from (4.5.12)) a'P(a/ax'P) where a'P = (3VPlaxv)(0)av. Figure 4.7 Relations between q9 and f and the maps they induce. 140 MANIFOLDS The action offieonanarbitrary vector inTUB" now follows directly since fioislinear: }.0(aY(a/am) =a"}.0(a/ax") =a"(8f“/8x")(O)(8/By“). (45.10) The Jacobian matrix gives arepresentation ofthelinear map between TPM andTf(p)N. Equation (4.5.10) expresses thechain rule ofdifferentiation and establishes theequivalence ofdefinitions (4.2.3) and (4.5.3) forthe tangent map onTPIB". IfAeTOIR" isregarded asanordered pair, A=(O,a)with a=ZQ‘=1a"e, inthenatural basis forIR"then Ais equivalent tothederivation a"(8/8x")|@. The effect offloonthis derivation isgiven in(4.5.10). The derivation ontheright-hand side of (4.5.10) isequivalent totheordered pair (f(0), a"(8f“/8x")(0)e;,) where {e;,} isthenatural basis forBC". From (4.2.6) werecognise thisas(f(0), D,,f(O)), which istheform off*0A given in(4.2.3). Figure 4.7summarises therelationship between goandfandthemaps thatthey induce. Letusnext observe thatif1,0:U1(lR") —>UZ(lR") x"i> x'“=1p*‘(x1,...,x") (4.5.11) wemay infer from theabove that 1p*@(8/8x“) =(81/W/8x“)(0)(8/8x”‘). (4.5.12) Thetangent vector XatpeUM, thatwasrepresented inthechart (UM, go)byg0*pX =a"(8/Bx"), will have adifferent representation inthe chart (UM, 1,00go),since (1/1O¢)*pX =1/1*0O¢*pX=1/~=@(a"(9/9X”)) =a"(81pP/8x“)(0)(8/8x’F’) (from (4.5.12)) Ea’p(8/8x’F’) where a’F’=(81/1P/8x")(0)a”. ,1 - 7wlp1|R fwlfll Tl‘~|lof)lDl|Rmz ‘PM; ‘limp: {PM Q-p Tflpl/V l\.]Jofllp)= tiowltpi lR'"\plpl n IiIR i /\°/ /6p f(p) M t Nf Figure 4.7Relations between tpandfandthemaps they induce. TANGENT VECTORS 141 This representation of the same tangent vector X E TM at p in a different chart should be distinguished from the tangent vector fX e Tf(p)M. The latter is induced from a differentiable germ f:M—>M at p: the former from a change of coordinates in the neighbourhood of p E M. The relation between the natural (or chart- induced) components {a'P} of X in the basis {(3/ax'P)} at p to the natural components {a') of X in the basis {(3/3xv)} of a different chart about p, may be recognised as a Gl(n, 1R) basis-induced transformation. (Recall coordinate transformations are invertible.) Historically this was one of the characterisations of a `contravariane vector. It prescribed how the components of a vector were to be related to a change of coordinates. In the previous section we motivated the definition of a tangent vector by considering differentiation along a curve. Having now defined tangent vectors we can return and define the tangent vector to a curve. If C: I —> M is a smooth curve with C(to) = po then the tangent vector to C at po is V pC C„(3/3t) (4.5.13) so for f .9;(M) 11,0(f) = (c*,0(alat))(f)— (313t)(f o C)(to). Thus the tangent vector to C at Po maps functions to their derivative along the curve at po, as was anticipated by the choice of notation in (4.4.3) (C*),(3/3t) = ((aC/30)(t0)(3/3X) 1 E TM. As an illustration consider C: (0,1) —› 1R 2 given by Cl(t) = a sin bt C2(t) = a cos bt a, beE. If {(313x1), (3/3x2)) is a natural basis for Tpu1F12 then C*0(3/3t) = (3Ci lat)(to)(alax1). From the above we have (3/3t)C1(t0) C1(t0) = ab cos bto = bC2(t0) (3C213t)(t0) C2(to) = --ab sin bto = —bC1(t0) (4.5.14) 4.6 Vector Fields So far tangent vectors have been associated with points on the manifold. By smoothly assigning a tangent vector to each point we define a vector TANGENT VECTORS 141 This representation ofthesame tangent vector XeTPM atpina different chart should bedistinguished from the tangent vector f,pX eTfg,)M. The latter isinduced from adifferentiable germ f:M—> Matp:theformer from achange ofcoordinates inthe neighbourhood ofpeM.The relation between thenatural (orchart- induced) components {a'P} ofXinthebasis {(8/8x'P)} atptothe natural components {a"} ofXinthebasis {(8/8x")} ofadifferent chart about p,may berecognised asaGl(n, 1B)basis-induced transformation. (Recall coordinate transformations areinvertible.) Historically thiswas one ofthecharacterisations ofa‘contravariant’ vector. Itprescribed how thecomponents ofavector were toberelated toachange of coordinates. Intheprevious section wemotivated thedefinition ofatangent vector byconsidering differentiation along acurve. Having now defined tangent vectors wecanreturn anddefine thetangent vector toacurve. IfC:I—>Misasmooth curve with C(tg) =pgthen thetangent vector toCatpgis 1/5,5 c.,,(a/at) (45.13) soforfe@(M) Vim=<C*..(@/@1>><f> =(8/@1>(f<» Cm)- Thus thetangent vector toCatpgmaps functions totheir derivative along thecurve atpg,aswasanticipated bythechoice ofnotation in (4.4.3) (C,.),0(8/St) =((8C"/8t))(tg)(8/8x)‘ eT,,nM. (4.5.14) Asanillustration consider C:(0,1) —>B2given by C‘(t) =asinbt C2(t)= acosbt a,belB. If{(8/8x‘), (8/8x2)} isanatural basis forTPUIBZ then C,,,n(8/81) =(8C"/8t)(tg)(8/8x‘). From theabove wehave (8/8t)C‘(1g) EC‘(tg) =abcosbtg=bC2(l0) (SCZ/8t)(tg) EC2(tg) =—absinbtg=—bC‘(tg). 4.6Vector Fields Sofartangent vectors have been associated with points onthemanifold. Bysmoothly assigning atangent vector toeach point wedefine avector 142 MANIFOLDS field. Thus a vector field maps functions to functions. In fact this is a convenient starting point for the definition of a vector field, it being a consequence that a vector field assigns a tangent vector to each point. A vector field X on a manifold M is a derivation on the algebra of smooth functions X : 5-,(M) (M) X(Xf + pg) = AX(f) + p,X(g) A, p E E; f, g E 5-e(M) X(fg) = X(f)g + f X(g). (4.6.1) (In the previous section we used capital letters to denote tangent vectors; in the following capital letters will be used for vector fields. Tangent vectors will henceforth be labelled by the point with which they are associated.) Whereas tangent vectors are derivations into E, vector fields are derivations that map the algebra of smooth functions into itself. A vector field X is called smooth if, for every smooth f E 9;(M), X(f) is smooth. The set of smooth vector fields on M will be denoted Ti(M). Given an X E TI(M) we may define a vector Xp E TM, for any p E M, by (Xf)(P) = Xpf. (4.6.2) It is clear from the derivation properties of X and Xp that this does indeed define a tangent vector. Since vector fields map functions to functions we may define a product in an obvious way. For X, YE TI(M) XY: ?1,(M)—> f X(Y(f)). (4.6.3) This composed mapping will not, however, be a vector field. It will not satisfy the Leibnitz property (4.6.1) required of a derivation. In fact (XY)(fg) = (XY)(f)g + f(XY)(g) + X(f)Y(g) + Y(f)X(g). From this it is clear that we can obtain a new vector field from the commutator of two vector fields [X, Y] = XY — YX. (4.6.4) Being the commutator of an associative product this bracket operation on vector fields is antisymmetric and satisfies the Jacobi identity [[X, Y], Z] + [[Y, Z], X] + [[Z, X], Y] = 0. (4.6.5) Smooth vector fields form a module (see Appendix A) over .9-,(M), and hence a vector space over E identified with the constant functions. The commutator then turns the vector fields into an (infinite-dimensional) Lie algebra. The commutator is also called the Lie bracket. 142 MANIFOLDS field. Thus avector field maps functions tofunctions. Infactthisisa convenient starting point forthedefinition ofavector field, itbeing a consequence thatavector field assigns atangent vector toeach point. Avector field Xonamanifold Misaderivation onthealgebra of smooth functions X;@(M) _>@(M) X(/If+H8)=/lX(f) +t4X(g) /1,/4613;/i g6@(M) X(f8) =X(f)8 +fX(8)- (4-6-1) (Inthe previous section weused capital letters todenote tangent vectors; inthefollowing capital letters will beused forvector fields. Tangent vectors willhenceforth belabelled bythepoint with which they areassociated.) Whereas tangent vectors arederivations into IR,vector fields arederivations that map thealgebra ofsmooth functions into itself. Avector field Xiscalled smooth if,forevery smooth fe@(M), X(f) issmooth. The setofsmooth vector fields onMwillbedenoted T1(M). Given anXeT‘(M) wemay define avector XpeTPM, forany peM,by (Xf)(P) =Xpf' (4~6-Z) Itisclear from thederivation properties ofXand Xpthat thisdoes indeed define atangent vector. Since vector fields map functions to functions wemay define aproduct inanobvious way. For X, YeT1(M) XY: @(M) Z> @(M) fii> X(Y(f)). (4.6.3) This composed mapping willnot, however, beavector field. Itwillnot satisfy theLeibnitz property (4.6.1) required ofaderivation. Infact (XY)(fg) =(XY)(f)g +f(XY)(g) +X(f)Y(g) +Y(f)X(g)- From thisitisclear that wecanobtain anew vector field from the commutator oftwovector fields [x,Y]=xv-YX. (4.64) Being thecommutator ofanassociative product thisbracket operation onvector fields isantisymmetric andsatisfies theJacobi identity [{x,Y],z]+[[Y,z],X]+[[2,X],Y]=0. (46.5) Smooth vector fields form amodule (see Appendix A)over °J’(M), and hence avector space over 1Bidentified with theconstant functions. The commutator then turns thevector fields into an(infinite-dimensional) Liealgebra. Thecommutator isalsocalled theLiebracket. VECTOR FIELDS 143 If f: M N is a smooth map between manifolds then, for any p E M, the tangent map f*p sends TM to Tf(p)/s/. If X and Y are smooth vector fields on M and N respectively, with Xi,. and Y), given by (4.6.2), such that Yfip)-= f*pXp Vp E M (4.6.6) then X and Y are said to be f-related. We will often simply write Y =-- f*X. This notation does not imply that any smooth map f: M ---> N enables a smooth vector field on M to be mapped to one on N. If f is not one to one, with f(p) = f(q) say, then for an arbitrary X, f„pX * f*0X. If f is not onto then smooth vector fields on N that are f-related to X c Tl(M) can differ outside the image of f. An important example is that of a smooth curve C / —> M. Different smooth vector fields on M can be tangent to all the points on the image of C. For the special case in which f is a diffeomorphism for every X e Ti(M) there is a unique Y E TI(N) such that Y = f*X. As we noted in the previous section it is common not to distinguish Xp E TM from its coordinate representation c io*),X. Thus if (U, cp) is a chart for the neighbourhood of p, with coordinate functions {xl, one identifies {(3/axi)1 p} with a basis for TM. If X E T(M) then in the neighbourhood of p we can express X as X = X1(3/3x'), where X' e 5",(M) are not distinguished from their representations in this chart. The elements (a/axt) form a basis for 7-1(U), the Y ,-module of smooth vector fields on U. They form the natural local basis or local coordinate basis. Since the ring of smooth functions is not a division ring there is no reason why the .5,-module TI(M) should have a basis, and in general it will not have. This is because for a general manifold there are no vector fields that do not vanish somewhere. (The two-sphere, for example, is such a manifold.) 4.7 The Tangent Bundle One way of formalising the way a vector field on an n-dimensional manifold M assigns a tangent vector to each point is to construct a new 2n-dimensional manifold TM by collecting together all the tangent spaces TM from all points of M: TM = U TM. (4.7.1) An element of TM is a tangent vector Xp, labelled by the point p and VECTOR FIELDS 143 Iff:M-—>N isasmooth map between manifolds then, forany peM, thetangent map ftpsends TPM toTm,,N. IfXand Yare smooth vector fields onMandNrespectively, with XpandYpgiven by (4.6.2), such that Yflp) =f*pXp GM then Xand Yaresaid tobef-related. Wewill often simply write Y=f*X. This notation does notimply thatanysmooth map f:M->N enables asmooth vector field onMtobemapped tooneonN.Iffis not one toone, with f(p) =f(q) say, then foranarbitrary X, f,pX ¢f*,,X. Iffisnotonto then smooth vector fields onNthat are f-related toXeT1(M) candiffer outside theimage off.Animportant example isthat ofasmooth curve C:I-—>M.Different smooth vector fields onMcanbetangent toallthepoints ontheimage ofC.Forthe special case inwhich fisadiffeomorphism forevery XeT1(M) there is aunique YeT‘(N) such that Y=f*X. Aswenoted intheprevious section itiscommon nottodistinguish XpeTPM from itscoordinate representation cp*,,X. Thus if(U,cp)isa chart fortheneighbourhood ofp,with coordinate functions {xi}, one identifies {(8/8x")[,,} with abasis forTPM. IfXeT‘(M) then inthe neighbourhood ofpwecan express XasX=X"(8/Bx’), where X‘e@(M) arenotdistinguished from their representations inthischart. The elements (8/8x‘) form abasis forT1(U), the97-module ofsmooth vector fields onU.They form thenatural local basis orlocal coordinate basis. Since thering ofsmooth functions isnotadivision ring there is noreason why the9-module T‘(M) should have abasis, andingeneral itwill nothave. This isbecause forageneral manifold there areno vector fields that donot vanish somewhere. (The two-sphere, for example, issuch amanifold.) 4.7TheTangent Bundle One way offormalising theway avector field onann-dimensional manifold Massigns atangent vector toeach point istoconstruct anew 2n-dimensional manifold TM bycollecting together allthetangent spaces T,,M from allpoints ofM: TM=UTPM. (4.7.1) P Anelement ofTMisatangent vector Xp,labelled bythepoint pand 144 MANIFOLDS its components in some basis for TM. Moreover the construction of TM must satisfy certain smoothness criteria with respect to these assignments. If a tangent vector Xi, E TM is represented in a local chart (UM, cpm), with coordinate maps (xj), by cpXp = yia/ax) then we define (x'(p), yl(p)) E 112" as the coordinates of a point in TM. That is, the chart (U m, çom) for M induces a chart (UTM, ÇOTM) for TM by (cPrm)(Xp) = (xi(P), Yl(P)) where cpm(p) = xt(p)e, and (.99m)Xp = yl(p)(alax i) i(97.)(p), {ei) being the natural basis for En. As we have remarked earlier a tangent vector to IR" is equivalent to an element of IF12": the derivative in the direction V at p being equivalent to (p, V). Thus PTM assigns to Xp the element of IR2" equivalent to (cpm).pXp E Tcp(p)IFIn Since M is a differentiable manifold it is possible to give a topology and differentiable manifold structure to TM. If (UTM, ÇOTM) is a local chart for TM, induced by (Um, cpm), then the map specifying a change of coordinates in TM: (çcTM) ° (T-A4): Fi2n E2n is given in terms of the map specifying a change of coordinates on M (41)2 0 (TV), :Rn ___, En x'(p) x"(p). The tangent map is ((cPm)2 (TV))1*((pm),(p): T(cpoi(p)E n T(9N)2(p)E n yka/axk yk((ax,i/axk))a/ax,i (where we are using summation convention) so that ((407-m)2° Y1)(q) = (x"(P), Y k(P)(af Vax k)(P)) (4.7.2) These maps define (see figure 4.8) a diffeomorphism (cp 7-m) 12 of (Trit4)1«uTm)1 n (U1-A4)2) onto (T.Tm)2((uTm)1 n (urm)2). 144 MANIFOLDS itscomponents insome basis forT,,M. Moreover theconstruction of TM must satisfy certain smoothness criteria with respect tothese assignments. Ifatangent vector XpeTPM isrepresented inalocal chart (UM. <;0M), with coordinate maps (x’), by<;0*,,X,, =y/8/8x/' then we define (x'(p), y/f(p)) eIRE”asthecoordinates ofapoint inTM. That is, thechart (UM, <;0M) forMinduces achart (UTM, <;0TM) forTMby (<t>m)(X,.) E(f(p), f(p)) Where ‘PM(P) =/\*"(P)et and(‘PM)*pXp =Yi(P)(3/ax’) i(¢,,,)(p)» {er} being thenatural basis forIR”.Aswehave remarked earlier atangent vector toIR"isequivalent toanelement of1R2": thederivative inthedirection Vatpbeing equivalent to(p,V).Thus <;0TM assigns toXptheelement ofIRZ"equivalent to(<pM),,,,X,, eT,,,(,,)lR". Since Misadifferentiable manifold itispossible togive atopology anddifferentiable manifold structure toTM. If(UTM, <;0TM) isalocal chart forTM, induced by(UM, <;0M), then themap specifying achange ofcoordinates inTM: (<;0TM)2 0(<;0}M)1 :1R2"—>1R2", isgiven interms of themap specifying achange ofcoordinates onM (wt);O(§01I41)111B"—>IR" f(p)*——>X"(P)~ Thetangent map is ((90/wlz °(‘P-i1'))1*<¢M).<p) YT(¢M)I(P)B" t’T(¢M)Z(P)B" y"8/Sxk l——> y"((8x"'/8x"))8/8x". (where weareusing summation convention) sothat ((§0T.M)2 Q((pTli/f)1)(‘xi? y’)(q) =(X"'(P), y"(P)(@X”/@x*)(p))- (4~7-2) These maps define (seefigure 4.8)adiffeomorphism (<;0TM),2 of (QOTM) |((Um)1 n(UTM)2) onto (‘PTM)2((UTM)| n(UrM)2)- 4|):g.ll' wt'67 Mrmh l'*Pml2 w7H)1Z Figure 4.8 THE TANGENT BUNDLE 145 From its construction U Tm is diffeomorphic to Um x IFin, but globally TM need not be a product manifold. A product manifold M x N is formed from ordered pairs of elements from the manifolds M and N. If {(Ua, (pa)} and {(Vb, vb)} are atlases for M and N respectively then an atlas for M x N is defined by the collection of charts (PaX 14 Ua x V b Fichm M +dim N 9) 1--* (9) a(P), 6(0). Such a collection of maps satisfies the criteria for being an atlas. The local product structure of TM allows the definition of a natural projection map : TM M, X,, p (4.7.3) which identifies the point on M to which the tangent vector in TM is attached. It is convenient to picture UTm, with its local product structure exposed, as a space over Um (see figure 4.9). All the tangent vectors at p are drawn as the space TM associated by the projection H to a point p of M. The local coordinate representative of n is usually given the same name, n E 2n E n xl (The inverse image set TM is sometimes denoted II -I(p) and U Tm denoted H -1(Um) although this notation should not be confused with the notion of an inverse map!). lp,X) t H IR2" TM u, Figure 4.9 The local product structure of the tangent bundle. The existence of a projection map makes TM into a fibred space, the elements related to p by II being the fibre over p. The manifold TM together with n is called the tangent bundle of M. We have here an example of a fibre bundle. Although in all fibre bundles the fibre spaces are fused together by giving the bundle the structure of a product manifold locally, bundles with different global topologies can be con- structed by relating fibres in overlapping neighbourhoods (UTm), n (uT,02 in different ways. This is like the difference between a cylindrical ribbon with a twist and one without a twist. In both cases the twist can be eliminated from any neighbourhood but is an essential characteristic distinguishing one ribbon from the other. THE TANGENT BUNDLE 145 From itsconstruction UTM isdiffeomorphic toUM><IR”, butglobally TMneed notbeaproduct manifold. Aproduct manifold M><Nis formed from ordered pairs ofelements from themanifolds MandN.If {(U,, r,0,,)} and{(V,,, 1p,,)} areatlases forMandNrespectively thenan atlas forM><Nisdefined bythecollection ofcharts <0.><wirUa><Vt——>1R‘ii"""*‘ii"‘” (1>,q)%>(¢..(1>), wt(q))- Such acollection ofmaps satisfies thecriteria forbeing anatlas. The local product structure ofTM allows the definition ofanatural projection map lI:TM——>M,Xpli>p (4.7.3) which identifies thepoint onMtowhich thetangent vector inTMis attached. Itisconvenient topicture UTM, with itslocal product structure exposed, asaspace over UM(seefigure 4.9). Allthetangent vectors at paredrawn asthespace TPM associated bytheprojection IItoapoint pofM.The local coordinate representative ofIIisusually given the same name, II:1R2"—>IR",(xi,yi)+—->xi.(The inverse image setTPM is sometimes denoted II'i(p) and UTM denoted II‘i(UM) although this notation should notbeconfused with thenotion ofaninverse mapl). T-PM U1-M : [R217 ‘Pm x,yl QK se-4-=2 '16I1 ><Q4--:.. Figure 4.9Thelocal product structure ofthetangent bundle. Theexistence ofaprojection mapmakes TMintoafibred space, the elements related topbyIIbeing thefibre over p.The manifold TM together with IIiscalled thetangent bundle ofM.Wehave here an example ofafibre bundle. Although inallfibre bundles thefibre spaces arefused together bygiving thebundle thestructure ofaproduct manifold locally, bundles with different global topologies canbecon- structed by relating fibres in overlapping neighbourhoods (UTM), F‘:(UTM)2 indifferent ways. This islikethedifference between a cylindrical ribbon withatwist andonewithout atwist. Inboth cases the twist canbeeliminated from anyneighbourhood butisanessential characteristic distinguishing oneribbon from theother. 146 MANIFOLDS A smooth section of TM is a Cc° map a: M TM (4.7.4) such that H o a = (id)m. Thus a(p) E TM for all p E M . It may be represented in local charts (UTM, rioni),(Um, qoki) by à(x) = (x`, y` a'(x)) (4.7.5) where the {a') are real functions on Ujir (see figure 4. 10). Thus a smoothly assigns a tangent vector to each point p E M. We may identify a smooth section a with a smooth vector field X by (X.f)(P) = a(P)f V f E (4.7.6) In this way every smooth vector field on M is equivalent to a smooth section of TM. If FTM is the space of smooth sections of TM we will henceforth use the above to identify Tl(M) with FTM. (p, X) y) f- LP TM a u, IR" Figure 4.10 A local section and its representation. 4.8 Differential 1-Forms The smooth vector fields on M form a module over the commutative ring of smooth functions, and hence inherit a vector space structure over Ili identified with the constant functions. We shall frequently need to distinguish maps that are linear with respect to the module structure from those that are only linear with respect to this vector space structure. Thus we refer to maps as being (M)-linear (or more simply Fi-linear) or JR-linear. A 1-form field (or 1-form on M) is an element of the module dual to Tl(M); that is, an ,9;-valued 9;-linear map on vector fields. A 1-form is smooth if it maps smooth vectors to smooth 146 MANIFOLDS Asmooth section ofTMisaC°°map 0:Mi> TM (4.7.4) such that II<>0= (id)M. Thus 0(p)e TPM forallpeM. Itmay be represented inlocal charts (UTM, (pTM),(UM, (pM) by 6(x) =(xi,yiE0i(x)) (4.7.5) where the{oi} arereal functions onUR» (see figure 4.10). Thus 0 smoothly assigns atangent vector toeach point peM.Wemayidentify asmooth section 0with asmooth vector field Xby (Xf)(P) =U(P)f VféWM) (4-7-6) Inthisway every smooth vector field onMisequivalent toasmooth section ofTM. IfFTM isthespace ofsmooth sections ofTMwewill henceforth usetheabove toidentify Ti(M) withFTM. Um iP'Xi ‘PTM (x,y) ________.__ ____-__---- -i-1>—ii1 K4 U U U /‘I M 1 D IR 1 [J X Figure 4.10 Alocal section anditsrepresentation. 4.8Differential l-Forms Thesmooth vector fields onMform amodule over thecommutative ringofsmooth functions, andhence inherit avector space structure over IRidentified with theconstant functions. Weshall frequently need to distinguish maps that arelinear with respect tothemodule structure from those that are only linear with respect tothis vector space structure. Thus werefer tomaps asbeing §'(M)-linear (ormore simply @-linear) orIR-linear. A1-form field (or1-form onM)isanelement of themodule dual toTi(M); thatis,an§-valued §—linear map onvector fields. A1-form issmooth ifitmaps smooth vectors tosmooth DIFFERENTIAL 1-FORMS 147 functions. A smooth 1-form on M will also be called a differential 1-form. The space of smooth 1-forms on M is denoted Ti(M). If X E TI(M) assigns Xp E TM to the point p then for we Ti (W) we define cop by (0)(x))(p) = cop(xp)- (4.8.1) Clearly co p is a linear map from TM to E, that is, an element of the dual space TM. Elements of T*pM are called co-vectors or 1-forms at p. Thus co smoothly assigns an element of TM to every point p of M. In analogy to the construction of TM we may collect together all the cotangent spaces and form a new space T*M = U T*pM (4.8.2) Like TM the space T*M inherits a manifold structure from that of M, with a natural projection from T*M to M. With this structure T*M becomes the cotangent bundle. We may identify a smooth 1-form on M with a smooth section of T*M. So if FT*M is the space of smooth sections we have a natural equivalence between elements of FT*M and T1 (M). For every f E 5-,(M) we may associate an element df E TI(M) by the rule X(f) = (df)(X) V X E Tl(M). (4.8.3) That is, df E FT*M assigns (df) p E T*pM to the point p with Xp(f)= (df) p(Xp). (4.8.4) The element (dfl p which maps TM to E is related to f*p which maps TM to Tf(p)11:1 : in fact they are naturally isomorphic. If g is a real-valued function on E, A1-4g(A), then from (4.5.4) (f*pXp)(g) = X p(g f). By the chain rule dg Xp(g f) = X(f) Thus f*pXp E Tf(p)F1 is equivalent to the ordered pair (f(P), Xpf) = (AP), (df)p(Xp)). The existence and linearity of f*p ensures that (4.8.3) really does define a 1-form. Despite this natural isomorphism we shall distinguish the maps (df)p and f*p. If x( is one of the coordinate functions and (3/3x0 is a vector from the natural local basis then (4.8.3) gives dx1(3/3.0 = (ax/ax') = (51,. (4.8.5) DIFFERENTIAL 1-FORMS 147 functions. Asmooth 1-form onMwill also becalled adifferential 1-form. The space ofsmooth 1-forms onMisdenoted T,(M). If X§Ti(M) assigns Xpe TPM tothepoint pthen forweTl(M) we define cupby (w(X))(P) =wp(Xp)‘ (4-8-1) Clearly cupisalinear map from TPM toIR,that is,anelement ofthe dual space T’§,M. Elements ofT";,M arecalled co-vectors or1-forms atp.Thus tosmoothly assigns anelement ofT*;,M toevery point pof M.Inanalogy totheconstruction ofTMwemay collect together allthe cotangent spaces andform anewspace T*M=oT=;,M (4.s.2) Like TMthespace T*M inherits amanifold structure from that ofM, with anatural projection from T*M toM.With thisstructure T*M becomes thecotangent bundle. Wemayidentify asmooth 1-form onM with asmooth section ofT*M. SoifFT*M isthespace ofsmooth sections wehave anatural equivalence between elements ofFT*M and T1(M). Forevery fe§’(M) wemay associate anelement dfeTl(M) bythe rule X(f) =(df)(X) VXe Ti(M). (4.8.3) That is,dfeFT*M assigns (df),, eT";,M tothepoint pwith X,.<f>=<<1f>,.<X,.>- <4-84> Theelement (df)P which maps TPM toIRisrelated tof*,,which maps TPM toTfg,)lB :infact they are naturally isomorphic. Ifgisa real-valued function onIR,/II-—>g(/I), thenfrom (4.5.4) (f*pXp)(g) :Xp(g Bythechain rule dgXp(g°f)=Xp(f) E(f(P))- Thus f,,,,X,, eTfg,)IB isequivalent totheordered pair (f(p), Xpf)=(f(p), (df)p(Xp))‘ Theexistence andlinearity offij,ensures that(4.8.3) really does define a1-form. Despite thisnatural isomorphism weshall distinguish themaps (df),, andfrp. _ Ifxiisoneofthecoordinate functions and(8/8x/) isavector from thenatural local basis then (4.8.3) gives dxi(8/8x/i) =(8xi/8x/i) = (4.8.5) 148 MANIFOLDS Thus {dx1} is a local basis for Ti(M) naturally dual to the basis {(3/ax')}. In some coordinate neighbourhood, for any f EFfe(M), df can be expanded in a local basis df = df(a/a.V)dx 1, giving the classical expression df = (3fl3xi)dx 1 (4.8.6) from (4.8.3). It is worth emphasising that in this expression the chi are not 'infinitesimal increments of the coordinates' but linear mappings on the tangent vectors. By evaluating this expression on a vector tangent to some curve we obtain the derivative of f along the curve: in this way df encodes the way in which the value of f changes as the point in M begins to move. The components of a 1-form with respect to the natural basis {dx`}, associated with the chart (Um, cp), are used to coordinate the bundle T*M. If a c Ti(M) with a = a„ c1.0, a, c .Ti(M), then cr is associated with the smooth section p p:M ---> T*M represented in a local chart by OP) (x(P), a'M(P)). If f: M N is a smooth map then we have already defined the pull-back map f* that takes a smooth function g on N to a smooth function pg on M, f*g = g o f. Thus rg is evaluated at p by using f to send p from M to N where it is evaluated with g. In the same spirit we can define the pull-back of a 1-form co on N to a 1-form rco on M. If XP E TPM we define (f*(n)pXp = cof(p)(f"Xp). (4.8.7) We need to check that for a smooth assignment of Xp to TM and a smooth co on N this rule assigns (f*co)p smoothly to T*pM. This can be seen from the local coordinate expression for (4.8.7). Firstly we note that for g E 3-,(M) (f*(gco))pXp = (gw)f(p)(f*pXp) = (g o f)(p)wf(p)(f.pXp) = (f*g)(p)(f*(n),,X p thus f*(gw) = (f*g)(f*co). (4.8.8) If {x'} i = 1, . . m and {y} j = 1, . . n are local coordinates for M and N such that the coordinate representation of f is given by yi = f(x'), then if X = X1(alax') f„pXp = X1(p)(afilax9(p)(3/3_01Rp) 148 MANIFOLDS Thus {dxi} isalocal basis forT1(M) naturally dual tothebasis {(8/8x')}. Insome coordinate neighbourhood, foranyfe@(M), dfcan beexpanded inalocal basis df= df(8/8xi)dxi_ giving theclassical expression df=(Sf/8xi)dxi (48.6) from (48.3). Itisworth emphasising that inthisexpression thedxiare not‘infinitesimal increments ofthecoordinates’ butlinear mappings on thetangent vectors. Byevaluating thisexpression onavector tangent to some curve weobtain thederivative offalong thecurve: inthiswaydf encodes theway inwhich thevalue offchanges asthepoint inM begins tomove. The components ofa1-form with respect tothenatural basis {dxi}, associated with thechart (UM, cp),areused tocoordinate thebundle T*M. IfareT1(M) with a=ozgdx”, age@(M), then tris associated with thesmooth section p p:M——> T*M represented inalocal chart by Xi‘(P)*>(Xi‘(P)» v/,t(p))- Iff:M—> Nisasmooth map then wehave already defined the pull-back map f*that takes asmooth function gonNtoasmooth function figonM,fig=gof.Thus figisevaluated atpbyusing fto send pfrom MtoNwhere itisevaluated with g.Inthesame spirit we candefine thepull-back ofa1-form wonNtoa1-form fiw onM.If XpeTPM wedefine (f"‘w),,X,, =wf(,,,(f,,,X,,). (4.8.7) Weneed tocheck that forasmooth assignment ofXptoT,,M anda smooth wonNthisrule assigns (f’iw),, smoothly toT’j,M. This canbe seen from thelocal coordinate expression for(4.8.7). Firstly wenote thatforge‘J*(M) U*(gw))pXp :(gu))f(P)(f*pX[J) = °f)(p)wf(p)(f*pXp) = thus fi(gw) =(fig)(f*w)- (4-8-8) If{xi} i=1,....mand{yl}j=1,....narelocal coordinates forM and Nsuch that the coordinate representation offisgiven by yi=f7(xi). then ifX=Xi(8/8xi) f*,,X,, =Xi(P)(@fi/@Xi)(1>)(@/@yi)|r<p> DIFFERENTIAL 1-FORMS 149 and for dx] E 17()N dXi(f*pXp) = 10(p)(3Plaxi)(p) = (afi lax')(p) dx 1(Xp). It follows from (4.8.8) that if co = w dx/ rco = (col f)(3P13.,V) dxf (4.8.9) and the smoothness of f and the component functions coi ensure that rco is smooth. If WE T*pM then we can use any chart for the neighbourhood of p to represent co, using the natural local basis. Given two different charts we can compare the representations of co by using the pull-back of the map that relates the charts. Suppose (II cp) is a chart for the neighbour- hood of p with q(U) = Ul. Given a diffeomorphism : U1(Fin) --* U2(1P) we have a new chart (UM, p o cp). If tp is specified by p: U1(11") --> U 2(111") xP x'P = ip(x', . . x") then we have the inverse map U1 (R) P XP = 111-1P(X1 1 , . . . , n). In §4.5 we showed that if X E TM is represented in the (Um, (p) chart by = Xv(a/axv)1 q,(p) then the representative in (Um, 4'o cp) is (tp cp)X = Xv(4 013xv)(cp(p))(alax'f 1)!()(p). When representing a 1-form we have to remember that the pull-back map acts in the opposite direction to the map itself. Since chart maps are invertible co E TM is represented in (U cp) by cpT-(p1 Tw, with cpco = w, dxvi T(p) say. In the chart (Um, 4'0 cp) the representation of co is Op. cp)(7;,10;)(p) co, where ° (p)(uLT*)(p)a) = 114::)(p)Vo lpi (I) = /P(plogt)(p)(ovdxv = co voip- = w dfml (4,)(p) from (4.8.9). Thus whereas the components of the tangent vector X are transformed with the Jacobian matrix representing ip, the components of the 1-form co transform with the inverse matrix since DIFFERENTIAL 1-FORMS 149 andfordxieT*)g,)N dx1<r..X,.> =X'<p><~=>f//~=>x'><p> =(eff/ex*><p> d»<‘<X,.>- Itfollows from (4.8.8) thatifw=at,dx/I fiw =((1)!-°f)(5fi/5Xi)dXi (4.8.9) andthesmoothness offandthecomponent functions wjensure that fiwissmooth. IfweT’§,M then wecanuseanychart fortheneighbourhood ofpto represent w,using thenatural local basis. Given twodifferent charts we cancompare therepresentations ofwbyusing thepull-back ofthemap that relates thecharts. Suppose (UM, (p)isachart fortheneighbour- hood ofpwith (p(UM) =U1.Given adiffeomorphism w:U1(lB”)_> U2(lR”) wehave anewchart (UM, 1/1<>ip).If1/1isspecified by wll/1(1R")—> Uz(1R”) xl‘i—->x’“ =1p(xi, ...,x”) then wehave theinverse map w"1U1(1R")—> U103”) x'“t——>x“ =1/1‘ii‘(x’i, ...,x'”). In§4.5 weshowed that ifXeTPM isrepresented inthe(UM, (p)chart by (P*pX : then therepresentative in(UM, we(p)is (w°<P)*,,X =X"(9wi‘/9X”)(<P(P))(9/5X'i‘)l(Mtg)- When representing a1-form wehave toremember that thepull-back map acts intheopposite direction tothemap itself. Since chart maps areinvertible weT‘§,M isrepresented in(UM, (p)by(p;,(i,,‘,iw, with ‘P$<i=i‘" =“’vdxvi¢<p> say. Inthe chart (UM, wemp)the representation ofatis (‘V°(lii)(_1l1l6:>)(P) ‘"»Where (‘V°‘P)(_vi»$)<p>“’ =iii(It/i»;)(p)iii<;l1Pi; ii’=ll’<_t»i§)(p>“’vd"v =wv(@w*‘”/@x'”)((v °<P)(P))dX'“l(¢.6>(p> =witdxi”|<w)<p> from (4.8.9). Thus whereas thecomponents ofthetangent vector Xare transformed with theJacobian matrix representing zp,thecomponents of the1-form wtransform with theinverse matrix since 150 MANIFOLDS (a tpo/ax a")(atir = That is, the components of a) transform contragradiently to those of X. The behaviour of the change in the components of ai induced by changing the coordinate basis is the historical characterisation of a covariant vector (see figure 4.11). wv=(!7) iR Figure 4.11 Different representations of a covariant vector field on Um. 4.9 Tensor Fields In Chapter 1 we introduced the tensor algebra associated with an arbitrary vector space. We may now apply this to the particular case when that vector space is the cotangent space at any point of a manifold. Thus elements of Ps(rpM) are called tensor fields at p of covariant degree r and contravariant degree s. It is purely for convenience that we have selected the cotangent space rather than the tangent space, the notation of Chapter 1 having been chosen such that taking the arbitrary vector space V to be TM gives the conventional labelling for mixed tensors. It is for this reason that it was convenient in Chapter I to think of elements of V as acting on V* rather than the other way around. Clearly we have T(TM) = T sr (T pM). Whereas the cotangent space at any point is a real vector space the set of 1-form fields forms an Fi-module. In the same way as we constructed the tensor product of vector spaces we may construct the tensor product of the .9;-module of 1-form fields with itself and the dual module of 150 MANIFOLDS (S1/1“/8x“)(81p‘i“/6x"') =0:. That is.thecomponents oftotransform contragradiently tothose ofX_ The behaviour ofthechange inthecomponents ofwinduced by changing thecoordinate basis isthehistorical characterisation ofa covariant vector (seefigure 4.11). UM U1 ‘P \lJo‘P U2 ‘l~\Q\//J E F \\\>\/Q I i _6 I_6°’"‘“’(wl “V-“(ml IR Figure 4.11 Different representations ofacovariant vector field onUM. 4.9Tensor Fields||> InChapter 1weintroduced thetensor algebra associated with an arbitrary vector space. Wemay now apply thistotheparticular case when that vector space isthecotangent space atany point ofa manifold. Thus elements ofTf(T“§,M) arecalled tensor fields atpof covariant degree randcontravariant degree s. 1tispurely forconvenience thatwehave selected thecotangent space rather than thetangent space, thenotation ofChapter 1having been chosen such thattaking thearbitrary vector space VtobeT‘;,M gives the conventional labelling formixed tensors. Itisforthisreason thatitwas convenient inChapter 1tothink ofelements ofVasacting onV*rather than tneother wayaround. Clearly wehave T§(T’;,M) =T§(T,,M). Whereas thecotangent space atanypoint isarealvector space theset of1-form fields forms an@-module. Inthesame way asweconstructed thetensor product ofvector spaces wemay construct thetensor product ofthe@-module of1-form fields with itself and thedual module of TENSOR FIELDS 151 smooth vector fields. Elements of the tensor product module, P(M), are called tensor fields of covariant degree r and contravariant degree s. We identify Tg(M) with F.4,(M). Thus an element of T(M) smoothly assigns an element of T(TM) to each point p in M. As for the case of vector and 1-form fields this way of regarding tensor fields is formalised in terms of a fibre bundle, the bundle of mixed tensors TM. Thus TM = U p Ts,.(T*pM), with a coordinate system induced from that of M. The natural projection of the bundle maps each tensor field to the point in M at which it is attached. Smooth sections can be defined in an obvious way, allowing the identification of the set of smooth tensor fields T(M) with the space of smooth sections FP,M If (U, T) is a chart for some neighbourhood of M, with chart maps fx"), then {(3/3.0} and {dzi} are bases for Tl(U) and Ti(U) respec- tively. Thus locally any tensor field T c T(M) can be written as T =  hcixil 0 clx` , 0 . . . 0 dx', O (a/axii) 0 (a/axJ2) 0 . . . 0 (a/axh). (4.9.1) This is just a formula from §1.5 rewritten with &VI replacing e'. and (a/axh) replacing X1i. The summation convention is employed. The indices are staggered in anticipation of the introduction of a metric tensor field when we shall use the raising and lowering conventions introduced in Chapter 1. Whereas one can always use a local coordinate basis in which to expand tensor fields such a basis is not always the most convenient. In particular, when we have a metric tensor it is often useful to employ a suitably adapted basis. The submodule of T(M) formed by all totally antisymmetric covar- iant tensor fields forms the exterior algebra of differential forms, A(M), under the exterior product of (1.2.2). We shall identify A0(M) with g;(M). Thus a smooth differential form is associated with a smooth section of the exterior bundle AM -= Up A(T*pM). Whereas an element of the exterior algebra of an arbitrary vector space is called an exterior form, the term differential form is reserved for an element of the ; -module A(M). If /3e fArM, section of the bundle of exterior r-forms we may use a local coordinate basis to write = E ,(dx Pi) A (del A . . A (c1x14) (4.9.2) 1 „ fi = ToPp,p, ..x(c1x11') A (dx"2) A  . A (dX4') equivalently where the summation convention is used. These formulae are trans- cribed from §1.2 with the substitution of dx . for el`. TENSOR FIELDS 151 Smooth vector fields. Elements ofthetensor product module, Tf(M), arecalled tensor fields ofcovariant degree randcontravariant degree s. Weidentify T3(M) with ‘J(M). Thus anelement ofTf(M) smoothly assigns anelement ofTf(Tj,M) toeach point pinM.Asforthecaseof vector and1-form fields thisway ofregarding tensor fields isformalised interms ofafibre bundle, thebundle ofmixed tensors TjM. Thus T§M =L1,,T§(T‘§,M), with acoordinate system induced from thatofM. Thenatural projection ofthebundle maps each tensor field tothepoint inMatwhich itisattached. Smooth sections canbedefined inan obvious way, allowing theidentification ofthesetofsmooth tensor fields Tf(M) with thespace ofsmooth sections l"TjM. If(U,qt)isachart forsome neighbourhood ofM,with chart maps {xi}, then {(8/8xi)} and {dxi} arebases forTi(U) and T1(U) respec- tively. Thus locally anytensor field TeT§(M) canbewritten as T=T,,,2_ I_,,lt.Ii2- -~f:dxil ® dxiz ® ®dxi'®(a/ax/1) ®(a/aw) ® ®(a/ext). (4.9.1) This isjust aformula from §1.5 rewritten with dxil replacing ei'and (8/8x/i‘) replacing X,-,.The summation convention isemployed. The indices arestaggered inanticipation oftheintroduction ofametric tensor field when weshall usetheraising and lowering conventions introduced inChapter 1.Whereas onecanalways usealocal coordinate basis inwhich toexpand tensor fields such abasis isnotalways themost convenient. Inparticular, when wehave ametric tensor itisoften useful toemploy asuitably adapted basis. Thesubmodule ofT§(M) formed byalltotally antisymmetric covar- ianttensor fields forms theexterior algebra ofdifferential forms, A(M), under theexterior product of(1.2.2). Weshall identify Ag(M) with @(M). Thus asmooth differential form isassociated with asmooth section oftheexterior bundle AM=L1,,A(T’§,M). Whereas anelement oftheexterior algebra ofanarbitrary vector space iscalled anexterior form, theterm differential form isreserved foranelement ofthe 9-module A(M). If,6eFA,M, section ofthebundle ofexterior r-forms wemayusealocal coordinate basis towrite /t=2/i..,.....,.,<dx"i)x<dxe>x---A(M) (4.92) #1$u2$- --Fr _ equivalently B=..,1,(dXi“)/\(dXi")/\- .-/\(am where thesummation convention isused. These formulae aretrans- cribed from §1.2withthesubstitution ofdxi“fore"". 152 MANIFOLDS Given a smooth map f between two manifolds the induced maps on the tangent and cotangent spaces can, to some extent, be extended to tensor fields. 1ff:M-->N then we extend the map f*p to an II-linear map on contravariant tensors at p f*p:Ts(r;,M)—> Ts(T )N) Xi X20 ... Xs f*pXi ®fX 2 0 ... f *pX, Xe TM. (4.9.3) As for the case of vector fields we cannot in general use this map to obtain a smooth tensor field on N from one on M. We have, of course, the obvious generalisation to f-related contravariant tensor fields. The smooth map f does, however, give rise to a map p which enables smooth 1-form fields on N to be pulled back to smooth 1-forms on M. This pull-back map may be extended to an 1R-linear map on smooth covariant tensors on N : Tr(N) --> Tr(M) col (02 . 0 cor f*(02 0 f*wr co' E TI(N). (4.9.4) For such a definition to make sense it is important that we have (4.8.8), that is f*(gco) = (rg)(rco) g E E Ti(N). For E Tr(N) and {Xi} E TM I = 1, . . r we have (N)p(Xi, X2, . Xr) = 1fip)(f*pX1, f*pX2, . . In general the smooth map f: M —> N does not induce a map on smooth contravariant tensor fields on M; nor on mixed tensor fields, the maps ft,, and f*p acting in opposite directions. For the special case of a diffeomorphism, however, there is an induced map on smooth vector fields as was noted in §4.6, and the problem of the maps acting in different directions is readily overcome since diffeomorphisms are in- vertible. If op : M —> N is a diffeomorphism then we define Cp by Cp:P*(M)--> Pr(N) _ cp-i*(01 cp--1*(02 0 0 r W cp*Xi 0 . . . Ocp*X, W' E Ti(M), X E Ti(M). (4.9.5) Again we require (4.8.8) for consistency. Equivalently 152 MANIFOLDS Given asmooth map fbetween twomanifolds theinduced maps on thetangent andcotangent spaces can, tosome extent, beextended to tensor fields. Iff:M—> Nthen weextend themap f..,,toan1P1-linear map oncontravariant tensors atp f.,,;T‘(T";,M) —>T‘(T’}g,,N) X,®X2®...®X, +—>f.,,X, ®f,.,,X2®...®f.,,X, X,ET,,M. (49.3) Asforthecase ofvector fields wecannot ingeneral usethismap to obtain asmooth tensor field onNfrom oneonM.Wehave, ofcourse, theobvious generalisation tof-related contravariant tensor fields. The smooth map fdoes, however, give rise toamap fiwhich enables smooth I-form fields onNtobepulled back tosmooth 1-forms onM. This pull-back map may beextended toanIR-linear map onsmooth covariant tensors onN f*1Tr(N)‘> TAM) wi®w2®...®w’t;> fiwi®f*wi®...®fiw’ wieT,(N). (4.9.4) Forsuch adefinition tomake sense itisimportant thatwehave (4.8.8), thatis J°i(8w) =(f*8)(fiw) 86~"7'(N), weT1(N)- Forfie T,(N) and{X,~} eT,,M i=1,...,rwehave (f*B)p(Xlv X2~ "'"IX1) :fif(p)(.f*pXl1f*pX21 '''vf*pXr)' Ingeneral thesmooth map f:M—> Ndoes notinduce amap on smooth contravariant tensor fields onM;noronmixed tensor fields, the maps f,,,,andff,acting inopposite directions. Forthespecial caseofa diffeomorphism, however, there isaninduced map onsmooth vector fields aswas noted in§4.6, and theproblem ofthemaps acting in different directions isreadily overcome since diffeomorphisms arein- vertible. Ifmp1M—>Nisadiffeomorphism thenwedefine tiby <?>1Ti(M)—> Ti(N) <i>(wi®wi®...®w’®X1®...®X,) =1p‘i*wi® qfi*w2 ®...®1p'i*w’ ®1p..X, ®...®q>*X, wieT,(M), X,eTi(M). (4.9.5) Again werequire (4.8.8) forconsistency. Equivalently TENSOR FIELDS 153 (k( w ' (02 0 . wr Xi - 0 Xs)(171, Y2, -  , yr, as) col 0 co2 0 ... cor 0 X I 0 ... 0 X,(40-1*Y1, cr l*Y2,  - (12-1*11,- cral,   92* oes). Example 4.2 For the smooth map R2 R2 P (p(p) (4.9.6) (a, b) OP), 922(P)) = (a cos t + b sin t, b cos t - a sin t) with t a constant, the inverse is given by ço-i: R2 F--> T-1(p) (a, b) ((ço-1)1(p), (cp-1)2(p)) = (a cos t - b sin t , b cos t + a sin t). For a vector field Y, CpY = cp*Y . Taking Y = x2(3/3x) + xy(3/3y), with x and y the standard coordinates on 1112, gives p Yp = x2(p){(41/3x)(p)(3/3x)1 9,(p) + (3922/8x)(p)(3/3y)1 9,(p)) + x(p).Y(P){( 3491/aY)(P)(3/3x)Lp(p) + (aT 2/3Y)(3)(3/aY)19,0)} (40*N(p) (x 2 COS t xy sin t)(p)(alax)S TC, ) (xy cos t - x2 sin t)(p)(3/3y)l q,o,) We may use cp-1 to express the coordinates of p in terms of those of p(p), giving (cp*Y)op) = (x2 cos t - xy sin t)(cp(p))(alax)1 9,(p) + (xy cos t - y2 sin t)(cp(p))( 3 /3y)1 go). SO cp*Y = (x 2 COS t - xy sin t)(3/3x) + (xy cos t - y2 sin t)(3/3y). We consider now a 1-form cr = x 2dx + xy dy (p‘œ)40) = (T -I*c)To) = cri*œp = x2(p)t(a(T-1)'/ax)(cp(p))dx1,0) (3(0P-1)1/aY)(49(P))41(p)} (xY)(P){( 8(49-1)2/ax)(9)(P))dx((gp) (3(7)-1)2/aY)( T(P))4199(p)} = {x2(p)cost + (xy)(p) sin tIdxiq,(p) + {(xy)(p) cos t - x2(p) sin t}dylip(p) = (x2 COS t - xy sin t)(99(p))dx1, (p) + (xy cos t - y2 sin t)(cp(p))dyi go) TENsoR FIELDS 153 ¢(w'®wi®...®w’®X,®...®X,)(Y,,Y2,..., Y,,txi,...,tr‘) =cui®w2®...®w'@X,@... ®X;(<P_i*Y1» <P_i*Y2i ---7<P_i*Y,, <P*IYi~ ---,<P*tY‘)- (4-9-6) Example 4.2 Forthesmooth map <r=1Bi—>1Bi P‘—’<P(P) (a,b)i—-> (q2i(p), q22(p)) =(acost +bsin t,bcost —asin t) with taconstant, theinverse isgiven by </1“=1R2—>1Ri P‘—’<P'i(P) (a,b)i—-> ((q;F1)1(p), (q;"‘)2(p)) =(acost —bsint, bcost +asin t). Foravector field Y,¢Y=q1,Y. Taking Y=x2(8/8x) +xy(8/8y), with xandythestandard coordinates on1R2,gives ¢».,.Y,.=Xi(p){(@<P‘/@x)(p)(@/8x)lW) +(94%/@x><p><@/@y>l.i,)i +X(P)Y(P){(9<Pi/9)')(P)(9/9X)iW) +(9<P2/9y)(P)(3/9)’)|¢(p)} (<P*Y)¢(p) = (xicost +xysint)(p)(8/8x)l,,,g,, +(xycost —xisint)(p)(8/8y)],,,g,, Wemay useqfitoexpress thecoordinates ofpinterms ofthose of q1(p), giving (<P*Y)-ptp) =(X2005! -X)’$in!)(<P(P))(9/9X)l<p(p) +(xycost —yisint)(q1(p))(8/6y)],,,(,,). so q1,Y =(xzcost —-xysint)(8/6x) +(xycost —yisin t)(8/8y). Weconsider now a1-form tr=xidx +xydy (¢“)¢tt=> =(¢"'*“)¢tt=> =9”_i*% =X201)l(9(<P'i)i/9X)(<P(P))dXi¢(p) +(9(<P'i)i/9y)(<P(P))d)'|¢(p)} +(X)')(P){(9(<P_i)2/9X)(<P(P))dXlW) +(9(<P‘i)2/9)’)(<P(P))d)’l¢(,,)} ={x2(p)cost +(xy)(p) sint}dx|,,,(,,, +{(Xy)(11)¢<>St —X2(P)$int}dy|6(,.> =(xzcost —xysint)(<p(p))dxl,,,g,, +(xycost —yisint)(¢(p))dy|W, 154 MANIFOLDS as in the previous example, so epa = (x 2 cos t — xy sin t)dx + (xy cos t — y 2 sin t)dy. We have (§3a)((sPY)( /3) = (x2 cos t — xy sin t)2(p) + (xy cos t — y 2 sin t)2(p) = {(x cos t — y sin t)4 + (x cos t — y sin t)2(y cos t + x sin t)21(p) = (x4 + x2Y2)(50-1(P)) = (oe(r)(49-1(P)) = (Œ(Y) ° T -1)(P) SO (3œ)(e07) = 4.10 Exterior Derivatives In §4.8 we associated with every f cFfi(M) an element df E FT* M. Thus we have an operator mapping functions to 1-forms. We may extend this operator to an Fi-linear map on FAM: d : FARM —> FAR +1M (4.10.1) with the properties: df(X) = Xf X E rAm, f E 5, M (4.10.2a) d(cr A /3) _ dœ A 0 + (-1)P cr A clfi cr E rApA4 , f3 E FAM (4.10.26) dd --- d2 = 0. (4.10.2c) The operator d is called the exterior derivative. Its existence and uniqueness are most easily demonstrated using a local chart and the properties of the exterior algebra. In any coordinate neighbourhood of M an element of FAM can be expressed in a local natural basis. Since d is Fl-linear it is sufficient to consider its effect on an element of the form co = g dxl. A . . g E 5,(M). From properties (4.10.26) and (4.10.2c) dco = dg A dX1 A   A dX`k with dg given by property (4.10.2a). So for the assumed form of u) we have the unique form for da). The defining properties of d enable do) to 154 MANIFOLDS asintheprevious example, so (pa=(xicost —xysint)dx +(xycost —y2sint)dy. Wehave (¢’¢Y)(¢’Y)(P) =(x2cost —xysint)2(p) +(xycost —y2sint)2(p) ={(xcost —ysint)“ +(xcost —ysint)2(y cost +xsint)2}(p) =<x“+x*y*><¢-1<p>> =<a<Y>><¢-1<p>> =(¢Y(Y)0<P'i)(P) so (¢>¢Y)(¢’Y) =¢’(¢Y(Y))- 4.10 Exterior Derivatives In§4.8 weassociated with every fe@(M) anelement dfeFT*M. Thus wehave anoperator mapping functions to1-forms. Wemay extend this operator toanIR-linear map onFAM 2 d:FA,,M i> I“A,,+1M (4.10.1) with theproperties: df(X) =Xf XeFAM, fe9M (4.10.2a) d(a,(B) =da,(B +(—1)Pa/Adfi a/eI"A,,M,BeI"AM (4.10.2b) ddEdz=0. (4.10.2c) The operator discalled theexterior derivative. Itsexistence and uniqueness aremost easily demonstrated using alocal chart andthe properties oftheexterior algebra. Inanycoordinate neighbourhood of Manelement ofPAM canbeexpressed inalocal natural basis. Since d isB-linear itissufficient toconsider itseffect onanelement oftheform w=gdxi',(...,(dxi* ge@(M). From properties (4.10.2b) and(4.10.2c) da) =dg/(dxi',\. ../\dXii‘ with dggiven byproperty (4.10.2a). Sofortheassumed form oftowe have theunique form fordw.The defining properties ofdenable dwto EXTERIOR DERIVATIVES 155 be evaluated on a set of vector fields, for any CO E ['AM. We consider first a 1-form, it being sufficient to assume w = g dx x, g E (M) thus dw(X,, X2) = (dg A dX)(Xi, X2) = {dg(X1)dx(X2) — dg(X2)dx(X1)} from the definition of the exterior product. From property (4.10.2a) 2 dw(X,, X 2) = X1(g) dx(X 2) — X2(g) dx(X 1) = X i(g dx(X2)) — gX i(dx(X2)) — X2(g dx(X1)) + gX2(dx(X1)) = X i(w(X2)) — X2(o)(X1)) + g[X2, X i](x). Using this property once again in the last term gives 2 dw(X,, X 2) = X1(w(X2))— X2(w(X1)) — w([X,, X 2]). It follows that for any a E rAiM (da)(X, Y) = (1/2){X(a(Y)) — Y(a(X)) — œ([X, Y])}. (4.10.3) Similarly if a E FA2M (da)(X, Y, Z) = (1/3){X(a(Y, Z)) + Y(a(Z, X)) + Z(a(X, Y)) — a([X, Y], Z) — a([Y , Z], X) — a([Z, X], Y)) V X, Y, Z E FTM. (4.10.4) For the general case of a E FA,M r (da)(Xo, X1, ..., Xi) = 1 (xo, . . . . Xi)) r+ 1 1=0 1 , E xd, kk, ..., Xi) T + V Xo, X1, Xr E FTM (4.10.5) where means omit this term from the argument list. An important property of d is that it commutes with the pull-back map f* : FAN —> ['AM induced from a diffeomorphism f: M N. First observe that if g E Ff°(M), Xe ['TM, then (f*dg)(X) = dg(f *X) = (fX)(g) (by 4.10.2a) = X(f* g) = d(rg)(X) (using property (4.10.2a) again) giving f*dg = d(f* (4.10.6) EXTERIOR DERIVATIVES 155 beevaluated onasetofvector fields, foranywePAM. Weconsider firsta1-form, itbeing sufficient toassume w=gdx x,ge@(M) thus dw(X1t X2)=(dg/\dX)(X1t X2) =%{d8(X1)dX(Xz) —d8(Xz)dX(X1)} from thedefinition oftheexterior product. From property (4.10.2a) 2dw(X1, X2) =X1(g) dx(X2) _X2(g) dx(XI) =X1(gdX(Xz)) TgX1(dX(Xz)) -X2(gdX(X1)) +8X2(dX(X1)) =X1(w(X2)) *X2("1(X1))+ 8[X2, X1l(x)- Using thisproperty once again inthelastterm gives 2dw(X1, X2)=X1(w(X2))_ X2(w(X1)) _w([X1, X21)- Itfollows thatforanyoreI‘/\,M (dot)(X, Y)=(1/2){X(a(Y)) —Y(ot(X)) —a([X, Y])}. (4.10.3) Similarly iforeFAZM (da)(X, Y,Z)=(1/3){X(ot(Y, Z))+Y(ot(Z, X))+Z(ot(X, Y)) -¢Y([X, Y],Z)-¢Y([Y, Z],X)—¢Y(lZ» X],Y)) VX, Y,ZeFTM. (4.10.4) Forthegeneral caseoforeI“/\,M I’1 , /\(da)(X@. X1.....X.)=$2<-1>'X,-(am. ....X,-.....X.» /=0 1 . - /\+L Z(-1)/""0t([X,-, Xk],X0,...,x,,...,x,,,...,x,) r+1Us/<k$r vxg,x,,...,X,eFTM (410.5) where X,means omit thisterm from theargument list. Animportant property ofdisthat itcommutes with thepull-back map fi:FAN —>PAM induced from adiffeomorphism f:M—>N.First observe thatifge9(M), XeFTM, then (fidg)(X) =dg(f*X) =(f*X)(8) (by4-10-2") =X(fi8) =d(fig)(X) (using property (4.10.2a) again) fidg=d(fig) (4.10.6)giving 156 MANIFOLDS Now consider d{r(gdx" A dx" A   A dx4)} = dff*(gd.,ci.) A f*(dx'2) A . . A f*(dx4)) = d{r(gdx`') A d(f*X12) A   A d(f*xik)} = d(f*(gdx9) A d(f*x") A  A d(f* = d((f*g)(f*dxi.)) A f*(dx'2 A . . A drsk) from above as d2 = 0 = d(g) PdX`' P(dX I2 A  - A dx`k) as d(f* dx') = dd(rx') = 0 = rdg A f*dXil Ar(dX 12 A    A dx") = f*d(gclxi , A dx'2 A . . . A dX10. It follows since d and f- are R-linear maps that f*d = d (4.10.7) f* on arbitrary elements of rAm. 4.11 One -Parameter Diffeomorphisms and Integral Curves In many situations in theoretical physics one is concerned with situations that can be described in terms of 'flows on a manifold'. This technical term is borrowed from what is perhaps the simplest case to visualise, the laminar flow of a fluid around a smooth surface. The motion of a fluid around a vortex is another familiar example of a flow. If each element of the medium experiencing such a flow is followed in time it traces out the image of a curve. Hence for a smooth flow one can establish a correspondence between local fluid flow and a local vector field. The notion of a flow in time is naturally associated with a bijective mapping, the flow taking a neighbourhood U(p) of a point p on a manifold M to a neighbourhood U(p') in some fixed interval of time. For some fixed interval t we describe such an evolution by q: U(P'). (4.11.1) For a chosen U(p) we have a diffeomorphism for t e /p, where /p C R is an open interval about 0. To describe what happens in an arbitrary time interval we define cp in terms of cp, by W C (I x M) ---> M, (t, p) T(t, p) = cp,(p) (4.11.2) where, for each t e I, (pi is a local diffeomorphism from some U(p) c M to U(p')C M. Conversely, for every U(p) C M there is an 156 MANIFOLDS Now consider d{f*(gdxi1Adxi1 A...Adxi*)} =d{fi(8dXi‘) /\f*(dXi’) /\---/\fi(dXi‘)} =d{f’"‘(gdxi') Ad(f*xi1) A...Ad(f*xi*)} from above =d(f*(8dXi‘))/\d(f*Xi’) /\---/\d(f*Xi‘) asdz=0 =d((f*8)(f*dXi‘)) /\f*(dXii /\---/\dxi) =d(f‘ig)Af‘idxi1Afi(dxi* A...Adxit) asd(f*dxi) =dd(f”ixi) =O =fidg/xfidxii /\f*(dXii/\- --/xdxii) =f*d(gdxi'Adxi1 A...Adxik). Itfollows since dandfiareIR-linear maps that fid=dfi (4.107) onarbitrary elements ofPAM. 4.11 One-Parameter Diffeomorphisms andIntegral Curves Inmany situations intheoretical physics oneisconcerned with situations that canbedescribed interms of‘flows onamanifold’. This technical term isborrowed from what isperhaps thesimplest case tovisualise, the laminar flow ofafluid around asmooth surface. The motion ofafluid around avortex isanother familiar example ofaflow. Ifeach element ofthemedium experiencing such aflow isfollowed intime ittraces out theimage ofacurve. Hence forasmooth flow one canestablish a correspondence between local fluid flow and alocal vector field. The notion ofaflow intime isnaturally associated with abijective mapping. theflow taking aneighbourhood U(p) ofapoint ponamanifold Mto aneighbourhood U(p’) insome fixed interval oftime. Forsome fixed interval twedescribe suchanevolution by (1),:U(p)?> U(p’). (4.11.1) For achosen U(p) wehave adiffeomorphism forteI,,, where 1,,CIRisanopen interval about O.Todescribe what happens inan arbitrary time interval wedefine tpinterms oftp,by <PrWC(1><M)—>M»(tiP)e—><P(1»P)=¢>,(p) (4-11-2) where, for each teI,tp,isalocal diffeomorphism from some U(p) CMtoU(p’) CM.Conversely, forevery U(p) CMthere isan ONE-PARAMETER DIFFEOMORPHISMS AND INTEGRAL CURVES 157 J1,, C / such that (p, is a diffeomorphism from U(p) to U(p') for t E I. Motivated by the example of fluid flows, in which the configuration of fluid elements at any time can be obtained from the successive compo- sition of evolution maps, we demand that (PI2 ° CPti QP11+12 V ti, t2 El such that t1 + t2 E and that To(P) = P VpEM. (4.11.3) In particular, TT' = cp_,. Families of diffeomorphisms of this type are called local one-parameter diffeomorphisms on M. A one-parameter family of local diffeomorphisms gives rise to a vector field on M. For every p E M the map cp defines a curve q(p) starting at p €10): lp M t Ti(p). To say that the curve starts at p means that cpo(p) = p. Using the definition (4.5.13) we have a tangent vector defined at every point of the image of the curve. By taking the set of all such curves we define a tangent vector at each point of M. Since different curves have image points in common it is necessary to check that this rule gives an unambiguous assignment of tangent vectors. Suppose that y9,0(p) = cp,(p') for some (to, p) and (4, p'), then (MO — 97(t-1o+16(P') = (4 7/-6 ° (Pt6)(P') = Tr--/6(Tr6(0) = = (93/-6 ° Tto)(P) = 49(--(6+(0(P) by (4.11.3) again. Thus if the curves T(p') and co(p) have image points in common then ço(p') is a reparametrisation of cp(p). The parametrisa- tions merely differ by the addition of a constant and so =-- cp(p),(i_o_o, and the tangent vectors agree where the image points coincide. Hence the one-parameter family of local diffeomorphisms defines a tangent vector at each point of M; the smoothness of cp, ensures that the assignment of tangent vectors is smooth and we have a smooth vector field. For the example of a fluid flow this vector field is everywhere tangential to the flow lines. In the above we showed how a one-parameter family of local diffeomorphisms defined a set of curves, enabling a vector fi eld to be introduced that was everywhere tangential to these curves. We now show how the argument can be reversed. If X is a vector field on M then a curve C : / M, t p(t), is called an integral curve of X if X is C-related to (aiat). That is, if C is specified by C : t xm = (4.11.4) by (4.11.3) ONE-PARAMETER DIFFEOMORPHISMS AND INTEGRAL cuRvEs 157 1,,CIsuch that (,0,isadiffeomorphism from U(p) toU(p’) forteI,,. Motivated bytheexample offluid flows, inwhich theconfiguration of fluid elements atanytime canbeobtained from thesuccessive compo- sition ofevolution maps, wedemand that (p,29(;0,, =(;0,,+,, Vtl, tzelsuch thatt1+t2eI andthat (pg(p) =p VpEM. (4.113) Inparticular, <;0,'i=<p_,. Families ofdiffeomorphisms ofthistype are called local one-parameter diffeomorphisms onM.Aone-parameter family oflocal diffeomorphisms gives risetoavector field onM.For every peMthemap godefines acurve (;0(p)starting atp iiipiil” —)M (4.11.4) t+—+ <P,(P)- Tosaythat thecurve starts atpmeans that (pg(p) =p. Using the definition (4.5.13) wehave atangent vector defined atevery point of theimage ofthecurve. Bytaking thesetofallsuch curves wedefine a tangent vector ateach point ofM.Since different curves have image points incommon itisnecessary tocheck that this rule gives an unambiguous assignment oftangent vectors. Suppose that <;0,,,(p) = rp,,i,(p') forsome (tg,p)and(tg,p’), then <t>,(P') =<t><,_t.,+,.'.(P') =(</>1-it°</>4)(p') by(4-11-3) =wt-4(¢>4(p')) =</>1-t,(<t>.,(p)) =(wt-itO</>..,)(P) =(Pt-ta+t.,(P) by(4.11.3) again. Thus ifthecurves rp(p') and(p(p)have image points incommon then rp(p') isareparametrisation of(p(p). The parametrisa- tions merely differ bythe addition ofaconstant and so(p(p'),, I(P(P).(,_,;,+,,,), and thetangent vectors agree where theimage points coincide. Hence theone-parameter family oflocal diffeomorphisms defines atangent vector ateach point ofM;thesmoothness ofgo, ensures that theassignment oftangent vectors issmooth andwehave a smooth vector field. Fortheexample ofafluid flow thisvector field is everywhere tangential totheflow lines. Inthe above weshowed how aone-parameter family oflocal diffeomorphisms defined asetofcurves, enabling avector field tobe introduced that was everywhere tangential tothese curves. We now show how theargument canbereversed. IfXisavector field onM then acurve C:I—>M,t+—>p(t), iscalled anintegral curve ofXifX isC-related to(8/8t). That is,ifCisspecified byC:t+—>x" =)l"(t), 158 MANIFOLDS giving C*(3/8t) = A4(t)(3/axm)1 4,), and in local coordinates X = Mal 3.0), then C is an integral curve of X if AP(t) = PLO, t(t), An(t)) (4.11.5) for ft. = 1, . . n. It follows from the theory of ordinary differential equations that solutions to (4.11.5) always exist, being uniquely deter- mined by the initial conditions x0(p) = AP(0). The smoothness of the fP ensures that such solutions are not only smooth functions of t, for t in some interval I C R, but are also smooth functions of the initial point xP(p), for p in some neighbourhood U C M. Thus if C: / —> M and C' : I' ---> M are integral curves of X starting at p we must have I' C I say, with C equal to C' on the restriction to I'. By taking the largest such interval we have a uniquely determined maximal integral curve of X starting at p. Example 4.3 Suppose X = x(alay) — y(313x) E [-TIFF. Let C: 1 — lB2, t 1-3 W(0, A2(t)) be an integral curve of X that starts at the point (a, h) E R2. Solving = _A2, A2 = Al subject to this condition gives: = a cos t — b sin t, )1,2(t) = b cos t + a sin t. Here we may take I = IR, the maximal integral curve mapping the whole real line into the circle, the curve being periodic with period 2. A vector field whose maximal integral curves starting at p are defined on all of IR, for every p E M, is called complete. In general this will not be the case, the domain of the maximal integral curves depending on which point they start at. Introducing a suggestive notation we denote by p(p) the maximal integral curve of X E ['TM starting at p cp(p): I,, --> M t cp,(p). If to E with 99,0(p) = q then setting h: + to gives a curve ii)(q) =- cp(p). h. The images of Ip(q) and p(p) coincide, as do their tangent vectors since the reparametrisation merely involves the addition of a constant. Thus zp(q) is certainly an integral curve of X, starting at q. If /p = (a, b) then J1 = (a —to, b — to) and since a <O < b we have —to E Jq, giving tp_10(q) = p. If tp(q) were not maximal, with .1,1 C Iv then reversing the argument would contradict II, being the maximal domain of integral curves starting at p. So maximal integral curves with image points in common are all related by repara- metrisations that translate the domain of definition along the real line. It then follows that if cpt is defined by ePt :P Tr(P) V p with t E 158 MANIFOLos giving C,(8/St) =/1“(t)(8/8x“)|M,,, andinlocal coordinates X=f“(8/ 8x“), then Cisanintegral curve ofXif /l"(t)=f"(/1‘(t), ...,/1"(t)) (4.11.5) forp=1,...,n.Itfollows from thetheory ofordinary differential equations that solutions to(4.11.5) always exist, being uniquely deter- mined bytheinitial conditions x“(p)=/l"(0). Thesmoothness oftheft‘ ensures that such solutions arenotonly smooth functions oft,fortin some interval ICIR,butarealso smooth functions oftheinitial point x"(p), forpinsome neighbourhood UC M.Thus ifC:I—>Mand C’:I’—->Mareintegral curves ofXstarting atpwemust have I’CI say, with Cequal toC’ontherestriction toI'.Bytaking thelargest such interval wehave auniquely determined maximal integral curve of Xstarting atp. Example 4.3 Suppose X=x(8/8y) —y(8/8x) el"TlR2. Let C:I—->IBZ, Il—> (/li(t),/12(t)) beanintegral curve ofXthat starts atthepoint (a, b)elB2. Solving /11=—/12, /I2=/lisubject tothis condition gives: /li(t) =acost bsint, /l2(t) =bcost +asin t.Here we may take I=IR,themaximal integral curve mapping thewhole reallineinto the circle, thecurve being periodic with period Zn. Avector field whose maximal integral curves starting atparedefined onallofIR,forevery peM,iscalled complete. Ingeneral thiswillnot bethecase, thedomain ofthemaximal integral curves depending on which point they start at.Introducing asuggestive notation wedenote byqt(p) themaximal integral curve ofXeFTM starting atp ¢(1>)1Ip —>M t*—> ¢n(1>)- IftgeIpwithqi,,,(p) =qthensetting h:J,,—>Ip t+—>t+tg gives acurve 1/1(q) =qt(p) 9h.The images of1p(q) andq2(p) coincide, asdotheir tangent vectors since thereparametrisation merely involves theaddition ofaconstant. Thus tp(q) iscertainly anintegral curve ofX, starting atq.If1,,=(a,b)then J,,=(a—tg, b—tg)and since a<0<bwehave —tgeJ,,, giving 1/1_,,,(q)= p.If1p(q) were not maximal, with J,CIq,then reversing theargument would contradict 1,, being themaximal domain ofintegral curves starting atp.Somaximal integral curves with image points incommon areallrelated byrepara- metrisations thattranslate thedomain ofdefinition along therealline. It then follows thatifqt,isdefined by (17,117!-i)q),(p) Vpwith teI,, ONE-PARAMETER DIFFEOMORPHISMS AND INTEGRAL CURVES 159 then cp, is an invertible map satisfying (4.11.3). Above each point p E M we erect the fibre /p and denote the space formed by all the fibres by W. If Ii,, C I Vp then W C (/ x M), and cp is defined by 49: (t, Tr(P). Each 419, is an invertible map on some domain contained in M. Furthermore these maps are smooth, the solutions to the differential equations for an integral curve being smooth functions of the starting point. It follows that every smooth vector field X on M generates a one-parameter family of local diffeomorphisms: each point of M being mapped along an integral curve of X. In local coordinates the trans- formation p q(p) is represented by cpP(t, xi(p), . . xn(p)) = 1, . . n where cp0(0, xl(p), . xn(p)) = x(p) and (Mt, + t 2, xl(p), . . xn(p)) = cpn(t 2, q)1(t1, xl(p), xn(p)), 992(t1, xl(P),   xn(P)),  cpn(ti, xl(p), . . xn(p))). (4.11.6) We may use the smoothness of the functions (PP in the variable t to obtain a linear approximation of TP for small t cpn(t, xl(p), xn(p)) = cpn(0, xl(p), xn(p)) + apn(0, xl(p), .. xn(p)) + . . . (4.11.7) where VA denotes the derivative with respect to t. Since qv(p) is an integral curve of X, starting at p, if in local coordinates X = fn(alaxn) we have q(0, xi(p), .. xn(p)) = xn(p) and cpn(0, xl(p), . . xn(p)) = fn(p). (4.11.8) Thus for t sufficiently small (4.11.6) may be approximated by x(p) H—* x(p) + t fn(p) + . (4.11.9) Example 4.4 If x coordinates IR then a smooth vector field on 1R is X = x2(3/3x). If cp(t, p) is the maximal integral curve starting at p we require ONE-PARAMETER DIFFEOMORPHISMS AND INTEGRAL CURVES 159 then rp,isaninvertible map satisfying (4.11.3). Above each point peM weerect thefibre 1,,anddenote thespace formed byallthefibres by W.If1,,CIVpthen WC(I><M),andrpisdefined by ¢rW——> M.(I,P)e——>¢i(P)- Each rp,isaninvertible map onsome domain contained inM. Furthermore these maps aresmooth, thesolutions tothedifferential equations foranintegral curve being smooth functions ofthestarting point. Itfollows that every smooth vector field XonMgenerates a one-parameter family oflocal diffeomorphisms: each point ofMbeing mapped along anintegral curve ofX.Inlocal coordinates thetrans- formation pi—-> (p,(p) isrepresented by x"(p)i—->(p#(t,xi(p),...,x"(p)) 1t=1,...,n where ¢“(0»f(p)» --4x”(P)) =X“(P) and ¢"(n+rt.Xi(p), ---.x"(P)) =wilt» ¢i(l1»Xi(P)» ---IX"(p)). <‘P2(l1-Xi(P), ---iX"(P)),~ --. rp"(t,, xi(p), ...,x"(p))}. (4.11.6) Wemay usethesmoothness ofthefunctions rp“inthevariable tto obtain alinear approximation ofrp“forsmall t ¢“(t,xi(P). ---,X"(P)) =¢"(0,x‘(P), ---.X"(P)) +t¢~(0,xi(p), ...,x"(p)) +...(411.7) where rp“denotes thederivative with respect tot.Since rp(p) isan integral curve ofX,starting atp,ifinlocal coordinates X=f“(8/ax") wehave ¢i‘(0»Xi(P)- ~-'1X"(P)) =Xi‘(P) and ¢i‘(0,Xi(P)- ---tX”(P)) =fi‘(P)- (4-11-3) Thus fortsufficiently small (4.11.6) may beapproximated by x“(p) t———> x"(p) +tf"(p) +.... (4.11.9) Example 4.4 Ifxcoordinates IRthen asmooth vector field on1BisX=x2(8/8x). If (p(t,p)isthemaximal integral curve starting atpwerequire 160 MANIFOLDS (0(t, 14= coqt , 13)2 13)= P. The solution is cp(t, p)= 131(1 — tp). If p > 0 we must have t E (-00, p-1), if p = 0, tE (—cc, cc) whilst for p <0, t E 00). The domain W = Up/p is the region of IR2 bounded by hyperbolae in the bottom-left and upper-right quadrants. This is shown in figure 4.12. We can verify that indeed Tr,. cp,,= pA1 — tip) (PIPPI,P) = 1 — t2p/(1 — tip) 1 — (t1 + t2)p =It+12(p). We have shown in figure 4.12 the effect of one of the local diffeo- morphisms çot. Figure 4.12 This diagram illustrates the effect of a local diffeomorphism (pt. 160 MANIFoLDs ¢>(t.P)=¢>(t»11)i ¢>(0.p)=11- The solution isqo(t, p)=p/(1 —tp). Ifp>0we must have te(—°°,p_i), ifp=O,te(—°@, 9°)whilst forp <O,te(p‘i, 9°).The domain W=U,,1,, istheregion of1R2bounded byhyperbolae inthe bottom-left andupper-right quadrants. This isshown infigure 4.12. We canverify thatindeed cog9q0,,=q0,,,,,. _P/(1—tip) _ p _<P»,(<P»,P) 1_,2p/(, _hp, ,_(,1+mp ¢>.,+,,(1>)- Wehave shown infigure 4.12 theeffect ofone ofthelocal diffeo- morphisms q0,. \\\\\‘\\\\\.\ \\\\)\\\\v.V'V.V.V.v QQOOQ‘$33314:»$~2~:~~r \No‘f‘~3 -121 1; W W /% Figure 4.12 This diagram illustrates theeffect ofalocal diffeomorphism q9,. ONE-PARAMETER DIFFEOMORPHISMS AND INTEGRAL CURVES 161 If we modify the above example by restricting X to the manifold M consisting of the open interval (1, co), then the maximal integral curve starting at p has domain i'p =(p — 1, 13). So in this case cp, is defined somewhere if t E ( — 1, 1). The domain W' = 1..1n4 Vp,e Ai is shown in figure 4.12. Cf(p) Figure 4.13 The geometrical interpretation of the commutator [X. Y] of the vector fields X and Y. Exercise 4.1 Let X and Y be vector fields with p(p) and tp(p) the respective integral curves starting at p, and q), and tp, the associated local diffeomorphisms (see figure 4.13). For t sufficiently small and positive a one-parameter family of local diffeomorphisms is given by C, = ocp o o 99v„ with C(p): t 1—> C 1(p) a smooth curve starting at p. If Co(p) is the tangent vector to C(p) at the point p show that CAP) = [X, Y]p. Hint: For f E.61,(M)f ocp, = f + tXf + t 212X2f + 0(t3), where X2f = X(Xf). 4.12 Lie Derivatives In §4.4 we motivated the concept of a tangent vector by introducing differentiation of functions along a curve. Having arrived at the defini- tion by which a vector field is a derivation on the algebra of smooth ONE-PARAMETER DIFFEOMORPHISMS AND INTEGRAL CURVES 161 Ifwemodify theabove example byrestricting Xtothemanifold M consisting oftheopen interval (1,99),then themaximal integral curve starting atphasdomain 1,,=(p'i -1, p'i). Sointhis case qr),is defined somewhere ifte(—1, 1).The domain W’=LJ,,I;, Vp_eM is shown infigure 4.12. W,/;(\p,/;(p)) Y X / W,/;lPJ \ P (pip) Figure 4.13 The geometrical interpretation ofthecommutator [X,Y]of thevector fields XandY. Exercise 4.1 LetXandYbevector fields with <p(p) and1/1(p) therespective integral curves starting atp,andmp,and1/1,theassociated local diffeomorphisms (see figure 4.13). Fortsufficiently small andpositive aone-parameter family oflocal diffeomorphisms isgiven byC,=1/1_\/, 9qJ_\/, 9I/IV, °(P\/,, with C(p) :tt—>C,(p) asmooth curve starting atp.IfCg(p) is thetangent vector toC(p) atthepoint pshow thatCg(p) =[X,Y],,. Hint: For fe@(M)f<> mp,=f+tXf+ti/2X2f +O(t3), where Xif=1‘-'(Xf)- 4.12 LieDerivatives In§4.4 wemotivated theconcept ofatangent vector byintroducing differentiation offunctions along acurve. Having arrived atthedefini- tion bywhich avector field isaderivation onthealgebra ofsmooth 162 MANIFOLDS functions we showed in that section how the action of any vector field on a function is the derivative of that function along a curve; namely the integral curve of the vector field. That is, if cp is an integral curve of X, starting at p d (Xf)(P) = —dt(f ° T(P))( 0) V f E = lim t-1 {f(q),(p)) - f(p)}. (4.12.1) Since X is a smooth vector field then associated with the curve p(p) starting at p, is the local diffeomorphism cp, on the neighbourhood of p. It is instructive to rewrite the above in terms of the pull-back cp*, (Xf)(P) = urn t-1 {(Cf)(P) f(P)}. (4.12.2) This form of the derivative, X on f, suggest a generalisation to a derivative on an arbitrary tensor field Te FTM. We may use the map associated with the vector field X, to map T c!„(p) back to the point p where it can be compared with T. If the limit as t tends to zero of the difference between the two tensors divided by the parameter t exists. it is called the Lie derivative at p of T with respect to X, denoted by (ZxT)(p) (see figure 4.14) (Yxn(P) = lim r1 {(CP-tnp Tp} Figure 4.14 This diagram illustrates the vectors used in the definition of the Lie derivative of a vector field Y with respect to the vector field X (tangent to some integral curve). Vp E Al. (4.12.3) 162 MANIFoLDs functions weshowed inthat section how theaction ofanyvector field onafunction isthederivative ofthatfunction along acurve; namely the integral curve ofthevector field. That is,if(pisanintegral curve ofX, starting atp <Xf><p>=g(to¢<p>><0> weaw) —umF1{to00>—f(p)}. <4-12-1)—i—>0 I Since Xisasmooth vector field then associated with thecurve <p(p) starting atp,isthelocal diffeomorphism (p,ontheneighbourhood ofp. Itisinstructive torewrite theabove interms ofthepull-back (pi (Xf)(P) =1,1,1}Vi{(<P’?f)(P) -f(P)}- (4-12-2) This form ofthederivative, Xonf,suggest ageneralisation toa derivative onanarbitrary tensor field TeI"T§M. Wemay usethemap <,?2_,,associated withthevector field X,tomap T,,,(,,, back tothepoint pwhere itcanbecompared with T,,.Ifthelimit asttends tozero of thedifference between thetwo tensors divided bytheparameter t exists, itiscalled theLiederivative atpofTwith respect toX,denoted by(§£XT)(p) (see figure 4.14) (§£XT)(p) =1,13;t-1{(<;?2_,T),, -T,,} VpeM. (412.3) Ywiipi Xwlnl ‘l7riPl XP llp_,,Y),, Yr P Figure 4.14 This diagram illustrates thevectors used inthedefinition of theLiederivative ofavector field Ywith respect tothevector field X (tangent tosome integral curve). LIE DERIVATIVES 163 The definition of (0, is given in (4.9.5). In the particular case of fE -j'(A/) = (P=!* f = 92*ff and we have Yxf = Xf V f E -64-'(111). (4.12.4) It follows from (4.9.5) that Yx is a derivation on the algebra of tensor fields Zx(SOT)=YxS0T+SOIxT (4.12.5) in particular if f E .9;(M) x(fT) = (Xf)T + f xT. (4.12.6) From (4.9.6) we may deduce that Yx commutes with contractions: „ X1, . . Xs)) = (2xT)(c r 1, . . a,, X1, . . Xs) + E T(œ, ...,Zxa'k,   . Œ, X1,   Xs) k=1 + E T(cei, ar, .x1,   ., Yxxk,   , (4.12.7) k=1 Applying the general definition of a Lie derivative to a vector field Y gives (GY)(p) = - Yd. For any fE Ff'(M) (YxY)pf = l iton t-1{R€P-t)*(Y qmp))1f Ypf) = lim t-1 op)(9)*-J) Ypfl Now from (4.12.2) = f + tXf + 0(t 2) hence (4.12.8) (2xY)pf = limt -1 {Yg,r(p)[f — tXf + 0(t2)] — Ypf} = — Yp(Xf) + lim t -1 {Y,(p)f — Ypf} = —Yp(Xf) + Jim t-1 {(17f)(T1(P)) (Yf)(P)) Since Yf E ?i(M) we see from (4.12.1) that the last term is Xp(Yf), LIEDERIvATIvEs 163 The definition of<p_, isgiven in(4.9.5). Intheparticular case of fEWM) ¢>-1f= <t>I§*f= <P’?f andwehave .§EXf= Xf Vfe @(M). (4.12.4) Itfollows from (4.9.5) that SEXisaderivation onthealgebra oftensor fields .§£X(S® T)=.§£XS®T+S®.§£XT (4.12.5) inparticular iffe@(M) .§£X(f1")= (Xf)T +f§£XT. (4.12.6) From (4.9.6) wemay deduce thatSEXcommutes with contractions: $X(T(CY1, ...,(Yr, X1, ..., = ($XT)(a1, ...,a’,, X1, ...,Xx) I+ZT(or1,...,.§EXozk, ...,6),,x,,...,x,) k=l +ZT(or,,...,6),,X1,...,.s2XX,, ...,x,). (4.12.?) k=1 Applying thegeneral definition ofaLiederivative toavector field Y gives <em<p> =1,i;,1;§{<¢-..Y>,. -Y,» <4-12-8) Foranyfe@(M) <:Bm,.t=151,1;t*t1<¢>_.>.<Y..,.,.>>1r —no : [-1 {Y¢p,(p)(¢i*—rf) _ Now from (4.12.2) <P‘if= f+lXf+ 002) hence (.§£XY),,f =l,i_r).rit'i {Y,,,,(,,)[f —tXf+O(t2)] —Ypf} =—Y,,(Xf) +l,i_r)0nt'i{Y,,,,(,,,f —Y,,f} =_Yp(Xf) +l,L1,',1"i {(Yf)(¢>i(P)) -(Yf)(P)}- Since Yfe@(M) weseefrom (4.12.1) that thelastterm isX,,(Yf), 164 MANIFOLDS therefore (2xY)„f= x ,(Yf) - Y„(xf) ((3xY)f)(p)= (x (Y1) - Y(xf))(p) vp or 2xY= [X, Y[ (4.12.9) where [X, Y] XY — YZ is the commutator of the two vector fields. From this example we note that 2fx Px where f EY,(M). We say that x is not .7,-linear in X. This reflects the fact that the Lie derivative along a curve depends on the parametrisation of the curve and not just on its image. When using a coordinate basis to evaluate x on tensor fields with contravariant components it is useful to note from (4.12.9) that 2(3,a.e)(alax0 = [(a/ax:), (alax0] = O. The properties of x that we have established are sufficient to deter- mine it completely; it being the unique type-preserving derivation on tensor fields that satisfies (4.12.5), (4.12.6), (4.12.7) and (4.12.9). A consequence of these uniqueness properties is 2y1 — 2[x. Y]  (4.12.10) The commutator of two derviations that commute with contractions is a derivation that commutes with contractions. Certainly both sides of (4.12.10) agree when evaluated on a function, so we need only confirm that they agree when evaluated on an arbitrary vector field. This follows from the Jacobi identity (4.6.6). Whereas the established properties of the Lie derivative completely specify it, these being used in any practical calculation, the definition (4.12.3) conveys the geometrical significance: xT = 0 if and only if the tensor field T is invariant under the local diffeomorphisms generated by X. Example 4.5 We shall evaluate ZxY for X, YEFIE2 given by X = x(a13y) — y(alax), Y = x2(alax) + xy(alay). First we shall apply the definition (4.12.8) directly. In the first example of §4.11 we found the integral curves of X starting at p = (a, b). This gives the diffeomorphism go, (a, b) op t(a , b) = (a cos t — b sin t , b cos t + a sin t). We have already computed q_ Y, in example 4.2 (the map cp there being called cp_, here). So (4.12.8) becomes 164 MANIFoLDs therefore (§£xY),.f= X,.(Yf) —Y,.(Xf) ((§£xY)f)(p)= (X(Yf) —Y(Xf))(P) VP or §£XY= [X,Y] (4.12.9) where [X,Y]EXY—YZisthecommutator ofthetwovector fields. From thisexample wenote that§£,~X¢f§£X where fe@(M). Wesay that 58),» isnot @-linear inX.This reflects thefact that the Lie derivative along acurve depends ontheparametrisation ofthecurve andnotjustonitsimage. When using acoordinate basis toevaluate 58,,» ontensor fields with contravariant components itisuseful tonote from (4.12.9) that §E(g,g,»,(8/ox/i) =[(8/dxi), (8/ox/i)] =0. The properties of58,,»that wehave established aresufficient todeter- mine itcompletely; itbeing theunique type-preserving derivation on tensor fields that satisfies (4.12.5), (4.12.6), (4.12.7) and (4.12.9). A consequence ofthese uniqueness properties is [§£X, £6)/] = y]. The commutator oftwoderviations thatcommute with contractions is aderivation that commutes with contractions. Certainly both sides of (4.12.10) agree when evaluated onafunction, so.weneed only confirm thatthey agree when evaluated onanarbitrary vector field. This follows from theJacobi identity (4.6.6). Whereas theestablished properties of theLiederivative completely specify it,these being used inanypractical calculation, thedefinition (412.3) conveys thegeometrical significance: §£XT =0ifandonly ifthetensor field Tisinvariant under thelocal diffeomorphisms generated byX. Example 4.5 We shall evaluate §£XY forX,Yel"TlR2 given byX=x(8/8y)- y(6/ox), Y=x2(6/ox) +xy(6/By). First weshall apply thedefinition (4.12.8) directly. Inthefirst example of§4.11 wefound theintegral curves ofXstarting atp=(a,b).This gives thediffeomorphism (p, (a,b)1+ (p,(a, b)=(acost —bsint, bcost +asint). Wehave already computed (p_,,Y, inexample 4.2(the map (pthere being called <p_,here). So(4.12.8) becomes LIE DERIVATIVES 165 YxY = limt' [(x2 cos t — xy sin t — x2)(3/8x) + (xy cos t — y2 sin t — xy)(3/4)1 = lim t-1(cos t — 1)[x2(3/3x) + xy(3/4)1 — lim t t[xy(alax) + y2(3/3x)] /—o —xy(a/ax) — y2(3/ay). We now evaluate YxY more practically, using the derived properties of 2xY = X(x 2)(3/ax) + x 2Yx(3/3x) + X(xy)(a/3y) + xyY x(a/ay) = X(x2)(3/ax) — x 2Y(/ax)X + X(xy)(a/ay) — xyY (alay)X = X(x2)(alax) — x 2(a/ay) + X(xy)(3/3y) + xy(a/ax) since 2(,a,)(3/4) = 0 = —xy(a/ax) — y2(3/4). Since the Lie derivative is a derivation on the tensor algebra it is also a derivation on the exterior algebra of differential forms. There are a number of useful properties of the Lie derivative acting on differential forms. First, since the exterior derivative on forms commutes with yo* for any smooth map cp, it follows that ,Txd = dYx. (4.12.11) This is a very useful property for calculations involving Lie derivations of covariant tensor fields expressed in a natural coordinate basis. In Chapter 1 we gave the definition of the interior operator on exterior forms with respect to a vector from the dual space. The interior operator i x on a differential form co, with respect to a vector field X, is naturally defined to satisfy (ixcOp Thus the graded derivation i x is ,GY-linear in X. Since x commutes with contractions it follows that [2,C, = YI (4.12.12) When acting on differential forms the Lie derivative can be expressed in terms of the exterior and interior derivatives x = d ix + ix d V X E FTM. (4.12.13) The equality of these expressions is most readily seen by noting that LIEDERIVATIVES 165 SEXY =l,i1_:r.)t‘i [(x2cost ——xysint —x2)(8/8x) +(xycost~yzsint —xy)(8/8y)] =l,1_ITOIt'i(cost -—1)[x2(8/Bx) +xy(8/8y)] —lirgt'isint[xy(8/8x) +y2(8/8x)] =—xy(8/8x) —y2(8/8y). Wenow evaluate SEXY more practically, using thederived properties of 55x >< SEXY =(x2)(8/8x) +x2SEX(8/8x) +X(xy)(8/8y) +xySEX(8/8y) =X(x2)(8/ex) “x2§£(a/ex)X +X(x)’)(8i8)’) _x)’§£(a/ey)X =X(x2)(8/8x) —x2(8/8y) +X(xy)(8/8y) +xy(8/8x) since SE(g,g,,(8/8y) =0 =—xy(8/8x) —y2(8/8y). Since theLiederivative isaderivation onthetensor algebra itisalso aderivation ontheexterior algebra ofdifferential forms. There area number ofuseful properties oftheLiederivative acting ondifferential forms. First, since theexterior derivative onforms commutes with g0* foranysmooth mapgo,itfollows that This isavery useful property forcalculations involving Liederivations ofcovariant tensor fields expressed inanatural coordinate basis. In Chapter 1wegave thedefinition oftheinterior operator onexterior forms with respect toavector from the dual space. The interior operator iXonadifferential form w,with respect toavector field X,is naturally defined tosatisfy (IA/(I))p =IXFUJP. Thus thegraded derivation iXisW-linear inX.Since SEXcommutes with contractions itfollows that When acting ondifferential forms theLiederivative canbeexpressed in terms oftheexterior andinterior derivatives SEX=diX+iXd VXeFTM. (4.12.13) The equality ofthese expressions ismost readily seen bynoting that 166 MANIFOLDS both are derivations on the exterior algebra, commuting with d and agreeing on functions. For if f E Fi(M) (d ix + ix d)f = ix df = df(X) = Xf = xf. If û E FARM and FAM then (d ix + ix d)(a A13) = d[ixœ p + (-1)Pa' A ix r6] + ix[dcr A p + (-1)Pa A clfi] = dixa A + (-1)P-lixa A Cilq (-1)" da A ix13 + A dixi6 + ixdœ A p + (-1)P+1da A ixfi (-1)Pixa A dfi + cr A ixdfi = (dix + ix d)cr A fl + CV A (dix ix d)P- It is straightforward to see that (d i x + ix d) commutes with d since d2 = O. The existence of a local coordinate basis for M ensures that the above properties are sufficient to establish (4.12.13). Example 4.6 For Xe FTIF1 2 and a e FT*1R2 we shall evaluate Yxcv, first from the definition then, as will always be done in practice, from the established properties of 2x. We take X = x(3/4)— y(313x), œ = x2 dx + xy. dy. As was noted in the previous example we may use earlier examples to proceed from the definition Yxœ = hm t-1 [(x2 cos t — xy sin t — x2)dx + (xy cost — y2 sin t — xy)dy] = —xy dx — y2 dy. alternatively, xœ = X(x 2)dx + x22xdx + X(xy) dy + xyIxdy = X(x2)dx + d(Xx) + X(xy)dy + xyd(Xy) = —2xy dx + x 2 d(—y) + (x 2 — y2) dy + xy d(x) = —xy dx — y 2 dy. For the vector field Y of the previous example (2xoe)(Y) = —x 3y — xy3 cr(Ixr) = — x 3y — xy3 whereas cr( Y) = x 4 + x2y2. 166 MANI1=oLDs both arederivations ontheexterior algebra, commuting with dand agreeing onfunctions. Foriffe @(M) (<1ix+ix<1)f=ixdf=<1f(X)= Xf=-"3Xf- IfaeI"/\,,M and,BePAM then (dix+ixd)(¢YAt3) =dlixa/\/3 +(“1)i"YAix 5] +ixlda/\fi +(_1)i"Y/\ dfil =diXa/Afi +(—1)P‘iiXa/Ad/3 +(—l)P da/AiXfi +a/AdiXfi +iXdaA,B +(—1)P*idaAiX/J’ +(—1)PiXaAd/J’ +aAixdfi =(diX +iXd)aAfi +aA(diX +iXd)/3. Itisstraightforward toseethat (diX+iXd)commutes with dsince dz=0.Theexistence ofalocal coordinate basis forMensures thatthe above properties aresufficient toestablish (4.12.13). Example 4.6 ForXeI"TlR2 andael"T*lB2 weshall evaluate SEXa, firstfrom the definition then, aswillalways bedone inpractice, from theestablished properties ofSEX.Wetake X=x(6/Sy) —y(6/6x), a=x2dx+xydy. Aswasnoted intheprevious example wemayuseearlier examples to proceed from thedefinition SEXa =lingt‘i[(x2cost —xysint—x2)dx +(xycost —yisint —xy)dy] I E—xy dx—y2dy. alternatively, SEXa =X(x2)dx +xi'SEXdx +X(xy) dy+xySEXdy =X(xi)dx +xid(Xx) +X(xy)dy +xyd(Xy) =—2xy dx+xid(—y) +(x2—yz)dy+xyd(x) =—xy dx—yzdy. Forthevector field Yoftheprevious example (5-'3x¢Y)(Y) =T163)’ -Xyi a(SEXY) =—x3y —xy3 whereas a(Y) =x‘+xzyz. LIE DERIVATIVES 167 These are indeed related by Ifx(œ(Y)) = (Z;(001 craxn- 4.13 Integration On Manifolds The differential forms derive a certain prominence amongst the tensor fields on a manifold from the fact that they give rise to a theory of integration, generalising the Riemann integral in IFIr. We recall that such integrals may be defined as the limit attained by a Riemann sum of terms, each consisting of a measure associated with some (usually cubical) subdivision of a domain multiplied by the value taken by the function to be integrated at some point within each cell of the subdivision. We shall assume that the reader is familiar with the methods of evaluating multiple integrals in FIr by means of iterated integrals. The classical notation for a Riemann integral suggests a natural definition for the integral of an r-form on IRr over an oriented domain. With such a definition a mapping from IFir to an n-dimensional manifold M enables an r-form on M to be integrated: we use the map to pull it back to IRr where the integration is defined. Properly formulated the above idea gives the theory of integration of differential forms over oriented chains. Let [0, 1]r be the set of points p E IFIr that satisfy 0 ak(p) 1, k = 1, . . r in any natural chart {i'} for IRT. Thus [0, 1]r is the unit cube in IRr. Introduce Qr for the natural 'volume' r-form da' A da2 A . . . A dur which serves to orient [0, 11r. An oriented r-cube on an n-dimensional manifold M is the pair (Cr, QT) where Cr is a C map Cr: [0, l]r M. (To say that CT is C' on the closed set means that there is a C' map Cer between open sets containing the domain and image of Cr such that Cr is obtained from Tr by restriction.) In a local chart (U, x) we may represent the map Cr : p E [0, hr1-->qEM by its components, W, . . A') x(q) = Ar(al(p), . . u(p)) i = 1, . . n. (4.13.1) Every oriented r-cube gives rise to 2r oriented (r — 1)-cubes called its oriented (r — 1)-faces. Each face is defined by restricting the map Cr to points p for which al(p)= e, where E = 0, 1. Denoting the (r — 1)- faces by Cro-,) [0, 11r-I M we have then Cro-.;)(al(P),   , 01-1(P), cri+l(P),   , u(p)) = Cr(ul(P),   - 01-1(P), e, a1+1(P),  - ar(P)) j = 1, . . r; E = 0, 1 (4.13.2) LIEDERIVATIVES 167 These areindeed related by §EX(w(Y)) =(§5Xw)(Y) +w(§ExY)- 4.13 Integration OnManifolds The differential forms derive acertain prominence amongst thetensor fields onamanifold from thefactthatthey giverisetoatheory of integration, generalising theRiemann integral inIR’.Werecall thatsuch integrals may bedefined asthelimit attained byaRiemann sum of terms, each consisting ofameasure associated with some (usually cubical) subdivision ofadomain multiplied bythevalue taken bythe function tobeintegrated atsome point within each cell ofthe subdivision. We shall assume that the reader isfamiliar with the methods ofevaluating multiple integrals inlR’bymeans ofiterated integrals. The classical notation foraRiemann integral suggests a natural definition fortheintegral ofanr-form onlR'over anoriented domain. With such adefinition amapping from IR’toann-dimensional manifold Menables anr-form onMtobeintegrated: weusethemap topull itback tolR’where the integration isdefined. Properly formulated theabove idea gives thetheory ofintegration ofdifferential forms over oriented chains. Let[0,1]’bethesetofpoints pelR’ that satisfy 0Eo"(p)E1, k=1,...,rinanynatural chart {oi} forlR’.Thus [0,1]’istheunit cube inlR’.Introduce Q’forthenatural ‘volume’ r-form doiAdoz A Ado’ which serves toorient [0,1]’. Anoriented r-cube onan n-dimensional manifold Misthepair (C’, Q’)where C’isaC“map C’:[0,1]’—> M.(Tosaythat C’isC°°ontheclosed setmeans that there isaC°°map <6’between open setscontaining thedomain and image ofC’such that C’isobtained from S6’byrestriction.) Inalocal chart (U,x)wemay represent themap C’zpe[0,1]’t—>qeMbyits components, (/li,...,/1’) xi(q) =/li(oi(p), ...,o’(p)) i=1,...,n. (4.13.1) Every oriented r-cube gives riseto2roriented (r—1)-cubes called its oriented (r—1)-faces. Each face isdefined byrestricting themap C’to points pforwhich o/(p) =s,where s=0,1.Denoting the(r—1)- faces byC[,f,i, :[0,1]"i —>Mwehave then Ci,1i)(0i(P)» ---,v"‘i(P). o"*i(P). ..-,9’(P)) =C’(0i(P)~ ~~--Ui'i(P)- 8.0iii(P)» -~-,9’(P)) j=1,...,r;s=0,1 (4.13.2) 168 MANIFOLDS Each (r — 1)-face may be given a unique orientation S -2, induced from the orientation of Cr: Q( = (_ )E+1 i(alacog-2 r j = 1, . . r; E = 0, 1 (4.13.3) from which it follows that the faces labelled by e = 0, 1 have opposite induced orientations. An oriented 1-cube has two oppositely oriented 0-faces (its end points or vertices) each of which is assigned an orientation + or —. We may recursively define k-faces of Cr, for k --= r — 2, r — 3, . . 0; these being the k-cubes obtained by similarly restricting the (k + 1)-cubes. The 2r 0-faces (or vertices) of Cr are the 0-cubes obtained by restricting the map cr with all ai(p) equal to zero or one. For b i E IF1 the finite sum EibiCri, that maps some set {Cri, 52;} of oriented r-cubes into M, is called an oriented r-chain (with real coefficients). The oriented r-cube (Cr, QT) has a boundary (r — 1)-chain denoted by a(Cr, QT) which is defined as a(CT, Qr) = E E (C, ) (4.13.4) i=1 e=0,1 The boundary operator 3 extends naturally to all r-chains: 3(E bi(Cri, S22)) = E Q;). It follows directly from the definition of Cr-2 that 33 = 0 since the (r — 2)-faces cancel pairwise. From an r-form a, defined on the image of CT, we can use the map Cr to 'pull back' a to [0, lir. The r-form (Cr)* cr has the representation hdcril A do '2 A   A do", h E9;(Er). The orientation Q r of Cr is now used to define Er = ±1 by Qr -= Er dat' A dal' A   A da l'. We define the integral of Cr*a over [0, 11r in terms of the Riemann integral of h Cr*Cr = Er h dal' . . do". fto,ir [o,iir This may be evaluated as the iterated integral Er Jfi fi(f. h(al , a 2, .. a') dal)da2 . o J0 0 . . dar . We may now define the integral of an r-form on M over an oriented r-cube = (Cr). (4.13.5) Jc itou' This definition is extended to include 0-forms by defining the integral 168 MANIFoLDs Each (r—1)-face may begiven aunique orientation S2},’_§,i), induced from theorientation ofC’: -_:(_‘1)E+1 i(3/ao1)Q, :1, ...,V;E=0,1 52I from which itfollows that thefaces labelled bye=0,1have opposite induced orientations. Anoriented 1-cube hastwo oppositely oriented 0-faces (its end points orvertices) each ofwhich isassigned an orientation +or—.We may recursively define k-faces ofC’,for k=r—2,r—3,...,0;these being thek-cubes obtained bysimilarly restricting the(k+1)-cubes. The 2’0-faces (orvertices) ofC’arethe 0-cubes obtained byrestricting themap C’with alloi(p) equal tozero orone. Forb,~elBthefinite sum E,~b,~C,’, that maps some set{C,’, S2,’} oforiented r-cubes into M,iscalled anoriented r-chain (with real coefficients). The oriented r-cube (C’, Q’)hasaboundary (r—1)-chain denoted by8(C’, S2’)which isdefined as ates9')=22(C(52),9553)). (4.13-4) i=1 €=O,l Theboundary operator 8extends naturally toallr-chains: 6(2)b,(c;,op)=2b,-8(C,’, 9;). I I Itfollows directly from thedefinition ofC"2 that 88=0since the (r—2)-faces cancel pairwise. From anr-form ct,defined ontheimage ofC’,wecanusethemap C’to‘pull back’ ato[0,1]’.The r-form (C’)*a hastherepresentation hdoil Adoi=A. ..Adoi', he@(lB’). The orientation Q’ofC’isnow used todefine 2,=i1by Q’=1;,do’)Adoi1A. ..Adoi'. Wedefine theintegral ofC’*a over [0,1]’interms oftheRiemann integral ofh IC'*a =e,Ji hdoil ...ClUi'. ll’ 1°11’ 0. . This may beevaluated astheiterated integral l 1 1 e,Ji... h(oi, oz,...,o’)doi)do2 ...do’.0 00 Wemay now define theintegral ofanr-form onMover anoriented r-cube Jica =,f,0,1,,(C’)*a. (4.13.5) This definition isextended toinclude 0-forms bydefining theintegral INTEGRATION ON MANIFOLDS 169 of a 0-form over a 0-cube to be the difference between the values of the 0-form taken at the two end points. If q. : [0, hr [0, hr is a smooth reparametrisation that preserves orientations and C'r = Cr o ço then fc" = LT) a -=o.iy (Cr oTr a= f 49*(Cr* a) [ [my -= I Cr*a -= ICr*a cp[0,1y since the last equality follows from a change of variable a 1—* = T(a) in the iterated integral. Hence and we say that the oriented r-cubes Cr and C' r are equivalent. The integral of a over the r-chain C =- Xib,C; is defined to be a= b1 J ,  fcr The culmination of this treatment of r-form integration over oriented r-chains is the elegant generalisation of Stokes's theorem afforded by this formalism. For any smooth r — 1 form 13 defined in the range of the r-chain C(r 1) we have fcdP = (4.13.6) The definitions are such that this follows immediately from the result in Fir. First we observe that (4.13.6) will hold for an arbitrary chain if it is true for any r-cube; then we use definition (4.13.5) to relate the integrals to Riemann integrals. Since C*d = dC* the proof of (4.13.6) reduces to that of Stokes's theorem in Br. Since the Riemann integral can be written as a repeated integral the proof finally rests on the fundamental theorem of calculus; the integral of a real function is the anti-derivative. An immediate consequence of Stokes's theorem is the generalisation of the rule for 'integration by parts' to exterior products of forms on a manifold. If E FArM, /3e FAqM then d(a A - da A + (-1)r a A c1/3. Consequently for some (r + q + 1)-chain C fc d(a A (3) = fc dœ A + (-1)r f c a A cif3 = fac a A (4.13.7) by Stokes's theorem. If a c = 0 or a, A /3 = 0 on a C we have the simple result INTEGRATION 0NMANIFOLDS 169 ofa0-form over a0-cube tobethedifference between thevalues ofthe 0-form taken atthetwoendpoints. Ifrp:[0,1]’t—>[0,1]’isasmooth reparametrisation thatpreserves orientations andC"=C’9rpthenI-=I4 C” (Ciwtp) =l,,_,,.<C'-<t>>*4=j,,,,,¢*<c'*4> Ef C’*a =I C’*a 4110-11’ l0~1l' since thelastequality follows from achange ofvariable ot—>o’=go(o) intheiterated integral. Hence I(Y=[ (YC17 Ct andwesaythat theoriented r-cubes C’and C"areequivalent. The integral ofaover ther-chain C=Z,-b,-C; isdefined tobe Luz =;b,- fqa. The culmination ofthistreatment ofr-form integration over oriented r-chains istheelegant generalisation ofStokes’s theorem afforded by thisformalism. Foranysmooth r—1form [3defined intherange ofthe r-chain C(rE1)wehave [C616=feta. (4.13.6) Thedefinitions aresuch that thisfollows immediately from theresult in IR’.First weobserve that (4.13.6) willhold foranarbitrary chain ifitis true forany r-cube; then weuse definition (4.13.5) torelate the integrals toRiemann integrals. Since C*d=dC* theproof of(4.13.6) reduces tothat ofStokes’s theorem inIB’.Since theRiemann integral canbewritten asarepeated integral theproof finally rests onthe fundamental theorem ofcalculus; theintegral ofareal function isthe anti-derivative. Animmediate consequence ofStokes’s theorem isthegeneralisation oftherule for‘integration byparts’ toexterior products offorms ona manifold. IfareI"/\,M, fieI"/\,,M then d(¢Y/\l3) =do‘/\l3 +(—1)' “A95- Consequently forsome (r+q+1)-chain C Ld(aAfi) =LdaAfi+ (—1)’LaAdfi= Lcd/([3 (4.137) byStokes’s theorem. IfSC=0oraAfi=0onSCwehave thesimple result 170 MANIFOLDS tda' = (-1)r+1 fc a A df3. (4.13.8) Example 4.7 We consider the chain C: C: [0, 112 --->IF13 (r, a) (sin 7TT cos 2a, sin 777 sin 2ra, cos ITT). If (r, 0, cp) are the standard polar coordinates for F13 then this map sends (r,o -) to the point on the unit sphere with polar coordinates (1, 7TT, 2ua). The spherical polar coordinates 09, (p) do not cover the sphere, there are coordinate singularities at 9 = 0, it and ço = 0, 2 17. (see figure 4.15). Thus the C chain C is a diffeomorphism from the interior of its domain onto its image, whilst the boundary of the cube is mapped onto the points at which the coordinates are singular. We will integrate the 2-form co -= r3 sin0 de A dcp over C. Note first that co is smooth on the whole of E3. This can be seen by changing to Cartesian coordinates that cover all of IR3, giving co = x dy A dz + y dz A dx + z dx A dy. We have C*c10 = rdr, C*4 = 2uda giving C*co = 2u2 sin (ntdr A da and I (it CC0 * = 2u2 i sin (ur)dr)da = 4u. .f[o,ij2 * o o Figure 4.15 The two-sphere as a two-chain. 170 MANIF0LDs Icdfl’/\fi =(_1)r+1‘[c(Y/\ Example 4.7 Weconsider thechain C: C:[0,1]2——> IR3 (I,o)ti> (sinrrrcos2rro, sinrrrsin2rro, cosarr). If(r,6,(p)arethestandard polar coordinates forIR3then thismap sends (r,o) tothepoint ontheunitsphere with polar coordinates (1,rrr, 2rro). The spherical polar coordinates (6,rp)donotcover thesphere, there are coordinate singularities at6=0,11and rp=0,211(see figure 4.15). Thus theC°°chain Cisadiffeomorphism from theinterior ofitsdomain onto itsimage, whilst theboundary ofthecube ismapped onto thepoints atwhich thecoordinates aresingular. Wewillintegrate the2-form co=r3sin6 d6Ad(p over C.Note first that coissmooth on thewhole of1R3.This canbeseen bychanging toCartesian coordinates thatcover allofIB3,giving co=xdyAdz +ydzAdx +zdxAdy. We have C*d6 =7TdI, C*d<p =2rrdo giving C*w =2rr2sin(11r)drA doand 11 I C*w =Zrrif sin(rrr)dr)do =4rr.10.112 00 6_0 0 \P=° T Figure 4.15 Thetwo-sphere asatwo-chain. INTEGRATION ON MANIFOLDS 171 In the above example it is tempting to say that we have integrated 'over the surface of the unit sphere', although we can so far attach no meaning to this statement, our integrals of forms being over chains. However, a class of chains (a member of which was considered in the example above) can be put into correspondence with subsets of an oriented manifold N, such that we can unambiguously refer to integra- tion over the subset. An oriented r-cube Cr is said to parametrise a region S of an oriented r-dimensional manifold N if Cr([0,1]") = S, Cr is a diffeomorphism on the interior of its domain and the orientation of the cube is compatible with that of the image. That is, if {(3/3°1} is an oriented basis for the cube then {C„p(a/aaa)} is positively oriented with respect to the orientation of N for all points p for which C a non-singular linear transformation. (These conditions are met in the above example with N the 2-sphere with orienting 2-form co.) We can certainly parametrise a region S with more than one r-cube, the crucial result being that if co is an r-form on N which is parametrised by both Cr and C' r then f co) = f crw. It is therefore meaningful to define =„) f is c wr where C' parametrises S. Although we shall not prove the above we observe that it is certainly reasonable. On the interior of their domains Cr and C' r are invertible, and hence (Cr)-10 C' ' is an orientation- preserving diffeomorphism between the interiors of the domains. We have already shown that integrals are invariant under changes of chain that are related by orientation-preserving diffeomorphisms, and so to prove the above result it is necessary to show (as one would expect) that the boundary does not contribute to the integral. (Such an argument shows that parametrising cubes can be a little more general than defined here.) An r-chain C --= EX", parametrises a region S if the image of C is S, each Cr, parametrises its image and the images of the interiors of the cubes are non-intersecting. Again one can show that the integrals of any smooth r-form over any two parametrising chains are equal. The proof that one can parametrise certain regions (for example, compact mani- folds and compact manifolds with boundary) is not simple and we refer the interested reader to the literature. 4.14 Metric Tensor Fields A metric tensor field g on manifold M is a section of a second -rank tensor bundle over M. Restricted to a point p E M it provides a metric INTEGRATION 0NMANIFOLDS 171 Intheabove example itistempting tosaythat wehave integrated ‘over thesurface oftheunit sphere’, although wecansofarattach no meaning tothisstatement, ourintegrals offorms being over chains. However, aclass ofchains (amember ofwhich wasconsidered inthe example above) can beputinto correspondence with subsets ofan oriented manifold N,such that wecanunambiguously refer tointegra- tion over thesubset. Anoriented r-cube C’issaid toparametrise a region Sofanoriented r-dimensional manifold NifC’([0,1]’) =S,C’ isadiffeomorphism ontheinterior ofitsdomain andtheorientation of thecube iscompatible with that oftheimage. That is,if{(8/8o”)} isan oriented basis forthecube then {C,,,(8/8o“)} ispositively oriented with respect totheorientation ofNforallpoints pforwhich C,,, isa non-singular linear transformation. (These conditions aremet inthe above example with Nthe2-sphere with orienting 2-form cu.)Wecan certainly parametrise aregion Swith more than oner-cube, thecrucial result being that ifcuisanr-form onNwhich isparametrised byboth C’andC’’then fcrfl) =jg.-cu. Itistherefore meaningful todefine l.w=l.- where C’parametrises S.Although weshall notprove theabove we observe that itiscertainly reasonable. Ontheinterior oftheir domains C’and C” areinvertible, and hence (C’)'i 9C” isanorientation- preserving diffeomorphism between theinteriors ofthedomains. We have already shown that integrals areinvariant under changes ofchain that arerelated byorientation-preserving diffeomorphisms, and soto prove theabove result itisnecessary toshow (asonewould expect) that theboundary does notcontribute totheintegral. (Such anargument shows thatparametrising cubes canbealittle more general than defined here.) Anr-chain C=ESC’, parametrises aregion Siftheimage ofCisS, each C’,parametrises itsimage andtheimages oftheinteriors ofthe cubes arenon-intersecting. Again onecanshow thattheintegrals ofany smooth r-form over anytwoparametrising chains areequal. The proof that onecanparametrise certain regions (for example, compact mani- folds andcompact manifolds with boundary) isnotsimple andwerefer theinterested reader totheliterature. 4.14 Metric Tensor Fields Ametric tensor field gonmanifold Misasection ofasecond-rank tensor bundle over M.Restricted toapoint peMitprovides ametric 172 MANIFOLDS tensor on the space TM. If g is a symmetric positive-definite non- degenerate metric tensor field the manifold is said to be a Riemannian manifold. If g is a symmetric but indefinite non-degenerate metric tensor field the manifold is said to be a pseudo-Riemannian or (semi- Riemannian) one. For the special case of signature (p, 1) a pseudo- Riemannian manifold is called Lorentzian. Let us develop the description of a (pseudo-) Riemannian metric in a local chart (Um, (pm). If {dx-P} is a local basis for 1 forms for T*pM we may write the tensor field g as g = g dxP 0 dxv (4.14.1) where the n(n + 1)/2 real-valued functions ga, = g(313.0, ataxy) satisfy gpv = gyp (1, y = I, . . n). A g-orthonormal basis {Xa} of TM is one that satisfies g(x a, xb) = nab = ±1 a, b = 1, . . n. (4.14.2) An ordered basis of local vector fields defines a local frame on M and an ordered basis of 1-forms a local co-frame. The components nab of g in a g-orthonormal co-frame are real constants and we may write g = nabea 0 et' where {ea} E FT*M is a g-orthonormal co-frame satisfying ea(Xb) = Va, b =1, ...,n. (4.14.3) Fields of frames are sometimes called moving frames. As described in Appendix A the metric tensor enables TM and T*pM to be related. If a E FT*M then a' c FTM is defined by g(tr, X) = a(X) V X EFTM. (4.14.4) The contravariant (pseudo-Riemannian) metric g* is a tensor field on M that when restricted to a point p E M provides a metric on the vector space T*pM, defined by g*(cr, 16) = g(ef, ,(3) V cy, E FT*M. (4.14.5) In a local chart we may write g* = gPvalaxv 0 Waxy = nabx, o x b where gvo = gV E 5-e(M) and gPvgv = 6tP4 ?Jahn& = The Gl(n, E) elements ea° relating natural and g-orthonormal co-frame fields, 172 MANIF0LDs tensor onthespace T,,M. Ifgisasymmetric positive-definite non- degenerate metric tensor field themanifold issaid tobeaRiemannian manifold. Ifgisasymmetric butindefinite non-degenerate metric tensor field themanifold issaid tobeapseudo-Riemannian or(semi- Riemannian) one. Forthespecial case ofsignature (p,1)apseudo- Riemannian manifold iscalled Lorentzian. Letusdevelop thedescription ofa(pseudo-) Riemannian metric ina local chart (UM, <pM). If{dxii} isalocal basis for1forms forT’;,M we may write thetensor field gas g=g,,,dx" ®dx” (4.14.1) where then(n+1)/2real-valued functions g,,,=g(8/8x", 8/8x’) satisfy g,,,=g,,,(11,v=1,...,n).Ag-orthonormal basis {Xa} ofT,,M isone thatsatisfies g(x,,X,,)=1),,=i1 Ll,6=1,...,I1. (4.14.2) Anordered basis oflocal vector fields defines alocal frame onMand anordered basis of1-forms alocal co-frame. The components r7,,,,ofg inag-orthonormal co-frame arerealconstants andwemay write g=r7,,,,e” ®ei’ where {e“}eFT*M isag-orthonormal co-frame satisfying e“(Xb) =dfi Va, b=1,...,n. (4.14.3) Fields offrames aresometimes called moving frames. Asdescribed in Appendix Athemetric tensor enables T,,M and T’;,M toberelated. If aeFT*M then EreFTM isdefined by g(a, X)=a(X) VXe FTM. (4.14.4) Thecontravariant (pseudo-Riemannian) metric g*isatensor field onM thatwhen restricted toapoint peMprovides ametric onthevector space T’§,M, defined by g*(a,6)=g(a,B) Va,66FT*M. (4.14.s) Inalocal chart wemaywrite g*=g"’8/Sxi‘ ®8/8x’ =n"i’X,, ®X), where g"i‘=gi”e9*(M) and g'“'g..p =6-,‘; '7“i’m,. =6‘:- The Gl(n, IR)elements e,'jrelating natural andg-orthonormal co-frame fields, METRIC TENSOR FIELDS 173 (4.14.6) ea = ea'clx0 are now functions on M. Some authors refer to the co-frame {ea) as an n-bein, others reserve the term n-beins for the n 2 functions e E It should be noticed that, unlike the natural co-basis, in general de' 0, a =1, n. If the 1-form co is written locally as co = comdxt` = coae° then the metric dual is c7) = w3/3xP = coaX„, where 0 1 = rya), and a). =abWb. Similarly, if locally X = 13.,va = a, then = aea where 4 = g m„v and a -= ?lobe (see Appendix A). The index notation is doing double duty here, the Greek and Roman alphabets indicating that the components are with respect to a natural and orthonormal basis respectively. The symbols (0 = co(dxP) and co° = ca(ea) obviously represent different functions on M. Thus it is potentially hazardous when working with components to give i and a a numerical value. Clearly a safer (but rarely used) procedure would be to write unambiguously = ca(alaxP)dxa = w(X)e' X = cl.,0(X)313xP = ea(X)X a. We discussed in Chapter 1 how to use a metric on co-vectors to construct a metric on p-forms. That procedure can now be generalised to construct a metric on differential forms. If M is an n-dimensional orientable manifold with a fixed atlas, specifying a positive orientation say, then one may smoothly assign an orientation to TM for all p E M. Equivalently, if (Ua, yoa) and (Ub, cpb) are any overlapping charts in this atlas, with coordinate functions {.,rg} are {yi'} respectively, then the real-valued function f on Ua fl Ub, defined by dX1 A dX2 A . . clx" = fdy 1 A dy2 A . . . A dyn, is everywhere positive since f is just the Jacobian of the transition map between charts. Thus we are assured of a non-vanishing n-form on any orientable differential manifold. If such a manifold admits a (pseudo-)Riemannian metric tensor field then a canonical choice of orienting n-form is z = el A e2 A . . . A en where {ea} is a g-orthonormal moving co-frame. We may now extend the construction of the Hodge map given earlier to M with *1 = z. This enables the domain of the Hodge map to be generalised to sections of AM. If p: M N is a smooth diffeomorphism between (pseudo)- Riemannian manifolds M and N such that the metric tensor fields gm on M and gN on N are related by gm _ 99*g N then cp is said to be a smooth isometry. As a special case if M = N then p is a smooth isometry of M. If {cp,) is a set of such maps on M METRIC TENSOR FIELDS e”=e,‘jdx" (4.14.6) arenow functions onM.Some authors refer totheco-frame {eii} asan n-bein, others reserve theterm n-beins forthen2functions e,'je97(M). Itshould benoticed that, unlike thenatural co-basis, ingeneral de“E0, a=1,...,n. Ifthe1-form atiswritten locally asw= wudx“ =w,,e“ then the metric dual is(Ti=w"8/8x” =w"X,,, where to"=g“"w,, and rpi‘=r;“i’w,,. Similarly, if locally X=F8/8x/i =§“X,,, then X=§,,dx“ =§,,e”where 5,,=g,,,§’ and5,,=17,,,,§i’ (seeAppendix A). The index notation isdoing double duty here, theGreek and Roman alphabets indicating that thecomponents arewith respect toanatural and orthonormal basis respectively. The symbols wt‘=w(dx“) and wi‘=w(e“) obviously represent different functions onM.Thus itis potentially hazardous when working with components togive itandaa numerical value. Clearly asafer (but rarely used) procedure would beto write unambiguously at=w(8/8x“)dx” =w(X,,)e" X=dxi‘(X)8/Sx” =e“(X)X,,. Wediscussed inChapter 1how touseametric onco-vectors to construct ametric onp-forms. That procedure cannow begeneralised toconstruct ametric ondifferential forms. IfMisann-dimensional orientable manifold with afixed atlas, specifying apositive orientation say,then onemay smoothly assign anorientation toT,,M forallpeM. Equivalently, if(U,,, rp,,)and(U,,, (pg)areanyoverlapping charts inthis atlas, with coordinate functions {xi} are{yii} respectively, then the real-valued function fonU,OU,,,defined bydxiAdxi A...dx"= fdyiAdy2A Ady", iseverywhere positive since fisjust the Jacobian ofthetransition map between charts. Thus weareassured ofa non-vanishing n-form onanyorientable differential manifold. Ifsuch a manifold admits a(pseudo-)Riemannian metric tensor field then a canonical choice oforienting n-form is2=eiAeiA...Ae" where {e“} isag-orthonormal moving co-frame. Wemay now extend the construction oftheHodge map given earlier toMwith *1=2.This enables thedomain oftheHodge maptobegeneralised tosections of AM. If(p:Mt—>Nisasmooth diffeomorphism between (pseudo)- Riemannian manifolds MandNsuchthatthemetric tensor fields gMon MandgNonNarerelated by st)=<t>*g~ then (,0issaid tobeasmooth isometry. Asaspecial case ifM=N then (pisasmooth isometry ofM.If{(p,~} isasetofsuch maps onM 174 MANIFOLDS then they form the isometry group of M under composition. The set of vector fields {KJ that generate these isometries are known as Killing vectors. Because the commutator of Lie derivatives is the Lie derivative with respect to a commutator of vector fields, in the neighbourhood of any point in M the Killing vector fields form a Lie algebra under the commutator; [K„ K11 = c,,kKk where {c, 11'} are the structure constants in this basis. The isometry group defines a Killing symmetry of the (pseudo)-Riemannian structure on M; the metric tensor field satisfying Kg = 0 for any vector field K in the algebra of Killing vectors. In general a (pseudo)-Riemannian manifold will admit no isometries, and hence possess no Killing vectors. Furthermore, there is a maximum number, n(n + 1), of Killing fields that can exist for any metric on M. Example 4.8: Euclidean Manifolds The topological space whose points consist of the n-tuples in En may be given a manifold structure by adopting an atlas consisting of the identity chart that assigns a unique element of IRn to each point. On any open sets U, V on this manifold one may adopt 'local curvilinear coordin- ates', Tu : U IR", cpv : V IR" provided Tu o ço-,» is smooth and 1: 1 with a non-zero Jacobian on U n V. This manifold has a natural Riemannian structure. In a global chart {x', 11:1") the metric tensor field takes the form g = Et,i, „dx`Odxl. The manifold IRn with this Riemannian structure is a model for an n-dimensional Euclidean man- ifold. Any n-dimensional Riemannian manifold isometric to this one under a (smooth) diffeomorphism provides a model for the space of Euclid. Such manifolds admit In(n — 1) rotational isometries (the integ- ral curves of the Killing vectors lying on an (n — 1)-sphere) together with n translational isometries (with the Killing vectors having open integral curves). The group of these isometries is known as the Poincaré group of n-dimensional Euclidean space. Some of the ideas in this chapter are illustrated in Appendix B where the familiar vector calculus of three-dimensional Euclidean space is reformulated. Bibliography Abramhams R, Marsden J and Ratiu T 1983 Tensor Analysis and Applications (New York: Addison-Wesley) Bishop R L and Goldberg S 11980 Tensor Analysis on Manifolds (New York: Pitman) Clarke C 1979 Elementary General Relativity (London: Edward Arnold) 174 MANIF0LDs then they form theisometry group ofMunder composition. The setof vector fields {K,-} that generate these isometries areknown asKilling vectors. Because thecommutator ofLiederivatives istheLiederivative with respect toacommutator ofvector fields, intheneighbourhood of anypoint inMtheKilling vector fields form aLiealgebra under the commutator; [K,, K,-]=c,-,-i‘Kk where {c,»,»"} arethestructure constants inthis basis. The isometry group defines aKilling symmetry ofthe (pseudo)-Riemannian structure onM;themetric tensor field satisfying §£Kg=0 foranyvector field Kinthealgebra ofKilling vectors. Ingeneral a (pseudo)-Riemannian manifold will admit noisometries, and hence possess noKilling vectors. Furthermore, there isamaximum number, %n(n +1),ofKilling fields thatcanexist foranymetric onM. Example 4.8:Euclidean Manifolds Thetopological space whose points consist ofthen-tuples inIR"maybe given amanifold structure byadopting anatlas consisting oftheidentity chart that assigns aunique element of1B"toeach point. Onanyopen sets U,Vonthismanifold one may adopt ‘local curvilinear coordin- ates’, (pg: Ui——> 1B",(pvz Vi——>lB" provided (pg9(p{,iissmooth and1:1 with anon-zero Jacobian onUOV.This manifold has anatural Riemannian structure. Inaglobal chart {xi,lB'i} themetric tensor field takes theform g=2,-=1, __,,dxi®dxi. The manifold 1B”with this Riemannian structure isamodel forann-dimensional Euclidean man- ifold. Any n-dimensional Riemannian manifold isometric tothis one under a(smooth) diffeomorphism provides amodel forthespace of Euclid. Such manifolds admit §n(n —1)rotational isometries (the integ- ralcurves oftheKilling vectors lying onan(n—1)-sphere) together with ntranslational isometries (with theKilling vectors having open integral curves). Thegroup ofthese isometries isknown asthePoincaré group ofn-dimensional Euclidean space. Some oftheideas inthischapter areillustrated inAppendix Bwhere thefamiliar vector calculus ofthree-dimensional Euclidean space is reformulated. Bibliography Abramhams R,Marsden JandRatiu T1983 Tensor Analysis andApplications (New York: Addison-Wesley) Bishop RLandGoldberg SI1980 Tensor Analysis onManifolds (New York: Pitman) Clarke C1979 Elementary General Relativity (London: Edward Arnold) METRIC TENSOR FIELDS 175 Dodson C T J and Poston T 1977 Tensor Geometry (London: Pitman) Hawking S and Ellis G 1973 The Large Scale Structure of Space —Time (Cam- bridge: Cambridge Unversity Press) Kobayashi S and Nomizu K 1963 Principles of Differential Geometry (New York: Interscience) Poor W A 1981 Differential Geometric Structures (New York: McGraw-Hill) Thirring W E 1978 A Course in Mathematical Physics: 2. Classical Field Theory (Heidelberg: Springer) METRIC TENSOR FIELDS 175 Dodson CTJandPoston T1977 Tensor Geometry (London: Pitman) Hawking SandEllis G1973 TheLarge Scale Structure ofSpt1C€~—Tlm€ (Cam- bridge: Cambridge Unversity Press) Kobayashi Sand Nomizu K1963 Principles ofDifferential Geometry (New York: Interscience) Poor WA1981 Differential Geometric Structures (New York: McGraw-Hill) Thirring WE1978 ACourse inMathematical Physics: 2.Classical Field Theory (Heidelberg: Springer) 5 Applications in Physics 5.1 Galilean Spacetimes Since the time of Aristotle the evolution of the language for physics has to a large extent been governed by the choice of an appropriate event space. One may formulate the Galilean relativistic description of physics in terms of a four-dimensional fibre bundle in which each fibre is a Euclidean three-space and the projection is onto a one-dimensional oriented Euclidean time manifold. Events in this Galilean bundle are assigned a standard time point by this projection and the one- dimensional Euclidean metric on the base may be used to measure time differences between such events. Such elapsed times are unambiguous up to an arbitrary scaling corresponding to a choice of time units. If the time difference is zero the events are considered to be simultaneous and it is then possible to use the standard Euclidean metric on the corresponding fibre to define their spatial separation. A family of curves, members of which intersect each fibre only once such that each point of every fibre lies on one and only one curve foliates the bundle. Any two non-simultaneous events that lie on the same curve can be regarded as having the same spatial position with respect to this family. Each such family defines a coordinate system. The Galilean bundle is provided with a preferred class of families of curves; the trajectories of freely falling particles moving with uniform Newtonian velocities. They define the class of inertial reference systems. This dynamical structure endows the bundle with a preferred parallelism. We shall return to its mathematical formulation when we encounter the Newtonian connec- tion. (The bundle may be given alternative parallelisms, for example, Applications inPhysics 5.1Galilean Spacetimes Since thetimeofAristotle theevolution ofthelanguage forphysics has toalarge extent been governed bythechoice ofanappropriate event space. One may formulate theGalilean relativistic description ofphysics interms ofafour-dimensional fibre bundle inwhich each fibre isa Euclidean three-space andtheprojection isonto aone-dimensional oriented Euclidean time manifold. Events inthisGalilean bundle are assigned astandard time point bythis projection and the one- dimensional Euclidean metric onthebase may beused tomeasure time differences between such events. Such elapsed times areunambiguous uptoanarbitrary scaling corresponding toachoice oftime units. Ifthe time difference iszero theevents areconsidered tobesimultaneous and itisthen possible touse the standard Euclidean metric onthe corresponding fibre todefine their spatial separation. Afamily ofcurves, members ofwhich intersect each fibre only once such that each point ofevery fibre liesonone and only one curve foliates thebundle. Any twonon-simultaneous events thatlieonthesame curve canbe regarded ashaving thesame spatial position with respect tothisfamily. Each such family defines acoordinate system. The Galilean bundle is provided withapreferred class offamilies ofcurves; thetrajectories of freely falling particles moving with uniform Newtonian velocities. They define theclass ofinertial reference systems. This dynamical structure endows thebundle with apreferred parallelism. Weshall return toits mathematical formulation when weencounter theNewtonian connec- tion. (The bundle may begiven alternative parallelisms, forexample, GALILEAN SPACET1MES 177 one might single out those reference frames in which particles have a uniform velocity when falling freely in some Newtonian gravitational field.) In addition to the maximal set of six Euclidean Killing vectors on each fibre and the time translation symmetry, the existence of the preferred class of inertial frames endows the Galilean bundle with another three-parameter symmetry group corresponding to the transformation between inertial frames that differ by a uniform Newtonian three- velocity. The complete 10-parameter Galilean group is the relativistic group for Galilean physics (see figure 5.1). IR3 IR' t(p) t'(p)  Figure 5.1 The Galilean bundle with a Euclidean three-space assigned an arbitrary time coordinate by projection. The existence of the above structure for Galilean relativistic spacetime is a basic tenet of Newtonian dynamics. Physical descriptions prior to the introduction of a lorentzian relativistic' structure for spacetime implicitily assume such a time-preferred fibre pattern for the spacetime manifold. Two clocks at rest in a Galilean inertial system may assign different time parameters and even run at different rates relative to each other. However, it is a fundamental postulate of Galilean relativistic physics that the behaviour of all good clocks is independent of their relative state of motion. (By a good clock one means a clock that is robust and whose behaviour in external fields of force can in principle be compen- sated for.) It is further assumed that all good clocks may in principle be synchronised in an inertial system and used to calibrate the evolution rates of all physical processes. In Newtonian physics observers may also be equipped with measuring rods as well as clocks synchronisable with a hypothetical universal time. Rigid rods are used to construct rigid pieces GALILEAN SPACETIMES 177 onemight single outthose reference frames inwhich particles have a uniform velocity when falling freely insome Newtonian gravitational field.) Inaddition tothemaximal setofsixEuclidean Killing vectors oneach fibre andthetime translation symmetry, theexistence ofthepreferred class ofinertial frames endows theGalilean bundle with another three-parameter symmetry group corresponding tothetransformation between inertial frames that differ byauniform Newtonian three- velocity. The complete 10-parameter Galilean group istherelativistic group forGalilean physics (seefigure 5.1). --56. ____.__-_.tiIR“ f(p) I Hp) 3 Figure 5.1The Galilean bundle with aEuclidean three-space assigned anarbitrary time coordinate byprojection. The existence oftheabove structure forGalilean relativistic spacetime isabasic tenet ofNewtonian dynamics. Physical descriptions prior to theintroduction ofa‘Lorentzian relativistic’ structure forspacetime implicitily assume such atime-preferred fibre pattern forthespacetime manifold. Two clocks atrestinaGalilean inertial system may assign different time parameters andeven runatdifferent rates relative toeach other. However, itisafundamental postulate ofGalilean relativistic physics that thebehaviour ofallgood clocks isindependent oftheir relative state ofmotion. (Byagood clock onemeans aclock thatisrobust and whose behaviour inexternal fields offorce caninprinciple becompen- sated for.) Itisfurther assumed thatallgood clocks may inprinciple be synchronised inaninertial system and used tocalibrate theevolution rates ofallphysical processes. InNewtonian physics observers may also beequipped with measuring rods aswell asclocks synchronisable with a hypothetical universal time. Rigid rods areused toconstruct rigid pieces 178 APPLICATIONS IN PHYSICS of apparatus such as standard metres, telescopes, oscilloscopes etc and the Newtonian description of phenomena relies fundamentally on such a framework. However if, as Einstein did, one builds a world picture based on a spacetime geometry with a Lorentzian-signatured metric structure such 'commonsense' operations as length and time measurement cannot be taken as primitive concepts. Thus a more appropriate notion of a clock is required and one must relinquish measuring processes based on extended rigid structures since they are strictly undefined as primitive operations. With any new set of measurement definitions associated with classical observers in a refined spacetime picture we must expect to be able to recover in some approximation the valuable global Newtonian spacetime notions. Einsteinian relativity has sharpened the notion of a good clock and made redundant the concept of a preferred time projection. Physical clocks that approximate the ideal clocks of a non-Galilean description measure the elapsed time between events in IR4 as a function of their relative motions, and it is only for clocks moving with uniform relative Newtonian velocities, small compared with the Newtonian velocity of light, that the notion of elapsed time between events can be divorced from the relative state of motion of the measuring clocks. Such a reformulation is often referred to as a relativistic description. In the following we are motivated towards one particular relativistic formulation: that inherent in a reformulation of Maxwell's equations on a four-dimensional manifold possessing a Lorentzian metric structure and a Poincaré isometry group. We shall follow the historical path that led Einstein to this elegant (and physically more accurate) world structure by examining one of the most successful of all physical theories: classical electrodynamics. 5.2. Maxwell's Equations and Minkowski Spacetime Physical theories are usually formulated in terms of quantities with physical dimensions. The assignment of a physical dimension to a quantity often follows from its operational definition in terms of some measuring process, a coherent choice of units often facilitating the expression of a physical law. Our mathematical introduction of tensor fields is based upon an underlying manifold where chart coordinates and components of all tensors may be regarded as physically dimensionless numbers. However, in order to compare such a tensor field description with a physical theory written in terms of dimensioned quantities one must effect a transformation. If a physical theory is formulated in terms of tensors over the real field one may restore all physical dimensions 178 APPLICATIONS INPHYSICS ofapparatus such asstandard metres, telescopes, oscilloscopes etcand theNewtonian description ofphenomena relies fundamentally onsuch a framework. However if,asEinstein did, onebuilds aworld picture based ona spacetime geometry with aLorentzian-signatured metric structure such ‘commonsense’ operations aslength and time measurement cannot be taken asprimitive concepts. Thus amore appropriate notion ofaclock isrequired and one must relinquish measuring processes based on extended rigid structures since they arestrictly undefined asprimitive operations. With anynewsetofmeasurement definitions associated with classical observers inarefined spacetime picture wemust expect tobe able torecover insome approximation thevaluable global Newtonian spacetime notions. Einsteinian relativity hassharpened thenotion ofa good clock and made redundant the concept ofapreferred time projection. Physical clocks that approximate the ideal clocks ofa non-Galilean description measure theelapsed time between events inIR‘ asafunction oftheir relative motions, anditisonly forclocks moving with uniform relative Newtonian velocities, small compared with the Newtonian velocity oflight, that thenotion ofelapsed time between events can bedivorced from the relative state ofmotion ofthe measuring clocks. Such areformulation isoften referred toasa relativistic description. Inthefollowing wearemotivated towards one particular relativistic formulation: that inherent inareformulation of Maxwell’s equations onafour-dimensional manifold possessing a Lorentzian metric structure andaPoincare isometry group. Weshall follow thehistorical path that ledEinstein tothiselegant (and physically more accurate) world structure byexamining oneofthe most successful ofallphysical theories: classical electrodynamics. 5.2.Maxwell’s Equations andMinkowski Spacetime Physical theories areusually formulated interms ofquantities with physical dimensions. The assignment ofaphysical dimension toa quantity often follows from itsoperational definition interms ofsome measuring process, acoherent choice ofunits often facilitating the expression ofaphysical law. Our mathematical introduction oftensor fields isbased upon anunderlying manifold where chart coordinates and components ofalltensors may beregarded asphysically dimensionless numbers. However, inorder tocompare such atensor field description with aphysical theory written interms ofdimensioned quantities one must effect atransformation. Ifaphysical theory isformulated interms oftensors over thereal field one may restore allphysical dimensions MAXWELL'S EQUATIONS AND MINKOWSKI SPACETIME 179 appropriately as follows. The dimensionless tensor field equations de- scribing the theory are initially expressed in a local chart with dimen- sionless spacetime event coordinate maps, say (t, x1, x', x3). Chart transformations are then performed to some standard coordinates with assigned physical dimensions. If necessary, new tensors with physical dimensions can be defined by scaling dimensionless ones by some constant parameter with appropriate dimensions. The numerical values chosen for such dimensioned parameters establish the choice of units for the system. If one wants to work with coordinates having the standard dimensions of time and length, say (t, x 1, x2, .x3), one may introduce three standard dimensioned units such as c, a standard speed, h a standard unit of action and a reference mass m0. The restoration of physical units follows from the simple chart transformations t = oC2 Ih)t X = (M. oCIOX k k = 1, 2, 3. (5.2.1) A dimensionless tensor field will have components with dimensions when referred to a basis induced from a local chart with dimensioned coordinates. It is a fundamental property of matter that it can exert a long-range influence on other matter by both the effect of its mass (the gravitation- al interaction) and its electrical charge (the electromagnetic interaction). The latter is a property that comes in two opposite varieties or polarities that are responsible for the 'attractive' and 'repulsive' forces of elec- trostatic interaction. (No analogous 'repulsive' long-range Newtonian gravitational interaction between matter has been observed.) After the pioneering efforts of Faraday and Maxwell the electromagnetic interac- tion between matter is described in terms of an intermediary physical field. This field was originally conceived to consist of a pair of vector fields (E, B) on Euclidean IR 3 parametrised by a universal time t. If we denote by the (time-dependent) function p —> IR the electrical charge density in C M-3 and by j the (time-dependent) vector field on IR 3 describing the charge crossing normally a unit area (the current density in A m-2) then, in mxs dimensioned units, (mass in kilogrammes (kg), time in seconds, length in metres (m)) the electric E and magnetic B vector fields satisfy Maxwell's equations: div E = pie° curl E = —aBlat curl B = peuj + —1 —aE div B = 0 (5.2.2) C 2 at We are assuming that the sources (p,j) exist in a free space or 'vacuum' environment. If E = E,(/ax') e f TB' then by (3E/at) one means (a E ,/3t)(3/3x where, in the chart (Lc', x 2, x3) for IR3, the Euclidean MAxwELL’s EQUATIONS AND MINKOWSKI SPACETIME 179 appropriately asfollows. The dimensionless tensor field equations de- scribing thetheory areinitially expressed inalocal chart with dimen- sionless spacetime event coordinate maps, say(t,xi,x2,x3). Chart transformations arethen performed tosome standard coordinates with assigned physical dimensions. Ifnecessary, new tensors with physical dimensions can bedefined byscaling dimensionless ones bysome constant parameter with appropriate dimensions. Thenumerical values chosen forsuchdimensioned parameters establish thechoice ofunits for thesystem. Ifonewants towork with coordinates having thestandard dimensions oftime and length, say(§,xi,x2,x3), onemay introduce three standard dimensioned units such asc,astandard speed, ha standard unit ofaction and areference mass mg. The restoration of physical units follows from thesimple chart transformations t=(mgcii/ft); xi‘=(mgc/h)x" k=1,2,3. (5.2.1) Adimensionless tensor field will have components with dimensions when referred toabasis induced from alocal chart with dimensioned coordinates. Itisafundamental property ofmatter thatitcanexert along-range influence onother matter byboth theeffect ofitsmass (thegravitation- alinteraction) anditselectrical charge (the electromagnetic interaction). Thelatter isaproperty thatcomes intwoopposite varieties orpolarities that areresponsible forthe‘attractive’ and ‘repulsive’ forces ofelec- trostatic interaction. (No analogous ‘repulsive’ long-range Newtonian gravitational interaction between matter hasbeen observed.) After the pioneering efforts ofFaraday andMaxwell theelectromagnetic interac- tion between matter isdescribed interms ofanintermediary physical field. This field wasoriginally conceived toconsist ofapair ofvector fields (E,B)onEuclidean IR3parametrised byauniversal time t.Ifwe denote bythe(time-dependent) function p:IR3 ->IRtheelectrical charge density inCm‘3 andbyjthe(time-dependent) vector field onIR3 describing thecharge crossing normally aunit area (the current density inAm'2) then, inMKS dimensioned units, (mass inkilogrammes (kg), time inseconds, length inmetres (m)) theelectric Eandmagnetic B vector fields satisfy Maxwell’s equations: divE =p/cg curlE =-88/8; 1SEdivB =0 curlB =ugj+———. (5.2.2) C2St Weareassuming thatthesources (p,j)exist inafreespace or‘vacuum’ environment. IfE=§,~(8/8;’) eFTIR3 then by(SE/Sf) one means (SQ,-/8t)(8/Sxi) where, inthechart (xi,xi.x3) forIR3,theEuclidean 180 APPLICATIONS IN PHYSICS metric tensor field has the representation g = Z=idx'Odx'. The con- stants E0, po and c = (EOM) -112 ensure thif the equations are dimen- sionally coherent. They are assigned dimensions as follows Luoi = IVIL Q2 [en] = [T2Q21 ML3 The functions ( E „ B ,):111 3 —> 11=1, each depending on the time para- meter t, will be called the MKS Cartesian components of the electric and magnetic field respectively. The Cartesian components of the electric field have dimensions [ML/T 2Q], with MKS units of N C-1, whilst those of the magnetic field have dimensions [M/TO] with MKS units of Teslas (or Wb m -2). The structure of this system of coupled partial differential equations permits one to construct a remarkable synthesis between the fields (E, B). This may be achieved by reformulating the system in terms of a pair of tensor equations on the event manifold 11:1 4 endowed with a particular metric structure. Instead of associating the Cartesian compo- nents of E, B with vector fields on 111 3, they are used to construct a 2-form Eon 114. Using a local chart (t, x 2, X 3) we define F = B + dt A E (5.2.3) where B = B ,dx2 A dX 3 B ,dx 3 A dx1 + B /CIX 1 A dX 2 E = E1dx1 + E,dx2 + E3dx3. In a similar way we unify the components of the current and charge density to construct the 3-form j: = cpo./ A dt + (pIcE0)dx1 A dx 2 A dx3 (5.2.4) where J = lidx 2 A dx 3 ± i2dX 3 A dx1 + 13dx 1 Ad. The {j,} are the components of the vector current j. The choice of dimensioned coefficients ensures that j and F have the same dimen- sions, namely [h/Q]. Note that, for any form f, df and f have the same physical dimensions: the exterior derivative does not change the physical dimensions of the form on which it acts. The metric tensor field adopted on IR4 is given in this chart by g = —c2dtOdt + (5.2.5) — Hence (cdt, dx') is an orthonormal co-frame with respect to this g. In terms of the Hodge map * associated with this Lorentzian-signatured metric Maxwell's equations may be expressed elegantly as the exterior equations 180 APPLICATIONS INPHYSICS metric tensor field hastherepresentation g=Z,i=,d£i®dli. The con- stants sg,ugand cE(E0,u0)_1/Z ensure that theequations aredimen- sionally coherent. They areassigned dimensions asfollows ML TZQZIH0I= IE0]=W The functions (Q,-,Q,-):lR3—> IR,each depending onthetime para- meter t,willbecalled theMKSCartesian components oftheelectric and magnetic field respectively. The Cartesian components oftheelectric field have dimensions [ML/TZQ], with MKS units ofNC'i, whilst those ofthemagnetic field have dimensions [M/TQ] with MKS units ofTeslas (orWbm'2). The structure ofthissystem ofcoupled partial differential equations permits one toconstruct aremarkable synthesis between thefields (E,B).This may beachieved byreformulating thesystem interms ofa pair oftensor equations ontheevent manifold IR‘endowed with a particular metric structure. Instead ofassociating theCartesian compo- nents ofE,Bwith vector fields onIR3,they areused toconstruct a 2-form EonIR‘.Using alocal chart (f,1i,12,13) wedefine P=Q+dfA§ (52.3) where Q=Bidxi Adxi+Qtdxi Adxi+Qsdli Adxi L=itdxi +izdxi +£3dl3~ Inasimilar way weunify thecomponents ofthecurrent and charge density toconstruct the3-form 2: Q=¢H0LA (IL+(P/¢50)d£i A(I12A(Iii (5-2-4) where L=lidlz A(IL3‘I’lzdli A(Iii‘I’lzdii Adiz- The {j,-} arethecomponents ofthevector current j.The choice of dimen§ioned coefficients ensures that 2andLhave thesame dimen- sions, namely [ti/Q]. Note that, foranyform f,dfandfhave thesame physical dimensions: theexterior derivative does notchange thephysical dimensions oftheform onwhich itacts. Themetric tensor field adopted onIR‘isgiven inthischart by g=—cZd;®d; + (5.2.5) Hence (cdf, dgi‘) isanorthonormal co-frame with respect tothisg.In terms oftheHodge map *_associated with this Lorentzian-signatured metric Maxwell’s equations may beexpressed elegantly astheexterior equations MAXWELL'S EQUATIONS AND MINKOWSKI SPACETIME 181 d* F = (5.2.6) dF = O. (5.2.7) One further and desirable simplification can be made: the set can be written entirely in terms of dimensionless tensors. First it is trivial to define dimensionless forms F and j by scaling each with any convenient parameters having the dimensions [h/Q]. We choose to write F = (eolh)F j = (eolh)1 where e0 is the elementary charge on the electron. In general, equations involving the Hodge map make reference to a specific metric. The equations (5.2.6) and (5.2.7), however, remain unchanged if we replace g by Ng where A is any positive-definite real-valued function on IR4. This foams since F is a 2-form in four dimensions. It is convenient for us to exploit this freedom here to rescale g by any constant with the dimensions of [1_ ]2 and use a dimensionless metric tensor field g L -2g. We shall denote the Hodge map associated with g as simply * and rewrite the Maxwell equations: d*F = j (5.2.8) dF = O. (5.2.9) One is of course free to use either dimensioned or dimensionless coordinates in extracting component equations from this set. We have spelt out in detail the straightforward manner in which one can make contact with the conventional MKS dimensioned field and source compo- nents. Henceforth we shall work with dimensionless coordinates and tensors. It is worth stressing that although we have built up these equations from the traditional Cartesian-oriented approach the equa- tions are now fully tensorial on the four-dimensional manifold with metric tensor g. We have extricated ourselves from a particular chart including a particular time map. This is a major achievement and may be regarded as the cornerstone development in Einstein's 'relativistic' world view. A metric such as g that has a signature with one minus sign is called Lorentzian. A four-dimensional manifold with Lorentzian metric will be called a spacetime. Tangent vectors in a Lorentzian spacetime may be classified into spacelike (positive-norm), timelike (negative-norm) or null (zero-norm) vectors. The tangent space is said to possess a light cone structure conferred on it by such a metric. Furthermore, timelike tangent vectors may be classified into future-pointing and past-pointing. If Xp is assigned a future-pointing role then —X,, is defined to be past pointing at p. If this assignment can be made unambiguously over the MAxwELL’s EQUATIONS ANDMINKOWSKI SPACETIME 181 djf =2 (5.2.6) d_F=0. (5.2.7) One further anddesirable simplification canbemade: thesetcanbe written entirely interms ofdimensionless tensors. First itistrivial to define dimensionless forms Fandjbyscaling each with anyconvenient parameters having thedimensions [h/Q]. Wechoose towrite F=(eg/h)f I=(50/792 where egistheelementary charge ontheelectron. Ingeneral, equations involving theHodge map make reference toaspecific metric. The equations (5.2.6) and(5.2.7), however, remain unchanged ifwereplace gbyZ/lgwhere Aisanypositive-definite real-valued function onIP14. This follbws since Fisa2-form infour dimensions. Itisconvenient for ustoexploit thisfreedom here torescale gbyanyconstant with the dimensions of[L]2 and useadimensionless metric tensor field g= L'Zg. Weshall denote theHodge map associated with gassimply *and rewrite theMaxwell equations: d*F =j (5.2.8) dF=0. (5.2.9) One isofcourse free touseeither dimensioned ordimensionless coordinates inextracting component equations from thisset.Wehave spelt outindetail thestraightforward manner inwhich one canmake contact withtheconventional MKSdimensioned field andsource compo- nents. Henceforth weshall work with dimensionless coordinates and tensors. Itisworth stressing that although wehave built upthese equations from thetraditional Cartesian-oriented approach theequa- tions arenow fully tensorial onthefour-dimensional manifold with metric tensor g.Wehave extricated ourselves from aparticular chart including aparticular time map. This isamajor achievement andmay beregarded asthecornerstone development inEinstein’s ‘relativistic’ world view. Ametric such asgthat hasasignature with oneminus sign iscalled Lorentzian. Afour-dimensional manifold with Lorentzian metric willbe called aspacetime. Tangent vectors inaLorentzian spacetime may be classified into spacelike (positive-norm), timelike (negative-norm) or null (zero-norm) vectors. The tangent space issaid topossess alight cone structure conferred onitbysuch ametric. Furthermore, timelike tangent vectors may beclassified into future-pointing andpast-pointing. IfX,,isassigned afuture-pointing role then —X,, isdefined tobepast pointing atp.Ifthisassignment canbemade unambiguously over the 182 APPLICATIONS IN PHYSICS whole manifold then the spacetime is said to be time orientable. It would be rather difficult to interpret physical phenomena on a manifold that was not time orientable. The spacetime modelled on TO with metric as in (5.2.5) is called Minkowski spacetime. Thus Minkowski spacetime admits a chart with coordinates (t, x1, x2, x3) in which the metric tensor field is given by 3 g = —dtOdt + dx'Odx'. (5.2.10) = We observe that the vector field (aim has a negative norm whilst (3/3x has a positive norm for i = 1, 2, 3 g((3/at), (a/a0) = —1 g(alaxt, 3/3x') = 1 (no sum). Minkowski space M possesses a 10-parameter group of isometries. In a chart in which the metric is given by (5.2.10) these isometries are generated by the following Killing vector fields To = (3/30, Tk k = 1, 2, 3 K3 = X1(alaX2) — X2(313X1) K2 = X 3(alaX1) — x1(3/3x3) (5.2.12) K1 = x2(3/3x3) — x3(3/ax2) Bk = t(alaX k) Xk(313t) k = 1, 2, 3. The isometry group of Minkowski space is called the Poincaré group. The vectors T = 0, 1, 2, 3, generate translations; the integral curves being open lines. The K i = 1, 2, 3 generate rotations; the integral curves lying on the surface of a sphere. The Bk, k = 1, 2, 3, generate boosts, the integral curves being open, forming hyperbolae. Exercise 5.1 Verify that if X is any of the vector fields in (5.2.12) then xg = 0. The structure of Maxwell's equations motivated the introduction of Minkowski space. In fact the form of Maxwell's equations arrived at, (5.2.8) and (5.2.9), is immediately valid in any Lorentzian spacetime (one not necessarily having the large number of isometries present for Minkowski space). Such a generalisation is the essence of Einstein's incorporation of arbitrary gravitational interactions into the underlying geometry of spacetime. (5.2.11) 182 APPLICATIONS INPHYSICS whole manifold then thespacetime issaid tobetime orientable. It would berather difficult tointerpret physical phenomena onamanifold thatwasnottime orientable. The spacetime modelled onIR‘with metric asin(5.2.5) iscalled Minkowski spacetime. Thus Minkowski spacetime admits achart with coordinates (t,xi,x2,x3)inwhich themetric tensor field isgiven by 3 g=-moat +2dxi®dxi. (5.2.10) i=l Weobserve that thevector field (8/St) has anegative norm whilst (8/Sxi) hasapositive norm fori=1,2,3 g((8/St), (8/8t)) =-1 j _ (5.2.11) g(8/8x’, 8/8x‘) =1 (nosum). Minkowski space Mpossesses a10-parameter group ofisometries. In achart inwhich themetric isgiven by(5.2.10) these isometries are generated bythefollowing Killing vector fields Tg=(3/3!), TX=(8/Gxii) k=1,Z,3 K3=xi(8/8x2) —xi(8/Sxi) K,=x3(8/Sxi) -xi(8/8x3) (5.2.12) K,=xi(8/8x3) —x3(8/8x2) Bk={(3/axi‘) +Xi‘(8/3!) l<=1,Z,3. The isometry group ofMinkowski space iscalled thePoincare group. The vectors T,,,/.1=0,1,2,3, generate translations; theintegral curves being open lines. The K,-,i=1,2,3generate rotations; theintegral curves lying onthesurface ofasphere. The Bk,k=1,2, 3,generate boosts, theintegral curves being open, forming hyperbolae. Exercise 5.1 Verify thatifXisanyofthevector fields in(5.2.12) then gxg =0. The structure ofMaxwell’s equations motivated theintroduction of Minkowski space. Infact theform ofMaxwell’s equations arrived at, (5.2.8) and (5.2.9), isimmediately valid inany Lorentzian spacetime (one notnecessarily having thelarge number ofisometries present for Minkowski space). Such ageneralisation istheessence ofEinstein’s incorporation ofarbitrary gravitational interactions into theunderlying geometry ofspacetime. OBSERVER CURVES 183 5.3 Observer Curves The classical physical interpretation of the components of a tensor field on spacetime is associated with the notion of an observer curve. To introduce the notion of local observer time into the spacetime manifold M we exploit the lightcone structure of the Lorentzian metric. A curve C whose image passes through p E M is said to be timelike at p if its tangent vector is timelike there. Next consider the physical interpreta- tion of the parametrisation of C: [0, 1]—> M. If (t, xk) are local chart maps for M we represent C parametrically by the equations t(p) = C°( -1), xk(p) = Ck(r) and we restrict ourselves to monotonic functions of r that make C a future timelike curve: g(C.ar, < 0. (5.3.1) The length of C is defined to be the real number s = foLg(C*3„ c,3011/2d.r. (5.3.2) Under a change of parametrisation r r'(r) mapping [0, 1]—> [0, 1] with (a ri/a r) >0 V r then C3, )--> (a Tr')(Ca r,) and dr (a r/aildf , so we see that the integral is invariant under such a reparametrisation. A parameter T is said to provide a proper-time parametrisation for C if c*ar) = —1. (5.3.3) An ideal observer is defined to be a proper-time parametrised future-pointing timelike curve on spacetime. The observer image is represented as a history or world line on the manifold. Elapsed time between events on the world line, as measured by such an observer curve, is determined by the difference between the affine parameter assigned to each event. It is a fundamental assumption that there exist standard clocks that operationally determine such an affine parametrisa- tion along their histories. For such curves (5.3.2) implies that the time between events linked by an observer curve is equal to the length of world line linking them; it is measured by a standard clock accompany- ing the ideal observer. This time measure is often called the proper time measured by C. It does appear that many natural processes (for example, decaying particles) can be used as standard clocks registering proper time. Once one is convinced of the existence of microscopic natural clocks for proper time, macroscopic clocks (assemblies of micro- scopic clocks) can then be synchronised using light signals, or any other physical mechanism that supports a formulation in terms of a locally Lorentzian geometry. Once this definition of a good clock is adopted it becomes evident that there is no unique proper time interval between two events that can be joined by a family of timelike observer curves. OBsERvER CURVES 183 5.3Observer Curves Theclassical physical interpretation ofthecomponents ofatensor field onspacetime isassociated with thenotion ofanobserver curve. To introduce thenotion oflocal observer time into thespacetime manifold Mweexploit thelightcone structure oftheLorentzian metric. Acurve Cwhose image passes through peM issaid tobetimelike atpifits tangent vector istimelike there. Next consider thephysical interpreta- tionoftheparametrisation ofC:[0,1]—> M.If(t,xi‘)arelocal chart maps for Mwe represent Cparametrically bythe equations t(p)—C°(r), xi‘(p) =Ci‘(r) and werestrict ourselves tomonotonic functions of‘L’thatmake Cafuture timelike curve: g(C.,8,, C18,) <0. (5.3.1) Thelength ofCisdefined tobetherealnumber I s=L]g(C..8,, C*8,)|i’Zdr. (5.3.2) Under achange ofparametrisation 1:i—>r’(r) mapping [0,1]—>[0, 1] with (8r’/8r) >0Vtthen C..8, i—>(8,r’)(C.'.8,i) and dri—> (81:/8r’)d1:', soweseethat theintegral isinvariant under such areparametrisation. Aparameter 1:issaidtoprovide aproper-time parametrisation forCif g(c..a,, c.6,)=-1. (53.3) An ideal observer isdefined tobeaproper-time parametrised future-pointing timelike curve onspacetime. The observer image is represented asahistory orworld line onthemanifold. Elapsed time between events ontheworld line, asmeasured bysuch anobserver curve, isdetermined bythedifference between theaffine parameter assigned toeach event. Itisafundamental assumption that there exist standard clocks that operationally determine such anaffine parametrisa- tionalong their histories. Forsuch curves (5.3.2) implies that thetime between events linked byanobserver curve isequal tothelength of world linelinking them; itismeasured byastandard clock accompany- ingtheideal observer. Thistime measure isoften called theproper time measured byC.Itdoes appear that many natural processes (for example, decaying particles) canbeused asstandard clocks registering proper time. Once one isconvinced oftheexistence ofmicroscopic natural clocks forproper time, macroscopic clocks (assemblies ofmicro- scopic clocks) canthen besynchronised using light signals, oranyother physical mechanism that supports aformulation interms ofalocally Lorentzian geometry. Once thisdefinition ofagood clock isadopted it becomes evident that there isnounique proper time interval between twoevents that canbejoined byafamily oftimelike observer curves. 184 APPLICATIONS IN PHYSICS Each curve will in general measure a different time interval since each curve has a different arc length. A timelike vector field V is called a world velocity (or four-velocity) vector field if g(V , V) = —1. As an example consider the vector field V = k(a t + &axe) (5.3.4) J= in a local chart (t, x1) in which the Minkowski metric g takes the form (5.2.10). The field is labelled by real constants k, y1, v2, v3. V is a velocity vector if k = [1 — ( y1)2 _ (y2)2 _ (v3)21-112 (5.3.5) What observer curve C has a tangent vector that coincides with V at each p on its image? For this we require 3 C*arlp = (at/a r)atlp E(axj/ar)axilp = v1„ i= 1 that is ((3t/ar), (ax)/ar)) = k(1, vi). These equations fix the paramet- risation of C up to an additive constant for r. For C labelled by the triplet y = (y 1, y2, 3) E IR3 a family of observer curves through the origin of the (t, xk) chart has the representation t(p) = kr, x1(p)= kviT or Op) = v1t, j = 1, 2, 3. For arbitrary constant t.7 the vector field V is a Killing field. We define a stationary observer to be a proper-time parametrised integral curve of a timelike Killing vector. Thus the vector field V, with arbitrary constant ty, yields a three- parameter family of stationary observers in Minkowski space. Any global Minkowski space chart in which the metric takes the form (5.2.10) is often referred to as an inertial chart. The chart maps define a global co-frame of exact 1-forms. The vector field V defines a congru- ence of ideal observers, each ideal observer being an integral curve of V. One often sees the phrase 'an inertial frame' or 'an inertial system' in this context. Care will be exercised in not adopting this phrase too readily: we have not assigned a frame of vectors along any observer curve so cannot at this stage, strictly speaking, make reference to an observer's inertial frame. However, the frame {at, 3,1} associated with the inertial chart is an example of an inertial frame along the integral curves of 3,. We shall return to the general definition of observer frames after we have introduced the concept of vector transport. The equation of the world line of a stationary observer in an inertial chart suggests that the triplet y be identified with the components of a Newtonian velocity three-vector. However, we would prefer to identify such a notion in the context of a general observer, not necessarily a stationary one. Since we now contemplate arbitrary observers we concentrate on TM rather than the whole history of the arbitrary 184 APPLicATioNs INPHYSICS Each curve willingeneral measure adifferent time interval since each curve hasadifferent arclength. Atimelike vector field Viscalled aworld velocity (orfour-velocity) vector field ifg(V, V)=—1.Asanexample consider thevector field 3 v=i<(a,+Z6/6,.) (53.4) 1-1 inalocal chart (t,xi)inwhich theMinkowski metric gtakes theform (5.2.10). The field islabelled byreal constants k,vi,vi,03.Visa velocity vector if kZ[1_(Ul)2 _(U2)2 _(,,3)2]-i/2_ (5_3_5) What observer curve Chasatangent vector that coincides with Vat each ponitsimage? Forthiswerequire 3 c..a,|, =(at/at)a,j, +2(&)xl/E)r)E),.,I,, =vj, (=1 that is((6)1/Gr), (E)xl/61)) =k(1, vi).These equations fixtheparamet- risation ofCuptoanadditive constant forr.For Clabelled bythe triplet u=(vi,vi,v3)eIR3 afamily ofobserver curves through the origin ofthe (t,xi‘) chart has the representation t(p) =kr, xl(p) =kv/ir orXi(p) =vit,j=1,2,3. For arbitrary constant uthe vector field VisaKilling field. Wedefine astationary observer tobea proper-time parametrised integral curve ofatimelike Killing vector. Thus thevector field V,with arbitrary constant u,yields athree- parameter family ofstationary observers inMinkowski space. Any global Minkowski space chart inwhich themetric takes theform (5.2.10) isoften referred toasaninertial chart. The chart maps define a global co-frame ofexact 1-forms. The vector field Vdefines acongru- ence ofideal observers, each ideal observer being anintegral curve of V.One often sees thephrase ‘aninertial frame’ or‘aninertial system’ inthiscontext. Care willbeexercised innotadopting thisphrase too readily: wehave notassigned aframe ofvectors along anyobserver curve socannot atthisstage, strictly speaking, make reference toan observer’s inertial frame. However, theframe {E),,E),,,}associated with theinertial chart isanexample ofaninertial frame along theintegral curves of6),.We shall return tothegeneral definition ofobserver frames after wehave introduced theconcept ofvector transport. The equation oftheworld lineofastationary observer inaninertial chart suggests that thetriplet vbeidentified with thecomponents ofa Newtonian velocity three-vector. However, wewould prefer toidentify such anotion inthecontext ofageneral observer, notnecessarily a stationary one. Since wenow contemplate arbitrary observers we concentrate onT,,M rather than thewhole history ofthearbitrary OBSERVER CURVES 185 observer world line. A point p E M together with a future-pointing timelike vector with norm —1 will be called an instantaneous observer at p. Let C be such an instantaneous observer associated with the general observer C and let A be any timelike future-pointing 1-chain (not necessarily another observer) with tangent vector A, at p. We wish to define the Newtonian velocity of A observed by C at p. Since E TM we have a unique orthogonal decomposition (5.3.6) where t E IR and g(P , C) = O. This latter condition implies f, --= —g(/t, t) since g(C, = —1, hence A = P — g(A, C)C. (5.3.7) The Newtonian velocity of A observed by C at p is now defined with respect to this orthogonal decomposition as i; = P/, or = t) (5.3.8) showing that u depends on both A and C. The vector P is spacelike and is said to lie in an instantaneous three -space of C at p. This is defined as the orthogonal complement of C in TM. We next consider the case of a null 1-chain F observed by C. The condition g(F, = 0 inserted into r = P — g(1; , C)e gives, with the aid of g(P , P) -= g(1; , P), = t(t - N) (5.3.9) where —= —g(F, C) and N (g(1' , C)/g(F, P))P is called the energy that C observes for r at p whilst N is the spatial direction observed for r. Note that N is spacelike with g(N , N) = 1. It is a fundamental result that there exist propagating solutions to Maxwell's equations corres- ponding to the phenomenon of electromagnetic waves. Such waves propagate in vacuo without dispersion and have null vector fields associated with them. Thus null curves may model the flow of electro- magnetic radiation, or photons. The images of timelike future-pointing curves are models for either massive point particles or the streamlines of mass—energy flows. A point particle of mass m is modelled by a future-pointing curve p with g(p, = —m2. Then p=P+ZC implies therefore implies g(p, p) ce2g(v, u) = V m2. Hence `6 and P may be expressed in terms of iy as (5.3.10) ep, ± m2 = z2; p = ZA = [1 — g(v, v) ]"2 (5.3.11) OBSERVER CURVES 185 observer world line. Apoint peM together with afuture-pointing timelike vector with norm -1willbecalled aninstantaneous observer atp. LetCbesuch aninstantaneous observer associated withthegeneral observer CandletAbeanytimelike future-pointing 1-chain (not necessarily another observer) with tangent vector Aatp.Wewish to define theNewtonian velocity ofAobserved byCatp.Since A,CeT,,M wehave aunique orthogonal decomposition A=P+aC 63$ where eIRand g(P, C)=0.This latter condition implies =—g(A, C)since g(C, C)=-1,hence A=P-g(/I,C)C. (53.7) The Newtonian velocity ofAobserved byCatpisnow defined with respect tothisorthogonal decomposition asv=P/‘E, or v=—P/g(/I, C) (5.3.8) showing that vdepends onboth AandC.The vector Pisspacelike and issaidtolieinaninstantaneous three-space ofCatp.This isdefined as theorthogonal complement ofCinT,,M. Wenext consider thecase ofanull1-chain Fobserved byC.The condition g(F,F)=Oinserted intoF=P—g(F,C)Cgives, withtheaid Ofg(P,P)=s(1’FP), r‘=<a(c-N) (53.9) where ‘EE—g(F, C)andNE(g(F, C)/g(F, P))P. Eiscalled theenergy thatCobserves forFatpwhilst Nisthespatial direction observed for F.Note thatNisspacelike withg(N, N)=1.Itisafundamental result that there exist propagating solutions toMaxwell’s equations corres- ponding tothephenomenon ofelectromagnetic waves. Such waves propagate invacuo without dispersion and have null vector fields associated with them. Thus nullcurves maymodel theflow ofelectro- magnetic radiation, orphotons. Theimages oftimelike future-pointing curves aremodels foreither massive point particles orthestreamlines ofmass—energy flows. Apoint particle ofmass mismodelled byafuture-pointing curve pwith g()O,p)=—mi. Then p=P+ECimplies g(P,P)+mi=‘Ei;P=E“ therefore implies g(P,P)=‘<?»ig(v. v)=‘éi—mi- (5-3-10) Hence EandPmaybeexpressed interms ofvas _ m<6-——i,,_g(y,0),”, (53.11) 186 APPLICATIONS IN PHYSICS mv P — (5.3.12) [1 _ ev, v)]1/2 Clearly if m 0, g(y, y) = 1 — (mre)2 < 1: that is, massive particles are observed to have bounded Newtonian velocities. If, for example, t = aIp and A = + i()aj in an inertial Minkowski chart then g(t, A) = —i and A = P + (-Oa I gives P = 11(1-)3. Hence in this chart u = (T) (1))a p is the Newtonian velocity of A observed by C at p. If the projection onto the instantaneous three-space orthogonal to e at p is effected by the projection operator Ilp:TpM (e)pi , = (1 — { t()} 't® op, then the Newtonian length of any space- like vector V E TM observed by C is defined as (galp V, Hp VW'. If W is a second spacelike vector in TM then the Newtonian angle between V and W observed by C is given by g(Hp V, FIp W) cos 0 — (5.3.13) [g(Hp V, Hp V)g(Hp W, Hp W)]1/2. The presence of the projector li p in these formulae, defined by the observer curve, means that the Newtonian length and angles specified in this way depend on the observer as well as on the vectors being observed. For the general future-pointing vector A=P+ZC we see that V has Newtonian length (g(v, v))"2 = [g(p, PA 1/2 k.. e If );.. is null, g(P,P) = and hence all null vectors are always observed to have unit length Newtonian velocities. We have already noted that g(v, v) < 1 if y is the Newtonian velocity of a particle with m O. If g(v, v) << 1 we may expand (4.3.11), (4.3.12) using the binomial expansion = m + v) + . . . (5.3.14) P = mv + . . (5.3.15) These formulae reinforce our identification of the instantaneous energy and three-momentum for a point particle. We see that the Newtonian kinetic energy of such a particle differs from the relativistic energy by the constant m. This difference between Newtonian and Einsteinian relativistic kinematics has had a profound effect in the subsequent development of relativistic physics. The images of different observer curves may be related by a diffeo- morphism of spacetime: in particular a diffeomorphism from the isometry group. We here consider a 'boost' diffeomorphism from the Poincaré group. We first compute part of an integral curve of the 'boost' vector field X = xla, + taxi (5.3.16) passing through a point Po with coordinates APPLICATIONS INPHYSICS mvP [1_g(U, U),,,2. (5.3.12) Clearly ifmE0,g(a, o)=1—(m/‘<€)i <1:that is,massive particles areobserved tohave bounded Newtonian velocities. If,forexample, C=8,1,, andA=x/(i:)8,,|,, +t(i:)8,|,, inaninertial Minkowski chart then g(C, A)=—i and A=P+i(i:)8,|,, gives P=xl(i:)8,,. Hence inthischart o=(x/(r)/t(r))8,1],, istheNewtonian velocity ofAobserved byCatp. Iftheprojection onto theinstantaneous three-space orthogonal toC atpiseffected byfhe projection operator II,,:T,,M —>(C),f, II,,=(1—{C(C)} ‘iC® C),,, then theNewtonian length ofanyspace- likevector VeT,,M observed byCisdefined as(g(II,, V,II,,V))iii. If Wisasecond spacelike vector inT,,M then theNewtonian angle between VandWobserved byCisgiven by 6650= giniv’ niwi . (5.3.13)[g(II,,V, II,,V)g(II,,W, II,,W)]iii The presence oftheprojector II,,inthese formulae, defined bythe observer curve, means that theNewtonian length andangles specified in this way depend onthe observer aswell asonthevectors being observed. For thegeneral future-pointing vector A=P+‘ECwesee that VhasNewtonian length (g(a, o))iii =[g(P, P)]iii/‘E. IfAisnull, g(P, P)=‘éiandhence allnullvectors arealways observed tohave unit length Newtonian velocities. Wehave already noted that g(a, u)<1if oistheNewtonian velocity ofaparticle with mE0.Ifg(a, 0)<<1we may expand (4.3.11), (4.3.12) using thebinomial expansion ‘E=m+§mg(v, v)+... (5.3.14) P=mo+ (5.3.15) These formulae reinforce ouridentification oftheinstantaneous energy andthree-momentum forapoint particle. Weseethat theNewtonian kinetic energy ofsuch aparticle differs from therelativistic energy ‘Eby theconstant m.This difference between Newtonian and Einsteinian relativistic kinematics hashadaprofound effect inthesubsequent development ofrelativistic physics. The images ofdifferent observer curves may berelated byadiffeo- morphism ofspacetime: inparticular adiffeomorphism from the isometry group. Wehere consider a‘boost’ diffeomorphism from the Poincare group. Wefirstcompute part ofanintegral curve ofthe‘boost’ vector field X=x18, +t8,,i (5.3.16) passing through apoint pgwith coordinates OBSERVER CURVES 187 (((po), -xl(Po), x2(P0), x3(P0)) in an inertial chart. We shall take p o to lie outside the 'light cone of (0, 0, 0, 0)', defined as the set L of points p satisfying 3 ( (p))2 - (t(P))2 = 0. This ensures that at p o X is timelike. For definiteness we shall assume t(po)> 0, xl(po) > 0 j = 1, 2, 3. The integral curve is given parametri- cally as t(p) = A°(r), xi(p) =A 1(r) j = 1, 2, 3, where the functions A":[0, co) --> lB, t = 0, 1, 2, 3 satisfy dA1/dr = A°, dA°/dr = A 1 dA 2/dr = 0, dA 3/dr = 0. Thus the curve is given by the solution A 1 (r) = (0) cosh T AC/(0) sinh A°(r) = A°(0) cosh r + A 1(0) sinh r A2(T) = A2(0) A3(r) = A3(0). Eliminating r between Al(r) and A°(r) gives part of a hyperbola through Po and p (Ai(T))2 _ (Ao( T))2 _ (A1(0))2 _ (A0(0))2 (5.3.18) If we relabel the functions AP with coordinate names (with po specified by 2 = 0), equations (5.3.17) may be rewritten as xl(po) + vt(p o) t(p) = _ (5.3.20) (1 v2)1/2 where cosh T = 11(1 - V 2)112 and sinh r = v/(1 — v 2)1/2 >0. These famil- iar equations relate the point po to the point p labelled by the parameter v = tanh r along the boost orbit (5.3.18). For a fixed v we have a diffeomorphism, generated by X, that may be used to relate two observer fields. Define the map Tv: M M, p p' t(p') = (t(p) + vx1(p))1(1 v2)112 xi(p') = (Op) + vt(p))1(1 — v 2)112 .0(p') = .0(p) k = 2, 3 (5.3.17) xl(p) = (1 _ 0112 t(p0) + vxl(po) (5.3.19) OBSERVER CURVES 187 (t(pO.)> x1(pO)> x2(p0)v x3(p0)) inaninertial chart. Weshall take pgtolieoutside the‘light cone of (0,0,0,0)’,defined asthesetLofpoints psatisfying 3E(x'(p>>2 -(i(p>>2=0. [=l This ensuresthat atpgXistimelike. Fordefiniteness weshall assume t(pg) >0,x/(pg) >0j=_1, 2,3.The integral curve isgiven parametri- cally 85f(p)=/10(1). X'(p) =A/(I) j=1,2,3,where thefunctions A“:[0, 09)—>IR,,u=0,1,2,3satisfy dAi/dr =A0,dA°/dr =A‘ dAi/dr =0,dA3/dr =0. Thus thecurve isgiven bythesolution Ai(r) =Ai(0) coshr +A°(0) sinhr A°(r) =A°(0) cosh ‘E+Ai(0) sinhr 7 2 (5.3.17) /P(T)=/I(0) A3(r) =Ai(0). Eliminating rbetween Ai(r) andA°(r) gives partofahyperbola through pgandp (/li(f))i —(/I°(I))i =(/li(0))i -(/i°(0))i~ (5-3-13) Ifwerelabel thefunctions A”with coordinate names (with pgspecified byr=0),equations (5.3.17) mayberewritten as xi(p)=i (5.319) t(p)=' (53.20) where coshr =1/(1—vi)iii andsinhr =v/(1—vi)i’i >0.These famil- iarequations relate thepoint pgtothepoint plabelled bythe parameter v=tanh ralong theboost orbit (53.18). Forafixed vwehave adiffeomorphism, generated byX,thatmay be used torelate twoobserver fields. Define themap (PtIM—>M.P—>P’ t(p’) =(t(p) +vxi(p))/(1 —vi)iii Xi(P') =(Xi(P) +vt(P))/(1 -vi)i’i x"(P') =Xi(P) /<=2.3 188 APPLICATIONS IN PHYSICS then :3 ,11, 1--> (at + yax) — y2)1/2. Thus the fixed parameter y can be identified as the Newtonian velocity of (q)ail as measured by a jp, for all p'. Note that for all r, V = tanh r < 1. It is of interest to note that since two successive diffeomorphisms of the above type parametrised by T1 and r2 respec- tively produce a diffeomorphism parametrised by T1 ± T2: coon° Tr, =(Pr, + we obtain as Newtonian velocity parameter y12 corresponding to T1 r2 v 12 tanh(ri + r2) tanh ri + tanh + 1 tanh ri tanh r2 1 + v v2 For all y1, y2 < 1, u12 = < 1, that is successive 'boost' transform- ations applied to observer curves can never give rise to observer curves with a Newtonian velocity in excess of 1 relative to all observers. 5.4 Electromagnetism In §5.2 we used the structure of Maxwell's equations to motivate the introduction of a four-dimensional Lorentzian spacetime. We here examine some further properties of these equations. If a is a p-form on U, U C M, satisfying the equation da = 0 it is said to be closed on U. Then there exist some region W C U for which = d13, for 13 a (p — 1)-form on W. The p-form a is then said to be exact on W. It is an important result that the global topology of U determines whether or not all closed forms are exact on U. For our local discussion, however, we can assert that the Maxwell equation dF = 0 implies that in some neighbourhood of every point on M there exists a 1-form A such that F = dA. Clearly given such an A there exists an equivalence class satisfying the same condition. Two members of this class differ by an exact 1-form a where A E Fi(U). The freedom to choose a 1-form potential from such a class is known as local electromagnetic gauge invariance. Two potentials in this class are said to be co-homologous. In a local Minkowski chart (t, x k) we may write 3 A = E A kdxk + cpdt k =1 and hence relate the real-valued functions Ak, cp to some electro- 188 APPLICATIONS INPHYSICS then ((pv)*ia1ip ii>(a1+ vax)Ip'/(1— v2)i/2' Thus thefixed parameter vcanbeidentified astheNewtonian velocity of(cp,,),.8,|,,i asmeasured by8,|,,- forallp’.Note that forallr, v=tanhr<1. Itisofinterest tonote that since two successive diffeomorphisms oftheabove type parametrised byr,and T3respec- tively produce adiffeomorphism parametrised byr,+T22 <Pr,°<Pr,=<P1,+1. weobtain asNewtonian velocity parameter 12,2corresponding toT1+T2 on=tanh(r1 +T2) tanh T,+tanh 1'2 v,+U2 :1+tanhr,tanh'r2 :1+vlvzi Forall12,,v2<1,12,2=U21<1, that issuccessive ‘boost’ transform- ations applied toobserver curves cannever give risetoobserver curves with aNewtonian velocity inexcess of1relative toallobservers. 5.4Electromagnetism In§5.2 weused thestructure ofMaxwell’s equations tomotivate the introduction ofafour-dimensional Lorentzian spacetime. We here examine some further properties ofthese equations. Ifatisap-form onU,UCM,satisfying theequation da=0itis saidtobeclosed onU.Then there exist some region WCUforwhich or=dfi,for[3a(p—1)-form onW.The p-form oristhen said tobe exact onW.Itisanimportant result that theglobal topology ofU determines whether ornotallclosed forms areexact onU.For our local discussion, however, wecan assert that theMaxwell equation dF=0implies that insome neighbourhood ofevery point onMthere exists a1-form Asuch that F=dA.Clearly given such anAthere exists anequivalence class satisfying thesame condition. Two members ofthisclass differ byanexact 1-form dAwhere Ae‘J(U).The freedom tochoose a1-form potential from such aclass isknown aslocal electromagnetic gauge invariance. Two potentials inthisclass aresaidto beco-homologous. Inalocal Minkowski chart (t,xi‘)wemay write 3 A=2/tkdx" +out I<=1 and hence relate thereal-valued functions AX, cptosome electro- ELECTROMAGNETISM 189 dynamic 'vector' and 'scalar' potentials. Introducing a local potential A means that (5.2.9) is satisfied identically and the other equation (5.2.8) becomes d*dA = j. (5.4.1) The above equation may be written in terms of the Laplace—Beltrami operator. To define this we need to introduce the co-derivative. On a general n-dimensional (pseudo-) Riemannian manifold we define the co-derivative 6: FApM FAIM _ by 6 = *-1cl*n. (5.4.2) (Recall from (1.1.2) that if cp is a p-form ncp cp') = (+1)Pcp.) Since on p-forms ** (-1)p(p _ p) det g det gl det g (5.4.3) det gi it follows immediately that 6 has the property 66 = 0, in common with d. The signs in the definition of the co-derivative are chosen to ensure that it is the adjoint operator to the exterior derivative, with respect to a certain inner product on differential forms on a compact Riemannian manifold. If M is a compact Riemannian manifold OM = 0) then a symmetric product on p-forms is defined by ' A *13 a, /3E FApM. (5.4.4) An 'integration by parts' gives, with Stokes's theorem and the compact- ness of M, (T, dip) = (No, 10 FAp _ 1M. That is, 6 is the adjoint of d with respect to this product. The Laplace—Beltrami operator is defined by A= —(c16 + (5c1). (5.4.5) Note that since d(6) increases (decreases) the degree of a form by one the Laplace—Beltrami operator preserves the degree of a form. With our conventions the Laplace—Beltrami operator has negative eigenvalues on a compact Riemannian manifold. In terms of the product of (5.4.4): (cP, AT) = —(9), d649) (cP, = —(6q', 6cp) — (dcp, cl(p). ELEcTRoMAGNETIsM 189 dynamic ‘vector’ and‘scalar’ potentials. Introducing alocal potential A means that (5.2.9) issatisfied identically andtheother equation (52.8) becomes d*dA =j. (5.4.1) The above equation may bewritten interms oftheLaplace—Beltrami operator. Todefine thisweneed tointroduce theco-derivative. Ona general n-dimensional (pseudo-) Riemannian manifold wedefine the co-derivative 6:FA,,M——>l“/\,,_1M by5=*'id*17. (5.4.2) (Recall from (1.1.2) that if(pisap-form mpE(p"=(—1)P(p.) Since on p-forms detg*=_1P("—P)__i*iIldetglEnn_,detg (54.3) ldsigl itfollows immediately that <5hastheproperty 66=0,incommon with d.The signs inthedefinition oftheco-derivative arechosen toensure thatitistheadjoint operator totheexterior derivative, with respect toa certain inner product ondifferential forms onacompact Riemannian manifold. IfMisacompact Riemannian manifold (SM =0)then a symmetric product onp-forms isdefined by (L1/,B)EIM(.Y A*f3 a,BeFA,,M. (5.4.4) An‘integration byparts’ gives, with Stokes’s theorem andthecompact- nessofM, (<P-d1l1)=(<5<P,1l1) (PEP/\,,M. 1l/6F/\p-1M- That is,<5isthe adjoint ofdwith respect tothis product. The Laplace—Beltrami operator Aisdefined by AE-(65+6d). (5.4.5) Note that since d(6) increases (decreases) thedegree ofaform byone theLaplace—Beltrami operator preserves thedegree ofaform. With our conventions theLaplace—Beltrami operator hasnegative eigenvalues on acompact Riemannian manifold. Interms oftheproduct of(5.4.4): ((1%Aw)="((1%d<5<P)—(P.6d<P) =—(5<P- 64>)—(d<P-d<P)- 190 APPLICATIONS IN PHYSICS The positivity of the Riemannian metric ensures that the right-hand side is negative-definite, thus so are any eigenvalues. The equation (5.4.1) can be written in terms of A as (A + dS)A = — *j. (5.4.6) It is possible to select a representative potential from the class of co-homologous 1-forms such that SA = 0. Such a choice is called selecting a Lorentz gauge. In this gauge the potential satisfies a Helmholtz wave equation: AA = —*j. (Note that the potential is not uniquely fixed by the Lorentz gauge condition. If A is changed to A' = A + dA, A c 9;(M), then SA ' = (SdA = 0 also if  is chosen to be harmonic, that is satisfy Ail. = 0.) Let us examine some solutions to Maxwell's equations in a region of Minkowski spacetime free of sources. Suppose we seek a solution to (5.4.1) of the form A = f dt, f E (M) using a polar chart (t, r, B, yo) in which g = —dtOdt + drOdr + r 2d00d0 + r 2 sin2Odcp04. We shall look for a static 'spherically symmetric' solution satisfying the symmetry condition 2' KF = 0 where the timelike Killing vector is Ko = (3/3t) and the rotational Killing vectors take the form K1 = sin Tao + cot 0 cos CO K2 = — COS cpae + cot Osin pa, (5.4.7) K3 = a. This can be achieved if the function f involves only the coordinate map r. A convenient orthonormal co-frame is {di ., dr, rd0, r sin 0 dcp). Then dA = afdr A dt = arfel A eo, so if *1 = el A e2 A e3 A eo then *dA = (ar)fe3 A e2 = (30fr2  s-in 0 dcp A de. Thus d*dA = a r(a Jr2)dr A sin 0 drp A de. This is zero if f = kir for some constant k. The solution A = kdtlr yields the electric 2form F = dA = —(kIr2)dr A dt. This is the Coulomb solution. The frame-dependent electric field 1-form E iF = (kIr2)dr gives the electric field vector -E.- = (k/r2)3 r, the integral curves of which give the familiar radial Coulomb pattern associated with a stationary charge in this frame. For a general F we define f c*F as the electric charge Q contained in the interior of the sphere which is the image of C. (If the charge is non-zero then this S 2 cannot be the boundary of a source-free region!) (Restoring dimensioned variables, is2* F = (Ed juoinQ 190 APPLICATIONS INPHYSICS Thepositivity oftheRiemannian metric ensures thattheright-hand side isnegative-definite, thus soareanyeigenvalues. Theequation (5.4.1) canbewritten interms ofAas (A+65)/1=-*1". (5.4.6) Itispossible toselect arepresentative potential from theclass of co-homologous 1-forms such that 6A=0. Such achoice iscalled selecting aLorentz gauge. Inthis gauge the potential satisfies a Helmholtz wave equation: AA=—*j. (Note that thepotential isnot uniquely fixed bytheLorentz gauge condition. IfAischanged to A’=A+dA,Ae‘f(M), then 6A’=6dA=0also ifAischosen tobe harmonic, thatissatisfy AA=0.) Letusexamine some solutions toMaxwell’s equations inaregion of Minkowski spacetime free ofsources. Suppose weseek asolution to (5.4.1) oftheform A=fdt, fe‘f(M) using apolar chart (t,r,6,rp)in which g=—dt®dt +dr®dr +rid6®d6 +risini6drp®d(p. Weshall look forastatic ‘spherically symmetric’ solution satisfying the symmetry condition SEX,F =0where thetimelike Killing vector is Kg=(8/St) andtherotational Killing vectors take theform K1= sinrpéig +cot6cosrp8,, K2=—cos(p8g +cot6sin(p8,,, (5.4.7) K3=8,,,. This canbeachieved ifthefunction finvolves only thecoordinate map r.Aconvenient orthonormal co-frame is{dt,dr,rd6, rsin6d(p}. Then dA=8,fdrAdt =8,fei Aei’, so if *1=eiAeiAe3Ae° then *dA =(8,)fe3 Aei=(8,)fri sin6drpA d6.Thus d*dA =8,(8,fri)drA sin6d(pA d6.This iszero iff=k/rforsome constant k.The solution A=kdt/r yields theelectric 2--form F=dA=—(k/ri)drAdt. This is theCoulomb solution. The frame-dependent electric field 1-form EEig,F= (k/ri)dr gives theelectric field vector E=(k/ri)8,, the integral curves ofwhich give the familiar radial Coulomb pattern associated with astationary charge inthisframe. Forageneral Fwedefine jg*F astheelectric charge Qcontained in theinterior ofthesphere which istheimage ofC.(Ifthecharge is non-zero then thisSicannot betheboundary ofasource-free region!) (Restoring dimensioned variables, IS;E=(50//10)"Q ELECTROMAGNETISM 191 determines a charge Q in Coulombs.) A class of 2-chains will determine the same electric charge. We define an equivalence relation on 2-chains as follows: C1 C2 iff CI = C2 ± 3E, where E is any source-free region. Equivalent chains are said to be homologous. Since in source-free regions *F is closed, Stokes's theorem ensures that the charge Q only depends on the class of chain chosen. As an example, we take C to be the 2-chain in Minkowski spacetime whose image is the sphere t -= constant, r = constant. Then for the Coulomb solution IC *F = kJ.c sin Od0 A ckp ,r12 2ki sin @MI° dcp = 471k. Since a Lie derivative with respect to a Killing vector K commutes with the Hodge map, 21c* -= *2K, and all Lie derivatives commute with d, we may deduce that if F satisfies the Maxwell equations with source 3-form j then EKF satisfies them with the source EKI. The existence of an underlying isometry group of spacetime is often used implicitly in constructing new solutions of Maxwell's equations from simpler ones. If we recall the definition of the Lie derivative, and compare it with the elementary textbook calculation used to construct the electric dipole solution as a limit of two equal and opposite Coulomb solutions, we indeed expect the following potential to provide a source-free solution k „ A = (-11).T k (wax—u ,)t = — (Z (a/ax1)r)dt — dt. The vector field Pax 1) represents a Minkowski space Killing vector in an inertial chart. Since the Lie derivative commutes with d, F = —.L(a/axi)(—ur A lit) r2 is the field of a static electric dipole with moment 0. In general for positive integers p, q, r a 'p, q, r'-type electric multipole solution fol- lows from Poincaré covariance as q F = {YP {Y q r {----r A r2 j, k = 1, 2, 3. There is one further symmetry of Maxwell's equations that deserves mentioning. A spacetime is said to admit local conformal isometries, generated by a vector field C, if the metric g is such that Icg = Ag (5.4.8) ELEcTR0IvIAGNETisIvI 191 determines acharge QinCoulombs.) Aclass of2-chains willdetermine thesame electric charge. Wedefine anequivalence relation on2-chains asfollows: C1EC2 iffC,=C2+62,where Zisanysource-free region. Equivalent chains aresaid tobehomologous. Since insource-free regions *Fisclosed, Stokes’s theorem ensures that thecharge Qonly depends ontheclass ofchain chosen. Asanexample, wetake Ctobe the 2-chain inMinkowski spacetime whose image isthe sphere t=constant, r=constant. Then fortheCoulomb solution L*F =kLsin6d6A d(p it/Z Zn =Zkfo sin6d6f0 d(p=411k. Since aLiederivative with respect toaKilling vector Kcommutes with theHodge map, SEX* =*SEX, andallLiederivatives commute with d,wemay deduce thatifFsatisfies theMaxwell equations with source 3-form jthen SEXF satisfies them with thesource SEXj. The existence of anunderlying isometry group ofspacetime isoften used implicitly in constructing new solutions ofMaxwell’s equations from simpler ones. If werecall thedefinition oftheLiederivative, andcompare itwith the elementary textbook calculation used toconstruct theelectric dipole solution asalimit oftwo equal and opposite Coulomb solutions, we indeed expect thefollowing potential toprovide asource-free solution k IA=(%);e(,,,,.,7tit =-%(;e,,,,,.,r)tit =-i‘rl,<it. The vector field (<9/Eixi) represents aMinkowski space Killing vector in aninertial chart. Since theLiederivative commutes with d, _P /<F -‘ ?.§E(a/3,,i,(7(IIt‘A isthefield ofastatic electric dipole with moment It.Ingeneral for positive integers p,q,ra‘p,q,r’-type electric multipole solution fol- lowsfrom Poincaré covariance as F={§5(a/ar')}i’{§5(a/at/)}‘i{§5(a/a.~*)}'{%d'A 91] i,j,k=1,2,3. There isonefurther symmetry ofMaxwell’s equations that deserves mentioning. Aspacetime issaid toadmit local conformal isometries, generated byavector field C,ifthemetric gissuch that 192 APPLICATIONS IN PHYSICS for some scale function A. For any n-dimensional space (n even) it then follows that if F E FA,,,M then c(*F) = *(2 cF). (5.4.9) Hence in a spacetime (n = 4) with a metric g, if such a C exists and the Maxwell 2-form F solves Maxwell's equations with source j then cF will be a solution, in the same metric, with source 2cj. In particular if j = 0 the source-free Maxwell equations exhibit a local conformal covariance in spaces admitting conformal isometries. Clearly, as a special case, all Killing vectors generate such symmetries, corresponding to the zero scale function. It turns out that in Minkowski space there are five further vector fields which are given in an inertial chart, with their scale functions, below D = .0(313x 0) AD 2 = g(D, D)(313.0 — 2xD AK,, = -4X I., (5.4.10) ft = 1, 2, 3, 0. These vector fields along with the 10 Killing vectors generating the Poincaré group, generate the 15-parameter local conformal group of Minkowski space. The source-free Maxwell equations are said to be conformally covariant in Minkowski space. Such a symmetry will gen- eralise to any space with a metric admitting local conformal isometries and the vector C in (5.4.8) is referred to as a conformal Killing vector of the metric g. The local conformal symmetry may generalise to a global symmetry if the topology of the spacetime manifold can accommodate a complete conformal Killing vector field. In §5.2 our introduction to Minkowski spacetime was motivated by the elegant reformulation of Maxwell's equations into a four- dimensional form. We now reverse the argument and show how these four-dimensional electromagnetic fields can be broken down into electric and magnetic fields in the instantaneous three-space of an arbitrary observer. Given any velocity vector field V, whose integral curves coincide with a set of observer curves, we use the Minkowski metric to define the associated dual 1-form V and write any F uniquely as F=ÉAV+B (5.4.11) where B is a 2-form satisfying i vB = 0 and È a 1-form satisfying iE = 0. (Note: a/at = —cit.) One refers to B E FA,M as the magnetic 2-form associated with V and F, and E efAIM as the associated electric 1-form. The electric field observed by this class of observers is E = ivF. (5.4.12) The magnetic vector field observed by this class can be related to F as 192 APPLICATIONS INPHYSICS forsome scale function AC.For anyn-dimensional space (neven) it then follows thatifFel'A,,,2M then $(‘(*F) =-(saga). (5.4.9) Hence inaspacetime (n=4)with ametric g,ifsuch aCexists andthe Maxwell 2-form Fsolves Maxwell’s equations with source jthen SECF willbeasolution, inthesame metric, with source SEC]. Inparticular if j=0the source-free Maxwell equations exhibit alocal conformal covariance inspaces admitting conformal isometries. Clearly, asa special case, allKilling vectors generate such symmetries, corresponding tothezero scale function. Itturns outthatinMinkowski space there are fivefurther vector fields which aregiven inaninertial chart, with their scale functions, below D=x'i(é)/Bx”) Ag=2 1<,,=g(D,D)(6)/ox") -2x,,D AX,=—4x,, (5.4.10) jt=1,2,3,0. These vector fields along with the10Killing vectors generating the Poincaré group, generate the15-parameter local conformal group of Minkowski space. The source-free Maxwell equations aresaid tobe conformally covariant inMinkowski space. Such asymmetry willgen- eralise toanyspace with ametric admitting local conformal isometries andthevector Cin(5.4.8) isreferred toasaconformal Killing vector ofthemetric g.The local conformal symmetry may generalise toa global symmetry ifthe topology ofthe spacetime manifold can accommodate acomplete conformal Killing vector field. In§5.2 ourintroduction toMinkowski spacetime was motivated by the elegant reformulation ofMaxwell’s equations into afour- dimensional form. Wenow reverse theargument andshow how these four-dimensional electromagnetic fields canbebroken down intoelectric and magnetic fields intheinstantaneous three-space ofanarbitrary observer. Given any velocity vector field V,whose integral curves coincide with asetofobserver curves, weusetheMinkowski metric to define theassociated dual 1-form Vandwrite anyFuniquely as F=EAii+B (5.411) where Bisa2-fofm satisfying i,,B=0and Ea1-form satisfying IVE =0.(Note: 8/6! =—dt.) One refers toBeFAQM asthemagnetic 2-form associated with Vand F,and EeTA,M astheassociated electric 1-form. Theelectric field observed bythisclass ofobservers is ,\_, E-IMF. (5.4.12) The magnetic vector field observed bythisclass canberelated toFas ELECTROMAGNETISM 193 follows. We use the velocity vector to define a metric k on the instantaneous three spaces g = - (DV + (5.4.13) We may factor the volume four-form as *1 = V A 1. (5.4.14) Any p-form w can be '3 + 1 decomposed' with respect to the velocity vector V: = a + VA/3 (5.4.15) with iva = 'Vie = O. If is the Hodge map associated with g then *(0 = --(T`ce) A V - (5.4.16) Applying this result to (5.4.11) gives *F = A -17 È. (5.4.17) But i = 0 so i v* F = - B. We define the vector field B = as the magnetic field associated with V; hence in terms of F B = -iv* F. (5.4.18) If {Ya} is a frame on the instantaneous three-space, orthonormal with respect to g, then the electric and magnetic field components in such a basis are given in terms of F as É(Ya) = (i vF)(Ya) = 2F(V, Y a) (iU3)(Y0) = -(iv*F)(Ya) = -2*F(V, Y a). As an example consider the Coulomb solution: F= q —dr A dt r2 r2 = x2 + y2 + z2 with observer curves tangent to V = (3/3t) and W = y((a/3t) + y = (1 - v 2)-112. With respect to V: E = –7,-(3/3r) B = O. 1.4 On the other hand, since rdr = xidx1 + x2dx2 + x3dx3, W observes E' - Par) + vx (3/30) -:12Y( B' = q" (x2(3I3x3) – x3(3I3x2)) r3 instead of E and B at p. ELEcTRoIvIAGNETisM 193 follows. We use thevelocity vector todefine ametric gonthe instantaneous three spaces »~..»-C.g- V®V +g. (5.4.13) Wemayfactor thevolume four-form as -1-vAA1. (5.414) Any p-form atcanbe‘3+1decomposed’ with respect tothevelocity vector V: ~ to=tr+VAB (5.4.15) withiva=ivfi=0.IfQistheHodge mapassociated with gthen *6»=-(ea)AI7-re. (5.416) Applying thisresult to(5.4.11) gives *F=-(at-2),, if+ai. (5.4.17) Butiv(fE) =0SOiv*F=-at-2.WedefinethevectorfieldB=$2-2as themagnetic field associated with V;hence interms ofF ,\_, B——lV*F. If{Y,} isaframe ontheinstantaneous three-space, orthonormal with respect tog,then theelectric andmagnetic field components insuch a basis aregiven interms ofFas Em)=(ivF)(Y@) =2F(v.Y.) (fB)(Y..) =-(iv*F)(Y..) =-2*F(V, Ya)- Asanexample consider theCoulomb solution: F=i,drAatr ri=xi+yi+zi with observer curves tangent toV=(8/8t) and W=j/((8/8t) + v(8/8xi)), y=(1—vi)'iii. With respect toV: E=?:i(a/er) B=0. Ontheother hand, since rdr=xidxi +xidxi +xidxi, Wobserves _ IE’=—‘]l’((a/er) +lo/at)) I‘ r a'=%i—{xi(8/8x3) -x3(8/8xi)) instead ofEandBatp. 194 APPLICATIONS IN PHYSICS It is worth stressing that although observers in Minkowski space experiencing arbitrary motion do not have world lines that can be naturally associated with the Poincaré group (their world lines are not integral curves of Killing vectors) the local definition of electric and magnetic fields for such observers follows as before since only a local frame and its dual are of relevance. During the historical development of classical electromagnetism it became apparent that a number of related properties could be assimi- lated into a single idea once the spacetime description of Maxwell's theory was recognised. These properties became particularly succinct in terms of a second-rank tensor known as the Maxwell stress tensor. Historically the components of this tensor, with respect to a basis with physical dimensions, were associated with the properties of mechanical systems. This was a consequence of the role played by such components in equations which coupled together the behaviour of fields and matter. We shall discuss such equations later. At this point we shall be content with introducing this tensor in the guise of a 3-form associated with every Maxwell field and arbitrary vector field, and proving that such a 3-form associated with a conformal Killing vector is closed in source-free regions. Define for any vector field V and Maxwell solution F the 3-form rv = ;fiyFA*F — iv*F A (5.4.19) Applying the exterior derivative and using Maxwell's equations for F produces dry = 1{di vF A *F — ivF A j — div*F A F}. (5.4.20) Recall the identity 2x = dix + id V X: hence di yF = vF (5.4.21) as dF = 0. Similarly di y*F = 2v*F — iv]. Inserting this in (5.4.20) gives dry = '1{1 vF *F — 2v*F A F — iyF + F}. (5.4.22) If C is a conformal Killing vector then eF A F = * CFA F A *2cF =YcF A *F. Hence specialising to the case of a conformal Killing vector CITC = HiCFAi friCi A F. Since i c(j A F) = ic.i A F — AicF and, being a 5-form, IA F is zero we have drc = iCFA1. (5.4.23) For each conformal Killing vector these equations describe a 'local conservation equation' in a source free region (j = 0). The identification 194 AFPLicATioNs INPHYSICS Itisworth stressing that although observers inMinkowski space experiencing arbitrary motion donothave world lines that can be naturally associated with thePoincare group (their world lines arenot integral curves ofKilling vectors) thelocal definition ofelectric and magnetic fields forsuch observers follows asbefore since only alocal frame anditsdual areofrelevance. During thehistorical development ofclassical electromagnetism it became apparent that anumber ofrelated properties could beassimi- lated into asingle idea once thespacetime description ofMaxwell’s theory wasrecognised. These properties became particularly succinct in terms ofasecond-rank tensor known astheMaxwell stress tensor. Historically thecomponents ofthistensor, with respect toabasis with physical dimensions, were associated with theproperties ofmechanical systems. This wasaconsequence oftherole played bysuch components inequations which coupled together thebehaviour offields andmatter. Weshall discuss such equations later. Atthispoint weshall becontent with introducing this tensor intheguise ofa3-form associated with every Maxwell field andarbitrary vector field, andproving that such a 3-form associated with aconformal Killing vector isclosed insource-free regions. Define foranyvector field VandMaxwell solution Fthe3-form Iv=§{i,,FA*F—i,,*FAF}. (5.4.19) Applying theexterior derivative and using Maxwell’s equations forF produces dTy=%{di,,FA *F—ivFAj —di,,*FA F}. (5.4.20) Recall theidentity SEX=diX+iXd VX: hence diMF =SEVF (5.4.21) asdF=0.Similarly diV*F =SE,,*F —ivj.Inserting thisin(5.4.20) gives drv= %{SEvFA*F—SEv*FAF—i,,FAj+i,,jAF}. (5.4.22) IfCisaconformal Killing vector then SEC*FAF =*SEgFAF = FA*SEgF =SECFA *F.Hence specialising tothecase ofaconformal Killing vector dTc=_iIcFAl +iiclAF- Since ic(jA F)=icjA F—jAICF and, being a5-form, jAFiszero we have dtc=—iCFAj. (5.4.23) Foreach conformal Killing vector these equations describe a‘local conservation equation’ inasource freeregion (j=0).The identification ELECTROMAGNETISM 195 of a closed 3-form I with a local conservation law is appropriate in an arbitrary spacetime. For consider a region described by some 4-chain U whose boundary may be written 3U = + I2 fl (5.4.24) with the image of each /I a spacelike hypersurface (each tangent vector to X, being spacelike). For I closed U = = 0 (5.4.25) U by Stokes's theorem, thus = —In (5.4.26) In cases where U may be chosen so that f = 0 one recognises that the flux of through equals the flux of j through E 2 (see figure 5.2). Figure 5.2 This diagram illustrates the equation 3U = + E2 Fl. Suppose that we have a field system describing a simply connected source-free region U of Minkowski space. If r is the proper time of some inertial observer passing through this region then in an adapted chart fr, p', p2, p31 we take E i to lie in the hypersurface r(p) = cl, for some constant c1. If the electromagnetic field vanishes at large spatial distances from the observer then we may take H to complete the boundary of U such that the electromagnetic field vanishes on H. Thus in this case the flux of j through the instantaneous three-space is time independent. If we write in terms of a 2-form current j and an ELECTRoMAGNETIsM 195 ofaclosed 3-form 3with alocal conservation lawisappropriate inan arbitrary spacetime. Forconsider aregion described bysome 4-chain U whose boundary may bewritten au=2,+22+n (5.424) with theimage ofeach Z,aspacelike hypersurface (each tangent vector toZ,being spacelike). For,9closed ,LU§ =Jud} =0 (5.4.25) byStokes’s theorem, thus A2,} :I-22‘? _in} (5426) Incases where Umay bechosen sothat jg} =0onerecognises that theflux of,9through Z,equals theflux ofjthrough Z2(see figure *-\”:ij;;\I /\ '< /\ \1- \I\» \Fl /,\ \ \\/ I iv’ \ ‘:\/\ Figure 5.2This diagram illustrates theequation 8U=Z1+22+II. Suppose that wehave afield system describing asimply connected source-free region UofMinkowski space. If‘Z’istheproper time of some inertial observer passing through thisregion then inanadapted chart {'l',pi,pi,p3}wetake Z,tolieinthehypersurface t(p) =c,-,for some constant c,-.Iftheelectromagnetic field vanishes atlarge spatial distances from theobserver then wemay take TItocomplete the boundary ofUsuch that theelectromagnetic field vanishes onTI.Thus inthiscase theflux of§through theinstantaneous three-space istime independent. Ifwewrite §interms ofa2-form current §and an 196 APPLICATIONS IN PHYSICS associated 3-form density /3, j = j A dT i) with i 0,3,4 = 0 and = 0, then clearly X*$ = [3 and =LP (5.4.27) It is tempting to reinterpret the conservation of j -flux associated with U in terms of a local flow of current j and an associated variation of density p. Certainly the 3-form equation dj = 0 implies a local contin- uity equation. In the above chart we may write d when acting on j as d = d + dr A (a/a.0 where d is the exterior derivative associated with the instantaneous three-space. Hence (as dj3 = 0) dj (a/ar)P — O. If we express j and p in a basis adapted to 1: = 1dP2 A dP3 + 'j2dP3 A dP1 + 'i3dP1 A dP2 p = pdpi A dp2 A dp3 (5.4.28) is equivalent to 3 E(a:iitap ) — (3p/ar) = 0. (5.4.29) = The interpretation of this local continuity equation must, however, be treated with caution. If j is a closed 3-form on U then so is = + (IX where Jf is any smooth 2-form. If Jf is chosen such that faxX = 0 then j and j' both have the same flux through X, although will redistribute the local density. Returning to (5.4.23) we see that there are 15 closed 3-forms, one for each of the 15 conformal generators of the Minkowski space conformal group. It is instructive to examine the currents associated with some of these Killing vectors. If V is a timelike Killing vector field generating time translations along its open integral curve then, using (5.4.12) and (5.4.18) to define E and B with respect to such a field, we easily find: TV = A if A 'V + 4k-A ± IT A '<4). (5.4.30) The physically dimensioned components of the vector obtained by taking the metric dual of the 2-form E A k with respect to k was identified by Poynting as the local field energy transmitted 'normally' across unit area per second (that is the local field energy current). Similarly the '-dual of the 3-form ;(i" A E + B A *‘ if) may, after restoring physical dimensions, be interpreted as a local field energy density. Since, for example E A E = g(E, E)q, the signature of g ensures that this density is positive-definite. This interpretation has persisted although with the caveats above we would prefer to identify the oriented integral f ,Tv, in a source-free region of spacetime, as the (5.4.28) 196 APPLIcATi0Ns INPHYSICS associated 3-form density p,9=9Ad7: +pwith i(g,g,,,9 =0and i,g,g,,p =0,then clearly E*9 =,5and jig=L6. (5.427) Itistempting toreinterpret theconservation of9-flux associated with Uinterms ofalocal flow ofcurrent 9andanassociated variation of density p.Certainly the3-form equation d9=0implies alocal contin- uityequation. Intheabove chart wemay write dwhen acting on9as d=Q+ d7:ASE(g,g,) where Qistheexterior derivative associated with theinstantaneous three-space. Hence (asgp=0) Q9-.s2,,,,,,6 =0. (5.428) Ifweexpress and,6inabasis adapted to2: 3’=§1dP2 Adpi+§2dP3 /\dpi+fisdpi Adpz E=Pdpi AdpiAdpi (5.4.28) isequivalent to (6.9,-/apt) -(Sp/31.’) =0. (5.4.29) The interpretation ofthislocal continuity equation must, however, be treated with caution. If9isaclosed 3-form onUthen sois 9'=9+d?7{where Elfisanysmooth 2-form. IfElfischosen such that jgzflf =0then 9and9'both have thesame fluxthrough E,although 9' willredistribute thelocal density. Returning to(5.4.23) weseethatthere are15closed 3-forms, onefor each ofthe15conformal generators oftheMinkowski space conformal group. Itisinstructive toexamine thecurrents associated with some of these Killing vectors. IfVisatimelike Killing vector field generating time translations along itsopen integral curve then, using (5.4.12) and (5.4.18) todefine EandBwith respect tosuch afield, weeasily find: rV=—EABAV+§(EAfE+BA?B). (5.430) The physically dimensioned components ofthevector obtained by taking themetric dual ofthe2-form EAEwith respect togwas identified byPoynting asthelocal field energy transmitted ‘normally’ across unit area persecond (that isthelocal field energy current). Similarly theg-dual ofthe3-form §(EA EE+EA QB) may, after restoring physical dimensions, beinterpreted asalocal field energy density. Since, forexample EA SE=g(E,E)f1, thesignature ofg ensures that this density ispositive-definite. This interpretation has persisted although with thecaveats above wewould prefer toidentify theoriented integral fir,-, inasource-free region ofspacetime, asthe ELECTROMAGNETISM 197 field energy associated with the spacelike 3-chain E and f s2i vdTv as a power flux across an oriented spacelike 2-chain .52. Suppose we consider a spacelike Killing vector field X generating spacelike translations along open integral curves and decompose Tx according to TX = PX f7 (5.4.31) (bx with i vitx = i v'fix = O. The Maxwell stress 2-form tix may be used to identify mechanical Newtonian forces produced by a 'flow' of a Newto- nian field momentum density 3-form <6x. In an analogous manner one may construct torque forms (angular momentum currents) using a Killing vector field that generates rotations along closed integral curves. As promised we now relate the stress 3-forms to an associated second-rank tensor. Given any local frame {Xa} a = 0, 1, 2, 3 in spacetime, with natural dual co-frame {eb}, we may obtain 16 real functions Tab defined by *ÎX = Tbcec or Tab = (*Tx”)(Xb). These may be used to define a second-rank tensor T Tabea(Deb (5.4.32) which is referred to as the stress tensor. Exercise 5.2 Show that if Tx, A eb = TXh A ea then Tab = Tba: the stress tensor is symmetric. Show that if Tx, A ea = 0 then Tb' -= 0: the stress tensor is traceless. These properties are satisfied for the Maxwell stress tensor as follows directly from the definition. We shall meet these properties again at a later stage in the context of a Clifford representation for this tensor. Exercise 5.3 If F =-Fabe ° A eb show that g T b Fcd FacFc b . ab = Exercise 5.4 Use the three angular momentum 3-forms T K, to evaluate the torque on an electric dipole in a uniform static electric field. (Hint: calculate the total electromagnetic 2-form and use this in (5.4.19) where the Killing currents are computed with the aid of the rotational Killing vectors.) Bibliography Misner C, Thorne K and Wheeler A 1973 Gravitation (San Francisco: W H Freeman) ELEcTRoiviAGNETisivi 197 field energy associated with thespacelike 3-chain Eandjgzivdrv asa power fluxacross anoriented spacelike 2-chain Si. Suppose weconsider aspacelike Killing vector field Xgenerating spacelike translations along open integral curves and decompose TX according to 6,,=,1“V+<aX (5.431) with iv,uX =IVSEX =0.The Maxwell stress 2-form ,uXmay beused to identify mechanical Newtonian forces produced bya‘flow’ ofaNewto- nian field momentum density 3-form ‘EX. Inananalogous manner one may construct torque forms (angular momentum currents) using a Killing vector field thatgenerates rotations along closed integral curves. Aspromised wenow relate the stress 3-forms toanassociated second-rank tensor. Given any local frame {X0} a=0,1,2,3in spacetime, with natural dual co-frame {ei’}, wemay obtain 16real functions TM,defined by*rX, =TMe‘ orTM,=(*rX,)(Xg). These may beused todefine asecond-rank tensor T=TM,e"®ei’ (5.4.32) which isreferred toasthestress tensor. Exercise 5.2 Show that ifIX,Aej,=TX,Ae,then TM,=TM: the stress tensor issymmetric. Show that ifTX,Ae“=0then Tgi’=0:thestress tensor is traceless. These properties aresatisfied fortheMaxwell stress tensor asfollows directly from thedefinition. Weshall meet these properties again ata later stage inthecontext ofaClifford representation forthistensor. Exercise 5.3 IfF=%FM,e” Aei’show that Tab=_i.gabFCdFcd —FHEFCD‘ Exercise 5.4 Usethethree angular momentum 3-forms ‘CK,toevaluate thetorque on anelectric dipole inauniform static electric field. (Hint: calculate the total electromagnetic 2-form andusethisin(5.4.19) where theKilling currents arecomputed with theaidoftherotational Killing vectors.) Bibliography Misner C,Thorne Kand Wheeler A1973 Gravitation (San Francisco: WH Freeman) 198 APPLICATIONS IN PHYSICS Sachs R K and Wu H 1977 General Relativity for Mathematicians (New York: Springer) Schutz B F 1985 A First Course in General Relativity (Cambridge: Cambridge University Press) 198 APPLICATIONS INPHYSICS Sachs RKandWuH1977 General Relativity forMathematicians (New York: Springer) Schutz BF1985 AFirst Course inGeneral Relativity (Cambridge: Cambridge University Press) 6 Connections The differentiable structure on a manifold enabled us to define two important differential operators; the exterior and Lie derivatives. Whereas the former acted only on antisymmetric tensor fields (differen- tial forms) the latter acted on any tensor field. However, whilst reducing to the directional derivative on functions the Lie derivative is not a suitable generalisation to a 'directional derivative on tensors'. This is because the Lie derivative of a tensor at p, along a curve C, does not just depend on the tangent to the curve at p but on the behaviour of tangent vectors in the vicinity of p. This feature of the Lie derivative is reflected in the fact that E T is not 9-,-linear in the vector field X. Another differential operator, a tensor covariant derivative, will now be introduced. The introduction of this new structure is equivalent to choosing a parallelism for the manifold. The general notion of parallel- ism is easy to grasp. It is only necessary to recognise that in general there is no preordained way to map a vector at one point on a mani- fold to a new vector at another point. Defining a parallelism on a manifold requires specifying a rule that will provide a means of comparing vectors at different points by transporting one to the other along some prescribed path connecting the points. Whereas the parallel transport map will depend on the path chosen to connect the points we do not want it to depend on how the path is traversed. (Parallel transport depends on the route taken but not on how bumpy the ride!) Although this feature of path dependence of parallel transport does not accord with the intuitive Euclidean concept it is an essential feature, characterising the curvature of the manifold. Given a parallelism we can define a covariant derivative by comparing a vector with its parallel translate and taking a suitable limit. Conversely, by introducing a new rule for differentiating vectors, and establishing a linear connection, we can define a vector field to be parallel along a curve if its derivative with respect to the tangent vector is zero. Connections The differentiable structure onamanifold enabled ustodefine two important differential operators; the exterior and Lie derivatives. Whereas theformer acted only onantisymmetric tensor fields (differen- tialforms) thelatter acted onanytensor field. However, whilst reducing tothedirectional derivative onfunctions theLiederivative isnota suitable generalisation toa‘directional derivative ontensors’. This is because theLiederivative ofatensor atp,along acurve C,does not justdepend onthetangent tothecurve atpbutonthebehaviour of tangent vectors inthevicinity ofp.This feature oftheLiederivative is reflected inthefactthatSEXTisnotF-linear inthevector field X. Another differential operator, atensor covariant derivative, willnow beintroduced. The introduction ofthisnew structure isequivalent to choosing aparallelism forthemanifold. The general notion ofparallel- ismiseasy tograsp. Itisonly necessary torecognise that ingeneral there isnopreordained way tomap avector atonepoint onamani- fold toanew vector atanother point. Defining aparallelism ona manifold requires specifying arule that will provide ameans of comparing vectors atdifferent points bytransporting onetotheother along some prescribed path connecting thepoints. Whereas theparallel transport map willdepend onthepath chosen toconnect thepoints we donotwant ittodepend onhow thepath istraversed. (Parallel transport depends ontheroute taken butnotonhowbumpy theride!) Although thisfeature ofpath dependence ofparallel transport does not accord with theintuitive Euclidean concept itisanessential feature, characterising thecurvature ofthemanifold. Given aparallelism wecan define acovariant derivative bycomparing avector with itsparallel translate andtaking asuitable limit. Conversely, byintroducing anew rule fordifferentiating vectors, andestablishing alinear connection, we candefine avector field tobeparallel along acurve ifitsderivative with respect tothetangent vector iszero. 200 CONNECTIONS 6.1 Linear Connections A linear connection on a manifold M is a map V: F TM x f TM --)FTM that satisfies the following, Vfg c5;(M), V X, Y, Z E FTM: fx gyZ =JVKZ gV yZ (6.1.1) Vx(fY + gZ) = X(f)Y + fV xY + X(g)Z + gV xZ. (6.1.2) Thus Vx is a linear mapping on vector fields which is also s'i-linear in X: it is called covariant differentiation with respect to X. From these properties it follows that we can specify V by giving the components of the vector Vxa.Xb in any convenient basis {Xa}: Vx/Yb = FabeXc. (6.1.3) The n3 functions F ab`, where n = dim M, are known as the connection components, or connection coefficients in this basis. These coefficients can be used to define a set of 1-forms, the connection 1-forms, b = cb a ea (6.1.4) where {ea} is the co-frame dual to {X a}. Thus we can write (6.1.3) equivalently as vx”xb = wcb(xa)xc. (6.1.5) If { Ya} is a new basis, related to {X I,} by a general linear transform- ation Ya = AabXb, then Ti7K,Yb = A aP V x,,G4 bc X c) AlAbcFpcaXq AaPX),(Abc)X,. If the inverse transformation is given by A aAab = (5,b, then the connection coefficients Fab' in the basis {Ya} are given by rab, = AaPAbcrpaqA-1q, + Azxp(Abc)A-1,,. (6.1.6) Equivalently the connection 1-forms in this basis are given by CO' b = Aba(O r 0-1 ra ± A-1 qadAba (6.1.7) The 'inhomogeneous' term in this transformation represents a departure from the transformation of the components of a tensor, reflecting the fact that the map X, Y V xY is not Fi-linear in Y. As anticipated the covariant derivative of a vector field with respect to X, evaluated at the point p, depends only on the value of X at p. For if V is any vector field and {X,} is a basis in the neighbourhood of p then (V,Z)lp = Va(p)(VZ)1 1,. So if V vanishes at p then 200 CoNNEcTioNs 6.1Linear Connections Alinear connection onamanifold Misamap V:FTM ><FTM -_>FTM thatsatisfies thefollowing, Vf,g e@(M), VX, Y,ZeFTM: v,X,,,z =jVXZ+gVyZ (6.1.1) VX(fY +gZ)=X(f)Y +fVXY+X(g)Z +gVXZ. (6.1.2) Thus VXisalinear mapping onvector fields which isalso9-linear inX: itiscalled covariant differentiation with respect toX. From these properties itfollows that wecanspecify Vbygiving the components ofthevector VX,X,, inanyconvenient basis {X,,}: VXaXb =FabcXc. The n3functions I’M,‘, where n=dimM,areknown astheconnection components, orconnection coefficients inthisbasis. These coefficients canbeused todefine asetof1-forms, theconnection 1-forms, of’),=r,,,“e@ (6.1.4) where {e"} istheco-frame dual to{X,,}. Thus wecanwrite (6.1.3) equivalently as VX,X,, =w‘,,(X,,)X,. (6.1.5) If{Ya} isanew basis, related to{Xg} byageneral linear transform- ation Y,=A,,i’X,,, then VY,Yi> =A/iVx,,(AbCXt) =A,PA,,‘I’,,C’4X,, +A,,PX,,(A,,‘)X,. Iftheinverse transformation isgiven byA‘i,,“A,,i’ =65,then the connection coefficients F’M,’ inthebasis {Ya} aregiven by FM,’ =A/’A,,‘F,,C‘?A'i,,’ +A,,PX,,(A),‘)A‘iC’. (6.1.6) Equivalently theconnection 1-forms inthisbasis aregiven by 01'“), =A,,'4w’,,A“i,” +A‘i,,“dA,,‘?. (6.1.7) The ‘inhomogeneous’ term inthistransformation represents adeparture from thetransformation ofthecomponents ofatensor, reflecting the factthatthemap X,Y-> VXY isnot9?:-linear inY. Asanticipated thecovariant derivative ofavector field with respect toX.evaluated atthepoint p,depends only onthevalue ofXatp. ForifVisanyvector field and{X,,} isabasis intheneighbourhood of pthen (VvZ)|,, =V“(p)(VX,Z)|,,. SoifVvanishes atpthen LINEAR CONNECTIONS 201 (V vZ)! -= 0 V Z. Thus if X and Y are vector fields such that Xl p = Ylp then (VxZ)Ip = (V Z) p V Z. Hence for any Xp e TM we have a covariant derivative in the direction of Xp, V xp:FTM TM. Let C be a curve with tangent vector C. Then if Y is a vector field we may covariantly differentiate Y in the direction of the tangent vector at any point C(t) on the curve. An assignment of a vector Yc( f) to every Tc(oM is a (smooth) vector field along C if the map t i--> Ygof is a smooth function of t Vfe 5-,(M). Thus if C is a smooth curve Ve.Y is a smooth vector field along C. As will be seen below VY only depends on the value of Y along C, and so in fact any vector field along C can be covariantly differentiated with respect to the tangent vector to produce another vector field along C. (Some authors denote VÈY by DY/dt where t parametrises C.) A vector field Y along a curve C is said to be parallel along C if it satisfies the equations Velf = O. (6.1.8) If we expand Y = ra j in a local coordinate chart in which xi(p)= Ci(t) represents C, j =1, ..., n then e = c*(a/at) = ck(t)(a/ ax'). Hence Vc(r(a/ax0) = (CY1)(8/axi) + rV c(a/9x]). But CY/ = [C(a/at)]P = Ck(t)(ar/Irk) = d(P.C)/dt and Ve,(3/ax0 = Ck(t)V (ataxqa/ax i) = k( kim(a/axm). Thus (6.1.8) gives the fol- lowing differential equations for the components P.0 of Y on C: d —dt(YmoC) + (P.C)Ck(t)(F kimoC) = 0. (6.1.9) For given functions Ck(t) and connection components Fkim(C(t)) these equations are known to have a unique solution Ym(C(t)) specified by the choice of initial components Ym(C(0)). (It is because these equa- tions only depend on the components of Y along C that a vector field along C can be differentiated.) Because of the above uniqueness result a parallelism is established by the linear connection V. If Ygo) is any vector in Tc(o)M and Y is the unique vector field along C such that VcY = 0 then li c(() is called the parallel translate of Yc(o) along C. Let Y be any smooth vector field along C with li c(0)* 0, and f the smooth function such that (foC)(t)= t. For t sufficiently small Z is a smooth vector field on C (—f)"(V OnY Z Y + E (6.1.10) n! n=1 where (V)2Y = Ve-(Ve,Y) etc. We have n(---f)n-1(Ve.)"Y± f 07 On + IY VZ=VY— E (since Cf = 1) n n! =1 n=1 n! LINEAR CONNECTIONS 201 (VvZ)|,, =0VZ. Thus ifXandYarevector fields such thatX|,,=Y|,, then (VXZ)|,, =(VyZ)I,, VZ. Hence foranyX,,eT,,M wehave a covariant derivative inthedirection ofX,,,_VX,:FTM—>T,,M. LetCbeacurve with tangent vector C.Then ifYisavector field wemaycovariantly differentiate Yinthedirection ofthetangent vector atanypoint C(t)onthecurve. Anassignment ofavector YC(,) toevery TC'(;)M isa(smooth) vector field along Cifthemap ti—>YC(!)_f isa smooth function oftVfe g>(M). Thus ifCisasmooth curve Vg-Y isa smooth vector field along C.Aswillbeseen below V('_~Y only depends onthevalue ofYalong C,andsoinfactanyvector field along Ccan becovariantly differentiated with respect tothe tangent vector to produce another vector field along C.(Some authors denote VCY by DY/dt where tparametrises C.) Avector field Yalong acurve Cissaid tobeparallel along Cifit satisfies theequations v,-Y=0. (6.1.8) Ifweexpand Y=Y16,-inalocal coordinate chart inwhich x/(p) =Cl(t) represents C,j=1,...,nthen C=C..(6/St) =C"(t)(6/ dxi‘). Hence Vg-(Yl(6/6xl)) =(CYi)(6/6x/) +Yiv,-(a/ext). But CY/i=[c,(e/at)]Y/ =c'<(i)(aY//ext) =d(Yl<>C)/dt and V,-(a/ext) =C"(t)V(g,g,,t)(6/Sxi) =Ci‘(t)F,,,’"(6/6x’"). Thus (6.1.8) gives thefol- lowing differential equations forthecomponents Y/BC ofYonC: %(Y"'<>C) +(YI@c)ck(t)(r,,,~@c) =0. (6.1.9) Forgiven functions C"(t) andconnection components FX,-'"(C(t)) these equations areknown tohave aunique solution Y’"(C(t)) specified by thechoice ofinitial components Y"i(C(0)). (Itisbecause these equa- tions only depend onthecomponents ofYalong Cthat avector field along Ccanbedifferentiated.) Because oftheabove uniqueness result a parallelism isestablished bythelinear connection V.IfY('(0) isany vector inTC(0)M andYistheunique vector field along Csuch that Vg-Y =0then YC'(!) iscalled theparallel translate ofYC(0) along C. LetYbeanysmooth vector field along Cwith YC'(0) E0,andfthe smooth function such that (f@C)(t) =t.For tsufficiently small Zisa smooth vector field onC Z5Y4-2% figci Y (6.1.10) n=1 ‘ where (Vg-)iY =VC"(VC" Y)etc.Wehave I n—1 .n I n .n+1Vg-Z=v,-Y—Z”(_ii n,(V‘i Y+Z(_fi (Z?) Y(Since cf=1) ri=l ' ri=1 ' 202 CONNECTIONS - = n( fy - 1(V dn + y .r _ nn(V O n + ly E E` ( " n=2 n! n=1 n! . = (m 4_ 1)(_fr(v È)n + 1 y + .o ( _n17e..) n(n + 1 y E E‘ " =o. m., (m + 1)m! n=i n! So Z is parallel along C with Zc(o) = Y go), thus Z go must be the parallel translate of Yc(0) to C(t). Note that any vector field Y satisfying Ygo) = A can be taken in (6.1.10) to evaluate the parallel transport of A E Tc(o)M along C (see figure 6.1). TiorCl01 YC■ I :71" Ynt C(t) C(0) Figure 6.1 The parallel translation of Y along the curve C. A vector field Y is said to be parallel, or covariantly constant, (with respect to V) if it satisfies the equation V xY = 0 V X. This implies that Y is parallel along all curves and thus the parallel transport map is independent of the path along which such a Y is transported. Exercise 6.1 A connection on a two-dimensional manifold is specified in a local chart with coordinate maps (x1, x2) by rui = and F222 = —.X2 with all other connection components zero in this chart. Prove that for a, b E IR the vector field Y = a exp [(x1)2/2[(3/3x1) + b exp[—(x1)2/2](3/3x2) is parallel along the curve C: [0, 11 (xl(p) = sin t, x2(p) = cost). A linear connection enables us to define a 'straight line', generalising one of the intuitive properties of straight lines in Euclidean space. A curve C is an autoparallel (of V) if its tangent vector field is parallel 202 CoNNEcTioNs =_i"(—f)"' i(Vc)"Y +i(—f)"(Vc)"i ‘Y n=2 "I ":1 "I =_£0" +1)(-f)'"(Vc)'"iiY +i(—f)"(V,c)"i ‘Y:0 m=[ (I71 + "=| H. i SoZisparallel along Cwith Zcw) =YC(0), thus Zcg, must bethe parallel translate ofYC(0) toC(t).Note thatanyvector field Ysatisfying YC(0) =Acanbetaken in(6.1.10) toevaluate theparallel transport of AeTC(0)M along C(seefigure 6.1). Tioyriofiyrin"iVZYI;,;" You (If) Yrioi [(0) Figure 6.1Theparallel translation ofYalong thecurve C. Avector field Yissaid tobeparallel, orcovariantly constant, (with respect toV)ifitsatisfies theequation VXY =0VX. This implies that Yisparallel along allcurves and thus theparallel transport map is independent ofthepath along which such aYistransported. Exercise 6.1 Aconnection onatwo-dimensional manifold isspecified inalocal chart with coordinate maps (xi, xi)byl‘,,i =—x‘ andF222 =—xi with all other connection components zero inthischart. Prove thatfora,beIR thevector field Y=aexp[(xi)i/2](Z-3/Z-Bx‘) +bexp[—(xi)i/2](Z-3/Z-Bxi) isparallel along thecurve C:[0,1]ii> (xi(p) =sint, xi(p) =cost). Alinear connection enables ustodefine a‘straight line’, generalising oneoftheintuitive properties ofstraight lines inEuclidean space. A curve Cisanautoparallel (ofV)ifitstangent vector field isparallel LINEAR CONNECTIONS 203 along C. Such curves are given as solutions to the equation Vce = O. (6.1.11) (Autoparallels are more frequently called geodesics although we prefer to reserve this terminology for the autoparallels of a pseudo-Riemannian connection which will be discussed later. Students everywhere will be relieved to know that if by 'straight line' we mean autoparallel, then at least for the Riemannian connection 'straight lines' are (in a certain sense) the shortest curves connecting two points!) If an autoparallel C is given parametrically in a local chart by xi(p)= OW then the CI must satisfy the system of differential equations d dtCm (r kim°C)0(t)C1 = or + (1"kimoC)e'kev = 0. (6.1.12) It is important to note that the solution of (6.1.12) is a parametrised curve. Although a general reparametrisation of the solution will not change the image set on M of the reparametrised C, the corresponding map will not in general satisfy (6.1.12) and will not therefore be an autoparallel. If C is an autoparallel, with parameter t, then the reparametrised curve Coh is also an autoparallel if and only if h = at + b for a, b E E. For an arbitrary curve C we define the acceleration to be the vector field V ct on C. (Thus the acceleration Each autoparallel is fixed uniquely by specifying (Ci, Ci) for some initial t. That is, for every Xi,, e TM there is a unique maximal autoparallel starting at p in the direction of Xi,. Let yx, be this autoparallel. The exponential mapping at p, Expo, maps a subset of TM into M:ExpoXo = yx,(1). Clearly Exp o is defined on those Xo for which yy, is defined on [0, 1]. Since for A E IR yAxp(t) = yxp(t), if Expo is defined on Xp then it is also defined on .À.Xo for A E [0, 1[. It in fact follows from the nature of the differential equations (6.1.12) that for every p E M there is a neighbourhood of the origin in TM, N o, such that the exponential mapping is a diffeomorphism onto a neighbourhood of p, N. If such an .N.0 is star shaped then it is called a normal neighbourhood. (To say that X0 is star shaped means that if u E X( then /1./) E X0 VA E [0, 1].) A normal neighbourhood of p is the image of a normal neighbourhood in TM under the exponential mapping. For every q in a normal neighbourhood of p, Np, there is one and only one Q E TM such that q = ExpoQ. Thus if {Xi) is any basis for TM with Q = the mapping ql-->{V} provides a coordinate system for No (see figure 6.2). Such coordinates are called normal coordinates at p. LINEAR CONNECTIONS 203 along C.Such curves aregiven assolutions totheequation VCC =0. (6.1.11) (Autoparallels aremore frequently called geodesics although weprefer toreserve thisterminology fortheautoparallels ofapseudo-Riemannian connection which willbediscussed later. Students everywhere will be relieved toknow that ifby‘straight line’ wemean autoparallel, then at least fortheRiemannian connection ‘straight lines’ are(inacertain sense) theshortest curves connecting twopoints!) Ifanautoparallel Cis given parametrically inalocal chart byxi(p) =Ci(t) then theClmust satisfy thesystem ofdifferential equations éigcm+(r,,,-*"<>c)c'<(t)c1' =0 Of cm+(r,,,-"'<>c)c'<C1' =0. (6.1.12) Itisimportant tonote thatthesolution of(6.1.12) isaparametrised curve. Although ageneral reparametrisation ofthesolution will not change theimage setonMofthereparametrised C,thecorresponding map will notingeneral satisfy (6.1.12) and will nottherefore bean autoparallel. IfCisanautoparallel, with parameter t,then the reparametrised curve C<>h isalso anautoparallel ifand only if h=at+ bfora,belR. For anarbitrary curve Cwedefine the acceleration tobethevector field V@C onC.(Thus theacceleration Each autoparallel isfixed uniquely byspecifying (Ci, Ci)forsome initial t.That is,forevery X,,eT,,M there isaunique maximal autoparallel starting atpinthe direction ofX. Let j/X, bethis autoparallel. The exponential mapping atp,Exp,,p, maps asubset of T,,M intoM:Exp,,X,, =j/X,(1). Clearly Exp,, isdefined onthose X,,for which j/X,isdefined on[0,1].Since forAeIR)/;,XP(I) =j/X,(At), ifExp,, isdefined onX,,then itisalso defined onAX,, forAe[0,1].Itinfact follows from thenature ofthedifferential equations (6.1.12) that for every peMthere isaneighbourhood oftheorigin inT,,M, Ng,such thattheexponential mapping isadiffeomorphism onto aneighbourhood ofp,N.Ifsuch anNgisstarshaped then itiscalled anormal neighbouihood. (TosaythatNgisstarshaped means thatifveNgthen AveNg VAe[0, 1].)Anormal neighbourhood ofpistheimage ofa normal neighbourhood inT,,M under theexponential mapping. For every qinanormal neighbourhood ofp,N,,,there isoneandonly one QeT,,M such that q=Exp,,Q. Thus if{X,} isanybasis forT,,M with Q=2f’=,QiX,- themapping qi—>{Qi} provides acoordinate system for N,,(seefigure 6.2). Such coordinates arecalled normal coordinates atp. 204 CONNECTIONS Figure 6.2 This diagram illustrates the exponential map and normal coordin- ates. Exercise 6.2 If f ilk are the connection coefficients with respect to a normal co- ordinate basis at p show that ri,k(p) + riik(P) = O. Hint: Show that ?'„(t) = y' where y = y'Xi. 6.2 Examples and Newtonian Force To gain some insight into covariant derivatives we turn to Fin. This manifold has an absolute parallelism: the parallel-transport map is path independent. If {x'} are standard coordinates and X,, = E,c'ajp then the parallel translate of Xp at q is Xq = q. Thus in such a standard chart the connection is defined by Va,ai = 0. Such a connection is referred to as the standard connection on En. It is of interest to compute the standard connection for F12 in a polar chart (r, 0) related to the standard one by rcos0 0 < r < cc x2= r sin 0 < O 27r. (6.2.1) This induces a coordinate frame transformation: = (xl/r)a, + (x2/r)a2 (6.2.2) 204 CoNNEcTioNs TDM I,» 1/ P . I I \1 I \ \1/ I ,i \ \ / \ \ // 1, \ \// / \ \ \ ‘\ ‘ \\s\ 6 >~Q ___\“E 4 f \ M Figure 6.2This diagram illustrates theexponential map andnormal coordin- ates. Exercise 6.2 If1",,-i‘ aretheconnection coefficients with respect toanormal co- ordinate basis atpshow that F;/kkp) +F/tk(P) =0- Hint: Show thatj‘/f,(t) =oiwhere u=uiX,-. 6.2Examples andNewtonian Force Togain some insight into covariant derivatives weturn toIR”. This manifold hasanabsolute parallelism: theparallel-transport map ispath independent. If{xi} arestandard coordinates and X,,=E,-ci8,-|,, then theparallel translate ofX,,atqisX,=E,-ci8,~|,,. Thus insuch a standard chart theconnection isdefined byVg,8,- =0.Such aconnection isreferred toasthestandard connection onIR". Itisofinterest tocompute thestandard connection forlRiinapolar chart (r,6)related tothestandard oneby xi=rcos6 0<r<oc(62.1) xi=rsin6 0<6E27I. This induces acoordinate frame transformation: 8,=(xi/r)8, +(xi/r)82 (6.2.2) EXAMPLES 205 and ao = —x,a, + xia2. (6.2.3) Since al and 32 are parallel we have Va,(a.) = tar(xlir)iai [3,(x2/0]32 = 0 Var(39) = —[3,.(x 2)]31 + [3,.(x1)]32 = (1/03 0 V a,(30) = V2,(3,.) = (1/03 9. Writing V2,(30) as Frerar Fro°3e, etc we may read off the components of the connection in the polar chart; Fart' = Free = (1/r), foor = —r with all others zero. If a curve C is given in the natural chart as xl(p) = OW then V = .V(t)3 1. Let us evaluate the acceleration of the curve C: [0, 1] --> 1H2 given in the above polar chart by (r o C)(t) = p(t), (0 o C)(t) = e(t) for smooth real functions p and 0 of t. The tangent vector to C may be written (6.2.4) C = c*a, = pa, + 0a0 hence v  C = par + 030 + pva . + OV  So. But '7  a, = twarar + Ova,a, = (0/p)ae and v a = pva,a0 + Ov2a6 = (P/p)ao — Hence thethe natural 11:12 acceleration of C is VcC = (p — P 02)3, + (PO + 21)6)(1/Ma0 -With respect to the standard Euclidean metric on IR2 g = a1oa2 + a2oa2 = aroa, + (11r 2)aeoae (6.2.5) (6.2.6) and identifying the parameter t with Newtonian time, we recognise the orthonormal components of this acceleration in the polar frame as the radial and transverse components of Newtonian acceleration of a par- ticle moving in two dimensions under the influence of some Newtonian force. ExAMi=LEs 205 and 8g=—xi8, +xi82. (6.2.3) Since 8,and82areparallel wehave V618,) =l9,(Xi/0191 +l9,(Xi/0192 =0 Ve,(aa) =—I3r(x2)I3I ‘I’I3r(xi)Ia2 =(1/r)3a (614) Va,,(3e) =“Y3, Vg,,(8,) =(1/r)8g. Writing Vg,(8g) asF,g’8, +F,gii8g, etcwemay read offthecomponents oftheconnection inthepolar chart; Fg,”=F,gii'=(1/r), Fgg’=—r withallothers zero. Ifacurve Cisgiven inthenatural chart asxi(p) =Cl(t) then V¢C =X/i(t)8,». Let usevaluate the acceleration ofthe curve C:[0,1]_>lRigiven intheabove polar chart by (Y°C)(l)=/1(1),(9°C)(t)=9(1) forsmooth realfunctions pandG)oft.The tangent vector toCmay be written C=(2.8,=pa,+98,, hence VgC=pa,+68,,+pvga, +Gvgag. But vga,=pvga, +ov,,a, =(o/p)a,, and Vgag =pvgag +®Vg,ag =(6/P)30 EP93,- Hence thenatural lRiacceleration ofCis VCC =(p—p€)i)8, +(p(-9+2p9)(1/p)8g. (6.2.5) With respect tothestandard Euclidean metric onlRi g=81®82 +82®83 =8,®8, +(1/ri)8g®3g (62.6) andidentifying theparameter twith Newtonian time, werecognise the orthonormal components ofthisacceleration inthepolar frame asthe radial and transverse components ofNewtonian acceleration ofapar- ticle moving intwodimensions under theinfluence ofsome Newtonian force. 206 CONNECTIONS Exercise 6.3 Use the standard connection in IR3 to compute the orthonormal compo- nents of the Newtonian acceleration of the curve C : [0, 1] —* IR 3 given by (r o C)(t) = R(t), (0 o C)(t) = OW, (cp o C)(t) = 1,(t) where the maps (r, 0, q)) are standard polar coordinates in 113. The above examples in R2 and IR3 suggest that the Newtonian postulates describing the motion of a single point particle in space be rephrased in terms of the 'natural' connection as follows. (1)A free particle is one that moves along the trajectory described by an autoparallel of the natural connection in Euclidean space, para- metrised by universal time. (2)A point particle of inertial mass m moving in a non-autoparallel curve C, parametrised by Newtonian time, experiences a force 5, on C given by v c(mC). (6.2.7) In many problems in physics g arises as a restriction to C of a vector field on R3 determined from some field theory. If g, is prescribed, (6.2.7) may be used to determine a Newtonian trajectory. As an example, for motion of a particle under the gravitational force produced by a static spherically symmetric distribution of matter (with total gravitational mass M), we may use the Newtonian potential (13 = GMIr in a polar chart, where G is the Newtonian gravitational constant, to obtain = —mdif = (GMmIr 2)3,.. (6.2.8) For a particle with electric charge q the Newtonian Lorentz force is q{i" + iÈ/3}. In standard coordinates {x'} , E = Eidx' and B = B IC1X2 A dX3 B2d-V 3 A dx1 + B3dx1 A dx2 are 1-and 2-forms re- spectively on IR3, parametrised by Newtonian universal time. The metric duals are taken with respect to the Euclidean metric. Solutions of (6.2.7) for particle trajectories subject to these force laws give an excellent description of the behaviour of matter in gravitational and electromagnetic fields provided the motion never approaches Newtonian speeds comparable with 108 m 6.3 Covariant Differentiation of Tensors We have introduced the covariant derivative V,. as a map on vector fields. To extend the definition to its action on smooth 1-forms 206 CONNECTIONS Exercise 6.3 Usethestandard connection in1B3tocompute theorthonormal compo- nents oftheNewtonian acceleration ofthecurve C:[0,1]—>1B3given by(r<>C)(r) =R(r), (6<>C)(r) =(~)(r), (tp<>C)(r) =<I>(r) where the maps (r,6,qa)arestandard polar coordinates inB3. The above examples in1B2and 1B3suggest that theNewtonian postulates describing themotion ofasingle point particle inspace be rephrased interms ofthe‘natural’ connection asfollows. (1)Afreeparticle isonethatmoves along thetrajectory described by anautoparallel ofthenatural connection inEuclidean space, para- metrised byuniversal time. (2)Apoint particle ofinertial mass mmoving inanon-autoparallel curve C,parametrised byNewtonian time, experiences aforce §onC given by a=V¢(mC). (6.21) Inmany problems inphysics 9arises asarestriction toCofavector field on1R3determined from some field theory. If§isprescribed, (6.2.7) may beused todetermine aNewtonian trajectory. Asan example, formotion ofaparticle under thegravitational force produced byastatic spherically symmetric distribution ofmatter (with total gravitational mass M), wemay usetheNewtonian potential (D=GM/r inapolar chart, where GistheNewtonian gravitational constant, to obtain a=-mdTr> =(GMm/r2)8,. (62.8) Foraparticle with electric charge qtheNewtonian Lorentz force is §=q{E+i?§}. Instandard coordinates {xl}, E=E,-dx‘ and B=Bjdxz ,\dx3 +B2dx5 Adx‘ +B3dx‘ Adxz are 1-and 2-forms re- spectively onB3,parametrised byNewtonian universal time. The metric duals aretaken with respect totheEuclidean metric. Solutions of (6.2.7) forparticle trajectories subject tothese force laws give an excellent description ofthebehaviour ofmatter ingravitational and electromagnetic fields provided themotion never approaches Newtonian speeds comparable with108ms". 6.3Covariant Differentiation ofTensors Wehave introduced thecovariant derivative VXasamap onvector fields. Toextend thedefinition toitsaction onsmooth 1-forms COVARIANT DIFFERENTIATION OF TENSORS 207 pe FA1 M we define V,6' by (VA(Y) = —P(Vx17) + X(i6(Y)) X, Y EFTM. (6.3.1) If fE 9;(M) it follows from this that V x(f13) = f*V xl3 + (X.00. (6.3.2) If {Xa}, {eb} are dual bases it follows from eb(Xa)= ô that if cob, are defined by (6.1.5) then Vxoec = —(0`b(X2)0. (6.3.3) If for f E .5,(M) V xf ' X(f) (6.3.4) we note that (6.3.1) is equivalent to adopting the rule Vx(/6(11) = (V /3)(Y) + /3(V Y). (6.3.5) The covariant derivative is said to commute with contractions. Having defined the covariant derivative of 1-forms and vector fields we can extend the definition to arbitrary tensors by adopting this property of commuting with contractions Vx : FP,M ---> FTs,M V xT(X 1, „ X„ e', e5) —T(V xX 1, . . Xr, el, . . es) — . . . — T(Xl, . . „ X„ ..... V xes) + V x(T(Xl, „ Xr, el, . „ es)). (6.3.6) Such a covariant derivative satisfies the Leibnitz property V(TOW) = VxTOW + TOV xW (6.3.7) for all tensor fields T and W. That is, Vx becomes a type-preserving derivation on the algebra of tensor fields. If a mixed tensor has components Tal' bi, .,b, in any basis it is conventional to denote the components of V x,T in the same basis by Tal' 6,, ,b„k  For any Te FP,M the tensor field VTe rr,A4 defined by (VT)(X, X1, . X,, . . V X, Xi E FTM, e° E FT*M (6.3.8) is called the covariant differential of T. Thus starting with a rule that defines a transport of vector fields along curves we have extended the covariant derivative to an operator on general tensor fields. es) = (V xT)(Xi, . . ., X, el, . . „ e) COVARIANT DIFFERENTIATION OFTENSORS 207 fief‘/\1M wedefine Vxfiby (VXfi)(Y) =—fi(VXY) +X(fi(Y)) X,YEFTM. (6.3.1) Iffe@(M) itfollows from thisthat VX(ffi) =fVX5 +(Xf)fi- (6-3-Z) If{X,,}, {eh} aredual bases itfollows from e”(X,,) =65thatifw"care defined by(6.1.5) then VXae° =—w’,,(X,,)e". (6.3.3) Ifforfe@(M) Vxf EX(f) (6.3.4) wenote that(6.3.1) isequivalent toadopting therule VX(5(Y)) =(Vx5)(Y) +5(VXY)- (6-3-5) The covariant derivative issaid tocommute with contractions. Having defined thecovariant derivative of1-forms and vector fields wecan extend thedefinition toarbitrary tensors byadopting thisproperty of commuting with contractions VX:I‘TfMi>I‘TfM VXT(X1, ...,X,,e1,...,e‘) =—T(VXX1,...,X,,e1,...,e‘)— ... —T(X1, ...,X,,e1, ...,VXe‘) +VX(T(X1, ...,X,,e‘,...,e‘)). (6.3.6) Such acovariant derivative satisfies theLeibnitz property VX(T®W) =VXT®W +T®VXW (6.3.7) foralltensor fields Tand W.That is,VXbecomes atype-preserving derivation onthe algebra oftensor fields. Ifamixed tensor has components T"1~""""',,]_,__',,: inanybasis itisconventional todenote the components ofVXkT inthesame basis byT“1"--=“',,l_____,,n.,<. Forany TeI‘TfM thetensor field VTe I‘Tf,, 1Mdefined by (VT)(X,X1,...,X,,e1, ...,e’)= (VXT)(X1, ..., X,,e1,...,e‘) VX, EFTM, e“€FT*M (6.3.8) iscalled thecovariant differential ofT.Thus starting with arule that defines atransport ofvector fields along curves wehave extended the covariant derivative toanoperator ongeneral tensor fields. 208 CONNECTIONS 6.4 Curvature and Torsion Tensors of V Whereas the lack of-linearity in the map X, Y--->VxY prevents V itself from being identified with a tensor it may be used to construct two important tensors. First observe that for any function f E,5,(M): x(iY) = (Xf) Y + fVxY and x(fY) = [X, fY]= (Xf)Y + f[X, Y] V X, Y rTm. It follows that if we define T(X, Y) = V xY — V yX — [X, Y] (6.4.1) then T(X, fY)= fT(X, Y). Since T(X, Y) = —T(Y, X) by construction then T(X, Y) is .5,-linear in both arguments. Consequently associated with T is a type (2, 1) tensor field T known as the torsion tensor of V: T(X, Y, ,e) = 13(T(X, Y)). Associated with any local basis is a set of torsion 2-forms Ta Ta(X, Y) = ea(T(X, Y)). The torsion tensor can be written in terms of these 2-forms as T = 2PDX a. (6.4.2) (6.4.3) (6.4.4) If {ea} is any co-frame, in which the connection 1-forms are {Wa b} , then the torsion 2-forms are given by Ta = dea ± wan A eb. (6.4.5) This is called the first structure equation. It may be proved by contract- ing on a pair of arbitrary vectors. Using (4.10.3) have 2(dea (Dab A eb)(X, Y) = X(ea(Y)) — Y(ea(X)) — el[X, Y]) + b(X)e(Y) — co° b(Y)eb(X) X(ea(Y)) — Vxea(Y) — Y(ea(X)) + Vyea(X) — ea([X, Y]) by (6.3.3). The right-hand side may be simplified by using (6.3.1), producing Wab A eb)(X (de ' , Y) = ea(T(X, Y)) when (6.4.1) is used. Thus (6.4.5) follows from the definition (6.4.3). The second important tensor constructed from V involves two covar- iant differentiations. Again we note from the fundamental properties of 208 CONNECTIONS 6.4Curvature andTorsion Tensors ofV Whereas thelack of@-linearity inthemap X,Y-> VXY prevents V itself from being identified with atensor itmay beused toconstruct two important tensors. First observe thatforanyfunction fe§(M): VXUY) =(Xf)Y +fVxY and 5/3X(fY) E[X,/‘Y1=(Xf)Y +f[X,Y] VX, YeFTM. Itfollows thatifwedefine T(X, Y)=VXY —VYX —[X,Y] (6.4.1) then T(X, fY)=fT(X, Y).Since T(X, Y)=—T(Y, X)byconstruction then T(X, Y)is9-linear inboth arguments. Consequently associated with Tisatype (2,1)tensor field Tknown asthetorsion tensor ofV: T(X, Y,B)=/3(T(X, Y)). (6.4.2) Associated with anylocal basis isasetoftorsion 2-forms T“ T“(X, Y)=§e“(T(X, Y)). (6.4.3) Thetorsion tensor canbewritten interms ofthese 2-forms as T=2T“®X,,. (6.4.4) If{e“} isanyco-frame, inwhich theconnection 1-forms are{uflb}, then thetorsion 2-forms aregiven by T“=de“+w“,,,\ eb. (6.4.5) This iscalled thefirst structure equation. Itmay beproved bycontract- ingonapairofarbitrary vectors. Using (4.10.3) have 2(de" +cu“),Aeb)(X, Y) =X<e"<Y>> —Y<e"<X>> —@“([X,Y1)+w"t<X>@b<Y> —w"t<Y)eb<X> =X<@“<Y>> ~view) —Y<e<X>> +vY@"<X> —@“([X,Y1) by(6.3.3). The right-hand side may besimplified byusing (6.3.1), producing (de“+0)”),Ae“)(X, Y)=§e"(T(X, Y)) when (6.4.1) isused. Thus (6.4.5) follows from thedefinition (6.4.3). The second important tensor constructed from Vinvolves twocovar- iantdifferentiations. Again wenote from thefundamental properties of CURVATURE AND TORSION TENSORS OF V 209 V that for any tensor field U V xV iyU = JV xV yU + (Xf)V yU VjyVxU=JVyVxU V X, Y EFTM. If we define R(X, Y)U = V xV yU — V yV xU — V1. y1U (6.4.6) V U, X, Y, then again we have .?#-linearity and antisymmetry in X, Y. Furthermore, for any smooth function f on M R(X, Y)(fU) = fR(X, Y)U (6.4.7) and R(X, Y)f = 0. (6.4.8) Since Vx is a tensor derivation R(X, [Vs, Vy] — V1x , r is a type-preserving derivation on the algebra of tensor fields R(X, Y)(UOW) = R(X, Y)UOW UOR(X, Y)W (6.4.9) for all X Y, U and W. This derivation is called the curvature operator of V. The curvature operator may be used to define the (3, 1) curvature tensor R of V: R(X, Y, Z, /3) = fl(R(X, Y)Z). (6.4.10) Since R(X, Y) = —R(Y, X) we may introduce a set of curvature 2-forms R d,. by R = 2RdcOec® Xd. (6.4.11) In terms of the connection forms wab with respect to any co-frame {ea}: Rab = dWa b c A Wcb (6.4.12) This is the second structure equation. For verification we contract on an arbitrary pair of vectors: 2(dwab Wac A ( b)(X , Y) = x(wab(n) — Y(wab(x)) — wab([x, Y]) + c(X)wc b(Y) (ii (Y) b(X) = X(ea(V yXb)) — re° (V xXb)) e a(V Ix, yiX b) — Vxea(X,)ec(V yXb) + V yea (Xc.)ec(V xX b) X(ea(V yXb)) — Y(ea(V xX b)) — ea (V Ix, 11) — xea(V yXb) + V yea(VxX b) = ea(R(X, Y)X b) CURVATURE AND TORSION TENSORS orV 209 Vthatforanytensor field U VXVWU =fVXVYU +(Xf)VYU VWVXU =fVYVXU VX, YeFTM. Ifwedefine R(X, Y)U =VXVYU —VYVXU —VlxjYJU (6.4.6) VU,X,Y.then again wehave 9-linearity andantisymmetry inX,Y. Furthermore, foranysmooth function fonM R(X, Y)(fU) =fR(X, Y)U (6.4.7) and R(X, Y)f=0. (6.4.8) Since VXisatensor derivation R(X, Y)E[V/Y, Vy]—VlxjY]isa type-preserving derivation onthealgebra oftensor fields R(X,Y)(U®W) =R(X,Y)U®W +U®R(X, Y)W (6.49) forallXY,UandW.This derivation iscalled thecurvature operator of V.The curvature operator may beused todefine the(3,1)curvature tensor RofV: R(X,Y,Z,5)=B(R(X, Y)Z). (6.410) Since R(X, Y)=—R(Y, X)wemay introduce asetofcurvature 2-forms R46by R=2R“c®e‘®X,,. (6.4.11) Interms oftheconnection forms a>”,,with respect toanyco-frame {e”}: Rab =dc0”,, +(Ouc A(Och. This isthesecond structure equation. Forverification wecontract onan arbitrary pairofvectors: 2(da>”,, +c0”c,\c0‘,,)(X, Y) =X(w”t>(Y)) -Y(w”t(X)) -w“t([X» Y1)+w"¢(X)w‘t(Y) -w”.-(Y)w‘t(X) =X(eu(VYXb)) _Y(e”(VxXb)) _e“(V[x. Y]Xb) ‘VXea(Xc)ec(VYXb) +Vy@”(X@)@‘(VxXt) =X(eu(VYXb)) _Y(e”(VxXb)) _e“(V|x. Y]Xb) _VXeu(VYXb) +VY@“(Vx/Yb) =@"(R(X, Y)/Yb) 210 CONNECTIONS = R(X, Y, Xb, e°) = 2R° b(X, Y). By contracting the (3, 1) curvature tensor we obtain a (2, 0) tensor: the Ricci tensor. That is, Ric(X, Y) = R(Xa, X, Y, ea) (6.4.13) where the arbitrary bases {Xa} and {ea} are dual. For a general connection `Ric' has no particular symmetry properties. It is sometimes more convenient to work with the set of Ricci 1-forms {Pa}, elements of which are defined by Pb (6.4.14) hence Pa = Ric(X b, Xa)eb. (6.4.15) Because of their g'-linearity the torsion and curvature operators can be evaluated on tangent vectors: they do not require vector fields. By suitably extending a pair of tangent vectors to vector fields we can construct figures out of segments of integral curves, giving a character- isation of the torsion and curvature operators. Let Np be a normal neighbourhood of p with Xp, Y E TM. Each qeNp lies on one and only one (up to a linear reparametrisation) geodesic radiating from p. We define Xq E TqM by Xq = TqpX p where rqp is the parallel translation map along the autoparallel. This assign- ment of a tangent vector to every q E Np is smooth: we denote the resulting vector field by X. We similarly extend Yp to a vector field Y. We have constructed X such that G'2,,X = O V Zp E TM, thus T(Y, X)ip = [X, Ylp. From exercise 4.1 at the end of §4.11 we see that T(Y p, Xp) is the tangent at p to the curve formed from the integral curves of X and Y (see figure 6.3). In considering the curvature we extend Xp and Yp differently: this time to commuting vector fields X and Y. We could, for example, choose normal coordinates {x'} with = Xp and with 3 3 X= and Y= . ax' 3x 2 If cp(p) and ip(p) are the integral curves of X and Y respectively, starting at p, then we form the quadrilateral shown in figure 6.4. We denote the parallel translation map from TM to TqM, along the 3 ax' 3 3x2 Yp 210 CONNECTIONS =R(X,Y,X,,,e“) =2R",,(X, Y). Bycontracting the(3,1)curvature tensor weobtain a(2,0)tensor: theRicci tensor. That is, Ric(X, Y)=R(Xa, X,Y,e”) (6.4.13) where thearbitrary bases {X,,} and {e"} aredual. For ageneral connection ‘Ric’ hasnoparticular symmetry properties. Itissometimes more convenient towork with thesetofRicci 1-forms {Pa}, elements ofwhich aredefined by Pb I IXnRnb hence P,=Ric(X,, X,,)e". (6.4.15) Because oftheir 9*-linearity thetorsion andcurvature operators can beevaluated ontangent vectors: they donotrequire vector fields. By suitably extending apair oftangent vectors tovector fields wecan construct figures outofsegments ofintegral curves, giving acharacter- isation ofthetorsion andcurvature operators. LetNpbeanormal neighbourhood ofpwith Xp, YpeTPM. Each qeNp liesonone and only one (uptoalinear reparametrisation) geodesic radiating from p.Wedefine XqeTqM byXq=1:qpXp where 1:4,,istheparallel translation map along theautoparallel. This assign- ment ofatangent vector toevery qeNp issmooth: wedenote the resulting vector field byX.Wesimilarly extend Yptoavector field Y. We have constructed Xsuch that VZPX =0VZP eTPM, thus T(Y, X)lp =[X,Y]p. From exercise 4.1attheendof§4.11 weseethat T(Yp, X,,) isthetangent atptothecurve formed from theintegral curves ofXandY(seefigure 6.3). Inconsidering thecurvature weextend Xpand Ypdifferently: this time tocommuting vector fields Xand Y.Wecould, forexample, choose normal coordinates {xi}with 3 3 ax]P:Xp and 8x2P:Yp with 8 8X=l and Y=ifax] ax" If(p(p) and 1p(p) aretheintegral curves ofXand Yrespectively, starting atp,then weform thequadrilateral shown infigure 6.4. Wedenote theparallel translation map from TPM toTqM. along the CURVATURE AND TORSION TENSORS OF V 211 T I Y), , X',) Figure 6.3 Geometrical interpretation of the torsion tensor. Tps T „ T„T gi, ip Figure 6.4 Geometrical interpretation of the curvature tensor. curve shown, by Tv. If Zi, is any vector in TM then we calculate the parallel translate around the figure by using (6.1.10), dropping terms of order greater than t2: T9P ZP = {Z — tVxZ ± t2i2Vx2Z}q + 0(t3) T r Z = {Z — t(VxZ + V yZ) + t2 rq qp p /2(V x2 Z ± V y 2 Z ± 2V yVxZ).),. + 0(t3). CURVATURE AND TORSION TENSORS OFV 211 T" Q Y’ / P rir,,x,,i Figure 6.3Geometrical interpretation ofthetorsion tensor. Trqtquza Xr=w,<>\p,upi Y ‘stT/‘QTQP Z/J Tqnzn X '\Pl/7) S y 9-r Z/J P Ynstsrtrqtwzp Figure 6.4Geometrical interpretation ofthecurvature tensor. curve shown, byrqp.IfZPisanyvector inTPM then wecalculate the parallel translate around thefigure byusing (6.1.10), dropping terms of order greater than t2: rqPZP ={Z—tVXZ +t2/2VX2Z}q +O(t3) r,qrq,,z,, ={Z-t(vXz +V)/Z)+12/2(vX2z +vylz+2VyVXZ)}, +O(r3). 212 CONNECTIONS Proceeding around the loop we compare tpst-„Trq-cqpZp with Zp: T sT„Tr. T pZ —Z lim P " P P — ([Vy, Vx]Z)p = R(Yr, Xp)Zp. t2 since pc, fl = O. This expression shows that the curvature measures the path dependence of parallel translation. 6.5 Bianchi Identities Because of the way in which the torsion and curvature tensors are constructed out of V certain combinations of their covariant derivatives can be written back in terms of these two tensors. The resulting identities are called Bianchi identities. The (1, 1) tensor field (VxR)(Y, Z) is defined by (VxR)(Y, z)(w, = (VxR)(Y, Z, W, 16). For any X, Y, Z cl-TM consider the vector = {(VxR)(Y, Z) + (V yR)(Z, X) + (V zR)(X, Y)}(V) Z)}(V). X.Y,Z Here Yx . y z denotes the cyclic sum of X, Y, Z. Now (VxR)(Y, Z)= Vx(R67, Z)) — R(V xY, Z) — R(Y , VIZ), so we may write = {Axyz Bxyz}(V) X Y,Z where Axyz(V) = Vx(R(Y, Z))(V) = V x(R(Y, Z)(V)) — R(Y, Z)(17,117) B(V) = (R(VxY, Z) + R(Y, VxZ))(V). We may express A xyz in terms of the curvature operator Axyz(V) = Vx(R(Y, Z)(V)) — R(Y, Z)(VxV) = [Vx, [VY, Vz1 — V,JJV. For any operators P, Q, R we have the (Jacobi) identity P.Q.12[P,[Q, R]] = 0 hence A XYZ(V) = [VX, V1y 71IV. X,Y.I X,Y.Z 212 CONNECTIONS Proceeding around theloop wecompare r,,,r,,r,qrq,,Z,, with Zp: _r,rS,r,r Z—Z=(ivy.ml).=RmX,.>Zp» since [X,Y]=0.This expression shows that thecurvature measures the path dependence ofparallel translation. 6.5Bianchi Identities Because oftheway inwhich thetorsion and curvature tensors are constructed outofVcertain combinations oftheir covariant derivatives can bewritten back interms ofthese two tensors. The resulting identities arecalled Bianchi identities. The (1,I)tensor field (VXR)(Y, Z)isdefined by(VXR)(Y, Z)(W, /5) =(VXR)(Y, Z,W,B).ForanyX,Y,ZeFTM consider thevector 69={(VXR)(Y, Z)+(Vi/R)(Z, X)+(VzR)(X, Y)}(V) EX_%_Z{(VxR)(Y, Z)}(V)- Here S”X_KZdenotes thecyclic sum ofX,Y,Z.Now (VXR)(Y, Z)=VX(R(Y, Z))—R(VXY, Z)—R(Y, VXZ), sowemay write 01':Xv‘E£‘Z{‘4XYZ _Bxvzliv) where AXYZ(V) =VX(R(Y, Z))(V) =VX(R(Y> Z)(V)) —R(Y,Z)(VXV) BXYZ(V) =(R(VXY> Z)+R(Y»VxZ))(V)- Wemay express AXYZ interms ofthecurvature operator AXYZ(_V) =VX(R(Y» Z)(V)) —R(YtZ)(VXV) =IV» IVY= Vzl'V|Y.Z]]V' Foranyoperators P,Q,Rwehave the(Jacobi) identity 9)[P-IQ. Rll=0P.Q.R hence XI?/4XYZ(V) =:,v5f_ZlVx- Vlr.Z]IV- BIANCHI INDENTITIES 213 Writing out By in terms of V B XYZ(V) = (VVvYVZ VZVV yY V[VvY, ± V V Y VxZ VV.,ZV Y XB XYZ(V) Y =,,X,,z(VvvzVx VxVvyz Viv yz, xj + VxVv 7i, VvzrVx + Vi vzy, x])V ([V[y, zi, Vx] Z], X] + T(Y, Z), VX] V[T(Y, Z), xi) V. Using the Jacobi identity again gives Bxyz(V) = — f {[V, V Ir. — R(T(X, Y), Z)1V. X.Y.Z X.Y.2 Thus = — {R(T(X, Y), Z)}17 X.Y2 and since this is valid for arbitrary V: x.Y.Z{(VxR)(Y, Z) + R(T(X, Y), Z)} = 0 (6.5.1) V X, Y, Z E FTM. This is known as Bianchi's second identity. In a similar way we obtain an identity by covariantly differentiating the defining relation for the torsion tensor. We leave it as an exercise to prove Bianchi's first identity: {R(X, Y)(Z) — T(T(X, Y), Z) — T)(Y, Z)} = 0. (6.5.2) X.Y,Z Because of the inherent antisymmetry of the exterior product these identities assume an elegant expression in terms of the torsion and curvature 2-forms. If we exteriorly differentiate the second structure equation and replace clwac by Rac — —a k A a)kc then the second Bianchi identity is expressed as dRa b c A Rh c A (Oct, = O. (6.5.3) Similarly by applying d to the first structure equation and expressing do)°, back in terms of Rac and deb back in terms of T' gives the first Bianchi identity as d Ta + (Dab A Tb = Rab A eb. (6.5.4) BIANCHI INDENTITIES 213 Writing outBXYZ interms ofV BXYZ(V) =(vvxrvz —Vzvvxv _Vtvxr, zy+Vrvvxz —Vvxzvr —V[Y.vXz])V, X§;i,Bm<v> =XF£Z(VvyzVx -Vxvvyz —Vjvyz, X]+VxVv,y -VVZYVX +V[VZY_ X])V =X%Z([V[Y, Z]:VX] _V[[Y. z].X] +[VT(Y. zwVxl _V[T(Y. Z).x])V- Using theJacobi identity again gives X_%ZBXYz(V) =‘X‘5£Z{lVx» V|Y.Z]]—R(T(X, Y)»Z)}V- Thus OJJ=—X%Z{R(T(X, Y),Z)}V andsince thisisvalid forarbitrary V: X5{Z{(VxR)(Y, Z)+R(T(X, Y).Z)}=9 (6-5-1) VX, Y,ZeTTM. This isknown asBianchi's second identity. Inasimilar way weobtain anidentity bycovariantly differentiating thedefining relation forthetorsion tensor. Weleave itasanexercise to prove Bianchi’s first identity: Xf£Z{R(X, Y)(Z) —T(T(X, Y).Z)—(VXT)(Y, Z)}=0.(6.5.2) Because oftheinherent antisymmetry oftheexterior product these identities assume anelegant expression interms ofthetorsion and curvature 2-forms. Ifweexteriorly differentiate thesecond structure equation andreplace dw"C byR“,—to")Awk,then thesecond Bianchi identity isexpressed as dRHb +(Dal. AR(b _Ra(.A (Orb : Similarly byapplying dtothefirst structure equation and expressing dw"C back interms ofR“,anddebback interms ofTbgives thefirst Bianchi identity as dTa +(Dab ATb :Rab Aeh. 214 CONNECTIONS 6.6 Metric-Compatible Connections The introduction of a connection on a manifold does not require any metric properties, and so far we have assumed none. However, when introducing a connection on a pseudo-Riemannian manifold we can impose relations between the connection and the pseudo-Riemannian structure. Parallel translation gives a map between the tangent spaces of any two points connected by some curve. On a pseudo-Riemannian manifold it is natural to require that this parallel-translation map be an isometry between the two tangent spaces. That is, parallel translation preserves the lengths of all vectors. A connection such that parallel translation has this property is called metric compatible. Suppose that Y is a vector field parallel along the curve C. If V is metric compatible then the length of Y will be constant along C, that is C(g(Y, Y)) = O. Since for f E 5;(M) C(f)= Vcf, and V commutes with contractions C(g(Y, Y)) = V c(g(Y, Y)) = V cg(Y, Y) + 2g(V Y, Y). If Y is parallel along C then the second term is zero. Requiring that the length of all parallel vectors along C be constant gives V cg = O. For a metric-compatible connection this holds for all C, so V is metric compatible if and only if Vg = 0. (6.6.1) If {Xi} is any local basis then covariantly differentiating the functions = g(Xi, X1) gives X(g) = V xg(Xi, Xi) + g(cok i(X)Xk, X1) + g(Xi, wk(X)X) = V xg(X„ X1) + wk,(X)g ki + co' i(X)gik. If {Xi} is orthonormal then the functions gij are constant. So in an orthonormal frame the connection forms of a metric-compatible connec- tion satisfy the antisymmetry condition + w. , = 0 (6.6.2) where (op =g,kcok,. Since the Hodge dual is defined by the metric it follows that covariant differentiation with respect to a metric-compatible connection commutes with this operation. First observe that the volume n-form is parallel Vx*1 = 0 V X. (6.6.3) If {e°} is an orthonormal co-frame such that = el Ae2 A A en 214 CONNECTIONS 6.6Metric-Compatible Connections The introduction ofaconnection onamanifold does notrequire any metric properties. andsofarwehave assumed none. However. when introducing aconnection onapseudo-Riemannian manifold wecan impose relations between theconnection and thepseudo-Riemannian structure. Parallel translation gives amap between thetangent spaces of any two points connected bysome curve. Onapseudo-Riemannian manifold itisnatural torequire thatthisparallel-translation map bean isometry between thetwo tangent spaces. That is,parallel translation preserves thelengths ofallvectors. Aconnection such that parallel translation hasthisproperty iscalled metric compatible. Suppose that Yisavector field parallel along thecurve C.IfVis metric compatible then thelength ofYwillbeconstant along C,thatis C(g(Y, Y))=O.Since forfe@(M) C(f) =Vcf, andVcommutes with contractions C(g(Y, Y))=Vc(g(Y. Y)) =V@g(Y, Y)+2g(V@Y, Y). IfYisparallel along Cthen thesecond term iszero. Requiring thatthe length ofallparallel vectors along Cbeconstant gives V¢g =O.Fora metric-compatible connection this holds forallC,soVismetric compatible ifandonly if Vg=O. (6.6.1) If{X,-} isanylocal basis then covariantly differentiating thefunctions 81/: g(x,", X1)gives X(g!j) =VX8(Xt» X1)+8(wkt(X)X/o Xi)+g(x,", wkj(X)Xk) =Vx8(Xn X1)+wki(X)gkj +wkj(X)gik- If{X,-} isorthonormal then thefunctions g,-Iareconstant. Soinan orthonormal frame theconnection forms ofametric-compatible connec- tionsatisfy theantisymmetry condition where wj,Eg_,~kw",-. Since theHodge dual isdefined bythemetric itfollows thatcovariant differentiation with respect toametric-compatible connection commutes with thisoperation. First observe thatthevolume n-form isparallel vX*1=0 VX. (6.63) If{e”} isanorthonormal co-frame such that *1=e‘Ae2A ...Ae" METRIC-COMPATIBLE CONNECTIONS 215 then Vx*1 = a(X)ea A e2 A   A en +    + (—We' A    A 0J",(X)ea. Now &a(X)ea A e2 A A en = i(x)el A e2 A A en and so (6.6.3) follows from (6.6.2). It can now be seen from the definition (1.4.5) that if V is metric compatible Vx* = *Vx e X. (6.6.4) A metric-compatible connection is completely characterised by its torsion tensor. That is, the connection coefficients can be determined in terms of the metric and torsion tensors. For a metric-compatible V we have V u(g(V, W)) = g(V uV, W) + g(V, VW) for any vector fields U, V and W. By cyclically permuting U, V and W we obtain three such expressions. Adding the first two and subtracting the third gives U(g(V, W)) + V(g(W, U)) — W(g(U, V)) = g(V uV, g(1/VW) g(V vW, U) g(W, VU) — g(V wU, V) — g(U, VV). The definition of the torsion operator enables this to be rewritten as 2g(V uV, W) = U(g(V, W)) + V(g(W, U)) — W(g(U, V)) —g(U, [V, W]) + g(V,[W, U]) + g(W, [U, V]) —g(U, T(V, W)) + g(V, T(W, U)) + g(W, T(U, V)). (6.6.5) If {Xa) is an arbitrary basis then the structure functions Cab` of the basis are given by [X„, Xid = CabcXc. (6.6.6) If three different basis vectors are inserted in (6.6.5) then we can solve for the connection coefficients: rabP = IgcP{X,(gb,) X(g) Xe(gab) — Cbcdgad +ccadgbd + cabdged Axb, Jvc, 5—c a) + T(X,, Xa, b) +T(Xa, Xb, Xe)). (6.6.7) Here g`P is the inverse matrix to gab, gpcg`q = 6qp. If {ea} is the dual basis then " -Ca = ea = gabeb . There are two classes of bases in which this expression for the connection coefficients simplifies: in a coordinate basis the structure functions are zero, whilst in an orthonormal basis the metric components are constant. For the case of an orthonormal basis the above expression for the connection coefficients enables the connec- tion 1-forms to be given as 2wab = edix,,ix,(ded — Td) + ix,(dea — Ta) — ixo(deb — Tb). (6.6.8) METRIC-COMPATIBLE CONNECTIONS 215 then VX*1= —w‘,,(X)e“Ae3A ...Ae" +...+(—1)”e‘ A...A LU'1a(X)€a. NOW w1,,(X)€” A€2A ...A€" = a)‘l(X)€1Ae2A ...A€n and so(6.6.3) follows from (6.6.2). Itcan now beseen from the definition (1.4.5) thatifVismetric compatible vX*=*vX vx. (6.64) Ametric-compatible connection iscompletely characterised byits torsion tensor. That is,theconnection coefficients canbedetermined in terms ofthemetric andtorsion tensors. Forametric-compatible Vwe have VU(g(V, W))=g(VUV, W)+g(V, VUW) foranyvector fields U, Vand W.Bycyclically permuting U,Vand Wweobtain three such expressions. Adding thefirsttwoandsubtracting thethird gives U(g(V, W))+V(g(W. U))-W(g(U, Y)) =g(Vt/V. W)+g(V.Vt/W) +g(VvW. U)+g(W,VVU) _8(VwUi V)_8(U» Vwl/)~ Thedefinition ofthetorsion operator enables thistoberewritten as 2g(Vt/V, W)=U(g(V, W))+V(g(W, U))-W(g(U, Y)) —g(U. IV»WI)+g(V,IW»Ul)+g(W.[UtVI) —g(U, T(V, W))+g(V, T(W, U))+g(W, T(U, V)). (6.6.5) If{X0} isanarbitrary basis then thestructure functions Cab‘ ofthe basis aregiven by [X,], Xb] :C,,b"X£. Ifthree different basis vectors areinserted in(6.6.5) then wecansolve fortheconnection coefficients: rabp =%gcP{Xa(gbc) +Xb(gca) —Xc(gab) _Cbcdgad +Ccadgbd +Cabdgcd _T(Xbr Xrv Ya) +T(Xr> X11’ Yb) +T(X,,X,,,X,)}. (6.67) I-Iere g‘Pistheinverse matrix togab, gpcg“? =6;.If{e"} isthedual basis then X,,=ea=gabeb. There aretwoclasses ofbases inwhich this expression fortheconnection coefficients simplifies: inacoordinate basis thestructure functions arezero, whilst inanorthonormal basis the metric components areconstant. Forthecase ofanorthonormal basis theabove expression fortheconnection coefficients enables theconnec- tion1-forms tobegiven as Zwab =edixui/\/h(ded _Td) +i/“(deg _Ta) _'I/\/“(deb " 216 CONNECTIONS This formula is of great computational utility. From now on we will only consider metric-compatible connections. 6.7 The Covariant Exterior Derivative It is often convenient to work with sets of differential forms indexed with respect to some basis. The torsion and curvature forms provide an example. The Bianchi identities for these forms, (6.5.3) and (6.5.4), involve an exterior derivative plus 'correction terms' involving the connection 1-forms. Such combinations of terms can be efficiently encoded into a 'covariant exterior derivative'. Given a mixed tensor that is totally antisymmetric in some subset of r vectors we can associate a set of r-forms with any basis {Xj} with dual {e)}. Suppose that S is such a tensor of type (r + q, p). We define a set of r-forms -JP I, by S'i 'Pj, . . Xr) = S(XI, . . X„ . . X h, e',, . . elP). (6.7.1) We define the covariant exterior derivative D of the S'I  1P), j„ in terms of a connection V by (r + 1)DS'i  `r ),(X0, = i( - ni v x,s(x0 , , xr, xi, , , j=0 - E - kS(T(X,, Xk), X0,     , f(k,   , Xiq, e6, eiP). (6.7.2) The 'hat' above a symbol indicates that that term is omitted from the sequence. T is the torsion operator of V. It follows from the above rather cumbersome expression that , =    jai,. +   h . .   iq -(0L ), A S"    —0;), A s"    iP ji I,  (6.7.3) This can be verified by using (4.10.5). For the special case in which p = q = 0 the covariant exterior derivative reduces to the ordinary exterior derivative. We can then infer from (6.7.2) that eaAVx‘,=d— r A (6.7.4) 216 CONNECTIONS Thisformula isofgreat computational utility. From now onwewillonly consider metric-compatible connections. 6.7 TheCovariant Exterior Derivative Itisoften convenient towork with setsofdifferential forms indexed with respect tosome basis. The torsion andcurvature forms provide an example. The Bianchi identities forthese forms, (6.5.3) and (6.5.4), involve anexterior derivative plus ‘correction terms’ involving the connection 1-forms. Such combinations ofterms can beefficiently encoded intoa‘covariant exterior derivative’. Given amixed tensor that istotally antisymmetric insome subset ofr vectors wecanassociate asetofr-forms with anybasis {Xi} with dual {el}. Suppose thatSissuch atensor oftype(r+q,p).Wedefine aset ofr-forms S“-"P,-I jqby Si1"'iP]'1___jq(X1, ...,Xr) : S(Xl, ..-,X,, Xi], ...,Xjq, 6'), ...,6'-P). (67.1) Wedefine thecovariant exterior derivative DoftheS"'~~"'»=,-]___,-Q in terms ofaconnection Vby (V+1)DSi""i”jl___jq(XO, ..._,Xr) : ..., ...,X,, Xi], ...,Xjq, en, -..,ell’) /=0 -Z(-1)/+*s(r(x,, xk),X0,...,X,,...,X,,..., (ls/<k$r X,,X1,»-..,X,-q,e’),...,ell’). (6.72) The ‘hat’ above asymbol indicates thatthat term isomitted from the sequence. Tisthetorsion operator ofV.Itfollows from theabove rather cumbersome expression that DSil"'iPj1-~~ q =dS’."""’1l»~1., +‘”'l't./\5""’"""/l~--1.. +~~~+"""’i./\S'l""'_‘/1-.-1.. —(Ol‘j]AS'>""iPjJ_H]-q _ ..._(L)}-“jqASil"'iPjl'__jj. This canbeverified byusing (4.10.5). For thespecial case inwhich p=q=0thecovariant exterior derivative reduces totheordinary exterior derivative. Wecanthen infer from (6.72) that 6” Avxa = d_ T“ Aixn. THE COVARIANT EXTERIOR DERIVATIVE 217 (Alternatively this important relation can be verified on 0-and 1-forms; its general validity then following from the fact that both expressions are graded derivations.) Repeated application of (6.7.3) gives the following Bianchi identity for D, D2S1' A St, " /I L i — A Sii  R Sti  I q  '  11 —    —RJ`i, A Sti   in L. (6.7.5) It follows from (6.7.1) that under a change of basis the set of forms iPii _1, transform according to the classical tensor transformation rules. The ,Gi-linearity in the arguments of the right-hand side of (6.7.2) ensures that the DSit... ; transform like the S'i if- under a h  It/ 'lip change of basis. If S/ and 7" are sets of r-forms and s-forms respective- ly, labelled by the multi-indices I and J then, as may be seen from (6.7.3), D(SI A 7") = DS' A + (—WS' A DV. (6.7.6) The interior derivative with respect to a set of basis vectors maps a set of p-forms indexed with q indices into a set of (p — 1)-forms indexed with (q + 1) indices. The anticommutator of this operator with D gives a useful relation. If Lx. Dix. + ix.D (6.7.7) then Lx. maps a set of p-forms into a set of p-forms indexed by the extra index a. It acts as a derivation on exterior products Lx.(S1 A 7') =LES' A + A L x. (6.7.8) First we consider a set of 1-forms, A'' jg. For A any 1-form (6.7.4) gives ix.dA = VA — ebix.Vx&A + ix.TbixA. Using this in (6.7.3) gives = _ ebixavxbAi, lp xaTb i 011   + xn co ' „A lp 1,  (01, i Ai, 7, x„ r,, Il  7,J,. + in  i _g Now if A is any 1-form Vx,A = V xixAec — C p (X b)eP THE COVARIANT raxrrzruorz DERIVATIVE 217 (Alternatively thisimportant relation canbeverified on0-and 1-forms; itsgeneral validity then following from thefactthatboth expressions are graded derivations.) Repeated application of(6.7.3) gives thefollowing Bianchi identity forD, 31...,‘ _ t t...t_ _ i_ i...i_ _ DSI ”jI___jq—Rli\AS’ Pj]‘__/q+...+Rp,XASl ’,l___Iq _/V t...t_ __R’/1/\S‘ “I It ‘... —Ri‘jqASi""iPj|___j:. Itfollows from (6.7.1) that under achange ofbasis thesetofforms S"‘"‘*‘r,-,___,-q transform according totheclassical tensor transformation rules. The @-linearity inthearguments oftheright-hand side of(6.7.2) ensures that theDS"1'-~"r,-l,__,-q transform liketheS""~~‘r,-‘___,-q under a change ofbasis. IfS’andT’aresetsofr-forms ands-forms respective- ly,labelled bythemulti-indices 1and Jthen, asmay beseen from (6.7.3), D(SI AT!) =DSI AT] +(_1)'S1 A Theinterior derivative with respect toasetofbasis vectors maps aset ofp-forms indexed with qindices into asetof(p—1)-forms indexed with (q+1)indices. The anticommutator ofthisoperator with Dgives auseful relation. If LXI]EDix” +iXaD (6.7.7) then Lxamaps asetofp-forms into asetofp-forms indexed bythe extra index a.Itactsasaderivation onexterior products I_,X"(S, AT!) :LXESI AT] +S1ALXaT]. First weconsider asetof1-forms, A"1:‘""/>,-,___,-q. For Aany 1-form (6.7.4) gives iXndA : VXEA — €blXuVXhA + iXaTbiXbA. Using thisin(6.7.3) gives iXDAi|...i,,_ I H ll ' Q _ 1...; _b" 1...)", _-6- t...t_ _—VX/1' "1"....). ‘Y'X.VX./4' ”t.---/., +‘X.T‘Xi/4' "1.-4-1., III+ iXawl;i’Al] ll|,...lP/_l _ wlJi'iXnA||...1,1,.-.l';jl..'jq _- 1,,t...t_ ._ _ /,3 i...i_ ._ ,IX.“ /.A' P/l"'/I/Y"'!q+w]JlXnAl "11---/,1,<--/.1" Now ifAisany1-form VXbA =VXbiXrAe‘ —iXtAw‘,,(X,,)eP 218 CONNECTIONS so = VX,iXA w ca(X0ix,A and ebix,Vx,A = diRA — co` aixA (by (6.7.4) again) SO ix„DAii  - ip Ii'''), = Vx„Ali   II dixA II   'Pi, (DcaixrA hI ''ail  iPi,  t,i,...i„  ly  iy    i P j, ± (01' ir Recognising the right-hand side as containing Dixfil, enables this to be written as LA"""1 = (V ixTb A iXb)Ail  + coi(X a)Ail  ip 4 . . — coi,h(Xa)Ai. 1.   11q. We have obtained this expression for the   `Pi, 1-forms; but Lx„ Vx,, and ix„Tb A ix, are all derivations on exterior products of multi- indexed p-forms so it is consequently valid on arbitrary p-forms: Lx,Sli  iP 11 I, = X„, XTb A iXp)Si' ip /I  1, + w i,(X,),S`.   -   . . . — COJ' is(Xa)Sil  'I's    .iti  (6.7.9) If  = S(e., . e'P, Xji, . . Xj,) then this can be written as'lq Lx„Si,   ,i„ = V eiP, X1i, . . + A ixhS  iPii (6.7.10) For the special case of cp, any ,9;-valued p-form, this reduces to ixAT ± Di Jc 99 = V x„ ± ix,Tb A i x, T. (6.7.11) The definition (6.7.2) can be applied to 0-forms where there is the simplification that the torsion terms do not enter. Since gab = g(Xa, Xb) we have Dgab(X)= xg(Xa, Xb). Thus for a metric-compatible connec- tion Dgab = 0. (6.7.12) 218 CONNECTIONS so Ixavxh/1 = VXhIXflA — (()ca(Xb)iX‘A and e"iXaVXbA =diXnA —0)‘,,iXL_A (by(6.7.4) again) so iX..DA"""”/. ...); =VX.A"""’1l---1. _dix.A“""”1.---/t +wCfliX.A"""’ji-~14 +iXnTbiXbAi,...i,,j1H_]_q +iXnwi,!_'Ai, ...t‘,t,...i,,j‘ 44pjq _wi,l_riXaAi1... t,i....t,,j]m]_q ___ —iX.w"1.A"“""tl... t.1.-4-1. +”"'1.lX.A"""’tl-A 2.)...-it Recognising theright-hand side ascontaining DiXnA‘F---‘P,-I V__,-Qenables thistobewritten as I-X/4" '‘‘"1,._./'4 _ -6- e—(VX, +1x,,T /\1x,,)/1" "’)",.../,, +w"t,(Xa)A" 1' '”/,...j,, _ j,_ i...i_ ,_.4 4 "' w];(Xa)Al P]]...]_‘],...]q' Wehave obtained thisexpression fortheA‘)-< -"P,-1___,-q 1-forms; butLxa, VXBand iXaT" Aixb areallderivations onexterior products ofmulti- indexed p-forms soitisconsequently valid onarbitrary p-forms: i...iI-X.5‘ "1.1., =(VX, +lx,,Tb /\iX,,)Si] "’i’j, .../,,+0Ji‘t,(Xa)Si‘ ‘"?"""‘i"j, ...)',, ...- wt-,S(X,)s11---1»,, U-,___,q. (67.9) IfS"1‘-"5-l___,-4 =S(e’*, ...,e‘/=,X,-1,...,X,-Q)then thiscanbewritten as LXaSi1...i,,I_l mjq =vX,s(@'1, ...,61>,X,-1,...,X,-Q)+iXflT"Aixhstt--»>,,___,q. (6.7.10) Forthespecial caseofqa,any9-valued p-form, thisreduces to ixadqo +Dixflqo =Vxflqo +iXaT" Aixhqa. (6.7.11) The definition (6.7.2) canbeapplied to0-forms where there isthe simplification thatthetorsion terms donotenter. Since gab=g(X,,, X,,) wehave Dga,,(X) =VXg(X,,, X,,). Thus forametric-compatible connec- tion Dg,,,, =0. (6.7.12) THE COVARIANT EXTERIOR DERIVATIVE 219 It follows that if indices labelling a set of forms are raised or lowered with the components of the metric then this operation commutes with the covariant exterior derivative. If the volume n-form is expanded as ,‘1 = (n!)-1 e eii A . . . A et^ then ri,. in= n!*1(X,,, . . Xj. So for a metric-compatible connection = 0. (6.7.13) As anticipated the Bianchi identities (6.5.3) and (6.5.4) can now be written as DRa b = 0 (6.7.14) DTa = Ra b A eb  (6.7.15) In an orthonormal basis the connection 1-forms of a metric- compatible connection are antisymmetric: they satisfy (6.6.2). It follows that the curvature 2-forms satisfy an analogous relation. Moreover, because of the tensorial nature of the transformation of the curvature 2-forms under a change of basis this antisymmetry is maintained in an arbitrary basis. Using this antisymmetry the second Bianchi identity (6.7.15) can be contracted to obtain various other identities. We leave it as an exercise to prove the following contracted Bianchi identities: ixpixgix,DTa= — ixpix,Raa) (6.7.16) p,q,r,a ixpix,ixaDTa= ixiPp — ix,Pq (6.7.17) Tae" + D Tr= 2 (6.7.18) p,q,r P q p,q,r DTa= Pb A eb (6.7.19) 6.8 The Curvature Scalar and Einstein Tensor The existence of a metric tensor enables 'type-changing' of the (3, 1) curvature tensor to various other fourth-rank tensors. We will normally denote all such tensors by the same symbol, making it clear in the context in which it appears exactly which tensor is meant. Similarly the Ricci tensor can be related to a (1, 1) tensor which can then be contracted to a scalar. That is, the curvature scalar is given by = Ric(X a, Xa) (6.8.1) where as usual X° = gabxb. In terms of the Ricci 1-forms Pa, = (6.8.2) THE COVARIANT EXTERIOR DERIVATIVE 219 Itfollows that ifindices labelling asetofforms areraised orlowered with thecomponents ofthemetric then thisoperation commutes with thecovariant exterior derivative. Ifthevolume n-form isexpanded as *1=(nl)‘1s,-‘W,-fie‘) A...Ae'~ then 5,-I___,-H =n!*1(X,-1, ..., X,-H). So forametric-compatible connection D£i1___i” = Asanticipated theBianchi identities (6.5.3) and (6.5.4) cannow be written as DR“),=0 (67.14) DTa =Rab Aeb. Inanorthonormal basis the connection 1-forms ofametric- compatible connection areantisymmetric: they satisfy (6.6.2). Itfollows that thecurvature 2-forms satisfy ananalogous relation. Moreover, because ofthetensorial nature ofthetransformation ofthecurvature 2-forms under achange ofbasis thisantisymmetry ismaintained inan arbitrary basis. Using this antisymmetry thesecond Bianchi identity (6.7.l5) canbecontracted toobtain various other identities. Weleave it asanexercise toprove thefollowing contracted Bianchi identities: Sf’iXiXiXDT =2(iXiXR —iXiXR) (6.7.16)p‘q_,',, P7r '1 H4P’ P'‘"7 iXpiXqiXnDTa= IXqPp “" iXpPq ixpix lXDT,,€” + 9)ix iXDTr: IA/RP’. "’ P-q-r ”" Paw 4 iXnDT"= P),Aeb. (6.7.19) 6.8TheCurvature Scalar andEinstein Tensor The existence ofametric tensor enables ‘type-changing’ ofthe(3,1) curvature tensor tovarious other fourth-rank tensors. Wewillnormally denote allsuch tensors bythesame symbol, making itclear inthe context inwhich itappears exactly which tensor ismeant. Similarly the Ricci tensor can berelated toa(1,1)tensor which can then be contracted toascalar. That is,thecurvature scalar Qtisgiven by at=Ric(X,, X”) (6.8.1) Where asusual X”=g“"X,,. Interms oftheRicci 1-forms P,,, at=iX.P,,. (6.8.2) 220 CONNECTIONS In n-dimensions the Einstein (n — 1)-forms G, are defined by G c = R ab ab A e These may be related to the Ricci forms; we have Gc — Rab A iX,iXb*ea = R ba A i.riX,*ea (6.8.3) = ixb(Rba Aix r*ea) + Pa A ix,*ea = ixb(Rab A 1X°1X,*1) — Pa A ix,*ea = ixqix4Rab A 1X,*1) Pb A iX,*1} P a A ix,*ea. Now Rab A ix,*1 = 0, since it is an (n + 1)-form, so G, = — R*ec. + b A 1xbix,*1 — Pa A ixr*ea = R*ec — 2P° A *ec, and Pa A *e ix,Paeb A *ea, = ixsPa{+ix,(eb A *ea) grw*ea} = ix*Paf+gba*ec + gbe*eal = +ixP a*ea + ix,P a*ea- The contracted Bianchi identity (6.7.17) gives the antisymmetric part of the Ricci tensor in terms of the torsion, so Pa A *e — gt*e, + i rPc*ea + irixix,DTb*ea = — R*e c + *Pc + *ixix,DTb thus Gc = ec — 2*Pc — 2*ixix,DT5 or *-IGc = — 2P, — 2i xiXbDTb. (6.8.4) The set of Einstein forms are equivalent to a (2, 0) tensor. The Einstein tensor G is defined by G = *-IGc(Dec. (6.8.5) The antisymmetric part of the Einstein tensor is determined by the torsion. Using (6.7.17) once again gives ix,*-1Gc — ix,*-1Gb = —2ixhix,ixaDTa. (6.8.6) The covariant exterior derivative of the Einstein forms can also be related to the torsion. Writing GC = Rab A *eabc we have DGc = DR ab A *,,abc ' Rab A D*eabc. The first term is zero by the first 220 CONNECTIONS Inn-dimensions theEinstein (n—1)-forms Gearedefined by G.=RabAiX.*@""- (6.83) These may berelated totheRicci forms; wehave Ge=Rab/\ix,.lx"*e” =Rba/\iX"ix,.*@“ =iX"(Rba /\lx,*@”) _Pa/\ix,*@” =iX°(Rab /\lX"ix,*1) “Pa/\lx,.*¢’° =iX"{iX"(Rab /\lx,*1) —Pb/\lx.*1} _Pa/\lx,*¢’”- Now R0,,AiX(*1 =0,since itisan(n+1)-form, so Ge=—?R*eC +PbAiXiiX‘_*1 —P,AiX(*e" =—97t*eC —2P”A*e,,c and P”/\*¢’a¢=iX"Paeb /\*6“ :l,\"’P”{—lX.(@b /\*@t1) +8b¢*@a} =l,\"’P”{_8t»@*¢’¢ +8i¢*@..} =—iX”P“*e, +iX(P”*e,,. The contracted Bianchi identity (6.7.17) gives theantisymmetric part oftheRicci tensor interms ofthetorsion, so P”A*e,,,=—97t*ec +iXtPC*e,, +iXaiX(iXhDT"*e,, =—97t*ec +*P,+*iX(iXhDT" thus G,=?R*eC —2*P,. —2*iX(,iXhDT" or *"‘GC =£7te, —2P,.—2iX(iXhDT". (6.8.4) The setofEinstein forms areequivalent toa(2,0)tensor. The Einstein tensor Gisdefined by G=*"Gc®e°. (6.8.5) The antisymmetric part oftheEinstein tensor isdetermined bythe torsion. Using (6.7.17) once again gives lXh*—]G( —lX‘_*—1Gb :—2lXhlX(lxlDTa. The covariant exterior derivative oftheEinstein forms canalso be related to the torsion. Writing G‘=Ra),A*e“"“ we have DG‘ =DR“), A*e””‘ +Ra),AD*e“"‘. The first term iszero bythefirst THE CURVATURE SCALAR AND EINSTEIN TENSOR 221 Bianchi identity. In n-dimensions we can expand the Hodge dual as 1 = (n — 3)! e' A e'5 A A einix„ Now ix, ... is proportional to *1 contracted on n vectors, thus its Covariant exterior derivative is zero, so D*el1i2/3 = (n —1 3)! (P-■ A e" A A e"el' A T"A    A e" 1 Ti A e" A A   i (n — 4)! 4 = A *ei02`3 i4 thus DGc = Rab A pi A * eabc p (6.8.7) Equivalently this relation can be written in terms of the (2, 0) Einstein tensor. The divergence of G,V.G, is a 1-form defined by (V.G)(Y) = V G(X° , Y) (6.8.8) thus V.G = (ix.*-1Vx»p — wcp(Xa)ix.*-1Gc)eP = *-1(ea A V ,K,G p — p A G c)eP We may now use (6.7.4) to give V.G = * -1(DGp — Ta A ixGdeP and (6.8.7) then gives V.G = —*-1(Tq A iXgRab *eabdep. 6.9 The Pseudo-Riemannian Connection (6.8.9) Since a metric-compatible connection is completely characterised by its torsion tensor it follows that there is a unique torsion-free metric- compatible connection for any pseudo-Riemannian structure. This con- nection is called the pseudo -Riemannian connection. It is also sometimes associated with the names of Levi—Civita and Christoffel. From (6.6.7) we see that in a coordinate basis the condition of zero torsion is expressed as a symmetry of the connection coefficients, "ab1' = r baP  For this reason a torsion-free connection is often called 'symmetric'. The connection coefficients of the pseudo-Riemannian connection expressed    A Ti")iX  -  iX, siX,4*elli2i3 THECURVATURE SCALAR ANDEINSTEIN TENSOR 221 Bianchi identity. Inn-dimensions wecanexpand theHodge dual as *e"'=" =?—Te‘* Ae'§A...Ae'~1X‘ ...lX'_5lXu*8"'2'3.<4-3) ~ Now ix!"...iXMiXu*e‘1'?‘3 isproportional to*1contracted onnvectors, thus itscovariant exterior derivative iszero, so D*8"'3'3 = (T"A8'5A ...A8“ —8"/\ T“/\ ... A8"' +...+(-1)"-WA AT"~)iX,n...iX,5iXu*e""'="’ = T"A8'-‘A .../\€"'1X-In ...1Xl5*8"'Z'3,~1 :T14A*e11l:'1i4 thus DGC =Rab ATp A*€abCp. Equivalently thisrelation canbewritten interms ofthe(2,0)Einstein tensor. Thedivergence ofG,V.G, isa1-form defined by (V.G)(Y) =VXaG(X", Y) (6.8.8) thus V.G =(iXt*"VXaG,, —w‘,,(X,,)iX¢*“Gc)eP =*"(e“ AVXaG,, —cu‘),AG()eP. Wemay now use(6.7.4) togive V.G =*"(DG,, —T“AiXflG,,)eP and(6.8.7) then gives V.G =—*“(T‘? AiXqR,,,, A*e"",,)eP. (6.8.9) 6.9 ThePseudo-Riemannian Connection Since ametric-compatible connection iscompletely characterised byits torsion tensor itfollows that there isaunique torsion-free metric- compatible connection foranypseudo-Riemannian structure. This con- nection iscalled thepseudo-Riemannian connection. Itisalsosometimes associated with thenames ofLevi—Civita andChristoffel. From (6.6.7) weseethat inacoordinate basis thecondition ofzero torsion is expressed asasymmetry oftheconnection coefficients, F,,,,P =I"),,,P. Forthisreason atorsion-free connection isoften called ‘symmetric’. The connection coefficients ofthepseudo-Riemannian connection expressed 222 CONNECTIONS in a coordinate basis are often called the Christoffel symbols. For actual computations it is often most efficient to use an orthonormal basis. In such a basis there are, by (6.6.2), In(n — 1) independent 1-forms or 1n2(n — 1) independent connection coefficients. For a coordinate basis the zero-torsion condition cuts down the number of connection coeffi- cients to 4n2(n + 1). Thus in an orthonormal basis there are n2 fewer connection coefficients. Because of the Bianchi identities the curvature tensor of a torsion-free connection has extra symmetries. Equation (6.7.16) reduces to an expression of the `pairwise interchange' symmetry of the Riemann tensor. Equation (6.7.17) shows that for zero torsion the Ricci tensor is symmetric. For zero torsion the Einstein tensor is symmetric, by (6.8.6), and divergenceless by (6.8.9). We can use (6.7.4) to write the exterior derivative in terms of any torsion-free connection. Since any metric-compatible connection satisfies (6.6.4) we obtain a useful relation between the pseudo-Riemannian connection and the co-derivative (5 which was introduced in (5.4.2). If cp is a differential p-form then ix.Vx„cp is certainly a (p — 1)-form. Introducing the Hodge map and its inverse: = ir**-1Vx„(19 = ix.*Vx,* -1cP by (6.6.4) = *(Vxa* -IT A ea) (by (1.4.7)) = *(ea A V Mr -10 where i is defined in (1.1.2). We now use (6.7.4): x° x,(P = *cin*-4. The inverse of the Hodge map is given in (5.4.3). By considering the cases of even and odd dimensions separately it can be seen that this can be rewritten as ix.Vx.y) = —*-1d*ricp, that is ixSq) = —6(12. (6.9.1) From now on, unless we specify to the contrary, we shall restrict ourselves to the pseudo-Riemannian connection. For most of what follows it will be essential that the connection is metric compatible, whereas in most places torsion merely contributes extra terms. Exercise 6.4 An Einstein space is one for which Ric = cg for some constant c. Show that if, in three or more dimensions, Ric = fg for f E 5-,(M) then: (i)f = 91.1n (ii)Ge = (n n— 2)3t*ec (iii) d1 = O. 222 CONNECTIONS inacoordinate basis areoften called theChristoffel symbols. Foractual computations itisoften most efficient touseanorthonormal basis. In such abasis there are, by(6.6.2), §n(n —1)independent 1-forms or %n2(n —1)independent connection coefficients. Foracoordinate basis thezero-torsion condition cuts down thenumber ofconnection coeffi- cients to§n2(n +1).Thus inanorthonormal basis there aren2fewer connection coefficients. Because oftheBianchi identities thecurvature tensor ofatorsion-free connection hasextra symmetries. Equation (6.7.16) reduces toan expression ofthe ‘pairwise interchange’ symmetry ofthe Riemann tensor. Equation (6.7.17) shows thatforzero torsion theRicci tensor is symmetric. Forzero torsion theEinstein tensor issymmetric, by(6.8.6), anddivergenceless by(6.8.9). Wecanuse(6.7.4) towrite theexterior derivative interms ofany torsion-free connection. Since anymetric-compatible connection satisfies (6.6.4) weobtain auseful relation between thepseudo-Riemannian connection andtheco-derivative 6which wasintroduced in(5.4.2). Iftp isadifferential p-form then ix-1VX"<p iscertainly a(p—1)-form. Introducing theHodge map anditsinverse: iXtVXa<p =iXt**"VXu<p =ix-*VXa*“<p by(6.6.4) =*(Vx,,*"<P/\ 6'”) (by(1-4-7)) =*(e”/\VX,'7*_l<P) where 77isdefined in(1.1.2). Wenowuse(6.7.4): iX"VX,,(l7 =*dn*“<r~ The inverse oftheHodge map isgiven in(5.4.3). Byconsidering the cases ofeven andodddimensions separately itcanbeseen that thiscan berewritten asix-Vxutp =—*"d*r7q0, thatis iX@VXuq0 =—6<p. (6.9.1) From now on,unless wespecify tothecontrary, weshall restrict ourselves tothe pseudo-Riemannian connection. For most ofwhat follows itwillbeessential thattheconnection ismetric compatible, whereas inmost places torsion merely contributes extra terms. Exercise 6.4 AnEinstein space isoneforwhich Ric=cgforsome constant c.Show thatif,inthree ormore dimensions, Ric=fgforfe€'(M) then: (1)f=at/n (ii)G,=%at*e, (ta)dot=0. THE PSEUDO-RIEMANNIAN CONNECTION 223 Example 6.1 Let g be the metric tensor of a four-dimensional spacetime: g = —e°®e° + V,=,ekOek. In a local chart with coordinates (t(p), r(p), 0(p), cp(p)) a class of spherically symmetric metrics may be parametrised by functions H o, H1, H, of r(p) and a function .1 of t(p), by choosing a local orthonormal co-frame as e° = Hodt = eAHidr e2 = eq-12d0 e3 = ell, sin Odcp. As an example of using (6.6.8) verify that the connection 1-forms co al, of the pseudo-Riemannian connection are given in this basis by table 6.1. Hence construct table 6.2 for Grxeb where X, is a dual orthonormal frame: eb(X,)= k b. 6.10 Sectional Curvature A two-dimensional subspace S of TM will be called a tangent plane to M at p. If {X, Y} is any basis for S and Q(X, Y) = g(X, X)g(Y, Y) — (g(X, Y))2 (6.10.1) then Q(X, Y) = 0 if and only if g induces a degenerate metric on S. Such a tangent plane is called degenerate. If S is any non-degenerate tangent plane at p then the sectional curvature of M at p, along the plane section S, is K(S): g(R(X, Y)X, Y) K(S) = (6.10.2) Q(X, Y) Thus the sectional curvature at p is a real function of the tangent planes at p. Exercise 6.5 Verify that the definition of K(S) is independent of the basis chosen. For the case in which M is Riemannian the sectional curvature generalises the intuitive notions of curvature of two-dimensional sur- faces. If Xo is a normal neighbourhood of the origin in TM then Expp(X0 n S) is a two-dimensional Riemannian submanifold of M. Let WO be an open ball of radius r centred about the origin in xo n S, with r sufficiently small that Exp p is a diffeomorphism onto B(r), an open ball centred about p. Let si(r) be the area of WO and A(r) be THE PSEUDO-RIEMANNIAN CONNECTION 223 Example 6.1 Let gbethe metric tensor ofafour-dimensional spacetime: g=—e°®e°+EZ=1e“®e“. Inalocal chart with coordinates (t(p), r(p), 8(p), cp(p)) aclass ofspherically symmetric metrics may be parametrised byfunctions H0,H1.H2ofr(p) andafunction /1oft(p), bychoosing alocal orthonormal co-frame as en=Hodt e‘=e‘H1dr e2=e‘H2d6 e3=e‘H2 sinGdtp. Asanexample ofusing (6.6.8) verify that theconnection 1-forms w,,,, ofthepseudo-Riemannian connection aregiven inthisbasis bytable 6.1.Hence construct table 6.2forVxaeb where X,,isadual orthonormal frame: e"(X,,) =55. 6.10 Sectional Curvature Atwo-dimensional subspace SofTPM willbecalled atangent plane to Matp.If{X,Y}isanybasis forSand then Q(X, Y)=0ifandonly ifginduces adegenerate metric onS. Such atangent plane iscalled degenerate. IfSisanynon-degenerate tangent plane atpthen thesectional curvature ofMatp,along the plane section S,isK(S): g(R(X. Y)X,Y)KS=—ii——. 6.10.2() Q(X,Y) l) Thus thesectional curvature atpisarealfunction ofthetangent planes atp. Exercise 6.5 Verify thatthedefinition ofK(S) isindependent ofthebasis chosen. For thecase inwhich MisRiemannian thesectional curvature generalises theintuitive notions ofcurvature oftwo-dimensional sur- faces. IfN0isanormal neighbourhood oftheorigin inTPM then Exp,,(.N0 F)S)isatwo-dimensional Riemannian submanifold ofM.Let 973(r) beanopen ball ofradius rcentred about theorigin inN0F)S, with rsufficiently small that Expp isadiffeomorphism onto B(r), an open ballcentred about p.Let.sd(r) bethearea of973(r) andA(r) be Table 6.1 The torsion-free orthonormal connection forms cor,h = —coh„ for the metric of example 6.1. 0 1 2 3 a 0 0 —(H1H(H 1)e-Àe" — 011-1 0)ei —(A11-10)e2 —0.1H0e 3 0 —(1-111111-12)e-Àe2 —(H.V H ,H2)e-Àe3 2 0 —( cot 01H 2)e-Âe3 3 0 = dX/dt, H ciff„/dr. Table 6.2 Associated table of Levi —Cevita connection coefficients specified by V ,Geb in the dual bases satisfying el' (X,) = e° e' e2 e' —(1-0H0HI)e-2e' —(HUH01 -11)e—'e° 0 0 vxI —0.1110)el —(,i1H0)e° 0 0 VX2 —0.11-101e2 (1-11H1H2)e-Àe2 —(.1110)e° 0 —(1-12'1H,H2)e-xel vx, —0.11-101e3 (IV Hif12)e-À0 ( cot 01H2)e-Âe' —(.1110)e° — (H41H 11-12)e—"e' —( cot 01H 2)e-q-2 FIL dIf„/dr Table 6.1Thetorsion-free orthonormal connection forms w,,,=—w,,,, forthemetric ofexample 6.1. b 0 a1 2 3 U-JI\l'—*@0 —(H(,/H(,H,)e-le“ -(21/H.,)el -()1/H.,)e1 -(i/1-1.,)e-* 0 —(!-IQ/H,H2)e"*e2 —(H§/H|H3)e‘*e~‘ 0 —(cot6/H2)e**e»‘ 0 )1Eat/at,H},Ean,/at Table 6.2Associated table ofLevi-Cevita connection coefficients specified byVxueh inthedual bases satisfying e"(X,,) =6",. efl el e2 efl VXu VXI VX, VX,—(H{,/F_10H,)e"'le' —(H(,/lf10H,)e"‘e° O 0 —(/1/H0)e1 —(/1/H0)e° _0 O —(/1/H0)e2 (HQ/H,H2)e-‘el —()1/H(,)e° 0 —(H§/H,H2)e"‘e' —(/1/H(,)e~‘ (HQ/H,H2)e7‘e1 (cot6/HZ)e-/le“ —(/1/H(,)e“ —(H§/H,H;)e‘ —(cot6/H2)e-’~e3 15at/<11,H;EdH,,/dr SECTIONAL CURVATURE 225 the area of B(r). Thus ,s4(r) is determined by the Euclidean geometry of TM whilst A (r) is determined by the Riemannian geometry of Expp(x, n S). The sectional curvature is determined by a comparison of these two areas: K(S) = urn12 — A(r) (6.10.3) r2s4(r) The proof of these assertions can be found in, for example, Helgason (1978). Exercise 6.6 Take M to be the two-sphere with the standard metric induced from IR 3 (see figure 6.5). Calculate the sectional curvature using (6.10.2). Verify that (6.10.3) gives the same result. (Note that B(r) is a spherical cap with geodesic radius r (figure 6.5).) Figure 6.5 A manifold is said to have constant curvature if its sectional curvature is constant. Exercise 6.7 Show that M has constant curvature c if and only if Rab = (6.10.4) 6.11 The Conformal Tensor Two metric tensor fields g and g such that g = exp(2A)g for some function A are said to be conformally related. Whereas a conformal resealing of the metric will change the curvature it is possible to construct a tensor out of the Riemann tensor that is invariant under SECTIONAL CURVATURE 225 thearea ofB(r). Thus .s4(r) isdetermined bytheEuclidean geometry of TPM whilst A(r) isdetermined bythe Riemannian geometry of Exp,,(NU F)S).The sectional curvature isdetermined byacomparison ofthese twoareas: sfl —AK(S) =lim12 (6.10.3)H" r‘&fl(r) The proof ofthese assertions canbefound in,forexample, Helgason (1978). Exercise 6.6 Take Mtobethetwo-sphere with thestandard metric induced from IR (see figure 6.5). Calculate thesectional curvature using (6.10.2). Verify that(6.10.3) gives thesame result. (Note thatB(r) isaspherical cap with geodesic radius r(figure 6.5).)3 N Figure 6.5 Amanifold issaid tohave constant curvature ifitssectional curvature is constant. Exercise 6.7 Show thatMhasconstant curvature cifandonly if R“"=ce“". (6.10.4) 6.11 TheConformal Tensor Two metric tensor fields gand gsuch that g=exp(2/l)g forsome function Aaresaid tobeconformally related. Whereas aconformal rescaling ofthemetric will change thecurvature itispossible to construct atensor outoftheRiemann tensor that isinvariant under 226 CONNECTIONS such scalings. Let {ea} be a g-orthonormal co-frame, with dual {Xa}, and {?) a R-orthonormal co-frame, with dual {X—a }, where = exp(A)ea = exp(-4X a. (6.11.1) If t' is the pseudo-Riemannian connection of g with connection forms w—ab with respect to {ea) then from (6.6.8) (co—ab) = (Dab + Xb(y1)ea — Xa(yl.)eb. (6.11.2) Similarly the curvature forms Rab of t‘ in the {ea} basis are (Rab) = Rab — Xa(A)eb A dA - X,(i1),(`Weah. (6.11.3) We have used D X a(A.) = VxadA, which follows from (6.7.11). Contract- ing with X I gives the Ricci forms and curvature scalar of : exp(X)Fb = Pb ± (2 n)Vxbd + (n — 2)X bWdA + (2 — n)X,(X)Xc(A)e b — ixyxadÂeb (6.11.4) exp(2X)2I = — 2(n — 1)i xiNxbdil + (1 — n)(n — 2)X c(ii.)X`(X). (6.11.5) The conformal 2 -forms Cab are defined (in more than two dimen- sions) in terms of the curvature 2-forms and their contractions by Cab = Rab n —1 2 (P A eb Pb A ea) + (n — 2)1 (n —1)Rea A eb. (6.11.6) These 2-forms have the important property of being invariant under conformal scalings of the metric. That is, if Cab are the conformal 2-forms of g with respect to {ea} then Cab = Cab. (6.11.7) If the (3, 1) conformal tensor (or Weyl tensor) C is defined by C = 2CabOeb®Xa (6.11.8) then equivalently e' = C. (6.11.9) From their definition the conformal 2-forms Cab are manifestly antisymmetric under interchange of a and b. They also satisfy (for zero torsion) analogous identities to those for the curvature 2-forms, namely Cab A eb = ° (6.11.10) iX)XhCpq = iXpiXqCab  (6.11.11) VX,CIA A ea VX„Clil A eh ± Xh(X)ea AdA 226 CONNECTIONS such scalings. Let{e“) beag-orthonormal co-frame, with dual {X,,}, and{e"} ag-orthonormal co-frame, with dual {X,,}, where 21=exp(/1)e“ XI,=exp(—/1)X,,. (6.11.1) IfVisthepseudo-Riemannian connection ofgwith connection forms 05,},with respect to{e“} then from (6.6.8) (w’,,),) =0),),+X,,(/1)e“ —X,,(/1)e,,. (6.11.2) Similarly thecurvature forms RT,ofVinthe{e'7’}basis are (R1)=R,,,,+Vxbd/1Ae,, -vX,a,1Ae,, +X,,(,1)e,, Adi -X,,(/1)e,, A61-Xc(/1)X‘(/1)e,,,,. (611.3) Wehave /used DX,,(/1) =Vxad/1, which follows from (6.7;\11). Contract- ingwith X”gives theRicci forms andcurvature scalar ofV: 6xp(1)F,, =P,+(2--n)vX,<u +(n-2)X,,(1)a1 +(2-n)X,(/1)X‘(/1)e,, -1X,vX.d1e,, (6.11.4) exp(21)271 =at-2(n-1)iX.vX,a1 +(1-n)(n-2)Xc(/1)X‘(/1).(611.5) The conformal 2-forms Ca),aredefined (inmore than two dimen- sions) interms ofthecurvature 2-forms andtheir contractions by 1 1Cflb=Rab"rf2(P../\@b —Pb/\efl) + @aAeb- (6.11.6) These 2-forms have theimportant property of/b\eing invariant under conformal scalings ofthemetric. That is,ifCa), aretheconformal 2-forms ofgwith respect to{e“} then 6,},=c,,,,. (6.11.7) Ifthe(3,1)conformal tensor (orWeyl tensor) Cisdefined by C=2C“,,®e”®X,, (6.11.8) thenequivalently c=c. (611.9) From their definition the conformal 2-forms Ca), are manifestly antisymmetric under interchange ofaandb.They also satisfy (forzero torsion) analogous identities tothose forthecurvature 2-forms, namely Cab/\eb :0 lxaixhcpq :lxplxqcflb. THE CONFORMAL TENSOR 227 In addition there is the identity irCab = O. (6.11.12) A manifold is conformally flat if its metric is conformally related to a flat one. Certainly the conformal tensor must vanish for a conformally flat space. In fact in more than three dimensions a manifold is conformally flat if and only if its conformal tensor is zero (Eisenhart 1949). 6.12 Some Curvature Relations in Low Dimensions In two dimensions there is only one independent curvature form, which must be proportional to the volume form. We have Rab = Ieab. (6.12.1) Since there is only one tangent plane we write the sectional curvature simply as K. This is related to the curvature scalar by K = 2TL. (6.12.2) The conformal 2-forms are not defined in two dimensions. However, all two-dimensional manifolds are conformally flat (Eisenhart 1949). It is often useful to exploit this by adopting coordinates in which the metric is parametrised by the scale function that relates it to a flat metric. We can use the metric to relate the (3, 1) curvature tensor to a (4, 0) tensor, R =2RabOeab. Both factors in the tensor product are 2-forms, and it is often convenient to have a notation for the tensor obtained by taking the Hodge dual of either factor. We write *R = 2*R abOeab (6.12.3) and R* = 2Rab *0 eab (6.12.4) In three dimensions the dual of a 2-form is a 1-form, so R* = 2R abOec A ix,*eab = 2R abi xr*eab (Dec The first factor now involves the Einstein forms, which were given in (6.8.3). So if 2G,C)ec (6.12.5) we have R* = (6, or R = (6*-1. (6.12.6) T1-IE CONFORMAL TENSOR 227 Inaddition there istheidentity ix-C,,,, =0. (6.11.12) Amanifold isconformally flatifitsmetric isconformally related toa flatone. Certainly theconformal tensor must vanish foraconformally flat space. Infact inmore than three dimensions amanifold is conformally flatifand only ifitsconformal tensor iszero (Eisenhart 1949). 6.12 Some Curvature Relations inLow Dimensions Intwodimensions there isonly oneindependent curvature form, which must beproportional tothevolume form. Wehave Rab = %gi.E,,j,. Since there isonly one tangent plane wewrite thesectional curvature simply asK.This isrelated tothecurvature scalar by K=gar. (6.12.2) Theconformal 2-forms arenotdefined intwodimensions. However, all two-dimensional manifolds areconformally flat(Eisenhart 1949). Itis often useful toexploit thisbyadopting coordinates inwhich themetric isparametrised bythescale function thatrelates ittoaflatmetric. Wecanusethemetric torelate the(3,1)curvature tensor toa(4,0) tensor, R=2R,,,,®e”". Both factors inthetensor product are2-forms, anditisoften convenient tohave anotation forthetensor obtained by taking theHodge dual ofeither factor. Wewrite *R=2*R,,,,®e"” (6.12.3) and R*=2R,,,,®*e"" (6.12.4) Inthree dimensions thedualofa2-form isa1-form, so R*=2Rab®ec /\lx,.*@"h =2RabiX,*eab®e£: The first factor now involves theEinstein forms, which were given in (6.8.3). Soif ‘QE2G,®e‘ (6.12.5) wehave R*=‘Q,or R=‘§*". (6.12.6) 228 CONNECTIONS Now cfi*-1 = —Gcixjx,*—le`Oeab = (—R*e c + 2*Pc xh*—lecOeab by (6.8.4). To simplify the first term write ix)xh*-iece( _ = ixiXix b iy(ix,ix,* -11 A ec) 3i3Ox,*-11 = iX'{iXx i,*-11 A ec) iXh*-11gac} ± 3* —leba = j j (p A * leb) + 3* 'eha = gbcix,ix,*-11 + 2*-1eba = *-1eha. In exactly the same way we obtain Pcix„ix,* oelec = ix,Pc*-leac ix,Pc*-leh` Using the symmetry of the Ricci tensor, (6.7.17), gives pcixaixb*-iec = ,-1(ea A 'h b A Pa + Reba). so we have (g* = 2(191e b a Pa A eh — Pb A ea)®eab. Thus (6.12.6) shows that in three dimensions Rab = 1Re1,a + Pa A eh — Pb A ea. (6.12.7) The first immediate consequence is that the conformal 2-forms are identically zero in three dimensions. It also follows that in three dimensions any Einstein space is necessarily of constant curvature. Exercise 6.8 (i)Use the conformal scalings of (6.11.2)—(6.11.5) to show that if in n dimensions Y a = DP„ — [2(n — 1)] Idgi, A ea then I -7-a = exp(—)1) x [Y, + (n — 2)Xb bal  (ii)Show that Y, A eh — Yb b A ea = (2 — n)DCah and Y, A ea = O. In three dimensions Cab 0 and so in this case the tensor Y a0ea is conformally invariant. Thus the vanishing of Y, ea is a necessary condition for conformal flatness: in fact it is also a sufficient condition (Eisenhart 1949). In three dimensions the (2, 0) tensor SEY *Ya(Dea is conformally covariant, symmetric and traceless, by (ii). (iii)Show that in three dimensions D Y„ = O. In four dimensions there are useful identities involving the 'left and right' duals of the curvature tensor. Setting R± ± *-`R*) (6.12.8) we have R- = (Pp A eq — Po A — ;Reaq)OePq (6.12.9) 228 CONNECTIONS Now ‘§*" =—G,iX“iXb*'1e‘®e"" =(—9t*e, +2*P,)iX“iXb*"e‘®e“" by(6.8.4). Tosimplify thefirstterm write iX“iXh*“e‘e,. =ix-iX“iXh*“1e, =iX<(iX“iXh*'11Ae,) +3iXaiXb*"1 =iX"{iX,,(iX,,*7l1/\ec) _lX,,*_118@¢} +3*_1@1m =lx1ix.,(@@ AF161») *lX,iX,,*_11 +3*_1@tm =gbciX“iX,*_l1 +2*_1eba =*—1ebH' Inexactly thesame wayweobtain P(iXuiXh*"1e‘ =iXbP,.*"e,,‘ —iX“P,*“e,," +gt*“e,,,,. Using thesymmetry oftheRicci tensor, (6.7.17), gives P@lX,iX,.*71‘5( =*71(@a APb_61>APa+93666)- sowehave ‘QF1 =2(§@?.e,,,, +P,Ae),—PbAe,,)®e”". Thus (6.12.6) shows thatinthree dimensions Ra),=§9te,,,, +P,,Ae), —PbAe,,. (6.12.7) The first immediate consequence isthat theconformal 2-forms are identically zero inthree dimensions. Italso follows that inthree dimensions anyEinstein space isnecessarily ofconstant curvature. Exercise 6.8 (i)Usetheconformal scalings of(6.11.2)—(6.11.5) toshow thatifinndimensions Y,EDP,-[2(n~1)]-'<I@J1A 8,,then Y“,=exp(-/1.) X[Ya +(n _ (ii)Show that YaAe),—Y),Aea=(2—n)DC,,,, and Y,,Ae”=0. Inthree dimensions Ca),E0andsointhiscase thetensor Y,,®e" is conformally invariant. Thus thevanishing ofYa®e“isanecessary condition forconformal flatness: infactitisalso asufficient condition (Eisenhart 1949). Inthree dimensions the(2,0)tensor SEY E*Y,,®e“ isconformally covariant, symmetric andtraceless, by(ii). (iii)Show thatinthree dimensions DY" =O. Infour dimensions there areuseful identities involving the‘left and right’ duals ofthecurvature tensor. Setting R:E§(Ri*"R*) (6.12.8) wehave R“=(Pl,Aeq—PqAe),—§9te,,,,)®e"‘? (6.12.9) SOME CURVATURE RELATIONS IN LOW DIMENSIONS 229 and R+ = C + gie pq0ePq (6.12.10) where C = 2Cpq0ePq. These relations can be verified in exactly the same way as their three-dimensional analogues. 6.13 Killing's Equation In §4.14 we introduced Killing vectors, these being vector fields that generate local isometries on a pseudo-Riemannian manifold. Because the pseudo-Riemannian connection is determined by the metric structure there are several useful relations between Killing vectors and this connection. Indeed, Killing vectors are often characterised by being solutions of Killing's equation, which is a differential equation for a vector field involving the pseudo-Riemannian connection. It is convenient at this point to introduce the operator Ax Ix —Vx V X EFTM. (6.13.1) It immediately follows that Ax is a derivation on tensor fields that commutes with contractions, also satisfying Axf = 0 Vf e Y,(M). In particular Ax(g(Y, Z)) = 0 = (Axg)(Y, Z) + g(A xY, Z) + g(Y, AZ). For V metric compatible Ag = xg so the above becomes g(AxY, Z) + g(Y, AZ) = —(2xg)(Y, Z). Since for any vector field Y we have AY = [X, Y] — VxY, if V is torsion free then A x Y = —V EX, hence g(V EX, Z) + g(V zX, Y) = (g)(Y, Z). (6.13.2) If )--e is the 1-form related by the metric to X then it is often convenient to rewrite the above in the equivalent form zV + 1yVzX= (Yxg)(Y , Z). (6.13.3) If K is a Killing vector then (6.13.2) becomes Killing's equation: g(V EK, Z) + g(V zK, Y) = 0 V Y, Z E F TM. (6.13.4) The relation (6.13.3) is often useful in applications. Subsequently we shall need a related result for the 2-form d X. If V and Y are arbitrary vector fields then by (6.7.4) SOME CURVATURE RELATIONS INLOW DIMENSIONS 229 and R*=C+({97te,,q®e"‘1 (6.12.10) where C=2C,,q®eP‘?. These relations canbeverified inexactly the same wayastheir three-dimensional analogues. 6.13 Killing’s Equation In§4.14 weintroduced Killing vectors, these being vector fields that generate local isometries onapseudo-Riemannian manifold. Because thepseudo-Riemannian connection isdetermined bythemetric structure there areseveral useful relations between Killing vectors and this connection. Indeed, Killing vectors areoften characterised bybeing solutions ofKilling’s equation, which isadifferential equation fora vector field involving thepseudo-Riemannian connection. Itisconvenient atthispoint tointroduce theoperator AXEst,-vx vxerm. (613.1) Itimmediately follows that AXisaderivation ontensor fields that commutes with contractions, also satisfying AXf= 0Vfe9'*(M). In particular AX(8(Y, Z))=0=(Ax8)(Y. Z)+8(AXY, Z)+8(Y.AXZ) ForVmetric compatible AXg=§EXg sotheabove becomes 5'(AxY» Z)+8(Y,AXZ) =—($x8)(Y» Z)- Since foranyvector field Ywehave AXY =[X,Y]—VXY, ifVis torsion freethen AXY =_'VyX, hence g(VYX,Z)+g(VZX, Y)=(§EXg)(Y, Z). (6.13.2) IfXisthe1-form related bythemetric toXthen itisoften convenient torewrite theabove intheequivalent form iZVyX +IYVZX =(§EXg)(Y, z). (613.3) IfKisaKilling vector then (6.13.2) becomes Killing’s equation: g(VyK, Z)+g(VZK, Y)=0 VY, ZeFTM. (6.13.4) The relation (6.13.3) isoften useful inapplications. Subsequently we shall need arelated result forthe2-form dX. IfVand Yarearbitrary vector fields then by(6.7.4) 230 CONNECTIONS V vd = V vea A Vx„ + ea A V vVX„ = — ea(VvXb)e b A Vx, ± ea A V VVX„ = eb A Vr,x, + ea A V vVx, = ea A (V vVX„ = ea A (R(17, Xa) Vix„Vv V[v, VVvX„) = ea A (R(V, Xa) + V x:7 v — Vvvy)V since V is torsion-free, = ea AR(1/, xa)f + dV v — e° A Vyky (6.13.5) Now id i = ixeaVx — ea A iXVX„ = VX — ea A (iXVX„ iX„VX) VX -17 so that Vî ixdi ± -lea A (iXVX„ iX„VX -17). Using (6.13.3) we have Vx = Oxdf + lzyg(x, xa)ea. (6.13.6) This gives e° A Vv ;ea A ipxyd .2yg(V xy, Xb)eab = A (Vxa(ivd — i7) + ,Yyg(Vx,y, Xb)eab = did + liv(e° A VxA-1.7) — 1Vvdi-/ + 1Zyg(Vx,11, Xb)eab = did — 4V dî + yg(Vxy, Xb)eab (6.13.7) since d2 = 0. Using (6.13.6) once again dVry = d(Zyg(V, Xa)ea) = did-}7 + Xa)eb° + yg(V xy, Xa)eb a. (6.13.8) Returning now to (6.13.5) with (6.13.7) and (6.13.8) produces vd = 2ea A R(V, + V xyg(V, Xa)eba This can be expressed in terms of the curvature 2-forms as Vdî7 = 2Y4l/bRab + V x,Zyg(V, Xu)eba (6.13.9) where Y° = ea(Y) etc. Operating on this with the interior product gives an expression with the Ricci forms: ix,Vx,d -17 = —2Y°P„ + V x,,Tyg(Xe, Xa)e° — V x,Zyg(Xa, Xa)eb 230 CoNNEcTioNs v,,11Y =v,,e1AvX"Y +e“Av,,v,,uY =-@"(vVx,,)@b AvXflY +e"Avvvxu Y =-e*>Av,,,X,Y +e@AvVvX,Y =er/\(VVVX,, -VV,-X,,)? =eaA(R(V’ X11)+Vx,,Vv +V11/,x,,| —VvVx,,)l7 =e”A(R(V, X,,)+vX,v,, -vw)Ysincevistorsion-free. =e”AR(V, X,,)Y+avvY -atAVWY. (613.5) Now IXIIY=1X@~vX,Y -atAIXVXHY =VXY-6"A(1XvX“ +1X,vX)Y +VXY sothat vXY=;1XaY +get,((1XvX,Y +1X,vXY). Using (6.13.3) wehave vXY=§iXdY +g§e,g(X, X,,)e“. (613.6) This gives 6"AVVXHVY =§e"A1,XflVaY +;§e,g(vX,v, X,,)e”" =ieaA(Vx,,(lvd 7)—lvvxfl 17)+i§£Yg(Vx,V1Xb)6’“b =;a1,,dY +g1,,(@“ AvXaaY) -§vVaY +i~§£Yg(VX,,Vi Xb)e”b =;a1,,11Y -gvVaY +;§t2,g(v,,,,v, X,,)e“” (613.7) since d2=O.Using (6.13.6) once again av,,Y =ga1,aY +§11(§t2,g(v, X,,)@“) =;a1,,aY +gvX,§t,g(v_, X,,)e"“ +§§e,g(vX,v, X,,)e"“. (613.8) Returning nowto(6.13.5) with(6.13.7) and(6.13.8) produces vVaY =26“AR(V, X,,)Y+vX,§e,g(v, X,)e'“‘. This canbeexpressed interms ofthecurvature 2-forms as v,,aY =2Y"V”R,,,, +vX,§t,g(v, X,,)e"” (6.13.9) where Y“=e”(Y) etc.Operating onthiswiththeinterior product gives anexpression with theRicci forms: 1,¢vX,_dY =—2Y”P,, +vX,_§eyg(X<, X,,)@" -vX,§eyg(X@, X,,)@b KILLING'S EQUATION 231 or, by (6.9.1) bd = 21 1°P, — V x,2' yg(X`, Xa)ea + V a yg(Xa, X a)eb . (6.13.10) Obviously such expressions are particularly useful for vectors that generate symmetries. Exercise 6.9 A vector field K is called a conformal Killing vector if Kg = 2/1.g for some function A. Show that K satisfies (i) bk = nA (6.13.11) (ii) 6dk = 2KaPa ± 2(n — (6.13.12) Exercise 6.10 For some calculations one needs to be able to commute a Lie derivative past a covariant derivative. If D(Y) [my, VJ — (6.13.13) show that (i) D(Y) is a tensor derivation that commutes with contractions (II) DfX(Y) — fl) x(Y) (iii)D x(ITS = fp x(Y)S for f E g°(M) for any tensor field S (iv) Dx(Y)Z = D z(Y)X (since V is torsion-free) If Dx,(Mb= Mab c(Y)Xe show that MabP07) = le(VT yg(X,., Xb) — Vayg(Xb, Xa) + Vayg(X,„ Xe)). (6.13.14) Hint: starting from Dx(Y)(g(Xb, Xe))= 0 follow the procedure for solving for the connection coefficients given in §6.6. Bibliography Eisenhart L P 1949 Riemannian Geometry (Princeton, NJ: Princeton University Press) Helgason S 1978 Differential Geometry, Lie Groups, and Symmetric Spaces (New York: Academic) KILLINo’s EQUATION 231 or,by(6.9.1) 6dY_2Y“P,, VX;§Eyg(X‘, X,,)e“+vX,se,g(X", X,,)e”. (6.13.10) Obviously such expressions are particularly useful forvectors that generate symmetries. Exercise 6.9 Avector field Kiscalled aconformal Killing vector if§BKg =2/lgfor some function /1.Show thatKsatisfies (1)61?=n/1 (613.11) (11)sax_2K”P,, +2(n1)a)1. (6.13.12) Exercise 6.10 Forsome calculations oneneeds tobeable tocommute aLiederivative past acovariant derivative. If DX(Y) E[§£y, VX]—VD/IX] (6.13.13) show that (i)DX(Y) isatensor derivation thatcommutes with contractions (ii)D/x(Y) =fDx(Y) foffe WM) (iii)DX(Y)fS =fDX(Y)S foranytensor field S (iv)DX(Y)Z =DZ(Y)X (since Vistorsion-free) IfDXn(Y)X,, EM,,,,”(Y)X, show that Mat/’(.Y) =ii8Cp(Vx,§£Y8(X~ X6)“VX,§£Yg(-X1» X6) +Vxhgy/g(X(', X,)). (6.13.14) Hint: starting from DX”(Y)(g(X,,, X,)) =0follow theprocedure for solving fortheconnection coefficients given in§6.6. Bibliography Eisenhart LP1949 Riemannian Geometry (Princeton, NJ:Princeton University Press) Helgason S1978 Differential Geometry, LieGroups, and Symmetric Spaces (New York: Academic) 7 Gravitation 7.1 Lorentzian Connections As we noted in §6.2 the space IR" has a natural connection. This is defined such that a natural coordinate basis is parallel. We have already seen how Newtonian dynamics may be described with the natural connection on IF13. In Chapter 5 Minkowski spacetime was modelled on IR4, the natural coordinate basis being declared orthonormal with respect to a Lorentzian metric. Such a field of global orthonormal frames is parallel with respect to the natural Fi4 connection, and thus we may now recognise the class of inertial frames as consisting of all frames that are parallel with respect to this connection. More generally on any spacetime we may use the unique torsion-free metric-compatible connec- tion (the Lorentzian connection) to evaluate the acceleration of curves. If a particle of mass ti is modelled on a unit timelike curve C then the acceleration VC may be attributed to a four-force Ffi: = Vc(pC). For example, if C describes a particle of electric charge q moving in a background electromagnetic field described by the 2-form F then the force is given by the Lorentz rule,°-; = qicF. Hence C may be determined by solving the equation Vc(1,1) = RicF. (7.1.1) (Since the particle may radiate an electromagnetic field this equation should be coupled with the Maxwell field equations (the particle produces a source of electric current) to determine F properly.) It is instructive to compare a Minkowski four-dimensional description with our earlier Newtonian formulation. We may express F in terms of electric and magnetic fields observed by an inertial observer a„ Gravitation 7.1Lorentzian Connections Aswenoted in§6.2 thespace lR"hasanatural connection. This is defined such thatanatural coordinate basis isparallel. Wehave already seen how Newtonian dynamics may bedescribed with the natural connection onlR3.InChapter 5Minkowski spacetime wasmodelled on lR“, the natural coordinate basis being declared orthonormal with respect toaLorentzian metric. Such afield ofglobal orthonormal frames isparallel with respect tothenatural lR“connection, andthus we may now recognise theclass ofinertial frames asconsisting ofallframes thatareparallel with respect tothisconnection. More generally onany spacetime wemay usetheunique torsion-free metric-compatible connec- tion (the Lorentzian connection) toevaluate theacceleration ofcurves. Ifaparticle ofmass Itismodelled onaunit timelike curve Cthen the acceleration VCC may beattributed toafour-force 9:9*=V¢(uC). Forexample, ifCdescribes aparticle ofelectric charge qmoving ina background electromagnetic field described bythe2-form Fthen the force isgiven bythe Lorentz rule 9=qi?F. Hence Cmay be determined bysolving theequation v¢()tc) =q1}T". (7.1.1) (Since theparticle may radiate anelectromagnetic field thisequation should becoupled with the Maxwell field equations (the particle produces asource ofelectric current) todetermine Fproperly.) Itis instructive tocompare aMinkowski four-dimensional description with our earlier Newtonian formulation. We may express Finterms of electric and magnetic fields observed byaninertial observer 8,, LORENTZIAN CONNECTIONS 233 F = E A dt + B. Similarly we express the trajectory four-velocity e in terms of the Newtonian velocity vk, k = 1, 2, 3, with respect to the same inertial observer, as C = y(a, + vkak), where y (1 — vkvk)1/2. IT is straightforward to calculate Vc(PC) = C'(tiy)a, + e(tryvk)a k (7.1.2) and iëF = —yE &a, — yEvaj + yvJia,B. (7.1.3) We have written E = Eidx1 and used i dt = y, cît = —3„ dx/ = a,. If we write iakB = —EkimBian, (where Ekim is totally antisymmetric k, 1, m = 1, 2, 3 and £123 = 1) then in an inertial chart for Minkowski spacetime (7.1.1) becomes ()..tyv„,) = —qy(Em + vkEkin0) C(12y) = —qyEmv'n. Since C = C.ar, e(o= dad,- = y relates the inertial time variable t to the proper time r at points on the curve. Similarly C.(xk) = dxk/dr = yvk = (dtldr)v k, hence vk = (dxk 1dt). Setting pk =ttyvk, = ity gives the equations in the form d —dt(Pm) = —q(Em + v kEki„,BI) d —dt`e = —qE,v`. We see that the Newtonian equations of motion are recovered for vkvk << 1. For many practical calculations it is, however, often easier to use (7.1.1) directly without passing to an inertial chart. Example 7.1 Use the transformation from the inertial Minkowski coordinates (t, x, y, z) to the coordinates n, y', z'). t = sinhij , x = cosh j, y' = y, z' = z to express the Minkowski metric tensor in the form g = —VdnOdn + + dy'Ody' + dz'Odz' on a patch defined by n, y', z' < co. Verify that the only non-vanishing connection components in this chart are given by Va,aq = (1/)3n = Va,a and Va ,an = Show that C = solves (7.1.1) for a constant electric field expressed in the inertial chart as F = EOCIX A dt if '6 = —qEolm. Hence derive the hyperbolic orbit =- (6-1, n = (fir, y' = 0, z' = 0) and show that this asymptotes to a light cone. Note (fi is the norm of the constant four-acceleration of the particle: g(V V = (g2. LORENTZIAN CONNECTIONS 233 F=EAdt+B.Similarly weexpress thetrajectory four-velocity Cin terms oftheNewtonian velocity u",k=1, 2,3,with respect tothe same inertial observer, asC=1/(E),+u"E)k), where y“=(1—u"uk)"2. Itisstraightforward tocalculate V6616) =C(w)@. +6'"(wv*)@t (7-1-2) and ___d 1}‘?=-)/5,1116, -YE/6,+)/611,13. (7.1.3) ~ . .-V »-_,Wehave_yv_ritten E—Ejdxl andused i(_-dt=y,dt=-8,, dxi=E),-.If wewrite i5kB =—ek’"’B,E),,, (where ek,,,, istotally antisymmetric k,l, m=1, 2,3and em=1) then inaninertial chart forMinkowski spacetime (7.1.1) becomes C(“YU"l) Z_ql/(Em +Uk8klmBI) C"(a1/) =—q1/E...v’”- Since C=C18,, C(t)=dt/dr =yrelates theinertial time variable tto theproper time ratpoints onthecurve. Similarly C(x"') =dx"/dr =yo"=(dt/dr)v", hence v"=(dx"/dt). Setting p"=It)/u", E=,u)/ gives theequations intheform $6..)=-q(E,..+»*et,...B'> Cl .5% =—qE,-v’. Weseethat theNewtonian equations ofmotion arerecovered for vkvk <<1.Formany practical calculations itis,however, often easier to use(7.1.1) directly without passing toaninertial chart. Example 7.1 Usethetransformation from theinertial Minkowski coordinates (t,x,y, 2)tothecoordinates (5,11,y’,z’). t=Esinh 11,x=Ecoshn, y’=y, 2'=2toexpress theMinkowski metric tensor intheform g=—§2d17®d11 +d§®d§ +dy'®dy’ +dz’®dz' onapatch defined byOEE, 17,y’,z’<99.Verify that theonly non-vanishing connection components inthis chart_ are given by V558,, =(1/t§)8,, =Vanég and vane” =E85. Show that C=<98”, ‘fiegt, solves (7.1.1) foraconstant electric field expressed intheinertial chart asF=E0dxAdt if‘Q=—qEO/m. Hence derive thehyperbolic orbit (§= <9", 11=‘Qt,y’=0,2’=0)andshow that thisasymptotes toa light cone. Note ‘Qisthenorm oftheconstant four-acceleration ofthe particle: g(V(_-C, VCC) =‘Q2. 234 GRAVITATION 7.2 Fermi —Walker Transport If C is any geodesic of an arbitrary spacetime (V cC = 0) then if g(X, C) = 0 at any point on the curve then X remains orthogonal to C at all points if V cX = O. But if the acceleration field A c = VC along C is not zero then this property is lost. However, on any given C we may usefully define a new connection V in terms of V and the metric tensor field g. Acting on any vector field X restricted to C cX VEX + g(C, X)A c — g(A c, X)C. (7.2.1) This connection is called a Fermi—Walker or F-connection on C. Its construction manifestly depends on the parametrised curve C itself. An immediate consequence of the definition is that C(g(X, Y)) = g(X, cY) + g(t cX, Y) V X, Y on C(7.2.2) so V is compatible with the metric tensor g. If C is an observer curve (g(C, C) = —1) then g(Ac, C) = 0 and hence V C = 0: so a velocity vector is also F-parallel. For any vector field Y on C, C(g(Y, C)) = g(t cY , 0, so if Y is F-parallel c Y = 0) then the metric projection of Y on C (or the angle between Y and C) is preserved along C. In particular a g-orthonormal frame {Xa} at one point of C, with a timelike basis vector X0 = C, will remain orthonormal with X0 = C at all points along C if parallel transported with respect to the Fermi—Walker connection. Such an F-parallel frame is said to be non-rotating along C and gives one a way of determining whether any spacelike vector undergoes spatial rotation along C: spatial rotation being measured by the components with respect to the F-parallel basis on C. It is generally believed that in spacetime an F-parallel spacelike vector S satisfying the orthogonality condition g(S, C) = 0 along a timelike curve C models the behaviour of an ideal gyroscope (one that experi- ences no non-gravitational torques) on C. Three such mutually ortho- gonal gyroscopes (g(S„ Si) = 6,1) together with C then define a non-rotating frame along C. It is interesting to note that this concept of frame rotation is determined by the metric properties of spacetime. The relation of these properties to gravitational fields is explored in the next few sections. 7.3 The Einstein Field Equations The theory of Newtonian gravitation provides an excellent description for a large class of natural phenomena. The gravitational interaction between macroscopic distributions of matter is defined in terms of 234 GRAVITATION 7.2Fermi—Walker Transport IfCisanygeodesic ofanarbitrary spacetime (V¢C =0)then ifg(X, C)=0atanypoint onthecurve then Xremains orthogonal toCatall points ifV¢X =O.But iftheacceleration field Ac=V@C along Cis notzero then thisproperty islost. However, onanygiven Cwemay usefully define anew connection Vinterms ofVandthemetric tensor field g.Acting onanyvector field Xrestricted toC V¢XEvcx+g(C,X)/tc -g(A¢, X)C. (7.21) This connection iscalled aFermi—Walker orF-connection onC.Its construction manifestly depends ontheparametrised curve Citself. An immediate consequence ofthedefinition isthat C(g(X, Y))=g(X,YCY)+g(V¢X,Y) vx,Yonc(7.2.2) soViscompatible with themetric tensor g.IfACisanobserver curve (g(C, C)=—1)then g(A¢. C)E0andhence V¢C =0:soavelocity vector isalso F-parallel. For any vector field YonC,C(g(Y, C)) =g(V¢Y, C), soifYisF-parallel (V¢Y=O)then the metric projection ofYonC(ortheangle between Yand C)ispreserved along C.Inparticular ag-orthonormal frame {X,,} atonepoint ofC, with atimelike basis vector X0=C,will remain orthonormal with X0=Catallpoints along Cifparallel transported with respect tothe Fermi—Walker connection. Such anF-parallel frame issaid tobe non-rotating along Candgives oneaway ofdetermining whether any spacelike vector undergoes spatial rotation along C:spatial rotation being measured bythecomponents with respect totheF-parallel basis onC. Itisgenerally believed thatinspacetime anF-parallel spacelike vector Ssatisfying theorthogonality condition g(S, C)=0along atimelike curve Cmodels thebehaviour ofanideal gyroscope (one that experi- ences nonon-gravitational torques) onC.Three such mutually ortho- gonal gyroscopes (g(S,-, S,-)=6,-I) together with Cthen define a non-rotating frame along C.Itisinteresting tonote that thisconcept of frame rotation isdetermined bythemetric properties ofspacetime. The relation ofthese properties togravitational fields isexplored inthenext fewsections. 7.3TheEinstein Field Equations The theory ofNewtonian gravitation provides anexcellent description foralarge class ofnatural phenomena. The gravitational interaction between macroscopic distributions ofmatter isdefined interms of THE EINSTEIN FIELD EQUATIONS 235 a Newtonian force derivable most simply from a real scalar field on Newtonian spacetime. As originally formulated, no account is taken of the propagation velocity of this interaction. It is regarded as an instantaneous or static interaction. When Einstein introduced the special theory of relativity the notion of simultaneity became observer depen- dent. The recognition that Maxwell's equations of electromagnetism could be formulated as a set of tensor equations on a four-dimensional spacetime encouraged Einstein to reformulate all the basic laws of classical physics in terms of spacetime tensor fields. According to Einstein the Lorentzian metric of spacetime should also be governed by partial differential equations so that the geometry itself has a dynamical status along with the fields of matter. The idea that the matter and geometry of a spacetime form a mutually sustaining dyna- mical system found fruition in the general theory of relativity proposed by Einstein in 1916. Despite its title this theory proposes that there is an absolute spacetime arena in which the classical events of physics take place. This needs qualifying as follows. If g is any spacetime metric tensor field satisfying Einstein's equations on a manifold M then for cp : M —> cpM a diffeomorphism, cp*g will solve the diffeomorphic image of Einstein's equations on cpM. Any such manifold isometric to M under a diffeomorphism is regarded as describing the same physical phe- nomena. The choice of field equations was partly inspired by the need to recover Newton's laws of gravity in the limit in which propagation effects could be neglected and partly by the aesthetic desire to maintain a tensorial description of spacetime events in which the coordinates of such events were to be relegated to the labelling conventions adopted by different observers. The field equations involve the curvature tensor of the Lorentzian connection and tensors constructed out of various matter fields describing the sources of the gravitational field. There are many ways to formulate these field equations. In the early literature one finds the tensor components of the field equations written out in some local chart from the manifold atlas. There is some virtue in writing out the local equations in full tensorial form since as we shall show this often facilitates their solution and simplifies their presentation. One should, however, note that each local solution of the coupled system of field equations may in general be extended to the whole manifold in different ways. If the global properties of the spacetime manifold are constrained then the class of solutions that can be defined globally will be similarly constrained. Whereas in principle all the physical consequences of such a theory should follow from the Einstein equations for gravity together with the field equations for the matter tensors, an often used approximation models macroscopic 'test' particles that interact solely with gravitation by geodesic world lines. Let us first write Einstein's equations in terms of exterior forms on THEEINSTEIN FIELD EQUATIONS 235 aNewtonian force derivable most simply from areal scalar field on Newtonian spacetime. Asoriginally formulated, noaccount istaken of the propagation velocity ofthis interaction. Itisregarded asan instantaneous orstatic interaction. When Einstein introduced thespecial theory ofrelativity thenotion ofsimultaneity became observer depen- dent. The recognition that Maxwell’s equations ofelectromagnetism could beformulated asasetoftensor equations onafour-dimensional spacetime encouraged Einstein toreformulate allthebasic laws of classical physics interms ofspacetime tensor fields. According toEinstein theLorentzian metric ofspacetime should also begoverned bypartial differential equations sothat thegeometry itself hasadynamical status along with thefields ofmatter. The idea thatthe matter andgeometry ofaspacetime form amutually sustaining dyna- mical system found fruition inthegeneral theory ofrelativity proposed byEinstein in1916. Despite itstitlethistheory proposes thatthere isan absolute spacetime arena inwhich theclassical events ofphysics take place. This needs qualifying asfollows. Ifgisany spacetime metric tensor field satisfying Einstein’s equations onamanifold Mthen for (p:M—>rpMadiffeomorphism, (p*g willsolve thediffeomorphic image ofEinstein’s equations on(pM. Any such manifold isometric toMunder adiffeomorphism isregarded asdescribing thesame physical phe- nomena. The choice offield equations waspartly inspired bytheneed torecover Newton’s laws ofgravity inthelimit inwhich propagation effects could beneglected andpartly bytheaesthetic desire tomaintain atensorial description ofspacetime events inwhich thecoordinates of such events were toberelegated tothelabelling conventions adopted by different observers. The field equations involve thecurvature tensor of theLorentzian connection andtensors constructed outofvarious matter fields describing thesources ofthegravitational field. There aremany ways toformulate these field equations. Intheearly literature onefinds thetensor components ofthefield equations written outinsome local chart from themanifold atlas. There issome virtue inwriting outthe local equations infulltensorial form since asweshall show thisoften facilitates their solution and simplifies their presentation. One should, however, note thateach local solution ofthecoupled system offield equations may ingeneral beextended tothewhole manifold indifferent ways. Iftheglobal properties ofthespacetime manifold areconstrained then theclass ofsolutions that canbedefined globally willbesimilarly constrained. Whereas inprinciple allthephysical consequences ofsuch atheory should follow from theEinstein equations forgravity together with the field equations forthematter tensors, anoften used approximation models macroscopic ‘test’ particles that interact solely with gravitation bygeodesic world lines. Letusfirst write Einstein’s equations interms ofexterior forms on 236 GRAVITATION some neighbourhood of the spacetime manifold M. If {G,} are the Einstein 3-forms associated with the Lorentzian connection, given in (6.8.3), then Einstein's equations for g are KG, + r,(g, = 0 c = 0, 1, 2, 3 (7.3.1) where {r,(g, 0)) is a set of stress 3-forms determined in this co-frame by some choice of matter fields, denoted generically here by (13, and lc is some (positive) coupling constant. (The notation indicates that T, depends on g and (to rather than being contracted on these fields.) We shall supplement these equations with a set of matter field equations denoted collectively by (13) = 0. (7.3.2) We cannot choose the stress forms arbitrarily, since for zero torsion (6.8.6) and (6.8.7) reduce to DG„ = 0 (7.3.3) and Ga A eb = Gb A ea- (7.3.4) The matter stress forms defined with respect to {ea} determine the stress energy tensor field = *-1T,C)ea (7.3.5) Any matter model for Einstein's equations must therefore give rise to a symmetric second-rank stress tensor er = abea (Deb that is divergence- less: V.:I = 0. In many cases given a matter model there is a well defined procedure for generating such a stress tensor. Indeed the most economical way to summarise the whole coupled system is in terms of an action functional whose extremal equations generate the full set of field equations including the consistent stress forms. Although it is straightforward to set up a heuristic scheme for applying a variational calculus to obtain all the field equations it would take us too far afield to set up a decent formalism for this purpose. (The precise formulation of a variational scheme involving spinors requires particular care.) We shall be content in this chapter to give some examples of matter models in exterior form together with their associated stresses. Such matter models have featured prominently in many theoretical discussions of gravitational interactions with fields. Exercise 7.1 Show, by contracting (4.8.4) and using (7.3.1), that in n dimensions Einstein's equations can be written as ix.*-1Ta 2KP, = * -Lr, e n — 2 236 GRAVITATION some neighbourhood ofthespacetime manifold M.If{Gc} arethe Einstein 3-forms associated with theLorentzian connection, given in (6.8.3), then Einstein’s equations forgare KG, +rC(g, (D)=0 c=0,1,2,3 (7.3.1) where {rc(g, <D)} isasetofstress 3-forms determined inthisco-frame bysome choice ofmatter fields, denoted generically here by(D,andKis some (positive) coupling constant. (The notation indicates that I, depends ongand(Drather than being contracted onthese fields.) We shall supplement these equations with asetofmatter field equations denoted collectively by a@,o)=0. (733 Wecannot choose thestress forms arbitrarily, since forzero torsion (6.8.6) and(6.8.7) reduce to Do,=0 (7.33) and Ga Aej, =Gb Aea. The matter stress forms defined with respect to{e“} determine the stress energy tensor field a=*"'t,®e”. (73.5) Any matter model forEinstein’s equations must therefore give risetoa symmetric second-rank stress tensor YT=9,,,,e"®e" that isdivergence- less: V3 =0.Inmany cases given amatter model there isawell defined procedure forgenerating such astress tensor. Indeed themost economical way tosummarise thewhole coupled system isinterms ofanaction functional whose extremal equations generate thefullsetof field equations including theconsistent stress forms. Although itis straightforward tosetupaheuristic scheme forapplying avariational calculus toobtain allthefield equations itwould take ustoofarafield tosetupadecent formalism forthispurpose. (The precise formulation ofavariational scheme involving spinors requires particular care.) We shall becontent inthischapter togive some examples ofmatter models inexterior form together with their associated stresses. Such matter models have featured prominently inmany theoretical discussions of gravitational interactions with fields. Exercise 7.1 Show. bycontracting (4.8.4) and using (7.3.1), that inndimensions Einstein’s equations canbewritten as iX.1*"r,,2P=**' .—E ..K C T( n __ 2 el THE EINSTEIN FIELD EQUATIONS 237 The conditions that the stress tensor be symmetric and divergenceless are required for it to be equated to the Einstein tensor of a metric- compatible torsion-tree connection. In addition further 'energy' condi- tions are usually required to hold in order for the stress tensor to be physically reasonable. The weak energy condition is that 5-(V, V) 0 for all timelike V. This condition is motivated by assuming that an observer whose curve is tangent to V would interpret 7(V, V) as an energy density. The dominant energy condition is similarly motivated. This can be phrased as requiring that jv be a future-pointing non-spacelike vector for all future-pointing timelike V, where Iv = —*Tv for r v = raea(V). Alternatively one can impose conditions on the stress tensor by requiring that the corresponding (via Einstein's equations) Einstein tensor has certain properties, resulting in gravity being, in some sense, attractive. The condition on the stress tensor such that Ric(V, V) 0 for all timelike V is called the strong energy condition. Details of these energy conditions can be found in Hawking and Ellis. 7.4 Conservation Laws In Newtonian dynamics the total energy and momentum of a system may be defined to be certain dynamical variables that remain fixed as the system evolves. Such constants of the motion have their origin in the existence of certain symmetries of the equations of motion. Similarly in the dynamics of continuous media the vanishing divergence of the Newtonian energy—momentum tensor affords a succinct description of the equations of motion, and the associated constants of motion may be obtained by integrating densities constructed from the components of such a tensor. On a curved manifold, however, caution is required in correlating conservation laws to the existence of a divergenceless stress tensor. In general it is necessary for the spacetime metric to admit some kind of symmetry in order to construct conserved quantities. Let 5- be a symmetric (2, 0) tensor whose metric related (0, 2) tensor has components g ab in some orthonormal frame {Xa }. For any vector field V we have Yvg(Xa, Xb) + g(IvXa, Xb) + g(Xa, YvXb)= 0 since v[g(Xa, Xb)] = 0 for any orthonormal frame {Xa }. Hence since ab = ba and V is torsion free: vg(X,„ X Off ab *1 = —2g(Y vXa, X b)Ff ab*i = _2g(vvx. _ vxy, x b)?fab *1 = —{g(V VXa, X b) g(X a,vXb)}Ff ab*1 2g(Vxy, Xb) jab *1. THE EINSTEIN FIELD EQUATIONS 237 The conditions that thestress tensor besymmetric anddivergenceless arerequired forittobeequated totheEinstein tensor ofametric- compatible torsion-free connection. Inaddition further ‘energy’ condi- tions areusually required tohold inorder forthestress tensor tobe physically reasonable. The weak energy condition isthat g(V, V)E0 foralltimelike V.This condition ismotivated byassuming that anobserver whose curve istangent toVwould interpret §(V,V)asan energy density. The dominant energy condition issimilarly motivated. This can bephrased asrequiring that jvbeafuture-pointing non-spacelike vector for allfuture-pointing timelike V,where 17:,=—*rv for‘Ev=r,,e“(V). Alternatively one canimpose conditions onthestress tensor byrequiring that thecorresponding (via Einstein’s equations) Einstein tensor hascertain properties, resulting ingravity being, insome sense, attractive. The condition onthestress tensor such that Ric(V, V)E0 foralltimelike Viscalled thestrong energy condition. Details ofthese energy conditions canbefound inHawking andEllis. 7.4 Conservation Laws InNewtonian dynamics thetotal energy andmomentum ofasystem may bedefined tobecertain dynamical variables that remain fixed as thesystem evolves. Such constants ofthemotion have their origin inthe existence ofcertain symmetries oftheequations ofmotion. Similarly in thedynamics ofcontinuous media thevanishing divergence ofthe Newtonian energy—momentum tensor affords asuccinct description of theequations ofmotion, andtheassociated constants ofmotion maybe obtained byintegrating densities constructed from thecomponents of such atensor. Onacurved manifold, however, caution isrequired in correlating conservation laws totheexistence ofadivergenceless stress tensor. Ingeneral itisnecessary forthespacetime metric toadmit some kind ofsymmetry inorder toconstruct conserved quantities. Let9beasymmetric (2,0)tensor whose metric related (0,2)tensor hascomponents ‘J""insome orthonormal frame {X,,}. Foranyvector field Vwehave §£vg(X,,, X,,)+g(§£vX,,, X,,)+g(X,,, §£vX,,) =0since §£v[g(X,,, X,,)] =0forany orthonormal frame {X,,}. Hence since 9””=9"”andVistorsion free: -$Vg(Xa, Xb)gab*1 =-2g(.stvX,,, X,,)9”"*1 =—2g(VvX,, -v,,,v,X,,)9""*1 =—{g(VVXa1 Xb) +g(Xav VVXb)}gab*1+ 2g(VX,v' Xb)gab*1‘ 238 GRAVITATION Now g(V vX,, X1,) + g(Xa, VX,,) = 0 since V{g(X„, Xi,)} = 0 and so Xb),50b*1 = g(V xy, Xb)Tah*1 = V x{g(V, X b),Gfab)*1 — g(V, V x,X0.5ab*1 — g(V, X 1,)V x„.3"b*l. Now for any (n — 1)-form J we may write di = ea A VJ = ,K(e° Ai) — VX,ea A So introducingintroducing la = e" Ai we have di = Vx.ja — x„ea)(Xb)j b. Thus we have 2vg(Xa, X brTab*1 = V,v(g(V, — (V xt)(Xb)g(V, X c)Tb`*1 +(V xfa)(X0g(17, X,),5 bc — g(V, V x/ V 0,5 ab*, _ 1 g(V, X b)V x ff ab * 1 = cl{V bgab*ea} fea zv ,1, k Xb)1/,3"bc + g(V, V x/ V b)f ab ± v by "yo-j ab)*1 =- ci{17T al,* ea } — {V ' xe b (X a) bce c ± V X'e bg ab + X a (-7 ab)eb}(V)*1. We may write this in terms of the (n — 1)-form Jv = Vb ' 6 I ab*eb as gab*i = d jv _ (v.,7)(v)*i. 1Vvg)(Xa, Xb) (7.4.1) From this relation we conclude that if the spacetime admits a conformal Killing vector field C, 2cg = 24, then /10-aa*1 = di c — (V.3-)(C)*1. Hence a closed (n — 1)-form may be constructed out of a divergenceless traceless stress tensor in a spacetime with conformal isometries. If the vector field K is Killing (YIN = 0) then irrespective of the trace of 3- dJK = O. If Po, (P,) are Killing vector fields on four-dimensional spacetime generating open timelike (spacelike) integral curves then the integrals of the corresponding 3-forms over a spacelike 3-chain define the energy (momentum) contributed by 3- to E. Similarly if J, are three Killing vector fields that generate the closed integral curves corresponding to the orbits of the rotation group SO(3) then the corresponding integrals may be taken as defining the angular momentum in E. There is a useful analogy between solutions of Einstein's equations, coupled to matter, admitting symmetries and solutions to Maxwell's equations coupled to charged matter. The closed 3-forms constructed out of the stress tensor and the Killing vector are the analogues of the closed electromagnetic current 3-form. Maxwell's equations have the important property that one may define the total charge contained in a 238 GRAVITATION Now g(VVX,,, X,,) +g(X,,, V(/X,,) =Osince V{g(X,,, X,,)} =Oandso i~§£v8(X@» Xb)gab*1 :8(VX.V» Xt>)g“h*1 =Vx.{8(V» Xt>)g”b.i'*1 _8(V~ VX.Xt>)gab*1 -8(VtX1=)Vx,,g('b*1- Now forany(n—1)-form Jwemay write dJ=e“AVX“J =VX”(e"AJ) —VX“e"AJ. Sointroducing j“=e“AJwehave dJ=Vxuj” —(VX”e“)(X,,)j". Thus wehave %.§£vg(X,,, X),)g"b*1 =Vx,{8(V» X6)g”b*1) _(Vx/')(Xb)8(ViX@)gbC*1 +(VX“e")(X,,)g(V, X,.).°I""*1 "8(V~ VX,,Xb)gab*1 _8(V» Xb)VX,,gab*1 =d{V,,9"”*e,,} -{e@(vX,X,,)v,ab@ +g(V,v,,,X,,)a~b +vbvxaaflb}-1 :d{Vbgab*ea} _{VX"@b(Xa)gb¢e': '4'VX"ebgab "l"Xa(gab)eb}(V)*1' Wemay write thisinterms ofthe(n—1)-form JV=V".97,,,,*e" as i(~§£v8)(Xa1 Xt>)gab*1 =djv_(V'g)(V)*1- (7-4-1) From this relation weconclude that ifthe spacetime admits a conformal Killing vector field C,§£¢g =2/lg, then 19,941 =61¢-(v-a)(c)*1. Hence aclosed (n—1)-form may beconstructed outofadivergenceless traceless stress tensor inaspacetime with conformal isometries. Ifthe vector field KisKilling (§£Kg =0)thenirrespective ofthetrace of9' dJK = IfP0,(P,) areKilling vector fields onfour-dimensional spacetime generating open timelike (spacelike) integral curves then theintegrals of thecorresponding 3-forms over aspacelike 3-chain Edefine theenergy (momentum) contributed by9'to2.Similarly ifJ,-arethree Killing vector fields that generate theclosed integral curves corresponding to theorbits oftherotation group SO(3) then thecorresponding integrals may betaken asdefining theangular momentum inE. There isauseful analogy between solutions ofEinstein’s equations, coupled tomatter, admitting symmetries and solutions toMaxwell’s equations coupled tocharged matter. The closed 3-forms constructed outofthestress tensor andtheKilling vector aretheanalogues ofthe closed electromagnetic current 3-form. Maxwell’s equations have the important property that onemay define thetotal charge contained ina CONSERVATION LAWS 239 compact region by the integral of the 2-form *F, which is closed in any source-free region, over any closed 2-chain. (Electric charge may be defined by a de-Rham period.) Einstein's equations give rise to analo- gous 2-forms that are closed in source-free regions of spacetimes with symmetries. Einstein's equations imply that when the stress tensor vanishes the spacetime is Ricci flat. So if the spacetime admits a Killing vector K then, from (4.13.12) the 2-form *dk is closed. In such spacetimes we shall refer to *d k as a Komar form, the component expression having been introduced into general relativity by Komar [11]. 7.5 Some Matter Fields The Einstein—Klein--Gordon system The massive real scalar field cp E rAoM is taken to satisfy d*dcp = 1.22*cp + U'(cp)*1 (7.5.1) where y is some real parameter and U is a polynomial in cp. The stress forms in the local frame {X a} are given by Ta = (iadcp A *dcp + dcp A in*dT) (tii2T2 U)*ea (7.5.2) where i a ix,. The stress associated with a constant U is sometimes attributed to a 'cosmological term'. As we have remarked, in order to be consistently equated to the Einstein tensor, the stress forms should satisfy DT, = O. Taking the expression in (7.5.2) gives Dr„ = (Dix,dcp A *dcp + ixncicp A d*dcp — dcp A Di *dcp) (P2c19 tr)(149 A *ea. (7.5.3) Now we may use (6.7.11) (for zero torsion): Dra = A *dcp + ix,dcp A d*dcp — dcp A V *dcp + cicp A id*d(p) iX,CiegY 2T tr)*1. Since V is metric-compatible cicp A V x, *dcp = dcp A *V x,c1cp = V xcicp A *dcp and so the terms involving V cancel. Since dcp A i kcl*dcp = x Jcicp A d*dcp) + i x,dcp A d*dcp, and dcp A d*dcp is a 5-form in four dimensions DT, = Xa(T)(d*dT ,u2*cp — U'*1). Thus whenever the field equations (7.5.1) hold ar a =- O. CONSERVATION LAWS 239 compact region bytheintegral ofthe2-form *F,which isclosed inany source-free region, over any closed 2-chain. (Electric charge may be defined byade-Rham period.) Einstein’s equations give risetoanalo- gous 2-forms that areclosed insource-free regions ofspacetimes with symmetries. Einstein’s equations imply that when thestress tensor vanishes thespacetime isRicci flat. Soifthespacetime admits aKilling vector Kthen, from (4.13.12) the 2-form *dK isclosed. Insuch spacetimes weshall refer to*dK asaKomar form, thecomponent expression having been introduced intogeneral relativity byKomar [11]. 7.5Some Matter Fields TheEinstein—Klein—G0rd0n system Themassive realscalar field (peFAOM istaken tosatisfy d*d(p =ttzttp +U'((p)*1 (7.5.1) where tiissome realparameter and Uisapolynomial in(p.The stress forms inthelocal frame {X,,} aregiven by It=%(i..d¢>A *d¢>+d¢>Ai..*d9>) —(iH2<P2 +U)*@.. (7-5-2) where 1,,Eix”.The stress associated with aconstant Uissometimes attributed toa‘cosmological term’. Aswehave remarked, inorder tobeconsistently equated tothe Einstein tensor, thestress forms should satisfy Dru=0. Taking the expression in(7.5.2) gives D1,, =§(DiX"d(pA *dtp +iX”d(pA d*dq0 —d(pA DiX‘*d(p) —(uztp +U’)dq0A *e,,. (7.5.3) Now wemay use(6.7.11) (forzero torsion): D17=%(Vx,,d¢>A *d¢>+ix.d¢>Ad*d¢> —do/1VX,,*d¢> +dqvxix,,d*d¢>) -iX,d<P(uZ9> +U')*1- Since V is metric-compatible d(pA VX“*d(p = d(pA *V,,-"d(p = VX“dq0A *d(pandsotheterms involving Vcancel. Since d(pA iX"d*d(;0 = —iXd(d(pAd*d(;0) +iXnd(pAd*d(p, and d(pA d*dq> isa5-form infour dimensions Dr.=X..(<P)(d*d¢> —#2*¢>—U'*1)- Thus whenever thefield equations (7.5.1) hold Dr, =0. 240 GRAVITATION Exercise 7.2 Show that 3(X0, 1(0*1 = r o A e° and that for (7.5.2) 13 To A e° = (—E(X„(T))2 ± 412492 + that is, for a suitable potential U the weak energy condition is satisfied. The Einstein—Proca system The 'massive' real 1-form field  is taken to satisfy d*dii = —m 2*A (7.5.4) with u some real non-zero constant. The associated stress forms are Ta= ",(i„dA A *di2). — i„*dA A (Li) + ;1.4.2(0 A *A + A A ia*A). (7.5.5) It may be noted that an integrability condition follows by applying *d to (7.5.4): = o. (7.5.6) The Einstein—Maxwell system For the electromagnetic field 2-form F we have the curved space Maxwell equations d*F = 0 (7.5.7) dF = 0 (7.5.8) with associated stresses Ta = *F — ia*F A F). (7.5.9) The Einstein Yang—Mills system Let A = AiT` be a Lie-algebra-valued 1-form, A. rA1M and {P} a basis for some Lie algebra, with Lie bracket [T', Ti]. The Yang—Mills field strength is the Lie-algebra-valued 2-form F = dA + [A, A] = F,T' where the bracket between a Lie-algebra-valued p-form H and a Lie-algebra-valued q-form B is [H, B] = H, A Bi[r, Ti] = H AB — (-1)PqB A H and dA = dA,P. It is useful to define an exterior covariant derivative on the Lie-algebra-valued p-forms H: DH = dH + [A, H]. (7.5.10) 240 GRAvITAT|oN Exercise 7.2 Show that3T(X(,, X(,)*1 =toAe“andthatfor(7.5.2) .1 1 7 17axe" =(5Z(X..(¢))" +%tr</>"+U)*1 a=(l thatis,forasuitable potential Utheweak energy condition issatisfied. TheEinstein—Proca system The‘massive’ real1-form field Aistaken tosatisfy 6-6/1=-111*/$1 (7.5.4) with usome realnon-zero constant. Theassociated stress forms are T,=%(1,a/i A-<1/1-1,,»-<1/1A6/?1)+;1t1(1,,/31 A-=/It+AAi,*/1). (7.5.5) Itmay benoted thatanintegrability condition follows byapplying *dto (7.5.4): 6/it=0. (7.56) TheEinstein—Maxwell system For theelectromagnetic field 2-form Fwehave thecurved space Maxwell equations d*F =0 (7.5.7) dF=0 (7.5.8) with associated stresses r,,=§(i,,FA *F—i,,*FA F). (7.5.9) TheEinstein Yang—M illssystem LetA=A,»T' beaLie-algebra-valued 1-form, A,-EI‘/\1M and {Tl} a basis forsome Liealgebra, with Liebracket [T',Tl].TheYang—Mills field strength istheLie-algebra-valued 2-form F=dA+§[A,A]=F,-T" where the bracket between aLie-algebra-valued p-form Hand a Lie-algebra-valued q-form Bis [r1,B]=H,-AB,-[T", T/]=HAB -(—1)P‘lBAH anddA=dA,-T". Itisuseful todefine anexterior covariant derivative ontheLie-algebra-valued p-forms H: on=an+[.4,H]. (7.5.10) SOME MATTER FIELDS 241 From the definition of F we have the Bianchi identity DF = 0. (7.5.11) The field equation analagous to (7.5.7) is D*F = 0. (7.5.12) The system is coupled to Einsteinian gravity with the stress forms Z!!.(i0F; A *Fi — ia*Fi A F1). (7.5.13) The Einstein—Maxwell-charged scalar system In this case an electrically charged complex scalar field (1) couples to both gravity and electromagnetism. The Maxwell equations now have electric current sources j[g, d*F = j (7.5.14) dF = 0 (7.5.15) where the current 3-form is j = Im((1)*a(I)*) (7.5.16) and the U(1) exterior covariant derivative is defined by in terms of the 1-form A satisfying F = dA. Under the maps A 1--> A — dA, (1)1--> e'(1) for A any real function on M, 9)(1.1-->eac1). All electrically charged tensors and their U(1) covariant derivatives belong to some representation of the group U(1). The Maxwell stress forms are now supplemented by ra[g, A, 101 = Re(i aa 4) A *act.* + aci) A ia*aCD*) — 1(12012 U(1012))*ea (7.5.17) The U(1) covariant field equation for (1) is a*acto = 11 2*(1) + U'(1)*1 (7.5.18) with U' = dUld1 (1312- Exercise 7.3 Show that the total stress tensor, the sum of those in (7.5.9) and (7.5.17), satisfies Dra = 0 when the coupled Maxwell—Klein—Gordon equations, (7.5.14), (7.5.16) and (7.5.18), are satisifed. SoME MATTER FIELDS 241 From thedefinition ofFwehave theBianchi identity DF=0. (7.5.11) Thefieldequation analagous to(7.5.7) is D*F =0. (7.5.12) Thesystem iscoupled toEinsteinian gravity with thestress forms I,=2;(i,F, A*F,--i,,*F,-AF,-). (7.5.13) TheEinstein—Maxwell-charged scalar system Inthiscase anelectrically charged complex scalar field <1)couples to both gravity and electromagnetism. The Maxwell equations now have electric current sources j[g,<I>]: d*F =j (7.5.14) dF=0 (7.5.15) where thecurrent 3-form is j=Im(<I>*€D<I>*) (7.5.16) andtheU(l) exterior covariant derivative isdefined by QM)=d<I>+iA<I> interms ofthe 1-form Asatisfying F=dA. Under the maps Ai—>A—dl,<1)l—>e"l<I>forAanyrealfunction onM,€D<I>i—>e"’€D<I>. All electrically charged tensors andtheir U(l) covariant derivatives belong tosome representation ofthegroup U(1). The Maxwell stress forms are now supplemented by r,,[g, A,<I>]=§Re(i,,€D<I>A *€D<I>* +€D<I>A i,,*€D<I>*) -%(u’|<P|2 +U(|<P|2))*@..~ (7-5-17) TheU(l) covariant field equation for<1)is €D*€D<I> =uZ*<I> +U’<I>*1 (7.5.18) with U’=dU/d|<I>|Z. Exercise 7.3 Show that thetotal stress tensor, thesum ofthose in(7.5.9) and (7.5.17), satisfies Dr,=0when thecoupled Maxwell—Klein—Gordon equations, (7.5.14), (7.5.16) and(7.5.18), aresatisifed. 242 GRAVITATION Ideal-fluid stress In astrophysical problems one often models massive fluids on a timelike vector field. If V is a local vector field with g(V, V) = —1 each integral curve is considered to describe the world line of a massive fluid element. If the fluid has mass density specified by the 0-form p, the 3-form mass current is j = P.V (7.5.19) and the mass in a spacelike 3-surface E is 1E]. If the number of particles in the fluid remains constant then dj = 0. We examine the symmetric tensor field = pl/®V. (7.5.20) Since vx,s-GT = (x.p)Vcw + pv,x0 v + pv0v,07 then (Vxj.)(ea, ) = (Xap)VaV + p(Vxy)(ea)V + pVaVxy. The symmetric tensor field has divergence V. = V(p)V + pV.V V + pV vV but (Vx(pV))(ea) = V(p) + pV.V , hence V. ,GT = V .(pV)V + pV vV = —.5(pi 7)V + pV ,V = —(*dj)V + pV vV. Thus for.?-7 to be divergenceless the acceleration of V must be proportional to V. But if V is timelike with constant norm its accelera- tion is orthogonal to itself. So the divergence of 5' is zero if and only if dj = 0 and V is a geodesic vector field, VV = 0. Electrically charged fluid stress Suppose that each integral curve of V models the world line of an electrically charged fluid element. Let the charge density Pe of the fluid be (elra)p. Thus each world line may be taken to correspond to a point particle with electric charge e and mass m. The gravitational field equations are the Maxwell—Einstein equations where the Maxwell equa- tions have as 3-form current source = *(pe (7.5.21) The symmetric stress tensor for the system of electromagnetic fields and fluid is 242 GRAVITATION Ideal-fluid stress Inastrophysical problems oneoften models massive fluids onatimelike vector field. IfVisalocal vector field with g(V, V)=-1each integral curve isconsidered todescribe theworld lineofamassive fluid element. Ifthefluid hasmass density specified bythe0-form p,the3-form mass current is 1"=p*l7 (7.519) andthemass inaspacelike 3-surface Eisfzj.Ifthenumber ofparticles inthefluid remains constant then dj=0.Weexamine thesymmetric tensor field 8"=pv®v. (7.5.20) Since vX,8Y =(X,,p)v®v +pVX“V®V +pl/®VXuV then (6.197)(@"~>=(X.p>v"v +p<vX.v>(e">v +pv"vX.v- Thesymmetric tensor field ‘KThasdivergence V6:=V(p)V +pV.VV +pvvv but(VXn(pV))(e“) =V(p) +pV.V, hence v.8Y=V.(pV)V +pVj/V=-a(pY)v +pVVV =—(*dj)V +pVVV. -C, Thus for ‘Jtobedivergenceless the acceleration ofVmust be proportional toV.ButifVistimelike with constant norm itsaccelera- tionisorthogonal toitself. Sothedivergence of9‘iszero ifandonly if dj=0andVisageodesic vector field, VVV =0. Electrically charged fluid stress Suppose thateach integral curve ofVmodels theworld lineofan electrically charged fluid element. Letthecharge density pgofthefluid be(e/m)p. Thus each world linemay betaken tocorrespond toapoint particle with electric charge eand mass m.The gravitational field equations aretheMaxwell—Einstein equations where theMaxwell equa- tions have as3-form current source 1,=*(p,Y). (7.5.21) Thesymmetric stress tensor forthesystem ofelectromagnetic fields and fluid is SOME MATTER FIELDS 243 =3-(1,4)+ P%" ®l" where ,l (m) is the Maxwell stress tensor. If d*F = J e then from (7.5.9) Dr(m), = F A i Ve. Since .7 enjoys similar properties to the Einstein tensor (6, an argument analagous to that leading to (4.8.9) shows that V.T= *-1Draea . So for any X, G'..9-(m)(X) = *-1(F A ix./e). Repeatedly using (1.4.7) with ** = F A ixJ, = F A ix**Je = F A *(*Je A X) = Ne A )A *F = *FA 'le A X. so that *(F A iAle) = *(*FA *Je A X) = iX*(*FA *Je) = iXi**F = and V.5(m)(X) = i xi:4F = Thus V 3 0,0= iF= (eplm)i vF, by (7.5.21), so V.T= eP i vF + pV vV — (*dj)V. (7.5.22) As we noted before, if Vis of constant norm then its acceleration is orthogonal to itself, and ivF(V)= i vivF = O. Thus by equating to zero the components of V..61 parallel and orthogonal to V we see that ,V is divergenceless if and only if the particle number is conserved, dj = 0 and vy = --eivF. (7.5.23) We recognise this as the Lorentz force law equation for charged world lines. 7.6 The Reissner—Nordstriim Solution In principle one can take an assumed form of metric and matter fields, parametrised by a set of functions, and compute the Einstein and stress tensors to obtain equations for the unknown functions. The resulting equations will be non-linear coupled partial differential equations. If the assumed form of solution is not appropriately parametrised then these SOME MATTER FIELDS 243 where 50,4) istheMaxwell stress tensor. Ifd*F =J,then from (7.5.9) D‘l.'(M),, =FAiXaJ,. Since 9Tenjoys similar properties totheEinstein tensor ‘Q,anargument analagous tothat leading to(4.8.9) shows that V5=*"Dr,,e”. SoforanyX,V.§(M)(X) =*_‘(FAiXJ,). Repeatedly using (1.4.7) with **=-7): F/\iX]e =F/\iX**]e =FA*(*J@/J?) -(*J,.AX)A*F— *FA*J,AX. sothat *(FAix]@) =*(*FA *]@AY)=lx*(*FA *]@) =iXi:,j**F =—iXi;)€F and v.a,M,(X) =1,1,-AF =1.-,,F(X). Thus VFW) =i;~,,F =(ep/m)iVF, by(7.5.21), so V7?="—p1,“F +pVVV-(*a)")v. (7.5.22) "'1 Aswenoted before, ifVvis ofconstant norm then itsacceleration is orthogonal toitselfuarld iVF(V) =iViVF =0.Thus byequating tozero thecomponents ofV..°T parallel andorthogonal toVweseethat 5is divergenceless ifandonly iftheparticle number isconserved, dj=0 and vvv=-£17. (7.5.23) Werecognise thisastheLorentz force lawequation forcharged world lines. 7.6TheReissner—Nordstrom Solution Inprinciple onecantake anassumed form ofmetric andmatter fields, parametrised byasetoffunctions, andcompute theEinstein andstress tensors toobtain equations fortheunknown functions. The resulting equations willbenon-linear coupled partial differential equations. Ifthe assumed form ofsolution isnotappropriately parametrised then these 244 GRAVITATION differential equations will not admit a solution, whilst usually a very general form of trial solution merely results in intractable equations. Thus, in practice, such a 'brute force' approach is somewhat limited in obtaining physically interesting solutions to Einstein's equations: the generation of such solutions being a specialised pursuit. The imposition of symmetries on the fields is one obvious way of restricting the number of free parameters. We here consider a static spherically symmetric metric. A metric is stationary if it admits a timelike Killing vector. If, in addition, this Killing vector is orthogonal to a family of spacelike hypersurfaces then the metric is called static. We consider a metric tensor that can be written in a local polar spacetime chart (t, r, 0, cp) as g = —Ho(r)2dtOdt + H 1(r)2dr®dr + r 2d00d0 + r 2 sin 20c1cp0dcp. (7.6.1) The chart is specified by {0 0< 7r, 0 cp <2r, 0< t < oc) and r is bounded to keep H o and H1 real. This metric is invariant under an SO(3) group of transformations generated by the rotational Killing vectors given in (5.4.7). It is also static since 20,30g =-- 0 and (atat) is orthogonal to the hypersurfaces with t = constant. As we pointed out in Chapter 6 it is convenient to choose an orthonormal co-frame in which to compute the connection forms. Choosing the local co-frame: {e° = Hodt, el = Hidr, e2 = rd0, e3 = r sin (94) one computes the non-vanishing connection forms H 0' u , H0H1 e 1 rH 13 cot 0 e3. W23 = = (The co-frames here are a special case of those used to compute the connection forms given in table 6.1.) The curvature forms now follow from the definition (6.4.12): 23 Woi = = W12 = W 21 = (013 = —(031 = e°1 244 GRAVITATION differential equations will notadmit asolution, whilst usually avery general form oftrial solution merely results inintractable equations. Thus, inpractice, such a‘brute force’ approach issomewhat limited in obtaining physically interesting solutions toEinstein’s equations: the generation ofsuch solutions being aspecialised pursuit. The imposition ofsymmetries onthefields isone obvious way of restricting thenumber offree parameters. Wehere consider astatic spherically symmetric metric. Ametric isstationary ifitadmits a timelike Killing vector. If,inaddition, thisKilling vector isorthogonal toafamily ofspacelike hypersurfaces then themetric iscalled static. We consider ametric tensor that can bewritten inalocal polar spacetime chart (t,r,6,tp)as g=—HO(r)Zdl®dt +H1(r)2dr®dr +r2d6®Cl0 +r2sin29d<p®dtP. (7.6.1) The chart isspecified by{OE6<11,0E(p<211,0<t<99}and ris bounded tokeep H0and H1real. This metric isinvariant under an SO(3) group oftransformations generated bytherotational Killing vectors given in(5.4.7). Itisalso static since §E(a,a,,g =0and (8/8t) is orthogonal tothehypersurfaces with t=constant. Aswepointed outin Chapter 6itisconvenient tochoose anorthonormal co-frame inwhich tocompute theconnection forms. Choosing thelocal co-frame: {e°=Hodt, e‘=Hjdr, e2=rdt), e3=rsinQdrp} onecomputes thenon-vanishing connection forms H0’ as=—w =—ie° O1 10 HOHI 12(912=—w21 ="T1 13W13=-0131 =W6‘ l cot6 , (U23=“(U32 =‘i9’-r (The co-frames here areaspecial case ofthose used tocompute the connection forms given intable 6.1.) The curvature forms now follow from thedefinition (6.4.12): 1 1, R23ITil —‘»)e'3r~ H; R=(ib)’;,,@.0’H,H011, THE REISSNER—NORDSTR6M SOLUTION 245 1 (1 y R11 – —t--------I e– r1-11 111 RO2 H'0 =r1-1110e02 Taking *1 e0123 R,2 = rift HI - 03 R03= e . rHTH0 the Einstein forms are calculated: = A e3 – 2R13 A e1 – 2R31 A e2 2 fi 21. 1 ± r2 1 e123 irHi‘1111 r21-12i 1 ) Gi = 2(2 11('I 1 + (p3 e r2 r2H; G2 = –2J(r)e" 13 G3 = 2J(r)e °'2 where \' 1 ± 1 1 J(r)= k HI H I + rHiFlo rH The vaccuum equations Ga = 0 are now all satisfied by 1 )1/2 Ho = 71-1= 1 + -- r for some constant m. This solution has the property that for large r the metric looks like the metric of Minkowski spacetime. To illustrate the effect of the electromagnetic field on the geometry of spacetime consider a spherically symmetric static Einstein–Maxwell system. In the above chart we choose a gauge in which A = f(r)dt, ensuring that Y iciF = 0 for F = dA and KJ any Killing vector of the spherically symmetric static metric. The Maxwell 2-form is F = L(r)e l A e° where L(r)= f' l(H 0111). Integrating the differential equations d*F = 0 gives Lr2 = q for some constant q. From (7.5.9) the Maxwell stress forms follow simply n2 ' 0 q- I 0,3 1 013 1 - 012 T = -e1/3 , r = 1- e - , = -e , T3 = 2r4 --e 2r4 2r 4 2r4 THE REIssNER—NoRDsTRoM SOLUTION 245 1 1'R Zim 13 31 TH|(H1)e HI R02=_i_e02 rH§H0 1 1’R :_i_ 12 12 rH1(H|) 6 H1103 R=—,ie .O3rH{H0 Taking *1=em” theEinstein forms arecalculated: G0=—2R|3A€3—2R33/(61—2R31/(62 2 1' 1 1-4-1-1--—1 TH) H1 r2 r2H{ 6 G1: 2(2fl_ _L+i_)e023 rHfH(, r2 r2H§ G2:_2J(r)etl13 G3 :2J(r)e012 where H’’ H’ 1 1’J(r)=( O) 1+,°+ ( H1 HOHI FHIHO rH1 H1 Thevaccuum equations G“=0arenow allsatisfied by 1 /J\Dl/2H=—= 1—- n H1 (+ r/ forsome constant /.t.This solution hastheproperty that forlarge rthe metric looks likethemetric ofMinkowski spacetime. Toillustrate theeffect oftheelectromagnetic field onthegeometry of spacetime consider aspherically symmetric static Einstein—Maxwell system. Intheabove chart wechoose agauge inwhich A=f(r)dt, ensuring that i’KIF= 0forF=dAand K)any Killing vector ofthe spherically symmetric static metric. The Maxwell 2-form is F=L(r)e’ Ae”where L(r) =f’/(HOH1). Integrating thedifferential equations d='-F=0gives Lr2=qforsome constant q.From (7.5.9) the Maxwell stress forms follow simply 2 2 2 Zq I-0=L4eI23_ 1.1:L4et)23, T2:L4e0I3_ 1.3=__Ze(ll2~ 2r 2r 2r 2r 246 GRAVITATION The presence of the stress modifies the equations above to ict 2 ( 1 )' 1 + 1 + g- = 0 [rill WI/ r2 r2I-11 4r4 K( 1 1 2 1- r 0 + n2 + :L-- = 0 rt1110 r2 r21-1;) 4r4 a 2 KI(r) - --- = 0. 4r4 These equations are all satisfied by 1 = (1 + q22 H = (7.6.2) H1 r 4Kr 2 ! The electromagnetic 2-form field is F = (q1r2)el A e°, so we may interpret this solution as the gravitational field of a spherically sym- metric static electrically charged source. It is known as the Reissner- Nordstrom solution. In the above solution we have two arbitrary constants and q. The latter we have identified with a source of electric charge. The former may be identified with a Newtonian gravitational mass. However, classical gravitation is observed to give rise always to an attractive interaction between macroscopic masses. This feature implies that !I should be chosen to be a negative constant. The examples below are intended to convince the reader of this identification. Exercise 7.4 Consider the geodesic motion of an uncharged test particle in a spacetime metric described by the local orthonormal co-frame {e° = Fdx°, ek = F -idxk k = 1, 2, 3} with F a function of the three spatial coordinates. Show that the geodesic C: I —> M, T (x°(r), xk(r)) is determined by + 2X°F-IC(F) = 0 + [(02F3 + Xi X iFla iF - 2X'F -IC(F) = 0. For C timelike choose a proper-time parametrisation to replace these with g(C, C) = -1 + 4aiF2 + 2F-'(»1a i - XiX181)F = 0. 246 GRAVITATION Thepresence ofthestress modifies theequations above to 2 1' 1 1 3()-2+,,)+" =0 VH1 HI I’ r~[—]I 4r“l H’ 1 1 2I<(2 ,0—,+ ,2)+q4=01 4r 2rH;H0 r~ r-H q_K](r) —TIA ——0. These equations areallsatisfied by _1_ M qz I/2 I‘1U—E—(l-l-T-kw) The electromagnetic 2-form field isF=(q/r2)e1Ae°, sowemay interpret this solution asthegravitational field ofaspherically sym- metric static electrically charged source. Itisknown astheReissner- Nordstrom solution. Intheabove solution wehave twoarbitrary constants uandq.The latter wehave identified with asource ofelectric charge. The former may beidentified with aNewtonian gravitational mass. However, classical gravitation isobserved togive rise always toanattractive interaction between macroscopic masses. This feature implies that it should bechosen tobeanegative constant. The examples below are intended toconvince thereader ofthisidentification. Exercise 7.4 Consider the geodesic motion ofanuncharged test particle ina spacetime metric described bythelocal orthonormal co-frame {e0=Fdx‘), e"=F"dx" k=1,2,3} with Fafunction ofthethree spatial coordinates. Show that the geodesic C:I—> M,T>—-—> (x°(r), x"(r)) isdetermined by x°+2x°F"'C‘(F) =0 xi+[(x°)2F3 +xix,-F-1]a,»F —2x'F"*C(F) =0. For Ctimelike choose aproper-time parametrisation toreplace these with 8(C'1C’)=-1 xi+gs,-F1 +2F-'(xfx,a, —x'”x/a,-)F =0. THE REISSNER—NORDSTR6M SOLUTION 247 If now WI << 1 and F2 = 1 — h with h << 1 then these approximate to = 13ih. By comparing with Newton's law of motion for a slowly moving particle in a Newtonian gravitational potential 43, make the weak field identification = —h/2. Exercise 7.5 In the above metric (7.6.1), set q = 0 and make the coordinate transformation r = R — — + 2 16R to write it in the isotropic form (+ g= (4R + /12dtOdt 4R — 1 — 2—)4(dROdR + R 2d00d0 + R 2 sin 20dcpOdcp). 4R In a region where p, << 4R this is of the type considered in exercise 7.4, (change from standard R3 polar to R3 Cartesian coordinates.) Recall that for a point source of Newtonian gravity due to a mass M, the potential (13 = —GMIr where G is the Newtonian gravitational coupling constant. Hence from h = GMIr identify the constant in the Schwarzschild solution; pt = —2 GM. Exercise 7.6 In the metric in exercise 7.4 above verify that for h << 1, G° = —2(3 kakh)el A e2 A e3. For an ideal fluid of density p show that TO = pe 1 A e2 A e3 in the frame {X a} in which its velocity V = X o. Hence use the Newtonian Poisson equation V 2q) -= 477-Gp to relate our lc to the Newtonian coupling G by 1 K= 167TG. Exercise 7.7 Use the result of exercise 7.1 to rewrite Einstein's equations in the form — eJL = 80G*-11 ", In the absence of the electromagnetic field (q = 0) the Reissner- Nordstrom metric reduces to the Schwarzschild metric. That is, we have a vacuum spacetime with metric THE REIssNER—NoRDsTRoM SOLUTION 247 Ifnow Ix’I<<1andF2=1—hwith h<<1then these approximate to Y’ : Bycomparing with Newton’s law ofmotion foraslowly moving particle inaNewtonian gravitational potential (D,make theweak field identification (D=—h/2. Exercise 7.5 Inthe above metric (7.6.1), setq=0 and make the coordinate transformation _EaL"R 2+16R towrite itintheisotropic form _ 4R+,u)2g- (4R_# dt®dz 4+(1-%)(dR®dR +R2d6®d6 +R2511.26d¢®d¢). Inaregion where /1<<4Rthisisofthetypeconsidered inexercise 7.4, (change from standard R3polar toR3Cartesian coordinates.) Recall that forapoint source ofNewtonian gravity duetoamass M, thepotential (D=—GM/r where GistheNewtonian gravitational coupling constant. Hence from h=GM/r identify theconstant inthe Schwarzschild solution; /t=—2GM. Exercise 7.6 Inthe metric inexercise 7.4 above verify that for h<<1, G0=—2(E)kE9"h)e’ Ae3Ae3.Foranideal fluid ofdensity pshow that to=pe‘Ae2Ae3intheframe {X,,} inwhich itsvelocity V=X0. Hence usetheNewtonian Poisson equation Vzcp =4rrGp torelate ourK totheNewtonian coupling Gby 1I<=i16rrG' Exercise 7.7 Usetheresult ofexercise 7.1torewrite Einstein’s equations intheform P,—§e,9t =81rG*"rc. Intheabsence oftheelectromagnetic field (q=0)theReissner— Nordstrom metric reduces totheSchwarzschild metric. That is,wehave avacuum spacetime withmetric 248 GRAVITATION g . (1 2M )dtOdt + (1 2M ) 1 drOdr + r 2(dO®de r r + sin 20404) (7.6.3) where the coordinate r is restricted to be greater than 2M. Some properties of this spacetime can be understood by looking at the behaviour of local light cones in this chart, where for fixed (r, t) we have a standard 2-sphere. The tangent vector p(3/3t) + q(3/3r) has norm squared (1 - 2M/r) -1q2 - (1 - 2MIr)p2 and is therefore timelike if 2M < 1 - r The local directions determined by all such tangent vectors lie in the local light cones attached to each point on the 2-sphere at (r, t). These light cones appear to close as the coordinate r approaches 2M. Thus any incoming timelike or null curve will asymptote to r = 2M in the (r, t) chart. On the other hand, if one calculates the scalar curvature near r =2M it appears well behaved, suggesting that the Schwarzchild coordinates may cover only part of some Lorentzian manifold. If we introduce the Eddington-Finkelstein coordinates (T, r', 0, cp) where T = t + r + 2M log (r - 2M) and r' = r then it is straightforward to compute d T in terms of dt and dr and write the above metric in these coordinates as g = -(1-11-1)ciTOcIT + dT0dr' + dr'OdT + r' 2(dO0d0 r' + sin 20 dq504). (7.6.4) The region of spacetime covered by r E (2M, co) t e (-co, 00) is now covered by r' and T ranging over the same values. There now appears no reason to restrict r' to be less than 2M. Thus we may regard the original coordinates as describing only part of a Lorentzian manifold, the whole of which is covered by the new coordinates with T> O. Looking now in the (r', T) plane at the forward light cones for r' < 2M, in which lie the future directed timelike curves, a dramatic result is evident. No future-directed timelike (or null) curve from r' <2M ever reaches the region of spacetime with r' > 2M: all such curves are eventually focused to r' = O. Thus there exists a horizon at r' =-- 2M, no causal information of any kind being received by an observer outside the horizon from points within. Furthermore, all incoming timelike curves that enter the horizon eventually (in a finite proper time) strike the line r' = 0 where the curvature tensor becomes unbounded. Such events do not belong to a Lorentzian manifold and 9 P 248 GRAVITATION —l g=—(1—L?/I)dt®dt +(1-if/I) dr®dr +r2(d6®d6 +sin26d(p®d(p) (7.6.3) where thecoordinate risrestricted tobegreater than 2M. Some properties ofthis spacetime can beunderstood bylooking atthe behaviour oflocal light cones inthischart, where forfixed (r,t)we have astandard 2-sphere. The tangent vector p(8/St) +q(8/Sr) has norm squared (1—2M/r)“q2 —(1—2M/r)p2 andistherefore timelike if Ill<1-21. P r The local directions determined byallsuch tangent vectors lieinthe local light cones attached toeach point onthe2-sphere at(r,t).These light cones appear toclose asthecoordinate rapproaches 2M. Thus any incoming timelike ornull curve will asymptote tor=2M inthe (r,t)chart. Ontheother hand, ifone calculates thescalar curvature near r=2Mitappears well behaved, suggesting that theSchwarzchild coordinates may cover only part ofsome Lorentzian manifold. Ifwe introduce theEddington—Finkelstein coordinates (T,r’,6,tp)where T=t+r+2Mlog (r—2M) and r’=rthen itisstraightforward to compute dTinterms ofdtanddrandwrite theabove metric inthese coordinates as g=-(1-3¥)dT®dT +dT®dr’ +dr’®dT +r’2(d6®d6 F +sin26d(p®d(p). (7.6.4) The region ofspacetime covered byre(2M, 99)te(-99, 99)isnow covered byr’and Tranging over thesame values. There now appears noreason torestrict r’tobelessthan 2M. Thus wemay regard the original coordinates asdescribing only part ofaLorentzian manifold, thewhole ofwhich iscovered bythenew coordinates with T>0. Looking now inthe(r’,T)plane attheforward light cones for r’<2M, inwhich liethefuture directed timelike curves, adramatic result isevident. Nofuture-directed timelike (ornull) curve from r’<2Mever reaches theregion ofspacetime with r’>2M: allsuch curves areeventually focused tor’=0.Thus there exists ahorizon at r’=2M, nocausal information ofany kind being received byan observer outside the horizon from points within. Furthermore, all incoming timelike curves that enter thehorizon eventually (inafinite proper time) strike theliner’=0where thecurvature tensor becomes unbounded. Such events donotbelong toaLorentzian manifold and THE REISSNER-NORDSTR6M SOLUTION 249 prohibit any further extensions of the spacetime. For a spherically symmetric star of mass M and radius parameter r> M the Schwarzschild metric describes the unique spacetime in the vacuum exterior to the star. The spacetime inside the star will depend on its matter stresses. A star unfortunate enough to evolve to a radius parameter less than 2M is predicted to find all its atoms on doomed world lines and undergoes catastrophic gravitational collapse. (For an object whose Newtonian mass is n times the mass of the sun this radius is about 3n km.) One of the most celebrated theorems in the theory of gravitation asserts that under a number of reasonable assumptions such a phenomenon is not restricted to the idealised spherically symmetric metric discussed here. The physics of the collapse of matter to a singular state is one of the great challenges of contemporary research. Further details of the Schwarzschild geometry can be found in, for example, Hawking and Ellis [12] and Misner, Thorne and Wheeler [13]. These books give a more complete account of the possible extensions to the exterior Schwarzschild solution. 7.7 Gravitation with Torsion Einstein's theory of gravitation is written in terms of a metric- compatible torsion-free connection. There have been many attempts to generalise these equations. One direction is to maintain their form but to relax the requirement that the connection has zero torsion. One must then supplement them with further equations that determine the torsion tensor. They may be regarded as geometrical descriptions of interactions that depend on tensor (and spinor) fields other than the metric. One may also contemplate gravitational theories in which the metric compati- bility of the connection is relaxed although such approaches have attracted little attention so far. Needless to say the adoption of a particular connection for the geometrical description of physical phe- nomena depends on the physics of the situation. Sometimes (as in the case of theories with supergravity) a connection with a torsion deter- mined by a spinor field equation provides an elegant formulation of a theory. Rewriting the theory in terms of the Levi—Civita connection is always possible, but possibly at a cost of algebraic complexity. As a simple example of a model written in terms of a metric- compatible connection with torsion, consider a self-interacting real scalar field a' coupled to gravity according to the field equations [14] ;.a,2Ga = Ta[cr] Aa,4.ea (7.7.1) cd*da2 = 2Acr3*1 (7.7.2) THE REIssNER—NoRDsTRoM SOLUTION 249 prohibit anyfurther extensions ofthespacetime. For aspherically symmetric star ofmass Mand radius parameter r>MtheSchwarzschild metric describes theunique spacetime inthe vacuum exterior tothestar. The spacetime inside thestarwilldepend onitsmatter stresses. Astarunfortunate enough toevolve toaradius parameter lessthan 2Mispredicted tofind allitsatoms ondoomed world lines and undergoes catastrophic gravitational collapse. (For an object whose Newtonian mass isntimes themass ofthesunthisradius isabout 3nkm.) One ofthemost celebrated theorems inthetheory of gravitation asserts that under anumber ofreasonable assumptions such aphenomenon isnotrestricted totheidealised spherically symmetric metric discussed here. Thephysics ofthecollapse ofmatter toasingular state isoneofthegreat challenges ofcontemporary research. Further details oftheSchwarzschild geometry canbefound in,for example, Hawking andEllis [12]andMisner, Thorne andWheeler [13]. These books give amore complete account ofthepossible extensions to theexterior Schwarzschild solution. 7.7Gravitation with Torsion Einstein’s theory ofgravitation iswritten interms ofametric- compatible torsion-free connection. There have been many attempts to generalise these equations. One direction istomaintain their form but torelax therequirement thattheconnection haszero torsion. One must then supplement them with further equations that determine thetorsion tensor. They may beregarded asgeometrical descriptions ofinteractions that depend ontensor (and spinor) fields other than themetric. One may alsocontemplate gravitational theories inwhich themetric compati- bility oftheconnection isrelaxed although such approaches have attracted little attention sofar. Needless tosaytheadoption ofa particular connection forthegeometrical description ofphysical phe- nomena depends onthephysics ofthesituation. Sometimes (asinthe case oftheories with supergravity) aconnection with atorsion deter- mined byaspinor field equation provides anelegant formulation ofa theory. Rewriting thetheory interms oftheLevi—Civita connection is always possible, butpossibly atacostofalgebraic complexity. Asasimple example ofamodel written interms ofametric- compatible connection with torsion, consider aself-interacting realscalar field acoupled togravity according tothefield equations [14] %a2G" =—r"[a] +/1.0/‘*e" (7.7.1) cd*da2 =2/la3*1 (7.7.2) 250 GRAVITATION with da' Ta = ea A — a (7.7.3) Ta = adœ A *dœ + dcr A i"*da). (7.7.4) The non-vanishing real parameters A and c are coupling constants. (For A. = 0 this model is equivalent to a theory of gravitation proposed by Brans and Dicke REF [15].) The equation (7.7.3) involving the torsion may be solved for the connection forms (6.6.8): ibda (I) ab = Q ab (-)e a (iadleb (7.7.5) in terms of the torsion-free connection forms Qab. It is an interesting exercise to rewrite the above system of equations in terms of the Einstein forms associated with the torsion-free connection. In such a reformulation the torsional effects due to the scalar field coupling to gravity may be interpreted as an additional contribution to the stress forms. In addition c becomes replaced by c — 6. Bibliography Adler R, Bazin M and Schiffer M 1975 Introduction to General Relativity (New York: McGraw-Hill) O'Niel B 1983 Semi-Riemannian Geometry with Applications in Physics (New York: Academic) Thorpe J A 1975 Proc. Symp. in Pure Mathematics vol XXVII, p425 250 GRAVITATION da T"=8"A7 (7.73) with 1'“=§c(i”da A*da +daAi”*da/). (7,7,4) The non-vanishing realparameters Aandcarecoupling constants. (For A=0thismodel isequivalent toatheory ofgravitation proposed by Brans andDicke REF[15].) The equation (7.7.3) involving thetorsion maybesolved fortheconnection forms (6.6.8): ibda iada (Dab =Q0’, +T 8,,'-T 6;, (7.7.5) interms ofthetorsion-free connection forms Qab. Itisaninteresting exercise torewrite theabove system ofequations interms ofthe Einstein forms associated with thetorsion-free connection. Insuch a reformulation thetorsional effects due tothescalar field coupling to gravity may beinterpreted asanadditional contribution tothestress forms. Inaddition cbecomes replaced byc—6. Bibliography Adler R.Bazin MandSchiffer M1975 Introduction toGeneral Relativity (New York: McGraw-Hill) O’Niel B1983 Semi-Riemannian Geometry with Applications inPhysics (New York: Academic) Thorpe JA1975 Proc. Symp. inPure Mathematics volXXVII, p425 8 Clifford Calculus on Manifolds The first three chapters of this book are purely algebraic. They deal with tensor, exterior and Clifford algebras of an arbitrary vector space. In the following chapters when dealing with manifolds, and applications in physics, we have assimilated the material of Chapter 1 by taking that vector space to be the cotangent space. We shall now similarly incorpo- rate Chapter 2. In Chapter 2 we identified the Clifford algebra with the vector space of exterior forms with the product given in (2.1.7). Hence on a pseudo-Riemannian manifold M we have the structure of a Clifford algebra on each fibre of the exterior bundle. The exterior bundle equipped with this multiplication in the fibres will be called the Clifford bundle C(M). The situation is that we have a vector bundle with two different rules for turning it into an algebra bundle; so we shall freely interchange the terms Clifford bundle and exterior bundle (for a pseudo-Riemannian manifold) depending on which aspect we wish to emphasise. Similarly we may sometimes refer to 'Clifford forms' to emphasise that we are thinking of the differential forms as elements of a Clifford rather than exterior algebra. Just as one can develop an efficient exterior calculus of differential forms with the exterior derivative (and more generally the covariant exterior derivative) and Hodge dual, one can efficiently calculate using the covariant derivative V and Clifford multiplication (equation (2.1.19) relating the Hodge dual to Clifford multiplication). Unlike the exterior algebra the Clifford algebra is not Z-graded. So Clifford multiplication of differential forms will naturally involve us with inhomogeneous differential forms; that is, sums of differential forms of different degrees. Certain equations involving forms of differing degrees can be conveniently expressed in terms of Clifford products. The utility of being able to Clifford multiply differential forms really becomes apparent when we come to spinor fields (these carrying Clifford Calculus onManifolds The first three chapters ofthisbook arepurely algebraic. They deal with tensor, exterior andClifford algebras ofanarbitrary vector space. Inthefollowing chapters when dealing with manifolds, andapplications inphysics, wehave assimilated thematerial ofChapter 1bytaking that vector space tobethecotangent space. Weshall now similarly incorpo- rateChapter 2. InChapter 2weidentified theClifford algebra with thevector space ofexterior forms with the product given in(2.1.7). Hence ona pseudo-Riemannian manifold Mwehave thestructure ofaClifford algebra oneach fibre oftheexterior bundle. The exterior bundle equipped with thismultiplication inthefibres willbecalled theClifford bundle C(M). The situation isthat wehave avector bundle with two different rules forturning itinto analgebra bundle; soweshall freely interchange theterms Clifford bundle and exterior bundle (for a pseudo-Riemannian manifold) depending onwhich aspect wewish to emphasise. Similarly wemay sometimes refer to‘Clifford forms’ to emphasise thatwearethinking ofthedifferential forms aselements ofa Clifford rather thanexterior algebra. Just asonecandevelop anefficient exterior calculus ofdifferential forms with theexterior derivative (and more generally thecovariant exterior derivative) andHodge dual, onecanefficiently calculate using thecovariant derivative VandClifford multiplication (equation (2.1.19) relating theHodge dual toClifford multiplication). Unlike theexterior algebra theClifford algebra isnotZ-graded. SoClifford multiplication ofdifferential forms will naturally involve uswith inhomogeneous differential forms; that is,sums ofdifferential forms ofdifferent degrees. Certain equations involving forms ofdiffering degrees canbe conveniently expressed interms ofClifford products. The utility ofbeing able toClifford multiply differential forms really becomes apparent when wecome tospinor fields (these carrying 252 CLIFFORD CALCULUS ON MANIFOLDS representations of the Clifford—as opposed to exterior—algebra). An inspection of many calculations involving spinors in theoretical physics reveals that often the components of a vector (or co-vector) are saturated with a set of y-matrices that generate a Clifford algebra. (Indeed there is even a special notation for such objects!) It is conceptually, as well as notationally, simpler to work directly with the Clifford algebra of differential forms. In this chapter we shall frequently use the notation, and results, of Chapter 2. In particular we shall juxtapose differential forms to denote their Clifford product. 8.1 Covariant Differentiation of Clifford Products If tt is an arbitrary inhomogeneous differential form and A an arbitrary 1-form on a pseudo-Riemannian manifold M then (2.1.7) gives — A A +  If V is the pseudo-Riemannian connection then V x(iii43) = ivAI + i AV x(I), since V x commutes with contractions, and VA = VA since V is metric compatible. Hence V(A) = V x/1(13 + AV x(1) (8.1.1) and it follows that V. is a derivation on Clifford products. (This does not require zero torsion.) Adding and subtracting equations (2.1.7) and (2.1.8) gives us relations that permit A A 4) and i A) to be expressed in terms of Clifford products: A cI) + (VA = 2A A cr, (8.1.2) — VIA = . (8.1.3) For {ea} a local orthonormal co-frame we denote ea A eb by eab . Then (8.1.3) gives [ebc, ea] = 2o( eb _ nabee) (8.1.4) where the left-hand side is a Clifford commutator and ?lab are the orthonormal components of the metric. So if we use the connection 1-forms to introduce the 2-form a Xbc(X) be c r__ xea A ea (8.1.5) we can write (6.3.3) as V xea = [a x, ea] . (8.1.6) If we introduce an orthonormal multibasis fei) for FAM then, since an 252 CLIFFORD CALCULUS ONMANIFOLDS representations oftheClifford—as opposed toexterior—algebra). An inspection ofmany calculations involving spinors intheoretical physics reveals that often the components ofavector (orco-vector) are saturated with asetofy-matrices that generate aClifford algebra (Indeed there iseven aspecial notation forsuch objects!) Itis conceptually, aswell asnotationally, simpler towork directly with the Clifford algebra ofdifferential forms. Inthischapter weshall frequently usethenotation, and results, of Chapter 2.Inparticular weshall juxtapose differential forms todenote their Clifford product. 8.1Covariant Differentiation ofClifford Products If<1)isanarbitrary inhomogeneous differential form andAanarbitrary 1-form onapseudo-Riemannian manifold Mthen (2.1.7) gives A<I>=A,\<I)+i,;<I>. IfVisthe pseudo-Riemannian connection then VX(iA-(D) = ivX,;g> +ii-Y_X<I>, since VX commutes with contractions, and VXA =VXA since Vismetric compatible. Hence VX(A<I>) =VXA<I> +AVX<I> (8.1.1) anditfollows that VXisaderivation onClifford products. (This does notrequire zero torsion.) Adding andsubtracting equations (2.1.7) and (2.1.8) gives usrelations that permit A,\<I>andi,;<I> tobeexpressed in terms ofClifford products: Ad)+<b"A=2AAd) (8.12) Ad)-<D"A=2i,;<I>. (8.13) For{e"}alocal orthonormal co-frame wedenote e“Aehbye“”.Then (8.1.3) gives [e"‘, e"]=2(17“e" —17""e‘) (8.1.4) where theleft-hand side isaClifford commutator and17”"arethe orthonormal components ofthemetric. Soifweusetheconnection 1-forms tointroduce the2-form UxE-iwbr(X)ebr =ivxeu A9,; (8-1-5) wecanwrite (6.3.3) as Vxe“ =[oX, e"]. (8.l.6) Ifweintroduce anorthonormal multibasis {e'} for1'/\M then, since an COVARIANT DIFFERENTIATION OF CLIFFORD PRODUCTS 253 exterior product of mutually orthogonal 1-forms is the same as a Clifford product V xel = [ax, . (8.1.7) If we expand an arbitrary differential form as (I) = 4),e' then V xci) = (X4)/)e1 + [ax, 43] (8.1.8) If S is any invertible element of the Clifford algebra and E" SeaS-1 then it follows from (8.1.6) that V xEa = [E x, Ea] with Ex = So-xS-1 + VxSS-1. If s E ±F± then fea' = se's') is another orthonor- mal frame. If aX denotes the expression in (8.1.5) computed with the connection forms in this new basis then a'x = saxs + Vxss 1. (8.1.9) Certainly the two sides of this expression can only differ by an element of the centre. Since ax is a 2-form and s E ±f± then saxs-1 is a 2-form and we need only check that Vxss-i is a 2-form. For s E ±f± we can write s = xix 2 . . . Xh where the x1 are 1-forms such that (x 1)2 = ±1, then V551 = (V xx1x2 . . . xh + xlV xx2 . . xh + . . . ± x1 xh-iv xxh)Rxh)-1 (x2)-1(x1)-1)] = xxi(x1)-1 xi[V xx2(x2)-1]0c1)-1 ±    xt xh-i[vxxh(x.h)-1(xi Since (x 1)2 is a constant V x.x` anticommutes with x' and hence with (0-1 = x14.02. So Vxxi(xi)1 = ;(vxxi(xi)-1 (xi)-1Vxxi) = Vxx' A (Xi)-1. It follows that Vxss-I is a 2-form. If {el} is an orthonormal multibasis for TAM then differentiating (8.1.7) expresses the curvature operator as R(X, Y)el = el for gtxy = V Ps r — Va x [ax, ay] a[x,Y]  (8.1.10) Since the curvature operator is SF-linear then for any 4) E TAM R(X, Y)4) = 43] . (8.1.11) It can be verified that gt xy is unchanged if ax Sa + V x.S.S-1 for any invertible S. The forms xy are certainly related to the curvature 2-forms Rab; we now establish the exact relationship. Differentiating (8.1.5) and using (8.1.7) gives Vxay = X(tobc( + [ax, ay], and hence XY = .14 {X(Wbc( Y)) Y(Wbc(X)) Wbc([X, Y])) ebe ± [ax, ay] . y))6,bc Referring to (4.10.3) we can simplify the first three terms: gt xy = ',doob,(X, Y)et'c + [ax, ay]. To recognise the last term we will use the COVARIANT DIFFERENTIATION or-"CLIFFORD PRODUCTS 253 exterior product ofmutually orthogonal 1-forms isthesame asa Clifford product Vxe’ =[cx.e']. (8.l.7) lfweexpand anarbitrary differential form as<1)=<I>,e' then VX<I> =(X<I>,)e' +[0x, <I>]. (8.1.8) lfSisanyinvertible element oftheClifford algebra andE“ESe”S‘1 then itfollows from (8.1.6) that VXE“ =[ZX,E”] with Ex= S0xS'1 +VXSS". Ifseif’then {e“'Ese”s“} isanother orthonor- malframe. Ifo’xdenotes theexpression in(8.1.5) computed with the connection forms inthisnew basis then o’x=soXs" +VXss'1. (8.1.9) Certainly thetwosides ofthisexpression canonly differ byanelement ofthecentre. Since oxisa2-form andse:1“ then soXs" isa2-form andweneed only check that VXss'1 isa2-form. Forse:1“ wecan write s=xlxz ...x“where thex‘are1-forms such that (x’)2 =i1, then VXss"1=(VXx1x2...x" +x'VXx2 ...x"+... +xl...x"_1VXx")[(x")_' ...(x2)_1(x1)'1)] =VXx1(x1)'1+ x1[VXx2(x2)_1](x')"+... +x1...x"_1[VXx"(x")_1](x' ...x"")‘1. Since (x‘)2 isaconstant Vxxi anticommutes with x‘and hence with W)“ =X"/(Xi)? 50VxX'(X’)T1=i(VxX'(X’)“ "(Xi)_1VxXi) = Vxx’ A(x’)'1. Itfollows that Vxss” isa2-form. If{e1} isanorthonormal multibasis forPAM then differentiating (8.l.7) expresses thecurvature operator asR(X, Y)e' =[97tXy, e']for 97ixY=VxUY_VYUx— iaxiayl —U[X.Y]- (8-1-10) Since thecurvature operator is9-linear then forany(I)ePAM R(X, Y)<l> =[97txy, <I>]. (8.1.11) Itcanbeverified thatQltxy isunchanged ifox>——> SoXS“‘ +VXSS" forany invertible S.The forms Qltxy arecertainly related tothe curvature 2-forms Rab; we now establish the exact relationship. Differentiating (8.1.5) and using (8.l.7) gives Vxoy =§X(w,,C(Y))e"” +[aX, oy], andhence QRXY : — _wbc(iXr YD}ebc +iUXv CY] ' Referring to(4.10.3) wecan simplify thefirst three terms: Qltxy = §dw,,,.(X, Y)e"‘ +[cx. oy]. Torecognise thelastterm wewillusethe 254 CLIFFORD CALCULUS ON MANIFOLDS following useful relation: [eab, ea] = 2rede. _ 2qadecb 2nbcead _ 2nacebd (8.1.12) We can use this and the antisymmetry of the connection forms, to write Eux, ay] = ,i(wab(X)w bc(Y) wab(Y)cobc(X))eaa = (W baA Wa eV() y)ebc So we have y)e,bc = _I; 9.LXY = 12 Rbc(X) 4 'X. bcr- bc (8.1.13) This can be rewritten, using the `pairwise symmetric' Bianchi identity for zero torsion (6.7.16), as &AT = lea(X)eb(Y)Rab (8.1.14) Exercise 8.1 Use (2.1.7) and (2.1.8) to show that (for zero torsion): Rabeb = Pa Pea = Rabeba = R. (8.1.15) (8.1.16) (8.1.17) 8.2 The operator 0 Many equations in physics can be elegantly formulated in terms of the exterior derivative d and the co-derivative 6. In Chapter 6 we showed how these operators could be expressed in terms of the pseudo- Riemannian connection. We now define an operator 0 on FAM by ' eaVx.  (8.2.1) Thus from (6.7.4) and (6.9.1) we have = d — (8.2.2) with 45 defined in (5.4.2). The operator 0 is sometimes called the Hodge de-Rham operator. Unlike d and O separately, 0 is not a homogeneous operator on differential forms; whereas d increases the degree of a form by one, 6 decreases the degree by one. The square of 0 is homogeneous for since d and 6 are nilpotent 02 = A (8.2.3) where A is the Laplace—Beltrami operator of (5.4.5). 254 CLIFFORD CALCULUS onMANIFOLDS following useful relation: [e"", efd]=21]"de“' —21]“de‘b +21]""e"d —21]““ebd. (8.1.12) Wecanusethisandtheantisymmetry oftheconnection forms, towrite i0x> Uyl=i(wab(X)wbc(Y) _wab(Y)wbc(X))eac =i(wba /\wac)(Xr Y)?!”- Sowehave gtxy =%R,,c(X, Y)el" =—§iXiyR,,ce"‘ . (8.1.13) This canberewritten, using the‘pairwise symmetric’ Bianchi identity for zero torsion (6.7.16), as Qtxy =%e"(X)e"(Y)R,,,, . (8.1.14) Exercise 8.1 Use(2.1.7) and(2.1.8) toshow that (forzero torsion): R",,e” =P“ (8.1.15) Pae“ =Qt (8.1.16) R.,,,e'"' =at (23.1.17) 8.2Theoperator ¢l Many equations inphysics canbeelegantly formulated interms ofthe exterior derivative dandtheco-derivative 6.InChapter 6weshowed how these operators could beexpressed interms ofthe pseudo- Riemannian connection. Wenow define anoperator gzlonPAM by dze“VXa . (8.2.1) Thus from (6.7.4) and(6.9.1) wehave gl=d—6 (8.2.2) with<5defined in(5.4.2). Theoperator gziissometimes called theHodge de-Rham operator. Unlike dand6separately, gziisnotahomogeneous operator ondifferential forms; whereas dincreases thedegree ofaform byone, 6decreases thedegree byone. The square ofgzlishomogeneous forsince dand6arenilpotent glz=A (82.3) where AistheLaplace—Beltrami operator of(5.4.5). THE OPERATOR Ø 255 We can trivially rewrite the pair of Maxwell equations d * F = J, dF = 0 as ØF =j (8.2.4) where j = J. As an example of manipulating Clifford expressions we now re-express the Maxwell stress tensor in terms of Clifford products and evaluate its divergence. The stress tensor is related to the stress forms by ‘61 = *ir a 0 ea = *-1ra(Xb)eb 0 ea. For a four- dimensional Lorentzian spacetime ** = and the stress tensor com- ponents are ffba = ib* Ta. From (7.5.9) 2ra = iaF A *F — ia*F AF F. First we use (8.1.2) to exchange the exterior products for Clifford products: 4ra = iaF *F + *Fi aF — ia*FF — Fi a*F. Now we use (8.1.3) 8ra = (eaF — Fea)*F + *F(e aF — Fea) — (ea* F — *Fea)F — F(ea*F — *Fe) - Finally we use (2.1.19) to write the Hodge dual in terms of the volume 4-form z: Ta = FeaFz . We have used F> = —F since F is a 2-form and z(1) = scicoz. Once again we use (8.1.3) to obtain the stress tensor components ba = (Fe aFeb ebFeaF) . (8.2.5) When covariantly differentiating the stress tensor the derivatives of the co-frames in the above components will cancel the derivatives of the tensor basis, hence (Vff)a = 4('Ç xfeaFec + Fe aV xFec + ecV xrFeaF + e`Fe aV xf) . We want to use the Maxwell equations (8.2.4) to simplify this, but the terms VF and e` do not all occur in the right order to write them as Ø. The above expression is certainly a 0-form, so by applying the homogeneous projector (cf (2.1.12)) Y o we do nothing. Under this projector, factors in the Clifford product can be cyclically permuted (2.1.17). (We cannot, of course, then remove the projector.) So we have .ff)a = 1&)0(0Fe0F + V xfecFea) . Since F.> = —F, then V xfe` = —(0F):4. We can insert this in the above and then use j'oc13 = f0t to obtain (Vfla = o(g(Fe aF). We can now THE or-ERA'roR¢ 255 We can trivially rewrite the pair of Maxwell equations d*F=J,dF=0as ¢F=j (8.2.4) where j=—*_'J. Asanexample ofmanipulating Clifford expressions wenow re-express the Maxwell stress tensor interms ofClifford products andevaluate itsdivergence. The stress tensor isrelated tothe stress forms by5=Flt, ®e”=*’1r,(X,,)e" ®e“. For afour- dimensional Lorentzian spacetime **=-17,andthestress tensor com- ponents are9],,=i,,*ta.From (7.5.9) 27/'a=laF/\*F_ln*F/(F. First weuse (8.12) toexchange theexterior products forClifford products: 4r,,=i,,F*F +*Fi,,F-i,,*FF-Fi,,*F. Now weuse(8.1.3) 8rd=(e,,F —Fe,,)*F +*F(e,,F —Fea) —(e,,*F —*Fe,,)F —F(e,,*F —*Fe,,). Finally weuse(2.1.19) towrite theHodge dual interms ofthevolume 4-form 2: ta=§Fe,,Fz . Wehave used F5=—Fsince Fisa2-form and2(1)=<I>"z. Once again weuse(8.1.3) toobtain thestress tensor components 9',“=§(Fe,,Fe,, +ebFe,,F) . (8.2.5) When covariantly differentiating thestress tensor thederivatives ofthe co-frames intheabove components willcancel thederivatives ofthe tensor basis, hence (V-5),, =§(VXcFe,,Fe‘ +Fe,,VX(Fe‘ +e‘VX(Fe,,F +e‘Fe,,VX(F) . Wewant tousetheMaxwell equations (8.2.4) tosimplify this, butthe terms VXCF ande‘donotalloccur intheright order towrite them as The above expression iscertainly a0-form, sobyapplying the homogeneous projector (cf(2.1.12)) Efowedonothing. Under this projector, factors intheClifford product can becyclically permuted (2.1.17). (We cannot, ofcourse, then remove theprojector.) Sowe have (V-5),, =§9’0(¢Fe,,F +VX__Fe‘Fe,,) . Since F5=—F, then VX(Fe‘ =—(¢iF)§. Wecaninsert thisintheabove andthen use90¢ =Sf0<I>~5 toobtain (V-5),, =Sf0(¢iFe,,F). Wecannow 256 CLIFFORD CALCULUS ON MANIFOLDS use the Maxwell equations (8.2.4): Wo(Fjea)ea = using (2.1.18). Since j is a 1-form and F is a 2-form then Fj=jAF—iiF and so finally V.9- = —iiF (8.2.6) (We earlier obtained this result in the discussion of the electrically charged fluid stress in Chapter 7.) We have somewhat laboured the above calculation in order to illustrate some of the techniques that are useful in practice and to show how one can always interchange any exterior expression for a Clifford one and vice versa. 8.3 The Kahler Equation In 1928 Darwin [16] was experimenting with tensor equations in order to understand the properties of electrons. fle eventually made contact with Dirac's spinor wave equation (to be discussed later) but considered his method uneconomical. Apparently Landau and Ivanenko [17] had simi- lar intentions around the same time. These were perhaps precursors of the equation introduced in 1961 by Kahler [18] for a complex in- homogeneous differential form (I) on a pseudo-Riemannian manifold: 0(13 = MID iA(1) . (8.3.1) The term involving A describes the electromagnetic coupling to the Maxwell field F= dA. He was apparently motivated to develop a 'calculus of infinitesimals' in which relations of the form dxP A d.e = and dxPy dx v + dx v y dxP = 2gPv could co-exist on a pseudo- Riemannian manifold. Kahler recovered Dirac's solution describing the wave mechanics of a relativistic electron of mass ti in a hydrogen atom when he analysed (8.3.1) in flat Minkowski spacetime. It was a desire to find a first-order equation, such that the compo- nents satisfied the second-order Klein—Gordon equation, that motivated Dirac to formulate his celebrated equation in 1928 [19]. Because of (8.2.3), and since the Laplace—Beltrami operator is homogeneous, the p-form components 92p(4:11) of an arbitrary solution to (8.3.1), in the absence of an electromagnetic field, satisfy AWp(43) = ,u2Yp(cI)) . (8.3.2) However, an arbitrary complex differential form on spacetime has sixteen complex components; whereas a spinor of the complexified 256 CLIFFORD CALCULUS owMANIFOLDS usetheMaxwell equations (8.2.4): V9 :5f0(Flea)e” :3f1(Fl) using (2.1.18). Since jisa1-form and Fisa2-form then Fj=j,\F—i;F andsofinally vs=—i]=F. (82.6) (We earlier obtained this result inthediscussion oftheelectrically charged fluid stress inChapter 7.)Wehave somewhat laboured the above calculation inorder toillustrate some ofthetechniques that are useful inpractice and toshow how one can always interchange any exterior expression foraClifford oneandviceversa. 8.3TheKahler Equation In1928 Darwin [16]wasexperimenting with tensor equations inorder to understand theproperties ofelectrons. Heeventually made contact with Dirac’s spinor wave equation (tobediscussed later) butconsidered his method uneconomical. Apparently Landau andIvanenko [17]hadsimi- larintentions around thesame time. These were perhaps precursors of theequation introduced in1961 byKahler [18] foracomplex in- homogeneous differential form (Donapseudo-Riemannian manifold: ¢<t>=no-1/to. (83.1) The term involving Adescribes theelectromagnetic coupling tothe Maxwell field F=dA. Hewas apparently motivated todevelop a ‘calculus ofinfinitesimals’ inwhich relations oftheform dx“Adx”=0 and dx“\,dx”+dx"vdx"=2g*” could co-exist on apseudo- Riemannian manifold. Kahler recovered Dirac’s solution describing the wave mechanics ofarelativistic electron ofmass llinahydrogen atom when heanalysed (83.1) inflatMinkowski spacetime. Itwasadesire tofindafirst-order equation, such thatthecompo- nents satisfied thesecond-order Klein—Gordon equation, that motivated Dirac toformulate hiscelebrated equation in1928 [19]. Because of (8.2.3), andsince theLaplace—Beltrami operator ishomogeneous, the p-form components &f’p(<D) ofanarbitrary solution to(8.3.1), in theabsence ofanelectromagnetic field, satisfy As/,,(<t>) =tt1a>,,(<t>) . (83.2) However, anarbitrary complex differential form onspacetime has sixteen complex components; whereas aspinor ofthecomplexified THE KAHLER EQUATION 257 Clifford algebra has four complex components. Thus an arbitrary solution to (8.3.1) has more components than a solution to Dirac's equation. To understand the Kahler equation better, and its relationship to the Dirac equation, we examine the possibility of solutions lying in minimal left ideals—these carrying irreducible representations of the Clifford algebra. A set of four pairwise-orthogonal primitive idem- potents may be used to project an arbitrary element of the Clifford algebra into minimal left ideals. In flat Minkowski space we can always choose inertial coordinates {xa} in which ea = dxa, a = 0, 1, 2, 3 consti- tute an orthonormal basis. We can construct a set of globally defined primitive idempotents {P,} out of this parallel co-frame. The resulting idempotents will also be parallel, V xPi =0 V,Ver TM. Thus if OP, then cp, is in a minimal left ideal. If (13 satisfies (8.3.1) then multiplying (8.3.1) on the right by P, gives OTI = iAT, i = 1, 2, 3, 4 (8.3.2) since P, is parallel. Thus Kahler's equation decouples into four equiva- lent equations for elements lying in minimal left ideals. (If Kahler's equation was written in exterior form then the coupled equations for the homogeneous p-forms would not be very transparent.) A general solution of the Kahler equation has more degrees of freedom than a solution to the Dirac equation. This raises the question of the significance of (8.3.1) for the description of those particles in Nature (such as the electron–positron field) that are conventionally described by the Dirac equation. If one uses a spacetime 3+1 decom- position to perform a non-relativistic reduction then one obtains from (8.3.1) four copies of the Pauli–Schr&linger equation [20]. The wave mechanics of a particle described by such a system is indistinguishable from a non-relativistic description of an electron in an external electro- magnetic field except in one respect: all single-particle (quantum) states have an extra fourfold degeneracy. For example, if a beam of such hypothetical particles was passed through an inhomogeneous static magnetic field (a Stern–Gerlach experiment) it would be split into two components. This is what happens with electrons on atoms in a real experiment. Furthermore, no electromagnetic field could be devised that would split the degeneracy of each beam. However, a (powerful) inhomogeneous gravitational field would in general break the degenera- cy, producing four distinct beams in the field. Electrons described by the Dirac equation are not predicted to behave in this way. Although such an experiment has never been done with real electrons, our under- standing of the periodic table of the elements is based on the Pauli principle for electrons with two internal states rather than four. Without a major reformulation of this principle it is difficult to reconcile our current understanding of the quantum mechanics of electrons with the THE KAHLER EQUATION 257 Clifford algebra has four complex components. Thus anarbitrary solution to(83.1) hasmore components than asolution toDirac’s equation. Tounderstand theKahler equation better, anditsrelationship totheDirac equation, weexamine thepossibility ofsolutions lying in minimal leftideals—these carrying irreducible representations ofthe Clifford algebra. Asetoffour pairwise-orthogonal primitive idem- potents may beused toproject anarbitrary element oftheClifford algebra into minimal leftideals. InflatMinkowski space wecanalways choose inertial coordinates {x“} inwhich e”=dx”,a=0,1,2,3consti- tute anorthonormal basis. Wecanconstruct asetofglobally defined primitive idempotents {P,-} outofthisparallel co-frame. The resulting idempotents will also beparallel, VXP, =0VXeFTM. Thus if qa,E<1>P,- then <p,~isinaminimal leftideal. If<1>satisfies (8.3.1) then multiplying (8.3.1) ontheright byP,gives d(p,=tap,—iA(p,- i=1, 2,3, 4 (8.32) since P,isparallel. Thus Kahler’s equation decouples into four equiva- lent equations forelements lying inminimal leftideals. (IfKahler’s equation waswritten inexterior form then thecoupled equations forthe homogeneous p-forms would notbevery transparent.) Ageneral solution ofthe Kahler equation has more degrees of freedom than asolution totheDirac equation. This raises thequestion ofthesignificance of(8.3.1) forthedescription ofthose particles in Nature (such astheelectron-positron field) that areconventionally described bytheDirac equation. Ifoneuses aspacetime 3+1 decom- position toperform anon-relativistic reduction then one obtains from (8.3.1) four copies ofthePauli—Schrodinger equation [20]. The wave mechanics ofaparticle described bysuch asystem isindistinguishable from anon-relativistic description ofanelectron inanexternal electro- magnetic field except inonerespect: allsingle-particle (quantum) states have anextra fourfold degeneracy. For example, ifabeam ofsuch hypothetical particles was passed through aninhomogeneous static magnetic field (aStern—Gerlach experiment) itwould besplit into two components. This iswhat happens with electrons onatoms inareal experiment. Furthermore, noelectromagnetic fieldcould bedevised that would split the degeneracy ofeach beam. However, a(powerful) inhomogeneous gravitational field would ingeneral break thedegenera- cy,producing four distinct beams inthefield. Electrons described bythe Dirac equation arenotpredicted tobehave inthisway. Although such anexperiment hasnever been done with real electrons, our under- standing oftheperiodic table oftheelements isbased onthePauli principle forelectrons with twointernal states rather than four. Without amajor reformulation ofthis principle itisdifficult toreconcile our current understanding ofthequantum mechanics ofelectrons with the 258 CLIFFORD CALCULUS ON MANIFOLDS four copies of the Pauli-Schrbdinger equation obtained from (8.3.1). In an arbitrary curved spacetime (gravitational field) the Kahler equation will not decouple into four minimal left ideas (there will not be globally defined parallel primitives). Although the experimental significance of this is far from clear the fact that the degeneracy of the Minkowski space system can be broken would seem to lead to interpretational problems for the quantum theory. Exercise 8.2 Define in the usual Minkowski spacetime polar chart (t, r, 0, cp) the local 1-forms ST = ri-kO(rkYnkl(0,cp)) = kY'T(0,q2) + rdY' kn k = 0, 1, 2 . . . in terms of standard spherical harmonics satisfying 02(ro, r T) = 0. Verify that 0Srkn - (1r k) dr and that for any inhomogeneous differential form R independent of dt: Ø(RS) = (OR + R 1 - kdr)ST . Verify that a solution of Kahler's equation with a Coulomb 1-form potential A = (e1r)dt in this spacetime may be written = E E E REkm(r, 0, cp)Te(t) e=± k m. -k where Re,, = {f(r) + gE k(r)dr)S'kn and Te(t) = exp (itoFt)(1 + jail) and for each E, k the 0-forms f and g satisfy the ordinary differential equations: f' + (1 - r k)f e:g + (co - p)g = 0 g, + g + — (1 + k) e r2f - (co + p)f = O. Exercise 8.3 The 1-form harmonics S'kn may also be used to analyse Maxwell's equations OF = 0. First observe that the 1-forms cek„, = Zek(Ar)ST obey f212cr = -,12cr and the 2-forms 13Ek„, = ZEk(Ar)drS'kn obey 020 = -A213, where ZE k label the independent Bessel solutions of the equation p"(r) + -2p'(r) + (A2 (k2 r-- kip(r) = 7 A * 0. 258 CLIFFORD CALCULUS ONMANIFOLDS four copies ofthePauli-Schrodinger equation obtained from (8.3.1). In anarbitrary curved spacetime (gravitational field) theKahler equation willnotdecouple intofour minimal leftideas (there willnotbeglobally defined parallel primitives). Although theexperimental significance of thisisfarfrom clear thefactthatthedegeneracy oftheMinkowski space system can bebroken would seem tolead tointerpretational problems forthequantum theory. Exercise 8.2 Define intheusual Minkowski spacetime polar chart (t,r,6,tp)the local 1-forms 52"=r""¢(r"YZ"(9» 80))=/<Y2"(9, <1’)+rdY'/Z‘ /<=0,1,2... —k<m<k interms ofstandard spherical harmonics satisfying ¢l2(r"YZ’) =0.Verify that 1-/<¢s;'=(—T—ldrS;" andthatforanyinhomogeneous differential form Rindependent ofdt: . 1—k =(¢lR +R775‘?-‘ . Verify that asolution ofKahler’s equation with aCoulomb 1-form potential A=(e/r)dt inthisspacetime may bewritten kW=222Ri,..(r,0.¢>>T£(r) s=: km= —/< where Rim={f§(r) +g§(r)dr}SZ‘ and T‘(t) =exp(iw‘t)(1 +isdt) and foreach 5,kthe0-forms fand gsatisfy theordinary differential equations: f'+(1%")f-e—,2g+(w-tt>g=0 8'+(i:—,O8+e—;f—(w+u)f=0- Exercise 8.3 The 1-form harmonics S1."may also beused toanalyse Maxwell’s equations ¢lF=0.First observe that the1-forms aim=Zf.(Ar)SZ' obey ¢l2a= —A2a and the 2-forms ,6§,,,=Z§(Ar)drSZ' obey ¢l2,8= —A2,8, where Zitlabel theindependent Bessel solutions oftheequation p"(r)+%p'(r)+(/11- )p(r) =0 A7-0. THE KAHLER EQUATION 259 Writing F = Edt + B with E = imE and A' = itoB write the harmonic component Maxwell equations as the complex pair: OE = —itoB OB = —itHE and seek solutions of the form E = pk(r)Snk' for some 0-forms Hence construct the multipole expansions: E' = E wi(A'km1-17„)exp(iwct) = —(HOE' e,k,m 1311 = »92(131.0m)exp(iwEt) EH = --ØB" e,k,m where 111,, = Z(wr)1drS w 0 = Z(wr)drS w t 0 = e'23 and AL„ /3„, are any complex constants. Exercise 8.4 The stress tensor for the Einstein—Kahler coupled system (with A=0) is T = 192 0(4:VneaV 0:Pee b + cVnebV x,c13e`ea)ea 0 eb . Verify that VT = 0. Hint. Since the co-frames with contracted indices will not contribute to the divergence concentrate on the terms 4(7'. T)b = 590(V x,434-vieaV x,(Deceb + cVyieaV x:7 xrctieceb + (V'leaVill)V xfceb + ci;,'IebeaVxyx,41)ecea + (VtiebVx,(13Vxtea). Note Wo(Vx:13EgebVx,41)ecea) = 0 since 590(11-14--) = IV for any W. Using (8.3.1) and its iterate, At = Wc13, the above terms cancel with the aid of the relations d<V= (5(13:== d(I)q= —(dc13)" 6(13n= —(60:13)q Vx:bea= —(c1(13 + SOY . The last relation follows from (8.1.3). Pk  Ti-IE KAHLER EQUATION 259 Writing F=Edt+Bwith E=iwE and B=iwB write theharmonic component Maxwell equations asthecomplex pair: ;zlE=—iwB ;2iB=—iwE and seek solutions oftheform E=pk(r)Sj,” forsome 0-forms pk. Hence construct themultipole expansions: E’=2&n(Az..Hz.>exp<iw~> B’=§¢E' s.k,m B"=;&P.(B2..fi2.>exp<iw£r) E"=jaw" where 11;,=Z;(wr)81drs;" to¢0 (31,,=Z§(cor)drS’,§‘ to¢0 $1=emandAim, Bi,”areanycomplex constants. Exercise 8.4 Thestress tensor fortheEinstein—Kahler coupled system (with A=0) is T=§&f’O(<I>5"e,,VXr(1>e”e,, +<I>5"e,,VXr<I>e”e,)e” ®eb. Verify that V'T =O. Hint. Since theco-frames with contracted indices willnotcontribute to thedivergence concentrate ontheterms 4(V'T)b : \(:P0(VXa(I)§'7€“VX((I)€C€b ‘i’ (I)E'7€“VXuVX((I)€c€j, +<I>5"e”VX((1>VX”e‘ej, +<I>5"ebe,,VXuVXr(1>e‘e" +<1>5"e,,VX(,<I>VXfle”e"). Note &P0(VXa<I>*“"e,,VXrlI>e‘e”) =0since &P0(\I‘5) =‘I1foranyW.Using (8.3.1) anditsiterate, AID=MID, theabove terms cancel with theaid oftherelations d<I>§= —(d<I>)5" 6<I>§= (6<I>)5" dd)": —(d<I>)" 6(1)": —(6<I>)" VX"<I>e"= —(d<I> +6(1))". Thelastrelation follows from (8.1.3). 260 CLIFFORD CALCULUS ON MANIFOLDS 8.4 The Duffin—Kemmer—Petiau Equations After the success of the Dirac equation in describing the electron there were attempts made to find first-order equations suitable for describing integer spin particles. The Duffin—Kemmer—Petiau equations are an example [21]. The Kahler equation is not unique in being a first-order equation for an inhomogeneous differential form which iterates to the Laplace— Beltrami equation. For example, consider d0, — (50_ = ft0 (8.4.1) where 0, (1 ± 00. This corresponds to the Duffin—Kemmer—Petiau equation. Writing this in terms of Clifford products, eaVx0 + Ville" = 2,u0 we see that the second term prevents the decoupling of the equation into minimal left ideals in Minkowski space. Since d and (5 map even (odd) forms to odd (even) ones (8.4.1) is equivalent to d0, -= p0 _ 60_ = . As a consequence (50, = 0 and d0_ = 0 so any solution to (8.4.1) will also satisfy the Kahler equation for 0. In the massless case (8.4.1) exhibits the generalised gauge symmetry 0, 1---> 0, + dx_ _ -> (13 _ 5x, and describes what in the physics literature are often called antisym- metric tensor gauge fields. Bibliography Chisholm J S R and Common A K (ed) 1986 NATO AS! Series 183 260 CLIFFORD CALCULUS onMANIFOLDS 8.4TheDuffin—Kemmer—Petiau Equations After thesuccess oftheDirac equation indescribing theelectron there were attempts made tofind first-order equations suitable fordescribing integer spin particles. The Duffin—Kemmer—Petiau equations arean example [21]. The Kahler equation isnotunique inbeing afirst-order equation for aninhomogeneous differential form which iterates totheLaplace- Beltrami equation. Forexample, consider d<I>.,-6<I>_=no (84.1) where (DiE§(1it1)<I>. This corresponds totheDuffin—Kemmer—Petiau equation. Writing thisinterms ofClifford products, e”VX“<I> +VX”<I>e“ =2/.t<I> weseethat thesecond term prevents thedecoupling oftheequation into minimal leftideals inMinkowski space. Since dand6map even (odd) forms toodd(even) ones (8.4.1) isequivalent to d<I>+ =,u<I>_ 6<I>_ =—,u<I>+ . Asaconsequence 6(1),, =0andd<I>_ =0soanysolution to(8.4.1) will alsosatisfy theKahler equation for(D. Inthemassless case (8.4.1) exhibits thegeneralised gauge symmetry (IL,I——><I>+ +d;(_ <I>_I——><I>- +61+ and describes what inthephysics literature areoften called antisym- metric tensor gauge fields. Bibliography Chisholm JSRandCommon AK(ed) 1986 NATO ASISeries 183 9 Spinor Fields In §2.5 spinors (or semi-spinors) were defined as carrying irreducible representations of the Clifford algebra. Any such irreducible representa- tion is equivalent to that carried by a minimal left ideal of the Clifford algebra. We thus took any minimal left ideal as the space of spinors. The Clifford bundle of a pseudo-Riemannian manifold M has as fibre at p, the Clifford algebra of the cotangent space of M at p. Any minimal left ideal of this fibre algebra carries the spinor representation. If we could smoothly assign a minimal left ideal of the fibre algebra to each p in M then we would have a bundle over M with each fibre carrying an irreducible representation of the corresponding fibre of the Clifford bundle. Such a bundle of spinor spaces would be a sub-bundle of the Clifford bundle. For such a bundle to exist the topology of M would have to be severely restricted. Requiring the bundle of spinor spaces to be contained in the Clifford bundle is unduly restrictive. Therefore, rather than requiring that the spinor spaces be minimal left ideals of the Clifford algebra, we only require that they carry a representation equivalent to that carried by any minimal left ideal. Locally any bundle of spinor spaces will be isomorphic to a sub- bundle of the Clifford bundle, with fibres being minimal left ideals of the Clifford algebra. As we shall show, if any bundle of spinor spaces exists we can always form a bundle by patching together the minimal left ideals of the Clifford algebra in such a way that locally a spinor field may be represented by a differential form lying in a minimal left ideal of the Clifford algebra. 9.1 Spinor Bundles We assume first that the pseudo-Riemannian manifold M is even dimensional so that the real Clifford algebra is central simple. Thus Spinor Fields In§2.5 spinors (orsemi-spinors) were defined ascarrying irreducible representations oftheClifford algebra. Anysuch irreducible representa- tion isequivalent tothat carried byaminimal leftideal oftheClifford algebra. Wethus took anyminimal leftideal asthespace ofspinors. TheClifford bundle ofapseudo-Riemannian manifold Mhasasfibre at p,theClifford algebra ofthecotangent space ofMatp.Any minimal leftideal ofthisfibre algebra carries thespinor representation. Ifwe could smoothly assign aminimal leftideal ofthefibre algebra toeach p inMthen wewould have abundle over Mwith each fibre carrying an irreducible representation ofthecorresponding fibre oftheClifford bundle. Such abundle ofspinor spaces would beasub-bundle ofthe Clifford bundle. Forsuch abundle toexist thetopology ofMwould have tobeseverely restricted. Requiring thebundle ofspinor spaces to becontained intheClifford bundle isunduly restrictive. Therefore, rather than requiring that thespinor spaces beminimal leftideals ofthe Clifford algebra, weonly require that they carry arepresentation equivalent tothatcarried byanyminimal leftideal. Locally any bundle ofspinor spaces will beisomorphic toasub- bundle oftheClifford bundle, with fibres being minimal leftideals of theClifford algebra. Asweshall show, ifanybundle ofspinor spaces exists wecanalways form abundle bypatching together theminimal leftideals oftheClifford algebra insuch awaythatlocally aspinor field may berepresented byadifferential form lying inaminimal leftideal of theClifford algebra. 9.1Spinor Bundles We assume first that the pseudo-Riemannian manifold Miseven dimensional sothat thereal Clifford algebra iscentral simple. Thus 262 SPINOR FIELDS C(T*pM,g) = ht ( R) D(1R), where Jlit,.(1R) is the algebra of all order-r real matrices and the real central division algebra D must be either the real numbers 1R or the quaternions H. Any minimal left ideal of C(rpM,g) carries the spinor representation. Thus minimal left ideals are r-dimensional right D-modules, Clifford multiplication inducing a D-linear transformation. As we noted above we are not now going to require that our spinor spaces be identified with any minimal left ideal, only that they carry an equivalent representation. Thus our spinor spaces will be right D-linear spaces such that Clifford multiplication is D-linear. Let .1(M) be a bundle over M such that for each p E M the fibre above p is a right D-linear space carrying an irreducible repre- sentation of C(T*pM,g). Any such bundle will be called a (real) spinor bundle, sections being called spinor fields. If any spinor bundle exists then M is called a spin manifold. A discussion of the topological restrictions on M in order for it to be a spin manifold are beyond the scope of this book. However, the reason that there is some restriction will become apparent later. Whereas M may have no spinor bundle, it may also have many. These can be split into equivalence classes. Two spinor bundles 3(M) and 3'(M) are equivalent if and only if there is a diffeomorphism relating them such that fibres of 4(M) above p are mapped into fibres of J'(M) above p with the diffeomorphism commut- ing with Clifford multiplication. An equivalence class of spinor bundles constitutes a spinor structure for C(M). (This definition of spinor structure is equivalent to the more usual one to be found in, for example, Milnor [221.) Let us assume that M is a spin manifold with 4(M) a spinor bundle. Fibres of the Clifford bundle are isomorphic to the algebra of D-valued matrices. If { ea (a")} is a local orthonormal co-frame defined on the open neighbourhood U, of M then an isomorphism between C(rpM,g) and D-valued matrices may be given at each p c U „. in terms of the generators {ea (°)I p) and the constant matrices {7a} satisfying 7a7b 7b 7a = 2gabi (9.1.1) For a given choice of D-valued 7-matrices we may correlate a local orthonormal co-frame with a local basis of sections of J(M). On U, there is a local basis for spinor fields {6, (0) such that = b(;*)y. (9.1.2) (Note that we juxtapose symbols to denote the Clifford action of sections of C(M) on sections of .4,(M).) (Thus the basis {1)} trans- forms under Clifford multiplication just like the 'first column' of a matrix basis for the Clifford algebra.) Notice that (9.1.2) does not uniquely determine the spinor basis. If (1,Œ)') also satisfies (9.1.2) then PO is a non-zero function on U, such that 262 SPINOR FIELDS C(T’j,M,g) =JI/t,(lR) ®D(lR), where JI/t,(lR) isthealgebra ofallorder-r realmatrices andtherealcentral division algebra Dmust beeither the real numbers IRorthequaternions H.Any minimal left ideal of C(T‘§,M,g) carries thespinor representation. Thus minimal leftideals arer-dimensional right D-modules, Clifford multiplication inducing a D-linear transformation. Aswenoted above wearenotnow going to require that ourspinor spaces beidentified with anyminimal leftideal, only that they carry anequivalent representation. Thus our spinor spaces willberight D-linear spaces such thatClifford multiplication is D-linear. Let.9=(M) beabundle over Msuch that foreach peMthe fibre above pisaright D-linear space carrying anirreducible repre- sentation ofC(T*;,M,g). Any such bundle willbecalled a(real) spinor bundle, sections being called spinor fields. Ifanyspinor bundle exists then Miscalled aspin manifold. Adiscussion ofthetopological restrictions onMinorder forittobeaspin manifold arebeyond the scope ofthisbook. However, thereason that there issome restriction willbecome apparent later. Whereas Mmay have nospinor bundle, it may also have many. These canbesplit into equivalence classes. Two spinor bundles .9=(M) and.9=’(M) areequivalent ifandonly ifthere isa diffeomorphism relating them such that fibres of.9=(M) above pare mapped intofibres of.9=’(M) above pwith thediffeomorphism commut- ingwith Clifford multiplication. Anequivalence class ofspinor bundles constitutes aspinor structure forC(M). (This definition ofspinor structure isequivalent tothemore usual one tobefound in,for example, Milnor [22].) Letusassume that Misaspin manifold with .9=(M) aspinor bundle. Fibres oftheClifford bundle areisomorphic tothealgebra ofD-valued matrices. If{e"("l} isalocal orthonormal co-frame defined ontheopen neighbourhood U6,ofMthen anisomorphism between C(T’§,M,g) and D-valued matrices may begiven ateach peUainterms ofthe generators {e“(°')|p} andtheconstant matrices {y“} satisfying 3/“yb +3/by“ =2g“"1. (9.1.1) For agiven choice ofD-valued y-matrices wemay correlate alocal orthonormal co-frame with alocal basis ofsections of.¢(M). OnU, there isalocal basis forspinor fields {b,-ial} such that e“("‘)b§") =bj~")y}‘,. (9.1.2) (Note that wejuxtapose symbols todenote theClifford action of sections ofC(M) onsections of.9=(M).) (Thus thebasis {b§“l} trans- forms under Clifford multiplication just like the‘first column’ ofa matrix basis fortheClifford algebra.) Notice that (9.1.2) does not uniquely determine thespinor basis. If{b§“”} alsosatisfies (9.1.2) then f(")isanon-zero function onU,such that SPINOR BUNDLES 263 b' = Palb(,a). (9.1.3) On U,fi U„U 1113 there must be some local section of C(M), such that le) = s(13a)b"). But ea(0)br = by = s(13OE)b(a)ya = I S(Palea(a)Ma). SO ea(13)S(13aV)= S(Pcv) ea(a)bc" and ea(13) = ,s(1a)ea(os(130-1. (9.1.4) Thus certainly sWa) is in the Clifford group F. If the spinor bases are changed as in (9.1.3) then s(8a)' = f(43)S (Mf(a') It turns out that we can, in fact, always choose the local bases in (9.1.2) such that the Clifford elements s(P") relating them on overlaps are in +F. (It is a standard result that any F bundle is reducible to a J bundle since F/ ÷F see for example Kobayashi and Nomizu [23].) On triple overlaps U Ufi U Uy Uoy the Clifford elements relating spinor bases satis- fy the coherence condition stet/3)03Y) = s (9.1.5) If M is both space and time orientable then we may choose local orthonormal co-frames related on overlaps by an element of SO+(p,q). Then if J(M) is a spinor bundle we may choose local spinor frames, as above, related on overlaps by an element of _,F+. It is important to know that such local bases exist; we shall call them standard spinor frames. (Strictly speaking our definition of a spinor bundle is equivalent to the usual one only in the orientable case. Without orientability our definition is equivalent to what would usually be called a pinor struc- ture.) If M is any pseudo-Riemannian manifold then we can choose local orthonormal frames, related on overlaps by an orthogonal transform- ation, MS" ) say. We can choose an s(16") E such that x(030) On triple overlaps we must have s (0)03Y) = In general, we cannot choose the {s("13)} so as to eliminate all the minus signs in these relations. We can do this if and only if M is a spin manifold. In the case in which D = H we have required the spin bundle to have a right H-linear structure. Thus spinor fields can be multiplied by quaternions. This condition could be relaxed. We know that each spinor space is a right H-linear space, so locally any spinor bundle must have this structure. But we could consider the more general case in which spinor fields can be multiplied by sections of a non-trivial quaternion bundle, this multiplication commuting with the Clifford action. The existence of a spinor bundle without the H-linear structure is equivalent to the weaker condition of having a generalised spinor structure [24]. So far we have only considered bundles of real spinors for the case in which M is even dimensional. If M is odd dimensional with signature such that the Clifford algebra is reducible then the central idempotents SPINOR BUNDLES 263 be"=f<">bt"- (9.12) OnUM;EU,,U U5there must besome local section ofC(M), s‘/5"‘, such that bf/5‘=s‘/5“)bf-"'. But e“‘/“bf-"l =bjfilyj‘, =s‘/"’)bj"‘)/j‘, = 5‘/9"‘le”"’lb}"‘. Soe”(/ils‘/3‘”b§"‘ =s‘/3“)e”(")bf-“’ and MB)=s</*%"<“>s<#">". (9.1.4) Thus certainly s‘/3“) isintheClifford group F.Ifthespinor bases are changed asin(9.1.3) then sifm’ =fimsi/5“)f(“)'l. Itturns outthatwecan, infact, always choose thelocal bases in(9.1.2) such that theClifford elements s‘/3°‘) relating them onoverlaps arein:1".(Itisastandard result that anyFbundle isreducible toail"bundle since I"/il" ElPI*, seeforexample Kobayashi and Nomizu [23].) On triple overlaps U0,UU5UU,EUm, theClifford elements relating spinor bases satis- fythecoherence condition sw/i)s(l5Y) =s(@r)_ (9_1_5) IfMisboth space and time orientable then wemay choose local orthonormal co-frames related onoverlaps byanelement ofSO*(p,q). Then if.¢(M) isaspinor bundle wemay choose local spinor frames, as above, related onoverlaps byanelement of+1“. Itisimportant to know that such local bases exist; weshall callthem standard spinor frames. (Strictly speaking ourdefinition ofaspinor bundle isequivalent totheusual oneonly intheorientable case. Without orientability our definition isequivalent towhat would usually becalled apinor struc- ture.) IfMisany pseudo-Riemannian manifold then wecanchoose local orthonormal frames, related onoverlaps byanorthogonal transform- ation, Ai/3°‘) say.Wecanchoose ans(/3“) eII"such that)((s(f"")) =A99“). Ontriple overlaps wemust have s(“5)s(5Y> =ism). Ingeneral, we cannot choose the{s("‘5l} soastoeliminate alltheminus signs inthese relations. Wecandothisifandonly ifMisaspin manifold. Inthecase inwhich D=Hwehave required thespin bundle tohave aright H-linear structure. Thus spinor fields can bemultiplied by quaternions. Thiscondition could berelaxed. Weknow thateach spinor space isaright H-linear space, solocally anyspinor bundle must have thisstructure. But wecould consider themore general case inwhich spinor fields canbemultiplied bysections ofanon-trivial quaternion bundle, this multiplication commuting with theClifford action. The existence ofaspinor bundle without theH-linear structure isequivalent totheweaker condition ofhaving ageneralised spinor structure [24]. Sofarwehave only considered bundles ofrealspinors forthecase in which Miseven dimensional. IfMisodd dimensional with signature such that theClifford algebra isreducible then thecentral idempotents 264 SPINOR FIELDS ± z), with z the volume n-form, decompose the Clifford algebra into simple ideals. So if M is orientable the Clifford bundle splits into two bundles of simple algebras. In this case we can define spinor bundles exactly as above and show that there are standard spinor frames related on overlaps by an element of „F+. When the Clifford algebra is isomorphic to the algebra of complex matrices then certainly any bundle carrying an irreducible representation of the Clifford bundle has local bases related on overlaps by elements of the Clifford group. But in this case we cannot argue that they can be chosen in ,F+ (assuming orientability); rather they will be elements of F+ multiplied by uni- modular complex functions. The existence of such a bundle is equivalent to having a Spinc structure, this being a weaker condition than having a Spin structure. The case of the complexified Clifford bundle is like that just discussed. If we assume orientability then the existence of a bundle carrying an irreducible representation is equivalent to having a Spinc structure. In the following we shall assume that M is a spin manifold. Unless we specifically say otherwise we shall mean by spinor bundle a bundle carrying an irreducible representation of the Clifford bundle, or its complexification, such that we have standard spinor frames related on overlaps by an element of ,F+. For the case of odd dimensions, or the complexified case, this is a stronger requirement than that the bundle simply carry an irreducible representation of the Clifford bundle. 9.2 Inner Products on Spinor Fields In Chapter 2 we took the space of spinors to be any minimal left ideal of the Clifford algebra, projected by some primitive idempotent P. In §2.6 we constructed spin-invariant products on the space of spinors with values in the division algebra PC(V,g)P --- D. We now want to define spin-invariant products on spinor fields with values in D. Although we shall use the same notation as in §2.6 now our spinors need not lie in any minimal left ideal of the Clifford algebra, and the product will take values in D which is the 'standard' algebra isomorphic to PC(V,g)P for any primitive P. If we had an inner product defined on sections of the spinor bundle then we could use this product to establish local canonical bases (orthonormal, symplectic etc.). On overlaps these canonical bases would be related by transformations in the invariance group of the product. Conversely we can use a set of local bases related on overlaps by an element of ,F+ to define a ,F+-invariant product on spinor fields. For the sake of definiteness we assume that the (real or complexified) 264 SPINOR FIELDS §(1i z),with zthevolume n-form, decompose theClifford algebra into simple ideals. SoifMisorientable theClifford bundle splits into two bundles ofsimple algebras. Inthis case wecan define spinor bundles exactly asabove andshow thatthere arestandard spinor frames related onoverlaps byanelement of+1“. When theClifford algebra is isomorphic tothealgebra ofcomplex matrices then certainly anybundle carrying anirreducible representation oftheClifford bundle haslocal bases related onoverlaps byelements oftheClifford group. Butinthis case wecannot argue that they can bechosen in+1“ (assuming orientability); rather they will beelements of,1“ multiplied byuni- modular complex functions. Theexistence ofsuch abundle isequivalent tohaving aSpinc structure, thisbeing aweaker condition than having a Spin structure. The case ofthecomplexified Clifford bundle islikethat justdiscussed. Ifweassume orientability then theexistence ofabundle carrying anirreducible representation isequivalent tohaving aSpin‘? structure. Inthefollowing weshall assume that Misaspin manifold. Unless we specifically sayotherwise weshall mean byspinor bundle abundle carrying anirreducible representation oftheClifford bundle, orits complexification, such that wehave standard spinor frames related on overlaps byanelement of+1“. Forthecase ofodddimensions, orthe complexified case, thisisastronger requirement than that thebundle simply carry anirreducible representation oftheClifford bundle. 9.2Inner Products onSpinor Fields InChapter 2wetook thespace ofspinors tobeanyminimal leftideal oftheClifford algebra, projected bysome primitive idempotent P.In §2.6 weconstructed spin-invariant products onthespace ofspinors with values inthedivision algebra PC(V,g)P ED.Wenow want todefine spin-invariant products onspinor fields with values inD.Although we shall usethesame notation asin§2.6 now ourspinors need notliein anyminimal leftideal oftheClifford algebra, andtheproduct willtake values inDwhich isthe‘standard’ algebra isomorphic toPC(V,g)P for anyprimitive P. Ifwehadaninner product defined onsections ofthespinor bundle then wecould use this product toestablish local canonical bases (orthonormal, symplectic etc.). Onoverlaps these canonical bases would berelated bytransformations intheinvariance group oftheproduct. Conversely wecanuseasetoflocal bases related onoverlaps byan element of+1“ todefine a+1“-invariant product onspinor fields. For thesake ofdefiniteness weassume that the (real orcomplexified) PRODUCTS ON SPINOR FIELDS 265 Clifford algebra is isomorphic to the algebra of all (real or complex) matrices, with the involution ij similar to transposition. In this case for matrices as in (9.1.1) there is a matrix C, symmetric or skew, such that C 7aT C -1 = 7a . (9.2.1) If { b} is a standard spinor frame, satisfying (9.1.2), then a bilinear product on local spinor fields is specified by defining b)(Œ) = C171. (9.2.2) The product has been labelled with the subscript (a) since in principle we have a different product for each U„. We want to show that on Uo the products ( , )(0 and ( , ) (o) coincide, for then we have a well defined product on spinor fields. First we show that these local products are spin invariant. For any such local product then (suppressing the (or)- labelling) (b„ eab ) = (b„ = Cy = (c-17a) = ( 7aTc-1) by (9.2.1), so (by eabl) = —yaT,,C,T11 = = —ya,„(bk, b) = —(eab„ Thus for any spinor fields and m E FC(M) (ço, m = (rOcp, 1p) (,) and hence these local products are spin invariant, having as adjoint involution. On U, 43 the standard spinor frames are related by b,(P) = s(13a)br for s(ga)c ,F+. So on U (bn b(,P))(c)= (s(13")br, s(ga)br)(„)= (s(13c0s(1a)br, be`))(a) = (br, , 13(1"))(.) = (b'6) , b()15)) (0). Thus for any local spinor fields (cp, = (cp, tp)03). Hence we have a well defined product on spinor fields and so omit the neighbourhood labelling. We demonstrated the existence of a spin-invariant product on spinor fields by constructing one using a special basis. That construction does not, in fact, specify a unique product. For given local orthonormal co-frames and 7-matrices the standard local spinor frames are not unique. If the local orthonormal co-frames are related by A(ag) then the s(0) c ,F+ such that x(s) = A (43) is determined up to a sign. So if { b'} is also a standard spinor frame with br' = Palb;a 1 for a local function f(a) then on overlaps we must have PO = ±f(a). So a non-zero function f is defined on M by /I u = (sgnfolf (a), with br' = ±fbr. So if (ba")', br)' = (br, br), then for any spinor fields f2(q), 1p)' = (T, tp). It is easily seen that the choices of orthonormal co-frames, 7-matrices and matrix C cannot affect the spinor product by more than a conformal scaling. Thus this prescription determines a class of confor- mally related spin-invariant products. PRODUCTS ONSPINOR FIELDS 265 Clifford algebra isisomorphic tothealgebra ofall(real orcomplex) matrices, with theinvolution §17similar totransposition. Inthiscase for matrices asin(9.1.1) there isamatrix C,symmetric orskew, such that cwc-1 =—y“. (9.21) If{b§“’} isastandard spinor frame, satisfying (9.1.2), then abilinear product onlocal spinor fields isspecified bydefining (b§"‘, bj-“‘)(,,) =C,-jl. (9.2.2) The product hasbeen labelled with thesubscript (a/)since inprinciple wehave adifferent product foreach U,,..Wewant toshow thatonU,,5 theproducts (,)(,,)and(,)U,)coincide, forthen wehave awell defined product onspinor fields. First weshow thatthese local products arespin invariant. For any such local product then (suppressing the (ct)- labelling) (bit eabj) =(bi: b/0'21) :C1721 :(C-170); Z_(VaTC_l)1j by(9.2.1), so (bi:eabj) :—Va1i<C/:11 :TVZIC/:11 :_Vii(b/<1 bj):—(eabi1 bj)~ Thus foranyspinor fields and meFC(M) (go,mt/1)(,,) =(miftp, 1/1)(,,) andhence these local products arespin invariant, having §17asadjoint involution. On Uaj; the standard spinor frames are related by bf”)=s(5“lb§“l forsill“) E,I‘*. SoonUH); (bye, b;l3l)(a) :(stfialbywl, s((i¢r)b;v1))(a) =(S<fla>~”'s<th>b§a>, byn)(a) =(him, bj‘a))(cr) :(bin), biB))(p)- Thus foranylocal spinor fields (mp,1/1)(a) =(<;0,1/1)(j;). Hence wehave a well defined product onspinor fields and soomit theneighbourhood labelling. Wedemonstrated theexistence ofaspin-invariant product onspinor fields byconstructing oneusing aspecial basis. That construction does not, infact, specify aunique product. For given local orthonormal co-frames and y-matrices thestandard local spinor frames are not unique. Ifthelocal orthonormal co-frames arerelated byAW’) thenthe SW3) e,.I‘+ such that ;((s“’5)) =AW” isdetermined uptoasign. Soif {bf"l’} isalso astandard spinor frame with b§“" =f“’lb§“‘ foralocal function ff“)then onoverlaps wemust have fi/3)=iff"). Soanon-zero function fisdefined onMbyflu“ =(sgnfi"l)f(“l, with bi,” =ifbj"l. Soif(b§“l', b§"“’)' =(b§“l, bi“), then foranyspinor fields f2(tp, 1/1)’= (rp,1/1).Itiseasily seen that thechoices oforthonormal co-frames, y-matrices andmatrix Ccannot affect thespinor product bymore than aconformal scaling. Thus thisprescription determines aclass ofconfor- mally related spin-invariant products. 266 SPINOR FIELDS Although in the above we assumed for definiteness that 07 was similar to transposition in a total matrix algebra, the above construction obviously goes through similarly in general. We may analogously con- struct spin-invariant products with adjoint involution or, for the complexified algebras, or If, for M even dimensional, 4(M) is a bundle of spinors carrying an irreducible representation of the complexified Clifford bundle then we may define charge conjugation on spinor fields. Once again, although we know that we can do this locally, we have to check that we can do it globally. We therefore give the definition locally using a standard spinor frame and make sure that it is consistent on overlaps. From (2.7.9) we know that there is a matrix m such that ya* = m -lyam, with m* = +m -1. (9.2.3) On 1. a, the operator c(a) is defined by ipc(a) = (b;covi)c(Œ) = b;alinfilP  (9.2.4) If #(a) is the local operation on spinor fields that complex conjugates the components in the M' ) basis then we use the same symbol to denote the automorphism of the complexified Clifford algebra defined by (aip)*(a) = a#((otp#0") . (9.2.5) Thus if ab a) = b.a j, then a4(a*cr) = b;") a I,* . (Care is needed with the notation. By tip* we mean the complex conjugate of the components of a, whereas a* ji are the components of the Clifford element a*. The difference between these is the difference between * and #(a).) If rn(a") is the local Clifford form such that m ("W") = biwnti, then it follows from (9.2.3) that a#() -- m(Œ)-1 a* m(a") (9.2.6) SO (cutP)c(") = (b(rr)a = kiaii* = = m(a)a#(014a)lpi* = rn (a) a#01m(a")-1 tpc() = a* p). If we expand p as tp = MI31ipi then 'Lilc() = brm iiipi*, but ip = s(Pa9bY4ipi so pc(Œ) = s(OcY)* b(Ja')m fizpi* = b /4') m 1,1pz. since S (13a)* -= S66'11 for ,t(t3a) E ,F+. Hence the operations c(a) and c(13) agree on Uo and we have a well defined operation of charge conjuga- tion, denoted c. If M is odd dimensional the complexified Clifford algebra is semi-simple. In this case either * or Tr is a conjugate-linear involuntary automorphism of the simple component algebras. In the 266 SPINOR FIELDS Although intheabove weassumed fordefiniteness that§17wassimilar totransposition inatotal matrix algebra, the above construction obviously goes through similarly ingeneral. Wemay analogously con- struct spin-invariant products with adjoint involution §or,forthe complexified algebras, §*or511*. If,forMeven dimensional, .<l>(M) isabundle ofspinors carrying an irreducible representation ofthecomplexified Clifford bundle then we may define charge conjugation onspinor fields. Once again, although weknow thatwecandothislocally, wehave tocheck thatwecandoit globally. Wetherefore give thedefinition locally using astandard spinor frame andmake sure that itisconsistent onoverlaps. From (2.7.9) we know thatthere isamatrix msuch that 'y“*=m-1)/"m, withm*=imol. (92.3) OnU,theoperator c(a) isdefined by 1/W’=(b%“’w’)“"‘ =b$"’m,-.-w"*- (9-2-4) If#(a) isthelocal operation onspinor fields that complex conjugates thecomponents inthebi")basis then weusethesame symbol todenote theautomorphism ofthecomplexified Clifford algebra defined by (a't/t)#‘“) =a#(‘”tp#(“) . (9.2.5) Thus ifab?) =bj“)a,-,- then a#“’lbj”) =bj“)a,-,*. (Care isneeded with the notation. Byal-,-*wemean thecomplex conjugate ofthecomponents of a,whereas a*,~,- arethecomponents oftheClifford element a*.The difference between these isthedifference between *and#(a).) Ifmi“) isthelocal Clifford form such that m(")b§”l =bj")m,-,- then itfollows from (92.3) that a#(") =m("‘)"a*m(") (9.2.6) SO (aw)c(a) .:(b(a)aj‘_.¢I')c(a) :b((a')'nkI_aI_!_*wi“ :m(a)b(n)aji*wi* :m(a')a#(a)b(a)wi‘ :m(a)a#(a)m(a)“wt'(a) :a*wc(a)_ Ifweexpand 1/;asi/;=biflip‘ then i/fl”) =bjflm,-,-i/;"*, but1/;= s('3”)b§-")1/1" so jpcial :S(I3“’l*b;_°‘ln»lj,,1p‘-i :bjfllmjiwii since s"3")' =so“) fors‘/3"‘) e+1“. Hence theoperations c(a) and c([3) agree onU65andwehave awell defined operation ofcharge conjuga- tion, denoted c.IfMisodd dimensional thecomplexified Clifford algebra issemi-simple. Inthiscase either *or11*isaconjugate-linear involuntary automorphism ofthesimple component algebras. Inthe PRODUCTS ON SPINOR FIELDS 267 latter case we can define charge conjugation using ri* instead of *. In even dimensions we have spin-invariant products on the real spinor bundle with adjoint involutions and The automorphism ri is inner with a = zaz --' for z the volume form. Using a subscript to label the product by its adjoint involution we have IV, 04.'1 = (1P, z€P). (9.2.7) In the complexified case we have similarly = (tPc, 49) (9.2.8) and =(pzeP)  (9.2.9) For a semi-simple real Clifford algebra there is a product on the semi-spinors associated with either or „;71. When the real Clifford algebra is isomorphic to complex matrices then either or îj is associated with a complex bilinear product, the other being associated with a conjugate-linear product; the products being related by 'charge conjugation' defined using ij. For the bundle of complex semi-spinors in odd dimensions then either or ij is associated with a complex bilinear product; either or being associated with a conjugate-linear one. The products are related by 'charge conjugation' defined with either * or ri*. 9.3 Covariant Differentiation of Spinor Fields In a similar way to that used to show the existence of a spin-invariant product we can define covariant differentiation of spinor fields using a standard spinor frame. We will first follow this most direct approach. We may then observe that the spinor covariant derivative has certain properties. In fact these properties completely determine this covariant derivative as we will then show. For most purposes it is sufficient to know that a unique covariant derivative having these properties exists. It is customary to use the symbol V to denote covariant differentiation of spinor fields as well as of tensor fields; the meaning depending on what it acts on. We prefer to use a separate symbol S to denote covariant differentiation of spinor fields. Although we shall only really be con- cerned with the pseudo-Riemannian connection on M it should be apparent that the discussion here is equally applicable in the case of non-zero torsion. If {e')} is a local orthonormal co-frame then, from (8.1.5) and (8.1.6), we have Vxeao") = [a(;),ea()] where c4) = co(b7)(x)ebc(cr) PRODuCrs ONSPINOR FIELDS 267 latter case wecandefine charge conjugation using 17*instead of*. Ineven dimensions wehave spin-invariant products ontherealspinor bundle with adjoint involutions Eand517.The automorphism 17isinner with a"=zaz" forzthevolume form. Using asubscript tolabel the product byitsadjoint involution wehave (111.<P)§1,=(1/1.Z<P).§- (9-2-7) Inthecomplexified case wehave similarly (111,<P)s~=W‘,<P): (9-2-8) and (1/1, (p)§17* :(wcv z(p)§ * For asemi-simple real Clifford algebra there isaproduct onthe semi-spinors associated with either 5or§17.When thereal Clifford algebra isisomorphic tocomplex matrices then either 5or517is associated with acomplex bilinear product, theother being associated with aconjugate-linear product; theproducts being related by‘charge conjugation’ defined using 17.Forthebundle ofcomplex semi-spinors in odddimensions then either 5or517isassociated with acomplex bilinear product; either 5*or517*being associated with aconjugate-linear one. The products arerelated by‘charge conjugation’ defined with either * or17*. 9.3Covariant Differentiation ofSpinor Fields Inasimilar way tothat used toshow theexistence ofaspin-invariant product wecandefine covariant differentiation ofspinor fields using a standard spinor frame. Wewillfirst follow thismost direct approach. Wemay then observe that thespinor covariant derivative hascertain properties. Infactthese properties completely determine thiscovariant derivative aswewill then show. For most purposes itissufficient to know thataunique covariant derivative having these properties exists. It iscustomary tousethesymbol Vtodenote covariant differentiation of spinor fields aswell asoftensor fields; themeaning depending onwhat itacts on.Weprefer touseaseparate symbol Stodenote covariant differentiation ofspinor fields. Although weshall only really becon- cerned with thepseudo-Riemannian connection onMitshould be apparent that thediscussion here isequally applicable inthecase of non-zero torsion. If{e“(")} isalocal orthonormal co-frame then, from (8.1.5) and (8.1.6), wehave VXe”("l =[o§}’),e”("l] where of?)=§a)j,‘§'(X)e"‘("). We 268 SPINOR FIELDS can use this local orthonormal co-frame to define a standard spinor frame satisfying (8.1.2). We can introduce a covariant derivative SP of local spinor fields by defining SPW) = aPMŒ). (9.3.1) If tp(a) is an arbitrary local spinor field then SP is defined by Sip() = SP(Ma)lp`) = SPb;ovi + MoX(Ipi). (9.3.2) The components are D-valued functions and the above requires that we know how to differentiate these. Quaternionic or complex-valued functions are differentiated as ordered quadruples or pairs of real functions; that is, the algebra D has a parallel basis. Of course, we will want to show that if ip is a local spinor field defined on U3 then Sp = Sp. We will then have a well defined covariant derivative on arbitrary sections of the spinor bundle and can drop the label (a). First we show that a consequence of the definition (9.3.1) is that the local spinor covariant derivatives obey a 'Leibnitz' rule. If A is an arbitrary 1-form on M with lp(a') a local spinor field then SP(A ip(a)) = SP(AaeaWtp1) = SP(Aab41IJ') = X(24a)boypi + Aao-Pbytp' + Aaby,Xetp`) = X(Aa)ealp(a) + aPAip(a) + AWX(tp') = le) + Aanp(") + Ab;a1X(ipi) by (8.1.8), so SP(Atp(a)) = Vxkip(a) + ASPIp (a). Since this is true for all local Iii") and the 1-forms generate the Clifford algebra we have SP(aip(")) = Vxa/p(a) + aSnp(a) (9.3.3) for any Clifford form a. If now tp is any spinor field then on U,fi we have Sp = SP(bp) = 0 -np + 140)1C(Ip1). But on U„.fi we have Mo) = s(lia)ba.) for s(Pa")c +r ÷, so Snp = SP(s( 13)Ma)Ipi) = xs 1Pi s(l3")GPMaliPi s(&)Mc')X(P1) = xs(Pa)scsari s(ficoavsoar bv3)x(p1) . Hence from (8.1.9) we see that Sip = Sri') and we have a covariant 268 SPINOR FIELDS canusethis local orthonormal co-frame todefine 21standard spinor frame satisfying (8.1.2). Wecanintroduce acovariant derivative Sf?’of local spinor fields bydefining S§§’b§"l =05918;“. (93.1) If1,11“)isanarbitrary local spinor field then S§§')isdefined by SS?’w<“> =S$?’(b.‘-W‘) =S52"b£-“H/1‘ +bE“’X(w‘)- (9-3-2) The components 171’areD-valued functions andtheabove requires that weknow how todifferentiate these. Quaternionic orcomplex-valued functions aredifferentiated asordered quadruples orpairs ofreal functions; thatis,thealgebra Dhasaparallel basis. Ofcourse, wewill want toshow that if171isalocal spinor field defined onUafl then S§}")1,v =S§‘»”1p. Wewillthen have awell defined covariant derivative on arbitrary sections ofthespinor bundle andcandrop thelabel (a). First weshow that aconsequence ofthedefinition (9.3.1) isthat thelocal spinor covariant derivatives obey a‘Leibnitz’ rule. IfAisanarbitrary 1-form onMwith 171°‘)alocal spinor field then SS?’(A1/1”")=$5?’(A..@“b$“’1//‘) =$$?"(A..l>}"’1/;~‘.-w‘) =X(A.)b}“’1/W +A..<1S2”b§"’1/,“.w‘ +A..b§“’1/ZX(1t1‘) =X(A.)@"w<"> +v‘t’Aw<"’ +Ab%“’X(w') =VXA¢<"> +Aa3?>1p<“> +Ab§“’X(1p') by(8.1.8), so SS?’(Aw‘“’) =VXA111"’+ASS2"w‘“’- Since thisistrue foralllocal 171"”andthe1-forms generate theClifford algebra wehave S§§’(a1/xi”) =VXa1p(“) +aS§§’)1p(°) (9.3.3) foranyClifford form a.Ifnow 171isanyspinor field then onU,,5we have SSW=SS€’(b%"’w’) =view+b%‘*’X(1»‘)~ ButonU07,wehave bi!”=s<f’°'lb§°'l forsill“)e+1"*,so 55?“/I =5if)(S“’“)bi°°1P') ZVXs(fla)b(tx)wi +s(fla)a(\g()b(_rr)wi +s(fia)b(a)X(wi) =VXs<fl~)s(fia)*‘1j, +s(Ba)a(g1s(fla)*‘,j, +b(fi1X(,j,1)_ Hence from (8.1.9) weseethat Sigh): =Sffflwandwehave acovariant COVARIANT DIFFERENTIATION OF SPINOR FIELDS 269 derivative Sx defined on arbitrary spinor fields such that Sxip(") = Sp(a). We have shown the existence of this spinor covariant derivative by specifying it in standard local spinor frames. These standard spinor frames were also used to introduce a spin-invariant product. Suppose that ( , ) is any such D-valued product with (bcr), tia)) = CT,' for some constant matrix C. Then if 'tp and cp are arbitrary spinor fields with = bnpi on Ua, (SA), 49) Sx(P) = (aPIP 14X(P1), 49) ± (V, °PT ± t'X(991))- Since aP is a real 2-form ar = —aP for any .1 that is the adjoint involution of a spin-invariant product. Hence (Sp, (P) OP, Sx(P) = (ba')X(IPI), (P) ± (P, b a')X(491)) = (X0P9)iCV(Pk (V)ICTAIX(e) where the product is DI-linear in the first variable. Since (X(V.P))i = X((ipz)i) and the matrix C -1 is constant (Sx/P, (P) ± (1P, Sx(P) = X(P, cP)  (9.3.4) Thus, in this sense, the spinor covariant derivative is compatible with any spin-invariant product for which the standard spinor frames are a 'canonical' basis. In particular, for the complexified case, Sx is compati- ble with both a complex bilinear and a Hermitian product, related as in (9.2.8). Thus the covariant derivative commutes with charge con- jugation, Sx.ip` = (Sxv)c. (9.3.5) This follows directly from (9.2.4) since the matrix m is constant. Having defined a covariant derivative in a particular basis we have observed the properties (9.3.3), (9.3.4) and (9.3.5). We will now show how any covariant derivative satisfying these axioms is unique. Obvi- ously Sx should map spinor fields to spinor fields. We shall require Fi-linearity in X sfx = fsx (9.3.6) the 'Leibnitz' rule Sx(alp) = V )(alp + aSx4' V a E TC(M), V E El(M) (9.3.7) and compatibility with some spin-invariant product (Sx4', (P) (P, S99) = X(P, (P). (9.3.8) COVARIANT DIFFERENTIATION oFSPINOR FIELDS 269 derivative SXdefined onarbitrary spinor fields such that SX1/ti“) = S(a)q,(¢Y)_ XWe have shown theexistence ofthisspinor covariant derivative by specifying itinstandard local spinor frames. These standard spinor frames were also used tointroduce aspin-invariant product. Suppose that (,)isanysuch D-valued product with (b§"), bf”) =C5‘forsome constant matrix C.Then if171and <72arearbitrary spinor fields with 1»=b$°"1/on U. (Sm<11)+(111.Sx<P) _ )M_ =(OSW +bi"’X(1//')» <1>)+(1/1,99¢+biX(<z>'))- Since 0%’)isareal 2-form 0)?” =—o§!’ forany9that istheadjoint involution ofaspin-invariant product. Hence (5)118 <1>)+(1/1.Sx<P)=(bl-“’X(1//’), <1>)+(1/1,bE°”X(<1>’)) =(X(1t1"))"C.-1‘<1>" +(1//')"<7.-7.-‘X(<1>") where the product isD/I-linear inthe first variable. Since (X(q/))/' =X((1p")/A) andthematrix C”isconstant Thus, inthissense, thespinor covariant derivative iscompatible with anyspin-invariant product forwhich thestandard spinor frames area ‘canonical’ basis. Inparticular, forthecomplexified case, SXiscompati- blewith both acomplex bilinear andaHermitian product, related asin (9.2.8). Thus the covariant derivative commutes with charge con- jugation, Sxil/i =(Sx1J1)C- (9-35) This follows directly from (9.2.4) since thematrix misconstant. Having defined acovariant derivative inaparticular basis wehave observed theproperties (9.3.3), (9.3.4) and (9.3.5). Wewillnow show how anycovariant derivative satisfying these axioms isunique. Obvi- ously SXshould map spinor fields tospinor fields. Weshall require 9-linearity inX the‘Leibnitz’ rule SX(aq;) =Vxaqa +aSX1p VaEFC(M), V1716 l".¢(M) (9.3.7) andcompatibility with some spin-invariant product (SW1. <P)+(11/~SX<P) =XW» <P)- (9-3-8) 270 SPINOR FIELDS Given an S that satisfies these axioms, is it unique? Suppose that S'x also satisfied the axioms above. Then if Lx S'x — Sx we have Lx: f.9(M) —p fg(M) (9.3.9) Lfx — flax (9.3.10) Lx(aip) = aL xv (9.3.11) (T, LAO (LxcP, = O. (9.3.12) Equation (9.3.11) says that Lx commutes with Clifford multiplication, SO Lxtp = tppx for some D-valued function Px. Putting this in (9.3.12) gives ((p, IPPx) + (SoPx, = O. If the product is DJ-linear in the first variable then, since Px E D. (T, x + PV(p, V) = 0- (9.3.13) The D-linearity in tp ensures that cp, (cp, tp) maps FJ(M) x f(M) onto D, so we can choose cp and tp such that (p , tp) = 1. This shows that pix = —Px, and if this is substituted into (9.3.13) then we see that Px must be in the centre of D. If D is one of the central algebras R or H then we must have Px = 0. Similarly if D = C with j the identity involution. However, for the remaining case of D C and j complex conjugation then Px can be any imaginary function. Since the mapping X --> Px is required to be-linear (by (9.3.6)) then if Sx satisfies (9.3.6)—(9.3.8) then so does S'x, with Sp = Sp + iA(X)tp (9.3.14) for any real 1-form A. Thus if the spinors carry an irreducible representation of a (real or complexified) Clifford algebra that is isomorphic to complex matrices then requiring compatibility with a pseudo-Hermitian spinor product leaves the freedom to add an arbitrary U(1) term to the covariant derivative. We can remove this arbitrariness by also requiring (9.3.5) to hold. This is equivalent to requiring that the covariant derivative also be compatible with a complex bilinear product. Because the different spin-invariant products are related as in (9.2.7) and (9.2.8) then the spinor covariant derivative is simultaneously com- patible with all. Exercise 9.1 Show that if Sx satisfies (9.3.5)—(9.3.8) then there are standard spinor frames such that Sxt,') =- In §2.6 we used a D-valued spin-invariant product to map a spinor into the D-linear dual space. We will use the definition and notation of 270 SPINOR FIELDS Given anSthat satisfies these axioms, isitunique? Suppose that Sf‘, alsosatisfied theaxioms above. Then ifLXES’),—SXwehave LX:I‘9(M) ->l".¢(M) (9.3.9) LIX=fLX (9.3.10) LX(a1/1) =aLX1p (9.3.11) Equation (9.3.11) saysthat LXcommutes with Clifford multiplication, soLX111 =tppx forsome D-valued function px.Putting thisin(9.3.12) gives (<t>.1//PX) +((PPx~ 111)=0- Iftheproduct isD/'-linear inthefirstvariable then, since px6D. (<0.Wlpx+p’}<(<t>.111)=0- (9-3-13) The D-linearity in1/1ensures that tp,1p—> (Q7,1/1)maps l'51(M) XI"5i(M) onto D,sowecanchoose tpand1psuch that (go,1/1)=1.This shows that ply=—pX, andifthisissubstituted into (9.3.13) then weseethat pxmust beinthecentre ofD.IfDisoneofthecentral algebras Ror Hthen wemust have px=0.Similarly ifDECwith jtheidentity involution. However, fortheremaining case ofDECandjcomplex conjugation then pxcanbeanyimaginary function. Since themapping X—>pX isrequired tobe@-linear (by(9.3.6)) then ifSXsatisfies (9.3.6)—(9.3.8) then sodoes S},with S’X1p =Sxtp +iA(X)1p (9.3.14) forany real 1-form A.Thus ifthe spinors carry anirreducible representation ofa(real orcomplexified) Clifford algebra that is isomorphic tocomplex matrices then requiring compatibility with a pseudo-Hermitian spinor product leaves thefreedom toaddanarbitrary U(l) term tothecovariant derivative. Wecanremove thisarbitrariness byalsorequiring (9.3.5) tohold. This isequivalent torequiring thatthe covariant derivative alsobecompatible with acomplex bilinear product. Because thedifferent spin-invariant products arerelated asin(9.2.7) and(9.2.8) then thespinor covariant derivative issimultaneously com- patible with all. Exercise 9.1 Show that ifSXsatisfies (9.3.5)—(9.3.8) then there arestandard spinor frames such thatSxbi” =of('i'lb§"l. In§2.6 weused aD-valued spin-invariant product tomap aspinor intotheD-linear dual space. Wewillusethedefinition andnotation Of COVARIANT DIFFERENTIATION OF SPINOR FIELDS 271 (2.6.7) for spinor fields. When our spinor space was a minimal left ideal of the Clifford algebra then the D-linear dual space is naturally identified with a minimal right ideal, and for a spinor cp and dual spinor zp we have cop— in the Clifford algebra. Although the notation of simply juxtaposing the spinors is a slight liberty when the spinor fields are not in the Clifford algebra we still have a mapping taking a spinor and a dual spinor to the Clifford algebra; cp,1 1---> cp/p where (cP/V)P = 97(VP) 40,0 V pEF,95(M) . (9.3.15) If the adjoint spinor is defined with respect to a product with which Sx is compatible then we have the useful relation x(491—P) = S,119/7) 49Sx1P. (9.3.16) This follows by differentiating (9.3.15); using the Leibnitz property on the left-hand side and the metric compatibility on the right-hand side. The curvature operator of S is defined in the obvious way, S(X,Y) = [S x,Sy] — Stx,yi. (9.3.17) There is always a local basis in which Sxb, = axbi, and hence S(X,Y)b, = R xyb, where Jiy is defined in (8.1.10). Since the curva- ture operator is 9;-linear then for any spinor field S(X, Y)ip = gtxop. (9.3.18) Using (8.1.13) and (for zero torsion) (8.1.14) we can write this in terms of the curvature 2-forms giving S(X, Y)v = — xi yR„beabip (9.3.19) Or S(X,Y)Ip = _1,-,ea(X)eb(Y)R ab1P- (9.3.20) 9.4 Lie Derivatives of Spinor Fields Because the Clifford product involves the metric then unless the vector field V is Killing the Lie derivative 2 v will not be a derivation on Clifford products. It follows immediately that there can be no 'Lie derivative' on spinor fields such that the obvious analogue of the 'Leibnitz' rule (9.3.7) holds for arbitrary vectors. Although one could call any operator a lie derivative on spinor fields' the utility of such a definition depends on the consequent properties. So we can anticipate that any definition of a Lie derivative on spinor fields will really only be useful for Killing vectors. We shall notationally distinguish the Lie COVARIANT DIFFERENTIATION OFSPINOR FIELDS (2.6.7) forspinor fields. When ourspinor space wasaminimal leftideal ofthe Clifford algebra then the D-linear dual space isnaturally identified with aminimal right ideal, andforaspinor tpanddual spinor 171wehave (79171intheClifford algebra. Although thenotation ofsimply juxtaposing thespinors isaslight liberty when thespinor fields arenot intheClifford algebra westillhave _amapping taking aspinor and a dual spinor totheClifford algebra; <p,1/11—>(pr/1where (<1>1l1)p =<1>(1l1P) E<1>(1/1,19) V196F9*(M) -(9-3-15) Iftheadjoint spinor 171isdefined with respect toaproduct with which SXiscompatible then wehave theuseful relation v.t(¢17»>=SW11+em. (9116) This follows bydifferentiating (9.3.15); using theLeibnitz property on theleft-hand sideandthemetric compatibility ontheright-hand side. Thecurvature operator ofSisdefined intheobvious way, = [SX,S)/1 _ SIX’)/1. There isalways alocal basis inwhich SXb,- =oXb,~, and hence S(X,Y)b,- EQxyb, where Qixy isdefined in(8.1.10). Since thecurva- tureoperator is9'-linear then foranyspinor field Using (8.1.13) and(forzero torsion) (8.1.14) wecanwrite thisinterms ofthecurvature 2-forms giving S(X,Y)1,l1 =—}iXiYR,,,,e""1,u (93.19) OI' S(X,Y)1,v =§e"(X)e"(Y)R,,,,1/1. (93.20) 9.4LieDerivatives ofSpinor Fields Because theClifford product involves themetric then unless thevector field VisKilling theLiederivative 3,,will notbeaderivation on Clifford products. Itfollows immediately that there can beno‘Lie derivative’ onspinor fields such that theobvious analogue ofthe ‘Leibnitz’ rule (9.3.7) holds forarbitrary vectors. Although one could callanyoperator a‘Lie derivative onspinor fields’ theutility ofsuch a definition depends ontheconsequent properties. Sowecananticipate thatanydefinition ofaLiederivative onspinor fields willreally only be useful forKilling vectors. We shall notationally distinguish theLie 272 SPINOR FIELDS derivative operator on spinor fields from that on tensor fields by using the symbol x. We shall first parallel the initial treatment of the spinor covariant derivative by using a standard spinor frame. We shall show that for a Killing vector the Lie derivative of an orthonormal co-frame can be written as a Clifford commutator. Thus defining the Lie derivative of the associated spinor frame to be mulplication by the element that enters into that commutator ensures the 'Leibnitz' property. In (6.13.1) we introduced the operator Ay -=- — V v, satisfying A v(fcp)= fA ycp for any function f and differential form cp. Since A y is a derivation on the exterior algebra we have A vcp = A vea A ix,S) V cp FAM. We can use (8.1.2) and (8.1.3) to write the interior and exterior products in terms of Clifford products, producing A vg0 = [A vea A ea, cd x,A yea cp — ,(A vea cpq e „ + e acpn A yea). The Clifford commutator is a Clifford derivation. The 2-form A yea A ea can be written in terms of the exterior derivative of V. Since A,,, commutes with contractions and A vf = 0 for fE(M), if {ea} and {X„} are dual bases and AyXa= m„bXb for some matrix m a b then A veb = —mabea. Then A ve" A ea = —mbaeb A ea= mabeb A ea, using the antisymmetry of the exterior product, so A yea A ea = A yXa A ea. Now A vXa = [V,XJ — V vXa, so if V is torsion free A vXa = —VV. thus A ve" A ea = ea A VxY = eAVXV = d (by (4.7.4)). The remaining terms in the expression for A v in general prevent it from being a Clifford derivation. If written in terms of the matrix mab then only the symmetric part enters: JiixAve"çv — "i(AveaVe, + e acpqA yea) = +InlancP + À(mba mab)(ebcP"ea + eacP"eb) using the usual index-lowering convention. Since g(AyX,, Xb) = Mab and A v commutes with contractions 'nab mb„ = —A vg(Xa, Xn). The metric compatibility of V enables us to write A vg = E vg and A vcp = V + vg(Xa, X")cp — Xb)(ebcpqea + eacpqeb). (9.4.1) Thus, as expected, A. and hence Yy, is a Clifford derivation if and only if V is a Killing vector. 272 SPINQR FIELDS derivative operator onspinor fields from that ontensor fields byusing thesymbol EEX. Weshall first parallel theinitial treatment ofthespinor covariant derivative byusing astandard spinor frame. Weshall show that fora Killing vector theLiederivative ofanorthonormal co-frame canbe written asaClifford commutator. Thus defining theLiederivative of theassociated spinor frame tobemulplication bytheelement that enters into thatcommutator ensures the‘Leibnitz’ property. In(6.13.1) weintroduced theoperator AVE52,, —VV,satisfying Av(frp) =fAV172 foranyfunction fanddifferential form cp.Since AVisaderivation on theexterior algebra wehave AV(p:AV€aAlXA‘(p We can use (8.12) and (8.1.3) towrite the interior and exterior products interms ofClifford products, producing Avrp =§[A(,e" Aea, (pl+§iXuA (,e“(p —§(A(,e”cp"e,, +e,,rp’lA Ve”). The Clifford commutator isaClifford derivation. The 2-form Ave“ Ae,, canbewritten interms oftheexterior derivative ofI7.Since Av commutes with contractions andAvf =0forfe@(M), if{e“} and{Xa} are dual bases and A;/X,, =m,,"X,, for some matrix mnb then Ave” =—m,,”e”. Then Ave“ Ae, =——mb“e" Aen=make), Ae“, using theantisymmetry oftheexterior product, soAve“ Ae, =A_(7Y,,Ae”. Now AVX, E[V,X,,] —VvX,,, soifVistorsion free AvX,, =—VXuV, thus Ave“A6,,=e”AvTj/=6“Av,“V=av (by(4.14)). Theremaining terms intheexpression forAVingeneral prevent itfrom being aClifford derivation. Ifwritten interms ofthematrix ma”then onlythesymmetric partenters: §iXnA ve“<'p —j(AVe“cp"e, +e,,rp"A Ve“) =-imt."/P +i(mb.. +m..t)(@"f/>"@" +@“<P"@’) using theusual index-lowering convention. Since g(AvX,,, X,,)=m,,,, andAVcommutes withcontractions mnb +mba :_AVg(XuI Xb)' Themetric compatibility ofVenables ustowrite Avg =§£(,g and Awe=lidVite]+%§£v8(X.. X”)<t> —§§£vg(X,,, Xb)(e”(p"e“ +e”rp"e"). (9.4.1) Thus, asexpected, Av. and hence §£v. isaClifford derivation ifand only ifVisaKilling vector. LIE DERIVATIVES OF SPINOR FIELDS 273 If K is a Killing vector then the above simplifies to 2'0) = VicT (Pi (9.4.2) So if {ea} is an orthonormal co-frame we have, from (8.1.6) 2' Kea = [UK + dk, ea] (9.4.3) where 0K = liKwpgePq. Under Lie transport along the flow of an isometry an orthonormal frame undergoes an orthogonal transformation. The Lie derivative gives the infinitesimal transformation, representing the Lie algebra of the orthogonal group on the frame. Analogous to the way in which we introduced the covariant derivative we can define the Lie derivative on the associated standard spinor frame to be given by left multiplication by the element that appears in this commutator: that is WO; = (UK + If ji = 13,1p1 and l'iop = b,K(Ip') + Kb then, recalling the defini- tion of the covariant derivative, we have gKP = S + dkip. (9.4.4) Such a definition can (and will) be taken for the Lie derivative on spinors with respect to an arbitrary vector, but only in the case of Killing vectors is there a clear geometrical interpretation with Y having useful properties. When K is a Killing vector then, like SK, K satisfies a 'Leibnitz' property: K(0) = Kav + aZ KV (9.4.5) This follows from (9.4.2) and (9.3.7). If ip is the spinor adjoint to tp, with respect to any spin-invariant product, then for K Killing Zic(Ti) = K92ip + cl)g (9.4.6) If the Lie derivative is written using (9.4.2) then this follows from the analogous property of Sx, (9.3.16). Equations (6.13.13) and (6.13.14) give the commutator of a Lie derivative with a covariant derivative. We now obtain the analogous expression for the spinor operators. This will be useful for examining the covariances of spinor equations in the next chapter. Straight from the definition we have [IK,Sv] — SiKy] = S(K,V) — 1V vdk. The curvature of S is given in (9.3.20), and V vdk can be expressed as in (6.13.9) to give [YK,Sv] — S[Kyi = —,IVx,Yicg(V,X„)eba. (9.4.7) LIEDERIVATIVES oFSPINOR FIELDS 273 IfKisaKilling vector then theabove simplifies to .2/:,<¢>=vK¢>+[g<11?.¢>]. (9.4.2) Soif{e“} isanorthonormal co-frame wehave, from (8.1.6) §EKe” =[OK+§dK, e“] (9.4.3) where 0K=§iKa)pqeP‘?. Under Lie transport along theflow ofan isometry anorthonormal frame undergoes anorthogonal transformation. The Liederivative gives theinfinitesimal transformation, representing theLiealgebra oftheorthogonal group ontheframe. Analogous tothe way inwhich weintroduced thecovariant derivative wecandefine the Liederivative ontheassociated standard spinor frame tobegiven by leftmultiplication bytheelement that appears inthiscommutator: that is ea),=(OK+§dK)b,-. If1/1: b,-1/1" and$K1/1 =b,»K(17/') +$Kb,-1/1" then, recalling thedefini- tionofthecovariant derivative, wehave 2,4»=sK¢+(aka. (9.4.4) Such adefinition can (and will) betaken fortheLiederivative on spinors with respect toanarbitrary vector, butonly inthecase of Killing vectors isthere aclear geometrical interpretation with ..‘3~Ehaving useful properties. When KisaKilling vector then, like SK,¥Ksatisfies a‘Leibnitz’ property: .¥K(a1/2) E§EKa1/2 +a.¥K1/2. (9.4.5) This follows from (9.4.2) and (9.3.7). If1/1isthespinor adjoint to1/1, with respect toanyspin-invariant product, then forKKilling §£K((i0w) IgK(i01/7 +<P~gKl//- (9~4~6) IftheLiederivative iswritten using (9.4.2) then thisfollows from the analogous property ofSX,(93.16). Equations (6.13.13) and (6.13.14) give thecommutator ofaLie derivative with acovariant derivative. Wenow obtain theanalogous expression forthespinor operators. This willbeuseful forexamining thecovariances ofspinor equations inthenext chapter. Straight from thedefinition wehave igtosvl _S[K,V] =S(K,V) -iVvdk- The curvature ofSisgiven in(93.20), andVvdK canbeexpressed as in(613.9) togive [gK,Sv] — S[K‘\/I = —§Vxh$Kg(V,Xa)€b". 274 SPINOR FIELDS For the special case of K a conformal Killing vector with Kg = t€ (M) (9.4.8) the above simplifies to [WK,Sv] — Spcm — A V. (9.4.9) We can use the commutator of the Lie derivative with a covariant derivative to evaluate the commutator of two Lie derivatives, Ex YPP 1[x, 1111) = [x' x(crf — xV W[X, 1111P. From (9.4.1) (d p) — d -17Zxtp = — xg(X,„ Xa)d -kip + xg(X a, X b)(ebd -17 ea + ea d- eb)lp and since ebdi- ea + ea di7 eb = 2gabd-f — 2(ea A id eb A ird -17) then ,Txg(Xa, X b)(eb d ea + ead-fieb) = 1ff xg(Xa, r)di 7 — xg(Xa, Xb)ea A d . It follows from the definition of i7 that - x-f T xY + xg(Y, X a)ea Since the Lie and exterior derivatives on differential forms commute = d[X, Y] + d(Y xg(Y, Xa)ea) = d[Î1 + vx„Yxg(Y,x„)eba + Yxg(vxhy,xa)eba thus Wx(d-ftp) — di-4)N — d[X, Y]tp = VxhIxg(Y,Xa)e ba/P + 2xg(Vx,Y,Xa)e balp A id î. Returning now to the commutator of the Lie derivatives we use (9.4.7) to obtain [Wx, *yi gjx, yj 1-Txg(Vx,Y, Xa)e ba — xg(X a, X b)ea AiXbdY. The right-hand side may be simplified so as to exhibit explicitly the antisymmetry in X and Y: ea A ixbd î = ea A Vo i — ixb V x,V eac 274 SFINQR FIELDS Forthespecial case ofKaconformal Killing vector with §EKg =2/lg /1E@(M) (9.4.8) theabove simplifies to [.2eK,sv] -sum=-§d1,( 7 (9.4.9) Wecanusethecommutator oftheLiederivative with acovariant derivative toevaluate thecommutator oftwoLiederivatives, igx» 31/11/-’ -g(x,1'11/-’ _ _ 1 ~_1 ~ _1 '“"—l¥X.SYlw SIX.Yiw+.¥X(dYw) ..dY$Xw ..d[X.Ylw- From (9.4.1) §Ex(dY1P)_ dygxi/l =i7Xd Y1!’_iigxgixw Xald Y1/l L .2 .2+8§EXg(X,,, Xb)(ebd Ye“ +e“dYeb)1p andsince ebdYe“+e“dKeb=2g“b<1I7 -2(@“,(i,,4d'Y +eb,(IX.dI7) then 1 -V -4.,8§EXg(X,,, X,,)(ebdYe“ +e“dYe”) :%:£Xg(Xar Xa)d? _%‘5£Xg(Xa: Xb)ea Itfollows fromthedefinition ofYthat -V ,—~_¢ ZXY —.§EXY +§EXg(Y, X,,)e“. Since theLieandexterior derivatives ondifferential forms commute -V »\_,.§£XdY -d[X,Y] +d(.§£Xg(Y, X,,)e“) =d[XTY] +VXh§£Xg(Y,X,,)e"“ +.§£Xg(VXhY,X,,)eb“ thus N Ad endY1»)—dY¥Xw —d[X,Ylw =VX,§5Xg(Y,X@)@b"1P +$Xg(VXbY1Xa)eba1/J _%§EXg(Xn1Xb)ea /\ Returning now tothecommutator oftheLiederivatives weuse(9.4.7) toobtain i§EX1§EY]_ g(x,Y]=i~5£X8(VX,,Y» X11)?“ _ri§£x8(Xm Xi)?“ /\ix*dY- The right-hand side may besimplified soastoexhibit explicitly the antisymmetry inXandY: e“AiX»dl:7' =e“AV,\/4? —ixbVXtKe"‘ LIE DERIVATIVES OF SPINOR FIELDS 275 SO 2.Yxg(V XQ)eb° — .Txg(X„, Xb)ea A id = —Yxg(Xa, Xh)ixb V,c,-feac —2xg(X a, Xb)ix,V Use of Killing's equation, (6.13.3), produces the final result [gx, vl — *fx,Y1 = —14'xg(Xa, Xb)2 yg(Xb, Xe)eac. (9.4.10) If either X or Y is conformal Killing then the right-hand side vanishes. Exercise 9.2 Show that if {K,} is an algebra of Killing vectors in flat space then [Idk„ Idk i] = KJ]. Hint: write out the commutator of two spinorial Lie derivatives in terms of the curvature of S. 9.5 Representing Spinor Fields with Differential Forms When M is even dimensional we can take as spinor bundle any bundle carrying an irreducible representation of the real Clifford bundle C(M). For the special case in which M is topologically En with a flat pseudo-Riemannian metric then we have a spinor sub-bundle of the Clifford bundle. Let {ea} be a global parallel orthonormal co-frame. Then for some choice of constant y-matrices there is a global matrix basis {e11} for Clifford forms such that e° = )/e u. Elements of this matrix basis can be written as Clifford polynomials of the parallel co-frames with constant coefficients, and so are parallel. Then l(M) is a spinor sub-bundle of C(M) if the fibres of 4(M) are the minimal left ideals spanned by {e11}. Sections of J(M) (spinor fields) are in- homogeneous differential forms. The pseudo-Riemannian connection V induces a connection on J(M). In fact this is easily seen to be the spinor covariant derivative, generally denoted S, for this particular spinor bundle. We can of course always choose non-parallel co-frames, say ea = séas-1 for se with Vxe° = [axe] for Ox =V xss-1. The corresponding standard spinor basis is {b, = se ,i) satisfying V xbr = ub 1. If T denotes the involution of transposition in the matrix basis {e,,} and C is the Clifford element such that ci;'=g = CaTC-I then a spin- invariant product on sections of J(M) is given by (99, 1P) = J0(C-1(P'/V). (9.5.1) Notice that the 0-form projector J o gives a product with values in the LIEDERIVATIVES 0FSPINOR FIELDS 275 so 2:£,,g(vX,Y, X,)e"“ -.§£Xg(X,,, Xb)e“AiX1-dY =—5£x8(X@» X1119.»VX,Ye"—§tXg<X... Xb)iX,VX* Ye“- UseofKilling’s equation, (6.13.3), produces thefinal result igx, 31'] _g[X,Y] =_:i§£X8(X@» Xb)$Yg(Xb» Xe)?“ (9-4-10) Ifeither XorYisconformal Killing then theright-hand sidevanishes. Exercise 9.2 Show thatif{K,-}isanalgebra ofKilling vectors inflatspace then Hint: write outthecommutator oftwospinorial Liederivatives interms ofthecurvature ofS. 9.5Representing Spinor Fields with Differential Forms When Miseven dimensional wecantake asspinor bundle anybundle carrying anirreducible representation oftherealClifford bundle C(M). For the special case inwhich Mistopologically 1P1"with aflat pseudo-Riemannian metric then wehave aspinor sub-bundle ofthe Clifford bundle. Let {é"} beaglobal parallel orthonormal co-frame. Then forsome choice ofconstant y-matrices there isaglobal matrix basis {e,-I-} forClifford forms such that é”=yf-j-e,-I-. Elements ofthis matrix basis can bewritten asClifford polynomials oftheparallel co-frames with constant coefficients, andsoareparallel. Then 9(M) isa spinor sub-bundle ofC(M) ifthefibres of9(M) aretheminimal left ideals spanned by{en}. Sections of9(M) (spinor fields) are in- homogeneous differential forms. The pseudo-Riemannian connection V induces aconnection on9(M). Infact this iseasily seen tobethe spinor covariant derivative, generally denoted S,forthis particular spinor bundle. Wecanofcourse always choose non-parallel co-frames, saye”=sé“s“ forse+1“, with Vxe” =[oX,e”] forox=VXss‘1. The corresponding standard spinor basis is{b,-=se,1} satisfying VXb,-=0Xb,-. IfTdenotes theinvolution oftransposition inthematrix basis {e,-I-} and CistheClifford element such that as"=CaTC" then aspin- invariant product onsections of9(M) isgiven by (911/1)=5f@(C“<P§”1/1)- (9-5»1) Notice that the0-form projector S11,gives aproduct with values inthe 276 SPINOR FIELDS real numbers rather than the isomorphic algebra with e n as identity. For the special spinor bundle here this product accords with the general prescription of §9.2. Although for this particular spinor bundle the connections S and V coincide there is still a need to distinguish W' K from 2K. For K a Killing vector these are seen, using (9.4.2), to be related by Kip = YKip + lipak (9.5.2) The Lie derivative K does not induce an operator on the sub-bundle g(M): it does not preserve the minimal left ideals. The addition of the second term ensures that W K/p E J11(M) for all E F3(M). In the above we showed how in flat space we had a spinor sub-bundle of the Clifford bundle. This is a very special situation. In general a manifold can admit a spinor structure without the Clifford bundle having a spinor sub-bundle. The following exercise illustrates this point. Exercise 9.3 (i)Let I be any minimal left ideal of C2.0(1F1). Show that there is a unique vector a such that va = i , Vip E I. Hint: Take an orthonormal frame {e 1,e2} and construct a matrix basis using P, = ;(1± e'). Then if /0 = C2.0(l1:1)P, then I = I0S for some invertible S. Expand S in the previously constructed matrix basis and explicitly construct the a such that Pf Sa = P,S. (ii)Argue that the real Clifford bundle of a two-dimensional sphere does not contain a spinor sub-bundle of minimal left ideals (since there is no non-vanishing vector field on a sphere). The sphere does, however, admit a spinor structure. We have emphasised that we cannot in general find a spinor sub- bundle of the Clifford bundle, and thus cannot in general identify spinor fields with certain differential forms. However, we can if we wish always do this locally. For each open neighbourhood U, of M we can choose a local basis for the Clifford algebra {e, (;')Q,a)}. The local matrix frame {eV} commutes with the basis {On for the division algebra. On Ucrfi there is a local Clifford form S () such that er = SuWeV(S( 13(0)-1 and Q0/3) = sokoQq")(s(0 ,),-1. If /() is the minimal left ideal spanned by the first column of e(;) and D is the 'standard' division algebra with basis {qk} then /0) is a right D-module with the rule e;),qk —=e;VV,a ). If we can choose the Su3a) coherently, that is S (a13) 5.(13r) = S(") on Uor then we can define an equivalence relation between Pa) and /(0) on Uap to form a spinor bundle. Thus the 5(0) can be chosen coherently if and only if M is a spin manifold. If this is the case then for Ipp(`') E ipta) and E , p,q E Uo we define the equivalence relation by 03) iff p = q and cp,((') p(')(S(13a))-1 . (9.5.3) 276 SPINOR FIELDS real numbers rather than theisomorphic algebra with enasidentity, Forthespecial spinor bundle herethisproduct accords withthegeneral prescription of§9.2. Although forthisparticular spinor bundle theconnections SandV coincide there isstillaneed todistinguish é.-€Kfrom SEX.ForKaKilling vector these areseen, using (9.4.2), toberelated by .§£,<171=sew+gwak. (95.2) TheLiederivative §EKdoes notinduce anoperator onthesub-bundle .9>(M): itdoes notpreserve theminimal leftideals. Theaddition ofthe second term ensures that.§£,<171 eI‘.Sl>(M) forall171eI".Sl>(M). Intheabove weshowed how inflatspace wehadaspinor sub-bundle oftheClifford bundle. This isavery special situation. Ingeneral a manifold canadmit aspinor structure without theClifford bundle having aspinor sub-bundle. Thefollowing exercise illustrates thispoint. Exercise 9.3 (i)LetIbeanyminimal leftideal ofC2_(,(lB). Show that there isa unique vector asuch that 171aE171,V1716 I.Hint: Take anorthonormal frame {e‘,e3} andconstruct amatrix basis using P:E§(1ie‘).Then ifIOEC3_(,(lB)P,. then IEIOSforsome invertible S.Expand Sinthe previously constructed matrix basis and explicitly construct theasuch thatP,Sa =P+S. (ii)Argue that thereal Clifford bundle ofatwo-dimensional sphere does notcontain aspinor sub-bundle ofminimal leftideals (since there isnonon-vanishing vector field onasphere). The sphere does, however, admit aspinor structure. Wehave emphasised that wecannot ingeneral find aspinor sub- bundle oftheClifford bundle, andthus cannot ingeneral identify spinor fields with certain differential forms. However, wecanifwewish always dothislocally. Foreach open neighbourhood U,ofMwecanchoose a local basis fortheClifford algebra {e§f“‘Q)“l}. The local matrix frame {eifl} commutes with thebasis {Q‘k“l} forthedivision algebra. OnUafl there isalocal Clifford form Si“/3* such thateif”ES‘”°le§f’l(S(/3“l)'1 and Q)?”=S‘/3“’Qi.“'(S(/3"l)‘1. IfIi“)istheminimal leftideal spanned bythe first column ofeff”and Disthe‘standard’ division algebra with basis {qk} then Ii“)isaright D-module with theruleef-§”q(. Ee§§"Qi.°’. Ifwe canchoose theSill“) coherently, that isS“"/“Si/31’) ESW’ onU,,j,./, then wecandefine anequivalence relation between I9”andI‘/‘lonU“), to form aspinor bundle. Thus theSW” canbechosen coherently ifand only ifMisaspin manifold. Ifthisisthecase then for171j,“"e 1),“)and gaff‘6If‘.p,q6U“),wedefine theequivalence relation by 141*~<1:/1 iftp-4and<11?=1;."*<st#~>>*'. 19.5.3) REPRESENTING SPINOR FIELDS WITH DIFFERENTIAL FORMS 277 The resulting equivalence classes of differential forms form a bundle. On U. we may represent a section of this bundle by a differential form lying in the minimal left ideal Pa"), on Ufi we may choose a representa- tive form in /(g), these being related on Uo by the above relation. If a is an arbitrary Clifford form and q E D then for cp(I3) Ip(co we have acp(13)q atp(")q so, indeed, this bundle is a spinor bundle, carrying an irreducible representation of the Clifford bundle with a D-linear struc- ture. Although sections of this bundle are not differential forms, but rather equivalence classes of local differential forms, we may represent local sections with any differential form in the class. However, the connection V does not induce a connection on this bundle (in general). The pseudo-Riemannian connection will not preserve the minimal left ideals 10), and we need to distinguish between it and the spinor connection S. Although it can be convenient to represent a spinor field locally by a differential form this can never be more than a matter of taste. Given that the spinor bundle carries an irreducible representation of the Clifford bundle we can define spin-invariant products, covariant differ- entiation etc, and the properties of these do not depend on how we choose to represent spinor fields. Bibliography Geroch R P 1967 J.Math.Phys. 8 782 1968 J.Math.Phys. 9 1739 1970 J.Math.Phys. 11 11 Greub W and Petry H R 1978 Lecture Notes on Mathematics vol 675 (Heidel- berg: Springer) Isham C 1978 Spinor fields in 4-dimensional space—times Proc.R.Soc. A 364 591 Kosman Y 1971 Annuli di Matematica 25 317-95 Lee K K 1973 General Relativity and Gravitation vol 4 p 421 Penrose R and Rindler W 1984 Spinors and Space—Time vol 1,2 (Cambridge: Cambridge University Press) Petry H R 1984 Spin Structures on Lorentz Manifolds, Trieste ISAS-44184 Pressley A and Segal G 1987 Loop Groups (Oxford: Oxford University Press) REPRESENTING SPINOR FIELDS WITH DIFFERENTIAL FORMS 277 The resulting equivalence classes ofdifferential forms form abundle. OnU,wemay represent asection ofthisbundle byadifferential form lying intheminimal leftideal Ii“), onU5wemay choose arepresenta- tiveform in1(5), these being related onU073bytheabove relation. Ifa isanarbitrary Clifford form and qEDthen for(pip)~171"’) wehave acplfllq ~a171‘“lq so,indeed, thisbundle isaspinor bundle, carrying an irreducible representation oftheClifford bundle with aD-linear struc- ture. Although sections ofthis bundle arenotdifferential forms, but rather equivalence classes oflocal differential forms, wemay represent local sections with any differential form intheclass. However, the connection Vdoes notinduce aconnection onthisbundle (ingeneral). The pseudo-Riemannian connection will notpreserve theminimal left ideals Ii“), and weneed todistinguish between itand the spinor connection S. Although itcanbeconvenient torepresent aspinor field locally bya differential form thiscannever bemore than amatter oftaste. Given that the spinor bundle carries anirreducible representation ofthe Clifford bundle wecandefine spin-invariant products, covariant differ- entiation etc,andtheproperties ofthese donotdepend onhow we choose torepresent spinor fields. Bibliography Geroch RP1967 J.Math.Phys. 8782 E 1968 J.Math.Phys. 91739 E1970J.Math.Phys. 1111 Greub WandPetty HR1978 Lecture Notes onMathematics vol675(Heidel- berg: Springer) Isham C1978 Spinor fields in4-dimensional space—ti1nes Proc.R.Soc. A364591 Kosman Y1971 Annali diMatematica 25317-95 LeeKK1973 General Relativity andGravitation vol4p421 Penrose RandRindler W1984 Spinors and Space—Time vol1,2(Cambridge: Cambridge University Press) Petry HR1984SpinStructures onLorentz Manifolds, Trieste ISAS-44/84 Pressley AandSegal G1987 Loop Groups (Oxford: Oxford University Press) 10 Spinor Field Equations 10.1 The Dirac Operator The Dirac operator gets its name from its appearance in Dirac's wave equation for the electron. It is now usual to extrapolate the nomen- clature from this spacetime setting to mean by Dirac operator any operator of the form of that occurring in Dirac's wave equation. There is no clear concensus on how far this extrapolation is to go. We shall use the terminology as follows: if Sx denotes covariant differentiation with respect to X of sections of a bundle carrying an irreducible represent- ation of the (real or complexified) Clifford bundle then the Dirac operator on sections is $ eaSx,. The co-frame {ea} is dual to the arbitrary tangent frame {X„}. Sometimes mathematicians use the terminology more liberally to mean by Dirac operator any operator of the above form where S- is any covariant derivative on sections of a bundle carrying any representation of the Clifford bundle. We will mostly be concerned with the Dirac operator on sections of a spinor bundle with the covariant derivative Sx of §9.3. The Dirac equation for a complex spinor field tp is SIP = PIP (10.1.1) where y is a complex constant. The nature of the manifold may restrict the eigenvalue y to certain real or imaginary values. In other cases we may only be interested in real or imaginary eigenvalues for physical reasons. If ST) is the standard spinor covariant derivative of §9.3.1 and A is a U(1) connection 1-form then a U(1)-covariant spinor derivative is given by SOp = ST )tp + qiA(X)tp (10.1.2) where q is the 'charge' coupling constant. The original equation of Dirac involved such a U(1)-charged covariant derivative Spinor Field Equations 10.1 TheDirac Operator The Dirac operator gets itsname from itsappearance inDirac’s wave equation fortheelectron. Itisnow usual toextrapolate thenomen- clature from this spacetime setting tomean byDirac operator any operator oftheform ofthat occurring inDirac’s wave equation. There isnoclear concensus onhow farthisextrapolation istogo.Weshall use theterminology asfollows: ifSXdenotes covariant differentiation with respect toXofsections ofabundle carrying anirreducible represent- ation ofthe(real orcomplexified) Clifford bundle then the Dirac operator onsections is,$Ee“SXa. The co-frame {e“} isdual tothe arbitrary tangent frame {X,,}. Sometimes mathematicians use the terminology more liberally tomean byDirac operator anyoperator of theabove form where SXisanycovariant derivative onsections ofa bundle carrying any representation oftheClifford bundle. We will mostly beconcerned with theDirac operator onsections ofaspinor bundle with thecovariant derivative SXof§9.3. The Dirac equation foracomplex spinor field 171is flip=71171 (10.1.1) where 74isacomplex constant. The nature ofthemanifold may restrict theeigenvalue utocertain realorimaginary values. Inother cases we may only beinterested inreal orimaginary eigenvalues forphysical reasons. IfS)?’isthestandard spinor covariant derivative of§9.3.1 and AisaU(1) connection 1-form then aU(l)-covariant spinor derivative is given by Sf\9l171 ES)?)171 +qiA(X)171 (10.1.2) where qisthe‘charge’ coupling constant. The original equation of Dirac involved such aU(l)-charged covariant derivative THE DIRAC OPERATOR 279 Exercise 10.1 Show that S(q)(X, Y)/p = So)(X, Y)tp + iqi xi yFlp where F = dA. In even dimensions the spinor representation of the complexified Clifford algebra induces a reducible representation of the even sub- algebra. If is proportional to the volume form with 2 = 1 then a complex spinor tp is reduced into 'Weyl' spinors lp-± carrying irreducible representations of the even subalgebra by tp± = ± p. (10.1.3) The projectors (1 ± Z) anticommute with members of the co-frame {ea} and are parallel. So if tp satisfies a massless (/.4 = 0) Dirac equation then so do the Weyl spinors ip-±. Such massless equations for the Weyl spinors are known in physics as Weyl equations. Spinors of the real Clifford algebras can also be subjected to the Dirac equation (10.1.1) (with i real). For signature (p, q) satisfying p — q = 0, 2 mod 8 the real Clifford algebra is a total real matrix algebra and the spinors are known in physics as Majorana spinors. In this case the Dirac equation may be known as a Majorana—Dirac equation. (Although the eigenvalue ,u in (10.1.1) can be taken to be any real constant such an equation can not be obtained from a variational principle. Without recourse to `anticommuting' parameters a variational principle will only give a Majorana—Dirac equation with zero eigen- value.) As we remarked at the beginning of §9.5, for the special case of a flat parallelisable manifold the Clifford bundle contains a spinor sub-bundle of minimal left ideals. The pseudo-Riemannian connection V induces the spinor covariant derivative on this sub-bundle. Thus in this case the operator 0, restricted to sections of this spinor sub-bundle, is a Dirac operator on spinor fields. One of Dirac's requirements for his equation for the electron was that the components of the field should satisfy a Klein—Gordon equation. As we have just noted above the operator 0, which squares to the Laplace—Beltrami operator, induces a Dirac operator on spinor fields in flat space. So this Dirac operator squares to the Laplace-Beltrami operator, acting on differential forms in the spinor sub-bundle. More generally, the square of the Dirac operator is known as the spinor Laplacian. We have = easx jebsxm = ftlebSx„V .(eaeb + ebea)S,G.SxhIP + -(e aeb ebea)Sx„Sx„IP = ftleaS + S x„S +-12eab[Sx, Sx,,11,0 xolp = 0eaS x:ti) + S x,S x. 11) + leabSix„. THE DIRAC OPERATOR 279 Exercise 10.1 Show thatS”?’(X, Y)171 ES‘°l(X, Y)171 +iqiXiyF171 where F=dA. Ineven dimensions thespinor representation ofthecomplexified Clifford algebra induces areducible representation oftheeven sub- algebra. IfEisproportional tothevolume form with E2=1then a complex spinor 171isreduced into‘Weyl’ spinors 1711'carrying irreducible representations oftheeven subalgebra by 171*E§(liE)171. (10.1.3) Theprojectors §(11“E)anticommute with members oftheco-frame {e“}andareparallel. Soif171satisfies amassless (71E0)Dirac equation thensodotheWeyl spinors 171?Such massless equations fortheWeyl spinors areknown inphysics asWeyl equations. Spinors oftherealClifford algebras canalsobesubjected tothe Dirac equation (10.1.1) (with Itreal). Forsignature (p,q)satisfying p—q=0,2 mod8 thereal Clifford algebra isatotal real matrix algebra andthespinors areknown inphysics asMajorana spinors. In thiscase theDirac equation may beknown asaMajorana—Dirac equation. (Although theeigenvalue 71in(10.1.1) canbetaken tobeany realconstant such anequation cannotbeobtained from avariational principle. Without recourse to‘anticommuting’ parameters avariational principle willonly give aMajorana—Dirac equation with zero eigen- value.) Asweremarked atthebeginning of§9.5, forthespecial caseofaflat parallelisable manifold theClifford bundle contains aspinor sub-bundle ofminimal leftideals. Thepseudo-Riemannian connection Vinduces the spinor covariant derivative onthissub-bundle. Thus inthiscase the operator 71,restricted tosections ofthisspinor sub-bundle, isaDirac operator onspinor fields. OneofDirac’s requirements forhisequation fortheelectron wasthat thecomponents ofthefield should satisfy aKlein—Gordon equation. As wehave just noted above theoperator 71,which squares tothe Laplace—Beltrami operator, induces aDirac operator onspinor fields in flatspace. SothisDirac operator squares totheLaplace—Beltrami operator, acting ondifferential forms inthespinor sub-bundle. More generally, thesquare oftheDirac operator isknown asthespinor Laplacian. Wehave (5)21/’ =@“5x,(@b5x,1l1) =;zlebSXb171 +§(e”e" +ebe”)SXaSXh171 +§(e”e" —e"e")SXflSXh171 =¢eaSx,,1/l +Sx,,Sx"1/’ +ieablSx,»Sx,,l1l’ =I4e”Sx,1l’ +Sx,SX"1/’ +i@abS(XmXt>)1/l +ie”bS[x,,.x,]1l’- 280 SPINOR FIELD EQUATIONS Now [X,, XI)] = i xni xhcle` X c, and so .1e"bS1x xhitp = _decS xp. gives = ix.Vx.ebSxhip + SkS,Hp + leabS(X a, Using (9.3.20) the curvature operator of S can be written in terms of the curvature 2-forms to give ,leabS(X„„ Xb)tp = Rpqe4." From (8.1.17) we have, for zero torsion, R ede'd = the curvature scalar, and so $2v = (sx. + ix,,vxbea)sx,ip — (10.1.4) Exercise 10.2 Analogously express the Laplace—Beltrami operator as 020 = (V x. ix,Vxbea)Vx(13 — ,14.*:1) — ,t4Recicriec1. 10.2 Covariances of the Dirac Equation and Conserved Currents Generally we expect equations formulated on pseudo-Riemannian mani- folds to have a covariance corresponding to any isometries. For exam- ple, in §5.4 we showed how the Lie derivative with respect to a Killing vector maps solutions to Maxwell's equations into new solutions. In the same way we may use the Lie derivative on spinors to obtain new solutions to the Dirac equation in spaces with isometries. For a vector field K we have wK$ = (vo. + l[dk, eaDsx, + eagok. If now K is a conformal Killing vector, with Kg = 24, then for A any 1-form 1 KA = V KA + k, A] + AA. This follows from (9.4.1) and the observation that for X,, a p-form eaXpea = (n — 2p)X p" (10.2.1) SO KS = Kea S — A$ + ea KS x„ = Kea S — + + ea S [K, — ea (CIA. A e a) by (4.4.9). Since K(ea (X b)) = 0 then Y iceaSx, + eaSIK, 0, and ea(dA A e) = ea A (a. A ea) + ia(dA A ea) = X a(.)ea — = (1 — n)(0. so [WK, = — ;(1 — n)dA. Since $(4) = dAv + APtp this may be 280 SPINOR FIELD EQUATIONS Now [X,, X,,]EiX“iXhde‘X,. andso %€ahS[X/Iyxhjw =_d€CSX(w. gives 5'21!’=ix"Vx,@h5x,ll’ T5x,5x'll’ +i@"bS(Xm Xblllh Using (9.3.20) thecurvature operator ofScanbewritten interms of thecurvature 2-forms togive ieabS(Xa1Xb)lf’ :~iRpqepqlf" From (8.1.17) wehave, forzero torsion, Rcdefd E-971, thecurvature scalar, andso 14°11=(S11+1X,vX»@~>sX.w -19111 <19-1.41 Exercise 10.2 Analogously express theLaplace—Beltrami operator as 712(1)=(vx.+I,,,vX1@")vX,<I> -gate-gR.,,<I>@~’. 10.2 Covariances oftheDirac Equation andConserved Currents Generally weexpect equations formulated onpseudo-Riemannian mani- folds tohave acovariance corresponding toanyisometries. Forexam- ple,in§5.4 weshowed how theLiederivative with respect toaKilling vector maps solutions toMaxwell’s equations into new solutions. Inthe same way wemay usetheLiederivative onspinors toobtain new solutions totheDirac equation inspaces with isometries. Foravector field Kwehave ems=(VKe"+§[dK,e"])SX0 +e".¥KSXn. Ifnow Kisaconformal Killing vector, with $Kg E2kg, then forAany 1-form $KA EVKA +§[dK, A]+AA. This follows from (9.4.1) and theobservation thatforX7,ap-form e,,X,,e“ =(n—2p)X]l (10.2.1) so .%K,$ E$Ke"SXa —A7?+e”.%KSXn Ei’Ke“SXu —All+,$i’K +e”S]K_Xn] —§e“(d}tAe,,) by(4.4.9). Since i’,(»(e“(X,,)) E0then $Ke”SXa +e“S]K,Xn] E0.and e“(d}.Ae,,) Ee”A(d}.Ae,,) +i"(d}.Ae,,) EXa(}t)e" —nd}.E(1—n)d}t so[gm ,8]E—A,$—§(1—n)d}.. Since ,$(A171) EdA171 +1,3171 thismay be COVARIANCES OF THE DIRAC EQUATION 281 written as IK + (n — = (10.2.2) If K is a Killing vector (A = 0) then K commutes with the Dirac operator and if tp satisfies the Dirac equation (10.1.1) then so does ZIA). For the massless case (p. = 0) we also have a covariance for K a conformal Killing vector: if S'tp = 0 then $[WK + 1(n — 1)41p = 0. Out of any two solutions to the Dirac equation we may construct a closed (n — 1)-form. For definiteness we take ( , ) to be a Hermitian- symmetric product on complex spinors with as adjoint involution. Then Re( , ) is a real-valued symmetric product. If we express an (n — 1)-form j as j = jaeaz, with z the volume n-form, then dj = eh A VxjlaeaZ) = eb A (VX,,jaeaZ jaVxbea(X,)ecZ). Now eb A (eaz) = eb A irz = A z) gabz = gabz, SO = (Via ixbVxbeaja)Z. (10.2.3) Taking j = Re(tp, eacp)eaz (10.2.4) gives = Re(S)op, ea(p)z + Re(tp, Scp)z = —Re($ip , yo)z + Re(tp, $cp)z where the covariant derivative S' is compatible with the spinor product. (This covariant derivative could contain a U(1) coupling.) Thus if Stp = pip, for II real, and similarly for 97, then dj = 0. In this way we obtain a conserved current (a closed (n — 1)-form) from any pair of solutions to the field equations. (Had we taken a spinor product with as adjoint involution then the form j would be closed for spinors satisfying the Dirac equation for an imaginary eigenvalue.) If ip is the adjoint to p with respect to the Hermitian-symmetric product then eacp)ea = )0(i p eacp)e° = 0(cpip ea)e° = )1(cpip). So the (n — 1)- form in (10.2.4) can be written as = Rei(cpip)z = *ReW i(cpip). (10.2.5) In particular, taking yo = iv) in (10.2.4) gives the U(1) current j = ie cop)eaz (10.2.6) This current would provide a source for the equation (such as Maxwell's equation) for any U(1) field entering into the spinor covariant deriva- tive. We now only consider the Dirac equation without a U(1) coupling. The presence of isometries, generated by a Killing vector K, ensures that if ip is a solution to the field equations then so is op. We thus have the associated closed currents COVARIANCES OFTHEDIRAC EQUATION 281 written as [§€K +§(n—1)}t,,8]E—}t,$. (10.2.2) IfKisaKilling vector (AE0)then £871 commutes with theDirac operator andif171satisfies theDirac equation (10.1.1) then sodoes £€K171. Forthemassless case (71E0)wealso have acovariance forKa conformal Killing vector: if$171=0then,$[§€K +§(n—1)lt]171 E0. Out ofanytwosolutions totheDirac equation wemay construct a closed (n—1)-form. Fordefiniteness wetake (,) tobeaHermitian- symmetric product oncomplex spinors with 517*asadjoint involution. Then Re(,) isareal-valued symmetric product. Ifweexpress an (n—1)-form 3as9Ej,,e"z, withzthevolume n-form, then dg:eh/\VX,,(l11e”Z) =eh/\(VX,,]'11e”Z +]11VX,,@”(Xe)@CZ)~ Now e"A(e“z) Ee"AiX1zE—iX1(e" Az)+g“"z Eg“"z, so d}E(Vxnja +iX1>VXbe"ja)Z. (10.2.3) Taking ,3»ERe(171, e,q1)e”z (10.2.4) gives <13=R¢(5x,1l1»@“‘P)Z +R¢(1/1. I$<1>)1=—R¢($1/1. <1>)Z+R90/1, I$<1>)1 where thecovariant derivative SXiscompatible with thespinor product. (This covariant derivative could contain aU(1) coupling.) Thus if $171E71171,for71real, andsimilarly for171,then d§»E0.Inthisway we obtain aconserved current (aclosed (n—1)-form) from any pair of solutions tothefield equations. (Had wetaken aspinor product with 5* asadjoint involution then theform }would beclosed forspinors satisfying theDirac equation foranimaginary eigenvalue.) IfJisthe adjoint to171with respect totheyHermitian-symmetric product then (1/1,@..<1>)@“ =9’@(1/»@1<1>)@" =9’@(<1>1//@..)@“ =9’1(<;vw)- $9the(H-1)- form in(10.2.4) canbewritten as 9=ReEf1(q1177)z =*Reff,(q1171). (10.2.5) Inparticular, taking q1=izpin(10.2.4) gives theU(l) current <9E(171,ie,,171)e”z. (10.2.6) This current would provide asource fortheequation (such asMaxwell’s equation) foranyU(l) field entering into thespinor covariant deriva- tive. Wenow only consider theDirac equation without aU(l) coupling. The presence ofisometries, generated byaKilling vector K,ensures that if171isasolution tothefield equations then sois£8,471. Wethus have theassociated closed currents 282 SPINOR FIELD EQUATIONS K = Re(V, eaWKIP)e aZ. (10.2.7) 10.3 The Dirac Equation in Spacetime In Chapter 5 Maxwell's theory of Electromagnetism was formulated in a Lorentzian spacetime. Together with relativistic mechanics this theory provides a good description of phenomena involving the electromagnetic interactions of charged matter. However, new phenomena sometimes occur (for example, when the energies involved in the interactions exceed certain critical values) that cannot be understood in terms of this theory. For instance, a faint green beam of light continues to liberate electrons from the surface of certain metals even when its intensity is reduced. Or, a strong magnetic field can be used to create pairs of particles. Furthermore, the very stability of atomic matter is not readily comprehensible in terms of a classical theory that predicts radiation from accelerating charged particles. For these and other reasons quan- tum mechanics was devised. Originally it provided an explanation of non-relativistic phenomena in domains in which classical mechanics was inadequate. The many-body version of this approach (in which the behaviour of a fixed but indefinite number of particles is accommo- dated) gave rise to a new formalism known as field quantisation. These methods were successfully extended to Maxwell's theory, in which the role of the classical field was replaced by some operator in an infinite- dimensional projective space of photon states. Historically it soon became clear that the classification of elementary particle types in Nature was intimately connected with the dynamical equations involving the respective field operators. Fields were clasified as bosons or fer- mions according to the observed behaviour of the respective many-body states. This classification was correlated according to whether they carried a representation of the rotation group SO(3) or its covering group SU(2). It was Dirac's famous equation for the electron—positron field that gave the impetus to the development of relativistic field quantisation and remains a cornerstone in the development of quantum field theory. As a single-particle theory (that is, where particle and antiparticle creation can be ignored to a first approximation) this equation gave a more accurate account of certain atomic spectra and the behaviour of electron beams in weak electromagnetic fields. Ingenious methods have since been invented to include the quantised radiation field in the theory. Some of the refined predictions of quantum electrodynamics provide examples of the most successful predictions in theoretical physics. 282 SPINOR FIELD EQUATIONS }KERe(171, e,,é€,;171)e”z. (10.2.7) 10.3 TheDirac Equation inSpacetime InChapter 5Maxwell’s theory ofElectromagnetism wasformulated ina Lorentzian spacetime. Together with relativistic mechanics this theory provides agood description ofphenomena involving theelectromagnetic interactions ofcharged matter. However, new phenomena sometimes occur (for example, when theenergies involved intheinteractions exceed certain critical values) thatcannot beunderstood interms ofthis theory. Forinstance, afaint green beam oflight continues toliberate electrons from thesurface ofcertain metals even when itsintensity is reduced. Or, astrong magnetic field canbeused tocreate pairs of particles. Furthermore, thevery stability ofatomic matter isnotreadily comprehensible interms ofaclassical theory that predicts radiation from accelerating charged particles. Forthese andother reasons quan- tum mechanics was devised. Originally itprovided anexplanation of non-relativistic phenomena indomains inwhich classical mechanics was inadequate. The many-body version ofthis approach (inwhich the behaviour ofafixed butindefinite number ofparticles isaccommo- dated) gave risetoanew formalism known asfield quantisation. These methods were successfully extended toMaxwell’s theory, inwhich the role oftheclassical field wasreplaced bysome operator inaninfinite- dimensional projective space ofphoton states. Historically itsoon became clear that theclassification ofelementary particle types in Nature wasintimately connected with thedynamical equations involving therespective field operators. Fields were clasified asbosons orfer- mions according totheobserved behaviour oftherespective many-body states. This classification was correlated according towhether they carried arepresentation oftherotation group SO(3) oritscovering group SU(2). Itwas Dirac’s famous equation fortheelectron-positron field that gave theimpetus tothedevelopment ofrelativistic field quantisation andremains acornerstone inthedevelopment ofquantum field theory. Asasingle-particle theory (that is,where particle and antiparticle creation canbeignored toafirst approximation) thisequation gave a more accurate account ofcertain atomic spectra andthebehaviour of electron beams inweak electromagnetic fields. Ingenious methods have since been invented toinclude thequantised radiation field inthe theory. Some oftherefined predictions ofquantum electrodynamics provide examples ofthe most successful predictions intheoretical physics. THE DIRAC EQUATION IN SPACETIME 283 Although it is beyond the scope of this book to enter into the realms of the quantum field theory of electrons and positrons it may be noted that such a formalism does require as an important ingredient a basis of solutions to the Dirac equation. These are put into correspondence with a basis of states used in the construction of the quantum theory. In Minkowski space a basis of such free-particle states may be labelled by the eigenvalues of a set of Lie derivatives with respect to a set of commuting Killing vectors. In recent years field theories on non-flat spaces have become in- creasingly relevant. We mention three examples. In order to study the behaviour of electrons in a superconducting toroid one must look at spinor fields on a space with a non-trivial topology. Phenomena assoc- iated with different types of boundary conditions on the electron field arise and may provide a geometrical interpretation of low-temperature electron states. Secondly, spinor fields on a dynamical string can be formulated in terms of a Dirac equation on a two-dimensional surface. Some believe that such a picture may underlie a viable model for all the basic forces in Nature. Finally we mention that in 1976 great excitement was generated by the construction of certain theories in which spin- fields were coupled to gravity in a manner that gave rise to new symmetries. Such supersymmetries were expected to ameliorate certain difficulties that arose when attempts were made to extend to gravitation the methods used to make successful quantum electrodynamical predic- tions. It is now thought that such effective-field theories are phenomenological remnants of a more general theory in which spinor fields in higher dimensions play a crucial role. In any phenomenological description of spinor fields and gravitation there is one aspect that deserves comment here. Although it is possible to construct a symmetric divergenceless stress tensor for a spinor field (this is given in the next section) it does not manifestly satisfy the positive-energy conditions mentioned in Chapter 7. This is analagous to the indefinite sign of the energy of a Dirac field in flat spacetime and is a reflection of the existence of antiparticle states in that case. This is one reason why a quantum interpretation is mandatory in order to give a cogent interpretation to Dirac's theory. In an arbitrary gravitational field, however, there is no natural way to define positive- and negative- energy states and the simple interpretational scheme used to interpret the quantum field theory in a flat space evaporates. It may be of course that the energy conditions are excessively restrictive when applied to spinor fields coupled to gravity, or that in a more fundamental theory of gravitation involving many fields no relevance should be attached to the stress properties of a single field. Although the resolution of this dilemma must await a more coherent synthesis of quantum field theory and geometry it is unlikely that the formulation and properties of spinor THE DIRAC EQUATION INSPACETIME 283 Although itisbeyond thescope ofthisbook toenter into therealms ofthequantum field theory ofelectrons andpositrons itmay benoted thatsuch aformalism does require asanimportant ingredient abasis of solutions totheDirac equation. These areputintocorrespondence with abasis ofstates used intheconstruction ofthequantum theory. In Minkowski space abasis ofsuch free-particle states may belabelled by theeigenvalues ofasetofLiederivatives with respect toasetof commuting Killing vectors. Inrecent years field theories onnon-flat spaces have become in- creasingly relevant. Wemention three examples. Inorder tostudy the behaviour ofelectrons inasuperconducting toroid one must look at spinor fields onaspace with anon-trivial topology. Phenomena assoc- iated with different types ofboundary conditions ontheelectron field arise andmay provide ageometrical interpretation oflow-temperature electron states. Secondly, spinor fields onadynamical string canbe formulated interms ofaDirac equation onatwo-dimensional surface. Some believe thatsuch apicture may underlie aviable model forallthe basic forces inNature. Finally wemention that in1976 great excitement was generated bytheconstruction ofcertain theories inwhich spin-§ fields were coupled togravity inamanner that gave rise tonew symmetries. Such supersymmetries were expected toameliorate certain difficulties thatarose when attempts were made toextend togravitation themethods used tomake successful quantum electrodynamical predic- tions. Itisnow thought that such effective-field theories are phenomenological remnants ofamore general theory inwhich spinor fields inhigher dimensions play acrucial role. Inanyphenomenological description ofspinor fields andgravitation there isoneaspect that deserves comment here. Although itispossible toconstruct asymmetric divergenceless stress tensor foraspinor field (this isgiven inthenext section) itdoes notmanifestly satisfy the positive-energy conditions mentioned inChapter 7.This isanalagous to theindefinite signoftheenergy ofaDirac field inflatspacetime andis areflection oftheexistence ofantiparticle states inthat case. This is onereason why aquantum interpretation ismandatory inorder togive acogent interpretation toDirac’s theory. Inanarbitrary gravitational field, however. there isnonatural way todefine positive- andnegative- energy states and thesimple interpretational scheme used tointerpret thequantum field theory inaflatspace evaporates. Itmay beofcourse that theenergy conditions areexcessively restrictive when applied to spinor fields coupled togravity, orthatinamore fundamental theory of gravitation involving many fields norelevance should beattached tothe stress properties ofasingle field. Although the resolution ofthis dilemma must await amore coherent synthesis ofquantum field theory andgeometry itisunlikely thattheformulation andproperties ofspinor 284 SPINOR FIELD EQUATIONS field equations on a manifold will cease to be important. The Dirac equation for a complex spinor (a Dirac spinor with unit charge) tp on spacetime is $tp + iittp = mtp (10.3.1) where we have explicitly exhibited the U(1) interaction with the electromagnetic 1-form potential A. The real eigenvalue m will be interpreted as a mass. The spinor field provides an electromagnetic current 1-form j, I = JI(iÇOi) (10.3.2) where is the spinor adjoint of tp with respect to the pseudo-Hermitian product whose adjoint involution is ri*. The Maxwell 2-form F = dA satisfies SF = j (10.3.3) with ô the co-derivative of (5.4.2). The electromagnetic current 1-form j is future-pointing and timelike for any spinor tp. The argument that this is so is algebraic. We first consider the charge density p = ietp). If we took the spinor adjoint as in (2.8.13) then the positivity of p would follow immediately. The fact that the spinor adjoint can be cast in this form follows ultimately from the positivity of the metric on the three-dimensional spacelike sub- spaces. It is instructive to argue the positivity of p directly from properties of the various spinor products. Let {ea} be a local orthonor- mal co-frame and ie 123 such that 22 = 1. Let u, be a spinor such that îu = EU, with E = ±1. Then u, carries a semi-spinor representa- tion of the subalgebra generated by {e', e2, e3}. Let ( ,) be the pseudo-Hermitian product associated with then (tie, = (Eîue, = e(u,, = ee'(u,, If a four-dimensional spinor tp is decomposed as tp = u, + u_ then = (u,, + (u u _)z*. We know from §2.7 that is the adjoint of a zero-index product on the semi-spinors of the three-dimensional subalgebra, whereas the product on four-dimensional spinors is of maximal index. Let us suppose that the product on four-dimensional spinors induces a positive-definite product on u, and a negative-definite product on u_. For three-dimensional semi-spinors we have (u,, = e(u,,iî..-`q*e°u,.) = = ee'(u f.) So the charge density p is diagonal in the three-dimensional semi- spinors: p = (u ±,ie"u „) + (u ie°u _). 284 SPINOR FIELD EQUATIONS fieldequations onamanifold willcease tobeimportant. The Dirac equation foracomplex spinor (aDirac spinor with unit charge) 1ponspacetime is ,$1p+iA1pEm1p (10.3.1) where wehave explicitly exhibited the U(l) interaction with the electromagnetic 1-form potential A.The real eigenvalue mwill be interpreted asamass. The spinor field provides anelectromagnetic current 1-form j, 7=9@¢$) (man where ifisthespinor adjoint of1pwith respect tothepseudo-Hermitian product whose adjoint involution is§17*. The Maxwell 2-form FEdA satisfies 8F=7 nos» with 6theco-derivative of(5.42). The electromagnetic current 1-form jisfuture-pointing andtimelike foranyspinor 1p.The argument that thisissoisalgebraic. Wefirst consider thecharge density pE(1p,ie°1p). Ifwetook thespinor adjoint asin(2.8.13) then thepositivity ofpwould follow immediately. Thefact thatthespinor adjoint canbecastinthisform follows ultimately from thepositivity ofthemetric onthethree-dimensional spacelike sub- spaces. Itisinstructive toargue the positivity ofpdirectly from properties ofthevarious spinor products. Let{e“} bealocal orthonor- malco-frame and 2Eiem such that 22E1.Letu,beaspinor such that Eu,Eeu,, with 8Ei1.Then u,carries asemi-spinor representa- tion ofthesubalgebra generated by{e‘,e2,e3}. Let (,)§., bethe pseudo-Hermitian product associated with5*then (u£1us')§* :(821451 ue')§* :E(ue> 2;:*ue’)§* :55I(ut-:1 ue')§>I<- Ifafour-dimensional spinor 1pisdecomposed as1pEu,+u_then (‘P1 1P).=* :(“+1 u+)§* T(“-1 M-)§*~ Weknow from §2.7that5*istheadjoint ofazero-index product onthe semi-spinors ofthethree-dimensional subalgebra, whereas theproduct onfour-dimensional spinors isofmaximal index. Letussuppose thatthe product onfour-dimensional spinors induces apositive-definite product onu,and anegative-definite product onu_. For three-dimensional semi-spinors wehave (us,ie°u,1) Ee(u£, i25"*e°u,1) Ee(u,, ie°2u,1) E1-:1-:’(u,, ie°u71). Sothecharge density pisdiagonal inthethree-dimensional semi- spinors: pE(u+, ie"u+) +(u_. ie°u_). THE DIRAC EQUATION IN SPACETIME 285 Now (u6, ie°146) = e(u e, = e(u e, zue). The volume 4-form z relates the products associated with and so that we have (146, ieuE) = E(u„ u,),r and p = (u t, ut). — (u u (10.3.4) Thus p is positive-definite or zero since the first product is positive- definite and the second negative-definite. To show that the charge density is positive-definite above we split the four-dimensional spinor into semi-spinors of the three-dimensional sub- algebra. This argument implies that g(j, V) is less than or equal to zero for all future pointing timelike vectors V and consequently that j must be a forward-pointing timelike or null vector field. It is instructive to rederive this result using the often useful Fierz rearrangement techni- que. To this end we will this time split the spinor into two semi-spinors of the even subalgebra. Let Ip± —= 1(1 ± iz)v, that is iztp± = ±V±, then (pe, atpe) = EE'(iztpe, aizvE") = EE'(pe, zazIpe) = — ee (.p,, (eve). So the components of j are diagonal in tp+ and 111—: ja = (V+, ieaV+) + (V+, ieaV+) ja+ (10.3.5) The norm of j, is given by = ea4'9(pE, ea46) = 1,76ea/Pe ifeaVe. Using (10.2.1) 4 eatpEe a = E(4 — 2p)(-1)P9 7p(pEir). p =0 Now op' 7pEy) = iztpe eiz = —inpfiztic = Tp-E and so only odd p enter into the sum. We have = i(Vizipt)iz = thus = —21,7EYIWiE)VE 2E/VEJI(iE)izVE = i'ji0PETIV-91PE = 0(1/ 6e )p e ea = —4(4)6 eaVE)(VE eaV) = So j,_ and j_ are both null and, since p 0, future pointing. Since the sum of two future-pointing null vectors lies in or on the forward light cone the current j is future pointing, timelike or null. Exercise 10.3 Consider the 1-form of (10.3.2) on an arbitrary even-dimensional Lorentzian manifold (not necessarily four dimensional). Show that the THEDIRAC EQUATION INSPACETIME 285 Now (us,ie°u,) Ee(u,, ie°ie123u,) Ee(u,, zu,). Thevolume 4-form zrelates theproducts associated with §17*and§*so thatwehave (u,,ieous) E£(u7, uE)§. and pE(u+, u+);1 —(u_, u_)5-. (10.3.4) Thus pispositive-definite orzero since thefirstproduct ispositive- definite andthesecond negative-definite. Toshow thatthecharge density ispositive-definite above wesplitthe four-dimensional spinor into semi-spinors ofthethree-dimensional sub- algebra. This argument implies thatg(7, V)islessthan orequal tozero forallfuture pointing timelike vectors Vandconsequently that must beaforward-pointing timelike ornull vector field. Itisinstructive to rederive thisresult using theoften useful Fierz rearrangement techni- que. Tothisendwewillthistime split thespinor into twosemi-spinors oftheeven subalgebra. Let171*E§(1iiz)171, thatisiz171i Ei171", then (171‘,a171") Eee’(iz171‘, aiz171f') E::::’(171‘, zaz171") E—::e'(171‘, a"171"). Sothecomponents ofjarediagonal in171*and171: j”E(171*, ie"171*) +(171', ie“171‘) Ejfi+ji. (10.3.5) Thenorm ofj,isgiven by -1‘?=(118.e"1/1*)(1/1*. @1111‘)=1'/7‘@“1/151/7 ‘@111? Using (10.2.1) 4 @”1l1’1T‘@1 =201E2P)(-1)”5f,.(¢’1l7‘)- Now P- (¢‘17‘)" =in/1‘17‘iz =—iz¢*i'51’t= -187: andsoonly oddpenter intothesum. Wehave 5f1(¢‘1T‘) =5f1(¢‘17‘il)il =—5f1(¢‘i7F‘)i->1 =—=*I5f1(1P"<7‘)il thus -1%=-2719.1117 7111*-2@W.(<1*1'11)iz11s =-417151111191111: =—4&fl.(¢‘171@.)17‘@"¢‘= -401‘,@..11t)(¢%@"1/1‘) =41'5- Soj,andj_areboth null and, since pE0,future pointing. Since the sumoftwofuture-pointing nullvectors liesinorontheforward light cone thecurrentj isfuture pointing, timelike ornull. Exercise 10.3 Consider the 1-form of(10.3.2) onanarbitrary even-dimensional Lorentzian manifold (not necessarily four dimensional). Show that the 286 SPINOR FIELD EQUATIONS density p is always positive semidefinite but that the argument for j being timelike or null only holds in 2, 4, 6 and 10 dimensions. We now consider the covariances of the Maxwell-Dirac equations under the isometry group of Minkowski space—the Poincaré group. We noted in §10.2 the covariance of the free (A = 0) Dirac equation under Lie derivatives with respect to Killing vectors. To analyse the covar- iances of the coupled Maxwell-Dirac system it is convenient to work with the finite diffeomorphisms rather than the Lie derivatives. This will also allow a discussion of the discrete orientation-changing transforma- tions. Let {xa} be global inertial coordinates for Minkowski space, such that {de} is a global orthonormal co-frame. We can label the diffeo- morphisms forming the Lorentz isometry group by a parallel element of the Clifford group. The diffeomorphism Tc(s) : M M is such that 7r*(s)dx° = sdxas-'. (10.3.6) If a is an arbitrary differential form then a = a Aix' with the multi-index / labelling a parallel basis for the exterior (or Clifford) algebra. Then ir*(s)a = (a .7r(s))sdx's -' (10.3.7) the components of the pulled-back form being composed with the diffeomorphism whilst the change in the basis is effected by Clifford multiplication. This suggests how we can induce an action of the diffeomorphism on a spinor field. Let {b d be a standard parallel spinor frame associated with the co-frame {dxa). Then if tp = tp`la„ we can define = (pl ir(s))sb (10.3.8) Since dxib, = Fb 1 for Fir constants, it follows from (10.3.7) that (10.3.8) satisfies z(s).(azp) = (7r*(s)a)(7r(s)Ip) Va E ['C(M). (10.3.9) If X is an arbitrary vector field we also have z(s)-Sxtp = (s)x(r(s)*V)- (10.3.10) This follows from (10.3.8) since Vxs = 0 and X(p1). 7r(s) = o it(s)). Since 7r(s) is an isometry, the pullback of the Clifford product of two forms is the product of the pulled-back forms. If {e°} and {Xa} are dual bases then so are {.7r*(s)e°} and {z„-1(s)X0), thus = $.:r(s). (10.3.11) It immediately follows that if tp and A satisfy (10.3.1) then so do ,n(s)-tp and .7*(s)A for 7r(s) any Lorentz transformation. The pullback map 286 SPINOR FIELD EQUATIONS density pisalways positive semidefinite butthat theargument forj being timelike ornullonly holds in2,4,6and10dimensions. Wenow consider thecovariances oftheMaxwell—Dirac equations under theisometry group ofMinkowski space——the Poincare group. We noted in§10.2 thecovariance ofthefree (AE0)Dirac equation under Liederivatives with respect toKilling vectors. Toanalyse thecovar- iances ofthecoupled Maxwell—Dirac system itisconvenient towork with thefinite diffeomorphisms rather than theLiederivatives. This will alsoallow adiscussion ofthediscrete orientation-changing transforma- tions. Let{x"} beglobal inertial coordinates forMinkowski space, such that {dx"} isaglobal orthonormal co-frame. We can label the diffeo- morphisms forming theLorentz isometry group byaparallel element of theClifford group. Thediffeomorphism rt(s) :M—>Missuch that rt*(s)dx” Esdx“s". (10.3.6) Ifaisanarbitrary differential form then aEa,dx' with themulti-index Ilabelling aparallel basis fortheexterior (orClifford) algebra. Then rt*(s)a E(a,9rt(s))sdx's" (10.3.7) thecomponents ofthepulled-back form being composed with the diffeomorphism whilst thechange inthebasis iseffected byClifford multiplication. This suggests how wecan induce anaction ofthe diffeomorphism onaspinor field. Let{b,-} beastandard parallel spinor frame associated with theco-frame {dx“}. Then if171E171"b,-, wecan define .rr(s)-171 E(171‘9rt(s))sb,-. (10.3.8) Since dx'b, EF],-bj forF],constants, itfollows from (10.3.7) that (10.3.8) satisfies rt(s)-(a171) E(rr*(s)a)(rt(s)-171) VaeFC(M). (10.3.9) IfXisanarbitrary vector field wealsohave ”($)'5x1/1 =S11;"(:)x(7T($)'l/))- (10-3-10) This follows from (10.3.8) since Vxs E0and X(171‘) @rt(s) E (rt{1(s)X)(171’9rr(s)). Since 1r(s) isanisometry, thepullback ofthe Clifford product oftwoforms istheproduct ofthepulled-back forms. If {§“} and{X,,} aredual bases then soare{rr*(s)e”} and{rr*"(s)X,}, tus rt(s)-,8 E,8-rt(s). (10.3.11) Itimmediately follows that if171andAsatisfy (10.3.1) then sodort(s)-1/1 and rr*(s)A forrr(s) any Lorentz transformation. The pullback map THE DIRAC EQUATION IN SPACETIME 287 commutes with the exterior derivative and, in the case of an orientation- preserving isometry, with the Hodge map and hence the co-derivative b. The pullback with an orientation-reversing isometry picks up a minus sign in moving past a Hodge dual, but since 45 involves two duals (or no choice of orientation) the pullback still commutes with it. So if F and j satisfy (10.3.3) then so do Jr*(s)F and n*(s)j. But j is a functional of the spinor field tp—to symbolise this we will here write j(v) for the 1-form determined by (10.3.2). Is it the case that j(Jr(s)-V) = Equation (10.3.2) involves the spinor adjoint with respect to a product whose invariance group does not contain the whole Clifford group, but only F. (This is the subgroup defined with the norm /2, so s = s-1 for S E +r.) The image under the vector representation of +F is the orthochronous Lorentz group. So if T€ ±F is such that x(T) is a reflection changing the time orientation, then n-(T)ip and Jr* (T)A will not satisfy the coupled Maxwell–Dirac equations given that tp and A do. We know that Lorentz transformations of the cotangent space extend to inner automorphisms of the real Clifford algebra and hence, by complex linearity, to inner automorphisms of the complexified algebra. These inner automorphisms will commute with complex conjugation, and so composing them with complex conjugation gives an outer automorphism of the complexified algebra. A spin transformation on each of a pair of spinors induces an inner automorphism on the Clifford elements formed with a spinor adjoint with respect to a spin-invariant product. That is, scpsip = x(s).(cpip—) if (and only if) s is in the invariance group of the spinor product used to define p. As we will see, if instead s is in the real subalgebra such that (scp, = (99, lp)* for a product on complex spinors then scpsip = x(s)(q)v)*. For the four-dimensional Lorentzian case that we are considering the space of complex spinors is the complexification of the real spinor space. The skew-symmetric product on real spinors with adjoint involution ij is extended by complex bilinearity to a product on complex spinors, ( , )1. In an appropriate basis, charge conjugation simply complex conjugates the spinor components and we have (So = (cP, 4))*,=.n. If we now define (go, — (icy' (10.3.13) then ( , ) certainly has Or as adjoint involution. The factor of i ensures that the product is Hermitian symmetric: (T, /P) = (icP c, = Vc)*:,=„ = (Vc, iT)*q = (iVc, T)in = (V, 9))* Since charge conjugation is involutory and the complex bilinear product in (10.3.13) is skew symmetric we have (10.3.12) THE DIRAC EQUATION INSPACETIME commutes with theexterior derivative and, inthecase ofanorientation- preserving isometry, with theHodge map andhence theco-derivative <5. The pullback with anorientation-reversing isometry picks upaminus signinmoving past aHodge dual, butsince <5involves twoduals (orno choice oforientation) thepullback stillcommutes with it.SoifFandj satisfy (10.3.3) then sodon*(s)F andrt*(s)j. Butj isafunctional ofthe spinor field 171—to symbolise thiswewillhere write j(171) forthe1-form determined by(10.3.2). Isitthe case that j(rt(s)-171) E1-t*(s)j(171)? Equation (10.3.2) involves thespinor adjoint with respect toaproduct whose invariance group does notcontain thewhole Clifford group, but only *F.(This isthesubgroup defined with thenorm 71,sosf"Es"for se*F.) The image under the vector representation of*1"isthe orthochronous Lorentz group. SoifTe*1"issuch that ;((T) isa reflection changing thetime orientation, then rr(T)-171 andrr*(T)A will notsatisfy thecoupled Maxwell—Dirac equations given that171andAdo. Weknow that Lorentz transformations ofthecotangent space extend toinner automorphisms ofthereal Clifford algebra and hence, by complex linearity, toinner automorphisms ofthecomplexified algebra. These inner automorphisms will commute with complex conjugation, and socomposing them with complex conjugation gives anouter automorphism ofthecomplexified algebra. Aspin transformation on each ofapairofspinors induces aninner automorphism ontheClifford elements formed with aspinor adjoint with respect toaspin-invariant product. That is,s<;0sT1d71 EX(s)-(171171) if(and only if)sisintheinvariance group ofthespinor product used todefine Aswewillsee, ifinstead sisintherealsubalgebra such that (sqo, s171)E(<71,171)*foraproduct on complex spinors then sq0§T71 EX(s)-(q11'71)*. Forthefour-dimensional Lorentzian case that weareconsidering the space ofcomplex spinors isthecomplexification oftherealspinor space. The skew-symmetric product onreal spinors with adjoint involution E17 isextended bycomplex bilinearity toaproduct oncomplex spinors, (,)5”. Inanappropriate basis, charge conjugation simply complex conjugates thespinor components andwehave (<1>‘»1l1‘);1, =(<141l1)*_§1,- (10-3-12) Ifwenow define (‘P1 E(iqprt 1/1);” then (,)certainly has517*asadjoint involution. The factor ofiensures thattheproduct isHermitian symmetric: (<11.41)=(i4>‘.41);.=—(i<1>.w‘)t, =(w%i<1>)*1, =011"»9121= (111,¢>)*- Since charge conjugation isinvolutory andthecomplex bilinear product in(10.3.13) isskew symmetric wehave 288 SPINOR FIELD EQUATIONS (çpC, /PC) = —(40, Ii))* (10.3.14) If tp— is the adjoint of y) with respect to ( , ) then for any three spinors (cf»TP-)*P = ((40175)Pe)c = Pc)(Pr = PTV' = Pc)*(Pc = —(pC, P)(Pc by (10.3.14). So (cpii)* p = —(cpc p c)p and (T )* = _q7c;7; c. (10.3.15) Consider now the element TE ±F with x(T) a time-orientation-changing reflection. Then Vq* = 7' 1 = —T-1, so Tcpcnp` = = T(cpi-p)*T-1 by (10.3.15). We now define a(T).1pc (10.3.16) and then have .(P-9 = e(T)(cP 17;)*. (10.3.17) The operation 3- is known as Wigner time reversal on spinors. It obviously satisfies Fi.(atp) = J-c*(T)a*(9 -.1p). (10.3.18) We now examine the covariances of the Maxwell—Dirac system under this operation. If A and y) satisfy (10.3.1) then so do —a*(T)A and 5-1). It follows from (10.3.17) that j(g.tp) = —.7*(T)j(tp) and hence —2r(T)A and .5.tp also satisfy (10.3.3). (Notice that whereas —77-*(T)A and a(T)-ip satisfy (10.3.3) they do not satisfy (10.3.1).) Plane-wave solutions play an important part in the physical interpre- tation of the free (A = 0) Dirac equation, and to these we now turn. If b is a parallel spinor then we look for a solution to (10.3.1), for A = 0, of the form tp = exp(if)b for f some real function. Then Pp = idfip and we require idftp = my). It follows that the 1-form df must be timelike, with (df)2 = —m2. (10.3.19) We can write the algebraic condition on b as ,21(1 + idf/m)b = b. (10.3.20) If s is a unit spacelike 1-form orthogonal to df and z is the volume 4-form then (zs)2 = —sz2s = s2 = 1 and zsdf = —zdfs = dfzs. So ;(1 + zs) is an idempotent orthogonal to ;(1 + idf/m) so that (1 + idf/m)(1 + zs) is primitive. With E and a taking the values ±1 a 288 SI>IN0R FIELD EQUATIONS (<09111‘)=—(¢>.111)’ (193-14) IfDistheadjoint of171with respect to(,)then foranythree spinors (¢>1/7)*p =((¢>1/71¢)‘ =((1/1.p‘)¢>)‘ =(1/1.p“)*¢>" =(1/1”Y19‘)*¢>‘ =—(1/1‘.p)¢>‘ by<19-3-14>. so<<1>171*p=—<<1»C17t>p and <4>11>*=-W7‘: <19-3-15> Consider now theelement Te*1“with ;((T) atime-orientation-changing reflection. Then T5"* ET5"E—T", so WTT1‘ =-T<1>"1i¢T* =T(4>17>*T-‘ by(10.3.15). Wenow define 9.171 Ert(T).171‘ (10.3.16) andthen have °J.<pfi1 =11*(T)(<7>17)*. (10.3.11) The operation 0isknown asWigner time reversal onspinors. It obviously satisfies .°I.(a171) Ert*(T)a*(‘J.171). (10.3.18) Wenow examine thecovariances oftheMaxwell—Dirac system under thisoperation. IfAand 171satisfy (10.3.1) then sodo—Jt*(T)A and 5-171. Itfollows from (10.3.17) that j(§-171) E—JT*(T)j(17J) and hence —1r*(T)A and9-171 also satisfy (10.3.3). (Notice thatwhereas —rr*(T)A and1r(T)-171 satisfy (10.3.3) they donotsatisfy (10.3.1).) Plane-wave solutions play animportant part inthephysical interpre- tation ofthefree (AE0)Dirac equation, andtothese wenow turn. If bisaparallel spinor then welook forasolution to(10.3.1), forAE0, oftheform 171Eexp(if)b forfsome realfunction. Then $171Eidf171 and werequire idf171 Em171. Itfollows that the1-form dfmust betimelike, with (df)3 =—m2. (10.3.19) Wecanwrite thealgebraic condition onbas §(1+idf/m)b Eb. (10.3.20) Ifsisaunit spacelike 1-form orthogonal todfand 2isthevolume 4-form then (zs)3 E—sz2s Es2E1and zsdf= —zdfs Edfzs. So §(1+ zs)isanidempotent orthogonal to§(1+idf/m) sothat §(1+ idf/m)§(1+ zs)isprimitive. With sand 0taking thevalues i1a THE DIRAC EQUATION IN SPACETIME 289 complete set of pairwise orthogonal primitive idempotents is given by {P „ = 4(1 + Eidf/m);(1 — azs)). (10.3.21) We can choose a basis of spinors such that each is an eigenspinor of one of these primitive idempotents. An inertial observer would use inertial coordinates {t, x, y, z} to interpret df(3/3t) as an energy and df(3/3x) as a component of momentum along the x-axis. If we assume that df and s are parallel then we can choose inertial coordinates ft, x, y, z) such that f = mt and s = dx. Then we can label plane-wave solutions by E and a, zp„ = exp(iEmt)b„ (10.3.22) where b„ = Peak°. for P„ = (1 + iEdt)(1 + adydzdt) (10.3.23) and we have chosen z = dxdydzdt. If we choose some parallel 13++ then we can build up the rest of the spinor basis by taking Clifford products. For example, we have dxP E0 = P -E-adx and dyP„ = P_„dy and hence dxdyP„ = P e_adxdy. So we may choose the basis as {13 ++, b__ = dxb ++, b_ = dyb ++, b,_ = dxdyb ++}. (10.3.24) If ( , ) has as adjoint then we may use the algebraic properties of this basis to work out the non-vanishing products, we have (bra, b ea') = (P eab ea Pb 5'0) 0 fog, P EC,'°.1* P 0'b ea') — (be,' P-coP cab ea') = ( 5 -Eef 5,a4b Ea> b 0-') Thus the only non-zero independent products are (b„, b,) and (13 +_, b__). If we choose the basis as in (10.3.24) then these are related for (b,_, b__) = (dxdyb„, dxb ++) = (b ++, dyb„) = (b„, b_ +). So by suitably scaling b „ we have (1)„, b„) = (b,_, b__) = 1. (10.3.25) Thus the two-dimensional subspaces with fixed £ are isotropic, whilst those with fixed a are unitary subspaces of maximal index. The &label of tp„ specifies the eigenvalue of the spinor Lie derivative in the 3/3t direction, gaiatiPec = iEmp. (10.3.26) Similarly a may be used to label the eigenvalue of the Lie derivative with respect to the vector ya/3z — za/ay that generates rotations about the x-axis, we have THEDIRAC EQUATION INSPACETIME 289 complete setofpairwise orthogonal primitive idempotents isgiven by {P7, E§(1+cidf/m)§(1 —ozs)}. (10.3.21) Wecanchoose abasis ofspinors such that each isaneigenspinor ofone ofthese primitive idempotents. Aninertial observer would useinertial coordinates {t,x,y,z}tointerpret df(8/St) asanenergy anddf(8/8x) asacomponent ofmomentum along thex-axis. Ifweassume that dfandsareparallel then wecanchoose inertial coordinates {t,x,y,2}such thatfEmtandsEdx.Then wecanlabel plane-wave solutions bysand0, 1717,,Eexp(ismt)b,,, (10.3.22) where bu,EPwbw for P7,,E§(1+isdt)§(1 +odydzdt) (10.3.23) and wehave" chosen zEdxdydzdt. Ifwechoose some parallel b++ then wecanbuild uptherest ofthespinor basis bytaking Clifford products. Forexample, wehave dxP,,, EP_,._,,dx anddyP_,,, EP_,,,dy andhence dxdyP,,, EP,_,,dxdy. Sowemay choose thebasis as {b+.,, b__ Edxb++, b_+ Edyb++, b+_ Edxdyb++}. (10.3.24) If(,)has517*asadjoint then wemay usethealgebraic properties of thisbasis towork outthenon-vanishing products, wehave (brow baa) =(peabeov Pe'a‘be’o’) :(beov Peo§n‘Ps'o'bs‘o') =(bear P—soPe’o’be’o') :6—££'6ao’(bso1 be'o')' Thus the only non-zero independent products are (b+7,b_+) and (b+_, b__). Ifwechoose thebasis asin(10.3.24) then these arerelated for (b+—1 b——) :(dxdyb++> dxb++) =(b++1 dyb++) =(b++~ b-+)- Sobysuitably scaling b+7wehave (b++, b_+) E(b+_,b__) E1. (10.3.25) Thus thetwo-dimensional subspaces with fixed 1:areisotropic, whilst those with fixed 0areunitary subspaces ofmaximal index. Thes-label of1717,,specifies theeigenvalue ofthespinor Liederivative inthe8/St direction, gs/atlllw =i~‘5mlPw- (10-3-26) Similarly 0may beused tolabel theeigenvalue oftheLiederivative with respect tothevector y8/82 —28/8y that generates rotations about thex-axis, wehave 290 SPINOR FIELD EQUATIONS *(y313z - z8/3y)1Pea Id(Ya/az zalay)vea = dydzpf = lidydzdtidttp„ = iEclydZdt/P„ 1(y8/8z -za/ay)Vea = OEC"Peo (10.3.27) The eigenvalues of ±i lead to the physical interpretation of an intrinsic spin of a half for the electron. More generally, the functional depen- dence of the components will contribute an orbital angular momentum, the eigenvalue of the Lie derivative being interpreted as the total angular momentum. 10.4 The Stress Tensor Although we have not done so the Dirac equation can be obtained from a variational principle. This ensures the existence of a symmetric stress tensor which is divergenceless when the field equations hold. We here simply present such a tensor and explicitly demonstrate (not so simply) that its divergence is zero for solutions to the Dirac equation. For definiteness we take ( , ) to be a Hermitian-symmetric spinor product with w as adjoint involution, then Re( , ) is real valued and symmetric. Let 'OE! ab = Re(tp, eaSxhip) + Re(tp, ebSAyp). (10.4.1) If S is compatible with the spinor product then Xi ab) = Re(S x4p, eaSmp) + Reetp, Vx.eaSxhIp) +Re(ip, e aS ,rS mp) + Re(S rzp, e bS mp) +Re(tp, VebSx„tp) + Re(p, e bSxSKtp).(10.4.2) Changing the order of the covariant derivatives eaSrSxhip = eaS(Xa, Xb)/P + eQS x„S x1P ± ea S pc xhilP = ea S(X a, X b)lp + S xottp — V x,e" S xnip +e` (V x2( b — V xhiYa)eaS xtp if V is torsion free. Now ec(Vx,Xu)ea —VXhec(Xa)e° = and e.c(Vx Xb) = —Vxfc(Xb). From (9.3.20) ea S(X a, X b)tp = 1,e1' R pop, and for zero torsion this can be written in terms of the Ricci forms, e"S(X a, X b)tp = P op, so 290 SPINOR FIELD EQUATIONS g(y8/Sz -28/6y)‘/“lea : _23/310‘/'w =§dydz171,,, =§idydzdtidt171,,, =§iedydzdt171,,, g(y8/82 —28/6y)wc0 =ii5<7ll’w- Theeigenvalues ofiéilead tothephysical interpretation ofanintrinsic spin ofahalf fortheelectron. More generally, thefunctional depen- dence ofthecomponents willcontribute anorbital angular momentum, theeigenvalue oftheLie derivative being interpreted asthetotal angular momentum. 10.4 TheStress Tensor Although wehave notdone sotheDirac equation canbeobtained from avariational principle. This ensures theexistence ofasymmetric stress tensor which isdivergenceless when thefield equations hold. Wehere simply present such atensor andexplicitly demonstrate (not sosimply) thatitsdivergence iszero forsolutions totheDirac equation. For definiteness wetake (,) tobeaHermitian-symmetric spinor product with 517*asadjoint involution, then Re(,)isreal valued and symmetric. Let °.T,,,ERe(171, e,,SXh171) +Re(171. e,,SX“171). (10.4.1) IfSiscompatible with thespinor product then Xu(gab) =R6(Sx"ll'» 615x711’) +R60/'» Vx"¢'u-Six,‘/') ‘l’Rfiili/» @11Sx"Sx,‘/') +R¢(SX"lll» ¢'I1Sx,,lP) +Re(171,V,\1-e,,SX“171) +Re(171, e,,SX1SX”171).(10.4.2) Changing theorder ofthecovariant derivatives @11Sx"Sx,,ll1 Z611-5(X“» Xbllll +¢’11Sx,.Sx"ll1 +¢’“S|x,,.x,,|ll’ =¢’“S(X111X11)ll' +SX,,>$w _Vx,¢’”Sx,lP +e‘(VX“X,, —VXhX,)e“SX]171 ifVistorsion free. Now e‘(VXhX,,)e" E—VXje‘(X,,)e“ E—VX,_e‘ ande‘(VX“X,,) E—VXne‘(X,,). From (9.3.20) e“S(X,, X,,)171 E]ePR7,,,171, and forzero torsion thiscanbewritten interms oftheRicci forms, e”S(X11-Xblill =ipbl/H 50 THE STRESS TENSOR 291 eaSrs xbv = eas(x a, xb)lp + S xbh — v ,Gec(xb)easx,11). (10.4.3) From (10.1.4) we have Sx.Sx„tP = — Vx„ec(Xa)Sx,IP + (10.4.4) We can rewrite Vrea as (Vx.ea)(Xc)e c = ec(Vx„Xa)ec = —Vx„ec(Xa)ec, and similarly Vx.eb = V,reb(X e)e` = —e b(Vx..X,)e =-- —V,rec(Xb)ec, collecting terms, X'(Tab) = Re(Sxp, e aS xhiP) + Re(S r 1P, e x„IP) —x„e(Xa)Re(V, ecS — x„ec(r)Re()P, e bS x,IP) —V x„ec P (b)Re(V, ea S x,1P) — V x„e c (X b)Re(V e cS +;Re(T, Pop) + Re(1P, Sx„SIP) +Re(, eb$211)) 1Re(', ebR1P). (10.4.5) If = abea 0 eb then 9-(Xa, Xt.) = Xa(Zfa b) + x„e`V ag + xf c(Xb)a = Re(S xP, e bS x„IP) + Re(S )01P, e US MP) +Re(V, SP) + Re(tP, eb$2) +1Re(tP, Pp) + ,14Re(tp, ebatp). Since the spinor product is symmetric with as adjoint then Re(p, Alp) = AT) for A any real 1-form and Vx„-GT(Xa, Xb) = —Re(PP, Sx„/P) + Re(V, Sp) + Re(ip, e b$27p). It follows that V..5 = 0 if $tp = mtp with m real. The above is seen to go through unaltered for real spinors with a spinor product whose adjoint involution is Had we taken a real- valued skew-symmetric spinor product on complex spinors with as adjoint then ,5 would be divergenceless for $tp = imp. For real spinors and a skew-symmetric product with as adjoint the stress tensor would be divergenceless if ,Sp = 0. Exercise 10.4 Show that the Maxwell—Dirac stress tensor is divergenceless when the coupled equations are satisfied. For the stress tensor of (10.4.1) the trace is given by aa = 2Reeip, $0. When the Dirac equation is satisfied we have „" = 2m(V, V). (10.4.6) THE STRESS TENSOR 291 e,,SX1SXh171 Ee“S(X,,, X,,)171 +SXb,$171 —VXne‘(X7,)e"SX(171. (10.4.3) From (10.1.4) wehave 51-5111/1 =fit/1—VX,,@‘(X..)SX.w +1909- (10-4-4) We can rewrite VX1e,, as(VX1e,,)(X,)e‘ Ee‘(VXuX“)e, E —VX“e‘(X“)e,, andsimilarly Vxiej, EVX1e,,(X7)e‘ E—e,,(VXtX,)e‘ E —VX1e‘(X,,)e,, collecting terms, X”(9111) =R@(5x~1//1 61518.1//) +R@(5x~1/1»@15x,1//) —VX,,@‘(X")R9(t/1, 6151.1/1) —VX,,@‘(X“)R9(t/1. 6151,11) -Vx,,@C(Xt1)Re(l/1-@”Sx.lP) _Vx,,@c(X1>lRe(ll’~ @1Sx"1/ll +iRe(l/1- PM/1) +Reillh Sxpgllll +Re(171, e,,,$2171) +§Re(171,e7,9t171). (10.4.5) If9E9,,,,e" ®e"then Vx,°~7(X”» X1)=X1(°~7”t) +Vx.@‘(X")~°7a= +Vx.@C(Xt=)°~7“@ =Re(SX.171, e7,SXfl171) +Re(SX-1171,e,,SXh171) +Reillh Sxpsl/ll +Reillli 61115211’) '1'iRe(l/1-P1111’) +iRe(lP1 61197111’)- Since the spinor product issymmetric with 517* asadjoint then Re(r71, A171) E—Re(171, Arp) forAanyreal1-form and Vx,,57(X”1Xt1) =_Re(7Sl/)1 S11,‘/1) +Reil/1- Sx,>$lP) '1'Reillh 611132111)- Itfollows thatV.9 E0if,S171Em171with mreal. The above isseen togothrough unaltered forreal spinors with a spinor product whose adjoint involution is517.Had wetaken areal- valued skew-symmetric spinor product oncomplex spinors with 5*as adjoint then 9would bedivergenceless for$171Eim171. Forrealspinors andaskew-symmetric product with5asadjoint thestress tensor would bedivergenceless if,$171E0. Exercise 10.4 Show that theMaxwell—Dirac stress tensor isdivergenceless when the coupled equations aresatisfied. For the stress tensor of(10.4.1) the trace isgiven by 9,,“E2Re(171, $171). When theDirac equation issatisfied wehave 9,,"=2m(t71, 171). (10.4.6) 292 SPINOR FIELD EQUATIONS Certainly for m zero the trace is zero. In general the spinor product will be pseudo-Hermitian and so for m * 0 the trace can still vanish. We have already noted in §7.4 that we can construct a closed (n — 1)-form from the stress tensor and a Killing vector, namely = abKbeaz where al, are the components of the stress tensor -3, given by (10.4.1), which is divergenceless when the field equations $tp = mip are imposed. In §10.2 we obtained by inspection a closed (n — 1)-form Jic for each Killing vector K. These two forms, .1K and IC, in fact differ by an exact form modulo the field equations, as we now demonstrate. We are going to have to recognise the exterior derivative of an (n — 2)-form when we see one, so first we note that if H = Habeabz then dH = 2{Xb(Hba) Vx,e(X b)Hba Vxbea(X`)Hbc}e az (dH)aeaz. (10.4.7) A fairly tedious calculation produces Re(tp, kSxntp) = eadktp) — (dH), + Re(tp, eaS Kip) — eak$v) + ;Re(tp, k e aPp) (10.4.8) where Hba = Re(, eb î<e, tp) — Re(tp, ebak tp). As well as frequently using the defining anticommutation relation of the Clifford algebra the calculation uses the fact that since the spinor product has ij as adjoint then Re(ip, Alp) = 0 for A any real 1-form. Thus for example Re(tp, eakebtp) = —Re(tp, e 5 Keap), as is necessary for Ha b = —Hba. (Although it is tedious we recommend that the reader verify (10.4.8), as it does help develop the calculational proficiency that unfortunately is sometimes required.) It follows from (10.4.8) that Reetp, eaS + Re(tp, k5 )0p) = 2Reetp, e(1404') — (dH)„ + Re(tp, ( k A e a)h). If we use the field equations, $îp = mtp, then Re(tp, (k A e„)$/P) = mRe(tp, (k A ea)Ip) = 0 since a real 2-form changes sign under the adjoint involution of the spinor product. Thus .1K= K modulo an exact form, modulo the field equations. Exercise 10.5 Repeat the analysis with a skew product whose adjoint is ,r;* with field equations = im 292 SPINOR FIELD EQUATIONS Certainly formzero thetrace iszero. Ingeneral thespinor product will bepseudo-Hermitian andsoformE0thetrace canstillvanish. We have already noted in§7.4 that wecan construct aclosed (n—1)-form from the stress tensor and aKilling vector, namely JKE9,,,,K"e“z where 9,7 arethecomponents ofthestress tensor 9, given by(10.4.1), which isdivergenceless when thefield equations ,$171E m171 areimposed. In§10.2 weobtained byinspection aclosed (n—1)-form 51,1foreach Killing vector K.These twoforms, JKand 97¢, infactdiffer byanexact form modulo thefield equations, aswe now demonstrate. We aregoing tohave torecognise theexterior derivative ofan(n—2)-form when weseeone, sofirst wenote that if HEH,,7,e“"z then <1”E2{X"(H1..) EVx,@"(X”)H11 EVx»@1(X‘)H11}@“Z E(dH)1@"Z-(10.4.1) Afairly tedious calculation produces R401.1?$1,,1»> =1114111»[email protected]?1»> -.-%<dH1.+Re(1/1.@..$1<1/1) —%R@(1/1. e..I?,$1/1) +%R@(1/1. T<'e.$1/1) (10-4-8) where H7,ERe(171, ebKe, 171)—Re(171, eb,K171). Aswell asfrequently using thedefining anticommutation relation oftheClifford algebra the calculation uses thefactthatsince thespinor product has517*asadjoint then Re(~171, A171) E0for A~any real 1-form. Thus for example Re(171, eaKe"171) E—-Re(171, e”Ke,,171), asisnecessary forH,,,E—H,,,,. (Although itistedious werecommend thatthereader verify (10.4.8), as itdoes help develop thecalculational proficiency that unfortunately is sometimes required.) Itfollows from (10.4.8) that Re(‘/J, Cask-171) + Re(l,ll. KSXHI/J) E2Re(1/1~@..é'?1<1l1) E§:(dH)1. +Re(1/h(1?x@1)$1/1)- lfweusethefieldequations, $171Em171.then R@(1».(1? Ae..>t11)= mR@(1». <1?A@114»)=9 since areal2-form changes signunder 517*.theadjoint involution ofthe spinor product. Thus JKE2.‘l=K modulo anexact form, modulo thefield equations. Exercise 10.5 Repeat theanalysis with askew product whose adjoint is5*with field equations $171Eim171. THE STRESS TENSOR 293 Example 10.1 Gravitational and Neutrino Waves Consider a spacetime in which the metric takes the form g = 2(du 0 do + do du — 2Hdu 0 du + dz 0 dz* + dz* ®dz) in coordinates (u, o, x', x2) with z x1 + ix2 and H a real function of u, z and z'. It is here most convenient to adopt a null basis. We choose the null co-frame {na} a = 1, 2, 3, 4 where n1 = du, n2 = do — Hdu, n3 = dz, n 4 = dz* and the duals are X, = 3/3u + H3/3v, X, = a/aU, X3 = 3/3z, X, = 3/3z*. The non-vanishing components of the metric are g = g2, = g34 = g43 = 2, or g'2=g21=g34=g43=4. Since the components of the metric are constant in this basis we can evaluate the connection forms by (6.6.8), the non-vanishing ones being co31 = —coo = 21-1,n1 W41 = —W14 = 2H z.n I. The only non-zero Ricci form is P, = 2H z,n1. We now adapt a spinor frame to this null co-frame. Let b1 be a spinor such that nib1 = n3b = O. We then form the spinor frame b2 = n2b,,b3 = n4b1, b4 = n2n4b,}. (We can represent the spinor b I by the differential form n1n3, this lying in a minimal left ideal of the complexified Clifford algebra. The other spinors {b,} are then seen to complete the basis for the minimal left ideal.) If {X„} is the frame dual to {n"} then, for ax defined in (8.1.5), we have ax, = ([1,1'13 + H „n4)111 with all other ax, zero. It follows that the spinors 13 1 and b3 are parallel. Hence if h, and h, are arbitrary complex functions of u and ip = h1(u)b, + h2(u)b3 then Pp = 0. To obtain Einstein's equations we now need to evaluate the spinor stress tensor. If ( , ) is the Hermitian-symmetric spinor product with 71* as adjoint then we can use the algebraic properties of the spinor frame to evaluate the products. For example, (131, b3) = (13,, n4b 1) = (n4b 1, 131)* = —(b 1, n3b ,)* = 0 since n3b = O. Also 13-, = n2b, and so nib-, = nIn2b = (1— n 2n1)b, = b and hence (3,, 13 1) = (n1b2, nib-) = —(b2, n1n1b2) = O. In this way we can show that the non-vanishing products are specified by the imaginary components (b 1, b2) = (133, 134). By suitably normalising b, we have (b b2) = (b3, 134) = i. The only non-zero component of the stress tensor of (10.4.1) is then = + THE STRESS TENSOR 293 Example 10.1 Gravitational andNeutrino Waves Consider aspacetime inwhich themetric takes theform g=2(du®du+dv®du—2Hdu®du+dz®dz*+dz*®dz) incoordinates (u,u,x‘,x2)with zEx‘+ixzandHarealfunction of u,zand2*.Itishere most convenient toadopt anullbasis. Wechoose thenull co-frame {n"} a=1, 2,3,4where n‘=du,n2=du—Hdu, n3=dz,n4=dz* and theduals areXl=6/6u +H6/60, X2=6/60, X3=6/62, X4=6/62*. The non-vanishing components ofthemetric are812=821: 834=843=2»OT812Z821: 834:843=i- Since the components ofthemetric areconstant inthisbasis wecanevaluate the connection forms by(6.6.8), thenon-vanishing ones being w3l=—wl3 =2HZn‘ w4l=—wl4 =2H:*n1. Theonly non-zero Ricci form isPl=2HZ»Zn1. Wenow adapt aspinor frame tothis null co-frame. Letblbea spinor such thatn‘bl =n3bl =0.Wethen form thespinor frame {bl, b3=n3bl,b3 =n‘bl,b4 =n3n4bl}. (We canrepresent thespinor blbythedifferential form n‘n3,thislying inaminimal leftideal ofthecomplexified Clifford algebra. The other spinors {bl} arethen seen tocomplete thebasis fortheminimal left ideal.) If{X,,} istheframe dual to{n”} then, foraxdefined in(81.5), wehave ax]=(HZn3 +H:»n4)n‘ with allother ax’. zero. Itfollows thatthespinors blandb;areparallel. Hence ifhland/13arearbitrary complex functions ofuand 1/1:hl(u)bl +h3(u)b_~, then $1/1=0.To obtain Einstein’s equations wenow need toevaluate thespinor stress tensor. If(,) istheHermitian-symmetric spinor product with 517*as adjoint then wecanusethealgebraic properties ofthespinor frame to evaluate theproducts. Forexample. (b1~b3):(b1~ ”4b1):(”4b1~bi)* :-(bu ”3b1)* =0 since n3bl =0.Also b3=n3bl andson‘b; =nlnzbl =(l—nZn')bl =blandhence (bl. bl)=(nlbl, n'b3) =—(b3. n‘n‘bZ) —0.Inthis way wecanshow that thenon-vanishing products arespecified bythe imaginary components (bl, b3)=(bl, bl). Bysuitably normalising bl wehave (bl- b2)=(b3~ bi) :i- Theonly non-zero component ofthestress tensor of(10.4.1) isthen fill=4Re(ih*lh’l +ih*;h'3). 294 SPINOR FIELD EQUATIONS Since for zero mass the spinor stress tensor is traceless we can write the Einstein equations as 2x-P, = *-1.re, and so the coupled system reduces to the equation = + ih*,h'2) 10.5 Tensor Spinors Starting with the spinor representation of the spin group we can build up higher-dimensionsal irreducible representations by forming tensor products. That is, tensor products of the spinor space and its dual space carry representations of the spin group, this space of tensors being decomposable into irreducible representation spaces. The covariant derivative on spinor fields induces a covariant derivative on these spin tensors and one can consider various field equations. We have already noted that elements of the Clifford algebra can be identified with (1, 1) tensors on the space of spinors. Certain higher-dimensional half-integral irreducible representations of the spin group can be found by taking the tensor product of tensors on the vector space V with the spinor space of C(V, g). Such objects can be thought of as spinor-valued tensors. As an example we consider a spinor-valued 1-form IF on spacetime. Then we can write this in any co-frame {ea} as tif Pa 0 ea (10.5.1) where each ipa is a spinor. We can think of IV as a mapping from vector to spinor fields: W(X) = P aen(X) V X E FTM . (10.5.2) Equivalently if {b,} is any standard spinor frame with va = tpb, then we can write III as = b, O/p1 (10.5.3) with the 1-forms ipi given by 1p' = tilaea . These spinors could carry irreducible representations of the complexified Clifford algebra, its even subalgebra or real subalgebra (Dirac, Weyl or Majorana spinors). Let us suppose that the p. are Weyl spinors, satisfying inpa = via. Then the tp, carry irreducible representations of the spin group S1(2, C). A 1-form is a tensor on the space of spinors, Clifford multiplication interchanging the semi-spinor spaces (since a 1-form anticommutes with the volume 4-form). So we may regard a spinor-valued 1-form as a degree-three tensor on the spinor space. If u and y are any two Weyl spinors, lying in the same semi-spinor space as the ipa, then we define 294 SPINOR FlELD EQUATIONS Since forzero mass thespinor stress tensor istraceless wecanwrite the Einstein equations asZKPC =*“r(, andsothecoupled system reduces totheequation KHZ.Z =Re(ih*lh'l +ih*3h’3). 10.5 Tensor Spinors Starting with thespinor representation ofthespingroup wecanbuild uphigher-dimensionsal irreducible representations byforming tensor products. That is,tensor products ofthespinor space anditsdual space carry representations ofthespin group, this space oftensors being decomposable into irreducible representation spaces. The covariant derivative onspinor fields induces acovariant derivative onthese spin tensors andonecanconsider various field equations. Wehave already noted that elements oftheClifford algebra canbeidentified with (1,1) tensors onthespace ofspinors. Certain higher-dimensional half-integral irreducible representations ofthespin group canbefound bytaking the tensor product oftensors onthevector space Vwiththespinor space of C(V, g).Such objects canbethought ofasspinor-valued tensors. Asanexample weconsider aspinor-valued 1-form ll!onspacetime. Then wecanwrite thisinanyco-frame {e“}as w=1/1,,®e” (10.5.1) where each 1/1,,isaspinor. Wecanthink of\I1asamapping from vector tospinor fields: \I1(X)=1/1,,e”(X) vxerm. (10.5.2) Equivalently if{bl} isanystandard spinor frame with 1/1,,=1/1f,bl then wecanwrite \Pas in=bl®1/1" (10.5.3) with the1-forms 1/1"given by1/1"=1/1f,e“. These spinors could carry irreducible representations ofthecomplexified Clifford algebra, itseven subalgebra orrealsubalgebra (Dirac, Weyl orMajorana spinors). Letus suppose that the1/1,,areWeyl spinors, satisfying iz1/1,, =1/1,,.Then the 1/1,,carry irreducible representations ofthespin group Sl(2, C). A 1-form isatensor onthespace ofspinors, Clifford multiplication interchanging thesemi-spinor spaces (since a1-form anticommutes with thevolume 4-form). Sowemay regard aspinor-valued 1-form asa degree-three tensor onthespinor space. Ifuand0areanytwoWeyl spinors, lying inthesame semi-spinor space asthe1/1“,then wedefine TENSOR SPINORS 295 IP(u, u) (u,zpa)e"v. (10.5.4) The brackets on the left-hand side signify that 111 is evaluated on u and v, whereas the brackets on the right-hand side are the spinor product of u and tp„ where the product has as adjoint involution. (The skew- symmetric complex bilinear product on Dirac spinors induces a non- degenerate product on each of the two spaces of Weyl spinors. If u = izu then the spinor product (u, lp„) will only involve ;(1 + iz)ip„.) It turns out [9] that irreducible SI(2, C) representations are carried by spin tensors that are totally symmetric in the covariant and contravariant arguments separately. It is therefore interesting to examine the condi- tion on IF such that (10.5.4) defines a mapping symmetric in u and v. In order to do this we will need the following: (u,v)w — (w, v)u = 4(u, w)v (10.5.5) for u, u and w any three Weyl spinors. To see this let a be another Weyl spinor and consider the expression (u, v)(w, a). Using It to denote the adjoint spinor we can write this as irviZia. Now we can expand v in7 as in (2.1.18) to give (u, v)(w, a) = it-Yo(v e)eA a =#o(14ev)îieAa. = (w, ev)(u, e A a). Now for u and v Weyl spinors and a any Clifford form (v, au) = (izv, aizu) = (v, zaz -lu) = (u, aqu) so (v, au) = 0 for a odd. In addition (v,au) = (a;.`v, u) = —(u, azv) so for a" = —a (c1.` = a) then (y, au) is symmetric (skew) in u and v. So (u, v)(w, a) — (u, w)(v, a) = (w, eA a) — (w v) and the first bracket on the right-hand side will only contain those e5, that are even under ri and under These are the 0-forms and the 4-forms, thus (u, v)(w, a) — (u, w)(v, a) = 2(w, v)(u, a) — 2(w, zu)(u, zŒ). Since v and a satisfy zv = —iv and za = —ia the terms on the right-hand side add up. We can use the skew symmetry of the product to rewrite the left-hand side, producing (w,u)(v, — (v, u)(w, a) = 4(w, v)(u, a). Since this is true for all a and the spinor product is non-degenerate (w, u)v — (v, u)w = 4(w, v)u. TENSOR SPINORS 295 ll'(u. 0)E(u.1/1,,)e"v. (10.5.4) The brackets ontheleft-hand side signify that ‘I1isevaluated onuand v.whereas thebrackets ontheright-hand sidearethespinor product of uand 1/1,,where theproduct has’g'asadjoint involution. (The skew- symmetric complex bilinear product onDirac spinors induces anon- degenerate product oneach ofthetwo spaces ofWeyl spinors. If u=izuthen thespinor product (u,1/1")willonly involve §(1+iz)1p,,.) Itturns out[9]that irreducible Sl(2, C)representations arecarried by spintensors thataretotally symmetric inthecovariant andcontravariant arguments separately. Itistherefore interesting toexamine thecondi- tionon\I1such that (10.5.4) defines amapping symmetric inuandv.In order todothiswewillneed thefollowing: (u.u)w -—(w,v)u=4(u, w)u (10.5.5) foru,uand wanythree Weyl spinors. Toseethisletabeanother Weyl spinor and consider theexpression (u,v)(w, a).Using ifto denote theadjoint spinor wecanwrite thisasiivfi/‘a. Now wecan expand vi?asin(2.1.18) togive (u,v)(w. a)=ZTff’(l(v We§l)e*‘a =E/’ll(W'e§lv) Tie"a' =(w,e§lu)(u, e"a). Now foruandvWeyl spinors andaanyClifford form (v,au)=(izu, aizu) =(v,zaz"u) =(v,a"u) so(0,au)=0foraodd. Inaddition (v,au)=(aetu, u)=—(u, a5v) soforas=—a(a5 =a)then (v,au)issymmetric (skew) inuandv.So (u,v)(w, a)—(u,w)(v, a)=(w,e§lv)(u, e"a) ~(w<—>v) andthefirst bracket ontheright-hand side willonly contain those efl that areeven under 17and under 5.These arethe0-forms and the 4-forms, thus (u,v)(w. a)—(u,w)(v, a)=2(w,v)(u, a)—2(w,zu)(u, za). Since vand asatisfy zu=—iu and za= —ia theterms onthe right-hand side addup.Wecanusetheskew symmetry oftheproduct torewrite theleft-hand side, producing (w,u)(v, tr)—(v,u)(w, a)=4(w, v)(u, a). Since thisistrueforall0/andthespinor product isnon-degenerate (w,u)u—(v,u)w =4(w, u)u. 296 SPINOR FIELD EQUATIONS This is just (10.5.5) with the spinors cyclically permuted. We can now use (10.5.5) and (10.5.4) to see that tis(u, y) — IF(ty, u) = 4(u, v)e"Tp„. Thus the spinor-valued 1-form is an irreducible spin tensor if it is 'traceless': ealpa ---- 0. (10.5.6) Exercise 10.6 Use the correspondence between 1-forms and (1, 1) spin tensors given at the end of §2.8 to label the components of a spinor-valued 1-form with one 'dotted' and two `undotted' indices. Show that the `tracelessness' condition is equivalent to symmetry in the two like indices. The spinor covariant derivative S. and the covariant derivative V x can be extended by the Leibniz rule to a covariant derivative, also denoted Sx, on spinor-valued 1-forms. In the obvious way SW = S xipa 0 + tp, 0 V xea . (10.5.7) (If any confusion is likely between the covariant derivative on spinor- valued 1-forms and that on spinors we can write the former as S3 2.) A representation of the Clifford algebra on spinor-valued 1-forms can be defined by aqi -= (ay) b) eh (10.5.8) so that we have a Dirac-like equation = mtP. (10.5.9) The pair of equations (10.5.6) and (10.5.9) are the Rarita —Schwinger equations for spin 3/2 [25]. Exercise 10.7 Show that (10.5.6) and (10.5.9) imply the 'Lorenz' condition (Sx,1F)(Xa) = 0. In Minkowski space we can pick a parallel co-frame such that (10.5.9) reduces to four Dirac equations. We can then find plane-wave solutions as in §10.3. If {b„} is the spinor basis of (10.3.24) then we have Dirac solutions as in (10.3.22) with the sign of the frequency correlated with the E labelling the basis spinors. By tensoring on four independent 1-forms to the two basis spinors with (say) E = +1 we can form eight linearly independent spinor-valued 1-forms. We can choose four of these satisfying the tracelessness condition (10.5.6). The eight spinor-valued 1-forms can be chosen as eigenstates of the Lie derivatives with respect to vectors generating time translations and rotations about the x-axis. The 1-form basis can be chosen to have eigenvalues of fi, —i, 0, 0) 296 SPINOR FIELD EQUATIONS This isjust (10.5.5) with thespinors cyclically permuted. Wecannow use(10.5.5) and(10.5.4) toseethat \I1(u, v)—\l‘(v, u)=4(u, v)e“1//H. Thus thespinor-valued I-form isanirreducible spin tensor ifitis ‘traceless’: @"tp,,=0. (10.5.6) Exercise 10.6 Usethecorrespondence between 1-forms and(1,1)spin tensors given at theendof§2.8 tolabel thecomponents ofaspinor-valued 1-form with one‘dotted’ andtwo‘undotted’ indices. Show thatthe‘tracelessness’ condition isequivalent tosymmetry inthetwolikeindices. The spinor covariant derivative SXand thecovariant derivative VX can beextended bytheLeibniz rule toacovariant derivative, also denoted SX,onspinor-valued 1-forms. Intheobvious way SXW =SX1/1,, ®e"+1/1,,®Vxe”. (10.5.7) (Ifanyconfusion islikely between thecovariant derivative onspinor- valued 1-forms andthat onspinors wecanwrite theformer asSf.) A representation oftheClifford algebra onspinor-valued 1-forms canbe defined by all!E(a1/;l,)® eh (10.5.8) sothatwehave aDirac-like equation ,8\I1=mm. (10.5.9) The pair ofequations (10.5.6) and (10.5.9) aretheRarita—Schwinger equations forspin3/2[25]. Exercise 10.7 Show that (10.5.6) and (10.5.9) imply the ‘Lorenz’ condition (5x,,‘V)(X") =0~ InMinkowski space wecanpick aparallel co-frame such that(10.5.9) reduces tofourDirac equations. Wecanthenfindplane-wave solutions asin§10.3. If{bm} isthespinor basis of(10.3.24) then wehave Dirac solutions asin(10.3.22) with thesign ofthefrequency correlated with the5labelling thebasis spinors. Bytensoring onfour independent l-forms tothetwo basis spinors with (say) 5=+1wecanform eight linearly independent spinor-valued 1-forms. Wecanchoose four ofthese satisfying thetracelessness condition (10.5.6). The eight spinor-valued l-forms canbechosen aseigenstates oftheLiederivatives with respect tovectors generating time translations androtations about thex-axis. The 1-form basis can bechosen tohave eigenvalues of{i,—i,0,0} TENSOR SPINORS 297 under the Lie derivative with respect to the rotation, whereas the spinor basis has eigenvalues —4i}. The four traceless spinor-valued 1-forms are then seen to have eigenvalues For the basis of (10.3.24) dxb„ = idtb„ = Eb„, dyb„ = ab_„, dzb„ = iEb_ 7(10.5.10) so a basis for positive-frequency solutions to (10.5.6) and (10.5.9) is {b „ 0 (dz — idy), b,_ O (dz — idy) — 2ib„ 0 dx, 13+, 0 (dz + idy) — 2ib,_ 0 dx, b,_ 0 (dz + idy)). (10.5.11) These are eigenstates of vIaz _ z3/3y' arranged in decreasing order of eigenvalues. A spinor-valued 1-form features in the theory of supergravity [8]. This theory involves a connection with torsion. As we remarked in §9.3 the definition of the spinor covariant derivative S,. in terms of the metric- compatible connection V does not rely on V being torsion-free. So in this case we could still adopt (10.5.7) as the definition of a covariant derivative on spinor-valued 1-forms. The field equation for the spinor- valued 1-form in supergravity, however, is most readily expressed in terms of another connection. If {Ta} are the torsion 2-forms of the connection V then a covariant derivative on differential forms is defined by ' Vx + lixT a A i)G. From (6.7.4) we see that t is just such that e"AVX= d. (10.5.13) If S. is the spinor covariant derivative associated with V then a covariant derivative S' x on spinor-valued p-forms is defined by gx1.11 ' Sx/P/ O el + 0 t xel (10.5.14) where el is a p-form basis. For W a spinor-valued p-form we may adopt the convention that for a any q-form a A 0 a A el. (10.5.15) The spinor covariant exterior derivative D maps spinor -valued p-forms to spinor-valued (p + 1)-forms: DW ea A 3 x.T. (10.5.16) If {b,} is a standard spinor frame associated with some orthonormal co-frame then we may expand W as W = b, 0 where the tiP are a set of p-forms. Then we can equivalently write the spinor covariant exterior derivative as TENSOR SPINORS 297 under theLiederivative with respect totherotation, whereas thespinor basis haseigenvalues {§i,—§i}. Thefour traceless spinor-valued 1-forms arethen seen tohave eigenvalues {§i, —§i, —§i}. Forthebasis of (10.3.24) dxbfl, =b-{-0, idtbw =sbw, dybll, =ob-lo,dzba, =isb-F,,(10.5.l0) soabasis forpositive-frequency solutions to(10.5.6) and(10.5.9) is {bll ®(dz—idy), b.,_ ®(dz—idy) —2ib.,., ®dx, btl ®(dz+idy) —2ib.,- ®dx,b.,_ ®(dz+id)/)}. (10.5.11) These areeigenstates of§El.a,aZ -way, arranged indecreasing order of eigenvalues. Aspinor-valued 1-form features inthetheory ofsupergravity [8].This theory involves aconnection with torsion. Asweremarked in§9.3 the definition ofthespinor covariant derivative SXinterms ofthemetric- compatible connection Vdoes notrelyonVbeing torsion-free. Soin thiscase wecould stilladopt (10.5.7) asthedefinition ofacovariant derivative onspinor-valued 1-forms. The field equation forthespinor- valued 1-form insupergravity, however, ismost readily expressed in terms ofanother connection. If{T“} arethetorsion 2-forms ofthe connection Vthen acovariant derivative ondifferential forms isdefined by fixEvx+g1XT" Ail". (10.5.12) From (6.7.4) weseethatVisjustsuch that e“A0,,=<1. (10.5.13) IfSXisthespinor covariant derivative associated with Vthen a covariant derivative SXonspinor-valued p-forms isdefined by slutESXIPI®e'+11,®0,.» (10.5.14) where e’isap-form basis. Forll!aspinor-valued p-form wemayadopt theconvention thatforaanyq-form a,\\P EllU[ ® HA8’. Thespinor covariant exterior derivative Dmaps spinor-valued p-forms to spinor-valued (p+1)-forms: owEe“A§,,,tp. (10.5.16) If{bl}isastandard spinor frame associated with some orthonormal co-frame then wemay expand ll!as‘I1=bl®1,0‘where the1p’areaset ofp-forms. Then wecanequivalently write thespinor covariant exterior derivative as 298 SPINOR FIELD EQUATIONS DIP = 131 0 + ePb1 0 wpq AV- (10.5.17) The Hodge dual of a spinor-valued p-form is defined in the obvious way, in analogy to (10.5.15). If N is a Clifford-valued q-form, N = nA C) e A for nA arbitrary Clifford forms and eA a basis for q-forms then we choose to define NIP = nAVI eA A el. (10.5.18) Having adopted these conventions we consider the equation e*DT = 0 (10.5.19) for a spinor-valued 1-form IF where e e° 0 e a. This equation is one of the field equations occurring in the theory of supergravity. Although it is usually known as the Rarita—Schwinger equation this equation is not obtained by simply putting m to zero in equations (10.5.6) and (10.5.9). The relationship between these equations is contained in the following exercise. Exercise 10.8 (i)Show that if IF is a spinor-valued 1-form then *(e*DIP) = Sx(ectp`) 0 ea — Hint: you will need A *eab) = gbcea gaceb. (ii)Show that if ço is a spinor field then e*D2cp = ebS(X b, Xa)cp 0 *e°. Hence show that if the Ricci and torsion forms are zero (10.5.19) has the 'gauge' symmetry 11"1--> W + Dep. Exercise 10.9 Consider the following equation for a spinor tp on spacetime: Sxtp — k- Stp = O vxE r Tm Note that this is equivalent to equating to zero a 'traceless' spinor- valued 1-form made from the covariant derivatives of tp. Since X and $ both anticommute with the volume 4-form this equation decouples into two equations for Weyl spinors. (i)If K is a conformal Killing vector with Jg = 2Ag show that if tp satisfies the above equation then so does g op - 14. This can be shown in the same way as for the analogous (but different!) result for the massless Dirac equation. (ii)By differentiating the equation obtain the integrability condition RI,. 1P — enSx„)SIP = 0- 298 SPINOR FIELD EQUATIONS ow=bl®d1/1"+§eP‘1b,»® wpqAqr‘. (10.5.17) The Hodge dual ofaspinor-valued p-form isdefined intheobvious way, inanalogy to(10.5.15). IfNisaClifford-valued q-form, N=n"®e,lforn"arbitrary Clifford forms ande,labasis forq-forms then wechoose todefine W =HA1/11 ® CA /(CI. Having adopted these conventions weconsider theequation e*D\I1 =0 (10.5.19) foraspinor-valued 1-form \I1where eEe”®ea.This equation isone ofthefield equations occurring inthetheory ofsupergravity. Although itisusually known astheRarita—Schwinger equation thisequation isnot obtained bysimply putting mtozero inequations (10.5.6) and(10.5.9). The relationship between these equations iscontained inthefollowing exercise. Exercise 10.8 (i)Show thatif\I1isaspinor-valued 1-form then *<e*I>\v> =SX,(@.1»*> ®ea—iv» Hint: youwillneed *(e‘A*e”b) =g"‘e" —g”‘e". (ii)Show thatiftpisaspinor field then e*D2(p Ee"S(Xl,, X,,)(p ®*e”. Hence show that iftheRicci andtorsion forms arezero (10.5.19) has the‘gauge’ symmetry \IJ+—~>\IJ+Dtp. Exercise 10.9 Consider thefollowing equation foraspinor 1/1onspacetime: sX¢- §)?,§¢=0 vxerm. Note that this isequivalent toequating tozero a‘traceless’ spinor- valued 1-form made from thecovariant derivatives of111.Since Xand,8 both anticommute with thevolume 4-form thisequation decouples into twoequations forWeyl spinors. (i)IfKisaconformal Killing vector with if-’Kg =2/lgshow that if1]) satisfies theabove equation then sodoes 3K1); —§,l1/1. This can be shown inthesame way asfortheanalogous (but different!) result for themassless Dirac equation. (ii)Bydifferentiating theequation obtain theintegrability condition Rbaq} _i(@nSx,. _ebSx,,)>$l/’ =0- TENSOR SPINORS 299 Clifford multiply to obtain the contracted conditions Pa + earl') Sp = 0 and atp + 3,2/p = O. Hence obtain the integrability condition CbP = O. (Note that P„ Aeb — Pb -bA a= eaPb ebPa for zero torsion.) (iii) If p = u + df v, for some function f and parallel Weyl spinors u and y, show that tp solves the above equation if Vxdf = X. Hence show that this equation has a `twistor' [9] solution with f = 2111ab-exh, where {xa} are inertial coordinates for Minkowski space. Exercise 10.10 When is a spinor a twistor? 10.6 The Lichnerowicz Theorem We anticipated in §10.1 that the eigenvalues of the Dirac operator will depend on the properties of the manifold. Whereas the spacetime Dirac equation involves a real 'mass' eigenvalue we will see below that the Dirac operator on a compact Riemannian manifold has only imaginary eigenvalues. The Lichnerowicz theorem [26], as we will now demons- trate, shows that if the curvature scalar is positive semidefinite then there are no zero eigenvalues. Let M be a compact Riemannian manifold. From §2.6 we know that `4* is the adjoint of a zero index Hermitian-symmetric product on Dirac spinors, ( , ). By integrating over M we introduce another Hermitian product im(V. (P)z where z is the volume n-form of M. The Dirac operator is anti-self- adjoint with respect to this product. To see this we need to recognise an exact form when we see one. To this end we write an (n — 1)-form J as J = jaenz and, for V torsion free, dJ = (V rja + ixbVxhe"j„)z by (10.2.3). Since t , ) has as adjoint involution with e".`'` = ea, (q), PP) = e"(1), S x„IP) {vx(e"cP, 111) — x„e a (X n)(eb 1P) — ($49, IP)}z. TENSOR SPINORS 299 Clifford multiply toobtain thecontracted conditions P11»+%@..6”w -1-$X.$w=0 and 91¢+3,821); =O. Hence obtain theintegrability condition C1111!’ : (Note thatP,,Ael,—Pl,Ael,=e,,Pl, —e,,P, forzerotorsion.) (iii)IftpEu+dfv, forsome function fandparallel Weyl spinors u andu,show thatzpsolves theabove equation ifVxdf EY.Hence show that thisequation hasa‘twistor’ [9]solution with fE§17,,l,x”x", where {x”} areinertial coordinates forMinkowski space. Exercise 10.10 When isaspinor atwistor? 10.6 TheLichnerowicz Theorem Weanticipated in§10.1 that theeigenvalues oftheDirac operator will depend ontheproperties ofthemanifold. Whereas thespacetime Dirac equation involves areal ‘mass’ eigenvalue wewillseebelow that the Dirac operator onacompact Riemannian manifold hasonly imaginary eigenvalues. The Lichnerowicz theorem [26], aswewill now demons- trate, shows that ifthecurvature scalar ispositive semidefinite then there arenozero eigenvalues. LetMbeacompact Riemannian manifold. From §2.6 weknow that ‘§*istheadjoint ofazero index Hermitian-symmetric product onDirac spinors, (,). Byintegrating over Mweintroduce another Hermitian product <1/1-<r>> E(M0/1»<P)Z where zisthevolume n-form ofM.The Dirac operator isanti-self- adjoint with respect tothisproduct. Toseethisweneed torecognise an exact form when weseeone. Tothisendwewrite an(n—1)-form Jas JEj,,e"z and, for Vtorsion free. dJE(Vxtja +iX»VXle“j,,)z by (10.2.3). Since (,)has‘§*asadjoint involution with e"?Ee”. <</>»$1/1)=<@“</ASm/1) =[M{vX,,<@"<p. 11>~vX,,@"<X.><@*<p. 11>-01¢.w)}z- 300 SPINOR FIELD EQUATIONS Now Vxfa(Xb ) = —ea(Vx/Vb) = —ix„Vx.eb, and so we may recognise an exact form in the integrand. By Stokes's theorem the integral of an exact form over a compact manifold is zero, thus (cP, $V) = 1,0). (10.6.1) Since it is anti-self-adjoint with respect to a Hermitian product the Dirac operator on a compact Riemannian manifold has imaginary eigenvalues. As a special case of the above we have OV, STP) = Since ( , is a (zero-index) Hermitian product the left-hand side is positive-semidefinite. Thus $21p = 0 .(=>$1p = 0. Using (10.1.4) to expand the spinor Laplacian gives ($/P, PP) = ixhVx„ea)Sx„V, IP) 1P) Since ((Sx„ + irSx,e a)Sx”1P, /P) = I m{Vr(Sx„V ,V) (Sr1P, Sx„1P) + ix,Vxheu(Sxv, v)Iz = —(SxdP, Sx„V) we have = (Sip, S) + ,2kp). (10.6.2) If 0 then all three terms are positive-semidefinite. If 2/I. > then there are no zero eigenvalues of the Dirac operator: if = 0 then = 0 <=> SAN! = 0 V X. When Ji is constant, such as for the standard metric on a sphere, then we obtain a lower bound for the eigenvalues of the Dirac operator. If $v = imp, with m real, then (m2 — 1R) "ti), = (Sp, Sx„V) and so m2 > 4Igt. The above arguments can be repeated with real spinors. From table 2.15 we see that the involution `j of the real Clifford algebra is the adjoint involution of a zero-index product; the product being either R-symmetric, C*-symmetric or fl-symmetric. 10.7 Killing Spinors Because of the importance of a knowledge of the geodesics on a manifold an interesting problem in general relativity is the determination 300 SPINOR FIELD EQUATIONS Now VXAe“(Xl,) E—e"(VXuXl,) E—iX“VX1el,, and sowemay recognise anexact form intheintegrand. ByStokes’s theorem theintegral ofan exact form over acompact manifold iszero, thus <<P-$111)=—<$<P, w>- (10-6-1) Since itisanti-self-adjoint with respect toaHermitian product theDirac operator onacompact Riemannian manifold hasimaginary eigenvalues. Asaspecial case oftheabove wehave <$w-$w>=—<$2w. 11>- Since (,) isa(zero-index) Hermitian product theleft-hand side is positive-semidefinite. Thus $21/JE0<I>SipE0.Using (10.1.4) toexpand thespinor Laplacian gives <$1P»>$1P> =‘((5% +lx"Vx,.@“)5x.1Pl1l)> +H901/J, 1/1)- Since <(SX,,+1.l»sX.@~)$X,1». 11>=fM{vX~<$X,,1».1»> —(5x'1l1~$1.1») +ix"Vx,@“(Sx,,1P,lP)}Z =“<5,\""1/J» Sx,,1/J) wehave ($11.$w>=<Sx"1P,Sx.,1P> +%<1/»_@Rw>- (10.6.2) If971EOthen allthree terms are positive-semidefinite. IfQR>0 then there arenozero eigenvalues oftheDirac operator: if91E0then ,$1p=0<:>SXip=0VX. When Qtisconstant, such asforthestandard metric onasphere, then weobtain alower bound fortheeigenvalues oftheDirac operator. If$01Eimip, withmreal, then (ml “igil<1/J»1/J)Z<5x"1/1» 5x,,1P> andso ml> Theabove arguments canberepeated with realspinors. From table 2.15 weseethat theinvolution I,‘ofthereal Clifford algebra isthe adjoint involution ofazero-index product; theproduct being either R-symmetric. C*-symmetric orF1-symmetric. 10.7 Killing Spinors Because oftheimportance ofaknowledge ofthegeodesics ona manifold aninteresting problem ingeneral relativity isthedetermination KILLING SPINORS 301 of first integrals associated with the geodesic equations. Such integrals may be identified with constants of the motion along geodesic curves. Killing symmetries play an important role in the search for such integrals. It was in this context that the notion of a Killing spinor naturally emerged [27]. Since then the same notion has been redisco- vered in the context of finding classical solutions to matter field equations in background geometries [28]. In particular, Killing spinors arise in the study of the residual supersymmetries exhibited by certain solutions to supergravity models. As we shall see the existence of such spinor fields imposes interesting constraints on the geometry of a manifold. A spinor field on some n-dimensional spin manifold M which, for some complex constant A, satisfies Sp Aktp (10.7.1) for all vector fields X, is said to be a Killing spinor. The name arises from the fact that such spinor fields can be used to construct conformal Killing vectors. An immediate consequence of (10.7.1) is that a Killing spinor is an eigenspinor of the Dirac operator, Sip = nAip. We have already noted in the section above that on a compact Riemannian manifold, A must be pure imaginary. Excluding the case in which the signature of the metric on M is (p, q) with p even and q odd then there is an Hermitian symmetric product on complex spinor (or semi-spinor) fields with as adjoint involution. Let ip be the adjoint spinor with respect to this product. Then a real 1-form k is given by k = j't(1071)- We can expand this in a basis {e} as K = J'oezir ea)e a = ca.) e 011)e" = euip)ea SO k* = (p, euv)* = (e ,)ea = eav)e" and k is indeed real. By differentiating (10.7.2) V xk = i(SxVii; + IPS = 1(A- k -11,175 + VÂXV)) ((m,e),371p) + (A:tzp, eatp))ea = ((p, Xe,45-4) + (p, A* ;i s eeatp))ea = 2Re(A)(V, IP)71( + 2i1m(A)(1P, (ea A SO X)tp)ea (Vx k)( 17) + y k)(X) = 4Re(X)(4', ip)g(X, Y). Using Killing's equation, (6.13.3), we have Kg = 4Re(A)(tp,v)g. (10.7.2) (10.7.3) Krtuwo SPINORS 301 offirst integrals associated with thegeodesic equations. Such integrals may beidentified with constants ofthemotion along geodesic curves. Killing symmetries play animportant role inthe search forsuch integrals. Itwas inthis context that thenotion ofaKilling spinor naturally emerged [27]. Since then thesame notion hasbeen redisco- vered inthe context offinding classical solutions tomatter field equations inbackground geometries [28]. Inparticular, Killing spinors arise inthestudy oftheresidual supersymmetries exhibited bycertain solutions tosupergravity models. Asweshall seetheexistence ofsuch spinor fields imposes interesting constraints onthe geometry ofa manifold. Aspinor field onsome n-dimensional spin manifold Mwhich, for some complex constant /1,satisfies SXI/J=,1X1p (10.7.1) forallvector fields X,issaid tobeaKilling spinor. The name arises from thefactthat such spinor fields canbeused toconstruct conformal Killing vectors. Animmediate consequence of(10.7.1) isthat aKilling spinor isaneigenspinor oftheDirac operator, $1/1E n/11/1. We have already noted inthesection above that onacompact Riemannian manifold. /1must bepure imaginary. Excluding thecase inwhich the signature ofthemetric onMis(p,q)with peven andqoddthen there isanHermitian symmetric product oncomplex spinor (orsemi-spinor) fields with E’-<asadjoint involution. Let ifbetheadjoint spinor with respect tothisproduct. Then areal1-form Kisgiven by 1?=e/>l(¢t7;). (10.7.2) Wecanexpand thisinabasis {e“} as 1?=e/u.<w17»'@..>@" =ffl1([email protected]>@" =<[email protected]>@" so I?*=(wl@1w)*@” =(emwe=(w.an/1)@" andKisindeed real. Bydifferentiating (10.7.2) vii?=smsxwti +0%)=fflt/U701? +0%)) =((1/1.ea/H71/1) +(H71/[email protected]))@" =((1/1./lei)?/1) +(1/Al*[email protected]))@" =2R¢(/l)(1P» W))?+Zi1m(/1)(1/1» (en/\Y)1l1)@” SO (Vxi()(Y) +(Vv1?)(X)= 4R¢(/1)(1P»1l1)8(Xl Y)- Using Killings equation, (6.13.3), wehave §£Kg E4Re(/l)(1p, 1/1)g. (10.7.3) 302 SPINOR FIELD EQUATIONS If we took a Hermitian-symmetric product ( , ) with 07* as adjoint (the signature does not have p odd and q even) then if lp is the adjoint with respect to this product then k = (10.7.4) is a real 1 -form. This satisfies Kg = —41m(A)(4', p)g. (10.7.5) Exercise 10.11 Show that if ip is a Killing spinor and K some Killing vector field then op is also a Killing spinor with the same eigenvalue A. The existence of Killing spinors on a Riemannian (as opposed to a pseudo-Riemannian) manifold necessitates interesting integrability con- ditions. We first note that the set of first-order differential equations for the components of tp given by (10.7.1) implies that if the spinor vanishes at some point p E M then it must vanish at all points that are arcwise connected to p [29, 30]. By differentiating (10.7.1) we may obtain an integrability condition involving the curvature. A straightforward cal- culation, using the zero torsion of V, gives S(X, Y)tp = —,1 2[;5e, ijp VX, YE rTm. This can be written in terms of the curvature 2-forms, using (10.3.20), as 4.a(X)eb( Y)Rabli) = —x2ea(x)eb(Y)fea ,eblv or Raop = —4X2eab (10.7.6) Clifford multiplying by ea produces the Ricci forms on the left-hand side: Pop = —4A2(n — 1)e op. Now if A is a real 1-form such that A 4 = 0 then certainly A2lp = 0. But A' = g(A, A) and so for a positive-definite metric we must have A = 0 for ti) non-zero. Thus the above integrability condition is that Pb = —4A2(n — 1)e b (10.7.7) and the manifold must be an Einstein space with curvature scalar given by = —4n(n — 1)A2. (10.7.8) So A must be either real or pure imaginary. We can use (10.7.7) and (10.7.8) to rewrite (10.7.6) in terms of the conformal 2-forms. Sub- stituting (10.7.7) and (10.7.8) into the definition (6.11.6) gives Cab = Rab 4A2eab and hence (10.7.6) becomes 302 SPINOR FIELD EQUATIONS Ifwetook aHermitian-symmetric product (,)with §17*asadjoint (the signature does nothave poddandqeven) then if10istheadjoint with respect tothisproduct then 1?=srl(i¢¢) (10.7.4) isareal1-form. This satisfies .§EKg=-41m(1)(¢, tp)g. (10.7.5) Exercise 10.11 Show that ifzpisaKilling spinor andKsome Killing vector field then &’K1p isalsoaKilling spinor with thesame eigenvalue A. The existence ofKilling spinors onaRiemannian (asopposed toa pseudo-Riemannian) manifold necessitates interesting integrability con- ditions. Wefirstnote thatthesetoffirst-order differential equations for thecomponents of1/Jgiven by(10.7.1) implies thatifthespinor vanishes atsome point peMthen itmust vanish atallpoints that arearcwise connected top[29,30]. Bydifferentiating (10.7.1) wemay obtain an integrability condition involving thecurvature. Astraightforward cal- culation, using thezero torsion ofV,gives S(X,l/)1/1 =-,12[2?, 17111 vx,YeFTM. This canbewritten interms ofthecurvature 2-forms, using (10.3.20), as %@”(X)@"(Y)Rab1/1 =—/l2@”(X)@"(Y)[email protected]@1]1l1 or R,,,1/1 =—4A2e,,,,1p. (10.7.6) Clifford multiplying bye“produces theRicci forms ontheleft-hand side: Pl,1/1 E—4/1Z(n —1)el,1/J. Now ifAisareal 1-form such that A1}:=0then certainly A21/1 E0. ButA2Eg(A, A)andsoforapositive-definite metric wemust have A=0for1;‘;non-zero. Thus theabove integrability condition isthat Pl,=—4}.2(n —1)€l, (10.7.7) andthemanifold must beanEinstein space with curvature scalar given by 91E—4n(n —l)A2. (10.7.8) So/1must beeither real orpure imaginary. Wecanuse(10.7.7) and (10.7.8) torewrite (10.7.6) interms oftheconformal 2-forms. Sub- stituting (10.7.7) and (10.7.8) into thedefinition (6.11.6) gives Cal,= R,,l,+4A2e,,l, andhence (10.7.6) becomes KILLING SPINORS 303 Cab = 0- (10.7.9) To go further we must make another assumption about M. A Riemannian manifold is locally symmetric if its curvature tensor is parallel. If M is locally symmetric then the conformal tensor is parallel and the conformal 2-forms satisfy VxCan = Ccbwc„(X) C„cw`b(X) VXET TM. (10.7.10) Differentiating (10.7.9) and using (10.7.1) and (10.7.10) gives {CpbcoP„(Xe) + CapcoPb(Xe)Iii) + ylC„beelp = 0. The first two terms vanish by (10.7.9), and so for A * 0 we have Cabe, = 0. From (10.7.9) we have e,Cabtp = 0 and so subtracting these gives icC„bli) = 0 and hence Cab = 0. (10.7.11) Together (10.7.7), (10.7.8) and (10.7.11) show that R „b = —4/12e ah, that is, M has a constant sectional curvature of —4/12. Hence the only locally symmetric Riemannian manifolds such that (10.7.1) has a solution for A* 0 are the standard sphere, in which case  is imaginary, or a hyperbolic space with A real, or a quotient of these spaces by a discrete group. 10.8 Parallel Spinors A spinor field ip is parallel if Skip = 0 V X E FTM. (10.8.1) Thus a parallel spinor is a special case (A = 0) of a Killing spinor. Not surprisingly M must be tightly constrained if it is to admit a parallel spinor. A discussion of parallel spinors necessitates a brief mention of Kahler manifolds. A tensor field J e F TIM is an almost complex structure on M if .12X J(J(X)) = —X V X EFTM. (10.8.2) A Riemannian manifold (M, g) with an almost complex structure J that is an isometry, g(JX, JY) = g(X, Y) V X, Y EFTM (10.8.3) and is parallel V xJ = 0 VXEFTM (10.8.4) KILLING SPINORS 303 c,,,,¢=0. (10.7.9) Tog0further wemust make another assumption about M.A Riemannian manifold islocally symmetric ifitscurvature tensor is parallel. IfMislocally symmetric then theconformal tensor isparallel andtheconformal 2-forms satisfy VXCa,, =C(l,w‘,,(X) +C,,(w‘l,(X) VXETTM. (10.7.10) Differentiating (10.7.9) andusing (10.7.1) and(10.7.10) gives {Cpbwpa(Xc) +Cnpwpb(Xc)}w +A-Ctiherw = The first two terms vanish by(10.7.9). and soforAEO wehave C,,,,e,.1/1 =0.From (10.7.9) wehave el.C,,l,1/1 E0andsosubtracting these gives i,.C,,l,1/1 E0andhence c,,,=0. (10.7.11) Together (10.7.7), (10.7.8) and(10.7.11) show that R,,,,=—4A2e,,,,, that is,Mhasaconstant sectional curvature of—4A3. Hence theonly locally symmetric Riemannian manifolds such that (10.7.1) hasasolution for AEO arethestandard sphere, inwhich case Aisimaginary, ora hyperbolic space with Areal, oraquotient ofthese spaces byadiscrete gI'Ollp. 10.8 Parallel Spinors Aspinor field zpisparallel if SXI/)=() VXEFTM. (10.8.1) Thus aparallel spinor isaspecial case (AE0)ofaKilling spinor. Not surprisingly Mmust betightly constrained ifitistoadmit aparallel spinor. Adiscussion ofparallel spinors necessitates abrief mention of Kahler manifolds. Atensor field JeI'T}M isanalmost complex structure onMif JZX EJ(J(X)) =—X VXE FTM. (10.8.2) ARiemannian manifold (M,g)with analmost complex structure J thatisanisometry. g(JX, JY)Eg(X, Y) VX, YEFTM (10.8.3) andisparallel VXJ E0 VXG TTM (10.8.4) 304 SPINOR FIELD EQUATIONS is called a Kahler manifold. A theorem due to Hitchin [31] states that a compact even-dimensional Riemannian spin manifold admitting a para- llel spinor is a Kahler manifold. For the special case of four dimensions a direct proof requiring orientability, but not compactness, can be found in [29]. It is possible to prove rather easily a result about parallel pure spinors on even-dimensional Riemannian manifolds. An even-dimensional Riemannian spin manifold admitting a parallel (complex) pure spinor is a Ricci-flat Kahler man- ifold. (10.8.5) The Ricci flatness is just a special case of (10.7.7). Pure spinors were introduced in Chapter 3. Recall from there that pure spinors are Weyl spinors (they carry a semi-spinor representation of the complexified even subalgebra). At each point p of M a non-vanishing pure spinor ip determines a maximal isotropic subspace jp+ of the complexified cotan- gent space by xtpp = 0 for X E rpMc iff X E j;. (10.8.6) We have rple = 0 4,-, where x* c $; if and only if x e. So a non-vanishing pure spinor field assigns a maximal isotropic subspace to the complexified cotangent space of every point. Let 4+ and j- be the spaces of complex differential 1-forms such that x ej+ if and only if E /p+. Given the subspaces 1+ and j -, determined by the pure spinor, we can define an almost complex structure J by Jx = ix V x E j+ (10.8.7) Jy = —iy Vy Ej—. (Note that we here think of J as an endomorphism of the cotangent (rather than the tangent) space.) Since it has eigenvalues ±i then J is certainly an almost complex structure, and since complex conjugation interchanges 4+ and j- it is a real tensor field. Since J preserves the isotropic subspaces j+ and j -, then to check that J is an isometry we need only consider the metric evaluated on an element of j+ and of j-. Let x E j+ and y E j— then g(Jx, Jy) = g(ix, —iy) = g(x, y) and so J satisfies (10.8.3). Since tit is parallel then the subspace j+ (and hence j-) is preserved under covariant differentiation. For if xtp = 0 and tp is parallel then V xxtp = 0 and hence V xx E j+ V x E 1±, V X e TM. Since covariant differentiation commutes with complex conjugation then it also preserves j-. Now if x E j+ we have Jx = ix and hence (V xJ)x +J(V xx)=iVxx. Since Vxx E j+ we have V xix = 0 and V xJx* = 0, hence V xJ = 0. Thus we have established (10.8.5). We can use the metric to construct a 2-form out of an almost complex structure satisfying (10.8.3). If J = Jahea Xb then the usual index- 304 SPINOR FIELD EQUATIONS iscalled aKahler manifold. Atheorem duetoHitchin [31]states that a compact even-dimensional Riemannian spin manifold admitting apara- llelspinor isaKahler manifold. Forthespecial case offour dimensions adirect proof requiring orientability, butnotcompactness, canbefound in[29]. Itispossible toprove rather easily aresult about parallel pure spinors oneven-dimensional Riemannian manifolds. Aneven-dimensional Riemannian spin manifold admitting aparallel (complex) pure spinor isaRicci-flat Kahler man- ifold. (10.8.5) The Ricci flatness isjustaspecial case of(10.7.7). Pure spinors were introduced inChapter 3.Recall from there that pure spinors areWeyl spinors (they carry asemi-spinor representation ofthecomplexified even subalgebra). Ateach point pofManon-vanishing pure spinor 1/1,, determines amaximal isotropic subspace jgofthecomplexified cotan- gent space by xt/1,, E0 forxET"j,,MC iffxE3;. (10.8.6) Wehave TQM‘: E.5;®Q;where x*E}; ifandonly ifxE}; Soa non-vanishing pure spinor field assigns amaximal isotropic subspace to thecomplexified cotangent space ofevery point. Let}*andQ‘bethe spaces ofcomplex differential 1-forms such that xe}* ifand only if x|l,e§;. Given thesubspaces 50*and j", determined bythepure spinor, wecandefine analmost complex structure Jby JxEix VxE§* (10.8.7) JyE —iy Vyejli (Note that wehere think ofJasanendomorphism ofthecotangent (rather than thetangent) space.) Since ithaseigenvalues iithen Jis certainly analmost complex structure, andsince complex conjugation interchanges }*and3'itisarealtensor field. Since Jpreserves the isotropic subspaces }*andQ1“, then tocheck that Jisanisometry we need only consider themetric evaluated onanelement of}*and of §l'.Letxe$* andy6}‘ then g(Jx, Jy)Eg(ix. —iy) Eg(x,y)andso Jsatisfies (10.8.3). Since 1pisparallel then thesubspace 5%"(and hence Q1‘)ispreserved under covariant differentiation. Forifx1/1E0and1,0is parallel then Vxxzp E0and hence Vxxe}* Vxe}*, VXE FTM. Since covariant differentiation commutes with complex conjugation then italso preserves 3". Now ifxE§* wehave JxEixand hence (VXJ)x +J(VXx) EiV)l»x. Since Vxxe}* wehave VXJx E0and VXJx* E0,hence VXJ E0.Thus wehave established (10.8.5). Wecanusethemetric toconstruct a2-form outofanalmost complex structure satisfying (10.8.3). IfJEJ,,”e" ®X,,then theusual index- PARALLEL SPINORS 305 lowering rule gives Jab = g(JX „, X b). If J satisfies (10.8.3) then g(JX„, X b)= —g(JX„, J'X b) = —g(X„, JX b) = —g(JX b, X0) and J,,,, The 2-form Q ijabeab (10.8.8) is called the Kahler 2-form. If x is any 1-form then Jx = —i,Q = J I(Qx). (10.8.9) We showed above that an even-dimensional Riemannian manifold admitting a parallel pure spinor is a Kahler manifold. In this case the Kahler 2-form can be constructed out of the spinor. If denotes the adjoint spinor with respect to the Hermitian spinor product whose adjoint involution is then a real 2-form F is given by F = (10.8.10) For any 1-form x i(Fx) = Yi{itPix Jo(ili»/-P-)x} = xe,,)ea — Yo(i/Pi)x and ao(itp/Tuea) = g(x, eax)) xea) = g(x, Yo(i/Pir)eax) = g(x, ea)Yo(ivi—p) — 1J„(ixtpiea). If now lp is a pure spinor and x c , as determined by (10.8.6), then the last term in the above vanishes. Thus for x E .Y+ I(Fx) = Y o(itp ir))x = tp)x. Since Op, tp) > 0 for zp 0 the Kahler 2-form Q related to the almost complex structure J of (10.8.7) is given by Q — 922(ivi) (10.8.11) (V, 1P) By only considering parallel pure spinors we have been able to use a basically algebraic argument to see directly that M must be a Kahler manifold. If M is even dimensional and orientable, with dim M 6 then if M admits a parallel spinor then it admits a parallel pure spinor. If M is orientable with ip parallel then the Weyl spinors ;(1 ± si)ip are also parallel where i is proportional to the volume form on M such that = 1. But for dim M 6 all Weyl spinors are pure and hence M is a Kahler manifold. Notice that we need to assume orientability but not compactness. In the above we have studied some of the conditions that are PARALLEL SPINORS 305 lowering rulegives J,,,,Eg(JX,,, Xl,). IfJsatisfies (10.8.3) then 8(JX,,. X1)=-g(/X.” JZX1) =-g(X..l JX1) =-8(JX1l XII) andJlll,E—J,,l,. The 2-form QE§J,,l,e”" (10.8.8) iscalled theKahler 2-form. IfXisany1-form then JxE—i,l-Q EHl(Qx). (10.8.9) We showed above that aneven-dimensional Riemannian manifold admitting aparallel pure spinor isaKahler manifold. Inthiscase the Kahler 2-form canbeconstructed outofthespinor. Ifll;denotes the adjoint spinor with respect totheHermitian spinor product whose adjoint involution is5*then areal2-form Fisgiven by F=a,(1¢?l7). (10.8.10) Forany1-form x -91(FX) =9’1{iWTX -5*0(iWT)X} =5f0(iWl7X@a)@“ E9’(>(i1lH77)X and 3(0([email protected]) :g(x, +i5f0(i1l11T(X@~ _eax)) Hjxerl) =g(-xi —2y0(iw’Fea-X) =g(x,@..)5f0(iWl7) -%5f(»([email protected])- Ifnow 1pisapure spinor andxe§*, asdetermined by(10.8.6), then thelastterm intheabove vanishes. Thus forxE,;V' 9115‘) =y1>(lW1l7)X =l(lP1ll0X- Since (1)1,ip)>0for1/1E0theKahler 2-form Qrelated tothealmost complex structure Jof(10.8.7) isgiven by Q=Eli). (10.8.11) (11%W) Byonlyconsidering parallel pure spinors wehave been abletousea basically algebraic argument toseedirectly that Mmust beaKahler manifold. IfMiseven dimensional andorientable, with dimME6then ifMadmits aparallel spinor then itadmits aparallel pure spinor. IfM isorientable with 1/1parallel then theWeyl spinors §(1iE)1/J arealso parallel where Eisproportional tothevolume form onMsuch that E3E1.ButfordimME6allWeyl spinors arepure andhence Misa Kahler manifold. Notice that weneed toassume orientability butnot compactness. Inthe above wehave studied some ofthe conditions that are 306 SPINOR FIELD EQUATIONS necessary for the existence of parallel pure spinor fields. The existence of compact Ricci flat manifolds was first demonstrated by Yau [32] following a famous conjecture by Calabi. When the very stringent necessary conditions for a parallel spinor are met one can sometimes appeal to the powerful Atiyah—Singer index theorem [33] to show that a parallel spinor does in fact exist. This theorem relates the differing numbers of 'left- and right-handed' Weyl solutions of the massless Dirac equation on a compact Riemannian manifold to a topological invariant. By the Lichnerowicz theorem we know that for a Ricci-flat compact Riemannian manifold the only such solutions are parallel spinors. Thus if the topological invariant is such that the difference between the number of left- and right-handed solutions is non-zero then there must exist parallel spinors. Exercise 10.12 Show that the almost complex structure on a Kahler manifold can be used to define a sub-bundle of minimal left ideals of the complexified Clifford bundle. Hence a Kahler manifold is a Spin' manifold. Show that the Riemannian connection induces a connection on this sub- bundle, and hence the Kahler equation can be restricted to a minimal left ideal. The importance of spinor fields in classical differential geometry has rarely been doubted. That they play an important role in many theories in physics is an act of faith shared by many physicists. In recent times a great deal of theoretical physics and differential geometry has become closely intertwined. The properties of Killing spinors are an example where both disciplines have gained mutual benefit from this interaction. In this book we have attempted to bring the amalgam of ideas that constitute Clifford algebras, differential geometry and the theory of spinors into a form that we hope will stimulate some readers to pursue such a synthesis further. 306 SPINOR FIELD EQUATIONS necessary fortheexistence ofparallel pure spinor fields. Theexistence ofcompact Ricci flatmanifolds was first demonstrated byYau [32] following afamous conjecture byCalabi. When thevery stringent necessary conditions foraparallel spinor aremet one cansometimes appeal tothepowerful Atiyah—Singer index theorem [33]toshow that a parallel spinor does infact exist. This theorem relates thediffering numbers of‘left- andright-handed’ Weyl solutions ofthemassless Dirac equation onacompact Riemannian manifold toatopological invariant. BytheLichnerowicz theorem weknow thatforaRicci-flat compact Riemannian manifold theonly such solutions areparallel spinors. Thus ifthetopological invariant issuch that thedifference between the number ofleft- andright-handed solutions isnon-zero then there must exist parallel spinors. Exercise 10.12 Show that thealmost complex structure onaKahler manifold canbe used todefine asub-bundle ofminimal leftideals ofthecomplexified Clifford bundle. Hence aKahler manifold isaSpinc manifold. Show that theRiemannian connection induces aconnection onthissub- bundle, andhence theKahler equation canberestricted toaminimal leftideal. The importance ofspinor fields inclassical differential geometry has rarely been doubted. That they play animportant roleinmany theories inphysics isanactoffaith shared bymany physicists. Inrecent times a great deal oftheoretical physics anddifferential geometry hasbecome closely intertwined. The properties ofKilling spinors areanexample where both disciplines have gained mutual benefit from thisinteraction. Inthisbook wehave attempted tobring theamalgam ofideas that constitute Clifford algebras, differential geometry and thetheory of spinors into aform that wehope willstimulate some readers topursue such asynthesis further. Appendix A Algebra In this appendix we have collected those algebraic results that we have referred to in the book. Thus the account here is very much tailored to our specific needs rather than giving a balanced view of the subject. The first few pages mostly define terminology that we have used. Although this is fairly standard the various morphisms' are used by different authors in slightly different ways, and there are some alternative terms that we have not listed. The section on algebras is much more dense, leading up to a proof of the structure theorem for simple algebras. Although the average reader will probably not want to plough through this exposition he will need to know the final result, and how it may be used to construct, for example, explicit representations of 7-matrices. The approach we have adopted is the historical one; more modern treatments prove the structure theorems for a wider class of rings than algebras over fields. We found useful the classic books of Albert (1961) [1] and Dickson (1960) [2], and the more modern book by Kochendorf- fer (1972) [3]. There are, of course, an abundance of books in which this material can be found, to suit all tastes. A group, G, consists of a set with a binary operation, or law of composition, that satisfies four axioms. Usually multiplicative notation is used to denote this group operation, the juxtapositioning of elements denoting their composition. In view of this notation we shall often refer to the law of composition as a product. The axioms are as follows. (i)For every a, b e G there is a unique c E G such that ab = c. (ii)The product is associative, (ab)c = a(bc). (iii)There exists an identity (or unit element), denoted 1, such that al = la = a V a E G. (iv)Every element a has an inverse a', aa-1 = aa = 1. When a group consists of a finite number of elements then this number is called the order of the group. In general the group product is Appendix A Algebra Inthisappendix wehave collected those algebraic results that wehave referred tointhebook. Thus theaccount here isvery much tailored to ourspecific needs rather than giving abalanced view ofthesubject. The first fewpages mostly define terminology that wehave used. Although this isfairly standard thevarious ‘morphisms’ areused bydifferent authors inslightly different ways, andthere aresome alternative terms that wehave notlisted. The section onalgebras ismuch more dense. leading uptoaproof ofthestructure theorem forsimple algebras. Although theaverage reader willprobably notwant toplough through thisexposition hewillneed toknow thefinal result, andhow itmay be used toconstruct, forexample, explicit representations ofy-matrices. The approach wehave adopted isthehistorical one; more modern treatments prove thestructure theorems forawider class ofrings than algebras over fields. Wefound useful theclassic books ofAlbert (1961) [1]andDickson (1960) [2],andthemore modern book byKochendorf- fer(1972) There are, ofcourse, anabundance ofbooks inwhich thismaterial canbefound, tosuitalltastes. Agroup, G,consists ofasetwith abinary operation, orlawof composition, thatsatisfies four axioms. Usually multiplicative notation is used todenote thisgroup operation. thejuxtapositioning ofelements denoting their composition. Inview ofthisnotation weshall often refer tothelawofcomposition asaproduct. Theaxioms areasfollows. (i)Forevery a,beGthere isaunique ceGsuch thatabEc. (ii)Theproduct isassociative, (ab)c Ea(bc). (iii)There exists anidentity (orunit element), denoted 1,such that a1ElaEa VaeG. (iv)Every element ahasaninverse a“', aa“ Ea"a E1. When agroup consists ofafinite number ofelements then this number iscalled theorder ofthegroup. Ingeneral thegroup product is 308 APPENDIX A not commutative, ab * ha. The set of elements that commute with all other elements is called the centre. A group for which the product of any two elements is commutative is called Abelian. Often additive notation is used to denote the law of composition in an Abelian group, in which case the identity is written as 0. A subset H, of a group G, which forms a group under the product of G is called a subgroup. Thus H is a subgroup if and only if uveliVu,yEH,u -IEHVuEH and E H. For example, the centre is a subgroup. We may form a subgroup H from any subset S of a group G by taking the set of all products that can be formed from elements of S and their inverses; this group is said to be generated by S. A subgroup enables a group to be decomposed into equivalence classes. If we have an equivalence relation on a set such that a is equivalent to h then we write a b. Equivalence relations satisfy a — a, a — b for b — a, and if a — b and b c then a — c. The set of all elements equivalent to an element a constitute the equivalence class of a, [a]. Any element of [a], such as a, is called a representative of the class. The equivalence classes of distinct elements are either identical or non-intersecting. If H is a subgroup of G then an equivalence relation on G is defined by a — b if b = ah for some h E H. The equivalence class of a is called the left coset of G, relative to H, generated by a. In an obvious way we define right cosets. For a special type of subgroup the cosets inherit a group structure. A subgroup H is called normal (or invariant) if ghg--1 c H VgE G, V h E H. The nota- tion H G denotes that H is a normal subgroup of G. It follows that the left and right cosets relative to a normal subgroup are equal. These cosets form a group under the product defined by [a][b] = [ab]. Since [a] = [ah] for h E H this definition only makes sense if H is normal. This group of cosets is called the quotient of G modulo 1-1, denoted GIH. We give an example. The set of integers (positive and negative) forms an Abelian group under addition, denoted Z. Any integer n generates a subgroup H. Thus H consists of the set {0, ±-n, -±2n, ±3n, . . .}. Any subgroup of an Abelian group is normal and so we can form the quotient, Z„ = ZIH. If m is any integer then m= qn +r, where 0 r < n, and so every element of Z is equivalent to a positive integer less than n. The class of the sum of two such integers is represented by their sum modulo a multiple of n. For example, Z, has two elements, [0] and [1], and [1] + [1] = [2] = [0]. (The notation Z„ will be used to denote any group isomorphic to these quotients. For example, the set {I, —1} forms a group under multiplication, isomorphic to Z2.) Roughly speaking a homomorphism is a mapping between groups that preserves the structure. Let q) be a mapping from G to G', then cp is a homomorphism if cp(ab)= cp(a)T(b). The product on the left-hand side is that of G whilst the product on the right-hand side is that of G'. If every element of G' is the image of some element of G under cp, then cp 308 APPENDIX A notcommutative. abEba.The setofelements that commute with all other elements iscalled thecentre. Agroup forwhich theproduct of any two elements iscommutative iscalled Abelian. Often additive notation isused todenote thelawofcomposition inanAbelian group, inwhich case theidentity iswritten as0.Asubset H,ofagroup G, which forms agroup under theproduct ofGiscalled asubgroup. Thus Hisasubgroup ifandonly ifuuEH Vu, uEH,u"EHVuEH and 1EH.Forexample, thecentre isasubgroup. Wemay form asubgroup Hfrom anysubset Sofagroup Gbytaking thesetofallproducts that canbeformed from elements ofSandtheir inverses; thisgroup issaid tobegenerated byS.Asubgroup enables agroup tobedecomposed into equivalence classes. Ifwehave anequivalence relation onaset such that aisequivalent tobthen wewrite a~b.Equivalence relations satisfy a~a,a~bforb~a,and ifa~band b~cthen a~c.The setofallelements equivalent toanelement aconstitute the equivalence class ofa,[a].Any element of[a],such asa,iscalled a representative oftheclass. The equivalence classes ofdistinct elements areeither identical ornon-intersecting. IfHisasubgroup ofGthen an equivalence relation onGisdefined bya~bifbEahforsome hEH. The equivalence class ofaiscalled theleftcoset ofG,relative toH, generated bya.Inanobvious waywedefine right cosets. Foraspecial type ofsubgroup thecosets inherit agroup structure. Asubgroup His called normal (orinvariant) ifghg"1E HVge G,VhEH.The nota- tion H<IGdenotes that Hisanormal subgroup ofG.Itfollows that theleftandright cosets relative toanormal subgroup areequal. These cosets form agroup under theproduct defined by[a][b] E[ab]. Since [a]E[ah] forhEH thisdefinition only makes sense ifHisnormal. This group ofcosets iscalled thequotient ofGmodulo H,denoted G/H. Wegive anexample. The setofintegers (positive andnegative) forms anAbelian group under addition. denoted Z.Any integer n generates asubgroup H.Thus Hconsists oftheset{0,in, i2n, i3n, ...}.Any subgroup ofanAbelian group isnormal andsowecanform thequotient, Z”EZ/H. Ifmisanyinteger then mEqn+r,where 0Er<n,andsoevery element ofZisequivalent toapositive integer lessthan n.Theclass ofthesumoftwosuch integers isrepresented by their sum modulo amultiple ofn.Forexample, Z3hastwoelements, [0]and[1],and[1]+[1]E[2]E[0].(The notation Z”willbeused to denote anygroup isomorphic tothese quotients. Forexample, theset {l,—1}forms agroup under multiplication, isomorphic toZ3.) Roughly speaking ahomomorphism isamapping between groups that preserves thestructure. Let (lobeamapping from GtoG’, then rpisa homomorphism ifrp(ab) E(p(a)(p(l7). The product ontheleft-hand side isthat ofGwhilst theproduct ontheright-hand side isthat ofG’.If every element ofG’istheimage ofsome element ofGunder cp,then qo APPENDIX A 309 is called surjective (or onto). If no two elements of G get mapped into the same element then yo is called injective (or one-to-one). A mapping that is both injective and surjective is called bijective. Groups that are related by a bijective homomorphism are called isomorphic, and we write G' -= G. In general a homorphism ço is not injective, and the set of elements in G mapped onto the identity of G' is called the kernel of ço (ker cp). The kernel of ço is a normal subgroup of G, and we have cp(G) G/ker cp. (Al) (This is known as the first isomorphism theorem.) This may be proved by introducing a map (LI, (13 : G/ker cp q(G) [a] cl)([a]) = cp(a). The proof consists of showing that not only does such a definition make sense, but (I) is a bijection. The following is usually known as the second isomorphism theorem. If N G and A G such that N A G then GIN GIA. (A2) A/N The conditions on the subgroups are just such as are required for this to make sense. The equivalence class of a in G given by N is written [a]N; [cilA being similarly defined. The proof of (A2) is established by introducing a map ço, cp:GIN-- GIA [allyq ([42]N) = [(I]A  Not only is such a map well defined but it is a surjective homomorphism with kernel AIN. Then (A2) follows from (Al). If H and K are two groups then there is a natural way in which the Cartesian product of these sets can be given a group structure. The Cartesian product set consists of ordered pairs of an element of H and an element of K. If (hl, lc]) and (h2, k2) are two such pairs then we may define their product by (h 1, k i)(h,, k2)= (h1h2, kik,). If G denotes the group formed by such pairs then G is the direct product of H and K, written G = H x K. An isomorphism from a group to itself is called an automorphism. If cp and p are automorphisms of G then their product may be defined by (cfv)(a) = cp(zp(a)). Under this product the set of all automorphisms of G forms a group, Aut G. If t is any element of G then we have a T in the automorphism group given by 1-(a) = tat -'. Such an automorphism is called an inner automorphism. APPENDIX A 309 iscalled surjective (oronto). Ifnotwoelements ofGgetmapped into thesame element then rpiscalled injective (orone-to-one). Amapping that isboth injective andsurjective iscalled bijective. Groups that are related byabijective homomorphism arecalled isomorphic, andwe write G’EG.Ingeneral ahomorphism cpisnotinjective, andtheset ofelements inGmapped onto theidentity ofG’iscalled thekernel of cp(kercp). Thekernel ofcpisanormal subgroup ofG,andwehave (p(G) EG/ker (p. (A1) (This isknown asthefirstisomorphism theorem.) This may beproved byintroducing amap CD, <1):G/kerrp E> cp(G) la]'—E>‘P(l@l) =(P01)- Theproof consists ofshowing thatnotonlydoes such adefinition make sense, but<1)isabijection. Thefollowing isusually known asthesecond isomorphism theorem. IfN(IGand A<1Gsuch that N¢A(IG then G/N——— EG/A. A2 A/N <> Theconditions onthesubgroups arejustsuch asarerequired forthisto make sense. Theequivalence class ofainGgiven byNiswritten [a]N; [a],l being similarly defined. The proof of(A2) isestablished by introducing amap rp, cp:G/N E> G/A la]/v '—'_> (P(lulu) :[ala- Notonly issuch amap well defined butitisasurjective homomorphism withkernel A/N. Then (A2) follows from (A1). IfHand Karetwo groups then there isanatural way inwhich theCartesian product ofthese setscanbegiven agroup structure. The Cartesian product setconsists ofordered pairs ofanelement ofHand anelement ofK.If(hl, kl)and(hl, kl)aretwosuch pairs then we may define their product by(hl, kl)(h3. kl)E(hjhz, kl/<2). IfG denotes thegroup formed bysuch pairs then Gisthedirect product of HandK,written GEHXK.Anisomorphism from agroup toitself iscalled anautomorphism. Ifcpand 1pareautomorphisms ofGthen their product may bedefined by(([J‘t/!)(d) Ecp(1/1(a)). Under thisproduct thesetofallautomorphisms ofGforms agroup. AutG. Iftisany element ofGthen wehave arintheautomorphism group given by r(a)Etat". Such anautomorphism iscalled aninner automorphism. 310 APPENDIX A Any automorphism that is not inner is called an outer automorphism. The ordered pairs consisting of an element of a group and an element of a group of automorphisms can be given a group structure other than that of direct product. If Q is a subgroup of Aut G then for col, w2 E Q, a 1, a2 E G we define (a1, coi)(a2, to2) = (aiwi(a,), wito,). With such a product we have (a, (o)' = (co -1(a-1), 0i'). The ordered pairs under this product form the semidirect product of G and Q, K say, written K = GC)Q. A ring has two binary operations, addition, denoted +, and multi- plication, denoted by juxtaposing elements. Under addition a ring forms an Abelian group, the additive identity being called the zero element. Multiplication is associative (unless specifically stated otherwise) and distributive over addition, (a + b)c = ac + bc c(a + b) = ca + cb. A commutative ring is one in which multiplication is commutative. The set of elements that commute with all other elements under multiplica- tion is called the centre. A ring need have no identity (or unit element), denoted 1, by which is meant a unit element under multiplication. For a ring with unit element an element a is called regular (or invertible) if it has a multiplicative inverse a', that is act' = = 1. A ring in which every non-zero element is regular is called a division ring. We have already noted that the integers, Z, form an Abelian group under addition; with multiplication they form a ring. Similarly with multiplica- tion being defined modulo n the group Z,, forms a ring. A field is a commutative division ring. (Sometimes a non- commutative division ring is called a skew field.) Familiar examples of fields are the rational numbers Q, the real numbers lR and the complex numbers C. For p a prime number then an example of a field with a finite number of elements is Z p. A field F is said to be of characteristic p if there is a prime number p such that a+a+a...+a=0 V a E F. p terms. In this case F contains Z p as a subfield. If there is no such p then F is said to be of characteristic zero, and in this case it contains the rational numbers as a subfield. We shall really only be concerned with the zero characteristic fields IFI and C. The complex numbers have the property of being algebraically closed, which results in the property that we shall observe of enabling any complex number to be written as a square. The real numbers do not have this property, no negative number being a square of a real number. A vector space over a field F, V, is a set (of vectors) with an operation of addition and a rule of scalar multiplication, which assigns a 310 APPENDIX A Any automorphism that isnotinner iscalled anouter automorphism. Theordered pairs consisting ofanelement ofagroup andanelement of agroup ofautomorphisms canbegiven agroup structure other than thatofdirect product. If£2isasubgroup ofAutGthen forwl,(U2EQ, al,aleGwedefine (al, wl)(a2, 0);)E(alwl(a2), wlwz). With such a product wehave (a,0))“ E(w“(a"), of‘). The ordered pairs under thisproduct form thesemidirect product ofGand Q,Ksay, written KEGQQ. Aring hastwo binary operations, addition, denoted +,andmulti- plication, denoted byjuxtaposing elements. Under addition aringforms anAbelian group, theadditive identity being called thezero element. Multiplication isassociative (unless specifically stated otherwise) and distributive over addition, (a+b)cEac+bc c(a+b)Eca+cb. Acommutative ring isoneinwhich multiplication iscommutative. The setofelements that commute with allother elements under multiplica- tioniscalled thecentre. Aring need have noidentity (orunitelement), denoted 1,bywhich ismeant aunitelement under multiplication. Fora ringwith unit element anelement aiscalled regular (orinvertible) ifit hasamultiplicative inverse a“, that isaa" Ea"‘a E1.Aring in which every non-zero element isregular iscalled adivision ring. We have already noted that theintegers, Z,form anAbelian group under addition; with multiplication they form aring. Similarly with multiplica- tionbeing defined modulo nthegroup Z,forms aring. Afield isacommutative division ring. (Sometimes anon- commutative division ring iscalled askew field.) Familiar examples of fields aretherational numbers Q,therealnumbers lRandthecomplex numbers C.Forpaprime number then anexample ofafield with a finite number ofelements isZp.Afield Fissaid tobeofcharacteristic pifthere isaprime number psuch that a+a+a...+aE0 VaEF.___-,€__s pterms. InthiscaseFcontains 2,,asasubfield. Ifthere isnosuchpthen Fis said tobeofcharacteristic zero, andinthiscase itcontains therational numbers asasubfield. Weshall really only beconcerned with thezero characteristic fields lRandC.The complex numbers have theproperty ofbeing algebraically closed, which results intheproperty that weshall observe ofenabling anycomplex number tobewritten asasquare. The real numbers donothave thisproperty, nonegative number being a square ofarealnumber. Avector space over afield F,V,isaset(ofvectors) with an operation ofaddition andaruleofscalar multiplication, which assigns a APPENDIX A 311 vector to the product of a vector with an element of the field. (In this context elements of the field are called scalars.) Under addition the vectors form an Abelian group, with multiplication by scalars satisfying the following: (i)(An)x = A(ux) (ii)(A + su)x = Ax + tx A.(x + y) = + Ay V A, E F, x, y E V. (iii)ix = x, where 1 is the unit element of F. If {x,) is a set of vectors such that x = E,A1x, for A' E F then x is said to be a linear combination of the xi. A set of vectors is called linearly dependent if any one vector can be written as a linear combination of the others. Conversely the set {x,} is linearly independent if /,Arx, = 0 implies that all A' are zero. A set of vectors {x,} is said to span V (or generate V) if any element of V can be written as a linear combination of the xi. A linearly independent spanning set is called a basis, or linear frame. Every vector space admits a basis, and when the vector space is spanned by a finite set any basis contains the same number of vectors, called the dimension of the vector space V, denoted dim V. Any vector can be written as a linear combination of the basis vectors, the uniquely determined scalar coefficients being termed the components of the vector with respect to that basis. If {e,} and {L} are distinct bases then the elements of one basis can be written as linear combinations of the other basis vectors, e- E f, = E Bige. Substituting either expression into the other gives E B/Aik = EAjkBkJ = ôjJ k =1 where the Krônecker 6,1 takes the value zero unless i = j when its value is one. Thus the coefficients relating the change of basis can be displayed as a non-singular n x n matrix, with entries in F. Such non-singular matrices form a group under matrix multiplication, the general linear group over F, Gl(n, F). In the above expressions we have chosen to position certain indices as superscripts, others as subscripts. It is often convenient to adopt the Einstein summation convention in which summation is implied over any repeated index, occurring once as a superscript and once as a subscript. Thus in the above expressions we APPENDIX A 311 vector totheproduct ofavector with anelement ofthefield. (Inthis context elements ofthefield arecalled scalars.) Under addition the vectors form anAbelian group, with multiplication byscalars satisfying thefollowing: (vi)(Mr=Max) (ii)(/l+u)xE/ix+ux /l(x+y)E/lx+/ly V/1,/tEF,x,yEV. (iii)1xEx,where 1istheunitelement ofF. If{xl} isasetofvectors such thatxEE,-}."xl for)1‘EFthen xissaidto bealinear combination ofthex,-.Asetofvectors iscalled linearly dependent ifanyonevector canbewritten asalinear combination of theothers. Conversely theset{x,-} islinearly independent ifZ,-/llxl E0 implies that all/1"arezero. Asetofvectors {x,-} issaid tospan V(or generate V)ifanyelement ofVcanbewritten asalinear combination ofthexl.Alinearly independent spanning setiscalled abasis, orlinear frame. Every vector space admits abasis, andwhen thevector space is spanned byafinite setanybasis contains thesame number ofvectors, called thedimension ofthevector space V,denoted dimV.Any vector canbewritten asalinear combination ofthebasis vectors, theuniquely determined scalar coefficients being termed the components ofthe vector with respect tothat basis. If{el} and{fl}aredistinct bases then theelements ofonebasis canbewritten aslinear combinations ofthe other basis vectors, fl er=EA/if; /=1 ll = 2 B]-"8,-. l=1 Substituting either expression intotheother gives 2B]_iAl_k =511 iEl 2A/(B/<" =5/0k=1 where theKronecker 6,1takes thevalue zero unless iEjwhen itsvalue isone. Thus the coefficients relating thechange ofbasis can be displayed asanon-singular nXnmatrix, with entries inF.Such non-singular matrices form agroup under matrix multiplication, the general linear group over F,Gl(n, F).Intheabove expressions wehave chosen toposition certain indices assuperscripts, others assubscripts. It isoften convenient toadopt theEinstein summation convention inwhich summation isimplied over any repeated index, occurring once asa superscript andonce asasubscript. Thus intheabove expressions we 312 APPENDIX A would simply omit the summation sign when using the summation convention. We shall frequently use this convention without further comment. When it is not clear from the context whether a sum is implied or not we shall explicitly state, for example, no sum. A subset U of a vector space V such that all linear combinations of vectors from U lie in U is called a vector subspace. The zero element and V itself are obviously vector subspaces, any other subspace being termed non-trivial. If S is any subset from V then all linear combina- tions of vectors from S form a vector subspace which is said to be generated, or spanned, by S. The dimension of the subspace generated by S is called the rank of the set. If U and W are subspaces of V then so is the intersection of these sets, u n W. This intersection is not empty since all subspaces contain the zero element: thus should we speak of non-intersecting subspaces we really mean subspaces that only intersect in the zero element. The sum of U and W, U + W, consists of vectors of the form x = u + w, u EU WE W. In general, such a decomposition of x into elements of U and W is not unique. It is, however, when u n w = O. In this case the sum is said to be direct, written U 10 W. (Later we shall reserve this notation for the direct sum of algebras, all vector space sums being direct unless stated otherwise.) For any subspace U there is a subspace W such that V = U W; W being called the complement of U in V. Obviously dim V = dim U + dim W. Any subspace U is a normal subgroup under addition. The quotient group V/U can be given a linear structure by defining Â[x] = [Aa], where the bracket denotes the equivalence class of x, with x — y if x = y + u for some u E U. With this structure V/U is called the linear quotient space of V modulo U. (In view of the additive notation the obsolescent term difference space might seem more appropriate.) A linear map between two vector spaces over the same field is a group homomorphism that commutes with scalar multiplication. That is, cp is a linear map from V to W if cP(Ax YY) = 40(x) ± PT(y) V x, y E V, E F. It follows that every linear map sends the zero element of V to that in W. A linear map may be completely determined by specifying its effect on some basis for V. The terms injective, surjective and bijective naturally apply to linear maps. A bijective linear map is called a vector space isomorphism. The kernel of a linear map is the kernel of the group homomorphism, and is readily seen to be a linear subspace. In an obvious way we can define addition of linear maps and multiplication by scalars such that the linear maps from V to W form a vector space, 1(V, W). Since any such linear map may be specified by a dim V x dim W matrix we have dim 2(V, W) = dim Vdim W. A linear 312 APPENDIX A would simply omit thesummation sign when using thesummation convention. We shall frequently usethis convention without further comment. When itisnotclear from thecontext whether asum is implied ornotweshall explicitly state, forexample, nosum. Asubset Uofavector space Vsuch that alllinear combinations of vectors from UlieinUiscalled avector subspace. The zero element and Vitself areobviously vector subspaces, anyother subspace being termed non-trivial. IfSisanysubset from Vthen alllinear combina- tions ofvectors from Sform avector subspace which issaid tobe generated, orspanned, byS.The dimension ofthesubspace generated bySiscalled therank oftheset.IfUandWaresubspaces ofVthen soistheintersection ofthese sets, Ufi W.This intersection isnot empty since allsubspaces contain thezero element: thus should we speak ofnon-intersecting subspaces wereally mean subspaces that only intersect inthezero element. Thesum ofUandW,U+W,consists of vectors oftheform xEu+w,uEU wEW. Ingeneral, such a decomposition ofxinto elements ofUand Wisnotunique. Itis, however, when UOWE0.Inthiscase thesum issaid tobedirect, written U69W.(Later weshall reserve thisnotation forthedirect sum ofalgebras, allvector space sums being direct unless stated otherwise.) Foranysubspace Uthere isasubspace Wsuch that VEU69W;W being called the complement of U in V. Obviously dimVEdimU+dimW.Any subspace Uisanormal subgroup under addition. The quotient group V/U canbegiven alinear structure by defining )t[x]E[Ax], where thebracket denotes theequivalence class of x,with x~yifxEy+uforsome uEU.With thisstructure V/U is called thelinear quotient space ofVmodulo U.(Inview oftheadditive notation the obsolescent term difference space might seem more appropriate.) Alinear map between two vector spaces over thesame field isa group homomorphism thatcommutes with scalar multiplication. That is, (pisalinear map from VtoWif <t*(/lx+uy)=Mitt) +u¢(>') Vt»yEV./ll#6F- Itfollows thatevery linear mapsends thezero element ofVtothatin W.Alinear map may becompletely determined byspecifying itseffect onsome basis forV.The terms injective, surjective and bijective naturally apply tolinear maps. Abijective linear map iscalled avector space isomorphism. The kernel ofalinear map isthekernel ofthe group homomorphism, andisreadily seen tobealinear subspace. Inan obvious waywecandefine addition oflinear maps andmultiplication by scalars such that thelinear maps from VtoWform avector space, §£(V, W). Since any such linear map may bespecified bya dimV><dimW matrix wehave dim§£(V, W)EdimVdim W.Alinear APPENDIX A 313 map from V to V will be called a linear transformation, or endomorph- ism, and we will also write End V for 2(V, V). Such linear transforma- tions can be multiplied by composing maps, (cpip)x = cp(tp(x)). With such a product End V has the structure of an algebra, about which more will be said later. Under mulitplication the non-singular linear trans- formations form a group, the automorphism group of V, Aut V. Of special importance is the vector space of linear mappings from the vector space V to the field F, known as the dual space, V*. When V is finite dimensional then dim V* = dim V. For each basis {e,} of V we may establish a natural dual basis {e*'} of V* such that eNe 1) = SI) Vi , j. (Note the conventional positioning of indices.) If arbitrary elements b and B in V and V* respectively are expanded in dual bases as b = be, B = (summation convention) then B(b) = B,b'. In particular, e*'(x) = x' expresses the components of x in terms of the corresponding natural dual basis action on x. Elements of V* are sometimes called co-vectors to distinguish them from elements of V, although for V finite dimensional this terminology is reciprocal since there exists a natural way to regard V as the dual to V* A vector space V is graded by an Abelian group G if V is expressible as a direct sum of subspaces that are labelled by elements of G. More precisely, V is a G-graded vector space if { V,I is a set of non- intersecting subspaces such that V = E,V, and k injectively assigns an element k(i) of G to each V,. G is called the group of degrees. Elements of V, are called homogeneous of degree k(i), denoted deg x = k(i) V x E V,. Since the zero vector lies in every subspace it is homogeneous of every degree. Paticularly when G = Z we will label the subspaces with elements of G. When the only element that is homogeneous of negative degree is the zero element we have a positive gradation. If we omit mention of the group G we shall mean by graded vector space a Z-graded space with positive gradation. A G-graded subspace of a G-graded space V admits a direct sum decomposition in terms of subspaces contained in the homogeneous subspaces of V. If V and W are G-graded spaces with homogeneous subspaces { V,} and {W1) then a linear map cp is called homogeneous of degree k if there is an element k E G such that p( V,) C W,+k Vi E G. It follows that the kernel of a homogeneous map is a graded subspace of V, whilst the image is a graded subspace of W. If U is a G-graded subspace of a G-graded V then the linear quotient V/U inherits a natural G-gradation, the equivalence classes being assigned the degree of a homogeneous repre- sentative. APPENDIX A 313 map from VtoVwillbecalled alinear transformation, orendomorph- ism, andwewillalso write EndVfor.5E(V, V).Such linear transforma- tions can bemultiplied bycomposing maps, (Q71/1)x E<p(1p(x)). With such aproduct End Vhasthestructure ofanalgebra, about which more will besaid later. Under mulitplication thenon-singular linear trans- formations form agroup. theautomorphism group ofV,AutV.Of special importance isthevector space oflinear mappings from the vector space Vtothefield F,known asthedual space, V*.When Vis finite dimensional then dimV*EdimV.Foreach basis {el} ofVwe may establish anatural dual basis {e*'} ofV*such that e*"(e,-) E6‘,- Vi, j.(Note the conventional positioning ofindices.) Ifarbitrary elements bandBinVand V*respectively areexpanded indual bases as bEb‘e, BEBle*' (summation convention) then B(b) EBl-b". Inparticular, e*‘(x) Ex‘expresses thecomponents ofxinterms ofthecorresponding natural dual basis action onx. Elements ofV*aresometimes called co-vectors todistinguish them from elements ofV,although forVfinite dimensional thisterminology isreciprocal since there exists anatural way toregard Vasthedual to V*. Avector space Visgraded byanAbelian group GifVisexpressible asadirect sum ofsubspaces that arelabelled byelements ofG.More precisely, VisaG-graded vector space if{Vl} isasetofnon- intersecting subspaces such that VEE,-V, and kinjectively assigns an element k(i) ofGtoeach V,-.Giscalled thegroup ofdegrees. Elements ofV,-arecalled homogeneous ofdegree k(i), denoted degx Ek(i) VxEV,-. Since thezero vector liesinevery subspace itishomogeneous ofevery degree. Paticularly when GEZ wewill label the subspaces with elements ofG.When theonly element thatishomogeneous ofnegative degree isthezero element wehave apositive gradation. Ifweomit mention ofthegroup Gweshall mean bygraded vector space a Z-graded space with positive gradation. AG-graded subspace ofa G-graded space Vadmits adirect sum decomposition interms of subspaces contained inthehomogeneous subspaces ofV.IfVand W areG-graded spaces with homogeneous subspaces {Vl} and {Wl} then alinear map tpiscalled homogeneous ofdegree kifthere isanelement kEGsuch that <p(V,-) CW,-ll. VZE G.Itfollows that thekernel ofa homogeneous map isagraded subspace ofV,whilst theimage isa graded subspace ofW.IfUisaG-graded subspace ofaG-graded V then the linear quotient V/U inherits anatural G-gradation. the equivalence classes being assigned thedegree ofahomogeneous repre- sentative. 314 APPENDIX A A bilinear mapping on V is a mapping on pairs of vectors which is linear in each argument separately. By bilinear form we mean a bilinear mapping on V with values in the field F. We shall also refer to such a mapping as a metric. Although this use of the word is not standard we adopt it due to its prevalent use in this sense for the applications we are interested in. (Such a metric will not in general satisfy the criteria for a distance function used to define a metric space!) A metric g is symmetric if g(x, y) -= g(y, x) V x, y c V and non-degenerate if g(x, y) = 0 V y implies that x = O. We shall be primarily concerned with the case of F = 11 with g symmetric and non-degenerate, and we now restrict ourselves to this situation. In this case g is said to be positive- definite if g(x, x)> 0 for all non-zero x. It is often convenient to choose a g-orthonormal basis, fe,), in which g(e„ ei) =nu where n u = ±1 if i = j or zero otherwise. The pattern of signs is known as the signature of g, and may be denoted (p, q) where there are p plus signs and q minus signs. The automorphism group or invariance group of a space with a metric is the subgroup of the group of non-singular linear transformations consisting of elements m such that g(m(x), m(y)) = g(x, y) V x, y E V. For a real-valued symmetric non-degenerate g of signature (p, q) the invariance group is called the orthogonal group, 0(p, q). Such a space will also more simply be called an orthogonal space. In particular, then, orthonormal bases are related by orthogonal transformations. The metric g can be used to associate with every element x E V an element 2 E V* by the rule that 2(y) = g(x, y) V y E V. We shall refer to such an 2 as the metric dual or adjoint of x (with respect to g). If the components of g in the basis {e,} are given by = g(e„ e i) and 2 is expressed in the dual basis as 2 = "X",e*' then iy1 = g,,xy. Since this must hold for all y it implies that Xi = g,ixi Frequently a lowering convention is adopted for indices in which gux`, such that if x = x'e, then 2 = x,e*`. The metric g:V x V --›11i naturally induces a metric g*: V* x V* --›11:1 by the rule g*(X, y) = g(x, y) V x, y E V. If the components of g* in the basis fe*11 are the numbers j g*u = g*(e*', e*/) then g* jk _ c5kThus the components of g* form the inverse of the matrix of components of g. The map - from V to V* is invertible and we denote its inverse by Thus if B E V* with B = Be*( then 13_ = Bte, where the index has been raised with the components of the metric, B' g*u B .,. For typographical reasons we shall use the same symbol to denote the 'lowering map' - and its inverse the 'raising map' _.., there being little scope for confusion so long as we state in which space the elements lie. 314 APPENDIX A Abilinear mapping onVisamapping onpairs ofvectors which is linear ineach argument separately. Bybilinear form wemean abilinear mapping onVwith values inthefield F.Weshall also refer tosuch a mapping asametric. Although thisuseoftheword isnotstandard we adopt itduetoitsprevalent useinthissense fortheapplications weare interested in.(Such ametric willnotingeneral satisfy thecriteria fora distance function used todefine ametric space!) Ametric gis symmetric ifg(x, y)Eg(y, x)Vx, yEVandnon-degenerate ifg(x, y) E0Vyimplies that xE0.Weshall beprimarily concerned with the case ofFEIB with gsymmetric and non-degenerate, and wenow restrict ourselves tothissituation. Inthiscase gissaid tobepositive- definite ifg(x, x)>0forallnon-zero x.Itisoften convenient tochoose ag-orthonormal basis, {el}, inwhich g(e,, e/-)E17,-,where 17,,Eilif iEjorzero otherwise. The pattern ofsigns isknown asthesignature ofg,andmay bedenoted (p,q)where there arepplus signs and q minus signs. The automorphism group orinvariance group ofaspace with ametric isthesubgroup ofthegroup ofnon-singular linear transformations consisting ofelements msuch that g(m(x),m(y)) E g(x, y)Vx, yEV.Forareal-valued symmetric non-degenerate gof signature (p,q)theinvariance group iscalled theorthogonal group, O(p, q).Such aspace will also more simply becalled anorthogonal space. Inparticular, then, orthonormal bases arerelated byorthogonal transformations. The metric gcan beused toassociate with every element xEVanelement YEV*bytherulethat f(y)=s(X-y) VyEV- Weshall refer tosuch an2?asthemetric dual oradjoint ofx(with respect tog).Ifthecomponents ofginthebasis {el} aregiven by g,-IEg(e,-, el)and 2isexpressed inthedual basis as2?E2?,-e*' then fly’Eg,-I-x"y/. Since thismust hold forally/0itimplies that 2,Eg,»,»x'. Frequently alowering convention isadopted forindices inwhich x,-Egl-Ix", such thatifxEx‘e,~then tiEx,»e*‘. The metric g:V ><V-> IR naturally induces ametric g*:V* XV*->IRbytherule s*(f-Y)Es(X-y) VX-yEV- Ifthe components ofg*inthe basis {e*'} are the numbers g*’lEg*(e*‘, e*l) then g,-I-g*/" E<5",-. Thus thecomponents ofg*form theinverse ofthematrix ofcomponents ofg.The map Vfrom VtoV* isinvertible and wedenote itsinverse by-.Thus ifBEV*with BEB,-e*" then B-EB‘e, where theindex hasbeen raised with the components ofthemetric. B’Eg*'lB,-. Fortypographical reasons we shall usethesame symbol todenote the‘lowering map’ ~anditsinverse the‘raising map’ -,there being little scope forconfusion solong aswe state inwhich space theelements lie. APPENDIX A 315 As well as real vector spaces we shall be interested in vector spaces over the complex field. In various ways the same Abelian group can be endowed with both an Fl-linear structure and a C-linear structure. When speaking of the dimension of such a vector space it is important to distinguish between the two linear structures, and when there is possibil- ity for confusion we use dimE and dimc to denote the dimensions associated with the different linear structures. Similarly we speak of JR-linear and C-linear transformations when there is possibility of confusion. If V is a real vector space then an endomorphism J such that J2 = —I, where I is the identity map, is called a complex structure on V. Such a J can only exist if V is of even dimension. A complex structure can be used to define multiplication of elements in V by complex numbers. For A + ip E C, A, yE IR, we define (A + ip)x = Ax + pfx V x E V. Such a C-linear structure turns V into a complex vector space V, the complex vector space associated with V (and J). We clearly have dim c = dim E V. There is another way in which a complex vector space can be fabricated out of a real vector space V. The ordered pairs of elements of V, V x V are given a real vector space structure by defining (xi, Yi) + (x2, Y2) = (x1 ± x2, Yi ± Y2) A(x, y) = ()Ix, ily) AcE. With this structure the ordered pairs form the external direct sum of V with itself, VCW. This direct sum space has a natural complex struc- ture, J:(x, y)—> (—y, x). The complex vector space associated with this complex structure is called the complexification of V, Vc. Thus Vc (VV), and dim c Vc = dim E V. An element of Vc is an ordered pair of elements from V. But since (x, y) = (x, 0) + i(y, 0) we shall write x + iy instead of (x, y). If then A + it E C this gives, as one would expect, (A + ip)(x + iy) = Ax —yy + i(Ay + !ix). If now we start with a complex vector space E then we automatically have an associated real vector space, ER, since IR is a subfield of C. This real vector space comes equipped with a natural complex structure, multiplication by i in E. With this complex structure E = (ER)c. A group homomorphism cp between complex vector spaces is called conjugate linear if cp(Âx) = yl.*cp(x) for A e C and A* denoting the complex conjugate. In particular, if cp is an JR-linear map on a real V that has complex structure J such that, TJ = —Jcp then cp is a conjugate linear map on V. APPENDIX A 315 Aswell asrealvector spaces weshall beinterested invector spaces over thecomplex field. Invarious ways thesame Abelian group canbe endowed with both anIR-linear structure andaC-linear structure. When speaking ofthedimension ofsuch avector space itisimportant to distinguish between thetwolinear structures, andwhen there ispossibil- ityforconfusion weusedimll-l and diml; todenote thedimensions associated with thedifferent linear structures. Similarly wespeak of lB-linear and C-linear transformations when there ispossibility ofconfusion. IfVisarealvector space then anendomorphism Jsuch thatJ2E—I,where Iistheidentity map, iscalled acomplex structure onV.Such aJcanonly exist ifVisofeven dimension. Acomplex structure can beused todefine multiplication ofelements inVby complex numbers. For/1+iuEC,/1,lielB,wedefine (/l+iu)x=/lx+/.tJx VxEV. Such aC-linear structure turns Vinto acomplex vector space V,the complex vector space associated with V(and J).We clearly have diml;V =ldimllv. There isanother way inwhich acomplex vector space can be fabricated outofarealvector space V.The ordered pairs ofelements ofV,V><Varegiven arealvector space structure bydefining (X1-Y1) +(X2-Y2) =(XI+X2»Y1+Y2) /1(x,y)=(/lx,/ly) 1612. With thisstructure theordered pairs form theexternal direct sum ofV with itself, V@V. This direct sum space hasanatural complex struc- ture, J:(x, y)—>(—y, x).The complex vector space associated with this complex structure iscalled the complexification ofV,VC. Thus V‘:E(VG) V),anddimCVC Edimll-lV. Anelement ofV‘:isanordered pair ofelements from V.But since (x,y)E(x,0)+i(y,0)weshall write x+iyinstead of(x,y).Ifthen /1+iueCthisgives, asonewould expect, (/1+iu)(x +iy)E/lx—uy+i(/ly +ux). Ifnow westart with acomplex vector space Ethen weautomatically have anassociated realvector space, ER,since IRisasubfield ofC.This real vector space comes equipped with anatural complex structure, multiplication byiinE.With thiscomplex structure EE(ER)? Agroup homomorphism cpbetween complex vector spaces iscalled conjugate linear if<p(/lx) E/1*<p(x) for/1ECand /1*denoting the complex conjugate. Inparticular, ifcpisan1R-linear map onareal V thathascomplex structure Jsuch that, cpJE—J<;0 then cpisaconjugate linear map onV. 316 APPENDIX A As we have remarked, the non-singular linear transformations on a vector space V form a group under multiplication, Aut V. If G is an arbitrary group then a representation of G is a homomorphism of G into Aut V, for some V. The vector space V is said to carry the representa- tion. The dimension of V is called the dimension of the representation. If this homomorphism is one-to-one then the representation is called faithful. If the image of G under the representation leaves no non-trivial subspaces of V invariant then the representation is called irreducible. If V may be decomposed into subspaces that are preserved under a representation of G then that representation is reducible, as it induces homomorphisms of G into the automorphism groups of these subspaces. If V and W carry representations cp and p respectively then these are termed equivalent if there is an isomorphism S, mapping V to W. such that the following diagram commutes for all gE G,xE V: (1)(g) X cp(g)x S S i.e. Scp(g)S-1 = p(g). P(g ) Sx p(g)Sx An algebra over the field F, al(F), consists of a vector space over F together with an algebra product, called multiplication, which satisfies aQ.b + uc) = /lab + mac Va, h, c E V/1., fiE F and similarly for multiplication on the right. We shall call the dimension of the vector space the dimension of the algebra. The algebra is associative if its product satisifes a(bc) = (ab)c. Thus equivalently an associative algebra .91(F) is a ring si that is a vector space for which a(ab) = a(crb)= (cea)b Va. hE1. V a E F. We may therefore apply the terminology defined for rings to algebras. A division algebra being, for example, a division ring that is an algebra. An algebra with a unit element that spans the centre is called central. When the vector space is graded by an Abelian group G and the algebra product satisfies deg (ab) = deg a + deg b then we have a G-graded algebra. Unless we further specify we shall mean by algebra si a finite-dimensional associative algebra over F. some arbitrary field; although in this book we shall only be concerned with the real or complex field. If the underlying vector space of an algebra si is the direct sum of two subspaces A, T then we will write I = + T. These subspaces need not be subalgebras, by which we mean a vector subspace that is closed under the algebra product. The centre is an example of a subalgebra. 316 APPENDIX A Aswehave remarked, thenon-singular linear transformations ona vector space Vform agroup under multiplication, AutV.IfGisan arbitrary group then arepresentation ofGisahomomorphism ofGinto AutV,forsome V.The vector space Vissaid tocarry therepresenta- tion. The dimension ofViscalled thedimension oftherepresentation. Ifthishomomorphism isone-to-one then therepresentation iscalled faithful. Iftheimage ofGunder therepresentation leaves nonon-trivial subspaces ofVinvariant then therepresentation iscalled irreducible. If Vmay bedecomposed into subspaces that arepreserved under a representation ofGthen that representation isreducible, asitinduces homomorphisms ofGintotheautomorphism groups ofthese subspaces. IfVand Wcarry representations (pandprespectively then these are termed equivalent ifthere isanisomorphism S,mapping VtoW.such thatthefollowing diagram commutes forallgEG,xEV: <t>(g) X———> <t>(s)X Sl l5 i-6-5<t1(g)5'1= p(s)- p(s) 5-Y———> t>(s)5X Analgebra over thefield F,s<1(F), consists ofavector space over F together with analgebra product, called multiplication, which satisfies a(kb+pc)Ekab+11ac Va,b,cEol,Vk,uEF andsimilarly formultiplication ontheright. Weshall callthedimension ofthevector space thedimension ofthe algebra. The algebra is associative ifitsproduct satisifes a(bc) E(ab)c. Thus equivalently an associative algebra :.2l(F) isaring $4that isavector space forwhich a(ab) Ea(ab) E(a'a)b Va. besl. Vere F.Wemay therefore apply theterminology defined forrings toalgebras. Adivision algebra being, forexample. adivision ring that isanalgebra. Analgebra with aunit element thatspans thecentre iscalled central. When thevector space is graded byanAbelian group Gandthealgebra product satisfies deg(ab) Edega +degb then wehave aG-graded algebra. Unless wefurther specify weshall mean byalgebra .94afinite-dimensional associative algebra over F.some arbitrary field; although inthisbook weshall only beconcerned with therealorcomplex field. Iftheunderlying vector space ofanalgebra s/listhedirect sumoftwo subspaces J5.‘Ethen wewillwrite sfE375+<6.These subspaces need notbesubalgebras, bywhich wemean avector subspace that isclosed under thealgebra product. The centre isanexample ofasubalgebra. APPENDIX A 317 For subspaces 33, 'T we define the product 31T to be the vector space spanned by all products of the bases for 33 and T. If (6 is some subspace such that si = (fi'6 ... (6 then '6 is said to generate si. A basis for (b will be termed a set of generators for si. In general the dimension of 93T will be less that the product of those of 31 and T. In fact we have If {c,} i =1, . . s is a basis for T then dim 31% = dim Oldim iff >d 1c1 = 0 for di E gi implies all di are zero. (A3) For if {bi} j = 1, . . r is a basis for 33 then 31T is spanned by the set of all products bic,. So dim 33T = rs if and only if these are all linearly independent, that is, if Exub,c, =_ 1-1 for /1.,) E F implies all Ao = O. For d, = E;,,Xubj this is just the statement of the result. As a special case we have, for some non-zero a E si, asi = si if and only if there is no non-zero b such that ab = O. The above result enables us to make the following simple observation, to which we will later refer. If there is an element b such that ab = 1 then b is the unique inverse of a. (A4) It is obvious that if a had an inverse then it would be unique. Given ab = 1 we have absi = szi. But ab si C asi so we must have asi = that is, from (A3), there is no non-zero d such that ad = O. Suppose there were a c such that bac c, that is bac — c = d where d O. This implies that abac — ac = ad. If, however, ab = 1 then the left-hand side is zero, whereas the right-hand side cannot be, so ab = 1 gives bac = c d c, that is, ba = 1. The structure of an arbitrary algebra may be understood in terms of certain building blocks of smaller algebras together with the rules for assembling them. One such way in which an algebra can be expressed in terms of others is as a direct sum. An algebra 91 is the direct sum of algebras 2/ 3 and T, si = 213, if we have a vector space direct sum and -33T = 1131 = O. This is obviously extended to sums of several algebras. An algebra that can be written as a direct sum of subalgebras is called reducible and the subalgebras are termed components. Reducible alge- bras contain invariant subalgebras, or ideals. A two-sided ideal, or simply an ideal, is a subspace I such that ALA C I. Obviously ideals are subalgebras. Thus the components of a reducible algebra are ideals. Suppose si = 21 + (C, then we define an equivalence relation in si by a — b if a = b + c where c E T. We denote the equivalance class of a APPENDIX A 317 Forsubspaces 973,<6wedefine theproduct 975%tobethevector space spanned byallproducts ofthebases for973and<6.If‘Qissome subspace such thatatE‘£19...‘Qthen ‘ftissaid togenerate st.Abasis for<5will betermed asetofgenerators forst.Ingeneral thedimension of913% willbelessthattheproduct ofthose of%and<6.Infactwehave If {cl} iE1,...,sis abasis for <6 then dim973‘€ EdimQ73dim‘€ iffEl-ldlcl E0ford,-E913 implies all dlarezero. (A3) Forif{bl} jE1,...,risabasis forQ73then 9/3%isspanned bythe setofallproducts bl-c,-. Sodim%‘€ Ersifand only ifthese areall linearly independent, thatis,if .. =() llrim>-_©"1;» forAllEFimplies allll,E0. Ford,EE;-ll,-l-bl thisisjustthestatement oftheresult. Asaspecial case wehave, forsome non-zero aEal,aslEatifandonly ifthere is nonon-zero bsuch that abE0.The above result enables ustomake thefollowing simple observation, towhich wewilllater refer. Ifthere isanelement bsuch thatabElthen bistheunique inverse ofa. (A4) Itisobvious that ifahad aninverse then itwould beunique. Given abElwehave absl Est.But absl Caslsowemust have aslEal, that is,from (A3), there isnonon-zero dsuch that adE0.Suppose there were acsuch thatbacEc,thatisbac-cEdwhere dE0.This implies that abac —acEad.If,however. abElthen theleft-hand side iszero, whereas the right-hand side cannot be, soabE1gives bacEcVc,thatis,baE1. The structure ofanarbitrary algebra may beunderstood interms of certain building blocks ofsmaller algebras together with therules for assembling them. One such wayinwhich analgebra canbeexpressed in terms ofothers isasadirect sum. Analgebra stisthedirect sum of algebras 93and<6,atE§B®‘6, ifwehave avector space direct sum and 93%E‘@973E0.This isobviously extended tosums ofseveral algebras. Analgebra that canbewritten asadirect sum ofsubalgebras iscalled reducible andthesubalgebras aretermed components. Reducible alge- bras contain invariant subalgebras. orideals. Atwo-sided ideal, or simply anideal. isasubspace Isuch that sllsl CI.Obviously ideals are subalgebras. Thus thecomponents ofareducible algebra areideals. Suppose atE93+<6,then wedefine anequivalence relation inatby a~bifaEb+cwhere cE‘€. Wedenote theequivalance class ofa 318 APPENDIX A by [a]. The elements of si form an Abelian group under the operation of addition; this group may be quotiented by defining [a] + [b] = [a + b]. The equivalence classes are made into a vector space by defining X[a] = [Aa] for A in F. The obvious way to try and make the equivalence classes into an algebra is by defining [a][b] = [ab]. If, however, c, d ET then [a][b] = [a + c][b + d] and so for consistency we would need [ab] = [ab + ad + cb + cd] that is, (ad + cb + cd) c T. This will be true for all a, b E ,9/ and c, d c if, and only if, is an ideal. When this is the case then what we have described is the quotient algebra of al modulo T, denoted sia. If si is a G-graded algebra with an ideal I which is a G-graded subspace then I wiil in fact be a G-graded algebra. As a vector space si/I inherits a natural G-gradation such that, if a is homogeneous, deg [a] = deg a. This makes ai/I a G-graded algebra since deg {[a][b]l = deg [ab] = deg ab = deg a + deg b = deg [a] + deg [b]. An algebra homomorphism is a linear transformation from an algebra al to an algebra -31 such that the multiplicative structure is preserved. That is, if cp is a linear transformation from si onto 91 then cp is an algebra homomorphism if q(ab) = q)(a)cp(b). When the linear trans- formation is a vector space isomorphism then we have an algebra isomorphism, two isomorphic algebras also being called equivalent, denoted si 33. An isomorphism from an algebra to itself is called an automorphism. If an algebra has a unit element then for any invertible s the mapping al-->sas-1 defines an automorphism, called an inner auto- morphism. An automorphism is readily seen to map the centre onto itself. If the automorphism is inner then individual elements of the centre are left invariant. The kernel of a homomorphism is the kernel of the linear transformation. If a is in the kernel of a homomorphism cp, 313 APPENDIX A by[a].The elements ofatform anAbelian group under theoperation ofaddition; thisgroup may bequotiented bydefining [a]+[b]E[a+b]. Theequivalence classes aremade intoavector space bydefining A[a] E[ha] forAinF. Theobvious waytotryandmake theequivalence classes intoanalgebra isbydefining lallbl=[abl- If,however, c,dE<6then lallbl=la+Cllb+dl andsoforconsistency wewould need [ab] E[ab+ad+cb+cd] that is,(ad+cb+cd)E<t%. This willbetrue foralla,bead and c, dE<t% if,andonly if,<6isanideal. When thisisthecase then what we have described isthequotient algebra ofstmodulo ‘ti,denoted st/‘ti. If sflisaG-graded algebra with anideal Iwhich isaG-graded subspace then IwillinfactbeaG-graded algebra. Asavector space sd/Iinherits anatural G-gradation such that, ifaishomogeneous, deg[a] Edega. Thismakes sd/IaG-graded algebra since dsgllallbll =dsglabl Edegab Edega +degb Edeg[a] +deg[b]. Analgebra homomorphism isalinear transformation from analgebra sfltoanalgebra %such that themultiplicative structure ispreserved. That is,ifcpisalinear transformation from sflonto 911then cpisan algebra homomorphism ifcp(ab) Ecp(a)cp(b). When thelinear trans- formation isavector space isomorphism then wehave analgebra isomorphism, twoisomorphic algebras also being called equivalent. denoted sflE93.Anisomorphism from analgebra toitself iscalled an automorphism. Ifanalgebra hasaunitelement then foranyinvertible s themapping aI—>sas‘1 defines anautomorphism, called aninner auto- morphism. Anautomorphism isreadily seen tomap thecentre onto itself. Iftheautomorphism isinner then individual elements ofthe centre areleftinvariant. The kernel ofahomomorphism isthekernel of thelinear transformation. Ifaisinthekernel ofahomomorphism cp, APPENDIX A 319 q2(a) = 0, then cp(bac)= cp(b)cp(a)cp(c)= 0 for all b, c and so the kernel is an ideal. In the same way as the analogous result for groups is proved, we may show that p(si) = si/ker cp. (A5) A different correspondence between algebras may be defined as follows. If u is a vector space isomorphism between si and sr such that (ab)" = bua" then si and si' are termed opposite algebras and we shall use sl.°P to denote the opposite to si. In general 54 .si. For the case when the opposite algebra is isomorphic to si then si' may be replaced with si in the definition above and we then speak of the mapping as an anti-automorphism. An anti-automorphism of particular interest is that which squares to the identity. We shall call this involutory anti-automorphism simply an involution. If 93 and % are algebras of dimension m and n then we have already described how to form a new algebra of dimension m + n, namely the direct sum. We now describe how an algebra of dimension mn may be formed, the tensor product. If si, 91, are algebras over F with dimensions mn, m and n respectively such that 33 has a basis {b,} i =1, . . m with multiplication table bib, = E B k has a basis {cp} p = 1, . . n with multiplication cpcq = Ecp,,c, then si is the tensor product of &A and %, si = 310%, if it admits a basis {a,p} i = 1, . . m; p = 1, . . n with multiplication given by a,pan = E B ijkC pqra kr k.r This criterion for si to be the tensor product of 93 and involves particular bases for &A and T, thus there is now an onus to show that it is in fact independent of the bases chosen. If we have bases as defined above then we can define a bilinear map 0:91 x 6,, cp1-->b,Oc p = a,p. If now {b;), {el)) are any bases for 31, then the bilinearity ensures APPENDIX A 319 q9(a) E0,then q9(bac) E<p(b)q9(a)q9(c) E0forallb,candsothekernel isanideal. Inthesame way astheanalogous result forgroups is proved, wemay show that q9(sl) Eat/ker tp. (A5) Adifferent correspondence between algebras may bedefined as follows. Ifuisavector space isomorphism between atandst’ u:sl———>sl’ aI——>a“ such that(ab)" Eb“a“ then sfands24’aretermed opposite algebras and weshall uses2l°Ptodenote theopposite tosf.Ingeneral sl°PEsl.For thecase when theopposite algebra isisomorphic toofthen sl’may be replaced with sfinthedefinition above and wethen speak ofthe mapping asananti-automorphism. Ananti-automorphism ofparticular interest isthat which squares tothe identity. We shall call this involutory anti-automorphism simply aninvolution. If973and<6arealgebras ofdimension mandnthen wehave already described how toform anew algebra ofdimension m+n,namely the direct sum. Wenow describe how analgebra ofdimension mnmay be formed, thetensor product. Ifsd,973,<6arealgebras over Fwith dimensions mn,mandnrespectively such that973hasabasis {bl} iE1, ...,mwith multiplication table b.-b,=2B...b. It <6hasabasis {cp} pE1,...,nwith multiplication CPCQ =Ecpqrcr then sfisthetensor product of973and<6,sfE973®<6, ifitadmits abasis {a,-P} iE1,...,m;pE1,...,nwith multiplication given by dipajq = 21B,-J-kCpq,t1k,. 1’ This criterion forsftobethetensor product of973and <6involves particular bases for973and<6,thus there isnow anonus toshow that it isinfactindependent ofthebases chosen. Ifwehave bases asdefined above then wecandefine abilinear map ®:973 ><<6E>sl bl,¢pI——>b,-®c,, Ealp. Ifnow {bf}, {c;,} areanybases for93,<6then thebilinearity ensures 320 APPENDIX A that the set of {bC)cl,' } are linearly independent, and hence a basis for ,94. Further, if and then we have bb ; = EB'imb'k CC = Ecp'grc',. (biioc)(b;oc) = Effimcpqrwkoer. So indeed the definition of the tensor product is independent of the bases for @ and (€. It should be stressed that the definition we have given for the tensor product algebra defines it only up to equivalence. This will be convenient later when we shall make use of the observation that if 01 and (t- are mutually commuting subalgebras of .91 with dim .94 = dim @dim then s4 = (330%. In the particular case that .94 has a unit element it will also be the unit element of @ and 'C. A familiar example of an n'-dimensional algebra is provided by the set of all n x n matrices (matrices of order n) with elements in F. The abstract algebra isomorphic to this will be termed a total matrix algebra, denoted /1/1„(F). Where no confusion is likely we will simply refer to such an algebra as a matrix algebra, and shall not exhibit the underlying field, writing A„. A basis for matrices of order n is obviously provided by all the elements with a unit in the ith row and jth column and zeroes elsewhere. We formalise this by defining an ordinary matrix basis to be fe,11 j, j = 1, . . n eifeki = 0 k eijejk = eik. The identity is the sum of the diagonal elements, that is I = ell+ +2Ie nn. Matrix algebras have the following simple but important property. Atn,(F)O.A4,,(F) = At,„,(F). (A6) The proof will consist of spotting how to label the basis. If {e} and {f po} are ordinary bases for An, and A„ then if we set e,,Of = E where I = (i — 1)n + p, J = (j — 1)n + q a basis for An,0.4/1„ is IELI) I, J = 1, . . mn. I f K = (k — 1)n + r, L = (1 — 1)n + s then 320 APPENDIX A thatthesetof{bl-®c;,} arelinearly independent, andhence abasis for sit.Further, if bj-b}=;B;,-lb}, and C}-Cit=EC}-q-C? then wehave r (b;®t~;,)(1>;-®¢;,) =§)B;-,,c;,,,,b;,®t~;. k,r Soindeed thedefinition ofthetensor product isindependent ofthe bases for93and<6.Itshould bestressed that thedefinition wehave given forthetensor product algebra defines itonly uptoequivalence. This willbeconvenient later when weshall make useoftheobservation that if613-and <6are mutually commuting subalgebras ofsitwith dimsfi Edim9Bdim<6 then atEi73®<6. Intheparticular case thatsithas aunitelement itwillalsobetheunitelement of613-and<6. Afamiliar example ofann3-dimensional algebra isprovided bythe setofalln><nmatrices (matrices oforder n)with elements inF.The abstract algebra isomorphic tothiswillbetermed atotal matrix algebra, denoted A/t,,(F). Where noconfusion islikely wewillsimply refer to such analgebra asamatrix algebra, andshall notexhibit theunderlying field, writing A/l,l. Abasis formatrices oforder nisobviously provided byalltheelements with aunitintheithrowandjthcolumn andzeroes elsewhere. Weformalise thisbydefining anordinary matrix basis tobe l,...,l1 e,"/ek[:0 =6,-k. The identity isthesum ofthediagonal elements, that isIEell+ ... +21e ,,,,.Matrix algebras have the following simple but important property. "'1/knt(F‘)®‘/““n(F) :‘A/tfl1Il(F)‘ The proof willconsist ofspotting how tolabel thebasis. If{ell} and {fpq} areordinary bases forA/t,,,andA/1,,then ifweset er/®lpq =E/1 where IE(i—1)n +p.JE(j— 1)n+qabasis forA/l,,,®Jl/1,, is{EU} I,JE1,...,mn.IfK=(k—1)n+r,LE(l—l)n+sthen APPENDIX A 321 E HE = 4CA pq)(e klOf „) = ôjke f ps = jk6 qr E IL = (50 - 1)n + q. (k -1)n + rE IL = 45JKE IL  One reason for the importance of matrix algebras is that any associative algebra can be imbedded in a total matrix algebra. If V is a vector space then the set of all linear transformations from V to V forms an algebra, the endomorphism algebra, End V. If M E End V and a E V then we will usually write the transform of a by M as Ma, with no brackets. The product of linear transformations M and N will be defined by (MN)a = M(Na). Occasionally it will be convenient to write the effect of a linear transformation as M:a —> am . In this case we will use the convention that LlAIN --= (CI M)V . Normally this latter notation will be reserved for involutions. Addition of linear transformations is defined in the obvious way, and it is clear that End V is a total matrix algebra. A representation of an algebra 54 is a homomorphism into End V, for some V. Representations of algebras are termed faithful, irreducible or equivalent using the obvious analogue to the case of group representa- tions. If ai is an algebra then it is certainly a vector space and thus the algebra End si is associated with it. We may put elements of si into correspondence with certain elements of End si as follows. For a E L(a) c End si is defined by L(a)d = ad V d E .94. It follows that L is a linear map from si into End 51 such that L(a)L(b) = L(ab). Thus L is a homomorphism, called the regular representation. If si has a unit element then the regular representation is faithful. For if L(a) = L(b) then L(a — b)d = 0 for all d, and taking d = 1 gives a = b. So, in this case, the set {L(a)} for all a E S61 forms an algebra, L(..si), equivalent to si. In an obvious fashion we define the mapping R such that R(a)d = da. Then R(a)R(b) = R(ba) and so for an algebra with unit element R(si) .94°P. Thus L(ai) and R(..4.) are subalgebras of End al which are also mutually commuting, for L(a)R(b)d = L(a)(db) = adb = R(b)L(a)d since is associative. What is more, if at has a unit element and Se End si commutes with all elements of L(si) then S must be in R(sii). APPENDIX A 321 EIJEKL :(ei/®fpq)(ekl®frs) :6jk6qreil®fps :6/k6qrEIL 6(/i —l)n+q.(kEl)n+rEIL =5tI<E1L- One reason fortheimportance ofmatrix algebras isthat any associative algebra canbeimbedded inatotal matrix algebra. IfVisa vector space then thesetofalllinear transformations from VtoV forms analgebra, theendomorphism algebra, EndV.IfMEEnd Vand aEVthen wewillusually write thetransform ofabyMasMa, with no brackets. The product oflinear transformations Mand Nwill be defined by(MN)a EM(Na). Occasionally itwillbeconvenient towrite theeffect ofalinear transformation asMza —>a”.Inthiscase wewill usetheconvention that a'”N E(a'”)”. Normally thislatter notation will bereserved forinvolutions. Addition oflinear transformations isdefined intheobvious way, anditisclear that End Visatotal matrix algebra. Arepresentation ofanalgebra biisahomomorphism into EndV,for some V.Representations ofalgebras aretermed faithful, irreducible or equivalent using theobvious analogue tothecase ofgroup representa- tions. Ifbiisanalgebra then itiscertainly avector space andthus the algebra Endsd isassociated with it.Wemay putelements ofbiinto correspondence with certain elements ofEndsd asfollows. Foraesd, L(a) EEndsi isdefined by L(a)d=ad VdEsQ. Itfollows that Lisalinear map from biinto Endsd such that L(a)L(b) EL(ab). Thus Lisahomomorphism, called the regular representation. Ifbihasaunitelement then theregular representation is faithful. ForifL(a) EL(b) then L(a—b)dE0foralld,and taking dE1gives aEb.So,inthiscase, theset{L(a)} forallaE56forms an algebra, L(s6), equivalent tobi.Inanobvious fashion wedefine the mapping Rsuch thatR(a)d =da.Then R(a)R(b) ER(ba) andsofor analgebra with unit element R(s&) E56°F’. Thus L(s&) and R(s&) are subalgebras ofEndsd which arealsomutually commuting, for L(a)R(b)d =L(a)(db) =adb =R(b)L(a)d since biisassociative. What ismore, ifbihasaunit element and SEEndsd commutes with allelements ofL(s6) then Smust beinR(s6). 322 APPENDIX A For (SL(a))1 = Sa and (L(a)S)1 = a(S1) so if S commutes with L(a) Sa = a(S1) that is Sa = R(S1)a Va. If, then, s61 is an n-dimensional algebra with identity then Ends4 is an n2-dimensional algebra with L(si) and R(A) as n-dimensional commut- ing subalgebras. If the dimension of L(.59.)R(A) were n2 then End ..94 would be the tensor product of L(.4) and R(si). Although in general this will not be the case it is in the following situation. If a is a central division algebra then L(a)oR(a) = End. (A7) If a is n-dimensional we need to show that dim {L(a)R(a)) = n2. The proof will require the following Lemma L(a)R(a) = L(a)u, + L(a)u 2 + + L(2)us where /41, . . us are in R(a) and the sums are direct vector space sums. Since the identities of L(2) and R(g) coincide we have R(2) c L(2l)R(). We pick a non-zero element of R(a), u, say, and form L(g)u 1. Then either L(2)u1 = R(2) or we can pick a u2 in R(a) that is not in L(a)ui, giving L(21)u2 n L(a)u, = O. For if crui = fiu2 where ci', L(g) then for 0 * 0 u2 = (13-1a)u1, which contradicts u2eL()ui. Proceeding in this manner completes the proof of the lemma. In the manner of the lemma we write L(a)R(a) = L(a)u, + + L(a)us. Since for u regular dim {L(g)u} = n we have dim (L(g)R(a)) = ns, with s n. Ifs <n then we may extend the set {u1, u2, . . us} to a basis for R(g) by choosing us . . un. Since, as we stated in the lemma, R(2) C L(a)R(a) us +1 = Ecriu; with cri E L(g). 322 APPENDIX A For (SL(a))1 ESa and (L(a)S)1 Ea(S1) soifScommutes with L(a) SaEa(S1) thatis SaER(S1)a Va. If,then, atisann-dimensional algebra with identity then Endsd isan n2-dimensional algebra with L(sfl) andR(sfl) asn-dimensional commut- ingsubalgebras. Ifthedimension ofL(s6)R(sl) were n2then Endsl would bethetensor product ofL(s4) and R(sfl). Although ingeneral thiswillnotbethecase itisinthefollowing situation. If9)isacentral division algebra then L(9I>)®R(2D) EEnd9I). (A7) If9)isn-dimensional weneed toshow that dim{L(9D)R(9))} En2.The proof willrequire thefollowing Lemma L(2D)R(9I>) EL(2l>)ul +L(2D)u2 +...+L(2l>)ul where ul,...,u,areinR(9D) and thesums aredirect vector space sums. Since the identities ofL(9D) and R(9D) coincide we have R(9D) CL(2D)R(9I)). Wepick anon-zero element ofR(9D), ulsay, and form L(2D)ul. Then either L(2D)ul ER(9Il) orwecanpick aH2inR(9D) that isnotinL(2D)ul, giving L(@)u2 FlL(9D)ul E0.ForifcxulEBu; where or,BEL(9D) then forBEO U2E(/3"a/)ul, which contradicts u2EL(9Il)ul. Proceeding inthis manner completes theproof ofthe lemma. Inthemanner ofthelemma wewrite L(2D)R(€t)) =L(9D)ul +...+L(9D)u,. Since foruregular dim{L(2D)u} Enwehave dim(L(9Il)R(9D)) Ens, with sEn.Ifs<nthen wemay extend theset{ul, ul,...,u,}toa basis forR(9Il) bychoosing alll,...,u,,.Since, aswestated inthe lemma, R(§D) CL(2t))R(9D) usllEEalu, with £1’,-EL(§D). APPENDIX A 323 However, since L(9)) and R(a) are commuting subalgebras [141, )3]= 0 VS E L(a) where the bracket denotes the commutator. In particular [us +1, /3] = 0 giving E[ce„ P]u, = O. Since the sum is direct, in the vector space sense, we must have 131 = 0 vpe L(a) = 1, s. That is, the a, are in the centre of L(a). But L(g) = a which is central, so the a, must all be multiples of the identity by the base field F. The expansion of u„ as a sum of the first s u, then contradicts their F-linear independence and so we must have s = n and the proof is complete. There is another reason for the prominent role played by matrix algebras. The structure of an important class of algebras may be given in terms of matrix algebras and division algebras. More generally the recalcitrant (or interesting) parts of an algebra may be collected together into a certain ideal such that the structure of the quotient modulo this ideal is given in terms of matrix and division algebras. The existence of this ideal will now be established. A non-zero element of an algebra is called nilpotent if some finite power of it vanishes. The smallest such power is called the index of that element. An algebra is called nilpotent of index v if v is the smallest integer such that all products of v terms vanish. We have already encountered the concept of a two-sided ideal; single-sided ideals are defined as follows. A left ideal of an algebra si is a subspace 1' such that AT C 2. Right ideals are defined in the obvious way. It follows that single-sided ideals are subalgebras, and so we may talk of nilpotent single-sided ideals. The sum of two nilpotent left ideals is a nilpotent left ideal. (A8) Let ga and T be nilpotent left ideals of index a and /3 respectively. Then + T is certainly a left ideal. If any element of + cf is raised to the power k then it will be a linear combination of terms of the form a = ala, a k where the a, are in or T. Suppose that p terms in this product are in 91, and that j is the largest integer such that al E3a. Then if a_1 e€ we set a_ 1a1 = a;, where a; E A, since 31 is a left ideal. Proceeding in this manner we can write a = b, . . b pr with the b, in 93 and r in T. Similarly, we have a =c, cgs where the c in T and s is in -A Here p + q = k. So if k = + - 1 then p < gives q )3, whereas q <13 gives p a and so we must have a = O. That is, 03 + is nilpotent with index no greater than one less than the sum of those of and T. Obviously this result is just as valid for right ideals. APPENDIX A 323 However, since L(<Zt)) andR(§D) arecommuting subalgebras [ul,,8]E0 V,BEL(€D) where thebracket denotes thecommutator. Inparticular [ul1,l,,8]E0giving Elia/it fllui : Since thesum isdirect, inthevector space sense, wemust have [a,-,,B]E0 V,BEL(§D)iE1,...,s. That is,the(1’l-areinthecentre ofL(€D). ButL(€D)E€D which is central, sotheoilmust allbemultiples oftheidentity bythebase field F.The expansion ofu,.,l asasum ofthefirst su,then contradicts their F-linear independence andsowemust have sEnandtheproof is complete. There isanother reason fortheprominent role played bymatrix algebras. Thestructure ofanimportant class ofalgebras may begiven in terms ofmatrix algebras and division algebras. More generally the recalcitrant (orinteresting) parts ofanalgebra may becollected together into acertain ideal such that thestructure ofthequotient modulo this ideal isgiven interms ofmatrix anddivision algebras. The existence of thisideal willnow beestablished. Anon-zero element ofanalgebra iscalled nilpotent ifsome finite power ofitvanishes. Thesmallest such power iscalled theindex ofthat element. Analgebra iscalled nilpotent ofindex vifvisthesmallest integer such that allproducts ofvterms vanish. We have already encountered theconcept ofatwo-sided ideal; single-sided ideals are defined asfollows. Aleftideal ofanalgebra sdisasubspace .55such that sdilfC.55.Right ideals aredefined intheobvious way. Itfollows thatsingle-sided ideals aresubalgebras, andsowemay talkofnilpotent single-sided ideals. Thesum oftwonilpotent leftideals isanilpotent leftideal. (A8) Let913and<6benilpotent leftideals ofindex orand/3respectively. Then 973+<6iscertainly aleftideal. Ifanyelement of973+<6israised tothe power kthen itwill bealinear combination ofterms oftheform a=alaz ...allwhere thea,-arein973or<6.Suppose thatpterms in thisproduct arein973,andthatjisthelargest integer such that 61,6973. Then ifa,--lE<6 weseta,--la, Eaj-,where dj-6%, since 973isaleft ideal. Proceeding inthismanner wecanwrite aEbl...bprwith the blin973andrin<6.Similarly, wehave aEcl...cqswhere thec,-are in<6 andsisinfli. Herep+qEk. SoifkEa+,B—lthenp<a gives qE,8,whereas q<,8gives pEorand sowemust have aE0. That is,973+<6isnilpotent with index nogreater than onelessthan the sum ofthose of973and<6.Obviously thisresult isjustasvalid forright ideals. 324 APPENDIX A If 2 is a nilpotent left ideal then .1 = + 2.4 is a nilpotent two-sided ideal. (A9) Firstly note that is indeed an ideal; for .942 C 2 since 2 is a left ideal and so si2s4. C 2sti, thus .1 is a left ideal. Similarly Asti C 1 and so 4.911 C 2.si C J, making .1 a right ideal. As well as being a left ideal 2.34 is nilpotent. For if x E 2,94, x = la for I E 2, a E A and Xk k - 1 a where l' = al, is in 2. So 254 is nilpotent with index less than or equal to that of 2. Since is the sum of two nilpotent left ideals, the previous theorem shows that 3) is nilpotent. These two results have been established for the purpose of proving the following. Every nilpotent left, right and two-sided ideal is contained in a unique maximal nilpotent ideal, the radical. (A10) Let X be a nilpotent ideal of largest dimension. If X' is any nilpotent ideal then, by (A8), X + X' is a nilpotent left ideal, and similarly it is a nilpotent right ideal and so an ideal. But X is of maximal dimension so we must have X' C X. If now 2 is a nilpotent left ideal then the above result and (A9) combine to give 2 C (2 + 2.91) C X. Similarly for right ideals. Before making the anticipated good use of the existence of the nilpotent radical it is necessary to establish some properties of other important elements of an algebra, the idempotents. A non-zero P is idempotent if P2 = P. An obvious example of an idempotent is the identity of a division algebra. The identity is the only idempotent in a division algebra. (All) Suppose P2 = P and P is not zero. Then P is invertible and p -1p2 =_ p-1P, that is P = 1. Of course, a unit element is a very special example of an idempotent. A more general example is provided by the diagonal elements of an ordinary matrix basis. A large class of algebras have an idempotent. Every non-nilpotent algebra contains an idempotent. (Al2) Obviously a nilpotent algebra cannot contain an idempotent. We will show that if an algebra does not contain an idempotent then in fact it must be nilpotent. Suppose that s4 contains an a such that slak = sua" for some power k. Then if = sak 1, is a left ideal of A, and hence an algebra, satisfying Ra = gi and hence Rb = where b c 01 is given by b = ak . So there must be some P E gi such that Pb = b, giving (P 2 — P)b = O. But Rb = 9.3 means that there is no non-zero x with xb = 0 and so 91, and thus sq, contains an idempotent. So if .54 does not contain an idempotent we must have 324 APPENDIX A Ififisanilpotent leftideal then 9=if+iftflisanilpotent two-sided ideal. (A9) Firstly note that9isindeed anideal; forslifCifsince ifisaleft ideal andsoslifsi Cifsil, thus 9isaleftideal. Similarly sisdC01and so9&4C.5554C9,making 9aright ideal. Aswell asbeing aleftideal ifs!isnilpotent. ForifxEifsi, xElaforIEif,aEAandx"Ell"‘‘la where l’Eal,isinif.Soifs!isnilpotent with index lessthanorequal tothatof.58.Since 9isthesumoftwonilpotent leftideals, theprevious theorem shows that9isnilpotent. These two results have been established forthepurpose ofproving thefollowing. Every nilpotent left,right andtwo-sided ideal iscontained in aunique maximal nilpotent ideal, theradical. (A10) LetNbeanilpotent ideal oflargest dimension. IfN’isanynilpotent ideal then, by(A8), N+N’isanilpotent leftideal, andsimilarly itisa nilpotent right ideal andsoanideal. ButNisofmaximal dimension so wemust have N’CN.Ifnow ifisanilpotent leftideal then theabove result and(A9) combine togiveifC(§£+ifsll) CN.Similarly forright ideals. Before making the anticipated good use oftheexistence ofthe nilpotent radical itisnecessary toestablish some properties ofother important elements ofanalgebra, theidempotents. Anon-zero Pis idempotent ifP2EP.Anobvious example ofanidempotent isthe identity ofadivision algebra. Theidentity istheonlyidempotent inadivision algebra. (A11) Suppose P2EPand Pisnotzero. Then Pisinvertible and P‘1PZE P“P, that isPE1.Ofcourse, aunit element isavery special example ofanidempotent. Amore general example isprovided bythediagonal elements ofanordinary matrix basis. Alarge class of algebras have anidempotent. Every non-nilpotent algebra contains anidempotent. (A12) Obviously anilpotent algebra cannot contain anidempotent. Wewill show thatifanalgebra does notcontain anidempotent then infactit must benilpotent. Suppose that ascontains anasuch that siak Esiak*1forsome power k.Then if973Esiak'1,91*»isaleftideal ofst,andhence analgebra, satisfying 911aE973andhence 973bE95 where bE973isgiven bybEa".Sothere must besome PE973such that PbEb,giving (P2—P)bE0.ButQltbE973means thatthere isno non-zero xwithxbE0andso973,andthus.91,contains anidempotent. Soifasdoes not contain anidempotent we must have APPENDIX A 325 dim (alai') < dim (Aak -1) for all powers k of all a e si. The finite dimensionality of A means that there must be a finite a such that Ace = 0; in particular aa + = O. Since this is true for all a, A is nilpotent. The existence of an idempotent enables an algebra to be written as a direct vector space sum of subalgebras. Let P be an idempotent in then 2(P) is defined to be the left ideal consisting of a E A such that aP = O. Similarly the right ideal R(P) is defined to consist of all a e A such that Pa = 0, and we define .4(P) = .T(P) n gt(P). The following theorem gives the two-sided Peirce decomposition of A. If P is idempotent in si then = PAP + P(P) + gt(P)P + J(P). (A13) All the terms in this sum are algebras. PA(P) consists of all a E si such that Pa = aP = a; P2(P) consists of all a E Si with Pa = a, aP = 0; R(P)P consists of all a E s4 with Pa = 0, aP = a and if a e 4(P) Pa = aP = O. So obviously these algebras are non-intersecting and what we need to show is that they span al. To see this we write a = PaP + P(a — aP) + (a — Pa)P + (a — Pa — aP + PaP) where each term in the sum lies in one of the subalgebras contained in the Peirce decomposition. Elements of J(P) are said to be (algebraically) orthogonal to P. An idempotent is called principal if there is no idempotent orthogonal to it. We can now go one step further from (Al2) with Every non-nilpotent algebra contains a principal idempotent. (A14) If ..91 is non-nilpotent then it certainly contains an idempotent. If u is a non-principal idempotent then there exists an idempotent y such that uu = vu = O. That is, y c 1(u). If P = u + y then P is idempotent with Pu = uP = u and Pv = yP = y. So if xP = 0 then xu = xP = 0, and if Px = 0 then ux = uPx = 0, that is, J(P) C 4(u). In fact .1(P) must be strictly contained in .1(u) for u e ,l(u) but not in l(P). If P is not principal then we set P' = P + w where w E .1(P). Since .1(P) C 1(u) if this process is continued it will eventually produce a principal idempo- tent since J(u) is finite dimensional. Of fundamental importance are the primitive idempotents. An idempotent is primitive if it can not be written as a sum of two orthogonal idempotents. The following could have formed an alternative definition of a primitive idempotent. P is the only idempotent of PAP iff P is primitive. (A15) If P were not primitive then P = u + y with u v = vu = O. So APPENDIX A 325 dim(Nak) <dim (Na"") forallpowers kofallaEN. The finite dimensionality ofNmeans that there must beafinite asuch that Na“ E0; inparticular a“+‘ E0. Since this istrue foralla,Nis nilpotent. The existence ofanidempotent enables analgebra tobewritten asa direct vector space sum ofsubalgebras. LetPbeanidempotent inN, then $(P) isdefined tobetheleftideal consisting ofaENsuch that aPE0.Similarly theright ideal 9l(P) isdefined toconsist ofallaEN such that PaE0,and wedefine 9(P) E$(P) O9l(P). The following theorem gives thetwo-sided Peirce decomposition ofN. IfPisidempotent inNthen NEPNP +P.%(P) +9t(P)P +9(P). (A13) Alltheterms inthissum arealgebras. PN(P) consists ofallaENsuch that PaEaPEa;P.%(P) consists ofallaENwith PaEa,aPE0; 97t(P)P consists ofallaEN with PaE0,aPEaand ifaE9(P) PaEaPE0.Soobviously these algebras arenon-intersecting andwhat weneed toshow isthatthey span N.Toseethiswewrite aEPaP+P(a—aP)+(a—Pa)P+(a——Pa—aP+PaP) where each term inthesum liesinoneofthesubalgebras contained in thePeirce decomposition. Elements of9(P) aresaid tobe(algebraically) orthogonal toP.An idempotent iscalled principal ifthere isnoidempotent orthogonal toit. Wecannow goonestep further from (A12) with Every non-nilpotent algebra contains aprincipal idempotent. (A14) IfNisnon-nilpotent then itcertainly contains anidempotent. Ifuisa non-principal idempotent then there exists anidempotent Usuch that uoEouEO.That is,oE9(u). IfPEu+Uthen Pisidempotent with PuEuPEu andPuEuPEu. SoifxPE0thenxuExPE0, andif PxE0then uxEaPx E0.that is,9(P) C9(u). Infact9(P) must be strictly contained in9(u) forvE9(u) butnotin9(P). IfPisnot principal then wesetP’EP+wwhere wE9(P). Since 9(P) C9(u) if thisprocess iscontinued itwilleventually produce aprincipal idempo- tentsince 9(u) isfinite dimensional. Offundamental importance are the primitive idempotents. An idempotent isprimitive ifitcan not bewritten asasum oftwo orthogonal idempotents. The following could have formed analternative definition ofaprimitive idempotent. Pistheonly idempotent ofPNP iffPisprimitive. (A15) IfPwere not primitive then PEu+1)with upEvuE0.So 326 APPENDIX A Pu = uP = u and Pv = vP = v and thus both u and v are in PAP. Conversely, if u is an idempotent in PAP then P — u is idempotent since P is the identity in PAP. Further, u(P — u) = (P — u)u = 0 and so P = (P — u)+ u, the sum of two orthogonal idempotents. The nomenclature is explained by the following. Every non-primitive idempotent is the sum of a set of pairwise orthogonal primitive idempotents. (A16) If P is not primitive then P = u + v, where u and v are orthogonal idempotents. Suppose that v is not primitive, then y = w + x with w and x orthogonal. Now v w = wv = w and vx = xv = x and so uw = uvw = 0, wu = wvu = O. Similarly ux = xu = 0 and so {u, w, x} are pairwise orthogonal idempotents. If we continue in this way then the process must terminate due to the finiteness of sei and we will arrive at a set of pairwise orthogonal primitives. Attention will now be focused on algebras whose radical is zero. It will transpire that we can completely determine the structure of all such algebras. An algebra whose radical is zero is called semi -simple. The first consequence of the definition is A semi-simple algebra has a unit element. (A17) If .91 is semi-simple then it is not nilpotent and so, by (A14), contains a principal idempotent P say. The Peirce decomposition of (A13) then gives where 91 = P(P) + 9t(P)P + .1(P). 93 is spanned by (P) and R(P). We shall show that these single-sided ideals are nilpotent and hence contained in the radical, which is zero by hypothesis. This will give = P.AP; but P is the identity in PAP, and hence of A. If P is principal then .1(P), which contains all elements orthogonal to P, can contain no idempotent and thus must be nilpotent. Since 9t(P) and 2(P) consist of all elements annihilated by left and right multiplication by P respectively 9(P)2(P) C 3(P). So if / E (P) and r E (P) then (rI) = 0 where a is the index of 3(P). Since (/r)a" +1 = l(rl) r = 0 then the ideal Y(P)9t(P) is nilpotent of index less that or equal to a + 1. Now Y(P)A = Y(P)(PAP + P(P) + 91,(P)P + J(P)) = Z(P)9(P)P + Y(P)5 6(P). Since Wt(P) is a right ideal 9i,(P)P C R(P) and since (P) = I(P) fl k(P) obviously J(P) C R(P) and so (P)si C I(P)R(P). In particular, Y(P)Y(P) C 9(P)94P). Since Y(P)94P) is 326 APPENDIX A PuEuPE uand PuEvPEvand thus both uand uareinPNP. Conversely, ifuisanidempotent inPNP then P—uisidempotent since Pistheidentity inPNP. Further, u(P—u)E(P—u)uE0and soPE(P—u)+ u,the sum oftwo orthogonal idempotents. The nomenclature isexplained bythefollowing. Ever non- rimitive idem otent isthe sum ofasetof - y p 4 - ppairwise orthogonal primitive idempotents. (A16) IfPisnotprimitive then PEu+u,where uand uareorthogonal idempotents. Suppose that uisnotprimitive, then uEw+xwith w and xorthogonal. Now owEwvEwand uxExu Ex and so uwEuvw E0,wuEwuu E0.Similarly uxExuE0andso{u,w,x} arepairwise orthogonal idempotents. Ifwecontinue inthiswaythen the process must terminate duetothefiniteness ofNandwewillarrive ata setofpairwise orthogonal primitives. Attention willnow befocused onalgebras whose radical iszero. It willtranspire thatwecancompletely determine thestructure ofallsuch algebras. Analgebra whose radical iszero iscalled semi-simple. The firstconsequence ofthedefinition is Asemi-simple algebra hasaunitelement. (A17) IfNissemi-simple then itisnotnilpotent andso,by(A14), contains a principal idempotent Psay. The Peirce decomposition of(A13) then gives NEPNP+€’73 where 973EP.§E(P) +9t(P)P +9(P). 973isspanned by.§E(P) and9i(P). Weshall show that these single-sided ideals arenilpotent and hence contained intheradical, which iszero byhypothesis. This will give NEPNP; butPistheidentity inPNP, and hence ofN.IfPis principal then 9(P), which contains allelements orthogonal toP,can contain noidempotent and thus must benilpotent. Since 9R(P) and .§E(P) consist ofallelements annihilated byleftandright multiplication byPrespectively 9i(P).§E(P) C9(P). SoiflE§£(P) andrE97l.(P) then (rl)" E0where aistheindex of9(P). Since (lr)"*' El(rl)"r E0then theideal .§E(P)9t(P) isnilpotent ofindex lessthat orequal toor+1. Now .§£(P)N =§£(P){PNP +P.S£(P) +97t(P)P +9(P)} =.:e(P)oi(P)P +..<£(P)9(P). Since 9t(P) is aright ideal 9t(P)P C9t(P) and since 9(P) E.S£(P) Fl9t(P) obviously 9(P) C9t(P) and so .S£(P)N .§E(P)9R(P). Inparticular, .E£(P).S£(P) C§£(P)9t(P). Since .E£(P)?R(P) 1S APPENDIX A 327 nilpotent of index -sa + 1 if x e 2(P) (x 2)"+ I = 0, and so 2(P) is a nilpotent left ideal, contained in the radical. In exactly the same way we show that R(P) is nilpotent and the proof follows. The primitive idempotents in a semi-simple algebra have the following important property. If P is an idempotent in a semi-simple A then PAP is a division algebra iff P is primitive. (A18) Suppose that PAP is a division algebra. Then P is the identity which, by (All), is the only idempotent in Ps4P. (A15) then ensures that P is primitive. To prove the converse we shall use the following Lemma If P is an idempotent of a semi-simple A then PAP is semi-simple. For suppose that y E X0, the radical of PAP. Then Ay is a left ideal of si. Since P is the identity in Ps4P (ay)1 = ayP(aPy) = ay(PaPy)". But PaP c PAP and so PaPy C 0. Thus if a is the index of X0, Ay is nilpotent of index a + 1. Since A is semi-simple Ay = 0 which, since A has a unit, gives y --= 0 and PsiP is semi-simple. Suppose now that P is primitive then PAP is semi-simple, by the above lemma, with unity P. If a is any non-zero element of PAP then PAPa is a non-zero left ideal of PAP; further it is not nilpotent since PAP is semi-simple. (Al2) ensures that PAPa contains an idempotent, but any idempotent in PA Pa is certainly idempotent in PAP for which P is the only idempotent since P is primitive ((A15)). That is, P c PAPa say P = ba for b E PAP. Since P is the identity in PAP this says that every non-zero a has a left inverse, and hence an inverse by (A4). The semi-simple algebras are not quite as 'simple' as the simple ones. An algebra that is not a one-dimensional nilpotent algebra is called simple if the only ideals are the zero ideal and the algebra itself. Simple algebras are certainly semi-simple. To see this we need only check that simple algebras cannot be nilpotent. Suppose that X is a nilpotent algebra, then XX is an ideal strictly contained in X. If this is not the zero ideal then X cannot be simple. If J■f,l■C is zero but the dimension of X is greater than one then any linear subspace of one less dimension is a non-zero ideal of X. The only exceptional case of a one-dimensional nilpotent algebra has to be excluded by the caveat in the definition. The study of semi-simple algebras may be reduced to the study of simple ones by the following. APPENDIX A 327 nilpotent ofindex Ea’+1 ifxE.§E(P) (x2)“*' E0, and so.E£(P) is anilpotent leftideal, contained intheradical. Inexactly thesame way weshow thatQt(P) isnilpotent andtheproof follows. Theprimitive idempotents inasemi-simple algebra have thefollowing important property. IfPisanidempotent inasemi-simple Nthen PNP isa division algebra iffPisprimitive. (A18) Suppose that PNP isadivision algebra. Then Pistheidentity which, by(A11), istheonly idempotent inPNP. (A15) then ensures that Pis primitive. Toprove theconverse weshall usethefollowing Lemma IfPisanidempotent ofasemi-simple Nthen PNP issemi-simple. Forsuppose that yEN0, theradical ofPNP. Then Nyisaleftideal ofN.Since Pistheidentity inPNP (<1y)“*‘ =eyP(aPy)" Eay(PaPy)“. ButPaPE PNP andsoPaPy CN0.Thus ifaistheindex ofN0,Nyis nilpotent ofindex Ea+1.Since Nissemi-simple NyE0which, since Nhasaunit, gives yE0andPNP issemi-simple. Suppose now that Pisprimitive then PNP issemi-simple, bythe above lemma, with unity P.Ifaisanynon-zero element ofPNP then PNPa isanon-zero leftideal ofPNP; further itisnotnilpotent since PNP issemi-simple. (A12) ensures that PNPa contains anidempotent, butanyidempotent inPNPa iscertainly idempotent inPNP forwhich Pisthe only idempotent since Pisprimitive ((A15)). That is, PEPNPa sayPEbaforbEPNP. Since Pistheidentity inPNP this says that every non-zero ahasaleftinverse, andhence aninverse by (A4). Thesemi-simple algebras arenotquite as‘simple’ asthesimple ones. Analgebra that isnotaone-dimensional nilpotent algebra iscalled simple iftheonlyideals arethezeroideal andthealgebra itself. Simple algebras arecertainly semi-simple. Toseethisweneed only check that simple algebras cannot benilpotent. Suppose that Nisanilpotent algebra, then NNisanideal strictly contained inN.Ifthisisnotthe zero ideal then Ncannot besimple. IfNNiszero butthedimension of Nisgreater than onethen anylinear subspace ofonelessdimension isa non-zero ideal ofN.The only exceptional case ofaone-dimensional nilpotent algebra hastobeexcluded bythecaveat inthedefinition. The study ofsemi-simple algebras may bereduced tothestudy ofsimple ones bythefollowing. 328 APPENDIX A An algebra is semi-simple iff it is simple or a direct sum of simple components. (A19) A direct sum of simple algebras is obviously semi-simple since the only ideals are smaller sums of simple algebras which are not nilpotent. To go the other way we shall use two lemmas. Lemma 1 If si has an ideal with a unit element then sa is reducible. Let 33 be an ideal of .94 and 1 be the unit in A. The Peirce decomposition of si is = 1,044 + lu(l) + 341g3)1a + 4(19). If 9'(1ga) = 190i16,1 + 44(10 + gl,(101 91 then g(4 3) C since 1g3 E which is a two-sided ideal of si. So if b E gl we have b = b1 + b2 with b1 E 92(1) and b2 E 3(1a). Then 61 93 = b1 since b2 is orthogonal to 130 but b1a = b so in fact we must have 1(4 3) = A. Since J(1 93) is orthogonal to 1g, it is orthogonal to A and so s9. = 338.1(191). Lemma 2 A non-zero ideal of a semi-simple algebra is semi-simple. Suppose that A is an ideal in a semi-simple 54, and that X is the radical of A. Then 93X91 C X since X is an ideal of 91 and .9431 C since A is an ideal of si. So sdP.X03).94 C glSa which is thus an ideal in sii; further it is nilpotent since it is contained in the radical of A. Since si is semi-simple 93X93 = O. Now (siXs61)3 C (X,9)X(s4X.99.) and &IX s4. C g3 so (stiNs4) 3 C 91N91, which we have shown is zero. That is, siXsti is a nilpotent ideal in a semi-simple si so siXsi = O. Since si has a unit element this gives X = 0 and 33 is semi-simple. We may now return to the proof of the theorem. If si is semi-simple but not simple then it has a non-zero ideal which, by Lemma 2, is semi-simple and hence has a unit. Lemma 1 then ensures that al is reducible. The components are certainly ideals and so semi-simple, and we may proceed to reduce them. If .54 is finite then we must arrive at an expression of si as a direct sum of irreducible components. The components are ideals, hence semi-simple, and irreducible hence simple. The reduction of a semi-simple algebra to simple components is unique up to an ordering of the components. (A20) Let s4 = @iC) . . .JT with the A, simple. The identity of .9i can be written as a sum of the identities in the 33 1 = ei0 . . . Se,. Suppose = ... CA, then k = kei+ ke2+ ...%kerVk =1, = ke, then Tk, Cse = ÇJ and the above sum must be direct: 328 APPENDIX A Analgebra issemi-simple iffitissimple oradirect sum of simple components. (A19) Adirect sum ofsimple algebras isobviously semi-simple since theonly ideals aresmaller sums ofsimple algebras which arenotnilpotent. To gotheother wayweshall usetwolemmas. Lemma 1 IfNhasanideal with aunitelement then Nisreducible. Let 93beanideal ofNand 1%betheunit in973.The Peirce decomposition ofNis IfEf’(1gl) E1glN1gl +].@§£(1g3) +97t(1gl)19l then Ef’(1gl) C973since lg,E918 which isatwo-sided ideal ofN.SoifbE973wehave bEbl+b2with blEEf’(1gl) andb2E9(19;l). Then b1g3Eblsince b2isorthogonal to1%, but blgEbsoinfact wemust have Ef’(1gl) E973.Since 9(19l) is orthogonal to19;,itisorthogonal to973andsoNE973C-B9(1@;l). Lemma 2 Anon-zero ideal ofasemi-simple algebra issemi-simple. Suppose thatQBisanideal inasemi-simple N,andthatNisthe radical of93.Then 97-INQB CNsince Nisanideal of973andN973C973 since QBisanideal ofN.SoN(%N97l)N CQBNQB which isthusanideal inN;further itisnilpotent since itiscontained intheradical of973. Since Nissemi-simple QBNQJ3 E0.Now (NNN)3 C(NNN)N(NNN) and NNN CQBso(NNN)3 CQBNQB, which wehave shown iszero. That is, NNN isanilpotent ideal inasemi-simple NsoNNN E0.Since Nhasa unitelement thisgives NE0andQBissemi-simple. Wemay now return totheproof ofthetheorem. IfNissemi-simple butnotsimple then ithasanon-zero ideal which, byLemma 2,is semi-simple and hence hasaunit. Lemma 1then ensures that Nis reducible. Thecomponents arecertainly ideals andsosemi-simple, and wemayproceed toreduce them. IfNisfinite thenwemust arrive atan expression ofNasadirect sum ofirreducible components. The components areideals, hence semi-simple, andirreducible hence simple. Thereduction ofasemi-simple algebra tosimple components isunique uptoanordering ofthecomponents. (A20) LetNE97il® ...@973, with the973,-simple. The identity ofNcanbe written asasumoftheidentities inthe971,»,1Ee,-(B...®e,. Suppose N=<6l®... ®<6,then<6kE<6kel+<6ke2+ ...<6ke,Vk=1,...,s- If<6l,-E<6le,- then <6,‘,CNe,E973,-andtheabove sum must bedirect: APPENDIX A 329 .911 = EAT bsi g=i so <€k is an ideal if and only if all the k, are ideals of 91i. But the a are simple, so = @, or (Ckj = O. If the k are irreducible then for a given k not more than one (C ki can be non-zero and it follows that the Cc are just the 91, up to a possible relabelling. The above two theorems determine the structure of semi-simple algebras in terms of simple ones. Before turning to the classification of these we consider representations of semi-simple algebras. Again the representation theory will reduce to that of simple algebras and so we consider this case first. All irreducible representations of a simple algebra are equiva- lent. (A21) If is any minimal left ideal of a simple al then we will show that any irreducible representation of .94 is equivalent to the representation on induced by the regular representation. Let p be some irreducible representation of si that maps .94 into End V, where V has no invariant subspaces under multiplication by p(si) . We first note that any minimal left ideal of End V, the pth column say, carries an equivalent representation to that carried by V. For if V is displayed as a 'column vector', with a basis {bk} consisting of zeroes except for a one in the kth row, then a basis for End V, {e il}, is formed by the arrays whose only non-zero element is a one in the intersection of the ith row and the jth column. Elementary rules of matrix multiplication then give eybk = bikb,. A basis for the pth column is {e kp) where k ranges over the order of the matrices, and e ge kp = bike,,,. So the pth column, for any p carries a representation equivalent to that carried by V. We introduce a linear transformation S that maps the minimal left ideal, 4, of .s4 into the pth column of End V: S4 = p(1)e pp. Since 4 carries an irreducible representation of sti then p(4) carries an irreducible representation of p(A) and so p(J)e pp certainly transforms irreducibly under p(.94). But this is a subspace of the pth column which transforms irreducibly, so either S is a vector space isomorphism or p(4)e pp = O. There must be some p for which this is non-zero, for otherwise we would have p(J) = 0, which cannot be since si is simple. So at least for some choice of p, S is a vector space isomorphism between the minimal left ideal .1 and the pth column of End V. If f E then the following diagram shows the equivalence of the representation carried by the pth column (and hence V) and that carried by 4: where s4T k,si C 21, APPENDIX A 329 .<t<e,,.<t =2N<6,,,»N where N<6,,,-N c971, i=1 so<6,,isanideal ifandonly ifallthe‘6),,areideals of9/3,-.ButtheQB,- aresimple, so<6,,,-E975,-or‘6),,E0.Ifthe<6,,areirreducible then fora given knotmore than one<6,,,-canbenon-zero anditfollows that the <6,,arejustthe975,-uptoapossible relabelling. The above two theorems determine the structure ofsemi-simple algebras interms ofsimple ones. Before turning totheclassification of these weconsider representations ofsemi-simple algebras. Again the representation theory willreduce tothat ofsimple algebras andsowe consider thiscase first. Allirreducible representations ofasimple algebra areequiva- lent. (A21) If9isanyminimal leftideal ofasimple Nthen wewillshow thatany irreducible representation ofNisequivalent totherepresentation on9 induced bytheregular representation. Letpbesome irreducible representation ofNthat maps Ninto EndV,where Vhasnoinvariant subspaces under multiplication by p(N). We first note that any minimal leftideal ofEndV,thepth column say, carries anequivalent representation tothat carried byV. ForifVisdisplayed asa‘column vector’, withabasis {bk} consisting of zeroes except foraoneinthekthrow, then abasis forEndV,{e,-,-}, is formed bythearrays whose only non-zero element isaone inthe intersection oftheithrow and thejthcolumn. Elementary rules of matrix multiplication then give e,-,-bk E5,-kb,-. Abasis forthepth column is{e,,,,}where kranges over theorder ofthematrices, and el,-ek,,E5,-lle,-,,. Sothepthcolumn, foranypcarries arepresentation equivalent tothatcarried byV. Weintroduce alinear transformation Sthat maps theminimal left ideal, 9,ofNintothepthcolumn ofEnd V: ss=p(9)e,,,,. Since 9carries anirreducible representation ofNthen p(9) carries an irreducible representation ofp(N) andsop(9)e,,,, certainly transforms irreducibly under p(N). Butthisisasubspace ofthepthcolumn which transforms irreducibly, soeither Sisavector space isomorphism or p(9)e,,,, E0.There must besome pforwhich this isnon-zero, for otherwise wewould have p(9) E0,which cannot besince Nissimple. Soatleast forsome choice ofp,Sisavector space isomorphism between theminimal leftideal 9andthepthcolumn ofEndV.IffE9 then thefollowing diagram shows theequivalence oftherepresentation carried bythepthcolumn (and hence V)andthatcarried by9: 330 APPENDIX A L(a) af s s p(a) kne pp p(a)p(f)e pp =p(af)e pp . Thus any irreducible representation of a simple algebra is equivalent to that induced on any minimal left ideal by the regular representation. We are now in a position to consider representations of semi-simple algebras. Irreducible representations of a semi-simple algebra are equivalent if and only if their kernels are the same. (A22) Equivalent representations must certainly have the same kernel, so what we need to show is that irreducible representations of a semi-simple algebra with the same kernel are in fact equivalent. A semi- simple algebra is the direct sum of simple ones, and so a representation can be irreducible only if the kernel contains all but one of the simple component algebras. Thus irreducible representations with the same kernel are irreducible representations of the same simple component algebra, and are thus equivalent by the preceeding result. We now return to the classification of algebras by studying the simple ones. The main result is given below. An algebra sti is simple iff ,91 = acmt, where g is a division algebra and ht a total matrix algebra. (A23) First we do the easy bit and assume si = gam,. Then .54 has an identity. Let b be a non-zero element of an ideal J, then b = Ewbue with at least one (bpq say) non-vanishing coefficient in a. But bpq = Ee,pbeq, and so pq-le ipbe qi = 1. That is 1 C .946.94 C J, giving .s4 C and thus si is simple. If now si is simple it has a unit element 1 = Z7=1Pi where the {Pi} are pairwise orthogonal primitive idempotents. If Ai/ -= P1.91P1 then the are certainly subspaces, and are in fact algebras since they are closed under multiplication. Multiplying two different algebras gives slijApk = AijP jP pS 4 pk = slijAjk(5.0 =Pi..9113;s4P05jp. 330 APPENDIX A L(a)fi-E-E->11)‘ Sl lS 10(0) p(f)epp?€_i)p(a)p(f)epp :p(af)epp' Thus anyirreducible representation ofasimple algebra isequivalent to thatinduced onanyminimal leftideal bytheregular representation. Wearenowinaposition toconsider representations ofsemi-simple algebras. Irreducible representations ofasemi-simple algebra are equivalent ifandonly iftheir kernels arethesame. (A22) Equivalent representations must certainly have thesame kernel, sowhat weneed toshow isthat irreducible representations ofasemi-simple algebra with thesame kemel areinfact equivalent. Asemi- simple algebra isthedirect sumofsimple ones, andsoarepresentation canbeirreducible only ifthekernel contains allbutoneofthesimple component algebras. Thus irreducible representations with thesame kernel areirreducible representations ofthesame simple component algebra, andarethus equivalent bythepreceeding result. Wenow return totheclassification ofalgebras bystudying thesimple ones. Themain result isgiven below. Analgebra Nissimple iffNE‘2D®A/l where 91>isadivision algebra andMatotal matrix algebra. (A23) First wedotheeasy bitand assume NE€Z>®A/l. Then Nhasan identity. Letbbeanon-zero element ofanideal 9,then bEE,~,,-b,-,~e,~,- with atleast one(b say) non-vanishing coefficient in91>.ButP41 bpq = ql‘ l andso 2b,,,,“e,-,,be,,,- =1. That is1CNbN C9,giving NC9andthusNissimple. Ifnow Nissimple ithasaunit element 1EEl‘-lP, where the{Pl} arepairwise orthogonal primitive idempotents. IfN,~,-EP,-NP, then the N,-,arecertainly subspaces, andareinfactalgebras since theyareclosed under multiplication. Multiplying twodifferent algebras gives 5411541»/< =Sdttptppsdp/< =911/541/<5Ip APPENDIX A 331 Now siPisi is a two-sided ideal, which is not zero since it contains Pi, and so the simplicity of si gives siPi.si = sti and hence ijpk = PAPk6ip = ilc(51p In particular si„ =isii for any j. Since P1 E .91 there must be elements eil, el; in sip and slip respectively, such that eijeJI = P. If we now define eij E by ei; = eilei; then P ke ti = e ijâik ei;Pk = This gives eiiepo = e4P;Ppepq = eifeig6ip = ezielje,ielq4 = eiiPiewSip = eiieiqôjp = e iq jp In particular the ed are idempotent. But ell c PialPi with P, primitive, so P1s4P1 contains only one idempotent, namely Pi, so we must have e = Pi. So the e,1 span a total matrix algebra Ait whose identity is = EP, = 1 the identity of si. Since for each k P,, is primitive, PksiPk is a division algebra with Pk as identity. Each si kk is an isomorphic copy of sill, say. For if a(') E S411 we define a(k)- — kk by a(k) = ekicrweik. Then for a,(1), swe 1ff(1)e)(k) = ekicr(1)0")elk = ekicr(1)PIP(1)eik since P1 is the identity in sill = ekiame Ike awe lk since the eq are a matrix basis and so (co1)p(1))(k) = a,(k)fi(k). This mapping from si ll to Aid, is obviously invertible and so indeed we have an isomorphism. By taking the direct sum of all elements in sill with their isomorphic images in all a i 1,1, we obtain another copy of si - a say. That is, if au) E we define a , E 9 to be APPENDIX A 331 Now NP,-N isatwo-sided ideal, which isnotzero since itcontains P,-, andsothesimplicity ofNgives NP,-N ENandhence = ti,-kéjp. Inparticular NllENl,-N,-l foranyj.Since PlENll there must be elements e,-l,el,inN,-landNl,-, respectively, such that el,-e,»l EPl.If We HOW define 8,, Gflij =6,~lel,- then P,,e,-,- Ee,-,6,-k el,-Pk =e,-,6,-ll. This gives eiierq =eitptppepq =ettetqétp =eileltetlelqétp Ee,-lPlel,,6,-,, :eilelqéir Ee,-,,<$,-,,. Inparticular theellareidempotent. ButellCP,-NP,- with P,primitive, soP-NP- contains onl oneidem otent, namel P,-,sowemust have 11 Y P Y e,-,-EP,-.Sothee,,~span atotal matrix algebra A/lwhose identity is 2911 =2P1 :1 theidentity ofN. Since foreach kPkisprimitive, PkNPl, isadivision algebra with Pk asidentity. Each Nll, isanisomorphic copy ofNll, say.Forif0/“)ENll wedefine 0/('9EN,,k by0/('9Eeklo/illelk. Then for0/<1), ,8“)ENll (,,,(1>g(1>)(k> =e,,,,,(11g(1>e,, =ekla/(nPlIB(l)elk since Plistheidentity inNll =9/<I¢1’me1t<e klt8(1)e1k since thee,-,-areamatrix basis and so(12/(‘l/3<1))”‘l Ealkl/ilk). This mapping from NlltoNkl,isobviously invertible andsoindeed wehave anisomorphism. Bytaking thedirect sum ofallelements inNllwith their isomorphic images inallNkk weobtain another copy ofNll, Q0 say.That is,if0/(1)ENllwedefine 0/E90 tobe 332 APPENDIX A a, = It is straightforward to see that a sill; further, elements of a commute with all the elements of At. For if a E a ae = Ea (k)e = a(Oe = e iia(ne lie = eilœwei = eiieJlawel; = e4c0-0 = Ee (k) = e ,ia. For every a E sa set a1(k) = ekiciejk. Then aii(k) = e"eliaepe 1k = ekiaii(1)elk so if a11 = ka,j(k) then a-1 E a. Further Eai;e4 = Ea4(k)e = Ea4(i)e4 i.j,k Eei,aefiei; = Eeipaeil = EPiaPj = a. Since this is true for every a E Si we have ai = aitt where a and At are as constructed in the proof. The expression of a simple .91 as si = cannot be unique. For if ei; is a matrix basis then so is erij = se11s-1 where s is any regular element of A. Then a = i,ia'ije with = Ee'oe'jk = Esekis-Jaseiks-' = s(s-las),1s-1 that is, a'q E SaS -1. It turns out though that the choice of a and At is unique up to an inner automorphism like this. Note that if sti = acxitt = watt then we must have a' = a. For if a E a' we can write a = Ei,jaue ,1 with the a E a, and if a, is to commute with Al then a' = + e22 + - + enn) = an since the identities in sa and At coincide. So if sa = aoht = 'am.' where At' = silts -I then we cer- tainly have a' = sas-t. If sa is simple such that sa = OA and .94 = a'OAC then there is an SE si such that Alt,' = sAts -1, at = sas-i. (A24) In view of the above comments it is sufficient to prove that Art' = Let {e,i} = 1, n be a basis for At and {e l,' q} p, q = 1, . . m be a basis for Ait'. Without loss of generality we assume m n. We write e'11 = Ec n-eti CUE a (i) i,j=1 332 APPENDIX A atE<1/(1)@t1/(2) ...@119’). Itisstraightforward toseethat EDENll; further, elements of9) commute withalltheelements ofM.ForiforEED a/9,, : = =B,-ltY(UBl,-B,-,- It : el.la(1)e ll.= 1].Z = It Forevery aENsetal,-("l Eek,-ae,-,.. Then alj(k) =ekleliaejlelk =ek1aI‘j(l)e1k soifal,EEta,-,<(") then al,E921. Further 2411191," =241100911 =Ea;/met] = 2:9,-,~£1E,-,-B,-j = : = £1. ll ll ll Since thisistrueforevery aENwehave NE9D®./l/l where 921andM areasconstructed intheproof. Theexpression ofasimple NasNE2D®./l/l cannot beunique. Forif ellisamatrix basis then soisej-,-Ese,»,-s_1 where sisanyregular element ofN.Then aE2,-,-aj-,-ej-, with _ _ —l -1_ -I -1al,—Xe’),-ae;-,, -zsek,-s ase,-ks —s(s as),-,»s k k thatis,£1]-,'ES@S_1. Itturns outthough thatthechoice ofEDandMis unique uptoaninner automorphism like this. Note that if NE<3J®Att E€D’®J1/l then wemust have 921’EED.Forifire‘.-E)’ wecan write orE2,’,-tr,-,-e,, withthetr,-,<EED,andiftristocommute withMthen orEa/ll(ell +en+...+e,,,,)Eallsince theidentities inNandM coincide. SoifNE9D®A/t E€D’®Jl/l’ where A/l’Es./l/ls“ then wecer- tainly have 921’Es9Ds". IfNisSimple such that NE§D®Jl/l and NE§D’®M' then there isansENsuch thatM’Esll/ls”, 921’Es9Ds". (A24) Inview oftheabove comments itissufficient toprove thatA/l’Esills"- Let{e,-,~} i,jE1,...,nbeabasis forMand{e;,,,} p,qE1,...,m beabasis forA/t’.Without lossofgenerality weassume mEn.We write I1 ell=261/91)" Cr/E93t.;=1 APPENDIX A 333 with at least one (c),1 say) of the cy not zero. If we set --1 a = c „e ipe'n (ii) and b = (iii) then a E enstien, b E Cisien with ab = Cle e' e pq lp 11 ql =E e e by (i) pq lp Cq _ q _ qi i,j=1 _ -1 — cPq cPq ell = en that is Also ab = e11. (iv) (ba)2 = b(ab)a = be iia by (iv) = ba so ba is an idempotent in e' 11s4e'11 = a'e'11; further it is not zero since a(ba)b = (ab)2 = e11, by (iv). But the identity is the only idempotent in a' so we must have If we now introduce and then ba = h --= i=1 g = Eei'ibeii j=1 hg = e11ae1e1be11 — i.j=1 i=1 = Eeilabeii since a E i=1 = e1e11e1 by (iv) i=1 = Eell i=1 APPENDIX A 333 with atleast one(c,,,,say) ofthec,-,-notzero. Ifweset 0= we'll and b=e'lle,,l (iii) then aEellNe’ll, bEellNell with abEc,§,,‘el,,e'lle,,l .. =C5419 1112 Ci/er/9 ql bl’ I,]El =Ct§<iCpq°11= ell thatis abEell. (iv) Also (ba)Z Eb(ab)a Ebella by(iv) Eba sobaisanidempotent ine'llNe'll E€D’e’ll; further itisnotzero since a(ba)b E(ab)2 Eell, by(iv). Buttheidentity istheonly idempotent in ED’sowemust have baEe’ll. (v) Ifwenow introduce h=Eetiaeii (Vi)i=1 and ;'=1 then II II (18=Zeilaeliejlbelj =Eetiaeiibeit 1;,-=1 i=1 II EEel-label, since aENe’ll i=l I1 =2911911911 bY(1") i=l H :2911 i=l 334 APPENDIX A that is hg = 1. (viii) So h must be the inverse of g ((A4)) and gh = 1. But gh = e1be11e11ae1 = i,]=1 i=1 = = by (v). Since ' L 1 we must have m = n, and hence .RE—At. In fact = E e,beipe,eglae4 p,q=1 = CibellaCi = by (NI). This completes the proof. A consequence of this theorem is the following which we will frequently use. If P is an idempotent in a simple si then P = E;._ ,P, where the P, are pairwise orthogonal primitives, and the uniquely determined r is called the rank of P. Two idempotents in si are similar iff they have the same rank. (A25) Any idempotent can certainly be written as a sum of pairwise orthogon- al primitives, this is (A16). To go further we shall use Lemma If P is idempotent in a simple A. then Ps4P is simple. Let9'1 be a non-zero ideal in PAP. Since9.3 is an ideal in PAP the left-hand side is contained in a But .9491,54 is an ideal in the simple and so the right-hand side gives PAP. Thus 91 = PAP. If si is simple then Ps4P is simple with identity P. If P = Er, =1P, with the P, primitive then PAP can be written as a tensor product of some division algebra and a total matrix algebra with the P. diagonal elements. The order of the matrices will then be r, which was shown in (A24) to be uniquely determined. It was also shown in (A24) that all matrix bases are similar, and so as a corollary all primitives are similar. If {P,} are pairwise orthogonal primitives then so are {sP,,s -1}, thus similarity preserves the rank of an idempotent. To see that having the same rank is sufficient for idempotents to be similar note that if P = P1 = with {P,) and {Q,) being different sets of pairwise orthogonal primi- tives then we can choose matrix bases with either the {P,} or the {Q,} 334 APPENDIX A thatis hgE1. (viii) Sohmust betheinverse ofg((A4)) andghE1.But 8hZ ejibeijettaeii ZEeiibeiiaeit |,j=1 IE1 n n I I IZ291159911 Z2911' by IE1 I=l Since 2,";le},E1wemust have mEn,andhence A/l’~Jl/l. Infact I1 —l __89118 Z2ejtlbelpeijeqlaelq P-q=l Thiscompletes theproof. Aconsequence ofthis theorem isthefollowing which wewill frequently use. IfPisanidempotent inasimple Nthen PE2,7:lP,-where theP,arepairwise orthogonal primitives, and theuniquely determined riscalled therank ofP.Two idempotents inN aresimilar iffthey have thesame rank. (A25) Any idempotent cancertainly bewritten asasum ofpairwise orthogon- alprimitives, thisis(A16). Togofurther weshall use Lemma IfPisidempotent inasimple Nthen PNP issimple. Let£73beanon-zero ideal in-PNP. Since 973isanideal inPNP the left-hand side iscontained in973.ButNQBN isanideal inthesimple N, andsotheright-hand sidegives PNP. Thus 973EPNP. IfNissimple then PNP issimple with identity P.IfPE21-lP,with theP,primitive then PNP canbewritten asatensor product ofsome division algebra and atotal matrix algebra with theP,asdiagonal elements. Theorder ofthematrices willthen ber,which wasshown in (A24) tobeuniquely determined. Itwasalsoshown in(A24) thatall matrix bases aresimilar, andsoasacorollary allprimitives aresimilar. If{P,} arepairwise orthogonal primitives then soare{sP,s“}, thus similarity preserves therank ofanidempotent. Toseethat having the same rank issufficient foridempotents tobesimilar note thatif PZiPiZiQi i=1 i=1 with {P,} and{Q,-} being different setsofpairwise orthogonal primi- tives then wecanchoose matrix bases with either the{P,} orthe{Q1} APPENDIX A 335 as diagonals, and (A24) then ensures the existence of an s:Q, = sP,s" The theorem above applies to simple algebras. However the first part may be seen to apply to the semi-simple case. For if P is an element of a semi-simple si then P = QICIQ2CD ... 0Q, where the Q, are in the simple components. P is idempotent if and only if all the Q, are idempotent. By the above theorem each Q, will have a unique rank and so the rank of an idempotent in a semi-simple algebra is uniquely determined. As a special case a primitive in a semi-simple algebra must be primitive in one of the simple components. Thus, of course, not all primitives, and hence all idempotents of the same rank, will be similar in a semi-simple algebra. A subset of all simple algebras is provided by the central simple ones; that is those simple algebras whose centre is generated by the identity. For these algebras we have the following important result. Every automorphism of a central simple algebra is an inner automorphism. (A26) If si is central simple then si = aagn where a is a central division algebra, and ,54°P = g°P0.4/VP. The existence of the involution of transposition on matrices shows that Atn°13 = At,, and so aogoPeht n,, by (A6). We are now in a position, at last, to make use of (A7), giving si0.91°P = End a0,ittn2, that is A.® SPP = A, where m is the dimension of si, and we have again used (A6). We extend any automorphism, t, on si to one on si0.91°P, T, by defining (ab)T = a`b Va E b E saw. In the 'uniqueness theorem', (A24), we essentially proved that all automorphisms of a total matrix algebra are inner. Thus for every X E 3/0SVP, X T = SXS-I where s c .940s4°P, that is at = sas' for a E si and b = sbs-1 for b E SPP. Thus s must commute with every element of se". Since si°P is central simple s must be in si, and so t is inner. So far we have assumed that all algebras are over some field, F, which has not warranted much attention; indeed we have usually simply referred to an algebra as si rather than as si over F. In a moment we shall assume a restriction on the choice of F. The situation for the simple algebras is also such that we may regard a simple algebra over F as an algebra over certain other fields. If .94 over F is simple then the centre is a commutative division algebra, that is, a field. In an obvious way si is an algebra over %, making al over central simple. In the following section we will examine involutions of a simple algebra si over F where F is assumed not to be of characteristic two. (As stated in the introduction for the purposes of this book F can be taken to be one of the zero characteristic fi elds IR or C.) If si over F has an involution T then the set of T-symmetric APPENDIX A 335 asdiagonals, and(A24) then ensures theexistence ofans:Q, EsP,~s" Vi. Thetheorem above applies tosimple algebras. However thefirst part may beseen toapply tothesemi-simple case. ForifPisanelement of asemi-simple Nthen PEQl€0Q2® ...G-DQ, where theQ,areinthe simple components. Pisidempotent ifand only ifalltheQ,are idempotent. Bytheabove theorem each Q,willhave aunique rank and sotherank ofanidempotent inasemi-simple algebra isuniquely determined. Asaspecial case aprimitive inasemi-simple algebra must beprimitive inoneofthesimple components. Thus, ofcourse, notall primitives, andhence allidempotents ofthesame rank, willbesimilar inasemi-simple algebra. Asubset ofallsimple algebras isprovided bythecentral simple ones; that isthose simple algebras whose centre isgenerated bytheidentity. Forthese algebras wehave thefollowing important result. Every automorphism ofacentral simple algebra isaninner automorphism. (A26) IfNiscentral simple then NE<ZZl®Jl/l,, where <22!isacentral division algebra, and N°PE<ZZl°P®Jl/l,,"P. The existence ofthe involution of transposition onmatrices shows that A/l,,°P EA/l,,, and soN®N°P E <ZZl®<ZZl°P®Jl/l,,z, by(A6). Wearenow inaposition, atlast, tomake use of(A7), giving N®N°P EEnd<ZZl®Jl/l,,z, that isN®N°P EA/l,,,where m isthedimension ofN,andwehave again used (A6). Weextend any automorphism, t,onNtooneonN®N°P, T,bydefining (ab)T Ea’b VaEN, bEN°P. Inthe‘uniqueness theorem’, (A24), weessentially proved that allautomorphisms ofatotal matrix algebra areinner. Thus forevery xEN®N°P, xTEsxs" where sEN®N°P, that isa’Esas'1 foraENandbEsbs" forbEN°P. Thus smust commute with every element ofN°P. Since N°Piscentral simple smust beinN,andsotis inner. Sofarwehave assumed that allalgebras areover some field, F, which hasnotwarranted much attention; indeed wehave usually simply referred toanalgebra asNrather than asNover F.IIIamoment we shall assume arestriction onthechoice ofF.The situation forthe simple algebras isalsosuch thatwemay regard asimple algebra over F asanalgebra over certain other fields. IfNover Fissimple then the centre <6isacommutative division algebra, that is,afield. Inan obvious wayNisanalgebra over <6,making Nover <6central simple. In thefollowing section wewillexamine involutions ofasimple algebra N over Fwhere Fisassumed nottobeofcharacteristic two. (Asstated in theintroduction forthepurposes ofthisbook Fcanbetaken tobeone ofthezero characteristic fields IRorC.) IfNover Fhas aninvolution Tthen thesetofT-symmetric 336 APPENDIX A quantities forms a subspace Yr. That is, aEYr if and only if aT = a. Similarly we define .9-7- to be the set of T-skew quantities, and then we have .94 = T. For if a c sa., a = + a T) + .1(a — a T). The sum is direct since if a = aT and a = —aT then a + a = 0 which (for characteristic not two) gives a = O. What is more, if the centre contains a T-skew q then al -= 12T ± q927.. If q is a non-zero element of the centre (of a simple algebra) then it has an inverse which is also T-skew. If a Eg T then a = qq-la, and (q-la)T = a Tq-1T = aq -1 = q-la. The T-symmetric quantities in the centre will form a subfield of T, say. We will refer to an involution as being an involution over `6, say, when ce is the subfield of the centre T left invariant by the involution. If si over F is simple with J and T involutions over then TJ is an automorphism of .9i over T. (A27) If T and J are involutions then TJ is certainly an automorphism of .94 over F. What we need to show is that it leaves elements in the centre invariant. The involutions T and J induce automorphisms of the centre, T. An element of T is T-symmetric if and only if it is J-symmetric. This is, in fact, sufficient to show that T and J induce the same automorph- ism on 'C. Let q be a non-zero J-skew element of ce then qqT is manifestly T-symmetric, and hence J-symmetric. But (qqr). = which since q is invertible, gives qTJ = _qT. SO (q + q T)i = —(q + qT). But q + qT is manifestly T-symmetric, and thus J- symmetric. Since any element that is both J-symmetric and J-skew must be zero we have qT = —q. We have shown then that any J-skew element of is also T-skew. But any element of T can be written as a sum of J-symmetric and J-skew parts and thus T and J coincide on Since T and J are involutions TJ must leave all elements of invariant. The observation that if sti over F is simple then si over is central simple gives (A26) a wider range of applicability than might at first sight be supposed. In particular, it enables us to prove the following. If si over F is simple and T is an involution over then J:a a is an involution over '6 iff there exists an s with s = +ST such that af = saTs-1. (A28) First the easy bit. If a = saTs -1 then La ai is an anti- automorphism. Furthermore aff = s(saTs-1)Ts-1 = S(ST)-lasTs-1, so if sT = ±s, J is an involution. Inner automorphisms leave all elements of the centre invariant. So if T is an involution over t then so is J. Conversely let J be an involution over (6, then JT is an automorphism over T ((A27)). (A26) then ensures the existence of a g such that a.T = g -1ag 336 APPENDIX A quantities forms asubspace EFT. That is,ae3}ifandonly ifaT=a. Similarly wedefine 51tobethesetofT-skew quantities, andthen we have 54=EFT+97. ForifaE54, a=§(a+a7)+§(a—a7). The sum isdirect since ifa=aTand a=—aT then a+a=0which (for characteristic nottwo) gives a=0.What ismore, ifthecentre contains aT-skew qthen .54=3}+q9’T. Ifqisanon-zero element ofthe centre (ofasimple algebra) then ithasaninverse which isalso T—skew. Ifaegy then a=qq_1a, and (q"‘a)T =aTq_lT =aq"1= q_‘a. The T-symmetric quantities inthecentre willform asubfield of<6,%say. Wewillrefer toaninvolution asbeing aninvolution over <6,say, when %isthesubfield ofthecentre <6leftinvariant bytheinvolution. If54over Fissimple with Jand Tinvolutions over %then TJisanautomorphism of54over <6. (A27) IfTandJareinvolutions then TJiscertainly anautomorphism of54 over F.What weneed toshow isthat itleaves elements inthecentre invariant. The involutions TandJinduce automorphisms ofthecentre, <6.Anelement of<6isT-symmetric ifandonly ifitisJ-symmetric. This is,infact, sufficient toshow that TandJinduce thesame automorph- ismon<6.Let qbeanon-zero J-skew element of<6then qqT is manifestly T-symmetric, and hence J-symmetric. But (qqT)' =—qqT’, which since qisinvertible, gives q”=—qT. So (q+qT)’ = —(q +qT). But q+qTismanifestly T-symmetric, and thus J- symmetric. Since anyelement thatisboth J-symmetric andJ-skew must bezero wehave qT=—q. We have shown then that any J-skew element of<6isalso T-skew. Butanyelement of<6canbewritten asa sum ofJ-symmetric andJ-skew parts andthus TandJcoincide on<6. Since TandJareinvolutions TJmust leave allelements of<6invariant. The observation that if54over Fissimple then 54over <6iscentral simple gives (A26) awider range ofapplicability than might atfirstsight besupposed. Inparticular, itenables ustoprove thefollowing. If.94over Fissimple and Tisaninvolution over %then J:a|—>a’isaninvolution over <6iffthere exists answith s=isT such thata’=saTs“. (A28) First the easy bit. Ifa’=saTS_1 then J:a>—>a]isananti- automorphism. Furthermore a”=s(saTs")Ts“ =s(sT)“asTs", soif ST=is,Jisaninvolution. Inner automorphisms leave allelements of thecentre invariant. SoifTisaninvolution over %then soisJ. Conversely letJbeaninvolution over %,then JTisanautomorphism over <6((A27)). (A26) then ensures theexistence ofagsuch that an:g~1ag APPENDIX A 337 a" = (g-'ag)T = g TaT(gT)-1 Since J is an involution a = = gr(grar(e)-1)7 -(gr)_i = g Tg-tag(gr)-1. Since this is true for all a we must have grg-1 -= A E T. If A = —1 then there is nothing left to do, if not then set s = g + gT = g(1+ A) and s will have the desired property. Obviously the choice of such an s is determined only up to multiplication by an element of the centre. A familiar example of an involution is provided by transposition of matrices. In some ordinary matrix basis we define T such that evil,. = e11. For some other basis fe) we define J by ey = e,. T and J are examples of what we shall call equivalent involutions. Two involutions, V and J, will be called equivalent if there is some automorphism S such that a' = asvs-■ ((as)v)s-.. If an inner S relates equivalent involutions J and V, related to some 'standard' involution T by aV = vaTV-1 J  T-1 a = ja , then j = ilsysT for some A. E In classifying the structure of algebras we showed first the existence of the radical. Semi-simple algebras were then defined to have zero radical. It was possible to determine the structure of a semi-simple algebra completely in terms of simple ones, whose structure was in turn given as a tensor product of a division algebra and a total matrix algebra. Most of the structure theorems for associative algebras were first given by J H M Wedderburn, and we shall refer to the expression of a simple ..9El such as a = aalt as the Wedderburn decomposition of A. It is all very well to be able to determine the structure of algebras whose radical is zero, but it would be rather limiting if it told us nothing about algebras with a radical. However, this is not the case. The most important result on the structure of algebras is known as Wedderburn's principal structure theorem. It states that (subject to certain caveats relating to the underlying field) any algebra is the vector space sum of its radical and the semi-simple algebra obtained from the quotient modulo the radical. We shall not need this result and so will not give the proof. This may be found in (for example) Albert [1], Kochendorffer [3] or, for the case of zero characteristic field, in Dickson [2]. As was stated in the introduc- tion to this Appendix we will really only be concerned in this book with algebras over the real field. For this case one can go further in determining the structure of all semi-simple algebras. The Wedderburn APPENDIX A 337 H’=(g“ag)T :TT T—lgH(g)- Since Jisaninvolution H=a”=g’(gTa’(gT)“)’(gT)“ =g’g"ag(g’)“- Since thisistrue forallawemust have gTg“ =/le<6. If/I=—1then there isnothing lefttodo,ifnotthen sets=g+gT=g(1+/1)ands will have thedesired property. Obviously thechoice ofsuch ansis determined only uptomultiplication byanelement ofthecentre. Afamiliar example ofaninvolution isprovided bytransposition of matrices. Insome ordinary matrix basis wedefine Tsuch that9,7,=e,~,~. For some other basis {eff} wedefine Jbyelf=9},-. Tand Jare examples ofwhat weshall callequivalent involutions. Two involutions, VandJ,willbecalled equivalent ifthere issome automorphism Ssuch thata’=aw“ E((a5) ")s*‘. Ifaninner Srelates equivalent involutions JandV,related tosome ‘standard’ involution Tby av=vaTv_‘ 1_ T-a—]a]', thenj =/lsvsT forsome /Ie<6. Inclassifying thestructure ofalgebras weshowed firsttheexistence of theradical. Semi-simple algebras were then defined tohave zero radical. Itwas possible todetermine thestructure ofasemi-simple algebra completely interms ofsimple ones, whose structure wasinturn given as atensor product ofadivision algebra andatotal matrix algebra. Most ofthestructure theorems forassociative algebras were firstgiven byJH MWedderburn, andweshall refer totheexpression ofasimple sdsuch assd=<.ZD®A/t astheWedderburn decomposition of$4.Itisallvery well tobeable todetermine thestructure ofalgebras whose radical iszero, butitwould berather limiting ifittoldusnothing about algebras with a radical. However, thisisnotthecase. The most important result onthe structure ofalgebras isknown asWedderburn’s principal structure theorem. Itstates that (subject tocertain caveats relating tothe underlying field) anyalgebra isthevector space sumofitsradical and thesemi-simple algebra obtained from thequotient modulo theradical. Weshall notneed thisresult andsowillnotgive theproof. This may be found in(forexample) Albert [1],Kochendorffer [3]or,forthecase of zero characteristic field, inDickson [2].Aswasstated intheintroduc- tiontothisAppendix wewillreally only beconcerned inthisbook with algebras over the real field. For this case one can gofurther in determining thestructure ofallsemi-simple algebras. The Wedderburn 338 APPENDIX A structure theorem reduces the classification of simple algebras over the reals to the classification of real division algebras. This had already been done by Frobenius in 1878. He showed that the only associative real division algebras are 11, C and H; the reals themselves, the algebra of complex numbers and the quaternion algebra. A proof may be found in Dickson [2] or Kochendorffer [3]. In view of this we now give a brief discussion of these algebras. Let si be a one-dimensional algebra over IR. Then a basis is provided by u where u2 = Au. If A = 0 then si is nilpotent of index two. If A 0 then it is invertible and if P = Au, P is an idempotent. For any a E si we have a = 1.4P, p c IR and I:a p clearly establishes an isomorphism between si and IR. The real algebra COR) is a two-dimensional algebra generated by i where i2 = —1. This real commutative algebra is not central. It has the well known involution of complex conjugation *:i The real quaternion algebra H(IR) has a basis {1, i, j, k} whose multiplication table is given in table Al. The algebra is generated by the subspace spanned by {i, j}, say. (We note here that the other four- dimensional real simple algebra At 2(IR) is generated by {a, 0} where a,2 = 1, p2 = 1 and c43 = — f3a . For example, a = e12 e 21 , e — e21.) The quaternions are not commutative but the algebra is central. In the given basis, {i, j, k} span the subspace of vector quaternions, whilst the identity spans the scalar quaternions. The involu- tion of quaternion conjugation, q, is defined to change the sign of the vector part of every quaternion. Then qg is self-conjugate and hence in the centre. By inspection q4 is seen to be strictly positive for non-zero q, say qg = A 2. Then q' Â-24 and indeed H is a division algebra. Suppose that T is some other involution, then (A28) ensures that qT = to-i where t = ±t. Since the only self-conjugate quaternions are in the centre, to get an involution distinct from conjugation we must have f = —t. In particular we define = kqk where k is one of the 'standard' basis vectors. This involution will be called a reversion since it leaves the generators {i, j} invariant, but of course reverses their order in products. By taking any vector quaternion t we have an involution given by qT = to-'. However, all such involutions are equivalent to reversion. Without loss of generality we can choose the defining t to satisfy t2 = —1. Then if t and k are linearly independent they generate H. To see this all we need to check is that the commutator [t, k], which is certainly a vector quaternion since it is anticonjugate, is not a linear combination of t and k. But t and k both anticommute with [t, k], which thus cannot be a linear combination of them. Since {k, t) generate H we may define an automorphism, G, by t1 = k, k G = t. This auto- morphism must be inner since H is a central division algebra and hence central simple. That is, t = gkg-1 for some g, and g-' = Â.--2g for some 338 APPENDIX A structure theorem reduces theclassification ofsimple algebras over the reals totheclassification ofrealdivision algebras. This hadalready been done byFrobenius in1878. Heshowed that theonly associative real division algebras areIR,CandH;thereals themselves, thealgebra of complex numbers andthequaternion algebra. Aproof may befound in Dickson [2]orKochendorffer [3].Inview ofthiswenow give abrief discussion ofthese algebras. Let.91beaone-dimensional algebra over IR.Then abasis isprovided byuwhere uz=Au.IfA=0then.91isnilpotent ofindex two. IfA9*0 then itisinvertible andifP=}t“'u, Pisanidempotent. Foranyae.91 wehave a=MP,yeIRandI:aI—>itclearly establishes anisomorphism between .91andIR. The real algebra C(lR) isatwo-dimensional algebra generated byi where i2=—1.This realcommutative algebra isnotcentral. Ithasthe well known involution ofcomplex conjugation *:il—>—i. The real quaternion algebra H(lR) hasabasis {1,i,j,k}whose multiplication table isgiven intable A1.Thealgebra isgenerated bythe subspace spanned by{i,j},say.(We note here thattheother four- dimensional real simple algebra M2(lR) isgenerated by{a,B}where 0/2=1, B2=—1 and afi=—['3a. For example, oz=e12+en, ['3=en—en.) The quaternions arenotcommutative butthealgebra is central. Inthegiven basis, {i,j,k}span thesubspace ofvector quaternions, whilst theidentity spans thescalar quaternions. Theinvolu- tion ofquaternion conjugation, qI—>q,isdefined tochange thesign of thevector part ofevery quaternion. Then qqisself-conjugate and hence inthecentre. Byinspection qqisseen tobestrictly positive for non-zero q,sayqq=A2.Then q'1=A-Zq andindeed Hisadivision algebra. Suppose that Tissome other involution, then (A28) ensures thatqT=tuft" where f=it.Since theonly self-conjugate quaternions areinthecentre, togetaninvolution distinct from conjugation wemust have f=—t.Inparticular wedefine q=kqk"1 where kisoneofthe ‘standard’ basis vectors. This involution willbecalled areversion since it leaves thegenerators {i,j}invariant, butofcourse reverses their order inproducts. Bytaking anyvector quaternion twehave aninvolution given byqT=tqt-1. However, allsuch involutions areequivalent to reversion. Without loss ofgenerality wecanchoose thedefining tto satisfy t2=—1.Then iftandkarelinearly independent they generate H.Toseethisallweneed tocheck isthat thecommutator [t,k],which iscertainly avector quaternion since itisanticonjugate, isnotalinear combination oftandk.Buttandkboth anticommute with [t,k],which thus cannot bealinear combination ofthem. Since {k,t}generate H wemay define anautomorphism, G,bytc=k,kc=t.This auto- morphism must beinner since Hisacentral division algebra andhence central simple. That is,t=gkg" forsome g,andg*'=lrzg forsome APPENDIX A 339 E Fi. So if s = A-1g then t = sld, which is the criterion for T to be equivalent to reversion. Table Al The quaternion algebra 1 1 1 j 1 k —1 k —1 1 1 —k —1 i k k 1 —i —1 Just as it is important to know that any positive real number can be written as a square of a positive number, and that any complex number can be written as a square, it will prove important to know that any reversion symmetric quaternion can be written as a square of a reversion symmetric quaternion. As we have remarked qq is a positive real number and so we may introduce a norm defined by 1q12 = qq. Reversion is related to conjugation by q = k-14k, and for any g we have q1 4/ tip, so if y = j)' then k-lyk 1Y 2 Writing 1 + q as 1 + q = q-lq + q = (1+ g -1)g gives g = (1+ q-1)-1(1 + g), for any g. In particular, if yo is a unit-norm reversion symmetric quaternion then Yo = (1 + Yo-I)-1(1 + yo) k-1(1 + k-Iy0k)k(1 + y o) 11 + Yc,112 ( 1 + yo )2 For any g we have = k-lcikkcA-1 = k-lqqk = q12, so from (i) ± yo2 = + YO)1(12 = 11 + yo12. Thus (ii) gives yo = x2, for the reversion symmetric x given by x = , 11+ yo 1 + yo Then for a reversion symmetric y of arbitrary norm we can write Y =1Y1Yo = {1Y1 12x)2, since any positive real number has a real square y — = (i) by (i) II + yo —11 since k2 = —1. (ii) root. APPENDIX A 339 /lelR. Soifs=/l'1g then t=sks, which isthecriterion forTtobe equivalent toreversion. Table AlThequaternion algebra 1 1 j k i j k " -1 k —j " " —k -1 i- -1 w=--->- :»='---—>- .... _. Justasitisimportant toknow thatanypositive realnumber canbe written asasquare ofapositive number, andthat anycomplex number canbewritten asasquare, itwillprove important toknow that any reversion symmetric quaternion canbewritten asasquare ofareversion symmetric quaternion. Aswehave remarked qqisapositive real number and sowemay introduce anorm defined by|q|2=qq. Reversion isrelated toconjugation byq=k"¢'jk, and forany qwe have q“=q/lq|2, soify=j?then k"yk _Y"=_T (1)lyl Writing 1+q as 1+q=q"q +q=(1+q")q gives q=(1+q")"(1 +q),foranyq.Inparticular, ifyoisaunit-norm reversion symmetric quaternion then Y0=(1+y@“)“(1 +yo) _k"(1+ k"yok)k(1 +yo) .—iijgl" by(1) il+Y0i 1+ 2=(-———y° j Sincek2=-1. (ii) i1+)’0—1| Forany qwehave jk'1qk]2 =k'1qkkqk“= k"qqk =qq=lq|2, so from (i) l1+yJ1|2=|k“(1 +y@)k|2 =I1+)’0i2- Thus (ii)gives yo=x2,forthereversion symmetric xgiven by x: 1‘i’ yo l1+y@l' Then forareversion symmetric yofarbitrary norm wecan write y=Iylyo ={[y|"2x}2, since anypositive realnumber hasarealsquare root. 340 APPENDIX A It will be useful to be able to identify the tensor products of these division algebras. Obviously ROE IF1, F1OC = C and IFIOH = H. The algebra UDC has a basis {1, i, j, ij} where i and j commute and i2 = j2 = _1. So if P = 1(1+ ij) and Q =1(1— ij) then P and Q are orthogonal idempotents such that 1 = P + Q. The algebra P(COC)P has P as identity, and since P is in the centre of COC we have P(COC)P = (COC)P, which is a two-sided ideal. Similarly for (COC)Q. Since P and Q are orthogonal COC = (C0C)PO(COC)Q. We may choose {P,iP) as basis for (COC)P and so have (COC)P = C. Similarly for the other ideal giving COC COC. (A29) The algebra COH has a basis {1, z, i, j, k, zi, zj, zk} where {1, z) is a basis for the complex subalgebra that commutes with the quaternion subalgebra spanned by {1, i, j, k}. COH may be generated by {z, i, j}. The subset {1, z) spans the centre which is thus isomorphic to C. If en = 2k(1 + zi) and e22 = .1(1 — zi) then ell, e22 are orthogonal idempotents with 1 = ell + e,2. If we choose e21 = jell = e22j and e12 = —je/2 = e iij then the eu form an ordinary basis for At2(IF1), so C(E)®H(11) C(111)0.4 2(11). (A30) We do not have to do any work to determine the structure of HOH. The quaternion algebra is a central division algebra and, since it has the involution of conjugation, H = H°P. So from Theorem 4 we have H(R)OH(F1) .M.4(1F1). (A31) Having completed our review of associative algebras we turn now to a generalisation of the concept of a vector space in which the field is replaced with a ring, or associative algebra, with unit element. A right R-module M. over the ring R is an additive Abelian group with a map from M x R M:(x, q)1--xq such that x(9192) = (xq 0.72 (i) x(qi + q2) = xqi + xq2 (ii) (x + y)q = xq + yq xi = x (iii) where 1 is the identity in R. 340 APPENDIX A Itwillbeuseful tobeable toidentify thetensor products ofthese division algebras. Obviously 1B®]R =IR,]R®C =Cand1B®H ==H. The algebra C®C hasabasis {1,i,j,ij}where iandjcommute and i2=jz=—1.SoifP=§(1+ij) and Q=§(1—ij) then Pand Qare orthogonal idempotents such that 1=P+Q.The algebra P(C®C)P has Pasidentity, and since Pisinthecentre ofC®C wehave P(C®C)P =(C®C)P, which isatwo-sided ideal. Similarly for (C®C)Q. Since PandQareorthogonal C®C =(C®C)P@(C®C)Q. We may choose {P,iP} asbasis for (C®C)P and sohave (C®C)P =C.Similarly fortheother ideal giving C®C =CGDC. (A29) The algebra C®H hasabasis {1,z,i,j,k,zi,zj,zk}where {1,z} isabasis forthecomplex subalgebra thatcommutes with thequaternion subalgebra spanned by{1,i,j,k}.C®H may begenerated by{z,i,j}. The subset {1,z}spans thecentre which isthus isomorphic toC.If e,,=§(1+ zi)and e22=§(1— zi)then en, en are orthogonal idempotents with 1=en+932. Ifwechoose 921=jeu =ezzj and 9,2=—je22 =—eHj then thee,-,-form anordinary basis forA/t2(]Pt), so C(]R)®H(]R) =C(1Pt)®A/t2(]R). (A30) Wedonothave todoanywork todetermine thestructure ofH®H. Thequaternion algebra isacentral division algebra and, since ithasthe involution ofconjugation, H=H°P. Sofrom Theorem 4wehave H(]R)®H(]R) =A/t4(]R). (A31) Having completed ourreview ofassociative algebras weturn now toa generalisation oftheconcept ofavector space inwhich thefield is replaced with aring, orassociative algebra, with unit element. Aright R-module M.over thering Risanadditive Abelian group with amap from MXR?>M:(x, q)iinrq such that r(qiq1) =(xqdqz (i) x(¢I1+ Q2)=X4111’ X412 (ii) (r+y)q=Xq+yq xl=x (iii) where 1istheidentity inR. APPENDIX A 341 The writing of the element from R on the right-hand side is of significance in (i) when R is non-commutative; in this case the above are obviously altered to give a left R-module. The notion of a linear map may readily be extended to apply to left (or right) R-modules. If I is a minimal left ideal in an algebra with unity, then I is an example of a left .94-module. If si is simple with sl = aatit then I is also a right 2-module, for multiplication on the right by a will preserve the I. In this case I is simultaneously a left si-module and a right 2-module, with the szi action being right 2-linear, and the a action being left si-linear. Thus for simple algebras we are lead to consider right H-modules. Although the concept of linear independence extends to modules, in general an R-module need have no basis. However, H-modules do have bases, the number of basis vectors determining the quaternionic dimen- sion, dim H. Thus, for example, if I is a minimal left ideal in si -= HOER, then dim HI = r, whereas dim BI = 4r. Bibliography Albert A 1941 Introduction to Algebraic Theories (Chicago: Chicago University Press) 1961 Structure of Algebras (Am. Math. Soc. Coll. Pub!. vol 24) Greub W 1978 Multilinear Algebra 2nd edn (Berlin: Springer) APPENDIX A 341 The writing oftheelement from Rontheright-hand side isof significance in(i)when Risnon-commutative; inthiscase theabove are obviously altered togive aleftR-module. The notion ofalinear map may readily beextended toapply toleft(orright) R-modules. IfIisa minimal leftideal inanalgebra with unity, 424,then Iisanexample ofa left42¢-module. If42¢issimple with 42¢=9D®A/l then Iisalso aright 91)-module, formultiplication ontheright by91)willpreserve theI.In thiscase Iissimultaneously aleft:21-module andaright 91)-module, with the42¢action being right 91>-linear, andthe91>action being left:21-linear. Thus forsimple algebras wearelead toconsider right H-modules. Although theconcept oflinear independence extends tomodules, in general anR-module need have nobasis. However, H-modules dohave bases, thenumber ofbasis vectors determining thequaternionic dimen- sion, dimH.Thus, forexample, ifIisaminimal left ideal in 42¢=H®A/l, then dimHI=r,whereas dimRI=4r. Bibliography Albert A1941 Introduction toAlgebraic Theories (Chicago: Chicago University Press) i 1961 Structure ofAlgebras (Am. Math. Soc. Coll. Publ. vol24) Greub W1978 Multilinear Algebra 2ndedn(Berlin: Springer) Appendix B Vector Calculus on E3 As an illustration of the methods of differential calculus it is useful to make contact with the elementary vector calculus of Euclidean 3-space. Such a space regarded as a manifold has the special property of admitting a class of global charts. We might call one such a chart a Cartesian chart since the coordinate maps {x'} i = 1, 2, 3 yield the familiar Cartesian coordinates x'(p) for p E fl3In such a global chart the Euclidean metric tensor is expressed as 3 g = Edx'Odx' i=1 The orthonormal frames {X} = (313x', 313x 2, 3/3x3) and co-frames {e'} = {dx1, dx2, dx3} are in this case naturally dual to each other. Observe also that dx' = 3/3x'. For some problems other non-global charts are useful. The familiar 'spherical polar' chart with coordinate functions (r, 0, cp) has co-domain 0 < r(p) < co 0 < p(p) 27T 0 < 0(p) <VT. The polar chart is related to the Cartesian chart on the overlap by the transformation of coordinates t)2 4_ (x2)2 ± (x3)2]1/2 If we tried to cover the whole surface r = constant (* 0), with a single coordinate chart there would arise an ambiguity in assigning r = [(xi) 2 + (x2)2 + (x3)211/2 [(x')2(0210 = sin -1 [(x1)2 + (x2)2 ± (x3)2r2 = cos -1 Appendix B Vector Calculus onIR3 Asanillustration ofthemethods ofdifferential calculus itisuseful to make contact with theelementary vector calculus ofEuclidean 3-space. Such aspace regarded asamanifold has the special property of admitting aclass ofglobal charts. Wemight callone such achart a Cartesian chart since thecoordinate maps {x'} i=1, 2,3yield the familiar Cartesian coordinates x'(p) forpelB3. Insuch aglobal chart theEuclidean metric tensor isexpressed as 3 g=2dx'®dx' x‘(p) e1R. i=l Theorthonormal frames {X,~} ={S/ox‘, E9/6x2, 8/6x3} andco-frames {e'}={dx‘, dxz, dx3} areinthiscase naturally dual toeach other. Observe also that (T?=8/Bx’. For some problems other non-global charts areuseful. The familiar ‘spherical polar’ chart with coordinate functions (r,8,tp)hasco-domain 0<r(p)<w O<(p(p)€21r 0<8(p)<1r. The polar chart isrelated totheCartesian chart ontheoverlap bythe transformation ofcoordinates r: [(xl)Z + (x2)Z + (x3)2]l/2 12 22l/2 Qzsin-1 [(x1)2 +(x2)2 +(x3)2]1/2 _COS_.___’£i_ (P [(xI)2 +(x2)2 +(x3)2]1/2' Ifwetried tocover thewhole surface r=constant (ab0),with a single coordinate chart there would arise anambiguity inassigning APPENDIX B 343 coordinates to the poles of the sphere. Such ambiguities can give rise to 'singularities' in subsequent calculations, these pathologies reflecting only an improper use of coordinates. In a polar chart we may write ax' ax ax ---d0x ax' ax' g = ar E(dr + 30 dO + ar —idcp)0( idr + id0 + t=1 acp 30 acp or, since = r sin 0 cos cp x2 = r sin 0 sin cp x3 = r cos 0 g = drOdr + r2d00d0 + r 2 sin 2040dcp. Similarly 3 g* = 1=1 3 = ERar/ax93, + (30/3x93 0 + (aqqaXI)adORarlaX r)a, 1=1 + (3013.03 e + (acp/ax9a q,] 3 3 1 a 3 1 a 3 =—® + 0 + ar ar r2 ae 30 r2 sin 20 cp 3. Hence an orthonormal co-frame in this chart is {E'} = {dr, rd0, r sin 0 dcp} with dual (orthonormal) frame (17,)ila la 1 tar' r 30' rsin 0 acid. The metric duals of dr, dû, dcp are the local vector fields a 3 1 3 = , = El) = a r r2 30' r2sin20 acp (Observe that points p with r(p) = 0, 0(p) = 0 are outside our working chart.) On the overlap U of a Cartesian chart and our polar chart, for f E 5-,(U) we may write df = (3f/ax9dx = (af/ar)dr + (af/30)d0 + (3f/acp)dcp. The metric dual of df is called the gradient of f, sometimes written grad f. On U grad f = cif = (afiaxi)atax , = 13 r)di + (3f130)de + (afiacp)d—cp af \ 3 1 ( af) a 1 ( af \ 3 = 3/.Jar r2\30 ae r2 sin 20k 399)a€P APPENDIX B 343 coordinates tothepoles ofthesphere. Such ambiguities cangive riseto ‘singularities’ insubsequent calculations, these pathologies reflecting only animproper useofcoordinates. Inapolar chart wemay write , . . . . . .‘8x’ 8x’ 8x’ 8x’ 8x’ 8x’ )= —d —d0 —d —d —d —dggist '+a0 +8(p ‘pl®(ar'+a0 6+a<p‘p or,since x‘=rsin0cos<p x2=rsin6sin<p x3=rcos0 g=dr®dr +r2d6®d6 +r2sin20d<p®d<p. Similarly 3 8*=;(9I,®9x,) 3 =2[(8r/8x'“)8, +(80/8x'i)8(, +(8(p/8x")8 ]®[(8r/8x")8,¢ i=1 +(a0/ems, +(a<p/ax")a,,] 8 8 18 8 1 8 8 _8r®8r +,Za0®a0 +,1sin1ea<p®a<p' Hence anorthonormal co-frame inthis chart is{E’} ={dr, rd6, rsin6d<p} with dual (orthonormal) frame 818 1 8Y~=—-—i—.{'} law r80’ rsin6 Stpi Themetric duals ofdr,d6,d(parethelocal vector fields ~ 8~ 18 ~ 1 8d=——,d6=——,d =i—.r 9? rz53 (p rzsin 26509 (Observe thatpoints pwith r(p) =0,6(p) =0areoutside ourworking chart.) Ontheoverlap UofaCartesian chart andourpolar chart, for fe@(U)wemay write df=(Sf/8x")dx" =(Sf/8r)dr +(Sf/86)d6 +(Sf/8(p)d(p. The metric dual ofdfiscalled thegradient off,sometimes written gradf. OnU gradf_df_(af/awe/ext -(Sf/8r)dr +(af/a0)d0 +(af/a¢)d<p _afa1afa 1afa 'i8r)8r J’rliaelae +r1sin1e(a<pla¢' 344 APPENDIX B In terms of the orthonormal basis { Y,} gradf — af )y, + 1( af )Y2 ± ar r 80 If Z is a vector field on U we may write Z = = ra/ar + ra/30 + (Paia(f) where r F(U). The 'rate of change of f' in the direction specified by the vector Z, or the directional derivative of f in the direction Z, is defined as Z(f). In terms of the vector field grad f Z(f) df(Z) = g(Z, crf) = g(Z, grad f). In three-dimensional Euclidean space it is customary to use a dot notation for the metric evaluated on two vectors, namely g(X, Y) -=- X.Y. This casts the expression for the directional derivative into the form Z(f) = grad f.Z. Let us explicitly compute the * map associated with the Euclidean metric. If (E') is any orthonormal co-frame with respect to this g then, with *1 = El A E2 A E3, we find *El E2 A E3, *E2 E3 A El, *E3 El A E2 *(E1 A E2) E3, *(E2 A E3) = E', *(E3 A El) = E2 *(EI A E2 A E3) = 1. Consequently, in this case, ** = 1 on all forms. The * map for Euclidean 11:13 establishes a relation between 2-forms and 1-forms. The metric dual, —, maps 1-forms to vector fields. Thus there is a corres- pondence given by the Euclidean metric tensor between 2-forms and vector fields on IR3. Given two vector fields in any g-orthonormal frame, X = Y =J, we have -(17 A — 2V)1:7-1 A -1>-2 (VV)-172 A k-3 (VV A VI. But since { } is an orthonormal co-frame *(1-; A = (VV.. 21) (VV ± V) 1-72 r T*E'. Hence the orthonormal components of the vector field *(I A k) correspond to the components of the cross or vector product of two vectors with orthonormal components (1), ('') respectively. Such a correspondence also enables us to make contact with the operation curl. 1 ( af Y3. r sin ario 344 APPENDIX B Interms oftheorthonormal basis {Y,-}iiP)tn)=—Y ——Y E —Y.gradf (Sr ‘+ra0 2+rsin6 sq»3 IfZisavector field onUwemay write Z=5'8/8x’ ={ya/at +gta/ae +gt/a/a<p where E‘,E’,59,§"’eF(U). The‘rate ofchange off’inthedirection specified bythevector Z,orthedirectional derivative offinthe direction Z,isdefined asZ(f). Interms ofthevector fieldgradf Z0")Edf<Z>=gtz,<5?)=g<Z.gram- Inthree-dimensional Euclidean space itiscustomary touse adot notation forthemetric evaluated ontwovectors, namely g(X, Y)E X-Y. This casts theexpression forthedirectional derivative intothe form Z(f) =gradf-Z. Letusexplicitly compute the*map associated with theEuclidean metric. If{E"} isanyorthonormal co-frame withrespect tothisgthen, with*1=E‘AE2AE3,wefind *E'=EZAE3, *E2=E3,\E‘, *E3=E‘,\E1 *(E1/\E2)=E3,*(E2/\E1)=E‘.*<E3/\E‘)=E2 *(E1/\ EZAE3) = Consequently, inthiscase, **=1onallforms. The *map for Euclidean 1R3establishes arelation between 2-forms and 1-forms. The metric dual, ~,maps 1-forms tovector fields. Thus there isacorres- pondence given bytheEuclidean metric tensor between 2-forms and vector fields on1R3.Given twovector fields inanyg-orthonormal frame, X=§"Y,-, Y=Q/Y1‘, wehave X/\ Y=(E162 *g2€1)'-Y1/\ Y2'1'(E2? —§3§2)Y2/\ Y3 +<s3c1— aw)?“ Yr Butsince {Ti} isanorthonormal co-frame *<>?AY/>=<s1:1— s1c1>Y.+<s1c*- eclir. +<5-“cl—em. err*1R% Hence theorthonormal components ofthevector field *(X,\Y) correspond tothecomponents ofthecross orvector product oftwo vectors with orthonormal components (5'), (C) respectively. Such a correspondence alsoenables ustomake contact withtheoperation curl. APPENDIX B 345 For a vector field V on UE FP we define — curl V = For example, in a Cartesian chart with V = 17 = 1»dx] dV = (al V2 – 321(1)dX1 A dX2 (32V3 - a3172)dX2 A dX3 + (a3v1 — 3iv3)dx3 A dx1 *d -17 = (311/2 – a2V1)dx3 + (a2v3 — 33v2)dxl + (a 3v1 — 81v3)dx2. Thus, indeed, the orthonormal components of *d i7 have the expected form for the components of the curl of the vector field with orthonormal components (v1, v2, v3). If we work in the polar chart with = Vra, + vea, + = viy, + v2y2 + v3y3 where 1/' = V', V2 = r1/8, V3 = r sin 01/9', then = V1E1 + V2E2 + V3E3 = V`dr + r 21/64:10 + r2 sin 20VTdcp where E' = dr, E2 = rd0, E3 = r sin Odcp. Hence dV = 381/rd0A dr + 3g,Vrdcp A dr + ar(r21/Nr A dO +39,(r21Mdcp A dO + ar(r2 sin 20 Vv)dr A cicp +9(r2 sin 20 V)d0 A cicp = [3r(r2V9) – aeVr]-irEl A E2 [80(r2 sin 29V) 1 –39,(r21/6)1 r2 sin 9E2 A E3 [39,1». –ar(r2 sin 20 V 1 cP)] r sin 0E3 A El SO *d = [a(r170) – a0V1(11r)E 3 + NV" –3,(r2 sin 29 V)]1/(r sin 0)E2 + [30(r2 sin 20 V(P) –9,(r2Ve)111(r2 sin 0)E'. The orthonormal components of *d fl once again provide the classical component expression of the curl of V, here in polar coordinates. The maps * and – also give a correspondence between vector fields APPENDIX B 345 Foravector field VonUER3wedefine curlV=*iTV. Forexample, inaCartesian chart with V=V/E9,-: V=Vldx/A av=(a,v1 -82V1)dx‘ Adxz+(a,v3 -E93V2)dx2,\dx3 +(E93V‘ —8lV3)dx3 Adx‘ *dl7=(a,v2~62V')dx3 +(azvi-83V2)dx1 +(a,v1-81V3)dx2. Thus, indeed, theorthonormal components of*dV have theexpected form forthecomponents ofthecurlofthevector field with orthonormal components (V1, V2,V3). Ifwework inthepolar chart with V=V'E9,+V969 +V¢8,, =v1Y,+WY,+V3Y3 where V‘=V’,V2=rV9, V3=rsin6V‘”, then V=V1E‘+ VZEZ+V3E3 =V’dr +r2V"d6 +r2sin26V‘”dq9 where E1=dr,E2=rd6, E3=rsin6dq9. Hence av=89V’d6,\dr +E9¢V’dq9,(dr +E9,(r2V")dr,\d6 +E9¢(r2V9)dq9,( d6+8,(r2 sin26V‘”)dr Ad(p +E99(r2 sin26V¢)d6 Ad(p 1 . =[3,(r2V9) —89‘/r];E1/\ E2 ‘i’[39(r2SlI126V¢) 1 -E9¢(r2V9)];2:1—5E2 AE3-l"[8,,V' -8,(r2 sin26V¢)] E3 AEl so *dv=[8,(r2V9) -89V'](1/r)E3 +[a,,v' —8,(r2sin 26V°’)]1/(rsin 6)E2 +[E99(r2sin26V‘”) —8¢(r2V9)]1/(r2 sin6)E‘. The orthonormal components of*dV once again provide theclassical component expression ofthecurlofV,here inpolar coordinates. The maps *and ~also give acorrespondence between vector fields 346 APPENDIX B and 0-forms on 1113. The 0-form div V associated with a vector field V is defined by (div V) = In a Cartesian chart * = Vldx2 A dX3 V2dX3 A dx1 + V3dx1 A dX2 d* = (3 10 + ay2 + 331/3)dX1 A dX2 A dX3. But in this case *1 = dxl A dX2 A dX3 so *d* V = aivi + a2v2 + a3v3. Exercise B1 Compute div V in the polar chart above. Thus the operations of grad, curl and div in IR 3 are seen to correspond to the application of the exterior derivative d to 0, 1 and 2 forms respectively followed by the metric correspondence relating such forms to their metric duals. It is a worthwhile exercise to verify the vector analysis identities grad (f7) = ( grad f)h + f( grad h) curl (fv) = (grad f) x u + f( curl v) div (fv) = g( gradf, v) + f div v div (v x u) = g(v, curl u). by associating differential forms of the appropriate degree with the functions f, h and vectors u, v. These relations all follow from the properties of the Hodge map, the Leibnitz rule for d and its nilpotency, d2 = 0. By composing the operator *d with itself one obtains a higher-order differential operator on forms. If f E 5,(R3) then in a Cartesian chart *df = if dx2 A dX3 a2fdX3 A dx + 33 f dx 1 A dX2 *d*df = (a; + a + ai)f this being the Laplacian operator on the function f. The Hodge map affords us an efficent way to calculate the Laplacian in any chart. The trick is to express forms in a coordinate (or natural) coframe prior to the action of d thus exploiting d 2 = 0 for each natural basis form, but to revert to the orthonormal co-frame prior to taking a Hodge dual. For example, in any polar chart df = a dr + GfdO + a,f dcp = 34E1 + (11r)a E 2 + 1/(r sin 0)3 (pf E3 346 APPENDIX B and0-forms on1B3.The0-form divVassociated with avector field Vis defined by (divv)=*d*V. InaCartesian chart *1?=V‘dx2Adx3 +V2dx3Adx' +V3dx'Adxz d*l7=(a,v1+ a,v2+83V3)dx1Adx2Adx3. Butinthiscase*1=dx1A dxiAdx} so *d*V=a,v1+ a,v1+a,v3. Exercise B1 Compute divVinthepolar chart above. Thus the operations ofgrad, curl and div in1R3are seen to correspond totheapplication oftheexterior derivative dto0,1and2 forms respectively followed bythemetric correspondence relating such forms totheir metric duals. Itisaworthwhile exercise toverify the vector analysis identities grad(f71) =(gradf)/1 +f(grad/1) curl(fv) =(gradf) ><v+f(curlv) div(fv) =g(gradf, v)+fdivv div(v><u)=g(v, curlu). byassociating differential forms oftheappropriate degree with the functions f,hand vectors u,v.These relations allfollow from the properties oftheHodge map, theLeibnitz rulefordanditsnilpotency, dz=O. Bycomposing theoperator *dwith itself oneobtains ahigher-order differential operator onforms. Iffe 9(lR3) then inaCartesian chart *df=81fdx2Adx3 +82fdx3Adx1 +83fdx1 Adxz *<l*df=(8%+8%+8§)f thisbeing theLaplacian operator onthefunction f.The Hodge map affords usanefficent way tocalculate theLaplacian inanychart. The trick istoexpress forms inacoordinate (ornatural) coframe prior totheaction ofdthus exploiting dz=0foreach natural basis form, but torevert totheorthonormal co-frame prior totaking aHodge dual. For example, inanypolar chart df=8,fdr +89fd6 +a¢f(l(P =8,fE1 +(1/r)89fE2 +1/(rsinl9)8q,fE3 APPENDIX B 347 *df = a rf E2 A E3 + (11r)a of E3 A + 1/(r sin Oa j El A E2. Or, reverting to a natural basis, *df = a rfr2 sin 0 de A dcp + sin 03 of dcp A dr + (1/ sin 0)3 9,f dr A de. Now apply d taking notice of the fact that dû A dû = 0 etc: d*df = (3 r(r2 sin 03J) + ae( sin 0 3 ef)± e(aV))clr A dO A dcp. But *(dr A dO A d(p) = 1/(r2 sin O)*(E' E2 A E3) = 1/r2 sin O. Thus finally 1 1 *d*df = —1 ar(r23 rf) + 38( sin 036f) + a2 f. r2 r2 sin 0 r2 sin 20 The notion of a Laplacian can be generalised to an operator on p-forms, in which case it is usually called more generally the Laplace– Beltrami operator. If cy E rAp(U) then Aœ E FA(U) is defined in Euclidean 3-space by Act' = (- 1)P i(d*d* – *d*d)a which reduces to the above Laplacian on 0-forms. The components of the Laplace–Beltrami operator on a 1-form give the 'vector Laplacian'. Many physical theories are formulated in terms of tensor fields satisfying field equations. Such field equations often arise as the result of setting to zero certain forms constructed out of d and * and other differential forms. For instance, the static Newtonian gravitational field in Euclidean 3-space devoid of matter is described in terms of a real function 4120 on 11V subject to the equation d*c14) = 0 or, after applying * (LI = 0. Solutions to this equation define a vector field X = cl419 called the Newtonian gravitational field. The integral curves of X describe lines of gravitational force. A massive (test) particle experiences 'Newtonian acceleration' in the direction determined by X To describe in more detail the interaction of this field with massive particles requires a formulation of Newton's laws of motion. Surprisingly we must wait until Chapter 6 before the notion of particle acceleration is defined. Suffice to say here that a massive particle is endowed with a parameter m, its inertial mass, such that it experiences the Newtonian gravitational 'force' mdc1). A smooth distribution of matter can generate a Newtonian gravitational field. If the distribution is specified by the mass density 0-form p E .5'(lR3), it acts as a source of Newtonian gravity according to Poisson's equation: d*dcl) = p*l. (NB Both sides of this equation E rA3(1R3).) APPENDIX B 347 *df=8,fE2 AE3+(1/r)8,,fE3 AE‘+1/(rsin 9)8,,,fE~' AE2. Or,reverting toanatural basis, *df=8,fr2 sin9d9A d(p+sin98,,fdrpA dr+(1/sin 9)8,,,fdr Ad9. Now apply dtaking notice ofthefactthatd9Ad9=0etc: d*df =(8,(r2 sin98,f) +89(sin1986f) +$é(8f,f))drAd9Ad(p. But *(drA d9Ad(p)=1/(rzsin 9)*(E‘ AE2AE3)=1/rzsin 9. Thus finally *d*df— L8 (28f)+i8 (sin98f)+#a2f _r2 rr ' r2sin96 9 r2sin29‘p' The notion ofaLaplacian can begeneralised toanoperator on p-forms, inwhich case itisusually called more generally theLaplace—- Beltrami operator. Ifael"/\,,(U) then Aael"/\,,(U) isdefined in Euclidean 3-space by Aa=(-1)1=+ 1(a*a* -*d*d)a which reduces totheabove Laplacian on0-forms. The components of theLaplace—Beltrami operator ona1-form give the‘vector Laplacian’. Many physical theories areformulated interms oftensor fields satisfying field equations. Such field equations often arise astheresult ofsetting tozero certain forms constructed outofdand *andother differential forms. Forinstance, thestatic Newtonian gravitational field inEuclidean 3-space devoid ofmatter isdescribed interms ofareal function <I>onIR3subject totheequation d*d<I> =0or,after applying * A<I>=O. Solutions tothis equation define avector field X=d<I> called the Newtonian gravitational field. The integral curves ofXdescribe lines of gravitational force. Amassive (test) particle experiences ‘Newtonian acceleration’ inthedirection determined byX.Todescribe inmore detail theinteraction ofthis field with massive particles requires a formulation ofNewton’s lawsofmotion. Surprisingly wemust waituntil Chapter 6before thenotion ofparticle acceleration isdefined. Suffice to sayhere that amassive particle isendowed with aparameter m,its inertial mass, such thatitexperiences theNewtonian gravitational ‘force’ md<T). Asmooth distribution ofmatter can generate aNewtonian gravitational field. Ifthedistribution isspecified bythemass density 0-form pe§'(lB3), itactsasasource ofNewtonian gravity according to Poisson’s equation: d*d<I> =p*1. (NB Both sides ofthisequation eI"/\_,(IR3).) 348 APPENDIX B Exercise B2 Obtain in the IR3 cylindrical polar chart with coordinates (r, cp, z) and orthonormal co-frames e' = dr, e2 = rclq9, e3 = dz the component equa- tion for the Newtonian potential (1), (1/03,(ra 10) + (1/r 2)32,(1) + aill) = p. 348 APPENDIX B Exercise B2 Obtain inthe1R3cylindrical polar chart with coordinates (r,tp,z)and orthonormal co-frames e‘=dr,e2=rdtp, e3=dzthecomponent equa- tionfortheNewtonian potential <I>, (1/r)8,(r8,<1>) +(1/r2)a;<1> +a§<I>=p. REFERENCES 349 References [1]Albert A 1961 Structure of Algebras (Am. Math. Soc. Coll. Publ. vol 24 [2]Dickson L 1960 Linear Algebras (Cambridge Tracts) (Cambridge: Cambridge University Press) [3]Kochendorffer R 1981 Introduction to Algebra (Groningen: Wolters- Noordhoff) [4]Jauch J M and Rohrlich F 1959 The Theory of Photons and Electrons (New York: Addison-Wesley) [5]Cartan E 1966 The Theory of Spinors (Cambridge, MA: MIT Press) [6]Chevalley C 1954 The Algebraic Theory of Spinors (New York: Columbia University Press) [7]Budinich P and Dabrowski L 1985 Math. Phys. 10 L7 Budinich P and Trautman A 1986 Lett. Math. Phys. 11 315 [8]van Nieuwenhuizen P 1983 An introduction to simple supergravity and the Kaluza —Klein program, in Relativity and Topology II (Les Houches) 1983 (Amsterdam: North-Holland) pp 825-932 [9]Penrose R and Rindler W 1986 Spinors and Space -time vol 2 (Cambridge: Cambridge University Press) [10]Adams J 1981 in Superspace and Supergravity ed S W Hawking and M Rocek (Cambridge: Cambridge University Press) [11]Komar A 1959 Phys. Rev. 113 934 [12]Hawking S and Ellis G 1973 The Large Scale Structure of Space —Time (Cambridge: Cambridge University Press) [13]Misner C, Thorne K and Wheeler A 1973 Gravitation (San Francisco: W H Freeman) [14]Dereli T and Tucker R W 1982 Phys. Lett. 110B 206 [15]Brans C and Dicke R H 1961 Phys. Rev. 124 925 Dicke R H 1962 Phys. Rev. 125 2163 [16]Darwin C G 1928 Proc. R. Soc. 118 654 [171 Ivenko D and Obukhov Y 1985 Ann. Phys., Lpz 42 59 [18]Kahler E 1962 Rend. Mat. 21 425 [19]Dirac P A M 1928 Proc. R. Soc. 117 610, 118 341 [20]Benn I M and Tucker R W 1983 Fermions without spinors Commun. Math. Phys. 89 341 [21]Duffin R J 1938 Phys. Rev. 54 1114 Kemmer N 1939 Proc. R. Soc. A 173 91 [22]Milnor J W 1963 Enseignement Math. 9 198 [23]Kobayashi S and Nomizu K 1963 Principles of Differential Geometry (New York: Interscience) [24]Benn I M and Tucker R W 1986 in Geometry and Spinors, Trieste Conf. 1986, Representing Spinors with Differential Forms [251 Rarita W and Schwinger J 1941 Phys. Rev. 60 61 [26]Lichnerowicz A 1964 Bull. Soc. Math. France 92 11 [27]Hughston L P. Penrose R, Sommers P and Walker M 1972 Commun. Math. Phys. 27 303 -8 [28]Duff M J. Nilsson B and Pope C N 1986 Phys. Rep. 130 1-142 Nilsson B 1986 Class. Quantum Gray. 3 141-5 REFERENCES 349 References 20: l_1:Albert A1961Structure ofAlgebras (Am. Math. Soc. Coll. Publ. vol24 1'2:Dickson L1960 Linear Algebras (Cambridge Tracts) (Cambridge: Cambridge University Press) 3:Kochendorffer R1981 Introduction toAlgebra (Groningen: Wolters- Noordhoff) 4:Jauch JMandRohrlich F1959 TheTheory ofPhotons andElectrons (New York: Addison-Wesley) 5:Cartan E1966 TheTheory ofSpinors (Cambridge, MA: MIT Press) 6:Chevalley C1954 TheAlgebraic Theory ofSpinors (New York: Columbia University Press) 7:Budinich PandDabrowski L1985 Math. Phys. 10L7 Budinich PandTrautman A1986 Lett. Math. Phys. ll315 8:vanNieuwenhuizen P1983 Anintroduction tosimple supergravity andthe Kaluza—l(lein program, inRelativity and Topology II(Les Houches) I983 (Amsterdam: North-Holland) pp825-932 9:Penrose RandRindler W1986 Spinors andSpace-time vol2(Cambridge: Cambridge University Press) 10:Adams J1981 inSuperspace and Supergravity edSWHawking and M Rocek (Cambridge: Cambridge University Press) 11:Komar A1959 Phys. Rev. I13934 12:Hawking Sand Ellis G1973 The Large Scale Structure ofSpace—Time (Cambridge: Cambridge University Press) 13:Misner C,Thorne KandWheeler A1973 Gravitation (San Francisco; WH Freeman) 14:Dereli TandTucker RW1982 Phys. Lett. ll0B 206 15:Brans CandDicke RH1961 Phys. Rev. 124925 Dicke RH1962 Phys. Rev. 1252163 _Darwin CG1928 Proc. R.Soc. ll8654 :Ivenko DandObukhov Y1985 Ann. Phys., Lpz 4259 Kahler E1962 Rend. Mat. 21425 Dirac PAM1928 Proc. R.Soc. ll7610, ll8341 Benn IMandTucker RW1983 Fermions without spinors Commun. Math. Phys. 89341 21:Duffin RJ1938 Phys. Rev. 541114 Kemmer N1939 Proc. R.Soc. A17391 Milnorl W1963 Enseignement Math. 9198 23:Kobayashi SandNomizu K1963 Principles ofDifferential Geometry (New York: Interscience) 241Benn IMandTucker RW1986 inGeometry andSpinors, Trieste Conf. 1986, Representing Spinors with Differential Forms 25:Rarita WandSchwinger J1941 Phys. Rev. 6061 26:Lichnerowicz A1964 Bull. Soc. Math. France 9211 27:Hughston LP.Penrose R.Sommers Pand Walker M1972 Commun. Math. Phys. 27303—8 28]Duff MJ.Nilsson BandPope CN1986 Phys. Rep. 130l—142 Nilsson B1986 Class. Quantum Grav. 3l41—5%l%l%lP—‘<°.°°.\|°\. 221 350 REFERENCES [29]Cahen M, Gott A, Lemaire L and Spindel P 1986 Killing spinors, in Geometry and Physics, Trieste Conf. 1986 [30]Lichnerowicz A 1986 Killing spinors according to 0 Hijazi, and applica- tions. in Geometry and Physics, Trieste Conf. 1986 [31]Hitchin N 1974 Adv. Math. 14 1-55 1321 Yau T 1978 Commun. Pure App!. Math. 31 339 -411 [33] Shanahan P The Atiyah-Singer Index Theorem. An Introduction (Springer Lecture Notes in Mathematics vol 638) 350 REFERENcEs [29] Cahen M,Gott A,Lemaire Land Spindel P1986 Killing spinors. in Geometry andPhysics, Trieste Conf. I986 [30] Lichnerowicz A1986 Killing spinors according toOHijazi. andapplica- tions. inGeometry andPhysics, Trieste Conf. 1986 [31] Hitchin N1974 Adv. Math. 141-55 [32] YauT1978 Commun. Pure Appl. Math. 31339-411 [33] Shanahan PTheAtiyah—Singer Index Theorem. AnIntroduction (Springer Lecture Notes inMathematics vol638) Index Abelian, 308 Acceleration, 203 Adjoint involutions, 67, 71 Algebra, 307, 316 Almost complex structure, 303 Alt, alternating map, 5 Angular momentum, 197 Anti-automorphism (algebra), 319 Anticommuting spinors, 103 Antisymmetric, 4 tensor gauge fields, 260 Atiyah—Singer index, 306 Atlas, 130 Automorphism, 3 group, 119, 308, 313 Autoparallel, 202 Basis (vector space), 311 Bianchi's first identity, 213 Bianchi's second identity, 213 Bijective, 309, 312 Bilinear covariants, 93 Bilinear form. 314 Bispinor, 100 Boost, 186 orbit, 187 Boundary, 125, 168 Brans—Dicke theory, 250 Calabi—Yau, 306 Central algebra, 316 Centre (ring), 310 Centre, 308 Chain rule, 135 Characteristic field, 310 zero, 310 Charge conjugate spinor, 96 conjugation (Dirac spinor), 287 conjugation (of spinor fields), 266 electric, 190 Charged scalar field, 241 Chart transformations, 131 Chiral spinor, 97 Ck map, 129 Christoffel symbols, 222 Clifford 2-forms, 252, 253 algebra, 23 algebra, (complexified), 60, 80 commutator, 50, 107 group, 42 group (Lie algebra of), 51 product (relation to exterior product), 24 sub-bundles, 276, 306 subgroups, 46, 71 Clock, 183 Closed forms, 188 Closed sets, 125 Co-derivative, 189 Coherence (on overlaps), 263 Co-homologous, 188 Commutative, 308 ring, 310 Index Abelian, 308 Acceleration, 203 Adjoint involutions, 67,71 Algebra, 307, 316 Almost complex structure, 303 Alt,alternating map, 5 Angular momentum, 197 Anti-automorphism (algebra), 319 Anticommuting spinors, 103 Antisymmetric, 4 tensor gauge fields, 260 Atiyah—Singer index, 306 Atlas, 130 Automorphism, 3 group, 119,308,313 Autoparallel, 202 Basis (vector space), 311 Bianchi’s firstidentity, 213 Bianchi’s second identity, 213 Bijective, 309, 312 Bilinear covariants, 93 Bilinear form. 314 Bispinor, 100 Boost, 186 orbit, 187 Boundary, 125,168 Brans—Dicke theory, 250 Calabi—Yau, 306 Central algebra, 316 Centre (ring), 310 Centre, 308Chain rule, 135 Characteristic field, 310 zero, 310 Charge conjugate spinor, 96 conjugation (Dirac spinor), 287 conjugation (ofspinor fields), 266 electric, 190 Charged scalar field, 241 Chart transformations, 131 Chiral spinor, 97 C"map, 129 Christoffel symbols, 222 Clifford 2-forms, 252, 253 algebra, 23 algebra, (complexified), 60,80 commutator, S0,107 group, 42 group (Lie algebra of),S1 product (relation toexterior product), 24 sub-bundles, 276, 306 subgroups, 46,71 Clock, 183 Closed forms, 188 Closed sets, 125 Co-derivative, 189 Coherence (onoverlaps), 263 Co-homologous, 188 Commutative, 308 ring, 310 352 INDEX Commutator of Lie and covariant derivative, 231 of Lie and spinor covariant derivative, 273 Complete vector field, 158 Complex conjugation, 41, 81, 95 structure, 116, 315 structure (on spinor space), 59 vector space, 315 Complexification, 315 Complexified Clifford algebra, 60, 80 Components (vector), 311 Conformal 2-forms, 226 group, 192 isometry, 191 Killing vector, 231 symmetry (of Maxwell's equations), 192 tensor, 226 Conformally flat, 227 related, 225 Conjugate linear map, 315 space, 44 Connection 1-forms, 200, 207 components, 200 Conservation laws, 237 Conserved currents (Dirac equation), 280 Constant curvature, 225 Continuous function, 125 map, 129 Contracted Bianchi identities, 219 Contraction map (on tensors), 17 Contragradient, 17 degree, 16 Contravariant, 141 degree, 16 Coordinate basis, 143 chart, 130 Coset, 308 Cotangent bundle, 147 Coulomb solution, 190, 193 Covariances of Dirac equation, 280 Covariant derivative, 200, 206 of spinor fields, 267 of tensor spinors, 296 of tensors, 199 Covariant degree, 16 Covariant differentiation (Clifford forms), 252 Covariant differential, 207 Covariant exterior derivative, 216 Cross product, 344 Curl, 345 Curvature, 199 constant, 225 forms, 209 operator, 209 operator (of spinor), 271, 279 operator as Clifford commutator, 253 scalar, 219 tensor, 208 Curve, 134 Decomposable, 3, 8 Degree, 2, 313 of tensor, 2, 16 Degree, (s) group of, 313 Derivation, 4, 127 Diffeomorphism, 129, 132 Differentiable manifold, 129 map, 129 structure, 131 Differential form, 146 Dimension, 311, 316 Dirac adjoint spinor, 92 equation, 278, 282 matrices, (see gamma matrix) operator, 278 spinors, 92, 104 stress tensor, 290 Direct product (group), 309 Direct sum, 3 algebra, 317 vector space, 312 352 INDEX Commutator ofLieandcovariant derivative, 231 ofLieandspinor covariant derivative, 273 Complete vector field, 158 Complex conjugation, 41,81,95 structure, 116,315 structure (onspinor space), 59 vector space, 315 Complexification, 315 Complexified Clifford algebra, 60,80 Components (vector), 311 Conformal 2-forms, 226 group, 192 isometry, 191 Killing vector, 231 symmetry (ofMaxwell’s equations), 192 tensor, 226 Conformally flat, 227 related, 225 Conjugate linear map, 315 space, 44 Connection 1-forms, 200, 207 components, 200 Conservation laws, 237 Conserved currents (Dirac equation), 280 Constant curvature, 225 Continuous function, 125 map, 129 Contracted Bianchi identities, 219 Contraction map (ontensors), 17 Contragradient, 17 degree, 16 Contravariant, 141 degree, 16 Coordinate basis, 143 chart, 130 Coset, 308Cotangent bundle, 147 Coulomb solution, 190,193 Covariances ofDirac equation, 280 Covariant derivative, 200, 206 ofspinor fields, 267 oftensor spinors, 296 oftensors, 199 Covariant degree, 16 Covariant differentiation (Clifford forms), 252 Covariant differential, 207 Covariant exterior derivative, 216 Cross product, 344 Curl, 345 Curvature, 199 constant, 225 forms, 209 operator, 209 operator (ofspinor), 271, 279 operator asClifford commutator, 253 scalar, 219 tensor, 208 Curve, 134 Decomposable, 3,8 Degree, 2,313 oftensor, 2,16 Degree, (s)group of,313 Derivation, 4,127 Diffeomorphism, 129, 132 Differentiable manifold, 129 map, 129 structure, 131 Differential form, 146 Dimension, 311, 316 Dirac adjoint spinor, 92 equation, 278, 282 matrices, (seegamma matrix) operator, 278 spinors, 92,104 stress tensor, 290 Direct product (group), 309 Direct sum, 3 algebra, 317 vector space, 312 INDEX 353 Directional derivative, 138, 344 Divergence, 221, 346 of Maxwell stress tensor, 256 Division algebra, 316 ring, 310 Dominant energy condition, 237 Dual space, 313 Duality rotation, 116 Duffin—Kemmer—Petiau equations, 260 Eddington—Finkelstein coordinates, 248 Einstein (n — 1)-forms, 220 field equations, 234, 236, 247 —Maxwell system, 240 space, 222 summation convention, 311 tensor, 220 —Yang—Mills system, 240 —Kahler stress tensor, 259 Electric charge, 190 Electrically charged fluids, 242 Electromagnetic radiation, 185 Electron, 181 Endomorphism, 313 Energy, 186 Energy conditions on the stress tensor, 236 Equivalent involutions, 337 representation, 316, 321 Eta (ij) on Clifford algebra, 23 on exterior algebra, 7 Euclidean manifolds, 174 vector space, 123 Even subalgebra, 39, 80 Exact, 188 Exponential map, 203 Exterior algebra (as quotient of tensor algebra), 5 derivative, 154 p-form, 5 product, 5 External direct sum, 315 bundle, 151 f-related vector fields, 143 Faces, 167 Faithful representation, 316, 321 Falling freely, 177 Fermi—Walker or F-connection, 234 Fibre, 145 Field, 310 algebraically closed, 310 characteristic of, 310 Fierz rearrangement, 98, 285 First structure equation, 208 'Flag' (null flag), 116 Flux, 195 Frame, 311 Galilean group, 177 -relativistic, 176 Gamma (y) matrix, 37, 86 Gauge invariance of electromagnetism, 188 General linear group, 311 Generalised spinor structure, 263 Generators, 308 algebra, 317 of a subgroup, 308 of a vector subspace, 312 Geodesics, 203 Germ, 137 Graded algebra, 316 subspace, 313 vector space, 313 Gradient, 344 Gravitation with torsion, 249 Gravitational mass, 206 Gravitational waves (with neutrinos), 293 Group, 307 representation, 316 Gyroscopes, 234 H-module, 60 Harmonic, 190 Hausdorff, 126 Directional derivative, 138,344 Divergence, 221, 346 ofMaxwell stress tensor, 256 Division algebra, 316 ring, 310 Dominant energy condition, 237 Dual space, 313 Duality rotation, 116 Duffin—Kemmer—Petiau equations, 260 Eddington—Finkelstein coordinates 248 Einstein (n—1)-forms, 220 field equations, 234, 236, 247 —Maxwell system, 240 space, 222 summation convention, 311 tensor, 220 -Yang—Mills system, 240 —Kahler stress tensor, 259 Electric charge, 190 Electrically charged fluids, 242 Electromagnetic radiation, 185 Electron, 181 Endomorphism, 313 Energy, 186 Energy conditions onthestress tensor, 236 Equivalent involutions, 337 representation, 316, 321 Eta(17) onClifford algebra, 23 onexterior algebra, 7 Euclidean manifolds, 174 vector space, 123 Even subalgebra, 39,80 Exact, 188 Exponential map, 203 Exterior algebra (asquotient oftensor algebra), 5 derivative, 154 p-form, 5 product, 5-INDEX 353 External direct sum, 315 bundle, 151 f-related vector fields, 143 Faces, 167 Faithful representation, 316, 321 Falling freely, 177 Fermi—Walker orF-connection, 234 Fibre, 145 Field, 310 algebraically closed, 310 characteristic of,310 Fierz rearrangement, 98,285 First structure equation, 208 ‘Flag’ (null flag), 116 Flux, 195 Frame, 311 Galilean group, 177 -relativistic, 176 Gamma (y)matrix, 37,86 Gauge invariance of electromagnetism, 188 General linear group, 311 Generalised spinor structure, 263 Generators, 308 algebra, 317 ofasubgroup, 308 ofavector subspace, 312 Geodesics, 203 Germ, 137 Graded algebra, 316 subspace, 313 vector space, 313 Gradient, 344 Gravitation withtorsion, 249 Gravitational mass, 206 Gravitational waves (with neutrinos), 293 Group, 307 representation, 316 Gyroscopes, 234 H-module, 60 Harmonic, 190 Hausdorff, 126 354 INDEX Hermitian, 41, 63, 84, 87, 90, 269, 300 conjugate, 92 Hodge de Rham operator, 254 Hodge map, 13, 15, 173, 180 and Clifford products, 28 Homeomorphism, 126 Homogeneous elements of a graded vector space, 313 linear map, 313 Homogenous, 2 Homologous, 191 Homomorphism algebra, 318 group, 308 Horizon, 248 Ideal, 10, 23, 317 fluid, 242 observer, 183 Ideal of an algebra, 317 Ideal, single sided, 323 Idempotent, 324 Identity, 307 ring, 310 Imbedded (submanifold), 133 Imbedding, 133 Immersion, 133 Index of inner product, 66, 76, 85 of nilpotent element, 323 Inequivalent involutions, 68 Inertial chart, 184 mass, 347 reference systems Infeld, 99 Injective, 309, 312 tangent map, 133 Inner, outer, 309 Inner automorphism algebra, 318 group, 309 Inner products (on spinor fields), 264 Instantaneous, 185 Integral curve, 157 Integration, 167 Interior derivative, 4 on Clifford algebra, 23 on exterior forms, 9 Interior multiplication, 11 Intrinsic spin, 290 Invariance group, 314 Invariant subgroup, 308 Invertible element (ring), 310 Involutions, 4, 336 Involutary anti-automorphism (see also 4 Involution classification of involutions in the real Clifford algebras, 78 equivalence of, 337 inequivalent involutions of real algebras, 68 on tensor product of algebras, 72 Irreducible representation, 316, 321 Isometry, 173 Isomorphism algebra, 318 group, 309 Isotropic coordinates, 247 subspace, 106 Jacobi identity, 142 Jacobian, 128 Kahler 2-form, 305 equation, 256 manifold, 304 Kernel, 309, 312, 318 Killing currents, 196 spinor, 300 vector, 174 Killing's equation, 229 Klein—Gordon field, 239 Komar form, 239 Laplace—Beltrami operator, 189, 254 Laplacian operator on spinors, 279 Left and right duals, 229 Left coset, 308 Left ideal, 323 354 INDEX Hermitian, 41,63,84,87,90,269, 300 conjugate, 92 Hodge deRham operator, 254 Hodge map, 13,15,173, 180 andClifford products, 28 Homeomorphism, 126 Homogeneous elements ofagraded vector space, 313 linear map, 313 Homogenous, 2 Homologous, 191 Homomorphism algebra, 318 group, 308 Horizon, 248 Ideal, 10,23,317 fluid, 242 observer, 183 Ideal ofanalgebra, 317 Ideal, single sided, 323 Idempotent, 324 Identity, 307 ring, 310 Imbedded (submanifold), 133 Imbedding, 133 Immersion, 133 Index ofinner product, 66,76,85 ofnilpotent element, 323 Inequivalent involutions, 68 Inertial chart, 184 mass, 347 reference systems Infeld, 99 Injective, 309, 312 tangent map, 133 Inner, outer, 309 Inner automorphism algebra, 318 group, 309 Inner products (onspinor fields), 264 Instantaneous, 185 Integral curve, 157 Integration, 167Interior derivative, 4 onClifford algebra, 23 onexterior forms, 9 Interior multiplication, 11 Intrinsic spin, 290 Invariance group, 314 Invariant subgroup, 308 Invertible element (ring), 310 Involutions, 4,336 Involutary anti-automorphism (see also5),4 Involution classification ofinvolutions inthe realClifford algebras, 78 equivalence of,337 inequivalent involutions ofreal algebras, 68 ontensor product ofalgebras, 72 Irreducible representation, 316, 321 Isometry, 173 Isomorphism algebra, 318 group, 309 Isotropic coordinates, 247 subspace, 106 Jacobi identity, 142 Jacobian, 128 Kahler 2-form, 305 equation, 256 manifold, 304 Kernel, 309, 312, 318 Killing currents, 196 spinor, 300 vector, 174 Killing’s equation, 229 Klein—Gordon field, 239 Komar form, 239 Laplace—Beltrami operator, 189, 254 Laplacian operator onspinors, 279 Left andright duals, 229 Left coset, 308 Leftideal, 323 INDEX 355 Left R-module, 341-2 Length (of a curve), 183 Levi—Civita antisymmetric symbol, 15 Lichnerowicz theorem, 299 Lie algebra of Clifford group, 51 Lie-algebra-valued p-forms, 240 Lie bracket Lie derivative on spinors, 271 on tensors, 161 Light-cone, 181 Linear connection, 200 dependence, 311 frame, 311 map, 312 quotient space, 312 space of linear maps, 313 transformation, 313 Local frame, 172 Locally symmetric space, 303 Lorentz force law, 243 Lorentzian Clifford algebra, 85, 113 connection, 232 manifold, 172 Lorenz gauge, 190 Lowering convention, 314 Majorana conjugate spinor, 95 Majorana spinor, 96, 104, 115 Majorana—Weyl spinor, 97, 104 Mass—energy, 185 Maximal integral curve, 158 isotropic subspace, 107 Maxwell stress (Clifford form), 255 Maxwell stress tensor, 194, 197 Maxwell's equations, 178, 181, 188 Clifford form, 255 Metric, 314 compatible, 214 compatible connection forms, 215 dual, 14, 314 on p-forms, 14, 27 tensor field, 171 topology, 126 Minimal left ideal, 55 Minkowski spacetime, 181-2 Mixed tensor, 16 Module, 340 Momentum, 186 Multi-index, 9, 27 Multilinear, 2, 16 Multipole, 191 n-form, 10 Natural basis, 143 dual basis, 313 local basis, Neighbourhood, 124 Neutrino waves (with gravity), 293 Newtonian acceleration, 205, 206 angle, 186 gravitational coupling, 247 length, 186 potential, 206 velocity, 185 Nilpotent, 323 Norm homomorphism, on Clifford group, 46 Non-associative algebra, 119 Non-degenerate metric, 314 Non-nilpotent algebra, 324 Non-rotating frame, 234 Normal coordinates, 203 neighbourhood, 203 subgroup, 308 Odd dimensions, 89, 92 of a group, 309 of a linear space, 313 of an algebra, 318 One-parameter diffeomorphism, 156 Open set, 125 Opposite algebra, 4, 319 Or a ring, 310 Orbital angular momentum, 290 Order, 2, 307 Ordinary matrix algebra, 320 Orientation, 14, 132 Left R-module, 341-2 Length (ofacurve), 183 Levi—Civita antisymmetric symbol, 15 Lichnerowicz theorem, 299 Liealgebra ofClifford group, 51 Lie-algebra-valued p-forms, 240 Liebracket Liederivative onspinors, 271 ontensors, 161 Light-cone, 181 Linear connection, 200 dependence, 311 frame, 311 map, 312 quotient space, 312 space oflinear maps, 313 transformation, 313 Local frame, 172 Locally symmetric space, 303 Lorentz force law,243 Lorentzian Clifford algebra, 85,113 connection, 232 manifold, 172 Lorenz gauge, 190 Lowering convention, 314 Majorana conjugate spinor, 95 Majorana spinor, 96,104,115 Majorana—Weyl spinor, 97,104 Mass—energy, 185 Maximal integral curve, 158 isotropic subspace, 107 Maxwell stress (Clifford form), 255 Maxwell stress tensor, 194, 197 Maxwell’s equations, 178, 181, 188 Clifford form, 255 Metric, 314 compatible, 214 compatible connection forms, 215 dual, 14,314 onp-forms, 14,27 tensor field, 171 topology, 126INDEX 355 Minimal leftideal, 55 Minkowski spacetime, 181-2 Mixed tensor, 16 Module, 340 Momentum, 186 Multi-index, 9,27 Multilinear, 2,16 Multipole, 191 n-form, 10 Natural basis, 143 dual basis, 313 local basis, Neighbourhood, 124 Neutrino waves (with gravity), 293 Newtonian acceleration, 205, 206 angle, 186 gravitational coupling, 247 length, 186 potential, 206 velocity, 185 Nilpotent, 323 Norm homomorphism, onClifford group, 46 Non-associative algebra, 119 Non-degenerate metric, 314 Non-nilpotent algebra, 324 Non-rotating frame, 234 Normal coordinates, 203 neighbourhood, 203 subgroup, 308 Odd dimensions, 89,92 ofagroup, 309 ofalinear space, 313 ofanalgebra, 318 One-parameter diffeomorphism, 156 Open set,125 Opposite algebra, 4,319 Oraring, 310 Orbital angular momentum, 290 Order, 2,307 Ordinary matrix algebra, 320 Orientation, 14,132 356 INDEX Oriented r-chain, 168 r-cube, 167 Orthochronous transformations, 47 Orthogonal group, 42, 314 idempotent, 325 Orthonormal basis, 314 Outer automorphism (group), 310 p-form, 5 Parallel, 201 along a curve, 201 spinor, 303 transport map, 202 vector field, 202 Parallelism, 199 Parametrise, 171 curve, 134 Parity-preserving orthogonal transformations, 47, 49 Period (of an automorphism), 119 Photons, 185 Physical dimensions, 178 Pierce decomposition, 325 Pin groups, 46 example of Pin(3, 1), 53 Pinor structure, 263 Plane-wave basis (for Dirac equation), 288 Poincaré, 178 group, 182 Polarities, 179 Potential, 188 Primitive idempotent, 325 Principal idempotent, 325 Proca field, 240 Product manifold, 145 Projection rNperator, 26 Proper time, 183 parametrisation, 183 Pseudo-Riemannian, 172 connection, 221 Pullback, 133 map, on functions, 133 on forms, 148 Pure spinors, 106, 108 Quantum theory, 282 Quotient algebra, 10, 25 Quaternion conjugation, 65, 73, 338 reversion, 339 Quaternions, 338 Quotient algebra, 318 group, 308 R-module, 340 Racah time reversal, 94 Radical, 324 Raising and lowering conventions, 19 Rank, 2, 312 of an idempotent, 334 of tangent map, 133 Rank-two spinor, 103 Rarita—Schwinger equations, 296 Reducible algebra, 317 representation, 119, 316, 321 Reflections, 43 Regular element (ring), 310 Regular representation (algebra), 321 Reissner—NordstrOm solution, 243 Representation equivalent, reducible, faithful, 316 of an algebra, 321 of a group, 316 Representative, 308 spinor, 108 Representing spinors, 275 Reversion (quaternions), 338 Ricci 1-forms, 210 tensor, 210 Riemannian, 172 Ring, 310 Rotational isometry, 174 Scalar field, 239 Schwarzschild metric, 247 Second structure equation, 209 Section, 146 of a tangent bundle, 146 Sectional curvature, 223 356 Oriented r-chain, 168 r-cube, 167 Orthochronous transformations, 47 Orthogonal group, 42,314 idempotent, 325 Orthonormal basis, 314 Outer automorphism (group), 310 p-form, 5 Parallel, 201 along acurve, 201 spinor, 303 transport map, 202 vector field. 202 Parallelism, 199 Parametrise, 171 curve, 134 Parity-preserving orthogonal transformations, 47,49 Period (ofanautomorphism), 119 Photons, 185 Physical dimensions, 178 Pierce decomposition, 325 Pingroups, 46 example ofPin(3, 1),53 Pinor structure, 263 Plane-wave basis (forDirac equation), 288 Poincare, 178 group, 182 Polarities, 179 Potential, 188 Primitive idempotent, 325 Principal idempotent, 325 Proca field, 240 Product manifold. 145 Projection operator, 26 Proper time, 183 parametrisation, 183 Pseudo-Riemannian, 172 connection, 221 Pullback, 133 map, onfunctions, 133 onforms, 148 Pure spinors, 106. 108INDEX Quantum theory, 282 Quotient algebra, 10,25 Quaternion conjugation, 65,73,338 reversion, 339 Quaternions, 338 Quotient algebra, 318 group, 308 R-module, 340 Racah time reversal, 94 Radical, 324 Raising andlowering conventions, 19 Rank, 2,312 ofanidempotent, 334 oftangent map, 133 Rank-two spinor, 103 Rarita—Schwinger equations, 296 Reducible algebra, 317 representation, 119,316, 321 Reflections, 43 Regular element (ring), 310 Regular representation (algebra), 321 Reissner-Nordstrom solution, 243 Representation equivalent, reducible, faithful, 316 ofanalgebra, 321 ofagroup, 316 Representative, 308 spinor, 108 Representing spinors, 275 Reversion (quaternions), 338 Ricci 1-forms, 210 tensor, 210 Riemannian, 172 Ring, 310 Rotational isometry, 174 Scalar field, 239 Schwarzschild metric, 247 Second structure equation, 209 Section, 146 ofatangent bundle, 146 Sectional curvature, 223 INDEX 357 Semi-direct product, 51 group, 310 Semi-orientation, 48 Semi-simple (algebra), 326 Semi-spinor representation, 55 Semi-spinors, 97 Signature, 314 Simple (algebra), 327 Smooth manifold, 131 Spacetime, 181 Span, 311 Spatial direction, 185 Special orthogonal group, 45 Spherical harmonics, 258 Spin(' manifold, 306 structure, 264 Spin groups, 46 example of spin(3, 1), 53 Spin-invariant products, 62 Spin manifold, 262 Spinor bundle, 261 covariant exterior derivative, 297 field, 262 frame, 263, 293 Laplacian, 279 representation, 55 structure, 262 Spinors, 54 Standard spinor frames, 263 Star map (see Hodge map) Static metric, 244 Stationary, 184 metric, 244 observer, 184 Stokes's theorem, 169 Stress energy tensor, 236 Stress tensor Dirac, 290 fluids, 242 Kahler, 259 Klein—Gordon, 239 Maxwell, 194 Proca, 240 Yang—Mills, 240 Strong energy condition, 237 Structure constants, 174 equations, first, 208 equations, second, 209 functions, 215, 280 Subalgebra, 316 Subgroup, 308 Submanifold, 133 Sum (vector space), 312 Summation convention, 311 Supergravity, 249, 296 Supersymmetry, 283, 301 Surjective, 309, 312 Symmetric metric, 314 Symmetrisation, 4 (see projection operators) Tangent, 136 bundle, 143 map, 138 plane, 223 space, 127, 137 vector, 136, 142 Tensor, 2 algebra, 2 algebra (mixed), 16 field, 150 product (of algebras), 319 spinors, 294 the group of all, 309 Time reversal (on spinors), 49 Topological manifold, 124, 127 space, 124 subspace, 125 Topology, 125 Torque, 197 Torsion 2-forms, 208 Torsion tensor, 208 Total matrix algebra, 320 Trace in Clifford algebra, 91 of a tensor, 18 theorems, 91 Translational isometry, 174 Translations, 182 Triality, 106, 117, 120 Semi-direct product, 51 group, 310 Semi-orientation, 48 Semi-simple (algebra), 326 Semi-spinor representation, 55 Semi-spinors, 97 Signature, 314 Simple (algebra), 327 Smooth manifold, 131 Spacetime, 181 Span, 311 Spatial direction, 185 Special orthogonal group, 45 Spherical harmonics, 258 Spinc manifold, 306 structure, 264 Spin groups, 46 example ofspin(3, 1),53 Spin-invariant products, 62 Spin manifold, 262 Spinor bundle, 261 covariant exterior derivative, 297 field, 262 frame, 263, 293 Laplacian, 279 representation, 55 structure, 262 Spinors, 54 Standard spinor frames, 263 Starmap (seeHodge map) Static metric, 244 Stationary, 184 metric, 244 observer, 184 Stokes’s theorem, 169 Stress energy tensor, 236 Stress tensor Dirac, 290 fluids, 242 Kahler, 259 Klein—Gordon, 239 Maxwell, 194 Proca, 240 Yang-Mills, 240 Strong energy condition, 237INDEX Structure constants, 174 equations, first, 208 equations, second, 209 functions, 215, 280 Subalgebra, 316 Subgroup, 308 Submanifold, 133 Sum (vector space), 312 Summation convention, 311 Supergravity, 249, 296 Supersymmetry, 283, 301 Surjective, 309, 312 Symmetric metric, 314 Symmetrisation, 4 8’,(seeprojection operators) Tangent, 136 bundle, 143 map, 138 plane, 223 space, 127, 137 vector, 136, 142 Tensor, 2 algebra, 2 algebra (mixed), 16 field, 150 product (ofalgebras), 319 spinors, 294 thegroup ofall,309 Time reversal (onspinors), 49 Topological manifold, 124, 127 space, 124 subspace, 125 Topology, 125 Torque, 197 Torsion 2-forms, 208 Torsion tensor, 208 Total matrix algebra, 320 Trace inClifford algebra, 91 ofatensor, 18 theorems, 91 Translational isometry, 174 Translations, 182 Triality, 106, 117, 120 358 Weak energy condition, 237 Wedderburn (structure theorems), Twisted vector rei Twistor, 299 equation, 298 Two-component formalism, Weyl U(1) covariant derivative, 241 of spinor, 270 exterior covariant derivative, 241 Unit element (ring), 310 Units, 181 Valence, 103 van der Waerden formalism, 99 Vector analysis in Euclidean 3-space, 342 Vector field, 141 representation, 42 twisted, 45 space, 310 subspace, 312 Volumn form, 14 eguatiow 279 Weyl spinor, 97, 100, 108 tensor, 226 Wigner time reversal, 94, 288 Witt basis, 107 Witt index, 66 World line, 183 X () the involutory anti-automorphism, 4 the involution on exterior algebras, 8 the involution on Clifford algebras, 23 Yang—Mills field, 240 Z(mod 2), 2, 22, 47 358 :: Twisted vector reg:t Weak energy condition, 237 Twistor, 299 Wedderburn (structure theorems), equation, 298 _ Two-component formalisnz, WeY1 equa-t1O.rrl|§ 279 Weyl U(1) spinor, 97,100, 108 covariant derivative, 241 tensor, 226 ofspinor, 270 Wigner time reversal, 94,288 exterior covariant derivative, 241 Witt basis, 107 Unitelement (ring), 310 Wittindex, 66 Units, 181 World line, 183 Valence, 103 Xi(E) vanderWaerden formalism, 99 theinvolutory anti-automorphism, Vector analysis inEuclidean 3-space, 4 342 theinvolution onexterior algebras, Vector 8 field, 141 theinvolution onClifford algebras, representation, 42 23 twisted, 45 space, 310 Yang-Mills field, 240 subspace, 312 Volumn form, 14 Z(mod 2),2,22,47