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[Ian_M._Benn,_Robin_W._Tucker]_An_introduction_to_(BookFi.org) xch OCR

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A published textbook by I. M. Benn and R. W. Tucker, not Phil's own work, kept in a Wedge World folder. It covers tensor and exterior algebra, Clifford algebras, spinors, pure spinors and triality, differentiable manifolds, connections and curvature, electromagnetism and gravitation, and spinor field equations such as the Dirac equation. The text is OCR of the front matter and start of chapter 1.

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KnIntroduction toSpinors andGeometry with Applications inPhysics IMBenn Faculty ofScience, University College ofTheNorthern Territory, Australia RWTucker Department ofPhysics. University ofLancaster. UK Adam Hilger, Bristol andNew York ©IOP Publishing Ltd1987 Allrights reserved. Nopart ofthispublication may bereproduced, stored inaretrieval system ortransmitted inanyform orbyanymeans, electronic, mechanical, photocopying, recording orotherwise, without theprior permission ofthepublisher. ToShian British Library Cataloguing inPublication Data TO Danlel and Edmund Benn, I.M. Anintroduction tospinors andgeometry with applications inphysics. 1.Spinor analysis I.Title II.Tucker, R.W. 512’.57 QA433 ISBN O-85274-169-3 ISBN 0-85274-261-4 (pbk) Library ofCongress Cataloging-inPublication Data Benn, I.M.(IanM.) 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 ISBN O-85274-169-3 ISBN 0-85274-261-4 (pbk) Consultant Editor: Professor RFSti-eater King’s College, London First published 1987 Paperback edition 1989 Published under theAdam Hilger imprint byIOPPublishing Ltd Techno House, Redcliﬂe 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 Contents Preface 1Tensor Algebra 1.1 Thetensor algebra 1.2 Theexterior algebra ofantisymmetric tensors 1.3 Theexterior algebra asaquotient ofthetensor algebra 1.4 TheHodge map 1.5 Themixed tensor algebra Bibliography 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 IFl"‘—>IR”ix >—>CD-I‘:-l\)i--* 13 16 20 21 23 28 39 42 54 62 80 85 105 106 106 117 122 123 124 1274.3 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 Maxwe1l’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—Nordstr6m solution Gravitation with torsion Bibliographyvii 129 134 1.36 141 143 146 150 154 156 161 167 171 174 176 176 178 183 188 197 199 200 204 206 208 212 214 216 219 221 223 225 227 229 231 232 232 234 234 237 239 243 249 250 viii CONTENTS 8Clifford Calculus onManifolds 8.1 Covariant differentiation ofClifford products 8.2 Theoperator 921 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 onIB3 References IndexPreface 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 1S-the introduction ofspaces with ‘exotic’ topologies and geo- metries informulating the basic laws ofNature. Consequently the student finds itnecessary topossess abroad knowledge ofmathematical techniques that encompasses such generalities aswell asthecomputa- tional skills necessary tousethisinformation. In‘this book wehave attempted toprovide aconcise but self- contained introduction tothebasic properties ofdifferential geometry éfindspinors accommodating some oftheneeds mentioned above. We ‘eelthatphysicists learn most rapidly byseeing new concepts spelled out insome C1€tElll: We have attempted ablend ofmathematics and theoretical physics which wehope willassist intheassimilation ofnew Elle-ias, andgive readers afeeling that they arecloser tothe‘nuts and botsoftheSLli)_|€C1I material. Inwriting anyintroduction toasubject as road asthiswehave hadtoface theproblem ofwhat prerequisites we 6Xpect our readers topossess. Fundamental toany appreciation of g'3I1S_Or methods isafirm familiarity with linear algebra. Thus ourbook ﬁgins with algebraic notions. Wehave tried toencapsulate theneces- Sa1'Y.concepts used inChapters 1and2into Appendix A.This should glovide areservoir ‘ofcompact information forthose who may findsome vecigrn Svocabulary inthese early chapters. Our emphasis here isonreal paces and their complexifications. Wefeel that thisapproach Svlgiilgs closest contact with what most physicists actually use‘when ducellgg with thecomplexified Clifford algebra ofspacetime. We_intro- spinor asanelement carrying anirreducible representation of r .3 x PREFACE some Clifford algebra. This emphasis ontheClifford algebras rather than thespin groups isslightly different from thatcommonly 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. The production ofour manuscript was greatly assisted with the aidofTEXnical 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 RWTuckerTensor Algebra This first chapter 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 3.COIlClS6 development ofrudimentary ideas isalsotobefound. The first section ofthischapter introduces thetensor algebra ofan arbitrary vector space. InChapter 2thiswillbethestarting point for ourconstruction oftheClifford algebra, which will bedefined asa quotient ofthetensor algebra. InChapter 4and subsequent chapters when beginning geometry wewill beinterested inthetangent space (and thecotangent space) ofamanifold. Wewillthen beable toapply thematerial ofthischapter immediately tothat vector space. Infactit \‘lV/Illbethecotangent space that istaken forthearbitrary vector space IAnticipating thiswehave (identifying thesecond dual space ofV with itself) written elements ofVasacting onV*,rather than theother Wayaround. Particularly important onmanifolds arethe totally antisymmetric Eensor fields; the.differential forms. In§1.2 weintroduce theexterior Orms onanarbitrary vector space. These will also play aprominent roleinour‘treatment oftheClifford algebra. Tofacilitate acomparison With theClifford algebra were-introduce theexterior algebra in§1.3 as 8quotient ofthetensor algebra. Only in§1.4 does 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 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 the scalar 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 ofdegreet rform ann’-dimensional vector space, T,(V). Wemay associate asetofrelements from Vwith atensor ofdegree r.Forx‘eV,i=1,...,randXyeV*,i=1,...,rwedefine (x1®x2® ...®x’)(X1,X2, ...,X,) =x1(X1)x2(X2) ...x’(X,). Inparticular, if{ei} isabasis forVthen thesetofalln’elements {e’1®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...(avE®*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 (a®b)(X1,X2,...,X,,X,+1,...,X,.,,) =a(X1...,X,)b(X,,+1,...,X,+,). Wemay take the(external) direct sum ofthevector spaces T,(V) for allrtoform aninfinite-dimensional vector space. The direct sum of such avector space with aone-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 Vand theidentity element; anyelement canbewritten asasum of tensor products ofelements from Vandtheidentity. Those tensors that aresimply aproduct ofvectors from Varecalled decomposable. By construction wehave thedirect sum vector space decomposition ‘JD :r(v)=Zoe)r,(v).PI Thetensor product issuch thatthetensor algebra isaZ-graded algebra; elements inT(V)that aresums 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 byi no=(—1)d‘*g“a. (1.1.2) This iscertainly anautomorphism since ifaandbarehomogeneous r7(a®b) =(—1)deg“®ba®b =(—1)d°g" “Ldegba®b (since thealgebra isgraded) =(—1)“‘ig”(—1)°""g"a(>1<)b andso i7(a®b) =T]62®T’]l). (1.1.3) Tosaythat 17isinvolutary means that 172=1,which indeed follows from (1.1.2). The homomorphism Z—>Z2 induces 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. Y“Thenotation a"isalsoemployed. 4 TENSOR ALGEBRA The tensor algebra isisomorphic toitsopposite algebrat andadmits aninvolutary anti-automorphism, orsimply aninvolution, Edefined on homogeneous elements by (x1®x2 ...®xP)¥ =xP®...®x2®x‘. (1.1.4) Itisstraightforward toseethat this really isananti-automorphism, namely (a®b)5 =b5®a5 such that$2=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 +l’]£Z®lXb. (1.1.5) IfxeVthen iXxEX(x), whilst forAinthesubspace spanned bythe identity iX2E0,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 17,with iX17E;—17iX, it follows that iXiy+iyix isaderivation onT(V). For xeVorthe subspace spanned bytheidentity (iXiy+1}/lX)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 try-k, which rearranges thesetofnumbers {1,2,...,p}such thattr,-k(i) =iifiaﬁj ork,try-k(j) =kand7T),-k(l() =j.Then adegree-p tensor Tissymmetric (antisymmetric) inthej,kentries if T(Xﬂjk(l),Xn-J_k(2), ...,X,Tj_k(p)) = +(—)T(X1,X2, ...,Xp).:1f- Atensor that issymmetric (antisymmetric) under allsuch inter- changes iscalled totally symmetric (totally antisymmetric). The totally antisymmetric tensors areparticularly important. Thesubspace oftotally 1'SeeAppendix A.--- THE EXTERIOR ALGEBRA orANTISYMMETRIC TENSORS 5 antisymmetric tensors inTp(V) isdenoted byAp(V), theelements of thisspace being called exterior p-forms, orsimply p-forms. The total antisynimetry ensures that ap-form isdetermined byitsevaluation on alldistinct combinations ofpvectors from abasis forV*.SoifVis n-dimensional and P denotes thenumber ofdistinct combinations ofpobjects chosen from n then l dimAp = Inparticular, theonly p-forms forp>n arezero, anddim/\,, =-'1.In analogy with thecase ofthetensor algebra itwill beconvenient to identify thefield Fwith aspace A0(V). Given anarbitrary element aeT,,(V), wedefine anew tensor siifg aeTp(V) by usea(X1,X2, ...,Xp) 1 Z;)_,25(U)a(Xa(i),Xa(2), ---,X09») VX," EV* (1-2-1) where the sum isover allpermutations o,5(0) being +1ifthis permutation iseven (i.e. aneven number ofpair interchanges rear- ranges theelements 1,2,...,pintotheorder o(1), 0(2), ...,o(p)) or —1ifthepermutation isodd (anodd number ofsuch interchanges). From thedefinition of52.45597 aweseethatitistotally antisymmetric and that sil.§£?T(s4§£?T a)=sz4§E9a. Hence siiﬁg isaprojection operator, sd§£9:Tp(V)—>Ap(V). Although ®:TS(V) ><T,(V)—> T_,+,(V), themap ®willnot map A,(V) XA,(V) into A,+,(V). Thus wedevise anew composition map interms of®and.9158? that does have thisproperty. Itiscalled theexterior pI'0dLtCl'l' andisdenoted byaAplaced between theelements ofA_,(V) andA,(V) A:/\_,(V) XA,(V) —-> /\,+,(V) a,b11> aAb=s§l§£’€T(a®b). (1.2.2) tThe 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 of$589”, asweshall doin thenext section. 6 TENSOR ALGEBRA Itfollows from thisdefinition that (61/\b)(Xla X2: '''Xsa X54-la -'-1X54-I) = 2£<@><a®b><X...... .....X.....>> V 1: 2£(S, [)a(Xj1, ..,XjS)b(Xkl, ....,Xk£) where thesum isover allpartitions of(1,2,...,s+t) into (j1,j2, ..., j,)and(kl, k2,...,k,),ande(s,t)isthesignofthepermutation (1, 2, ...3S+t)|W(_]'1,Jl2, ...,jS,k1,k2, ..., Theexterior product hasawell defined symmetry under theinterchange offactors such asaand babove. Tosee this weintroduce a permutation v, V (1,2,...,s,s+1, ...,s+t)+-—->(t+i, ...,t+s,1,2,...,r). Wewrite anypermutation oas0=rv,giving 8(0) =e(v)e(r). Inserting thisintheabove gives (Cl/\b)(X1, X2, ...,XS, XS_|_1, ...,X5-+t) 1 =my 2<‘I(<I)¢l(Xa(i>, ---,XU(S))b(XO(S+l)i ---,Xo(s+t)) e(v) = 21:5(T)a(X-r(.»+i)> ---»Xr(t+s))b(Xr(l)a ---,X110) :8(v)(b /\a)(Xlv ''‘vXs+t)' Atrivial combinatorial calculation gives e(v)=(—1)~“, andsowehave foranys-form aandt-form b Cl/\b:(-“1)s1b AQ. Theexterior algebra A(V)isformed bythedirect vector space sum of allthespaces ofp-forms Mw=Z@Mw)p=0 with multiplication given bytheexterior product. The exterior product isdefined onnon-homogeneous elements byextending .s.4i.§€E’J' tobe distributive over addition, ensuring that theexterior product is.Unlike thetensor algebra thisalgebra isfinite dimensional: wehave dim/\(V) =pig =2". (1.2.4)3= 2'55- 5-5; E,-,,.. .,.. 1191 *5‘ "E>‘€'."' _»§.:11-,,. iti it vi ‘QM '3‘-. iTHE EXTERIOR ALGEBRA OFANTISYMMETRIC TENSORS 7 The exterior algebra is,infact, associative. This willbeseen tofollow from the observation that ifsrilifﬁm =0then .Qi.§€§(a®m) = sl§£§(m®a) =0,VaeT(V),aswillnow beestablished. Let Ckbethegroup ofallpermutations ofkobjects. Then the subgroup ofCH, that only permutes thefirst sobjects isobviously isomorphic toCs,andweshall identify itassuch. LetHbeasetthat contains oneandonly oneelement from each leftcoset ofCH, relative toCs. Sofor each oeCH, o=hr, reCL,and hEH, with e(o) =r-:(r)e(h), andthen at§E5.’J'(m®a)(X1, ...,X,,.,) ——i- Z/1)Z ®X X_(S+l)! heH£( rEC5£(T)(m a)( 0(1)’ 1'" °(S+1'))' Forsome fixed IiletX;,(,-) =Y,-,then X,_,(,-) =X,,,(,-) =Y,(,-), andthus 2 €(r)(m®a)(Xo(1): --~'-aXo(s+t)) reCS Z EC 8(T)(”l®a)(YT(l)9 ''''''7Yr(s+t)) :2 €(T)l?1(YT(1), ...,Y.,(_,))a(Ys+1, ...,I/s+t). IECS =si§£€m(Y1,..., Y,)a(Y,+1, ...,Y,..,,). Soindeed xiii/’91 (m®a) =0ifsd.§£9m ==0.Similarly, itfollows that sﬂ§£?T(a®m) =0. Since, aswehave remarked, slid isaprojection operator, ifweset (1—ﬂ§E?T)(a®b) =m then a®b =sd§£’§(a®b) +m, with 951529 m=0.From thedefinition oftheexterior product wehave (aAb)Ac=s§l§£9' (si.§£97 (a®b) ®c) =s.i.§-£97 (a®b®c —m®c) since ®isassociative =s.i§£9' (a®b®c) from theabove result, which may beused once more togive (aAb)Ac=@4589" (a®s4..§f3E‘T (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 siipil isahomogeneous mapping ofdegree zero onthetensor algebra itfollows that T]isalso an automorphism oftheexterior algebra, thatis ,_ 8 TENSOR ALGEBRA "(Q/\9)=ll“/\175- (1-2-5) Similarly exterior forms arecalled even orodd according totheir Z2-gradation inthetensor algebra. The involution Ecommutes with ﬂit’? andsoitisalso aninvolution ofA(V). Taking thedefinition of s§l....‘E£’1J' and rearranging thepermutations gives the following simple expression for3;’acting onap-form to, co’?=(—1)L"’21o.>. (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, ofone lower 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®...®xP then iX1T=x1(X1)x2® ...®xP —x2(X1)x1®x3 ...®xP +x3(X1)x1®x2®x4 ...®xP +...+(—1)P'1xP(X1)x1®... ®xP-1 thatis (iX,T)(X2, ...,X,,)=Ze(v)T(X.,(1), ...,x,(,,) (1.27) where visanyoftheppermutations such that (1,2,...,r,...,p)—-—->(2,3,...,r—1,1,r,...,p). Substituting .vi$‘ZJ' Tinto(1.2.7) gives (iX,.@i§2aT)(X,, ...,x,,)=p.u§£ar(x,, ...,x,,). (1.2.8) From thedefinition wehave (s.i.§E€TiX]T)(X2, ...,Xp) = 2e(r)(x2® ...®xP)(X,.(2), ...,X,(p)) x2(X1)—~ 2e(r)(x1®x3® ...®xP)(X,(2), ...,X,(p)) +...+(_1i:l1;;EX1) ;e(r)(x1® ...®xP—1)(XT(2)a-;-:-" .-\ .21 ‘5.1£ $3£1‘sf,THE EXTERIOR ALGEBRA OFANTISYMMETRIC TENSORS 9 =B-57? 2e(o)(x‘® ...®xP‘)(X,,(,), ...,XU(p)) foroeCp =-(—p§!1—)!- (u§£ar)(x,, ...,Xp) andthus (sQ§£E’TiX1T)(X2, ...,Xp) =(psﬂ§E§T)(X1,...,X,,). (1.2.9) So(2.8) and (2.9) give iXs.4.§£9'= .vﬂ§£3iX. Thus iX:A,,—>/\,,_1, and iX(a Ab)=iX.vi.§£‘J(a®b) =.vi.§EET(iXa®b +na®iXb) andhence iX(aAb) =iXaAb +r)aAiXb. (1.2.10) Ifwe/\,,(V) then ﬂ§E9Yo= toand s5l§£§iXco= 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 (>9,itgenerates A(V) with theproduct A.Thus any element ofA(V)canbewritten asasum ofdecomposable forms, these being theones consisting ofproducts ofelementsfrom V.If{el} isany basis forthen-dimensional Vthenthe(§)p-forms ellAeizA...Ae"Pfor £1<£2<...<ip (pE1)form abasis forA,,(V). Itisoften convenient tolabel such p-forms byanordered multi-index, I=(i1,i2,...,i,,)withi1<i2<...<i,, with each index i,-varying from 1ton.Soiftoisanarbitrary p-form w=2 wi1i2...i'p 31‘/\e12/\ ---A91”l1<i2<. ..<lp =Ewlel where co,Ew,-1,-2 ,-PeFarethecomponents oftointhisbasis. Care must beexercised when using thesummation convention (see Appendix A)with ordered multi-indices. Since this convention operates with unconstrained summations onemay equivalently write 1 . . . a):;;O)l-;l2...l'pel1/\el2/\ ‘"' Aelp itbeing understood thatthecomponents aretotally antisymmetric inthe indices. If{fl} isanew basis for Vrelated to{el} byf‘=M",-er‘, {M1,-} eGl(n, F)i“, then wecaninduce acorresponding change inthe TThe group ofnXninvertible matrices with elements from F,seeAppen-..,Xrw) where reCp_1 dixA_ I ea 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 fl/(f2/\.../\]m=d€tl14€1/\€2/\... A6" where detM isthedeterminant ofthematrix {Ml}-} that relates the bases. Any n-form Qcanbeused toclassify frames {X,-}forV*.These frames fallintotwoclasses according tothesignofQ(X1, X2,...,X,,). Frames indifferent classes aresaid tobeofopposite orientation. The Gl(n, F)related frames {el}and{f‘}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®and411559". 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 usebold-face type todenote thequotient algebra anditsproduct, theuseofthesame symbols anticipating its isomorphism with theexterior algebra ofantisymmetric tensors already defined. A LetIbetheideal inT(V) consisting. ofsums ofterms oftheform a®x®x®b where xeVanda,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. The equivalence class that contains aisdenoted [a].The vector space structure ofA(V)isdefined by [a]+/1[b] =[a+lb] a,beT(V), /1eF (1.3.2) andthemultiplication which isdenoted byAisgiven by [a]A[b] =[a®b]. (1.3.3).<- .- -;1: me. :R'- ;i:._. -E-.2 - ‘I ._‘ --*1<f;'=,.,.. .:ii"i;;.5E li I lEXTERIOR ALGEBRA ASAQUOTIENT OFTHE TENSOR ALGEBRA The ideal IisaZ-gradedi‘ subspace ofT(V) and soA(V) inherits a natural Z-gradation given bydeg[a]=dega.The automorphism 17and theinvolution §preserve theideal Iandthey thus extend inanobvious waytoA(V) by tile]=[val [elf=[ail- Similarly interior multiplication preserves Iandsowemay define ix[a]=[iXa]. (1.3.5)(1.34) Ifx,ye Vthen 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, ix]/\1)’1=ix®)’1=1x /\)’l- More generally, itfollows that theideal Iisjustthekernel ofs4$9', and so[a]=[.v.i.SB9'a]. We have already seen, inproving thatAis associative, thatthiskernel isanideal. ToseethatitisinfactIwewill prove that x®a) ~xAto forxeV,weA(V). (1.3.7) Therecursive application ofthisresult gives x1®x2®. ..®xP ~stliﬁg (x1®x2®. ..®xP) =x1Ax2A. ..AxP. Wewillprove (1.3.7) byinduction onthedegree ofco.Itiscertainly true when coisa1-form; weassume itistrue forwofdegree lessthan p.Itissufficient toconsider thecase ofcodecomposable. Thedefinition ofAinvolves thepermutation ofthearguments intheevaluation, but this isobviously equivalent topermuting thefactors intheproduct. Thus from thedefinition ofAwehave 19 =-_i__ @(l <'I() <I(P)yAx‘--~ Ar” (P+1)!;E(v)y °®y ‘®--~®y where opermutes 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.‘ﬂit;14L 1i 1 i 12 TENSOR ALGEBRA THE HODGE MAP 13 interchanges bring the 0tothe second position. Soifv,isthe 1 permutation such that V?‘ (0,1,...,r,...,p)i>(r,0,1,...,?,...,p) where Fdenotes that rismissing from this sequence, then e(v,) =(——1)". Wecannow write anypermutation oaso=r,.v, for some r,where r,permutes thesetwith rremoved, then y“/\i/1 ---/xyp 1=__1__ 28(20)y0®yr0(1)® ___®yr@(p) (p+1)! to 1P _1 rr _ r,(0)® _o.® r,(r—l)® r,(r+l) +(p+1)!Zll( )y®%8(Tr)y y y ...®y"(P) 1 _ —W (y°®(y‘/\ ---Ax") P +2(“'1)r)’r®(}’0/\ /\)’1/\---/\)’p))-r=1 Substituting xforyogives 1 ‘D Ax/\yl2...p : (x®y12...p +2 (_1)ryr®(xAyl... r...p)) r=1 where ylz-"PEy1Ay2A Ayl’, and again thehatmeans that a term ismissing. A A Now y’®(xAy1 ’ F’)~y’®x®y1 " Psince (1.3.7) is assumed truefor(p—1)-forms ~—x®y’®y""'/9"” sincex®y+y®x~0 ~—x®(y’ Ayl'--1‘~-'1’) from (1.3.7) again, ~(_1)rx®yl2...p where thesigncomes from moving y’through r—-1terms. SoxAy1A ... AyP ~x®y‘2 P.Thus if(1.3.7) holds foratof degree lessthan pitisalsotrue when coisap-form. This completes the proof. Thus every equivalence class ofA(V)isrepresented byanelement of /\(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...ta‘?-'.fr2;- e‘- .-~.'-="!:;§. E» 1 1 1 I l 11 -i1 ,.1.4TheHodge Map When thevector space Vhasa(non-degenerate) metric gthen the Hodge dual, or*map, may bedefined onexterior forms. Since 1”1—’”1 1P 1"-P wehave dim/\,,(V) =dimA,,_p(V), andthus these twovector spaces areisomorphic. We may use the metric 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, Cl)=€1/\€2/\ Ae”. Since thedeterminant ofthematrix relating orthonormal frames isplus orminus one, depending ontherelative orientations, weseefrom (1.2.12) that there aretwopossibilities forw,differing byasign. The members ofag-orthonormal frame forVaresometimes called rt-beins inthephysics literature, generalising thefamiliar triad oforthonormal vectors inEuclidean three space. Some authors, however, associate this term with ther2elements {Nil} eGl(n, F)that relate anorthonormal frame toanarbitrary one{f"}, e’=N‘;-fl. Ifthecomponents ofgintheframe {el} are171‘,where 171‘=0ifiEj andforeach value ofi,17”Ei1,andthecomponents intheframe {f} aregilm, then it=Nu.N1;gW>- Hence det(171) =det(g”m)(det N)2. The components ofthemetric onthedual space form theinverse matrices, 17,3,-171" =atandgglglkm =at.(For further details seeAppen- dixA.)Soift=det(r),-J.-) =i1then, since det(m‘1) =(detm)"1 forall matrices rn, det(g§P) =t(detN)2. ButtoE(detN)f1Af2 A...Af", soifwewrite thesignofdetN as _detN “N_|detN| 14 TENsoR ALGEBRA then (U=H~ild@1(8i1))}112f1Af2/\ ---/\f”- (1-4-2) Iftheframes {el} and {f}arerelated byaGl(n, F)transformation thatpreserves theorientation, then ,uNE1. Ametric onVnaturally gives risetoametric onA(V).Westart by defining ametric gponthespace ofp-forms, Ap(V), foranyp>1. Since gpisdefined tobebilinear itissufficient tospecify itsaction on decomposable p-forms. IfAEa1A a/2A...Aa/Pand BEB‘AB2A ...AB1’then gp(A, B)Edet{g(0t‘, 181)}. (1.4.3) Itisconvenient todefine g0tosimply multiply thetwo 0-forms. Having defined ametric onthehomogeneous subspaces wedefine a metric GonA(V) byrequiring ittobediagonal inthehomogeneous subspaces. That is,if(I),‘I1eA(V)with, forexample, (1),,denoting the projection of(I1intothesubspace ofdegree p,then G(,\IJ)=g,,(,,,1i1,,). (1.4.4)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: *:/\,,(V) ——> A,,_,,(V) a|i> *a where *aisgiven implicitly by bA*aEg,,(b, a)co Vbe/\,,(V). (1.4.5) The standard n-form toisdefined asin(1.4.1). The definition may be completed bydefining themap ona0-form, *1E(1).This iscalled the volume n-form. Linearity extends the definition 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*bEbA*a Va, be/\,,(V). (1.4.6) Auseful calculus canbesetuprelating the*map totheinterior product. Wemay usethemetric gtoestablish anisomorphism (denoted byatilde) between VandV*.IfxeVthen themetric dual, x,isinV*;7 7—-Th -HT “-1.-‘-1 TM.7 7 HFt~_¢ THE HODGE MAP 15 given by y(f)=sot»y) VyEV- Itthen follows from thedefinition of*that *(/ix) =i;;*<1> reV.eA(v). (1.41) This formula canbeapplied recursively toadecomposable p-form to produce *(x1A)C2A AXp):l}".vl'_;(“r»--1...1}'1*1. Itisconvenient todisplay theaction of*onexterior products ofbasis vectors. Suppose that {e‘} and {X,-} aredual bases, with e"(X,~) E6*}. Wewilloften usetheshorthand IX}. E1);. Themetric dual, 5“,ofe“isg“"X2, EX“andwewrite iX~Ei”. Equation (1.4.8) takes thefollowing simple form fortheproduct ofp basis vectors *(e1Ae2A ...AeP) EiF’iP'1 ...i1*1. From thisitcanbeseen that thedual ofaproduct ofporthonormal 1-forms istheproduct oftheir complement inthebasis. Duals ofthe orthonormal basis forms canbeexpressed interms oftheLevi—Civita antisymmetric e-symbol. This isdefined such that 8:].-'2...i,, = +1(-1) if(i2,i2,...,in)isaneven (odd) permutation ofthestandard sequence (1,2,3,...,n). (1.4.9) With thesummation convention thevolume n-form canbewritten in theorthonormal frame {e’}as 1 .. .*1EZ;e,-1,-,__ ,-He"Ae‘1A ...Ae’~. (1.4.10) Ifthecomponents ofthemetric inthisorthonormal frame are17”we have *(@1'A91’/\ ---A31”) =--l———e’1"1---"Pr -e’1P+1/\ .../\e’~lp+]...l,;, where i'i2...i'_ _: [bi {biz " 5‘ "i,,.1...i.. 77'17 77%’ 5j1j2...jpi,,.1...i,,- Itissometimes necessary torearrange expressions such as €aA*(€b' A€b3 A...AQbf’). 16 TENSOR ALGEBRA This may beaccomplished byusing (1.4.8), forexample e“/\*(€"/\6‘)=6”Alc*@b =_iC(@a A*eb) Tgm*eb =—g“"i‘*1 +g“'*e” (since 6“/\*6”=s“”*1) :_gab*ec +g¢e=i=eb_ andsimilarly e“A*(€"/\6‘/\ed)Z8ab*(@C /\ed)'8“C*(@b /\ed)+8ad*(eb /\QC)- l.5TheMixed Tensor Algebra Just asthetensor product ®’V isthespace ofmultilinear mappings on V*><V*><,_,><V* (rtimes), thetensor product ofV*with itself, ®’V*, isthespace ofmultilinear mappings onV><V><...>< (F times). More generally wehave thevector space ofmultilinear mappings on V*XV*><___><v* >< V><V><...><V,JK V 24 L V, Y .. v rtimes SIIITICS thespace ®’V®‘V*. This space iscalled thespace ofmixed tensors of covariant degree randcontravariant degree s,T,.‘(V). (The assignment oftheterms covariant andcontravariant isamatter ofconvention. The way wehave indexed ourbases accords with theclassical component conventions.) Tensors inT,‘(V) will bereferred toasbeing oftype (rs)Itwillbeseen that wehave 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 VandVinno definite order. However, such tensor product spaces are naturally isomorphic tothe canonically ordered product. For example,‘ the ordered pairs V><V*arecertainly distinct from V*><V,theblllllﬁﬂf mappings onthese spaces being V*®V and V®V* I@5P@°1i"elY- HOW" ever, wemay define amap Q9by q9:V*®V l—-> 1/®V1 T|——> (pT whereTHE MIXED TENSOR ALGEBRA 17 Itiseasy toseethat rpdefines 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‘(X1-)E61,,then abasis forT,‘(V) isprovided bythen(’+~‘) elements {e"®e‘2® ...®e‘?®Xj-,®X,-,® ...®X,-5}. IfTisanyelement ofT,‘(V) then TETf]‘{§_"_"_‘,-f e‘1®e‘2® ...®e‘?®X;1® ...®X,-5 where thesummation convention isemployed. If{e"} isadifferent basis forV,with dual basis {X’,-}, then ife"EM‘,-el andX',-EN,-IX; itfollows from e"'(X',-) E61,-that MikNjk = Soifthetransformation coefficients arearranged into matrices Mand N,thetranspose ofNistheinverse ofM.Ifthecomponents ofTin thebasis labelled with aprime are 7-"J1--..I_§ ll.--If then i'_,_j5_ r r I t‘ t‘Y‘,-11]___,-r — T(Xi1, X52, ...,Xir, 6"‘, ...,6]‘) : .1 is .P .Pr q1""qM1,,...M,,Siv,,1...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: Ci:T1‘-(V)-——>Tiii(V) (¢r)(x, co)=T(w,X) VXe v*.weV. Tl?) ctr 1 1 13 TENSOR ALGEBRA 1 lthentry rm“-*""-i"'\_ C1¢T(av' vaa":):T(§9'-'9Xi1" 9;1‘?""e”"") kthentry (1-5-1) where {ei}isdual to{X,-}. _ _ _ Since thedual frames transform contragradiently thelinearity of‘T ensures thatthedefinition ofC1.isbasis independent. If,insome basis, Thasthecomponents i---i7-‘i1]...£fS then thecomponents ofCQTare 7"j1i--=_llElml1l_+1 ---.ls_I1. ..115-1"! lk+1 ...1, where the‘dummy’ index missummed over. Forthespecial case of TeT{(V) thecontraction C1maps Ttothefield F.Inthiscase the contraction map issometimes called thetraceof T,Tr.T. N When Vhasametric there isacanonical isomorphism ,between. V and V*. Similarly wecan useametric onVtodefine amapping between tensors ofdifferent contravariant and covariant degreps. For example, given atensor TeTf,(V) wecan define anSeT,_1(V) as follows: 1s(x,, ...X,_1;e1,...,es.est) ...- _1 ‘E1 '+1 s+l =T(X1a ---aX/(-19 e1aXka"'aXr—1>e ""5 61 *6] "' "8 .. - - - -1Inasimilar way wecould associate with Tatensor inT§.,1(V) or more generally atensor inT§(V) with p+qEr+s.We give an example. Given TeT2(V)wedefine SeT2(V)by s(w,Y,co)=T(W,at,"Y') vw,Yev*,wev.(1.5.2) If{e"}and{xi} aredual bases forVandV*respectively such that T=Tffe*'®e1®X,, S=Sf-j-e‘®e1®Xr then writing W,Yandcointhisbasis gives W"YlwkSf‘,- EW‘Ylw;,g‘i"gp;Ti’q- Since thismust hold forallW,Yandw 51;.=gqkgp].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 beambiguousI 1THE 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, TET,-j-"e"®e1"®Xk SES,-,~"e"® el®Xk then therelationship (1.5.2) between SandTrelates thecomponents by Siik =gqk3.vi'T1'qp' This cannow becompactly written as S,-J,"ET)“,-. (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),andthatSeT}(V) istraceless, where S(X,co)E T(X, 5)‘) VXe V*,coe V. Equivalently, T(X’, X,-)E0,where X‘Eg’lX,-. 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) 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- ible representations ofthe Clifford algebra give rise toirreducible representations ofthespin groups: spinors. Ifthereal vector space V with bilinear form gisanorthogonal space then wewish toimbed V and acopy ofthe real numbers asvector subspaces inthe real associative algebra C(V, g)insuch away that x2Eg(x, x),Vxe V. The square ofxdenotes itsproduct with itself inthisalgebra, andthe right-hand sideisarealnumber which liesinthevector subspace ofthe algebra spanned bytheidentity. IfSisanyinvertible element ofthe algebra andx’ESxS‘1 then obviously x'2Eg(x, x).Soifx’isinV wehave anorthogonal transformation. Those elements Ssuch that x’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 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-gradedt, 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 thatthis section willenable them todetermine allthespin-invariant products in whichever dimension iscurrently infashion, and, forexample, whether the charge 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 they-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 the1/-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. 1Grading isdiscussed inAppendix A.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 <_3(v,g)=T(V)/J. (2.1.1) The product will bedenoted V,satisfying [a]v[b] E[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 by1),Eandix andsoallofthese naturally induce operations (denoted bythesame symbol) inC(V, g).Ifx,yeVthen x®r=My+g(x,y)+%{(r+y)®(x +y)-s(X+Y»X+y) -X®x+g(x,X)—)’®)’+g(x,x)}~ Theterm inbrackets isinJandso x®}’ "'XAY -1-g(x, y). (2.1.2) More generally fortoap-form andxeVwehave x®c0 ~xAco+ixw. (2.1.3) Here YeV*isthemetric dual ofx,defined byIi?‘(y)Eg(x,y), VyeV.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 fortoofdegree lessthan orequal IOp—1,then itwillbetrue forallp-forms ifitholds fortotheproduct ofporthogonal 1-forms. Asweshowed intheproof of(1.3.7) itfollows from thedefinition oftheexterior product that ifx,y‘,iE1,...,p areinVthen x/\_V1A Ayp P =1/<p+1>(x<>i<>y1---P +2<—1>v*®<xAy1-~- t(21.4)r=1 where yl F...p : yl/\y2/\ .../\yr—1/\yr+1/\ I._Ayp- Since (2.1.3) isassumed truefortoofdegree p—1 orless yr®(xAy1.. ?...p) ~yr®(x®y1...?...p ____iYy1...?...p). Useof(2.1.2) gives 24 CLIFFORD ALGEBRAS AND SPINORS yr®(xAy1...?...p) ~2g(yr’x)y1...?...p _x®yr®y1...?...p Since they‘areassumed orthogonal wemay use(2.1.3) forcoa(p—1) or(p—2)-form toshow that yr®(x/\y1... ‘P...p) ,__,2g(yr’x)y1...? ...p_x®(yr/\y1_.. '2...p) _yrAiYy1... ';...p_ Wemay pulltheinterior derivative tothefront ofthelastterm anduse yr/\y1... ?...p :(_1)r-1y1...p toproduce yr®(x/\y1... F...p) ,___g(-yr, x)y1... r...p +(_1)rx®yl...p _(_1)rify1...p SO P . 2(__1)ryr®(xAy1...?...p) r=1 P /‘\ ~2(__1)rg(yr,x)y1... r...p +px®y1...p _pifyl...p "1 p_m ,___px®y1...p ___ Returning to(2.1.4) shows that if(2.13) istrue forwaq-form with qsp-1then itistrue fortoap-form. Thus (2.1.2) shows thatindeed (2.1.3) holds forallp-forms. Repeated useof(21.3) shows that an arbitrary tensor product isequivalent toasum ofexterior forms, for example x‘®x2®x3 ~x1®{x2 Ax3 +g(x2, x3)} ~x1Ax2Ax3 +g(x1, x2)x3 —g(x1, x3)x2 +g(x2, x3)x1. 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 such arelation andforpractical purposes weshall becontent with (2.1.3) andthefollowing other special case. Ifcoisanarbitrary p-form andxa1-form then w®x ~xA1700—iinw. (2.1.5) Fortoa1-form thisiscertainly true since itreduces to(2.1.1). Again we prove itsgeneral validity byinduction. Suppose that (2.1.5) holds forno ofdegree lessthan orequal top,then (yAw)®x ~(y®a)— i,w)®x by(2.1.3) ~y®(x A1700—i,17w)- xA17i,w +ifniyw by(2.1.5).THE CLIFFORD ALGEBRA 25 Asecond application of(2.1.3) gives (yAw)®x ~yA(xAnu)—i,,17w) +i,;x17co —xAiy-170) —iyifqw +xAiynw -ifiynw where wehave used nix=—iX17.Dropping theterms that cancel anda little rearranging gives (y/\w)®x ~XAn(yAw) —im(y/\w) 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 forQ(V, g).There isthus anatural way of introducing a.product, V,onthespace ofexterior forms that turns this vector space into analgebra, C(V,g)say, where C(V,g)=Q(V, g),If crandwareexterior forms then theexterior form crVcuisdefined by [a/]V[co]=[crV0)]. (2.1.6) Since [a/]V[cu]=[a/®w] theequivalence in(2.1.3) gives forxa1-form X\/(1): X/\C0+ Similarly (2.1.5) gives cuVx =xA17w— i,,17w. (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 that theClifford 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 qpisalinear mapping ofVinto asubspace ofC'(V, g),V’,which generates the algebra, andthat <P(x)/><P(y) +<P(y)w(x) =2a(x,y) Vx,yEV»(2-1-10)—-----mu-rung 26 CLiFFoRD ALGEBRAS AND srrnons The right-hand side isunderstood tocontain theidentity inC'(V, g). The mapping rpcanbeextended toahomomorphism II)from T(V) to C'(V, g): :T(V) --—> C'(V, g) (x®y) =‘P(x)A<P()’)- (Z1-11) Since V’generates C'(V, g),[T(V)] =C'(V, g).Itfollows from (21.10) and (2.1.11) that {x®x —g(x, x)}=0,and so(J) =0 where Jistheideal used toconstruct C(V, g).Thus if17isthemapping ofT(V) onto C(V, g)defined byrra=[a]then =1/101:, where rpis 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 of1/2isanideal 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 that if 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 with theproduct 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 thisproduct. Although the Clifford algebra isnotaZ-graded algebra thevector space ofexterior forms isaZ-graded vector space anditwillbeconvenient tousethe decomposition intoZ-homogeneous subspaces: C(v.g)=2SP...(Co/.gn (21.12)*0P... where nisthedimension ofVandtheprojection operators Sf’pproject out the homogeneous subspaces ofp-forms. IfAand Bare homogeneous ofdegree pandqrespectively then their Clifford product willnotingeneral behomogeneous; rather AB=srp.p(AB) +9"p+p_2(AB) +...+Sf‘p_ql(AB). (2.1.13) This follows directly from (2.1.7) and(2.1.8). Ifcpandrparearbitrary elements ofthealgebra then 9’@(<P1/1) =29’o(<Pptvp) (2-1-14)THE CLIFFORD ALGEBRA 27 where (ppEffpcp and(2.1.13) hasbeen used. Ifgpdenotes themetric onp-forms induced from g,asintroduced intheprevious chapter, then wemay introduce ametric oninhomogeneous forms, G,bydefining G(<;v.1/1)=Zsp(<Pp, WP) (2.1.15) that is,Gisdiagonal inthehomogeneous subspaces. This metric on forms canberelated toClifford multiplication G( W)=9’0(<P§1l1)- (3-1-16) From (2.1.14) the right-hand side isseen tobediagonal inthe homogeneous components ofcpand 1/1andsotoverify (2.1.16) allwe need tocheck isthat gp(rpp, 1/;p)= S1"0(<pf,ipp). Since both sides are linear in(ppand rppitsuffices toconsider thecase of(ppand ipp products of orthonormal 1-forms. If (pp=a1a2.. .aP and 1/1p=blbz ...bl’then from (2.1.7) 3)0((}9§1/Jp) =i5; ...lp; (b1b2 ... Ifthe{ai} and {bi} aresubsets ofanorthonormal basis then the right-hand sideiszero unless these setsarethesame uptoarelabelling. Since ip;...i5;(alaz ...a1’)=g(a1, a1)g(a2, a2)...g(aP, a1’) =gp(a1a2 ...a1’,alaz ...a1’) wehave verified (2.1.16). One trivial result thatisimportant forcalculations is 9’0(<W) =9%(w<P) (2-1-17) 21S 9°0(<P1/1) =Z5"@(¢>pwp) =E(—1)“”2lg,.(<r>p, 1/1,.) where [p/2] denotes theinteger part ofp/2, andtheresult follows from thesymmetry ofgp. Itwill sometimes beuseful toexpand anarbitrary element ofthe Clifford algebra inaG-orthonormal basis. If{e"} isag-orthonormal basis then {eA} isaG-orthonormal basis where themulti-index Atakes onallnaturally ordered sequences ofdistinct indices. We use the notation 12... _e P'=e1Ae2A...AeP=e1e2...eP‘. If80?“, eb)=naband nppdenotes theinverse matrix then weset ea=nppeb, giving eAanobvious meaning. Then 9"0(eieB) =6AB where 28 CLIFFORD ALGEBRAS AND SPINORS 6,43 denotes theKronecker function that takes thevalue zero, unless thesequences AandBarethesame inwhich case itsvalue isone. Ifa isanyelement oftheClifford algebra then wecanexpand inthisbasis a=2s;>p(a@,_,%)@A. (21.18) The Hodge dual ofaform may also berelated toClifford multi- plication. Thedefinition oftheHodge dual, (1.4.5), ofipp,*1pp, isgiven by(ppA*1pp=gp((pp, 1/1p)*1 forallp-forms (pp.Setting zE*1(2.1.16) enables thistoberewritten as(ppA*1/1p=9°0((p§i/1p)z =9’0((pp1p§)z. It immediately follows from (2.1.?) that 9’,,(cpp*ipp) =(ppA*1/1p andfrom (2-1-13) that5P@(<PpW§)Z =5i(¢>pw§)- Thus yn((pp*1'|Up) =9’i(<Pp1/IE1)giving *1};=11152. (2.1.19) Exercise 2.1 If{ea}, {Xb} areany dual bases, e“(X,,) =(32,and er,[3areany exterior forms, derive therelations "-1[P/21 . . . .6Y\/5 =Z%_(np1X,, ---1X,,p¢Y)/\(1";i ---1'§ép/3’)p=0 "-1LP/21. . . .a/Aﬁ =2—(-—l-))1——(iXa] ...lXHp7]pCY)\;(l";,, ...1";;,pﬁ). EO 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)ECpap(lB). One thing weknow about theClifford algebras istheir dimension. Since wehave identified theunderlying vector space with thespace of exterior forms thedimension ofCp_p(IB)is2"where p+q=n.Given abasis forVwecanrepeatedly use(2.1.?) toconstruct amultiplication table fortheClifford algebras, andinthissense weknow itsstructure completely. What wewould liketodoistorelate theClifford algebra to other ‘standard’ algebras. Inparticular wehave already seen that if Cp,p(lB) isnotsimple then wecan construct 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 OFTHEREAL cLiFFORD ALGEBRAS 29 Wewilldenote anorthonormal basis forVby{e‘,ff}fori=1,..., p,j=1,...,qwhere g(ei, e‘)=—g(f1, fl)=1.Itwillbeconvenient tosetz=e1Ae2A ...ePAf1A Afq. The two-dimensional algebra C0_1(lB) has asbasis {1,f}where f2=-1.Itisthus isomorphic tothealgebra ofcomplex numbers, C@,1(1B) =°3(1B)- (2.21) .Abasis forC1_0_(lB) is{1,e},and this algebra might notbeso immediately recognisable. IfP1=§(1+e)and P2=%(1 —e)then {P1, P2} isobviously anew basis. The multiplication table isgiven in table 2.1. Thus P1and P2each span mutually orthogonal one- dimensional subalgebras, each ofwhich isisomorphic tothefield IR,so that 01,003) =iaeaia. (22.2) Table 2.1 P1 P2 P1 P1 0 P2 0 P2 ‘Rather than simply determine thestructure ofC1,1(]B) we511311 take this opportunity todemonstrate some general features ofassociative algebras. Abasis is{1,e,f,z}where Z:_-3/\f= efSince 6andfare orthogonal. Themultiplication table isreadily completed (seetable 2.2). (For example, ez=eef=fsince eisofunitnorm.) Table 2.2 1 6 f Z N“-t~><'\>>-1 Nkopfbi-i\i~,Ni-—*<'m <‘bi—*N\i-, i-~<'b'\v-,l\i .Itis.straightforward toseethat theidentity spans thecentre. An Immediate consequence ofthisisthat C1p1(IB) isnotreducible. More genﬁrally. allCp,p(lB) have anidentity. Ifthealgebra were reducible 30 CLIFFORD ALGEBRAS AND SPINORS 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 the algebra isreducible. Thus either C111(lB) hasaradical oritissimple. Themultiplication table enables the two-dimensional Clifford algebras wehave already encountered tobe recognised assubalgebras. Both {1,e}and {1,2}span subalgebras isomorphic toIRGBIB, whereas thealgebra spanned by{1,f}isisomor- phic toC(18). Wecanusethepair oforthogonal idempotents inoneof theB6318 subalgebras towrite C1,1(IB) asasum oftwoleftideals. For example, ifP1=%(1+ z),P2=§(1— z)then C111(lB) =C1_1(lB)P1 + C1,1(lB)P2. Since fP1= eP1 and zP1= P1abasis fortheleftideal C1’1(]B)P1 is{P1, eP1}. Similarly abasis forC111(lB)P2 is{P2, eP2}. It isinstructive tolook atthemultiplication table forthealgebra inthis basis (seetable 2.3). Table 2.3 P1 eP1 P2 QP2 P1 P1 O 0 QPQ 6P1 eP1 0 O P2 P2 0 6P1 P2 0 8P2 0 P1 8P2 0 The leftideals C1,1(]B)P1 and C111(IB)P2 areboth minimal; they contain nosmaller leftideals. SoP1andP2areprimitive?‘ idempotents, forifP1=P+Qwhere Pand Qareorthogonal idempotents then C1’1(lB)P1 =C1,1(IB)P +C1,1(]B)Q. The sum must beadirect vector space sum. Forsuppose that bP=cQforsome bandc.Then since P isidempotent bP=bPP, but bPP =cQP =0since Qand Pare orthogonal. Thus b=c=0.SoifP1were notprimitive C111(IB)P1 would beasum oftwo smaller leftideals. Could C1_1(IB) contain any two-sided ideals‘? Suppose Iisatwo-sided ideal andthat aeI.Wecan write a=a1+a2where a1eC111(lB) P1, a2eC111(lB)P2. Now C111(lB)a1 isaleftideal which iscontained intheleftideal C1_1(IB)P1 since a1is.But this leftideal isminimal and soC1_1(lB)a1 =C1_1P1. Thus ifa1ab0there isabsuch that ba1= P1andsoba=P1+bag TThe notion of‘primitive idempotents’ isdiscussed in(A11)—(A19) ofApp- endix A.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 Q2ab0.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(lFl) andthequaternions, H(lB). Thequaternion algebra isadivision algebra whose only idempotent istheidentity andsowemust have c1_1(1B)=A/L2(lB). (22.3) Ofcourse wecould have obtained thisresult directly, forif{e1-1}, 1', j=1,2isanordinary matrix basis forA/|.2(lR) then asetofgenerators is {e,f}where e=e12 +e21, f=e12—e21. These generators anticom- mute andsatisfy e2=—f2=1. Abasis forC11,2(lB) is{1,fl,f2,2}and themultiplication table is given intable 2.4.This may berecognised asthemultiplication table of thestandard basis forthequaternion algebra byrelabelling fl=i, f2=j,z=k: Table 2.4 1 fl f2 Z 1 1 fl f2 Z f‘ f‘ —1 Z —f2 f2 fi Z 1 flZ Z f2 _fl _1 .C@1;,(lB) isgenerated byanorthonormal basis forV,{f1,f2,f3}. Since _z=f1f2f3 itwillcommute with these generators, andhence must 1116 in the centre. Furthermore, 22=1and so P1= 2(1+ Z),P2=§(1—z)areapair oforthogonal idempotents inthe Centre. Thus C11,3(IFi) isreducible, C1113(lB) =C11_3(lB)P1G)C11_3(lB)P2. A basis forC0,3(]B) isfl»fl»f2,f3,fl)”, f2f3, f3f1, 2}and since 2P1: P1» f1f2P1= ‘“f3Pi= f2f3P1= —f1P1= f3f1P1= “fzpi 3basis forC0,3(1B)P1 is{P1,f1P1,f2P1, f3P1}. Theresulting multiplicationtable isgiven intable 2.5. The identity inthisalgebra isP1.Again we have thequaternion algebra with astandard basis {P1, f1P1, f2P1, -f3P1}. The mapping i7isanautomorphism ofC1113(IFi), butmaps one 32 CLIFFORD ALGEBRAS AND SPINORS component algebra into theother since 172=—-2. Itthus establishes an isomorphism between these component algebras andso c1,,,(lR) =H(lB)(—BH(lB). (2.25) Table 2.5 P1 f‘Pt f2Pt f3Pt P1 Pl flpl fzpl .f3P1 flpl flpl _P1 "'f3P1 f2P1 f2P1 f2Pi f3P1 ‘P1 -JQP1 f3Pt f3P1 —f2Pt f‘Pl -Pl Itisunlikely that wewill recognise thesixteen-dimensional algebra C11,4(lB) bywriting outthemultiplication table. Anorthonormal basis for V{f1,f2,f3,f4} 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(lB) then we canexpress that algebra asthetensor product ofthese subalgebras. Such asetisprovided by{f1, 2,f2f3, f3f4}. 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, f1zf2f3 =f4, f12f3f4 =f2 and sof12f3f4f2f3 =—f3 and, indeed, wehave anew setofgenerators. The generators {f1, 2} mutually anticommute satisfying 22=—-(f1)2 =1.They therefore gener- ateanalgebra isomorphic toC1,1(B), that isA/t2(lB). Theanticommuting pair {f2f3;f3f4} both square tominus one, and sothey generate the quaternion algebra. (Inthestandard basis wemay choose {i,j}as generators.) Both A/t2(lB) and H(lB) arefour dimensional and sowe have C11_4(lB) =H(lB) ®Jl/l.2(IB). (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 the structure ofanarbitrary Clifford algebra tobedetermined. Infact, given that weknow thestructure of C111(lPl), C1111(lB) andC11p(lB) forq=1,2,3,4thefollowing determine thestructure ofalltherealClifford algebras: Cp+i.q(lB) 2Cq+1.p(IB) (2-2-7)THE STRUCTURE OFTHEREAL CLIFFORD ALGEBRAS 33 CP+1»¢?+1(1B) 2Cp.q(lB)®Ci,i(lB) (2.2.8) Cl».q+4(1B) =Cp,q(B)®C0,4(lB)- (22.9) Before demonstrating thetruth oftheabove assertion wehave toprove these relations. This willbedone bychoosing suitable generators. Aset of‘gen_61’ﬂt0rS fOrCp+1,q(lB) isprovided byanorthonormal basis forV, {EliPlferi=1, ---, P-l-1, j=1, ..., q.Alternatively, wecould generate thealgebra with {eP+1, eP+1e‘, eP+1fl}, i=1,___,P,1':1 ...,q.This follows since wecaneasily recover theoriginal generators from products ofthisset.The new generators aremutually anticommut- ing and for i=1, ..., p, (ep+1ei)2 = ep+1ez'ep+1ei = *(ep+'1)2(e')2 -1;Similarly (eP*1fl)2 =1.Sowehave asetofmutual- lyanticommuting generators, q+1ofwhich square toplus oneandp ofwhich square. tominus oneandso(2.2.7) indeed holds. Cp+1.q+1(lB) 15generated byfepﬂi 6’)fqﬁ, fl}fori=1, ..., p, j=1,.._., q.Anew setofgenerators are{eP+1, fqfl, @P+1fq+1e=' ep+1fq+1f’l with i=1, ---, P,j=1,..., q.(Although thenotation assumes p21 and qZ1 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+1](‘q+1ei)2 =ep+1]("q+1eiep+lfq+1er' =(ep+1)2fq+1e;'fq+1e1' :_(ep+1)20<‘q+1)2(ei')2 =(ei')2 = Similarly (Bf+1f"+1fi)2 =-1.Thus thesecond pair ofthesetgenerate C1,,q(R),. whereas thefirstpairobviously generate C1,1(lB). Theproduct ofthedimensions ofthese mutually commuting subalgebras isindeed thedimension ofCp11,p+1(lB) andwehave proved (2.2.8). The proof of(2.2.9) proceeds inthesame spirit. Anorthonormal basis forVprovides asetofmutually anticommuting generators for C1,,,1+,1(lR). Wepartition thegenerators into twosubsets, andform new generators outofthefirst subset andtheelements ofthesecond subset multiplied bytheproduct ofalltheelements inthefirst set.Ifthefirst setisofeven dimension, wewill then have two mutually commuting subsets ofgenerators. That is,wereplace thegenerators {€i’fJ"fq+1, fq+2,fq+3’ f-1+4} 1'=1,___,p;j= 1,___,q with theset tie‘,?f’;fi*‘.fi*2.fi+3.fi*4} 1"=1.....P;-I=1.....q Where 3=f‘?+1f‘1+2f‘?+3f‘?+‘*. Then 2‘f‘1+1 =—f‘¥+12 forexample andthe lastfour generators commute with thefirst p+q.Since Eel=elf 34 CLIFFORD ALGEBRAS AND SPINORS 2‘fl=f72‘ fori= 1,...,p,j=1, ...,qand22= 1wehave Cp,p(lB) andC1114(lB) asmutually commuting subalgebras. The dimensions ofthe algebras aresuch that wehave proved (2.2.9). Ofcourse wecould equally well have shown thatC1,14,p(lB) =Cpgp(lB)®C4,1,(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(lB) canbedetermined assuming q>p.Repeated useof(2.2.8) gives Cp,p(lB) ==C111p_p(lB)®C1,1(lFi)® ,..®C1_1(lR).'7' Y pterms Ifwesetq—p=4/I+1uwith pt<4then useof(2.2.9) shows that Cp,p(lB) =C11_p(IB) ®C1L.1(lB)® .C11,4(lFJi)®1C1,1(lB)® ...®C111(IFi). itterms pterms Since weknow thestructure ofalltheC11,p(lB) for1u<4,wehave expressed Cp_p(lB) asatensor product offactors ofknown structure. Now wedothesame thing assuming thatp<q;by(2.2.8) Cp,q(lB) ZCp ’0(B)®C1,1(B)® ...®C1, 1(lBJ).1-q L iii’ _ 7 7 qterms Now weuse(2.2.7) forthefirsttime: Cp,q(1B) 2C1,p-q-1(lB)®Ci,i(lB) ---Cl,1(B)' qterms Ifp—q==1or2then there isnothing lefttodo,andintheformer case wewillneed ourknowledge ofthestructure ofC1,1,(lB). Ifnotthen onemore application of(2.2.8) gives Cp,q(]B) 2CO,p—q—2(IB)®?1, ''_ Y __ q+1 terms Ifwesetp——q——2=4(:r+18,with 15'<4then (2.2.9) produces Cp1p(lR) =C11,p(lR)®C11_4(lB)® ...®C1,14(lB)®C1_1(lB)® ...®C111(1B)L _ ._ 2 __. _ _ WY 'Y trterms q+1terms Again wehave expressed thealgebra interms ofproducts ofalgebras whose structures areknown. Sowhat arethepossibilities forCp_p(lB)? Since C1,1(lB) =./1/|.2(lB) andat/t,,,(lFi)®./I/l,,(IPi) =A/l.,,,,,(IFi), repeated tensorTHE STRUCTURE OFTHEREAL CLIFFORD ALGEBRAS 35 products ofC1,1(lR) areisomorphic toatotal matrix algebra. Wehave seen that C11,4(lB) =H(IFi)®Jl/l.2(lB), and since H(lB)®H(lB) =A/t4(]B) thetensor product ofC1114(IB) 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 ofC111p(lB), [3<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 cp,q(IB) =~s4(1R)®./I/t.(1R) (2.2.10) where .94=C,HorHC-DH, andr2dimai =2P+‘1. Since C(IR)®H(1B) = C(lB)®J1/l2(lB), andaswehave already noted H(lR)®H (IR)=A/t4(IB) the second case would lead to(2.2.10) with .951.=C,1BorIR+1B.Soany realClifford algebra canbeexpressed asin(2.2.10) with 524=IR,C,H, IBG-)lB orHG-DH. Since weknow thedimension ofthereal Clifford algebras their structure ischaracterised bythealgebra st.The possibili- ties.for sﬂshow that thereal Clifford algebras areeither simple or semi-simple, inthelatter case being thedirect sum oftwoisomorphic simple components. Obviously thevalues ofpandqdetermine 524,in factfrom (2.2.8) itcanbeseen that sitisdetermined byp—q.Two applications of(2.2.9) give C10, q+8(lB) ZCp, q+4(B)®C0, 2Cp, q(lFl)®C0_ -4Cp,p(lB)®H(lB)®./I/t2(1B)®H(lB)®Jl/t2(1B) (by(2.26)) thus C1,,,11g(lB) =A/L16(lB)®Cp,p(lB). Soinfact ailisdetermined by P—qmod 8.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,t(1B) =c1_1(1a) =./l/t2(lB) and0,,11(lB)=c1_An),andsoby(22.8),Cs.0(lB) =C1,1(B)®CQ’ 1(lB) =./I/t2(lB)®C(lB). Wenow have thestruc- ttlre ofalltheClifford algebras, namely Cp_p(lB) =.ﬂ®Jl/I. where ailis given intable 2.6. Some ofthis table iseasy tounderstand and remember. Ifp+qiseven, then Cp_p(lB) iscentral simple, whereas for P+qoddthecentre isspanned bytheidentity and2.If22=—1then thecentre must beC,andthiswillbethecase ifp—qmod8 is3or7. IfZ2=1 then thecentre isisomorphic toIBGBIB and thealgebra is reducible. Itcanbechecked that 22=1forp—qmod8 equal to1or 5.The involution Ewillinduce aninvolution onthecomponents ofone 0f.the reducible algebras ifand only if25=2.The only reducible C11ffOI‘(l algebras occur when Vhasodd dimension and inthat case 2*”=—-25 and soeither Z,‘or517induce aninvolution onthesimple Components. 36 CLIFFORD ALGEBRAS AND SPINORS Table 2.6 p—qmod8 sf -t>-mo U1|-—*---0-\-_Jl\.> E51G-)(-BEGETIIE1 Ofparamount physical importance isthealgebra C3,1(1B). From table 2.6weseethat C3_1(1B) =Jl/l.4(IB) and sothealgebra admits an ordinary matrix basis {e,1,-}with 1',j=1,...,4.Itisinstructive to construct such abasis. This construction provides aconcrete example of Wedderburn’s structure theorem forsimple algebras. The identity isof rank four andfirst weseek asetoffour pairwise orthogonal primitive idempotents. Weseek anaandib which commute andsquare toone, forthen taking allsign choices theset{§(1 ia)§(1 ib)}consists of pairwise orthogonal idempotents. Forexample, if{ea}, a=0,1,2,3is anorthonormal coframe with (e°)2 =-1wechoose a=e =e02and setC3‘ P1: §(1+e1)(1 +e02) P2=i(1+ @1)(1 *B02) P.=i(1~e1>(1+em) P4=i(1- @1)(1-8”) where em=e0Ae2.These four primitives areall similar, forexample e-3P.<@Y*>'1 =P3 e°P1(e°)_1 =P4 (2.2.12) e03P1(e0s)-1 =P2_ Thus, e°3P1 CP2C3_1(]B)P1, e3P1 CP3C3_1(]B)P1 and e°P1 C P4C3_1(1B)P1 andweset(2.2.11) 911: P1 621: 603191 (2.213) 931: @3P1 941: GOP1.THE STRUCTURE orTHEREAL CLIFFORD ALGEBRAS 37 Ifthe{eh-} forj =1,...,4aregiven by 911=P1 en=(803)-1P2 en=(e3)_1P3 (2.2.14) 914=(e0)_1P4 then eljCP1C3_ 1(lB)PJ- and ell-e’,-1 =P1.Ifnow e,-I=e,-lelj than the e,-1doindeed form anordinary matrix basis. The resulting eifare tabulated II]table 2.7. Table 2.7 er)“--> l P1 303192 e3P3 "'€0P4 eospl P2 60123 _e3P4 e3P1 —-e°P2 P3 e03p4 BOP1 —-e3P2 e°3P3 P4 Any element ofC3_1(1R) canbeexpanded inthisbasis. Inparticular, theorthonormal 1-forms canbewritten as 6“=1'“,-,~e,-,~ (22.15)11Q where thearrays ofcomponents form areal representation (orMajo- rana representation) ofthefamiliar Dirac y-matrices. Inprinciple we candetermine these components from theformula Y3"=29kr@a9,='kk butitishere easier toproceed byinspection. From (2.2.11) €1=P1+P2—*P3—P4 soifthecomponents ya»arearranged asamatrix: @@CDr-\ CDCZ>+--*CD 63>-*c:>c::> >—*©CDO(Vii) = andagain from (2.2.11) e02=P1+P3""P2_P4 38 CLIFFORD ALGEBRAS AND SPINORS so e2=—e°P1 ——e°P_-, +e°P2 +e°P4 Z-941“ 923_‘932_914 similarly ea=—e2P1 —e2P3 +e2P2 +e2P4 :_e2e02Pl _e2e02P3 _62602112 _62802134 =e°P1 +BOP3 +e°P2 +e°P4 =941+923—932*914- Thus wehave 1-—*CI>C'_'.>C> C>1—*®CD CD<D>—=~CZ> ®®CD1—\(vi)= _ I--*C>C>CD CD1--‘QCJ CO1-*CD C>®CD1—*(~/E1)= _ Writing e3=e3(P1 +P2+P3+P4)gives 63Z931_942+913—924 andhence CD1--*CIZ><'.D l—*<Z'JC.DC> CDC>C>1— CDCJ1-—*C'.'D(16-’}-)= _ Since thealgebra C3,1(1B) iscentral simple, thetransposition canbe related totheinvolution Ebyaninner automorphism, namely aT=C'1a5C VaeC_~,_1(]B) (2.2.16) where Ccan bechosen such that C5=iC. The choice ofaCin (2.2.16) isdetermined uptoamultiple ofthecentre, andsowehave no choice inthesymmetry ofCunder .§.Forthebasis given intable 2.7we may take C=e1e2e3, andhave C5=—C. Since e1,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(1B) theClifford algebra wasisomorphicTHE STRUCTURE OFTHEREAL cLiFFORD ALGEBRAS 39 toatotal matrix algebra, generated byarealsetofDirac -y-matrices. As aless familiar example we now consider C4_0(]B) =H®J|/L2(lB). (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 z denotes thevolume 4-form with here 22=1.Thus apair oforthogonal primitive idempotents isgiven byP1=21-(1+2),P2=§(1—z).Since P2=e1P1e‘ wemay choose abasis forA/l2(]Pi) asfollows: 9r,='—> i P1 @1132 61191 P2 {P1, e23P1, e3i4P1, e24P1} isabasis forP1C4, 0(]B)P1. This isa canonical basis forthequaternion algebra. Replacing P1with P2gives a basis forPZC4, 0(lB)P2. Thus, aquaternion subalgebra ofC4,0(lB) that commutes with allthee,-J,isspanned by{1,e23,e34,e24}. 2.3TheEven Subalgebra The Z2-gradation oftheClifford algebra ensures that elements ofeven degree form asubalgebra, C:,',q(lB). That is,aeC;_q(]B) ifandonly 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{e’,fl} fori=1, ..., p+1, j=1,..., qare anorthonormal basis with (e‘)2 =——(fl)2 =1then asetofgenerators, with noredundant members, forC;,’+1_q(IB) is{eP+1e", eP*1f1‘} fori=1, ...,p,j=1,...,q.Since, forexample, eP+1e‘eP+1ef =—e‘ei wesee that products ofthis setproduce abasis for2-forms and sotheset generates C§+1.q(]B). Itisnothard toseethat there arenoredundant generators. These generators are mutually anticommuting with (eP+1el)2 =-1and(eP+1fl)2 =1thus C,;I,1_.,(1R) =c,,,(R). (23.1) $0ifC,,_q(1B) =.<2Q®./I/L, and C;§_q(IFi) =975®J|1l,» the algebra 913is obtained byrelabelling table 2.6. Since dimC;,q(R) =§dimCp_q(lB) it follows that r'2dim% =2"“ (see table 2.8). Whereas more than one value ofp—qmod8 cangive risetothesame dor93nocombination ofadand9/3isrepeated intable 2.8. Animportant example oftheeven subalgebra isprovided byC§_1(IB). From table 2.8weseethat thisalgebra isisomorphic tothealgebra of 40 CLiFFoRD ALGEBRAS AND sPiNoRs Table 2.8 p—q mod8 R4 93 IR IRC-BIB lR®lB -~.1O\tJ1-lkb->[\J+-ICD (“IQI525I !T1<’2It:g)Zt:ﬁ5H®H i_i___ .32‘.(5 1.»-1:5"W01-»-C/3complex matrices oforder two. The centre ofthealgebra isomorphic toC,isspanned by{1,2}where, asusual, z=ee. The involution Eleaves zinvariant and soinduces aninvolution on C§‘_1(lR) which issimilar totransposition. That is,if{sag}, cr,[3=1,2is anordinary matrix basis and theinvolution over C,t,isdefined by 80,5’=1-:50,then there isacEC§_1(lB) such that a‘=c‘1a5c VaeC§_1(]R) (2.3.2) with c5=:.l:c. The element cisdetermined uptoamultiple ofthe centre andsowecanhave only oneofthese signs. Infactitmust bethe minus signsince elements areinvariant under 5ifandonly ifthey arein thecentre, socmust bea2-form. Thus, although similar, tand E cannot beequivalent since cg=-—c.Equation (2.3.2) may benaturally extended todefine ronthewhole ofC3,1(lB). Ifjisany odd regular element ofC3_1(IR) then theinvolution Q9 defined by(Tb1-A a9=ja‘§j”1 VaeC3_1(lB) (2.3.3) willinduce aninvolution inC§1(]B). Since 24‘=-2thisinvolution must besimilar toHermitian conjugation inC§“_1(]B). That is,if*isthe involution over IRinC§'_1(]B) defined bysajf=£5,then there isa beC§_,(IP1) such that ag=b"‘alb VaeC§_1(1P1) (2.3.4) where bl=ib. Since bisonly determined uptoanelement ofthe centre, which isC,wecanhave either sign. This equation isnaturally extended todefine ‘LonC3,1(IB). Equations (2.3.2) to(2.3.4) show that transposition and Hermitian conjugation inC§_1(lB) differ byaninner automorphism ofC3_1(lB). This automorphism isnotaninner auto- morphism ofC;1(lP1). Wehave al=ua‘v*‘ (2.3.5)THE EVEN SUBALGEBRA 41 where v=bjc. The inner automorphism ofC3_1(lB), a—> vau'1, in- duces the involutary outer automorphism #onC{1(lB), where # complex conjugates thematrix components inthebasis {sag}. Itinfact follows thatwecanfindaunit-norm 1-form xsuch that a#=xax VaEC§'_1(]B) andxc=cx (2.3.6) foranappropriate choice ofcin(2.3.2). Forweknow that a#=vav‘1 forsome odd v,and since #2=1,v2liesinthecentre ofC§1(]B). Supposze that v=y+wz for the 1-forms yand w. Then v=y+wz+(yw—wy)z =yz+wz+2(yAw)z. The first two terms are0-forms, whilst thelastisa2-form, andsoforw#50wemust ha"? Y=AW,/IeIR.Thus v=(2—z)w, andsince A-zisinthecentre oftheeven subalgebra all‘=waw*1 foralleven a.Now 02ab0andso wz¢0,sowehave a#=xax'1 where x=w/(|w2j)1’2, giving x2=i1. Since sag“ =$0.5, xmust commute with the matrix basis, giving 1,,£,j., =0.Thus thesay;must lieintheeven subalgebra oftheorthogon- alcomplemeng tox,whereas C{1(IB) =Jl/l.2(1B), C3‘:0(]B) =Handsowe must have x+=1.We can choose the cof(2.3.2) tolieinthe subalgebra C2_1(IB) andthen xc=cx.Wegive anexplicit example. Abasis forC§1(lB) is{1,em, e02, e03, em, Q23, e31, Z},where wa usethepreviouslyintroduced notation. Inexactly thesame way aswe constructed amatrix basis forC3,1(1B), wecanconstruct thematrix basis given intable 2.9 for C;1(lB) where Pf=§(1+e02) and P;= §(1—e02). 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 e1commutes with thematrix basis andsquares toone, andwemay choose ittobethexofequation (2.3.6). This element canbeused together with theprimitives inthe even slubalgebra+ toform primitives inthefullalgebra. Forexample, if F1:§(1+x)P1> P2=2(1+x)P2'3 P3=%(1—x)P{“ and P4= §(1— x)P2+ then wehave asetofpairwise orthogonal primitives of C_3,1(IB). These aretheprimitives used toconstruct thematrix basis given intable 2.7. Notice that the involution that corresponded to transposition inthat matrix basis induces Hermitian conjugation inthe basis fortheeven subalgebra given here. Table 2.9 saﬁ-—) 1 Pf e°3P§ e°3Pf P; 42 CLIFFORD ALGEBRAS AND SPINORS 2.4TheClifford Group Those regular (that is,invertible) elements, s,such that sxs"1e V VxeV (2.4.1) form theClifford group, F.Itisstraightforward toseethat they do indeed form agroup. The vector representation ofF,X,maps Fintothe group ofautomorphisms oftheClifford algebra: ;g:l"———> AutC,,,q(IB) s1i> ;{(s) where ;((s)x =sxs“1. (2.4.2) Since 2s(x(S)x. x(S)y) =SxS“SyS‘1 +syS*w'1 =2g(x.y) Xclearly maps theClifford group into theorthogonal group. Ifnisthe dimension ofVthen therange of1depends onn.Ifniseven then x(F)=0(1),q) (2-4-34)Whereas fornodd ;((F) =SO(p, q). (2.4.3b) Let0beanyorthogonal transformation onV.Then since Vgenerates theClifford algebra, 0extends uniquely toanautomorphism ofthe algebra, that is,wedefine o(x2x2 ...xp)=ox10x2 ...oxp. Ifr1is even then theClifford algebra iscentral simple andallautomorphisms areinner, sointhiscase x(l") =O(p,q).Ifnisoddthen thecentre is spanned by{1,2},theidentity and thevolume n-form. Clearly, any orthogonal automorphism that does not leave the volume n-form invariant cannot beinner. However, anyautomorphism that does leave thecentre invariant isinner. ForifC2,,q(1B) issimple allautomorphisms over thecentre areinner. IfC2,,q(IB) isnotsimple then itisthesum of twocentral simple components Cp_q(IB) =C,,_2,(lB)P2(-3C2,’ q(lB)P2 where {P1, P2}areorthogonal idempotents that span thecentre. Ifois anorthogonal automorphism that leaves thecentre invariant then it induces anautomorphism onthesimple components, andthismust be aninner automorphism ofthecomponent algebras. That is,foranya, o(aP,-) =S,-(aP,-)S,~_1 where S,-S2” =P,-,theidentity inCp_q(IB)P,-, i=1, 2.IfS=S1+S2then Sr1= S;1+ S21,forSS"1= S1Sf‘+ S285‘ since S1S21= S2S1'l =0andP1+ P2=1.Now Oa=0(aP1) +0"(aP2) =S2aP2Sf‘ +S2aP2S21 =(S1+S2)(aP2 +aP2)(S1 +S2)'1= SaS“.THE CLIFFORD GROUP 43 Wehave shown thatfornoddanyorthogonal automorphism thatleaves thevolume n-form invariant isinner, thatis,X(l") =SO(p, q). Obviously theClifford algebra does nottransform irreducibly under thevector representation ofF,theZ-homogeneous subspaces being preserved. Infactthese spaces ofp-forms carry irreducible representa- tions. 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)= a,aab0.Then thereﬂection ofxintheplane orthogonal toyisgiven by Syx=x—2a'1g(x, y)y VxeV. (2.4.4) Ifwewrite ,,:av%>,+,. where risorthogonal toythen xiSyx ~r g(ay) y soSyindeed corresponds totheusual notion ofareﬂection. Itisreadily verified thatreﬂections areorthogonal transformations, for s(5,.x, 5,4)=g(x,x)+4@‘2s(x. y)2e(y, y)~4a"‘a(x, y)s(x, y) =g(x,X)- Thefollowing theorem hasalready been anticipated. Any orthogonal 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)=g(y, y)vi0andx—yisnotnullthen SW"T"g<ig(—Lyl;f)>»> (x'Y) L2_ls_(.ri..r) ,—_.s_(x. y)] _xtgtx.4)+go.2)—24042110‘Y) =X—(X—y) ifs(x, X)=g(x,y) iyi. 44 CLIFFORD ALOEi3RAs AND SPINORS Ifx-—yisnullthen x+ycannot besince xandyarenot.Then 5241»—X 2g(x’xi+y)(>1+y)=-yso+yrX+y) and soS,,S,+,,x =—S,,y =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. Ifoisany orthogonal transformation ofVthen, since ithasthesame length asy, oycanbetransformed 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 soo=u‘1v which isalso aproduct of reﬂections. Forrt=1relation (2.4.5) isobviously true andsowehave proved itsgeneral validity. Asastep towards writing anarbitrary element oftheClifford group inastandard form weobserve thefollowing. IfxEVandg(x, x)#50then xEFand;g(x) =17S,.. (2.4.6) Itissufficient toshow that X(x)y =—S,y foryEVsince Vgenerates thealgebra. Wehave x(x)y=Xxx“ ={2s(-Y. y)*MIX“ =—y+2s(X,y)X"1 andsince x2=g(x, x)450then “=~—i— andxx‘1=-— +2g(x’y)x- Si)’- Xg(x,x) Y ygo.X) Together (2.4.5) and (2.4.6) give acanonical form foranyelement of theClifford group. IfsEI‘then s=Ax‘...xhwhere /1isinthecentre 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) =Sxl...S24with heven, then X(s) =)((x1 ...x”). The kernel ofthevector representation is obviously thecentre andso(2.4.7) follows. Ifniseven then 17=1(2) where zisthevolume n-form andS,=;g(zx). Since zxisaproduct ofTHE CLIFFORD GROUP 45 n—1non-isotropic vectors itfollows that foranysEI‘,X(s) =;5(x1 ... x”),where hneed notnow beeven, andso(2.4.7) again follows. Ifniseven then theClifford algebra iscentral simple andsointhis case elements oftheClifford group areeven orodd. IfPi-is the subgroup ofFconsisting ofallelements that areeither even orodd, then fornoddF1“isanon-trivial subgroup. When nisoddthevector representation maps theClifford group onto thespecial orthogonal group and notthewhole orthogonal group. The twisted vector repre- sentation isintroduced tomap F1“onto O(p, q)fornodd aswell as even: tpzfi i> AutC,,_q(IB) s1——-—> tp(s) where <p(s)x =s'-'xs*1 forxEV. (2.4.8) Notice that (2.4.8) gives theaction of<p(s) onelements ofVbyClifford multiplication, and since Vgenerates thealgebra theaction onthe whole algebra isdefined: <P(Fi) =90>,q)- (3-4-9) Ifxisaregular element ofVthen xEl"1“ and foryEV<p(x)y = -—X(x)y =Sxy. Thus (2.4.9) follows from (2.4.5). Ifniseven then F1“=Fandifs"=sthen q0(s) =X(s). Ifs"=—s then <p(s)x =—sxs‘1 =szxz‘1s'1 =;((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 ofsisi,2s_=0forallxandsos__=0.Thus sisintheeven part ofthecentre which islB*.(Sometimes theClifford group isdefined differently. Itisdefined tobethegroup Gconsisting ofallregular s such thats"xs“ EV,VxEV.Itfollows that G=Pi.) The even elements intheClifford group form asubgroup 1”’.Inthis case the‘twisted’ representation andthevector representation coincide andwehave X(l“*) =SO(p, q). (2.4.10) Itfollows from (2.4.7) that ifniseven andsEI“then s=/Ix‘...x” where /IE1Bandhiseven. From (2.4.6) then ;g(s) =(—1)"S,.1 ...S24 which, since hiseven, isaneven number ofreﬂections. Hence inthis case X(I"+) =SO(p, q).Ifr1isodd then ;((l"*)C SO(p, q).Itagain follows from (2.4.7) thatifsEFthen X(s) =;g(x1 ...x")forsome x".If hwere Odd then X(x1 ...xh)=)((zx1 ...x")where 2isthevolume n-form which, fornodd, liesinthecentre. Ifhisoddthen zx‘...xh iseven andinI“soX(F+) =X(I‘) =SO(p, q). 46 CLIFFORD ALGEBRAS AND sPiNORs IfsEF’andniseven then sisaproduct ofaneven number ofnon-singular 1-forms whereas ifrtisodd, scanbewritten asaproduct ofnon-singular (n—1)-forms. (2.4.11) The case ofneven istaken care ofby(2.4.7). Fornoddwecanwrite s=/lxl...xl’with /1inthecentre. Since siseven ifhiseven then /IEIRands=i)t(x1z) ...(xhz). Byredefining xlthefactor ofiican beabsorbed. Ifhwere oddthen itwould beproportional tothevolume form, says=uzxl ...x"with tieIR.Once more, s=iju(zx‘) ... (zx") andwehave proved (2.4.11). The kernel ofthe‘twisted’ representation (and thevector representa- tion forneven) isIR*. Bysuitably ‘normalising’ elements. ofPiwe obtain asubgroup whose image under these representations isthesame asthat ofF1“,whereas thekernel issmaller. The norm homomorphism Aisagroup homomorphism: /1:1“ i> 1R* 51—-> )l(s) =sgs. (2.4.12) Ifsisinvertible then soiss5with (s5)"1 -=(s'1)§. IfSEFthen (sxs‘1)5 =sxs'1Vx EVso(s‘1)5xs5 =sxs‘1 orsgsx =xsgs. Since V generates thealgebra s’=‘sliesinthecentre. Ifsiseven oroddthen sgs iseven, andsoAdoes map Ftinto lR*.It1Sstraightforward toseethat M8182) =/l(Si)l(S2)- Wedenote thesubgroup ofF1’which consists ofthose elements whose norm isplus orminus oneby,1“; thesubgroup ofunitnorm elem-ents +1"? Wedefine ,1“ and +1“ similarly. The group ,I"i issometimes called PIN(p, q),21”’ called SPIN (p,q)and+1“ called SPlN+(p, q). IfS6F’thenS/(I/1(S)|)"2 E1-F’and<P{S/(I/1(S)l)"2} =20(8)andS0 indeed theimage of,1“ under (pisO(p, q)andthekernel consists of themultiplicative group formed byplus and minus one, which is isomorphic toZ2.Similarly ;g(,l“+) =SO(p, q)with kernel Z2. Wecanintroduce aslightly different norm, tt: ;.t:l"i —-—> lR* 31-—> j.t(s) =s‘5"s. (2.4.13) Obviously ,u(s) =i)t(s) depending onwhether siseven orodd andso theonly new subgroup isthegroup ofthose swith ju(s) =1,‘Ti. The various subgroups of1":that have been introduced can be arranged asfollows: E—-—> +Fi E ,1-+41,rA5_->+rP§I>.> ,r+. (2.414) I2-4,r+z Inthis last diagram (2.4.14) theappropriate mathematical symbolTHE CLIFFORD GROUP 47 here for isIi. Here +F",il,l"i denotes that +1“ isanormal subgroup of,1"? This iscertainly thecase, forif oE,1“ and sE,1“ then (sos”1)’l =sos“ since s”=is and A(sos'1) =1since )l(s) =i1.Ifwelook atall(four) quotients modulo ,1“ thisgives all(seven) quotients obtainable from thisdiagram. For example, i + ,.,,,.24F’/41“ Firstly consider +1“i/+1“. Ifthere arenooddelements ofunit norm then obviously ,1“ ~+1“, soassume thatoisoddwith /1(0) =1.Ifs_ isanyodd element in+1“ then s_—(s_o'1)o', where s_o"1 iseven with norm plus onesothat s._~o.Similarly ifs,isanyeven element s+~1andso+Fi/+1“ isthemultiplicative group ofplus andminus one, isomorphic toZ2.The argument above applies inexactly thesame wayto+l"i/+1“ and21"‘/+1“. Inthegeneral case ,1“ willcontain even elements with norms plus andminus one, ,1)/+ and _y+, andoddelements with both norms, ,)/" and L)/'. Itreadily follows that ,1":/,1“ hasfour elements [+'}/+1, [_)/*], [+)/'] and[_)/"]. 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“ =Z2><Z2.Invarious special cases thisquotient group can have lessthan four elements aswillbemade clear inthefollowing. The kernel oftpfrom FitoO(p, q)isthegroup ofplus andminus one, Z2,which iscontained inallthesubgroups in(2.6.14), andsothe kernel ofcprestricted tothese subgroups isthesame. Thus, forexample ‘P(Fi)= Pi/Z2 .2.F?(P(+Fi) +1“:/Z2 +ri Wehave already determined theimages ofiI"i and,1“ under tp,and now turn totheunit-norm subgroups. Ifxisanon-singular element ofVthen tp(x) =S,and/l(x) =g(x, x).Sotheimage ofunit-norm elements ofPiunder gocontains aneven number ofreﬂections inplanes orthogonal tonegative length, ‘timelike’, vectors. Such orthogonal transformations aresaidtobe‘orthochronous’; the subgroup oforthochronous transformations being denoted OT(p,q).ForxEV,;x(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. Thefollowing summarises theimages ofthe various subgroups under tp: 48 CLIFFORD ALGEBRAs AND SPINORS +Q9 :1“; —__> O(P2 q) (P +ri W) OT(pi (P +ri-_>O+(p, q) (24.15) (P ir+ ———> q) ‘P +F+ W SO+(p2 Ifthedimension ofViseven then theimage oftheClifford group under Xisthesame asunder tp.Ifqiseven then thevolume form isof unit norm, 2(2) =it(2) =1.Ashasalready been noted ifsisaneven element ofFthen X(s) =q2(s), whereas ifsisodd;g(s) =<;0(sz). Since, forqeven, /l(s2) =/l(s) andjLl(SZ) =it(s) theimages ofthesubgroups under Xare the same asunder qr).If,however, qisodd then )l(sz) =——)l(s) andj.t(sz) =—ju(s) andthus forsodd /l(s2) =ju.(s) and it(sz) =/l(s). Sointhiscase)g(.,I’i) =O+(p, q)and;g(+l"*) =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 -thefact that 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=PC-BQ. If oEO(p, q)then wedefine alinear mapping onP: m(o) :P—-—> P xI———> m(o)x =9’(ox) where 9’,21denote theprojections onto thesubspaces Pand Q.This mapping must beone-to-one, forifm(o)x =0then oxEQandsince o isanorthogonal transformation xmust bezero. Thus detm(o) #=0.If detm(o) >0then 0willbesaidtopreserve thespatial orientation ofV. Ofcourse forthisdefinition tomake sense itisnecessary toverify that thiscriterion does notdepend ontheparticular orthogonal decomposi- tionofVchosen. Ifx1,x2EPthen 8(xi» m(U)-Y2) =8(xi3 @(9'x2)) =8(x1= (H2) =8(‘7‘-7—1xi) 0352) =3(<1"“’X1) X2)=s(9’(v“X1)»I2) =s(m(<Y‘)X1,X2)~ Soifm(o)‘ denotes theadjoint map, with respect totheinducedTHE CLIFFORD GROUP 49 positive-definite orthogonal metric onP,wehave m(o)’ =m(o'1). 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 Zebuy) 2s(x.M) St‘C‘go.))”"s(y.y)Y’°”‘E” m(S,)x —x 2g(x’ M)u. sour) There arep—1linearly independent vectors inPorthogonal tou andthese areobviously eigenvectors ofm(S,) with eigenvalues one. A basis ofeigenvectors iscompleted byu,with 1.2:-2)--InW“)- thus detm(S,) g(v’ vl-T. g(”’ L‘) so/1y) ' The numerator isnegative-definite andsoreﬂections intimelike planes preserve spatial orientation. Any orthogonal 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 anyorthogonal 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. r -~.._ M52111; 4+4».q1 ‘-22 = U\ O : . . . . . . . ....I.............I.... ' . . . . .. I """""\-I-..., . H‘ 5mpQ‘;-__,_§_H “\,__ (24.16)I I . H‘ I 5“\ '"~\ \-4. S0lp.q)Y """"'-N.// i____/ ‘\..~ii_ji__-__ ,_...._.-.-._.._.-.i 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 the group ofallregular elements, C§,,(IR), 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§,,(IR) andFwhich give them amanifold structure. The exponential map isdefined ontheClifford algebra intheobvious way CO an expa =2——n—' aEC,,,q(IR). (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;f,,(lR). Thus thevector space oftheClifford algebra with theproduct ofClifford commutation can beidentified with theLie algebra ofC§_,,(IR). With thisidentification thevector representation of C§,,(IR), X,isseen tomap thegroup intotheautomorphism group of theLiealgebra; thiscorresponds totheadjoint representation of C,’f,,,(lR), Ad.Similarly ifwedefine ad:C (IR)i> EndC ,(IR)P-Q P4 a1—-> 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(IR). j The Clifford group isaLiesubgroup ofthegroup ofallinvertible elements anditsLiealgebra must beavector subspace oftheClifford algebra. Suppose that misintheLiealgebra ofF,then exp().m)x exp(—/lm) CV VxEV,V2EIR. (2.4.19) The standard group theory result, Adexp(Am) =exp(adAm), shows that thiscanhold forall)1ifandonly iftheClifford commutator ofm with xisinV.This canbeseen directly bydefining, forfixed mandx, theClifford-algebra-valued function f(l) =exp()Lm)x exp(~7Lm). Wethen have df(/1)/dﬁl =exp(Am)[m, x]exp(—}.m) andmore generally d”f(l)/dit” =exp(Am) (adm)”x exp(——/lm).THE CLIFFORD GROUP 51 Byexpanding f(/1) inaTaylor series about /I=0itcaneasily beseen thatf(/I)EVVAifandonly if [m,x]EV. Ifmiswritten interms ofeven andodd parts, m=m++m_ (hen from (2.3.7), formtobeintheLiealgebra ofFwemust have X/\U’l_:0 i,m_, EV VxEV. (2420) Ifthedimension ofViseven then theodd part ofmmust bezero whilst ifthedimension isoddm_canbeann-form which isthen inthe centre. Theeven part ofmhastobeasumof0-forms and2-forms The exponential ofaneven element will beeven whilst theone- parameter subgroup generated bythevolume form fornodd will consist ofelements that areingeneral neither even norodd Thus the I516algebra of§Ficonsist? ofthe2n(n —1)2-forms and theidentity. 11199 (exp/Im] exp/Im wemust have mg=—m ifmisintheLie algebra of+F—, similarly mg”=—mifmisintheLiealgebra of‘Ti. Thus theLiealgebra ofthese groups isthecommutator algebra ofthe 2-forms. The exponential map sends theLiealgebra into that component of thegroup which 1Sconnected totheidentity. This connected component asubgroup, soproducts ofexponentials arealso connected tothe 1 Conversely, every element ofthat component ofthegroup wicisconnected totheidentity canbewritten asafinite product of exponentials. Since 2-forms areeven under ijand odd under Ethe exponential maps theLiealgebra of+Fi into +F+, andsothismust 301118111 thecomponent of+Fi connected totheidentity. Wewillnow emonstrate that, except foroneexceptional case, +F+ isaconnected group. If5'5+1“ then s=a/xlxz x2”, with areII-1*and thexinon- singulag elements ofV.Bysuitably scaling awecanobviously arrange tlgﬁlt x’=E’=:1.Trhen thenorm ofsisgiven by)L(s) =,1(a)g1 _,_ 8,andsoifsE.,F wemust have aneven number ofnegative-norm vectors and a=i1. The negative-norm elements canbecollected at the left-hand side, for ifE’=1and .9I+1=-1 than we Write x’x’+1 =(x"x"+1x")x" Ex'ix'i+1 Where (x=r)2 :xixr+1xixzxr+1x)' =_1 The overall factor ofplus orminus onecanbeabsorbed byredefining1 . xandthus If5E+F+ S=0102 ...0"”where each ocanbewritten <1=xx x,yEVwiih x2=y-2=:1. (2.4.21) every element of+F" willbeconnected totheidentity ifandonly IFa+such products ofvectors are. Ify=ixthen 0=ixz and50f()1' 3, tobeconnected ~1must beconnected to+1.Foranindefinite -- 52 CLIFFORD ALGEBRAS AND SPINORS metric intwo dimensions theLiealgebra of+F” isspanned bythe volume 2-form 2,where 22=1.Inthis case exp(o:2) exp(B2) = VaﬁeIRsoif-1were connected to+1wewould in@XPl(¢Y+ﬁ)Zl 22fact be able to write itas an exponential. However, ex(62) =cosh6 +zsinh 19,and soexp(62) at-1forany I9.Thus in P ‘L tdRulin outthis exceptional case we this case ,F isnotconnec e. g always have apair oforthogonal vectors a,bwith a2=b2=i1. So (ab)2 =-1andsince exp(nab) =-1theidentity isconnected tominusW tillneed toshow that ageneralonebyaone-parameter subgroup. es oisconnected totheidentity. Weconsider three cases. Suppose firstly that xand yarelinearly independent, spanning an' 'Thorthogonal plane with positive- ornegative-definite metric. enwe have anorthonormal basis {x,it}where x2=a2=e,8=i1. Since 2=x2 we can write y=cos6x+sin6u, and xy= )’ e(cos6 +sinGexu) =eexp (e6xu). Wehave already shown that -1is 1dsoxisalsoconnected to+1 connected to+an y .I ithorthonormalIfxandyspan anon-degenerate orthogonal panew basis {x,it}with x2=-1,12 =ethen (xu)2 =1.Now wemust have y=cosh6x +sinh6u and xy=e(cosh6 +sinhQexu) =sexp (sI9xu). Again thefactthat -1isconnected totheidentity ensures that allsuch products xyare. Ifxandyspan anisotropic plane then welet{x,u}denote abasis in which itisanisotropic vector orthogonal tox.Then y=i(x+6a)and xy=+e(1 +Gexu). Since xuisnilpotent wehave xy=ieexp (élexu) and, since -1isconnected to+1, wehave demonstrated that xyis connected totheidentity. 2 We have shown that +F+ isaconnected Liegroup unless Vis two-dimensional with indefinite metric. Thus save forthisexceptional case .,F* isaconnected double covering ofthat component ofthe orthogonal group which isconnected totheidentity, anditfollows from thetopology oftheorthogonal group that +F+ issimply connected. Insuitably lowdimensions itisparticularly easy toidentify thespin groups, duetothefollowing: IfdimV E5then ifs"=isands5=is“ then sE,Fi. (2.4.22) Allweneed tocheck isthat ifxEVthen sxs" CVforsuch ans.If wesetx’=sxs_1 then x’*1=-x’andit’?=x’ifxEVandsiseven or oddunder both ijand5.Infiveorfewer dimensions theonly elements that areboth oddunder 17andeven under Earelinear combinations ofTHE CLIFFORD GROUP 53 1-forms and 5-forms. Soifn< - - n=5then the5-form isinthece:5ntilt‘ieofetSliltitaI;)€1,]::,/S S1?]?ZI1a_te(]Yl+I; where aisa1-form andba5-form then x’2=a2+b2+2ab. Now 2 and b2areboth 0-forms whereas abisa4-form But since 2'a 0-form and x’=sxs'1 then x'2isaO-form and thus ab=0xthls'3 either a=0orb=0.However, acannot bezero since ti‘atls automorphism cannot take anelement that isnotinthecentre Iltmnlfr CBHTFB, andS0b=0and(2.4.22) follows. InOt6 Needless tosay(2.4.22) does notgothrough insixdimensions For 1:1.::i::2.1i.i;.L;.:41.-:.:i If.11\‘>‘%“°‘4m‘ t )(@ "I"83456), where e12=elez 6tC,theI1 S"=sandsf?=s'1. However, se1s'"1 =—e23“56 andso F The results ofthissection willnow beillustrated bconsid I6;I algebra C3.1(]B)' InthiSCase FP=F.Anorthonorm}al basis efrmgl/th'e {ea}"I0’112’3Whm "(@°)’ =(@’)’=1.LetP=e0T=eggtheiis T@°T“ =-8“Pe°P"1 =e0 (2423) T@"TT1= 6‘ Pe’P‘1 =-e" i=123 II Thenorms ofthese elements areeasilyseent bAP=_ = /ll(T) =1 and M(T) =-1. SoPE+Fwhereas 6Tb )Fsiijsgbi) 1’ tat S16 +Fsuch that S’f=s ),(_,~)- 1 +' nowd —: 1) 1—-.Then s1=(sPT)(PT -1 ggmilgsrifgg)" S1€T, /;(s;1Pr) =1thusS1=e,(PT) wherel 2,E£4S26 i SUC lIa'[,_§‘Z,7=-S2 A(S):1thenS _ .‘ S36 1.Fsuchthat5§=_S )(sj):_1’ th2 _, 2-02 ,andif Wekno thth'i, 3 IenS3_031.)’ U2’U3 €+r+'Clifford Covfnmgtajtoi SI?vgefqrilps generate +1“, theLiebracket being a example 612,then ('e12)2 :ﬁlea(product olfztwospacelike 1-forms, forelements thus gengrate rotations a:deE<hP(i9:e. )=cos6)+sin6e12, Such togiw thefamir L_ ,n elifford commutators areseen etc Eleme tlalq 16alilgiebra ofthemtanon group» I612» 323]=2613 ‘ I1S S ¢ 9 - ex(601: uc asegenerate boosts ,with (em)? =1giving F‘B)cosh6 +sinh6e°1. The commutator oftwo boosts gives 3 I0ation, fo 1 01 02I .. stants are(lIClg:ITl'lIl1il1€3(IT lielooI<iii] 261112. The mmammg Structure con- rotation forexample [501 61218 &12tBzeélqmmutator ofaboost with 3 - ’ . 1 =6-egroup F’canbere _ n . cotliegrgig nflamfz group byllS1ng (2.4.22). This result+shows that +F+%sOU111 — + . thattheevensubI:il)gre1brIeIIgvili1ar'elementS'of C3’1(lB)' In§2'3ItwasShowntwobytwo matrices and SElS(}I£lOfPhlC tothealgebra ofallcomplex Consisting ofunit_n(;rm 61 + must bethesubgroup ofGl(2, C) Constructed amatrix basis fgtnélltsiﬁ Since wehave already exPlicitlY norm corresponds tothedt3.i()Itcanbedirectly verified that theeerminant. If{sag} isthebasis given in 54 CLIFFORD ALGEBRAS AND sPINORs table 2.9then £115=£22,£125=-£12, £215: —s21 and£225=en.Soif2 S:E Sag fag a,[i=l then 55%=(511522 *S'l2S21)(£11 +522) =det(5)1- Thus inthis four-dimensional Lorentzian case, ,F"=Sl(2, C), the group ofcomplex matrices oforder two with unit determinant. The group ofmatrices with determinants ofplus orminus oneisobviously isomorphic to,F+, .,F+ =2Sl(2, C).The PIN group, 2F,isobviously asubgroup ofGl(4, IR).However, since wehave identified 2F‘) asa matrix group itisconvenient toidentify ,Fasaproduct ofthismatrix group with adiscrete subgroup. Wehave already seen how anyelement of2Fcanbewritten asaproduct ofanelement of+F+ 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,Fthen itcanbeuniquely written ass=ot,oE2F+ and IEQ.Themultiplication oftwoelements s1ands2isgiven by 5152 =Uitiaztz =01510211-11112 =O1{X(t1)U2}t1t2' Now X(t1) acts on.,F+ asanouter automorphism and75(9) =£2.We canequivalently write elements of,Fasanordered pair ofanelement of,F* and anelement of£2with the multiplication defined by (o1, t1)(o2, t2)=(o2;((t1)o2, t1t2). Inthisform ,Fisrecognised asa semidirect product of2F+ and aZ2group ofautomorphisms, 2F=2F*®Z2. Wehave shown that 2F+ =,Sl(2, C)andthegener- atoroftheautomorphism group, x,sends otoxox. Aswasdiscussed in §2.1 wecanalways choose such anx,which complex conjugates the matrix components and thus 2F= iSl(2, C)®Z2, where the auto- 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 onSPINORS 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 onanyminimal leftideal The mapping into theendomorphism algebra ofanyminimal 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 ihte gueh an endomorphism algebra induced bytheregular representation will be called thespinor representation ofthenon-simple Clifford algebra, and till‘!ideal willbetermed thespinor space. The minimal leftideals W1 elI€I'_I‘I16(Il.S6I’!1l-Spl!’l0I‘ 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 t1l;eCit'§ni—(slpiIjor space. Thus thespinor representation ofa valem Sl2mi_Siino)rr agebra isreducible, being thesum oftwoinequi- Clifford alebpai drepresentations. The spinor representation‘ ofthe heftmum onuces aiepresentation ofanysubset byrestricting to _ pication on‘theideal byelements ofthat set.Inparticular it induces arepresentation oftheClifford group. Irreducible representations ofthe Clifford algebra induce irreducible representations oftheClifford group. (2.5.1) Th - - . . .Sent-3-818, thespinor representation of‘asimple Clifford algebra, orthe I P111011 representation ofanon-simple one, induces anirreducible epresentation oftheClifford group. This willalso becalled thespinor $1gig:-tspinpr representation. The proof. ofthe statement follows Clifford eyrom theobservation that non-singular vectors generate the bere1grgupotilnd theClifford algebra. InfacttheClifford group could _pace wit thesubgroup 2F— andthestatement would obviously stillbetrue. reéicegil irreducible representation oftheClifford algebra induces a 1erepresentation ofthe even subalgebra then that induced representation isthesum oftwoirreducible ones. Forsuppose thatIisa minimal leftideal oftheClifford algebra that splits into invariant 56 CLIFFORD ALGEBRAS AND SPINORS subspaces under leftmultiplication bytheeven subalgebra. LetWbe 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,,_q(]B) =C;_q(lFi) +C;_,(lB)x so C,,_q(IB)W =C;‘q(1B)W +xC;_q(lB)W CW+xW and CM(lB)xW =CM(]B)W. Since SCI and Iisaminimal left ideal wemust have S=1. If WF1X=Ythen Cpfq(IB.)YCY since Wand hence Xarepreserved-1:1 under leftmultiplication byCf,,,(]B). ButWisaninvariant subspace of tg'i--3 .§ minimal dimension and soeither Y=0,and Iisthesum oftwo invariant subspaces, orY=W=X=Iand transforms irreducibly.:.,"! Having shown that irreducible representations oftheClifford algebra induce arepresentation oftheeven subalgebra that iseither irreducible , orthesum oftwoirreducible representations, wewould liketoknow in which cases each possibility occurs. Suppose firstly that the even I 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 ofthat algebra. Suppose now thattheClifford algebra isreducible; thiscanonly occur if inodddimensions inwhich case theeven subalgebra issimple. Inthisit: case thesemi-spinor representations induce irreducible representations -.~K‘ '3 oftheeven subalgebra. ForletIbeaminimal leftideal (thesemi-spinor S space) andzdenote thevolume form. Then if,forexample, thekernel 3 ofthesemi-spinor representation isthesimple ideal CM(lR)(1 +z)the semi-spinor space isan eigenspace of the volume form, zqo=——<pVqiel. Since 2isodd and regular wehave Cp‘q(IB) = C,'§_,,(IB) +C;_q(IB)z and Cp_q(lR)I =C,”,“_,,(IB)I. SoIcan have noin-J? ii‘-2- variant subspaces under multiplication byC;_q(1B) since itisaminimal leftideal ofCpaq(IB). is The irreducible representations oftheClifford algebra caninduce a reducible representation ontheeven subalgebra even when that algebra i issimple. Thegeneral criterion isgiven bythefollowing. .-5SPINORS 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*(IR) Then- . P-‘I ' C,,_q(lB)P* isaleftideal and CM(1B)P+ =C;_q(]B)P+ +x(j;q(]B)P+ foranyoddregular x.Since P+isprimitive inC,1“,,(lB) then C,‘,f’q(IB)P+ isaminimal l€f'[.lCl68l oftheeven subalgebra andsothedimension of C,,,,,(]B)P istwice that oftheminimal leftideals ofC,‘,*,q(lB). Sothe mfinijmal leftld6EllS. ofthefull‘algebra aretwice thedimension ofthose _OifeSubalgebra if_3I1d_011_ly if_C,,,q(]F1)P+ isaminimal leftideal, that is,Hi andonly ifP.isprimitive inCp',q(.IFi). Ifaminimal leftideal ofthe u+algebra isPI'O]6Ct€(l outbyaprimitive ofthesubalgebra then the C,,_q(lR)-irreducible subspaces areobviously theeven andoddsubspaces. Just asthe1I'I'6dLlCIbl€. 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“. (253) Again thisfollows from thefact that +1“ generates C,',*q(1B). First we note thattheClifford algebra isgenerated bynon-singular vectors ofthe same norm. For if{(31,fi}with i'=1,_,_, P,j=1 qisan fmhOz_n°Yma} basis, andifp#50,then anew basis ofunit-norm vectors 15{B3\/26 +fl}. Thus C,‘,*_q(1B) isgenerated byproducts ofunit-norm vectors, and.such products arein+I“*. fEhel relationship between theirreducible representations oftheClif- pragebra anditseven subalgebra issummarised intable 2.10. The sngucturi ofCp,q(IB) isdetermined byp-—qmod8 where p+q=n_ the6lg.'[ different cases have been grouped inpairs. Forthefirst pair esemi-spinor representation ofthefullalgebra induces anirreducible representation ofthe subalgebra; whereas forthe second pair the Clifford spinor representation induces anirreducible even Clifford SplI1OI“ representation. Forthethird pair ofalgebras thespinor repre- S€It1)tE;IlOI1 splits into apair ofequivalent spinor representations ofthe suagtebra, ‘whereas inthe‘final case thespinor representation isthe suiliiotwoinequivalent S6II‘ll_-SplIlO.I‘ representations ofthesubalgebra. ftiltable 2.10 wegive thedimensions oftheirreducible representations 0E1eClifford algebra anditseven subalgebra. Wehave used C—-S/S toenote thattheirreducible representation oftheClifford algebra isa subalgebraCtCt—S/S Ct—-S/S andtseven C"—S gebraC (p—q)mod8fforda C*—$ representationC—S oftheC C—SCt—S representations andS/Sasemi-spnorirreducbe C—S/SC+—SC—S/SCt—S e210Dmensonsoftheentaton repres Dmensonaspinor ri/2)2n/2 _(2n/2) TabdenotesSPINORS 59 semi-spinor representation, C1—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 areIR-linear vector spaces, thedimensions ofwhich aregiven in table 2.10. Aswell asobviously being leftC,,,q(IR) modules thespinor spaces arealso right d-modules where 514isthealgebra given in table 2.8. Whereas, ingeneral, right multiplication willnotpreserve a leftideal itwillbepreserved under right multiplication byelements of ad.When theClifford algebra issimple 54isadivision algebra, whereas when theClifford algebra isnotsimple at=QDQDQD where Q2)isadivision algebra. Inthiscase thesemi-spinor spaces areright 223-modules. Itisan immediate consequence ofassociativity that leftmultiplication induces a 21>-linear transformation ontheminimal leftideals. Similarly theirre- ducible representations oftheeven subalgebra may beregarded as {ED-linear transformations where Q1)isoneoftherealdivision algebras IR, CorH.The dimensions ofthespinor and thesemi-spinor spaces regarded as2D-linear spaces canbefound from tables 2.8and2.10 since d(dimg,,) =dimﬁ where ‘iiiisad-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 2.Forexample, wemay define multiplication bytheimaginary unit byii/1= 1/)2, where 1/1liesina minimal leftideal. Alternatively, wecould define iip=-1/12. Although wehave already noted that allirreducible representations ofasimple algebra areequivalent, when representing asimple IR-algebra ona C-linear space thequestion ofequivalence needs treating carefully. Ifp and p’are representations ofany simple 1B-algebra sit,where s.€i(lB) =C(lR)(>9./1/L,,,(IPi), onIR-linear spaces Vand V’then there isan IR-linear transformation Sfrom V’toVsuch thatp'(a) =S‘1p(a)S for allaofvi.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)S Vaifand only ifiv’=p'(z)v'. Thus bydefining p(z)v =ivandp’(z)v' =—iv’ we gettwo complex-inequivalent representations ofasimple IR-algebra. This iseasily understood interms ofthecomplexified algebra. Regarded asacomplex vector space Vcarries anirreducible representation ofthe complexified algebra at‘:=si§l®C. The representation pextends by C-linearity tosic, p(ia)v =ip(a)v. Since C®C ==C6903, dcisreduci- bleanditsirreducible representations have askernel oneofthesimple ideals, andtheirreducible representations areequivalent ifandonly if thekernels arethesame. Ifp(z)v =ivthen p(1+iz)= 0and the kernel ofpisprojected bythe central idempotent §(1+iz). If, 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-modulet 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®Jl/L2 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(1Pi) =1H(lB). Here thespinor space isthealgebra itself. If{f1,f2}isanorthonormal basis then {1, fl,f2,flfz =z}isastandard basis forthequaternions. Wemay choose ascomplex structure right multiplication byzand define ia=azfor aeC0,2(lB). Then {1,f1}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>-(5?) p<f1>=(§’ 1%) pm”)-(Q per)-(3 I1)- Had weinstead chosen ia=—az then wewould have thecomplex conjugate matrices. These give acomplex-equivalent representation; we hav-p<a>* -p<r1a<f1>"1>_Regarded asacomplex vector space, Hcarries anirreducible repre- sentation ofthecomplexified algebra H®C. IfPi=§(1iiz)then Pi areprimitive idempotents inH®C and (H®C)Pi areminimal left ideals such that uiz =liui for all uie(H®C)Pi. Since P_=(f1)‘1P+f1 then right multiplication byflisaC-linear trans- formation between thetwo leftideals which obviously commutes with leftmultiplication andhence establishes theequivalence ofthese com- plex representations. Theeven subalgebra isisomorphic toC(IPi) andthespinor representa- tion ofC0,2(lB) induces areducible representation ofC({_2(IB), theeven andodd quaternions transforming irreducibly. These irreducible repre- TThe notion ofan‘H-module’ istobefound attheendofAppendix Awhere thequaternion algebra, H,isalsointroduced.SPINORS 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(IB) =A/L4(lB). Here thefour-dimensional spinor representation induces anirreducible representation oftheeven sub- algebra C§_1(lB) =C(IB)®Jl/L2(lB). Wemay choose asspinor space the minimal left ideal whose basis isthefirst column intable 2.7. By defining ii/2=zipforallspinors ipthespinor 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: 0(1)=(5 0(2)=ip(1) P(@‘2) =(3)1%) p(@°3) =ip(@”) P(@23) =(0-0) P(@°‘) =iP(@23) --CD~.._,©>-Me“) =(-0 P(@°2) =ip(@3‘)- The matrix representations ofthegenerators oftherotation group will berecognised asthePauli matrices (uptoconventional factors ofi). Defining iii:=-21/I 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,,_q(]B) —--—-> EndC,,_q(1R) aI——> 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 Qofthe Clifford algebra wecanusethiscorrespondence todefine arepresenta- tion p: 62 CLIFFORD ALGEBRAS AND SPINORS 'p":C,,_q(IR) i> EndC,,_q(IR) Cll—-—>5(a)=R(a§). (25.4) Indeed we have a representation since p"(a) p'(b) = R(a3)R(b9) =R(b§a3) =R([ab]5”) =p'(ab). Obviously the minimal right ideals transform irreducibly under thisrepresentation. Just asthe minimal leftideals may beregarded asright QB-modules these minimal right ideals canberegarded asleft9)-modules. i Aminimal right ideal isnaturally identified with the space of Q1)-valued SID-linear mappings onaminimal leftideal. ForifipeC,,_q(IR)P and<1)ePCM(IR)with Pprimitive then wemay write =1/»—->(w)=1/1 with (ip) ePCM(]R)P =El).Obviously (i,/Jq) =(ip)q forqEQD. Similarly theClifford algebra itself (orasimple component thereof) may beidentified with thespace of9.?)-valued linear transformations onthe Cartesian product ofaminimal leftideal andaminimal right ideal. For ifePC,,_q(IR) and 1/1eC,,_,,(IR)P then forany aeCp_q(IR) wemay write a(, 111)=(Paw giving 2 a(q, ii»)=qa(. 1/»).a(.wq)=“((1%v)qf<>IqE9% Iftheminimal leftideal carries thespinor representation pandthe minimal right ideal carries therepresentation 5then wemay induce a representation rontheClifford algebra (orasimple component) by defining T(S)(1P‘P) =[P(S)1//][?"(S)‘P] =S1P‘1’S’- Ifwechoose E=Ethen sf=s'1forse+1“ andtherepresentation r and thevector representation Xcoincide on+I"i. Inthis case the representations pand p’induce contragradient representations of+1“, andsince wehave seen how theminimal right ideal canbeidentified with thedual space oftheleftideal wecan construct aQB-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 onSPIN-INVARIANT PRODUCTS 63 thespinor space wemay useaproduct onthespinor space todefine an involution ontheClifford algebra bysending every element tothat which induces theadjoint linear transformation. Such aninvolution will betermed theadjoint involution. We shall construct aproduct of spinors tpandipwhich isthesame asthat ofSQ9andsipwhen se+l"+. The adjoint involution ofsuch aproduct will beeither 5orE17. Conversely, any product ontheminimal leftideal with EorE17as 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 EandZ17fall.Similarly, wecanclassify theproducts ontheminimal left ideals oftheeven subalgebra. Asacorollary inuptofivedimensions we can use (2.4.22) toexpress +1“ astheinvariance group ofsome product. Let$4besimple over IRand9besome involution. IfPisanyprimitive idempotent then P9=JPJ-1forsome element Jwith J9=5],8=i1. (2.6.1) For if leaves elements ofthe centre invariant and 9denotes transposition inamatrix basis inwhich Pisdiagonal then weare asosured (by(A23) ofAppendix A)ofaJwith Jif=iJsuch that fl"=J‘1a§J Vae524;inparticular, Pg=P=J'1P9P. Inthesame way ifthecentre isCwith $9inducing complex conjugation theargument can berepeated with Hermitian conjugation replacing transposition. Ifthen (P6s&Pthen J'1tp<5‘ ePatandwedefine (,):s&P Xs.tPi>>P.v1PE§D vi.111i-—>(iv.1/»)=J-‘cow (2-6-2) Ifaisanyelement ofatthen (‘P9aw)=(0%,#1) (2.63) and‘,9is‘theadjoint involution ofthisproduct. The minimal leftideal a¢P1Saright ‘ED-module. IfqeQBthen (vi,wq)=(viv)q (16-4) and tlitl: product isQB-linear inthe second entry. Ifwedefine Q’=Jq5J forqe@then jisreadily seen tobeaninvolution ofQB such that 64 CLIFFORD ALGEBRAS AND SPINORS Theinvolution jwillreverse theorder ofterms inaproduct; infact (<11.I/1)’=J"(cv. W”!=J"(1"<P5*I/VJ =J"v:"v>J*”1 =8J_’1P°’<P andthus (1/1.<22)=5(<P=1/1):’~ (2-6-6) Such aproduct willbecalled 9.51-symmetric or9191-skew aseisplus or minus one. Wemay usethis product todefine amapping from the minimal leftideal toitsdual space. IfL(siiP, 9))isthespace ofEZD-linear maps from s4PtoQ1)then wedefine ~:s4P---> L(s£P, ED) qr->trwhereaw)=(<P=1P)- <26-2) Weshall refer to6astheadjoint oftpwith respect to(,).We remarked intheprevious section that L(.si4P, QB)isnaturally identified with aminimal right ideal; elements acting ontheminimal leftideal by thealgebraic product. With thisidentification wehave Having chosen some arbitrary minimal leftideal onwhich todefine a product wecanobtain aproduct onanyother minimal leftideal. IfP andP’areprimitives then thesimplicity of$4ensures anelement Ssuch thatP’=SPS'1. Given theproduct of(2.6.2) wedefine {,}:s.4P’ ><a<lP'—> P’s¢P’ E9D’ Q/,[3i—-> {a/,B}=S(crS, )6S)S_1. (2.6.9) Wecanwrite thisas{a/,)6}=J"1ci/96 where J"1 =SJ‘1S5 andwhich satisfies P’?=J’P’J"1.Aninvolution on92)’equivalent totheinvolu- tionjonQ23isdefined bypl’=S(S“1pS)JS'1 forpe§D'. Itthen follows that {[3,ct}=s{a, ﬁ}l’and{<l’P= 5}=P"{‘1’»I8}- The product wehave constructed in(2.6.2) involves notonly the involution Ebutalsotheelement Jasdefined in(2.6.1). Obviously such anelement cannot beunique. Suppose that P9=J'PJ“'1 with J’?=s'J’. Then J"1JP =PJ"1J and soJ"1JP =PJ"1J =Asay, where Ae9.21.Since ,1i‘=J-1/13] =J'1J=‘*J"’9P9J =ss’J"1JP then )(1'=,,,.,ii_ (26.10)SPIN-INVARIANT PRODUCTS 65 If(ip,1/1)’=J"‘(p9i,11 then (Q9,iii)’=J"1JJ'1<p3ip andsince (tp,ip)eiib wehave (<10,I/1)’=7l(<P=¢)- (2-6-11) When EZZJ=IR,then jmust betheidentity andsowemust have s=s’ 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 Eistheadjoint 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 and denoted A.Suppose that (1/1,qo)=e(<p, 1/1)“. Then if(tp,ip)’=/l(<p, ip)with A=/1"then (1/1,t/P)’=8l(<P,111)“=8/l{l(<P, w)}"7l" =8/l{(<P, 1//)'}"/1'3 Since A=A“then (asdemonstrated inAppendix A)wecanset/I=,u)u" forsome ii.Thus 7Lq").'1 =,u(ii‘1q,u)"ii‘1 andweseethatifEisthe adjoint ofanHA-symmetric (orskew) product then itisalsotheadjoint ofanHI-symmetric (orskew-) product foranyjequivalent toreversion. IfEistheadjoint 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 QDI-symmetric orQM-skew product. Certain ofthese products can befurther labelled byasignature. First wenote that these products are non-degenerate; forif(J'1n?)ip =0Vipe .v(lPthen J‘1n§’ =0since the regular representation ofasimple adinduces afaithful representation on anyminimal leftideal. Consider now anon-degenerate Q2)!-symmetric product onaright 22)-module. Then ifthemapping from Q)into the j-symmetric quantities of91),q—> qlq, issurjective then there isan orthogonal basis ofunit-norm elements. Ifthismapping isnotsurjective butanyj-symmetric quantity canbewritten asiqiq, then there isan orthogonal basis ofelements normalised toplus orminus 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 (i/1,ip)=/1, then iftheproduct isQB!‘-symmetric A=/U.Ifwecanwrite it=qlqfor some qeQBthen i/1q'1 willhave unitnorm. The mapping q—>qiqisnot 66 CLIFFORD ALGEBRAS AND SPINORS asurjection from 2Dtothej-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 orHA-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 (Pi, ‘I1')=5:}- Example 2.1 Take $4=C4_0(IR) with EEE.Attheendof§2.2 weconstructed abasis forthis algebra. Let Pbewhat was there called P1, that isP= §(1+2).The division algebra Ps§lP EQ13isisomorphic tothequatern- ionalgebra, with standard basis {P,e23P, e3"'P, e2"'P}. Since P5=P,in thiscase theinvolution §induces aninvolution onQD.This isquaternion conjugation since, forexample, (e23P)5 =—e23P. AnH‘-symmetric product onsﬂPisgiven by (qr.ii»)=viv- AnH-linearly independent basis forQSIPis{P,e’P}. Wehave (P,P)=P (P,e1P) =Pe1P E0 (e1P, e’P) =Pe1e1P =P. Sothisbasis isinfactorthonormal, theproduct being ofindex zero. Thus farwehave shown how anyinvolution canbeputinto oneand only one class determined bytheQDI-symmetric or22)!’-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 Eand FRequivalent ifthere isanautomorphism ffsuch that aw=((a5’)<9’)(5’_1) forallaevi.Infactitfollows thatwith thisnotion of equivalence:SPIN-iNvARiANT PRODUCTS 67 Two involutions areequivalent ifand only ifthey arethe adjoints ofequivalent products. (2.6.12) Table 2.11 __i_i_____________i___ (1)IR-symmetric, ofindex v (2)IR-skew (only ineven dimensions) "3C-symmetric C-skew (only ineven dimensions) C'*-symmetric, ofindex v H"-symmetric, ofindex v HA-symmetric /*1/*i\r—*\K41/4"--1Q'\(J1k\n-/\h-/\Iu—/\\-/\u-/ Here products areequivalent ifthey areboth ofthesame one of seven main types and, where appropriate, ofthesame index. IfEand3715 areadjoints ofequivalent products then wecan introduce two QDIP symmetric orskew products, (,)1and(,)Kof(where appropriate) the same signature, with Eandfittheir respective adjoints. Both products admit acanonical basis ofthesame type, andanychange ofbasis may beeffected byleft multiplication byaregular element. Since both products arej-linear inthefirstvariable, andlinear inthesecond there must bearegular 0such that(tp,ip),=(otp, oip)K forallgoandipin theminimal leftideal, thatis _]—1q;.5P1)) =I<-l(0.(p)‘3l{O.,¢ :K'—1(piH0_‘.i{O.w -K"1<@%><@%>"1@t(o%>v.Ifweintroduce aninvolution 9defined by agE(0?"0)‘1a%(0i’{0) Vaeat then Pg=TPT'1 with T=(o*’fo)_1K. Wehave J"(p3ip =T"1<pgi/1, thatis (<11,1/1);=(Q9,w):r V92.weEP- B_u} (‘Piall’)! (a3<P= 1/1)J and ((1%("WT =(450-9» I//)r=(ag¢9 111); giving ((423 —a~’)<p, 1/1),=0.Since thisistrue forall1/Jes4P andthe product is non-degenerate (ct?—ag)<p =0Vtpe .viP and Q?=agVae94. Recalling the definition of ‘J we have ai’=o‘1(oao"1)i”o, that is,Eand ‘:7{areequivalent. Toprove the cfonverse wesuppose that E=Si’fI{9’ '1forsome automorphism 9’.Then i (.):.iiiP ><.¢xP—-> P.v.slP isaproduct with 3fasadjoint-involution wedefine {,}:.v.(tP9"‘ ><.@4P~‘1”"‘—> P9’_'_;i4P5’_’ 68 CLIFFORD ALGEBRAS AND SPINORS by{asI3}=(W9,55”)?» then {O6~16}=(viim"ﬁ")"”’ =(m*’°”w*’» ﬁ*’)“”=([m*’°’<<"’¢rl5’, ﬁ*’)5’C’ =(mfg, I3}- Similarly if (I3,<1’)=8(<I»5)"I11@I1 {BiCY}=055’,¢Y*’)”“l =8(a<i’,I>’5’)"““ =£{asﬁ}5”‘*’”’ =@{<1sﬁ}’ where jES"k9"1 isequivalent tok.Thus wehave aproduct on.<;(lP9’ -1 with Easanadjoint-involution ofthesame type astheproduct ons5lP, which hasflfasadjoint-involution. Asalready noted aproduct onany given minimal leftideal enables anequivalent product tobedefined on anyother minimal leftideal. Thus ifEE8’fl{9"1 then Eand ‘flfare adjoint-involutions ofequivalent products onanyminimal leftideal. Although forcomplex matrices notallautomorphisms areinner a corollary totheabove isthat ifinvolutions arerelated byEE9’?l{9"1 foranyautomorphism SPthen infactthere isaninner automorphism E such thatEEE?7{E'1. 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/L, C®./I/L, H®Jl/1., reven §r+2 §r+3 §r+2 rodd §(r+1) +3) +3) Since notallClifford algebras aresimple wenow consider involutions ofsemi-simple algebras that have two simple components. Let 66E93C-B<6 where QBand <6aresimple with QBE.<2(lP, <6E.<>(lQ for central idempotents P,Qwith PQEQPE0and P+QE1.Ifflfis aninvolution ofatthen Pi“, Q3” are central idempotents with Pi”Q9< EQi’<Pi’< E0andPi“+Q?“E1.SoatE.viP"’{€)atQi’< and, since theexpression ofasemi-simple algebra asasum ofsimple ones is unique uptoordering ((A1?) ofAppendix A)then either s.¢Pi’< E93and .v4Qi’< E<6,or.v.4Qi’< E83andsi4P% E<6.Intheformer case ‘iiiinduces aninvolution onthesimple algebras 975and<6andmay thus beclassified inthemanner already treated. Inthesecond case every element of973is sent to<6,andthiscanonly arise when <6isisomorphic totheoppositeSPIN-iNvARiANT PRODUCTS 69 algebra of973,<6E9B°P. There isinfactonly onesuch involution, upto equivalence. Ifatisthesum oftwo simple algebras, with fitand E involutions that donotpreserve these simple components, then 3%andEareequivalent. (2.6.13) If‘Elfand Eareasdescribed then E315isanautomorphism of$6that induces automorphisms onthe simple components 975and <6.We introduce anautomorphism 9’of.94bydefining b9’=b‘¥”<? Vbe9B c5’Ec Vce<6. Then 3’isanautomorphism ofsélsuch that forbe93bi’?Ebi”,which isin<6,andsob9’9*"1 Ebay’ Ebi“.Similarly force<6 cg’?Ec3’and soc5’<’9’_1 Ecw-1Ec”'13’<_1 Ecf“and wehave established theequiva- lence ofEand315. When .94E9J@<)./I/L€-)9.D®A/L where JI/Lisatotal matrix algebra andthe division algebra satisfies 92)E915°?’ then anyinvolution intheclass not preserving thesimple components will becalled a<22)-swap. (The real division algebras IR,CandHareisomorphic totheir opposite algebras.) Anysuch involution istheadjoint ofanon-degenerate product onthe leftideal formed from thedirect sum oftwominimal leftideals from the two different simple components. For letPbesome primitive idem- potent (necessarily inone simple component). IfEissome 9.2)-swap involution then QEP+P9isaE-symmetric idempotent. We may deﬁne (,)I~@4Q><~ﬁQ-—>Q~v1QE9l>@95 0-’,I3'—>(<165)=aft?- Such aproduct isnon-degenerate forifa/<5’)8 E0V)8then choosing )6to lieinonesimple component shows thatthecomponent oforintheother must vanish, andhence 0:E0.Itimmediately follows from (2.6.14) that (or,m)6) E(mga/, 6’)Vmeai (I5,I1’)=(0%I5)? (Mi,I3)=qf(¢\6I6)VqE@®9B» Wecanequally well introduce aproduct with different symmetry. Ifsis anyregular element lying inQB,ss‘1 E16,then letSEs—s5.This E-skew element of§.D(-BQD5’ hasaninverse given by S-1Es“—(F1)? SS_1 Ess'1+(s'1s)§’ E16,+19,;E1.(26.14) 70 CLIFFORD ALGEBRAS AND SPINORS Wemay now define {,}I@4Q ><~v~(iQ——>9F><-99-Fl a/,Bii> {a/,)8}ES'1a/ff}. This product satisfies {anmﬁ}={midi B} {ﬁr a/} :—S_l{a= {M1,5}=5"q’5{a@ I3}- 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 Esuch that sfEs"Vse+1“. Of course §and $17aresuch 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+qEniseven, sothat C,,_q(IR) iscentral simple. Then ifEisany involution there exists some Jsuch that a9EJa5J‘1 with J5EiJ.Ifse+1“ then s3EJs5J*1 EJs‘1J‘1; thus sfEs"1ifand only ifJs‘1 Es“J Vse +I‘+. Weknow (from the proof of(2.5.3)) that .,I"' generates theeven subalgebra, and so sfEs‘1 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}. IfZ5E—Zthen, since J5EiJ,either JEIB giving EEEorJEA2 forAeIRandforanya,a3Eza5z_1E a5”. If25E2and22E-1then thecentre ofC;,,(IR) isC(IR). IfJeC then, since Cisalgebraically closed, JE02E00?‘forsome oeCand a3’Eoo§a’§o"="‘0‘1 Eo(o"“1ao)‘5o"1 showing thatEisequivalent toE. If25Ezand 22E1 then C,',",,(IR) isreducible and thecentre is spanned bythe orthogonal idempotents {P+,P_} where PiE %(1i2).IfJisregular then JEAP, +)uP_ with A,iinon-zero reals. IfAandiiareboth ofthesame signthen there isnolossofgenerality in assuming them positive since multiplying Jbyanelement ofthecentre does notalter E.Inthiscase weset JE0213+ +1/2P_ E(UP+ +vP_)(0P,. +vP_)§ andEisequivalent to<§.Similarly ifAandiiareofopposite sign then, with no loss ofgenerality, we assume JEo2P,. —v2P_. IfSPIN-INVARIANT PRODUCTS 71 kEUP+ +vP_ then kzkg E(0P+ +'vP_)(P+ —P_)(oP+ +vP_) Eo2P, —v2P_ EJ. Thus ac?Ekzk5a5(k§)*1z‘1k_1 Ek2(k*1ak)§z'1k‘1 Ek(k‘1ak)§’lk‘1 andEisequivalent toEr). Forthecase inwhich 25E2wehave seen that therequirement that s5’Es_’Vs6,1“ only requires Etobeequivalent toeither .7;orEr).If, however, pE0then +1“ contains anodd element and requiring si’Es*1Vsr§ ,1“: uniquely determines EtobeE.For ifJEA+U/Z then therequirement that Jcommute with anodd element forces iito vanish. Similarly, ifqE0then ‘gigistheunique involution such that sfEs"1Vse‘Ti. Weturn now tothecase inwhich nisodd, with {1,2}spanning the centre ofCp_q(IR), andC;_q(IR) central simple. Since 2”EEzoneofthe involutions Z,‘and517willleave thecentre invariant, theother willnot. It follows thatifEisanyinvolution then either a3EJa-5]'1with J9‘EiJ, ora3‘EJa5’lJ*1 with J5"Ei]. Consider theformer case. Then requiring sfEs‘1Vse+1“ shows, exactly asbefore, that Jmust commute with elements oftheeven subalgebra. Then ifJ,istheoddpart ofJwecanwrite J-E(J,z'1)z and, since theodd element zcommutes with everything, JWwill commute with theeven subalgebra ifandonly ifJ,z'1 does. Since the even subalgebra iscentral simple J_must beproportional toz.Itthen follows thatJisinthecentre ofC,,_q(lFi) andEE’§. Inexactly the same way itfollows that ifs5EJs§’lJ "1and s3Es'1Vse+1“ then EE517. Wemay summarise asfollows: if+1“ Qf+1“thensg Es‘1Vse.,l"iiffEE§ if“Ti ¢,1” then sfEs‘1Vse ‘Ti iffE EEr). IfsfEs‘1VsE+1“ then Eisequivalent to5orEr);ifnE4 mod4 then either EE5orEEE1). (2.6.15) The involutions 5and E17induce thesame involution ontheeven subalgebra. This istheonly such involution that inverts elements of ,.I‘+. For ifEisany involution ofacentral simple C;_q(1R) then a5EJa5J‘1 forsome even J.Thus sfEs‘1Vse +1“ only ifEisin the centre, giving EEE.IfC;,“,q(IR) isnot central simple then a9EJa5]“ with Jeven ifE5leaves elements ofthecentre invariant, orodd ifE§induces anon-trivial automorphism onthecentre. IntheWu 72 CLIFFORD ALGEBRAS AND SPINORS former case wehave seen that only ifEE5issfEs‘1Vse +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 fjandE27onthe factors ofatensor product induce EorE17ontheproduct algebra. In thisway knowing theclass ofEandE17onthefactors enables theclass oftheinvolution ontheproduct tobedetermined, and §and §i7on arbitrary algebras canbeclassified byexplicitly classifying these involu- tions forafewlow-dimensional algebras. First weconsider involutions oftensor products. LetsitE93®Jl/L,, where 95isasimple algebra over IRandA/L,,isthe algebra oforder nreal matrices. LetEbeaninvolution on.94that induces involutions 9onA/tnand fiton95.We shall write this as EEf7{®9. If93E§D®Jl/l,,, andI"J{is€ZD‘<-symmetric orskew then Eis certainly either 23"-symmetric orskew, thesymmetry being determined bythat offitand9,inaway tobedetermined. IfQisprimitive in93 and Risprimitive in./1/in then PEQR isprimitive insit.If Q?”EKQK‘1 and R9ETRT‘1 then P3EJPJ“ with JEKT. Since J5’EKi‘<Tg, Jissymmetric ifKand Tareboth symmetric orboth skew, andskew ifKand Tareofdifferent symmetry. Thus ifeither i’l{ isQB"-symmetric with 9IR-skew, orfifisQD’<—skew with 9IR-skew, then Eis9)"-symmetric; otherwise itis91"-skew. Wenow investigate the signature inthecase inwhich Eis93*-symmetric. If(,)1istheproduct ons.(iPassociated with Ethen forb,-,b1-e95andma, m1,eJ1/1,, (b1m,,, b1-m11)1 EJ"m,,.5’b,-3b,-m1, ET‘1K“m,,gb,-3<b1m11 I T_1ma€’HmBKT1b13[{b1 since 95andA/t,,aremutually commuting subalgebras ofsi.So (bimaa :(rnaa rnf5')T(bi'i where theproducts ontheright-hand side arethose onthesubalgebras associated with theinvolutions indicated. Thus if,forexample, (,)1~is 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 (,)1hasabasis ofrr’+ss'positive-norm andrs’+sr'negative-norm vectors. Similarly itfollows thatif(,)1is9%-skew and(,)1;is€D’<-skew then (,)1admits anorthonormal basis with asmany positive- as negative-norm basis vectors. Wesummarise thesituation below. IfEis aninvolution onst,where sﬂE93®JI/l.,, with EEi7{®9 then:SPIN-INVARIANT PRODUCTS 73 (i)ifeither iiiis9)"-skew with 9IR-symmetric, orflfis 91*-symmetric with 9IR-skew, then Eis9?‘-skew; (ii)if‘THisQbk-skew and9isIR-skew then Eis99"-symmetric, ofmaximal index (ifany); (iii)if‘fitis9)"-symmetric, with signature r,s,and 9is IR-symmetric, ofsignature r’,s’then Eis99‘-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(IR) ®C(IR) EC(IR) G-)(]R) identity ®identity Eidentity 69identity identity ®conjugation EC-swap conjugation ®conjugation Econjugation G-Dconjugation. (2.6.17) Weusetheobvious notation foraninvolution onareducible algebra thatinduces involutions onthesimple component algebras. Theabove canbeverified bychoosing aspecific basis. If{1,i},{1,j}arestandard bases forthefactors andPl.E%(1iij)then {P,., iP,.} and{P_, iP_} arebases forthesimple components. Similarly C(lR) ®H(]R) EC(lR)®Jl/l2(IR) identity ®quaternion conjugation EC-skew identity ®reversion EC-symmetric complex conjugation ®quaternion conjugation EC*-symmetric, zero index complex conjugation C9reversion EC*-symmetric, index one. (2.6.18) If{1,2}and{1,i,j,k}arestandard bases forthefactors then {e,-1} is anordinary matrix basis where enE%(1+2i), e22E§(1—2i), 921=J9ii =922] and912=-1922 =_9iiJ- Finalliﬂ H(]R) ®H(IR) EA/t4(IR) conjugation ®conjugation EIR-symmetric, zero index reversion ®reversion EIR-symmetric, index two conjugation ®reversion EIR-skew. (2.6.19) Again thiscanbeverified byconstructing abasis. Itissufficient tonote 74 CLIFFORD ALGEBRAS AND SPINORS that aprimitive isgiven byPE§(1+iI)(1 +jJ)where {1,i,j,k}and {1,I,J,K}arestandard bases forthefactors. Abasis fortheminimal leftideal inJl/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. IfsitE93®<6 and EE?7{®9 then table 2.12 gives theclass oftheinvolution Einterms of theclasses offitand 9.These classes areencoded into thetypes of table 2.11, with the(43symbol denoting aninvolution onadirect sum of algebras inducing involutions onthecomponents, andthetypes (8),(9) and(10) (see table 2.13) being 9D-swaps for9)EB,HandC.The class offitdetermines therowofthetable whilst 9determines thecolumn, theclass ofEbeing given intheintersection. (The symmetry ofthe table reﬂects thefact that s>§l®93 E93®si.) For example, (2.6.16) is encoded into thefirst two rows andthefirst 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 ofthefactors would beareducible algebra with more than two components. Tousetheabove toclassify theinvolutions .1;-'and E17onarbitrary Clifford algebras weneed tobuild uptheClifford algebras from tensor products ofsmaller ones such that thestandard involutions onthe factors induce Zjand 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 §andE17oftheleft-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(IR): EEEn(>95 (2.6.20) %E§®& Cp, q+4(B) 2 Cp, ®C0, ZEE®E (2.6.21) 5'1EEn@511the (Seetabe2llfor910910101010810810 factors8810101099 the of tensorproductsUUOHS\D<‘ﬁ<I'lr)t'\1-—-<O\ frominvo[\ﬂ'("'\l/§P"'*(‘\1U\ productsinducednvoutionson51010 5(4355510 5 I 3 4(-3436931034 ontensor4 4 3(4334634104310 tions3 10 nofnvouC\1<—iEl'€*"ilﬁ[‘-\O s'f'cat'o scheme.)CHS<-1C\I("\Yl‘lfi\Dl""- I e2.13Thecaton\1—<(\l(‘*")ﬁ'l-f')\C‘l'*‘~ cass'fTabO0 GO 10 10 10 O\ G\ O'\0)EC-swapU? ‘~_|/ 9)EH-swap,\-_/ (8)EIR-swap, 76 CLIFFORD ALGEBRAS AND SPINORS Cp+4, 2Cp, gxgag nan) $11=En®Err Here wehave used thesame symbol todenote involutions ondifferent algebras. For example, in(2.6.21) theEontheleft-hand side isthe standard involution onCp_q+4(lB) whereas thesame symbol onthe right-hand side denotes firstly thestandard involution onCp_q(lR), and then thatonC0_4(IB). 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(lB) isassociated with aspinor product labelled byanindex then ifq#0 that index is maximal, whilst forq=0theindex iszero. Similarly, any index associated with E17ismaximal unless p=0inwhich case theindex iszero. (2.6.23) Table 2.14 Involution classes ofsome low-dimensional Clifford algebras. N 5 En C1_0l[B 1GB1 C2,0(IB 1(zer0 index) C3_0|[ 5(zero index) C4_0lB 6(zero index) C0,1' C0_2| 6 C03‘. 6('36 CM, 6(zero index) C1.1' 2 .'I!.5§.'i1i?€§§--.‘I!c,c- 1-‘O\\O--JDJL11O\-Ikl\JO0 Suppose that (,)isaproduct onsome leftideal with Easadjoint involution, andthat {s,-} isanorthonormal basis with rpositive-norm andsnegative-norm elements. Ifqab0then there isavector xwith x2=-1.Then {xs,-} isanew orthonormal basis fortheleftideal and (xs,-, xs,~) =(xgxs,-, 5,-)=(xzs,-, .9,-)=—(s,-, 2,-).Thus this basis hasr negative- andspositive-norm elements and soifthesignature iswellSPIN-INVARIANT PRODUCTS 77 defined wemust have r=s.Soifqi0theindex ismaximal. That itis infact zero forq=0can beverified byrepeated useof(2.6.22), together with table 2.14. The involution E17istreated inexactly thesame wa. Wecannow give theclass ofEandE17forallCp_q(lB). First wenote thatthisclass only depends onpmod8 andqmod 8:twoapplications of (2.6.21) and(2.6.22) enable thistobeinferred from table 2.13. Wehave given theclasses intable 2.15. Tocomplete thistable weuse(2.6.21) and (2.6.22) with table 2.14 tocomplete thefirst row and thefirst column. Then theclasses ofEandE17aresimultaneously entered inthe diagonals byusing (2.6.20) with themultiplication oftable 2.13. The third entry foragiven pand qintable 2.15 gives theclass ofthe involution that Eand E17induce ontheeven subalgebra. (The case of p=q=0isadegenerate case forwhich theclass is1.)Reference to thederivation of(2.3.1) gives C;“"’(R) 2C”'(B) (2.624) E=511- Thus theclass oftheinvolution onthesubalgebra isobtained from a relabelling oftheclassification ofE17. When p+qs5wecanusetheclassification oftheinvolution Eon theeven subalgebra toobtain +1“ asthegroup ofautomorphisms of theassociated product. This follows from (2.4.22). 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. For aC*-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=Othen wesimply write U(r) and Sp(r, H). The products associated with the21>-swap have thegeneral linear groups asauto- morphism groups. For, taking the product of(2.6.14), ageneral element oftheautomorphism group isS=s+s'1<f with sanyregular element of2D®A/L. Wehave arranged these spin groups intable 2.16. From this table wehave, forexample, forC3_1(IB) +1“ =Sp(2, C) whereas attheend of§2.4 wedemonstrated that +1“ =SL(2, C). These groups areisomorphic; infactwehave Sp(1,H)=sutz) Sp(2,IB)=SL(2,1B) (2.625) sp(2,c)=SL(2,c). Itcan beseen that intwo dimensions +1“ isisomorphic tothe orthogonal group. _mWNu+QﬁgECZD___Whe+5Quam66%“E55mEgg6%m5;}highs“;gigX5;} Qg“%_rﬂ83}Ami“;@Ejgwgnsﬁﬁxgrsg€_h%%:5@ *m_Hcfrgm6h%mEngCQNEWAmimmxgragmgag:5 ?%HOm_Q@NHQN‘ Q Edo_H__wr__gamé8:Qﬁawlgav>“OE:EdguegivmA8GGOGCUEWU55E3:8BgwlgQVAm__O_m:0E_UasE3:838_En__ASgnaw‘:HaU_%HUEE%%|<:At>U_5U:_B__O_:0:_;__;w|*QheUE®EE\$|QHQ>52:BhUE®EE\$_m_A: wNQ0Qgm__MWNﬂu©AwF@@®@N@W@@®NN@FmuN@®®N@W@MWH_m F®F5\‘®WW FN_®§5§@ “N@§® C©@®HF©@I_¢ Q® QE2IdﬁtpuwjwwarmsE2OEno_5_;_o>Ewemamall gaymgQENHi|lI HQ@MW_%5@©©in@MW_MU @W_~HHMW@ @F®§W@@®®NM5F®§5@NNAIWN@WN_W®N‘W@Nhm5@_©V@MWMUW@©¢NMW_Mm5Am©in@_@©©®©©@H_@©©®©©@H_®_M@©©®©©@ MW@ﬁ©QWmw F©@HH®HH@©©@HH®H_@©©®©©@H_®_H@ ﬁ@WAm_MW®QC@WAWHMWN N@®N @©®©@©®© ©©__u_ ©Q@@@@H@_wH®?_%®H %?QW©@_J@NH CW QAm?“ivWﬁnowa20:20E805ac=2;_O>EMo$320l __5>_wUs_wB®m_g5w552:COUUSUESE__O_;_O>E2:Earm“WB$ww_W_O2:MyUSNQﬁgBL_m§5%__W_REEDE22:U6m__O_;_O>EMOwgmﬁom__N2H__; l‘l‘‘ll‘‘‘ l 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 thereal field 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 onrt,anditwillbedenoted C,,,(C). The structure ofthealgebra canbedetermined asin§2.2, only here thesituation iseven simpler. Wehave C..+2(@) =CM?) <>i<>C2013) (2-T1) c,(c)=ceac (2-7-2) Czlc) =~M2(C)- (2'7'3) These aretheanalogues of(2.2.8), (2.2.2) and(2.2.3) andthey may be proved inasimilar way. They givethestructure ofallC,,(C). Ifniseven then . 2M2”/2(C) whereas ifnisodd Cn(([j) =)|/12i.._1)a(¢1f,‘)_ (2.7.4b) The structure oftheeven subalgebra follows from theanalogue of (2.3.1), namely c,;(c) =c,,_,(c). (21.5) Ifniseven then C,§(C) =./l5l,2n-*2—l(C)GB-/m.,2n»'2—1(C). (2.7.6a) whereas ifnisodd C,f(C) =./l/l.2{n—ll!2((:). (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 algebraTHE 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(t/C, gr?)=c(v,g)<>;<>cm). (2.7.?) For C(V, g)®C(lR) weshall write C‘3(V, g).Wemay define the conjugate-linear operation ofcomplex conjugation, *,onVC:if2eV‘: then z=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 will not, however, necessarily simply complex conjugate the components ofthese matrices. Thesituation isclarified below. Suppose that s.4(lB) =C(lB)®Jl/L,(IB) has some involutory auto- morphism, *,thatinduces anon-trivial automorphism onthecentre. Let 9/3betherealsubalgebra, that isae95ifandonly ifa=a*.Since any aesdcanbewritten asasum ofrealandimaginary parts itfollows that $4=C®?B. Sowehave C®9B =C®Jl/1,. Forthistobetrue must certainly besimple, and since theonly simple real algebras areiso- morphic to§ZJ®A/t with Qb=1B,CorHwemust have either 973=./I/t,or QB=H®J|/1,0. Bywriting sﬁ=C®Jl/L, forsome particular matrix sub- algebra A/L,wecan define another involutory automorphism #that leaves elements ofA/1,invariant andconjugates elements ofthecentre. If{e,-J,-} isanordinary matrix basis forA/L,then {e"1,-}isanother ordinary matrix basis, forA/t,’ say. Itfollows from the uniqueness ofthe Wedderburn decomposition ((A24) ofAppendix A)that e"§-,-=me,-,-m" forsome mesit.Soif Q:=-2?: Q0125 L;=l then I’ I’ ._ -1_ -1a*-2a’j;,'me,-J-m —m2a”j--e,--m Lj=l hf: thatis 41*=ma#m_1. (17.8)IJ |1 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'1(m*m)m. That is,mm* =m*m andp isinfactreal. Thedefining 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.) We summarise asfollows: 9.4=973®C with *anauto- morphism that conjugates Cand leaves QBinvariant, and at=Jl/t,®C with #anautomorphism that conjugates Candleaves A/L,invariant. The twoautomorphisms arerelated bya*=ma#m‘1. There aretwopossi- bilities for93,either 93=A/L,or93=H®A/t,,2; andtwopossibilities for m,either mm* =1ormm* E—1.These possibilities areinfactrelated. Ifail" C®?/3 =C®./I/L, with *and #automorphisms that conjugate thecentre and leave 973and A/L,respectively in- variant, then a*=ma#m"1 where wecan choose either mm* =1<=> 93=A/l,,ormm* =—1<:> 93=H®M,/2. (2.7.9) We now consider theproof. Since there aretwo and only two mutually exclusive possibilities for973,and similarly form,ifwecan prove that mm* =1<=>933=./1/t, then we must have mm* =-1<=>97$=[‘I®./l/l.,)2. Suppose firstly that 933=./I/l,.Then if{b,-,-} and {e,-,-}areordinary matrix bases for973and A/L,respectively then e,-I=sb,-J-s‘1 forsome sead.Ifwewrite a= I»! then a*=2aj§(sb,-J,-s'1)* =Ea:-‘J§~s*b,-I-s*‘1 if ii Q 5 =_2’a§j-s*s'1e,-’,~ss*'1 =s*s“a#(s*s‘1)*1. 1-] That is,wemay choose m=s*s‘1 giving m*=m‘1. Now the converse: weintroduce aC-conjugate-linear transformation onsélby defining a°=a*m =mai‘. Thus °preserves thecolumns ofA/t,. If m*=m‘1 then "isinvolutory and forany aest’! wewrite a= %(a+ac)+%(a—a°). Inparticular, theminimal leftideals of.94that arethecolumns ofA/L,with entries inCcan bedecomposed into eigenspaces of°.Since therealdimension ofaminimal leftideal ofsiis 2rthese eigenspaces arer-dimensional. Letipbeanelement ofoneof these eigenspaces. Then ifae9B mpiscertainly intheminimal left ideal ofR4andsince (at/1)“ =a*i/1° =at/1°itisinfactintheeigenspace. Hence these eigenspaces carry representations of9B.That is,if m*=m‘1 then irreducible representations ofsdinduce reducible repre- sentations of$3.But either 933=Jl/L,, inwhich case itsirreducibleE.4 5v.i: ii Y ...-.-...-......-....»--“J........._.,.i E i 2 =1 .!i IiTHE COMPLEXIFIED CLIFFORD ALGEBRAS 33 representations arer-dimensional, or93=H®Jl/t,,2 with 2!‘-dimensional irreducible representations. Thus m*=m"1 implies 93=Mr,The argu- ment oftheproof issummarised below. m=s*s_1=>mm*=1 :> U93:./l/L, <1irreducible representations of$4 induce reducible representations of 973. Hence mm* =1<:> 93=A/L, andsomm* =—-1<:> 923=H®A/tr/2_ Thefcomplexification ofthereal Clifford algebra associated with an even- imensional orthogonal space 1Sisomorphic tothealgebra of complex ITlElt.I'lC€S.. Complex conjugation (that leaves thereal Clifford algebra invariant) isequivalent totheautomorphism thatconjugates the components ofthese matrices when thereal algebra isatotal matrix algebra: inthis case complex conjugation will simply conjugate the comppnepts manappropriate basis. The algebra associated with the comp qxiication ofanodd-dimensional realorthogonal space isadirect sum otwomatrix algebras. Complex conjugation isequivalent tothe automorphism that conjugates thecomponents ofthese matrices ifand only iftherealalgebra isasum oftwototal matrix algebras. When the Heal algebra IS.thesum oftwo simple algebras whose Wedderburn decomposition involves thequaternions then complex conjugation in- uces anautomorphism onthesimple components ofthecomplexified algebra that isinequivalent toconjugating thematrix components. )2/hen therealalgebra isisomorphic tothealgebra ofcomplex matrices _encomplex conjugation ofthecomplexified algebra interchanges the simple components. The irreducible representations ofthecomplex algebras willagain be called spinor representations, orsemi-spinor representations when the algebra 1Sreducible, thespinor (orsemi-spinor) spaces being identified with minimal leftideals. These minimal leftideals areobviously of ailmplex dimension 2[”’2_l where [ri/2] denotes theinteger part ofn/2. Ienniseven.there. 1Sonly one such representation, uptoequiva- ence, whereas ifn1Sodd there are two inequivalent semi-spinor representations. II‘I'€dL1Cll)l6. representations ofC(VC, gc) 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 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”’2l; 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(2l‘”‘1)’2l), whilst thedimensions ofthose oftherealeven subalgebra aregiven in table 2.10. Weturn now toclassifying involutions ofthecomplexified algebras. TheC-linear involutions EandE17induce 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 ofEand 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 onnmod 8,and are given intable 2.17. The involutions Eand E17commute with complex conjugation, andsothey may becomposed with ittoform involutions E*and E1j*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) thatE*isthe adjoint ofazero index Hermitian-symmetric product ifand only ifTHE COMPLEXIFIED CLIFFORD ALGEBRAS q=0;otherwise anyindex ismaximal. Similarly Eij*istheadjoint ofa zero index product ifandonly ifp=O,otherwise maximal. Table 2.17 Classification ofinvolutions ofthecomplexified Clifford algebras. P1q=,, 5 57, Eonc;_,(ia) (ac 3€-)3 10 3 OO\10'\U1-l>UJl\)l—'* DJ-I-RLa-J DJ-ll;-Flr—\Q 10 4C-B4 4 4®4 4694 10 4 4 3 10 10 3693 3 3(-B3 Table 2.18 Theclasses ofE*andErj*onCp,q(lB) ®C andE*onC;,q(IPi) ®C. Pq 0 1 0 5 10 5 5®5 SC-B5 5 1 SC-35 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 andnotation 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 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 r“i"’+vb?“=211'” (28-1) where 1]isdiagonal with pentries ofplus oneandqofminus one. If p+q=nthen theorder ofthese marices is2l”’2l with thebracket denoting theinteger part. These matrices arealso usually assumed to have certain Hermiticity properties, andweshall examine thisshortly. Here wenote that thepresence ofsuch operations thatarenotC-linear issufficient toinfer that they-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 Cg, =A/1.2»/2(C). If{e“} isabasis for therealvector space thatgenerates CM(IR)then n/2 ea 2 Yijeij i,j=1 where {e,-,-} issome ordinary matrix basis forthecomplexified algebra. The arrays ofcomplex components, )4},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{ea}; 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 ofCp*1(]B), 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 =Jl/l2ii»—1)e(ll:) (‘B./l/lzip-i>i2(C) and thus Cf’, isisomorphic toatotal matrix algebra with C,,,(,(C) a subalgebra isomorphic tothedirect sum oftwoalgebras ofmatrices of half the order. Inasuitable matrix basis, therefore, C50 isthe subalgebra ofelements whose matrix components areblock-diagonal; thetwosimple component algebras having matrix components inonly theupper orlower blocks, thatisTHE CONFUSION OFTONGUES 87 ,- 0‘ 0 .y=(0E,-) i=1,...,p with 0’andZ’matrices oforder 2(P“)’2. From table 2.18 andtheremark attheendof§2.7 itfollows that the involution E*onC§_0 istheadjoint involution ofazero-index C*- symmetric product; that is,itisequivalent toHermitian conjugation. Thus wecanarrange abasis inwhich E*onCf,induces Hermitian conjugation onthediagonal blocks (but not, ofcourse, ontheoff- diagonal blocks), and insuch abasis ofand E‘areHermitian. If Iz’=/lel el’with)L=1or isuch that §2=1, andPl.=%(1i Z), then P,and P_aretheidentities inthesimple components ofC50. Since E=P1—P_then if)’/=/lj/1...j/Pthen all3)»But e0anticommutes with Zand soyocan only have off-diagonal components. Since also(e°)2 =—1wemust have ,y,:( 0T) (—T_1 0 forsome non-singular matrix T.Since e0anticommutes with alle‘we must infacthave E’=——T“oT. Ifwenow change basis sothat the components transform ya—>S}/“S” with 1|—iT ___1_ | I Szﬁli it) S1_\/2liT1—iT”1l then wearrive atthefollowing ‘standard’ representation ofthey- matrices: .l 0 - 0 ‘Y0"-=1(0 _|) y’=(0,- 00). (2.8.2) Here 0‘,and hence yl,areHermitian whilst 1/0ismanifestly anti- Hermitian. The case ofCfq may betreated similarly. Since Eij* is equivalent toHermitian conjugation incg,wearelead toa‘standard’ representation asabove, but with theofanti-Hermitian and the 1' removed from yo.(Inthis case e"denoting theone positive-norm vector.) Toillustrate further therelation between the'y-matrices andthemore abstract approach toClifford algebras that wehave pursued, we examine cg},inmore detail. First weshall choose amatrix basis fora simple component ofCgj,inwhich E*coincides with Hermitian conjuga- tion giving theHermitian {oi}. Wethen have from (2.8.2) astandard representation ofthe y-matrices and shall reverse theargument to 88 CLIFFORD ALGEBRAS AND SPINORS construct thematrix basis inwhich these arethecomponents ofthe {ea}. The reducible algebra Cg,isprojected intosimple components by the mutually commuting pair of central idempotents Pi= §(1iiem). Since e123Pi =$iPi, giving e12Pi =-Tie3Pi, ifwewant 0102 =io3then the{of} must bethecomponents ofthe{el}inC§‘,3_@P_. Tostart theconstruction ofthematrix basis weseek apairofmutually orthogonal idempotents thatareinvariant under theinvolution E*:these will form thediagonals ofabasis inwhich E*induces Hermitian conjugation. Wechoose 911 = + €3)P_ 822 : "“63)P_. andsince e22=eze1,122wemay complete thebasis with(2.83) 912=91192 =@2922 (2.8.4) 921=@2911: 92292 where 921’? =912- itF-H $.54...N<1.6’-PL = at-;~€a,5 then2 o‘,,.j,- =2s)a.e‘sj,). 1:1 Forexample, j (7112 231191321 'l'32191522 =_i51i@23£21 *“i*‘521@23522 since e123P_ =iP_ __-2 -2—18116 £21+18218 £22 where thee2hasbeen absorbed into £21and£22.From thedefinition Of£21 . (7112 =i(~‘511+ 522) :iP- where, werecall, P_istheidentity inthissimple algebra. Inthisway weconstruct thefollowing: (0i 01 10 i__ 2= 3:0-\_i 0) 0' (10) 0 (0_1). (2.8.5) Wemay usethese matrices in(2.8.2) toobtain astandard representa-THE. CONFUSION OFTONGUES 89 iion ofthey-matrices. Atthis point wereverse thereasoning and construct thematrix basis corresponding tothese components. From the diagonal yoweconstruct apairof(non-primitive) idempotents, U 1 %(l+i1’0)=01 %(l-iv“)=10_ 1 0 Putting (2.8.5) into (2.8.2) enables another pair ofidempotent matrices tobeconstructed 0 1 %(|+i1"r’)= 10 %(l-11/11") = 01- 1 0 Primitives areobtained from thefour products ofthese two pairs of idempotents, forexample 1 i(|*iY0)i(|-iY1Y2)=( °0 O Ifthen ‘;:jMr‘<G_.. 1'1 6 — if wehave 911=i(1_ie0)i(1 _i912) 622=§(l—ie°)%(1 +ieu) 933=%(1+i@0)i(1 _i912) 644=§(1+ie°)§(1+ie12).(28.6) Inexactly thesame waywetake products ofthe‘y-matrices toproduce a matrix ofzeroes except fora1inthe1',jentry, foralliandj.Asmay readily bechecked thisleads totheconclusion thattheremainder ofthe matrix basis must beasshown intable 2.19. Inodd dimensions there aretwo inequivalent representations ofthe y-matrices: these being thematrix components ofthe{e“} projected into either ofthesimple component algebras. Wenow show how a standard representation canbeconstructed forC51, where now pis even. Inthiscase Cg; =Jl/l2pr1(C) Q3./l/l2p='2(C) 90 CLIFFORD ALGEBRAS AND SPINORS THE CONFUSION OFTONGUES 91 WhereaS_C.e0 :M"2g’(C)' The ievelutien e-*isequivalent toHegmitian The diagonals inthematrix basis areasetofpairwise orthogonal conjugation onCM, whereas itswaps thecomponents ofCM. We primitive idempotents, andthese areallSimilan S0 choose amatrix basis {e,7}forCfj,inwhich E*coincides with Hermitian conjugation. IfPiarethecentral idempotents that project C,ii.i into 9‘0(9))') =9‘0(59ii-9-1) simple components, ande,-I-1“ =e,-I-Pi, then thee,-ff form matrix bases E. forsome S,thus forthese component algebras. Theinvolution ‘risdefined onCg,bythe requirement that itconjugate thecomplex factor andsatisfy thefollow- 9‘0(9)1)=9‘0(91'1‘)- ingproperties onthegenerators ofthereal subalgebra: e"=e’,1'=1, Bywriting thgidentity asaSum ofprimitives ...,p,eel=-—e°. Thus Tcoincides with E*onCej, andsoisindeed P . Hermitian conjugation inthebasis {e,;j}. Ifz=e1...epee then for i 1=911+922+---+9” With F=2"/2 p=2mod4, Pl.=§(1+i iz), whereas forp=0mod4, Pi= Wehave 50(el.l.) =1/(22/2)_ Thus. %(1iiz).Now certainly z=—z5*, andaswehave remarked E*swaps N thesimple components ofCg}, that isPie‘ =Pl, thus P;=Pi. It 3>0(a) :_% 2% follows that, asthenotation suggests, ’(induces Hermitian conjugation inthesimple component algebras inthebases thatwehave constructed, . thatis, {e,-,-1'}. Insuch bases e‘Pi arerepresented byHermitian matrices, whereas e°Pi isrepresented byanantiHermitian matrix. Infact for p=Omod4 e°Pi =$ie1 ...ePPi, whereas for p=2mod4, 1 e°Pi =Te‘...ePPi.1 33001) =“T7 Tra. (2.8.7) Inodd dimensions weletPl.denote thecentral idempotents. If,for example, {e,-,-*} isamatrix basis forthesimple algebra whose identity is Table 2.19 Amatrix basis forCE1. I P+then _ i P+=e11++...+e,,+ r'=2(”_1)/2. Since 9’0(P+) =§wehave 9’0(e,-,-‘) =1/[2(2("“1)'2)] giving 8’(aP )= 2 2:2 is:2 2isF {1/[2(2<"f1>’2)]} Tr(aP+)_ Thu, °*611 #323822 63633 —i61644 623611 622 181633 ‘@3644 3 ._' 1 _ 23.6611 IE822 633 6644 2(2(n_1)/2) + _ 161911 -639 22 @23923 944 '9t;—> H 3’O(a) =——-li[Tr(aP )+Tr(aP (2.8.8) ' 1 thIncalculating cross sections inquantum theory oneuses various trace A eorems forthe}/-ITl8.tI'lC6S. The following illustrative properties ofS0 algebra onto thesubSPace Spﬂnned bytheidentity- There istherefore a areequivalent tosome ofthemost imP°Htant- If{ala ---,61,1}iS8Set relation between theProjection oftheClifford algebra onto theSpace of ofLferms then 50001102 ---an)=0f0fFlOdd, and 5"@(a1a2 ...an) 0-forms, 300,and thetrace ofthey—matrices. Ineven dimensions any :Sue" '''Q2611)‘ These follow from the more general relations element canbeexpanded inamatrix basis nee’ 2SUM’ eypZSpe The e'f°Tm °°IT1P0I1@11i OfE1preduet ofnThe representation-independent operator trace, Tr,projects amatrix 2.12 I 1‘f01'1T1$, With fleven, canberelated tothat ofproducts ofn—2 terms, a:Zzaijeir I from (2.1.7) -.-...n..i.-.---1 L" y0(a1a2-- an)=5"0{a1 /((1:12 ...an)+i5-1(a2 ...a,,)} Since thea,-,-are(complex) 0-forms §2,112 Y.....-4.nuL'4 990(9) :_2_a1)'390(9i,=')- 5 Q2___an_lié.a }1; 1H andsince products canbereversed under S0,(2.1.17), =8011» a2)5“0(¢136l4 ---an)—g(ﬂ1, d3)Ef0(a2a4 ... 3)0(9ij) =5e0(91i9i)'9,»',-') =5e0(9i;9,i,-'9i'i) =5f0(9ri)5i)'- E an)+___+g(a1, an)_q>0(a2a3 ___an_1)_ i=3i0{15,£12a3...a,, —a2i,~,-]a3a4 ...an+...+ 1 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 ofa2W2]-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. IftpeC,,(C)P with Pprimitive then wemay form a matrix basis, {e,»,-},with ell=P,giving 1/1=2,-1//,-e ,1.Ifmp’=tpS“1 for some invertible Sthen 1;)’liesinthefirst column ofthematrix basis {ej-1} where e},=Se,j»S'1. Ifwewrite 1/1’=2,-1/Jjej-, then ifS-1= ZIp,qS;,,1e’,,q wehave 111;=E),-S,}1tp,-. 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, 1/3,isa‘row’ spinor which enables spin- invariant products tobedefined. Thus 1])istheadjoint oftpwith respect tosome spin-invariant product, itbeing anelement ofthedual space carrying acontragradient representation. From table 2.18 weseethat unless pisoddandqiseven theinvolution §17*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‘tbethe involution of Hermitian conjugation. (Inodd dimensions Tinduces Hermitian con- jugation inthesimple component algebras.) Then Tisrelated to517*as follows, aw=Aa*A '1 VaeCg, (2.8.9) with A5"* =A(equivalently AI=A). Ifqr)and 1/1areDirac spinors, lying inthefirst column inthematrix basis inwhich ‘risHermitian conjugation, then wemay define aspin-invariant product (<0.1/»>a»i=A-1¢¥'**w- <2-8.10)This product, which having §1)*asitsadjoint involution isinvariant under +1“, isaspecial case of(2.6.2). Assuch ittakes values inthe algebra ofcomplex numbers whose identity istheprimitive e11.Wecan trivially obtain aproduct with values intheunderlying complex field. POTif((P>w)§ﬁ* =<99»1/)>e11 then ((1%ll!)ITYUP» 1P)§n*- (2-8-11)THE CONFUSION o1=TONGUES 93 The adjoint of1/1with respect totheproduct in(2.8.10) istheDirac adjoint, thatis J»=A"11p§"* =1p“A"1. (2.812) The defining relation forA,(2.8.9), involves Hermitian conjugation which isdefined insome matrix basis, {ea}. If{ej-j} isanother matrix basis with ej-1-= Se,-I-S*1 then ejf=S““e,-,-SI. Thus ifSl=S"1 then ej-J,-I=ej»,-andtheinvolution Ialsoinduces Hermitian conjugation inthis basis. Soinfacttheinvolution I,andhence therelation (2.8.9), involves aclass ofbases theelements ofwhich arerelated byunitary transforma- tions. Suppose that weconsider that class ofmatrix basis forCg, in which em=—e°, e”=e‘,i=1,...,p.Equation (2.8.9) isequivalent toe“*=—A'1e"A, andsoinsuch abasis wemay choose A“ =ie°. This gives thefamiliar relation 1])=itpleo. (2.8.13) (The factor ofiisabsent inthecase ofCfq.) 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. Thenotable exception tothis restrictive definition oftheDirac adjoint isthebook byJauch and Rohrlich [4]. Intheabove weexcluded thecase inwhich pisoddandqiseven. In thiscase itis§*,rather than 517*, that istheadjoint involution ofa pseudo-Hermitian product. Unless piseven and qisodd inanalogy with (2.8.9) wemay define 615*=Ba*B"1 VaGCgq (Z.8.14) with B‘?=BI=B.Instead of(2.8.12) wedefine 1}]=B“1/15*. (2.8.15) Here theDirac adjoint isdefined with respect toa+1“-invariant product. ADirac spinor anditsadjoint areused toform theso-called bilinear Covariants. IfcpandtpareDirac spinors then, asexplained attheendof §2.5, thespinor representation gives rise toarepresentation r.We deﬁne T(S)(<P1l3) =S<P(~<'?/)- When, forexample, theDirac adjoint isdefined asin(2.8.12) then 'v(s)(<ptI1) =s(q>1Y1)s5"*. Thus forse‘Ti therepresentation rcoincides with thevector representation, thatis T(S)(<P1/7) =S(<P'l_1)-F_1- 94 CLIFFORD ALGEBRAS AND SPINORS When qisoddtheimage of“Ti 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 (pt/-1asa sumofp-forms, W=E9’@(<Pv@A§)@"A =2)sr,(q3@,,§<p)eA (by(21.17)). This gives thep-form components interms oftheproduct (2.8.l0), or (2.8.11), vi»=2<¢.@Ag(P>5P0(911)eA- (28.16)A Fortheparticular case ofCg,wehave 4%P@(1/11/7) =Trtwvi) 49’1(1/It/-1) =T1'(1/_1@“w)@a 4392(1/117)) =iT1'('f’@ab‘P)@ba (2-8-17) 49’3(W) =Tr(w@“Zw)@.iZ 49’4(1/1117) =—Tr(1Y1Z1/1)2 - The components ofthese homogeneous forms arethefamiliar scalar, vector, tensor, pseudo-vector andpseudo-scalar. Aswas noted above, thespinor representation on1/1induces therepresentation ronthese bilinears. Inparticular, thespinor representation of‘Ti induces the vector representation onthebilinears, theimage of*1“: under the vector representation being the group 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 OFTONGUES TheDirac adjoint isassociated with thepseudo-Hermitian product for which 517* or5*,istheadjoint involution. We also have thespin- invariant products forwhich theC-linear involutions E17andEarethe adjoints. From table 2.19 weseethat unless n=1mod8 or5mod 8the involution §17induces aninvolution onthesimple components ofthe reducible Clifford algebras. If9'denotes transposition insome matrix basis then, excepting thedimensions mentioned, wehave a5”=Ca*7C"”1 VaeCg, (2.8.18) with C5"=C‘7= iC. The symmetry ofCdetermines thesymmetry of thecomplex bilinear product defined by (Q9,1/1);” =C”“1q95'”1p. (2.8.19) Here goand1/1areDirac spinors lying inthefirst column ofthematrix basis inwhich FTisthetransposition. The symmetry ofthisproduct, for which E17istheadjoint involution, isgiven intable 2.1. The defining property ofC,(2.8.18), isequivalent to eag=-C_1e“C thematrix components ofwhich areusually taken asthedefinition of thecharge conjugation matrix. If1/1istheadjoint oft/1with respect to theproduct in(2.8.19) then {J=c—11/ﬁr =qﬂc-1. (2.s.20) This adjoint spinor isoften called theMajorana conjugate. Except forn=3mod8 or7mod8 theinvolution 5induces an involution onthesimple components ofthereducible Clifford algebras. Wemay define a5=DagD‘1 Va6CE’, (2.8.21) with D5=Dg=iD. This gives eag=D“1e"D. Thesymmetry oftheproduct defined by (‘P9wk=D“¢>iw (28-22)I Pu lsgiven intable 2.18. Weshall also use1/1todenote theadjoint with respect tothisproduct, specifying therelevant product whenever con- fusion islikely. In§2.7 wewere careful todistinguish theautomorphism *,referred to ascomplex conjugation, from theautomorphism #.Complex conjuga- tionleaves invariant therealsubalgebra generated bytherealorthogon- Ellspace with signature p,q,whilst #isdefined tocomplex conjugate thematrix components insome matrix basis. Thus thedefinition of# 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 ‘lVa. 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 real subalgebra isisomorphic tothealgebra ofcomplex matrices when p—q=3or7mod 8.Inthiscase complex conjugation ofthecomplex- ified algebra swaps thesimple components. Save forthis exceptional case weusethisrelation between thetwoautomorphisms todefine the charge conjugate spinor 1/1° 1/1°=1/1*m. (2.8.23) This canberewritten interms oftheDirac adjoint and thecharge conjugation matrix. Unless n=1or5mod 8,orpisoddwith qeven, wemay use(2.8.9) and(2.8.18) toproduce (ag7i*)’§-77 : *1)“§77CaJT§J'C_1A§7]_ Since complex conjugation commutes with the involution $11and T9=gt=#wehave a*=ma#m'1 with m=A'1*C (2.8.24) where wehave used AW =A.Weknow that wecanscale msuch that mm* =i1,which canbeaccomplished bychoosing Csuitably. With m given by(2.8.24) equation (2.8.23) becomes qr=cl/3? (2.s.25) Inexactly thesame way, except forthecase ofn=3or7mod8 orp even with qodd, (2.8.23) canbewritten as 1/1°=D1/-15 (2.8.26) where now 1/1isgiven 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=0,1,2mod 8,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 OFTONGUES 97 Thus aMajorana spinor isaneigenspinor ofthecharge conjugation Operation tp=iqﬁ. (2.827) This canbewritten interms oftheDirac andMajorana conjugates by using (28.25) 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 2’=zor2'=izensuring E2=1,z denoting thevolume n-form. IftpisaDirac spinor then itmay be decomposed into subspaces that transform irreducibly under theeven subalgebra, tp=111++tp_ where 1}/i=Pimp. (2.8.28) The semi-spinors 1/):arecalled 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 thereal central 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. Forp—q=7 mod8 Cp,q(IB) =C®./i/i2t~—1>12(]B) and C;‘q(]B) =./Vlgrn-1>r2(]B) where p+q=n.Wemay thus choose amatrix basis fortheClifford algebra inwhich theautomorphism rjsimply complex conjugates the components. The complexified algebra Cfjq isreducible, with 17inter- changing thesimple components. Complex conjugation, *,also swaps 98 CLIFFoRD ALGEBRAS AND SPINORS thecomponent algebras. The automorphism 17*willcertainly preserve thesimple components, andinasuitable basis weseefrom (2.6.17) that itcoincides with #,theoperation that complex conjugates thematrix components. ADirac spinor ipcanbedecomposed into spinors trans- forming irreducibly under therealeven subalgebra 1/»=1/1++1/1- withwt=in/11w"*)- (2-829) (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. Let a/,[3,tp,cpbeDirac spinors lying insome minimal leftideal projected bytheprimitive P.IfMandNarearbitrary elements ofthe Clifford algebra then 6’Mﬁ1T»N<r> =<2S@(Mﬂt/7N@xi)@”‘<P (by2-1-18) =5-’@A<P50(Ml6TI/N@Ag)- Theterms inthebrackets canbereordered using (2.1.13), andsince ¢=¢P 5zMj6171Ncp =6'e"‘<pS0(1I1Ne,, §Mj8)P. Now theterm inbrackets isinPC§,qP, which is‘isomorphic tothe algebra ofcomplex numbers with Pasidentity. That 1s,PXP =/1Pforit acomplex 0-form, giving S0(PXP) =/lS0(P) and S0(PXP)P =S0(P)PXP so 6./Mﬁtjl_1N(p =ri/e“’(pt'j)NeA ‘5M)8S0(P). Interms oftheproduct in(2.8.11) wehave <¢1’»Ml3>(1//»N<P)P =(as@A<P)(1P= N@AiMﬁ>5@(P)P whose 0-form component isthebasic Fierz formula (asMl3>(1l1»N<P) =<0-’=@A(1P=N@AEMl3>50(P)- (2-8-30) (The factor ofS0(P) arises from ournormalisation oftheeA.) o Theapproach tospinors thatwehave pursued isessentially algebraic. From theClifford algebra wecandefine thespin groups, andfrom theTHE CONFUSION OFTONGUES 99 1"epI‘€S€I1'[&lIiOI1S 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) oneintroduces ‘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 that the 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 for0.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. One canextend 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 §2.3 thatC§_1(lB) =C®JI/t2 andthatif{sag} isamatrix basis there exists a2-form crelating transposition, t,totheinvolution E a5=ca’c"1% 100 CLIFFORD ALGEBRAS AND SPINORS andal-form xthat squares tooneandcommutes with thesat,andc. Thus cmust have real components and, since itiscertainly anti- symmetric, wecanchoose itsuch that itscomponents form thestandard symplectic matrix, thatis C=2158,58,, (28.31) where thematrix ofcomponents cab,is 8,,=(_§’ (28.32) The even subalgebra ofthecomplexified algebra isthedirect sum of twoalgebras ofcomplex order-two matrices. C51’=A/lz(°3) <9MAC)- IfPi=§(1iiz), with zthe volume 4-form, then {s,,,j,P+} and {::,,,j,P_} arebases forthesimple component algebras. The complexified 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 willtransform 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 forC5,, {e,7}.We can schematically display thebasis wehave constructed asfollows: _ 80,519+ X£a,j;P_ ) e,-,-.(x8a8P+ ad/BP_ . (28.33) Itcanbechecked that thisisindeed anordinary matrix basis. Inthis basis thediagonal blocks arerelated bycomplex conjugation, asarethe off-diagonal blocks. If9denotes transposition inthisbasis then (80/ﬁPi)g :£BaPi' Butfrom thedefining property ofc,(2.3.2), sj,,,Pi =c'1sa55cPi =c'1sa.fPic (since ciseven) =c'1(£a5Pi)‘;’c. Similarly (x£,,5P:)g =x£5,,P;_l5 .1...-__.‘" E31THE CONFUSION OFTONGUES 101 and xs5,,P; =xc'1£a.55cP; =c"1xs§ﬁP;c (since xcommutes with c) =c"1Pisa5§xc (since xPi =Pix) =c‘1(xsa.j,»Pi)5c andso a5=cage“ VaeC5,. (2.8.34) IftpisaDirac spinor wecanwrite tpinterms ofitseven andodd parts asI/J=u+v.Ifweintroduce thenotation £a1P+ :bar (28.35)x8a'1P+ =bar 111611 M=u"‘b,,,, v=vdbd, M“,vf'eC. This accords with theconven- tional labelling since aandvcarry 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 xeh,.P_ =B8’ then Baby; =Og£11P+, Babjf} :0, Bébjg =O’5£11P+, Bdbﬁ = WC may define theMajorana conjugate of1})as 1])‘=tpiwl (28.37) (this isaspecial case of8.21), then(2.8.36) 1:1;=age” +vgc"1= a“’B“’c'1 +v‘5“B‘i’c"1. Wenow introduce Ba.=B°’c‘1 , (28.38)Ba, =Ba/C-1 giving, by(2.8.31), Ba.=c;,,1B" and B,-,.= c;,,1B”. Wecanwrite the Majorana conjugate as tp=u""'B,,, +u‘i’Bc-,, =u,,.B“ +v,,,B‘i’ where, forexample, ua.=uﬁcgj. That is,indices arelowered with the components ofthesymplectic matrix. Iftpisanother Dirac spinor written ineven andoddparts ascp=w+ythen 857$ :(uawa +UaJ’d)311P+- 102 CLIFFORD ALGEBRAS AND SPINORS Thus thisproduct ontheDirac spinors induces theSL(2, C)-invariant symplectic products ontheWeyl spinors. Sofarwehave relabelled the first row and the first column ofour matrix basis tofacilitate a correspondence with thetwo-component formalism. Any element ofthe matrix basis canbewritten asaproduct ofthefirstcolumn bythefirst row, e,7=e,-lelj», andsowecanapply thisrelabelling tothewhole basis £aBP+C_1: ba,Bjr_:; sP_c'1 =b-B" “"’ _1"” (28.39)xs,,BP+c =b,-,B,3 xsa.j9P_c‘1 =b,,.BB. 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 also have simple properties under theinvolution 5, forexample (b,,,B5)5 =—c"P+£a55 =—c‘1sa55cP+c“1 =—sj;,,P+c'1 =—b5B,,.. Similarly weobtain forthefullset (ba’B)B);: =_bBBa (b,-,.BB)’5 =—b5B,-,. (baB8)’ =“b;3Ba (b,,,B3)5 =—b3Ba..(28.40) Ifthen nisanyrealoddform n=n"’5b(,.Bj3 +n""3*b,,.Bj; n5=—n"f’bj,Bd. —n""5’b;;Ba,. Soifnisa1-form, even under 5,n85=-—n5““*. That is,thecomponents canbearranged asananti-Hermitian matrix (with different conventions thematrix ofcomponents isHermitian). Inparticular thebasis 1-forms, e“,canbeexpanded inthematrix basis as 6“ =UadBbd,Bﬁ +Uad38*ba,BB.II1 -...-....._-_-..mm»- 1THE CONFUSION OFTONGUES 103 Theanti-Hermitian matrices, om’, 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 :m"‘5ba,Bﬁ +"laB*bC'yBB. Requiring that mbeodd under 5isequivalent toitbeing a2-form. From (2.8.40) itfollows that this gives maf’ =mfg“. Obviously the components ofa0-form or4-form must form ananti-symmetric matrix, butwestillneed todisentangle thetwo. Wehave y0(ba*B)B) ='990’iB{5ba() =5P0(B"’¢"ba) :g0ii£l[3C_18a*lP+) Z990‘iC/3ai311P+)- Fortheprimitive s11P+ wehave Ef0(sHP+) =§,giving y0(ba*B,6) :$68011 also 924038513) :"500(51BC“’9a1P+Z)Z andsince P,,z =—iP+ itfollows that 3’4(b,,.B5) =§icj§,’z. Iftheinverse matrix isintroduced such that c5“’c'1a.), =55,,then, ifmat’ isanti- symmetric, maf’ =/lc“'3forsome complex A.Itthen immediately follows that9’0(m) =2ReA whilst 9’4(m) =-—2Im/lz. Inthis section wehave established contact with themost usual notations and nomenclature used forspinors inphysics. There is, however, yetonemore 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 part ofthisalgebra; 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. 501 ) MmmWWkMYWWUYMH____PPAS QPWmWLOEB Emb€hoAeW4591CyMMWhCYhParS0 umBV mmPA 9%wlRAMBHOﬁrHwlmOm 91l1 A8r9 0W3"4 1gm83PB___”€ HHEWL...‘ awn”_L___,__________r_ nm2L8M ImnAM“yCwCC 17’16IAMI 1 I HmLMIn_HA2791 GCmOPH €dM_ﬂT€V_mUW€dSWQpmm8SI€HWCS 3533H,I‘GmUﬁm€nd_mSMPHmmMgmEHmfLHMCI6_dH nA0raem1“kuM8791 W_bSCIbe8 91PmeM0haA47O9!M1M€ Gun GL _mU6gNdHbm3CWMg_dnbm3C(nd€dH2WI€mO€GItQEgOIt W0T1891 ISuO€tT0P )SS6IPyt_lSTeV I_____ Q93A2EUEVFf?gags:A2ﬁns:E?=_NEE__®E30% 6525E?ENEESIAwKD_UC:pawEWEE6:6KOUCSE?gags:ER? "HamE?pawE? 205%$0526V©_§®Q;HAmgwoEggU_W_5VNM65HAmiuAaoag$05w6%®6;HAmgwugagage2gigHAg? 5IHﬂu mg6AH8GUWUQ0 33$ >’8_w65_UEVE%m AwK®_uGcC_;w Aw$_Ué__% H55Qvawsvei gnaw5_v_&_E%wswiwiawv6_HEm_E%mXOU=1__=%wvm2UEW_E%wﬁ_véE\nwK®_U___$_=bAmA26Aw66QAw 8 Aﬁoiﬁx30>?m=Eo__m2VMQ®Q @_oE%m=mL_o__£\/UNAm_VN€205%_%§‘mc§o_d2VSEC;®g0_§ AﬁoﬁgmcaogzvNmAm_U_Nm€Egg_\n®>>|m=Eo__m_\/Qwavg®ﬁg AﬁoigNGEOTNEV2E_U_©€_\‘GEE“;_%§|m=ﬁo__m_€wﬁg®Egg $65“:HEEOFEVSQH__J_©_€205%%B|m=Eo_€2Vwavg®smug ?6E%_W=w__o?_>$2E_U_®_€ I I E H H H H H E AGagJ J J J‘miiJ_k@\_SE®w_m_U§h_6 Hr __gjwuggugKé\ QWU©_MwU S€_¢Ui ‘1‘_ 6HMO“Vw_5%__WEOHEG 3_*ﬁﬁg_v8_N_Uomm_w_HU:_UO(_QE_E_~>GTE%_C®:__W>|K2&EOQ82%00hmE;U©HEUOwwﬂsgoaE_NEW>E_:Ew_Cg__N>|§_mE832%WUEME;8H_W6Omm_NUEUOEE_E§E___aw©®:_@>|K®_QEOUsawismmE;UQHECOWWNHOSUOHQE_WEW>E_Eﬁ_w_Ug_§|v$_QE833%muHOEWEOWEQB2%?2£U_6®EBﬁ_2m5E__g$35832%NU“EVtaEéwHE;_U8ﬁOowm_WBECOEE_WEN>E___awU®5__w>|_NUH82%wummE?_U2_N6Om£HgwoaHE~_g>E|E%_U®:__w>|__N®_H82%muEocﬁgmnwag§_%__NBQOSBEBGoagesuE2sawNUHA3taE_$E§w___%__£05Ecogwehgg2:EAUSNhﬁm“SN“Mam$35gob___O_gE<Ndg_U__§m_ Pure Spinors andTriality This chapter contains some further properties ofClifford algebras and spinors. They may beregarded asmore advanced material and the presentation will beadapted 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 arepure. For thephysically interesting Lorentzian case thisgives acorrelation between Weyl spinors (orMajorana spinors) and null planes. Inthepositive-definite case maximal isotropic subspaces, andhence pure spinors, canbeputinto correspondence with complex structures. Inmore than sixdimensions notallspinors arepure. The possibility ofconstraining spinors tobe pure inphysical theories formulated inhigher dimensions has been investigated ([7], [8]). Let VbeanF-linear space with dim1.-V= 2r,and ganF-valued F-bilinear form with maximal index. (Here Fwillbeeither 1BorC.)We canexpress Vinterms ofmaximal (r-dimensional) totally isotropicPURE SPINORS 107 subspaces MandNasV=MC-BN.AWitt basis forVisformed from theisotropic bases {xi}forMand{yf} forNsuch that xfyl+y"'x"=(W. (3.1.1) Since gisofmaximal index theClifford algebra isatotal matrix algebra C(V, g)=A/t2r(F) (3.1.2) whilst thestructure oftheeven subalgebra isgiven by C*(V, g)=Jl/L2,»-1(F) G3./I/l2~1(F). (3.1.3 Let Ebethe2r-form with E2=1sothat theidempotents Pi= §(1iE)reduce C*(V, g)tosimple ideals. Interms ofaWitt basis for Vwemay choose 2’=[x1,y1][x2, yz]...[x",y’] (3.1.4) thebrackets denoting Clifford commutators. Let2Mbether-form product ofsome basis forMQSince Mistotally isotropic itsClifford algebra isjustitsexterior algebra A(M) andsothe r-form product ofadifferent basis will differ from 2Mbythedeter- minant ofthegeneral linear transformation relating thebases. Given the Witt decomposition V=MCBNwecanexpress anyelement ofC(V, g) interms ofproducts ofthexiandthey".Using therelations (3.1.1) the elements ofMcanbepositioned attheright-hand side ofanyterms so thatweseethatC(V, g)zM =C(N, g)zM =A(N)zM. Thus theleftideal C(V, g)zM hasthedimension oftheexterior algebra ofN,2’,and is hence aminimal leftideal. Wemay take thisminimal leftideal asthe space ofspinors. If1peC(V, g)zM then tp=BZM forBeA(N). Thus ‘Z1/1=B"§zM. Wehave I€zM =[x1,y1][x2,y2]...[x’, y’]x1x2. ..x' =lX‘,y‘lX1lX2,y2lX2 ~-~[xiy’lx’ andfrom (3.1.1) [xz’yilxi Zxiyixi :(1_yixi)xi :xi soEZM =2M. Thus ftp: B"zM and theeven and odd (under 27) Subspaces ofC(V, g)zM form the semi-spinor spaces ofthe even 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=/lPfor XeF.)Soifweuseamaximal totally isotropic subspace Mtodefine 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 ofz2~C(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=0VxeM.Ifnow uisanyelement of C(V, g)zM then u=BZM forsome Be/\(N). Foranyy’ENwecan 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 andonly ifxu=0VxeM.Because of(3.1.6) thiscanbecouched more generally. Aspinor uisarepresentative forTifandonly ifxu=0VxeT.(3.1.7) Given the totally isotropic Mthere isnounique Nsuch that V=MG)N.IfTisamaximal totally isotropic subspace with dim(T flM)=hthen wecanalways choose aWitt basis such that {xi} isabasis forMand{x1, ...,x”,y"+1, ...,y’}isabasis forT.Starting with abasis {x1, ...,xh}forT('1MtheWitt basis canbecompleted byaGram—Schmidt type ofconstruction. Ifweadapt theWitt basis in thiswaytotheisotropic subspaces MandTthen arepresentative forT isu=yh“ ...y"zM. Itwilloften beuseful tohave thiscanonical form forapure spinor. Ineven dimensions allelements oftheClifford group areeither even orodd. Thus, by(3.1.6), 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] FlT2)=rmod 2. (3.1.8) There issome seFsuch thatX(s).T2 =M.IfM1,u2arerepresentatives forT1and T2then zM— su2andifu=sulthen uisarepresentative forT=X(s). T1.Since siseither even oroddthen uand2Mbehave the same under 17ifand only iful and 1.42 do. Moreover, TOM=X(s).(T1 OT2)soitissufficient toprove that representativesPURE SPINORS 109 forTand Mareboth even orodd ifdim( TFlM)=rmod 2.Ifwe adapt aWitt basis toTand Mthen arepresentative uforThasthe canonical form u=y"*1 ...y’zM where dim(Tﬁ M)=h.Souand ZMareboth even oroddifr—h=0mod 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 isthat dim(T, OT2)=ror r——2.Ifthisisthecase then non-trivial linear combinations ofulandu2represent allTsuch that TFlT2=T1FlT2. (3.1.9) Asintheproof of(3.1.8) itissufficient toconsider representatives for TandM.Weadapt aWitt basis tothese subspaces. Anon-trivial linear combination ofrepresentatives forthese subspaces willbepure ifuis, where u=}lzM +y"+1...y’zM /l.eF. (3.1.10) Now x’u=0ifandonly ifi=1,...,hand2,’=,,+,A,-x1'u =0ifandonly ifA,=0Vj=h+1,...r,soifuispure, representing T’say, then T’F)M—TFlM.Ifthisisthecase then wecanchoose aWitt basis {x’,y"} i=1,...,rwith {x1, ...,x",y"+1’, ...,y”} abasis forT’. Inthisbasis representatives forT’willtake thecanonical form, soifu ispure MM +yh“ ...y’zM =uy"+1’ ...y"zM (3.1.11) h+lforsome iteF.Byrepeatedly using (3.1.1) theClifford products iny ...y’zM canbewritten interms ofexterior products, yh“ ...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. (3.1.12) Ifh+2=,¢rthen thiscanbeused toequate (h+2)-forms in(3.1.11): X1...x"(y”+'/\x”+1+...+ y’,\x’) =x1...x"(y"*1',\x"+1+...+ y"Ax’). (3.1.13) The {y”} canbewritten aslinear combinations ofthebasis {x",yi}. Since {x’,y”}isalsoaWitt basis wehave F yr»=yf+ZM‘7x1’+ N‘ z=h+1,..,r(3.1.14)j=h+l 110 PURE SPINORS AND TRIALITY where M‘?=—M1" and N"isalinear combination of{x1, xh}. Inserting (3.1.14) in(3.1.13) gives M5’=0Vi,j= h+1, ..., rand hence {x1, ...,x”,y"+1’, ...,y”} isjust anew basis forT;that is, T’=T.Sotheonly non-trivial case ish+2=r.Inthiscase (3.1.11) is seen tobesatisfied by yr—1f :yr—l +Ax)‘ +Nr—l yr! :yr _Axr—l +Nr. Here /Iisseen toparametrise allT’with T’QM=THM.Although ingeneral, aswehave stated, notallspinors arepure, insufficiently low dimensions theabove result canbeused toshow that allsemi-spinors arepure. Ifr€3then allsemi-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 thatifuj,u2are two such pure spinors representing T1and T2then dim(T1ﬁ T2)=- rmod2,whereas from (3.1.9) linear combinations ofulandu2willbe pure ifdim(T, F)T2)=rorr—2.Thus ifrQ3linear 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 give apractical way ofdetermining 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. Letiibethespinor adjoint touwith respect to thisproduct. Ifu1,u2represent T1and T2then T1FlT2aﬁ@ifandonly (U1, U2) = Suppose firstly that there issome xinT1F1T2.Then there issome y such thatxy+yx=1and (U1, '12)Z(uh (xy+}’x)“2) =(uh /Wuz) since xET2.Since thespinor product hasEasadjoint involution then (u1,xyu2)=(xu1, yu2) and this iszero ifxET1.SoT1FlT29*Q implies that (ul,u2)=0.Toprove theconverse weletMandNbeanyPURE SPINORS 111 twomaximal isotropic subspaces such that V=MGt)N.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’=ZN,andsince thespinor product isnon-degenerate wemust have (ZM, ZNZM) ¢0.But ZNZM isaspinor representing Nwhich was any maximal totally isotropic subspace notintersecting with M.Soifu,and n2represent T1andT2then (ul,u2)canonly bezero ifT,FlT2vi£5. IfEv=+(-)0 then for uany spinor 3’2,_p(ui3) = +(-)(—1)’9’p(u5) Eforallp. (3.1.17) Since E2=1 sr,,_,(w>) =sP,,_,,(aa)2' E=Ef’,,(u13'§) 2=v,(u(.%?L))x astheadjoint spinor isdefined with respect toaproduct with Easthe adjoint invo_l_t_rtion. Since EisaZr-form E5=(—-1)"z‘ and.9"2,_,,(uz3') = (—1)’9’p(u(z“v)) Eandtheresult follows. Ifv=BZM forBEA(N) then Ev=B’lzM and (-—1)’Zv =0”. So if0"=iv then §a”2,_,,(ut3) =+(—)3’p(uz3') Z. Ifu1,u2 represent T,and T2with dim(T, F)T2)=hthen 9’,,(u2i11)= Oifp<horp>2r——h,whilst 9’,,(u2iZ1) = ZTIQTZ. IfsEFthen S’,,(su2s'i21) =/l(s)s9’,,(u2£t”1)s'1 sowithout lossofgeneral- itywecanassume that ulrepresents Mwith u2representing some T with dim( TOM) =h.Inanadapted Witt basis weneed toconsider yh“ ...y’zMZM. Intheproof of(3.1.16) weshowed that (zM, y’zM) =0 unless y"=2N.Now zMzNzM =izM sowecanalways normalise the spinor product such that (ZM, zNzM)zM =ZMZNZM. The definition of u2z."i1 isthat u2iIZ1v =u2(u1, 0),sozM2'fMy’zM =(ZM, y’zM)zM. This is zero unless y’=ZN and for the normalisation just mentioned (ZM, zNzM)zM =zMzNzM. But zMy’z,-,4 =0unless y’=zNand so (zM2'M)y"zM =zMy’zM for allmulti-indices Iand soZMZM =2M. Hence yh“ ...y’zM2'M =y"+1 ...y’zM. The form oflowest degree in y"+1 ...y’zM isproportional tox1...x”,which isjusttheproduct of abasis forTﬁM.Since allpure spinors aresemi-spinors itfollows from (3.1.17) thatthere isnonon-vanishing p-form forp>2r—h. Asemi-spinor uispure ifandonly if3’,,(u£2') =0 Vpabr.(3.1.19) From (3.1.18) weseethat ifuispure then certainly 9’p(u'tZ) =0 Vpaﬁr,sowhat weneed todoistoshow that this condition ona Semi-spinor issufficient forittobepure. Any spinor ucanbewritten as H=BZM where BEA(N). There issome sEFsuch that SM=(1+b)zM where bEA(N) and 3’0(b) =0.Ifuisasemi-spinor then soissuandhence bmust beaneven element of/\(N). Suppose T112 PURE SPINORS AND TRIALITY thatff2(b) as0,then exp(—SP2(b)) EFF1/\(N). Now theClifford algebra ofNisjustitsexterior algebra andso 5”2[@XP*[—92(b))bl =9’@[@XP(-5’z(b))l9’2(b) +9’2[@XP(—9’2(b))l5%(b) =502(5) since SfU(b) =0.Thus exp(—5”2(b))su =(1+b’)zM where b’E/\+(N) with 5"0(b') =E-”2(b’) =0.Suppose thatthenon-vanishing homogeneous component ofb’oflowest degree isanh-form. Inanappropriate basis weassume that exp(—EJ’2(b))su =(1+hylyz ...yh+...)zM where theextra terms areofdegree horhigher. Multiplying byy’y"’ ...y”*1 willannihilate these other terms soif = h+l rr h+1 ___u-x ...xy...y exp( 3’2(b))su then v=(1+hy‘y2 ...y”)zM. (3.120) Now wecome tothepoint ofthisconstruction. Ifuisanyspinor and sEFthen 3’p(susﬁ) =)L(s)s8’,,(uiZ)s_1 and 8’p(uz7) =0Vpasr <=>9’,,(sus'z7i) =0Vpasr.Ifaisanyelement ofVthen aua"i1 =auita. By (2.1.7) and (2.1.8) auiia =g(a, a)(uzTi)" —2a,\i,,(u£Z)’l and SP,,(aua71) =(—-1)Pg(a, a)Sf’p(u'tZ) ——2(-—1)*”a Ai,2fi°,,(ut'l'). SoifSfp(ut7) = 0Vpasrthen 9’p(aua"iZ) =0VpasrVaEV.Thus ifthesemi-spinor u that westarted with satisfies 3’,,(ut7) =0Vp¢rthen thevwehave constructed in(3.1.20) alsosatisfies these conditions. Wewillnow show that thiscanonly hold ifA=0;that isexp(—-Ef2(b))su =2Mandhence uispure. Now visthesum oftwopure spinors and, aswehave already noted, apure spinor willsatisfy theconditions ofthetheorem. Soifu satisfies these conditions then /l9’,,{zM(yi...y’5zM) +y’...y”zM'2'M} =0 Vpasr. Aswenoted intheproof of(3.1.18) zMz'}",,, =2Mandso zM(yi...y”zM) +y1. ..y"zM2'M =zMy”...y1+y1...y"zM. Wenow rearrange these terms, remembering that hiseven: zMy” ...y1+y1...y”2M ={(x1y1) ...(x”y”) +(—1)"’2(y’x’) ... (y”x")}x”+’ ...x’. Now x’y’=2+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 thePURE SPINORS 113 (,~-h+2)-form component isnon-zero. Since h>2then inboth cases [here isanon-vanishing p-form with p<r,so Sf,,(uiZ)=OVpasr :)5",,(vi3')=OVpasr:>).=O. A5wehave already noted thisshows that uispure. Eight dimensions areinteresting asthelowest number ofdimensions inwhich notallsemi-spinors arepure. IfdimFV=8with F=IRorC andgisofmaximal index, then from tables 2.15 and2.17 weseethat (,)induces asymmetric product on the semi-spinors. Hence (u,eAu) =(eAu, u)=(u,eAf__u) and9’p(mTi) =0if isodd. Ifuisa semi-spinor then mi=Eu(z"u) =Eu1Zt"E’5 =Eui1E ineight dimensions, andsouii=(uzIr")". Soifuisanysemi-spinor then uil—Zp=0,4,89’,,(ut7). TheO-forms and8-forms arerelated by(3.1.17) soineight dimensions a semi-spinor uispure ifandonly ifEf0(ui2') =0,thatis,(u,u)=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 17,but allreferences to‘even’ and ‘odd’ spinors can be interpreted asreferring totheir behaviour under multiplication by (-1)’EFor areal (pseudo-) orthogonal space whose metric hasmaximal index the pure spinors ofthe real Clifford algebra have adirect geometrical interpretation. Fortheremaining real Clifford algebras we cannot apply theabove theory ofpure spinors directly. However, ifVis anyreal even-dimensional orthogonal space wemay correlate thepure spinors ofC’3(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 diInCM =2.Suppose that uand varerespectively even and odd Semi-spinors ofCC(V, g)representing T1and T2.Then because of (3.1.15) they areboth pure. From (3.1.8) weseethat dimC(T1 FlT2) must beodd (forrishere even, namely two). Hence dimC(T1 OT2)= 1.Ifzisthevolume 4-form ofVthen E=iz.Soifsuperscript c denotes theconjugate-linear charge conjugation operation (here involu- tory) and uiseven then u“isodd. The intersection ofthemaximal totally isotropic subspaces ofVCrepresented byuand u"isone 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*=llnforsome heC.Since complex conjugation isinvolu- tory, Amust satisfy A/1*=1,that is/lEU(1). There issome ].tEU(1) such that/l=uz,andifx=junitfollows that x*=x.Thus xisareal nullvector such that x(u+u")=O. (3.1.21) This real null vector isdetermined uptomultiplication byareal number. Suppose that urepresents Twhich has abasis {x,w}. Now x(wu°) =—wxu° =0since xu°=0:and certainly w(wu°) =0since wz=O.Thus wit“ and uboth represent T.Since thespace ofrepre- sentative spinors forTisonedimensional there issome itECsuch that wu°==Au.Wecannot have A=0since wdoes notlieinthesubspace represented byu“.SoifcuE/l'1w then wuc =u. (3.1.22) The charge conjugate ofthis isw*u =u".Soww*u =cou°=u,and since cuETwehave (ww* +w*w)u =u,andthus c0w* +aJ*co =1. (3.1.23) From the(complex) null1-form wwecanconstruct aunit 1-form a: aE0)+cu*. (3.1.24) Wehave because of(3.1.22) a(u+uc)=u+rt“. (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 FExAa (3.1.26) both determined uptoareal multiple. If1/JE u+u°then 1};isa Majorana spinor andbecause of(3.1.21) and(3.1.25) wecanequivalent- lythink oftherealforms xandFasbeing determined bytp. Thetheorems (3.1.18) and(3.1.19) enable therealforms xandFto beexpressed interms ofu,u“and their adjoint spinors. There isa freedom toscale thespinor product (,)whose adjoint isEbyacomplex number. IntheLorentzian case wecanalways choose aspinor basis such that charge conjugation simply conjugates thespinor components. Thus wecanrequire thatthespinor product satisfiesPURE SPINORS 115 (uh u2)* :(“101 I420) thisleaving only arealscaling freedom. Taking aspinor product which satisfied (3.1.27) weturn to(3.1.18). The intersection ofthesubspaces represented byuand u°isspanned bythereal 1-form x.So(3.1.18) tells usthat 9’1(iuﬁ°) isacomplex multiple ofx.The factor ofiis inserted toensure thatthis1-form isinfactreal. For 9’1(iuiZ°) =E-J”0(iui?°e,,)e“ =(u°,ie,,u)e" and 5”1(iuiZ°)* =—(u, ie,,u")e" (by(3.1.27)) =—(ie,,u, u°)e“ (since ’g'istheadjoint) =(u°,ie,,u)e" (since theproduct isskew) =5"1(iu£i°). andthus Sa"1(iuiZ°) =x (3.1.28) where xis,ofcourse, only determined uptoarealmultiple. Let{x,co} beabasis forT,represented byu,where wisthecomplex 1-form satisfying (3.1.22). Then toisdetermined uptothe addition ofa multiple ofx.From (3.1.18) weknow that iuiiisacomplex multiple of xw, sayiuﬁ=2exp(i6) xwforanappropriately scaled x.SoifG= 2(iuz.'I —iu°iZ°) then G=exp(il9)xw +exp(—it9)xw* and G(a> +00*)= cosBx+2isin6xAcuAw*.This Gwillbenothing other than theFof (3.1.26) ifinfact6=0.Wehave 20(0)+21*)=iu[ ] -iu°[ °] =rm-rm since wu=0andum“=u.Butifat,/3,tp,1/1areanyspinors then (W,(<P1l3)§l3) =((<Pi/3)0t»5) =(111,0-’)(<P,5)=—(0v»1P)(<P»/1’) =E01»(¢<i3)ﬁ)- and so(¢1T))§ =-1/15?. Thus 2G(w +01*)=iu£i° +(iut'?°)§ and since izu=uthen (ut'1"’)’l =—zuiJZ°z =—zu(EiT°) =—izu(i?§)° =—tu7°. andsouit"'°=9’1(uzTi°) +5P;,(uiZ°). Since 3-forms change signunder Ewe have G(w +00*)=S"1(imZZ°) =xby(3.1.28). That is,theFof(3.1.26) Canbewritten as F=Re(iu'11). (3.1.29) From (3.1.21) and (3.1.25) weseethat xandFcanequivalently be thought ofasbeing associated with theMajorana spinor 1/1=u+uc. Wecan also express xand Finterms oftpand itsadjoint. Since 116 PURE SPINORS AND TRIALITY u=izuwehave tp(ZT/J)=(-ma+iu°u'°)+i(u1§i°+(uu'°)5)1"‘; my where, since (got/1)? =—-tpqo, thefirst term isodd under Ewhilst the second iseven. Comparison with (3.1.28) and(3.1.29) shows that %ft1(1/»('E-T/1)) =x (11.30) -2sP2(¢(E'E')) =F. (31.31) Ifu’isrelated touby u’=exp(i6)u (3.1.32) then, from (3.128), weseethat u’determines thesame nulldirection as u.Ifu’determines the2-form F’then F’=Re(cos26iui2' —sin26uiZ). andsince u=izu F’=Re(cos2t9iuiZ —sin2t9ziu£i) =cos26F —sin26zF =exp(—2t9z)F. Since zF=—*Fweseethat the2-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]. Wenow consider thecase ofVareal even 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 (3.133) that is,Jisanisometry ofV.Wewill show that any such Jisin one-to-one correspondence with amaximal totally isotropic subspace of VP.Hence theone-dimensional space ofpure spinors ofthecomplex- ified Clifford algebra isinone-to-one correspondence with acomplex structure onVt. Suppose firstly that wehave such aJ.Then bycomplex linearity J defines atensor onVC.Define MCVCby ’tWe thank GSegal forpointing thisouttous.PURE SPINORS 117 xEM iffJx=ix (31.34) andyEM*iffy*EM.Then VC=M(9M*. IfJsatisfies (3.1.33) then g(x1, x2)=g(Jx‘, Jxz), and forx1,x2 EMwehave g(x1, x2)=0. Hence Misamaximal totally isotropic subspace ofVC.Conversely now suppose that wehave amaximal totally isotropic subspace M.Wecan define Jonelements ofMby(3.1.34). Requiring Jx*=(Jx)* defines J unambiguously onthewhole ofV’-7and, byrestriction, onV.Such aJ certainly satisfies J2=-1.ForanyaEV‘:wecanwrite a=at+a“ with atEM andatEM*. Then g(a, b)=g(a+, b‘)+g(a*, bt) and itfollows thatifJat=iatandJa‘=—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 dimFV= dim,r-S andtheindex ofgisthesame asthat ofh.IfS=StG9S", with Stand 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 = dim2.-Siwith theindex ofgthesame asthatofthemetric induced byh onS1.Wenow examine thepossibility ofthislatter situation occurring. Ifdim2.-V=nthen nmust beeven ifC+(V, g)istobereducible with S splitting into semi-spinor spaces. Then dim1.-S=2””andfordimFSP to beequal todimFV weneed 2"”=2n,which requires n=8.IfF=C then weseefrom table 2.17 that thesituation wearelooking fordoes occur ineight dimensions, with hbeing thespin-invariant spinor metric associated with theinvolution 5;‘.ForF=1Bthesituation depends onthe Signature ofg.Forgiven pandqthethird entry intable 2.15 classifies thespin-invariant product associated with E,hsay, ontheirreducible representation spaces oftheeven subalgebra. Ifthisentry is1G)1then the even subalgebra has two semi-spinor representations, with an IR-bilinear symmetric product on each. Such entries occur for (P,q)=(8,0), (O,8)or(4,4).From (2.6.23) weseethat inallthese cases theindex ofhisthesame asthat ofg.Actually alittle care is needed inreaching thisconclusion forthecase ofC4_4(lB). Weknow 118 PURE SPINORS AND TRIALITY that honShas maximal index, butwecould have hinducing a positive-definite product onStand anegative-definite one onS‘. However, ifxEVwith x2=1,then forvES‘there isauES’such that v=xu.Then h(u, 0)=h(xu, xu)=h(u, xzu), since theadjoint involution ofhisE,andtheindices ofthemetrics induced byhonSt and S"arethesame. Inthefollowing Vwill either beacomplex eight-dimensional vector space orareal eight-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<295+casi (3.21) Ifelements (D,ofEare decomposed into these subspaces as (D,-=x,+u,-+0,then abilinear form Bisdefined onEby B((I)1v (D2) =g(x11 x2) +h(u1a u2) +h(v1v U2) ' (We shall frequently decompose anelement (Dasabove, thesymbols x, uand0being reserved forthecomponents of(Dinthesubspaces V,S+ andS'.) Wecanintroduce atotally symmetric (3,0)tensor TonEinterms of theinner product h.Wedefine T(‘-D1, ‘D2,(D3)E/TW1, X2113) +h(“1> X3112) +h("2, X103) +h(l»l2, X3121) +h(l/I3, X1192) +/’l(T/(3, XZUI). Each term ontheright-hand side islinear ineach (D,-,thus Tisindeed multilinear. Byconstruction Tistotally symmetric. Wecan usethe bilinear Bandtrilinear Ttodefine abilinear map O: °I_E XE"-9 E SLlCh T(¢)1, (D2, (D3) 2 O(D2, (D3) The non-degeneracy ofBensures that 0isindeed well defined. Its bilinearity follows from thetrilinearity ofT.Since Tistotally symmetric (D10(D2=(D2Q(D1.If(D2and(D2areboth inthesame subspace, either V,StorS*, then T((D1, (D2,(D3)=0from (3.2.3) and hence (D1<>(D2 =0. ForxEV, btES+ andvES' wehave B(x0u,v)=T(x, u,v)=h(u, xv)=h(xu, v)=B(xu, v) andso x<1»u=xu (3.2.5) similarly xO0=xv. (3.2.6) Ifitistheadjoint ofuwith respect tohthen B(u0v,x)=T(u, 0,x)=h(xu, 0)=xiv I =9°0(tTt'xv) =9’0(xviZ) =3’0(x5P1(viIZ)) =B(x,9’1(v'u'))TRIALITY 119 so u0u=9’1(vi“Z). (3_2_7) Theproduct Qisnotassociative, forexample wehave xO(xOu)=xOxu=xzu=g(x,x)u (3.2.8) whereas xOx=0.The norm ofthespinor x0uisrelated tothenorms ofxanduby h(x1<> u,x20u)=g(x1,x2)h(u, u). (3.2.9) This follows from (3.2.5), (3.2.6) andthefactthat theadjoint ofhis5. The 24-dimensional vector space Eforms anon-associative algebra .951. under the<>product. The spinor representation oftheClifford group, ponS1,and the vector representation )5onVnaturally induce areducible representation YonEby Y(s).(x +u+0)EX(s).x +p(s).u +p(s).v. (3.2.10) Whereas gininvariant under )((s) VsEF,hisonly invariant under p(s) forsE,1"andso B((D1, (D2,) =B(Y(s).(D1, Y(s).(I>2) VsE,1". (3.2.11) Itreadily follows thatinaddition T((Dl, (D2,(D3)=T(Y(s).(D2, Y(s).(D2, Y(s).(D3) YsE,1".(3.2.12) From these lasttworelations wecaninfer from (3.2.4) that Y(s).((D1o (D2)=(Y(s).(D1) <>(Y(s).(D2) VsE+1"(3.2.13) thatis,Y(s) isintheautomorphism group ofthenon-associative algebra ad.Conversely itfollows that ifoisany automorphism ofatthat transforms Vand Sinto themselves then o=Y(s) forsome sE,1". (The starting point oftheargument isthat for1/JESthen 0.1;)=st/1for Some regular element softheClifford algebra.) The orthogonal space Vunder consideration has been carefully selected toensure that V,S+andS‘areallisometric. The existence 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(u0, uo)=1.Then alinear transformation r(u0) from VtoS'isdefined by 1.'(u0).x =x0uo. (3.2.14) Itimmediately follows from (3.2.9) that r(u0) isinfact anorthogonal transformation from VtoS'.The linear transformation t'(uO) is lllllquely extended toanautomorphism ofperiod twoonV(19S‘:that 120 PURE SPINORS AND TRIALITY is,ifvES‘such that v=r(u2).x forsome unique xthen wedefine r(u0).v =x.Finally wedefine r(u0) onStby 1.'(u0).u =2h(u, u0)u0 —u. (3.2.15) That is,r(u0) acts onStbysending utominus itsreﬂection inthe plane orthogonal touo.Thus r(u0) isanorthogonal transformation of St,andhence ofE.Inaddition, t(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. IfvEr(u0).a forsome athen 1:(u0).vt'(u0).x Eaxuo E2g(a, x)u0 —xauo E2h(r(u0).a, 1:(u0).x)u0 —xv since r(u(,) isanisometry from VtoS‘andso 1'(u0).vt(u0).x =2h(v, xu0)uO —xv=2h(xv, u0)u0 —xv=(t(uO).x)v. Since T(r(u0).(D1,r(u0).(D2,1:(u0).(D3) =T(t(u0).u1, t(u0).v2, t(u0).x3) +... itfollows from (3.2.3) that T(t(uO).(D1, t(u0).(I>2, r(u0).(D3) ET((D1, (D2,(D2,). (3.2.16) Whereas r(u0) isanorthogonal transformation ofEthat inter- changes Vand S‘, Y(s) isanorthogonal transformation ofEthat interchanges StandS‘.Ifx0EVisaunit vector, g(x0, x0)E1,then x0E+1"and Y(x0) isofperiod two, Y(x0)2 =1. Out ofthese two involutory transformations ofEweconstruct anorthogonal transforma- tionofperiod three. The triality map E(x0, uo)isdefined by E(x2, uo)EY(x0)r(u0). (3.2.17) ToseethatE(x0, uo)isofperiod three wewant toshow that T(l40)Y(x0)T("0) =Y(x0)T("0)Y(X0)- (3-2-18) Forexample, ifxEVthen T(“0)Y(X0)T(”0)-X ="7(”0)Y(X0)-(xuol =T040)-(xoxuo) E2h(x0xuO, uO)u0 —xoxuo E2h(x 0uo,x0Ou0)u0 —xoxug =2g(x, x0)u0 -—xoxuo (by(3.2.9)) Exxouo. Ontheother handTRIALITY 121 Y(X0)T("0)Y(x0)-X =Y(x0)T("0)-(xoxxu) =Y(x0)-(xoxxouo) =X30110- Thevalidity of(3.2.18) canbesimilarly demonstrated onelements from theother twosubspaces. Given (3.2.18) wehave 5(/Y0» ”0)3 I(Y(X0)T("0)Y(x0))(T("0)Y(x0)T(”0)) =(Y(x0)T("0)Y(x0))2 andsince both Y(x0) and'r(u0) areofperiod two E(x0, u0)3 =1. (3.2.19) Because Y(x0) andt(u0) both have these properties separately wehave B((I)1a (D2) :B('E(xO> uU)'q)1:v E"(xO:~ uU)'q)2) and T((D1, (D2, Q3) :T(E(x0, u0).(D1, E(x0, I/l0).(I)2, E(x0, U0).(p3). Thethree subspaces ofEarepermuted under E(x0, uo)asfollows: E(x0, u0).V CS’ E(x0, u0).S+ CS‘ E(x0, u0).S‘ CV.(3.2.22) Wehave focused onaVsuch that C(V, g)EC(S’, h)EC(S‘, h). The map E(x0, uo) isometrically permutes these three spaces. Any isometry between two orthogonal spaces uniquely extends toaniso- morphism between their Clifford algebras. LetNbetheisomorphism obtained from E(x0, u0): N:C(V, g))—-> C(S’, h)1:2» C(S‘, h)ii) C(V, g).(3.2.23) Because StC-BS‘isthespinor space ofC(V,g)themap Nenables any twoofthethree spaces V,S+andS‘tobetaken asthespinor space of theClifford algebra ofthethird! Forexample, S‘G)Vcanbetaken as thespinor space ofC(S+,h).Let<>denote theClifford product of C(S’, h).Then forxEVand111ES /l/(X1//) =A/(X)<>N011)- That is,ifuEStand1/1’ES’ES‘G)Vthen u<>1/1’EN((N‘1u)(N‘11//)). (3.2.24) Under thismultiplication byuthespaces S‘and Vareinterchanged; these being thesemi-spinor spaces ofC*(S+, h). Exercise 3.1 Show that ifVisacomplex vector space then C(V, g)EC(S, h)if d1m@V E2,4.Intherealcase what signatures canghave? (Remember thatthespinor inner product could beassociated with either Z5or51).) 122 PURE SPINORS AND TRIALITY Bibliography Chevalley C1954 The Algebraic Theory ofSpinors (New York: C0ll1lHbi& University Press) J --.-wt-v ll--—- i“ 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 that thereader hassome intuitive ideas about thismodel description ofourperceived three-dimensional world. (The term Euclidean space isnotsynonymous with Euclidean vector space. AEuclidean vector space isareal vector space with apositive- definite symmetric metric.) Atanearly age wealllearnt how a Cartesian coordinate system canbeintroduced toputpoints inEucli- dean space intocorrespondence with anordered triple ofrealnumbers, anelement ofB3.However, itisimportant that wedistinguish Eucli- dean three space from IB3.Euclidean space hasnopreferred coordinate system. Indeed weneed notofcourse even berestricted toCartesian coordinates. Despite ouremphasis onthedistinction between Euclidean three-space and1B3itisnonetheless inIP13that thefamiliar calculus of differentiation andintegration isintroduced. Through theintroduction Ofacoordinate system one may then apply thiscalculus toEuclidean Space. Itisthecorrespondence ofEuclidean space toIR”,through the introduction ofacoordinate system, that generalises toprovide the definition ofamanifold. This isdefined, inasense that willbemade precise, tobelocally likeIR".Because wecandefine differentiation and integration onIR"wecanextend these notions toamanifold. Unlike Euclidean space, foranarbitrary manifold wecannot choose SOme origin toputallpoints onthemanifold into aunique correspond- ence with points inIB".Forexample, wecould take thetwo-dimensional Outer surface ofahollow rubber ball. Whilst anycapoftheballcould beputintoone-to-one correspondence with points inaplane (bycutting thesection outand ﬂattening it),wecannot dothis with thewhole Surface. (Ifwesimply squashed theballthen twopoints onthesurface 124 MANIFOLDS would bemapped tothesame point ontheplane.) The fact that the surface islocally likeIR2issufficient 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, IR"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 onIR”weshow how asystem ofcoordinates onatopological manifold enables differentiation tobe defined, giving adifferentiable manifold. From itsintroduction inIR” 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 onIR". 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 Theusual definition ofcontinuity ofafunction f:U—>Wwhere Uand Waresubsets ofIRrelies onthenotion of‘nearness’ ofdifferent elements ofIR.Such ‘nearness’ ismeasured byaproximity function d:IR><]R—> IRwith theproperties: d(x, y)=d(y, x),d(x, y)=0if and only ifx=y,d(x, z)E..d(x, y)+d(y, z).(Note x,yElR.) A natural proximity function forthereal linethat hasthese properties is theabsolute value ormodulus map, (x,y)—>lx—ylandfissaid tobe continuous atxEIRifonecanfind apositive 5EIRforanypositive e belonging toIRsuch thatifd(x, y)<6then d(f(x), f(y)) <8.Thus one probes theneighbourhood oftheimage offinduced byaneighbour- hood about xinthedomain off.The firstgeneralisation ofthisidea to arbitrary sets consists ofdefining anew setcalled theneighbourhood.'TOPOLOGICAL MANIFOLDS 125 nbh(x, 5)CSifxES. This isthesetofelements yES such that d(x, y)<6,thatisasetofallpoints thatarewithin 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 thatdshould belinear ineither ofitsarguments. With thiscaveat inmind one refers tothe pair (S,d)asametric space. The defining properties oftheproximity function dofcourse remind oneoftheproperties ofdistances between points in.Euclidean space (for example, thetriangle inequality) and indeed it1Sworth noting that ifIR”isgiven. avector space structure one callP110056 (KI, y)=[g(x —y,x—y)]1’2 provided gisthepositive- definite Euclidean metric. Ifone does use theEuclidean metric to define dthen thesetnbh(x, 5)inEuclidean IR"looks likeanopen ball (open because oftheinequality d(x, y)< 6,VyEnbh(x, 6).The Lriangle inequality property ofdensures that allpoints yEnbh(x, 6) avesomeneighbourhoods that arecontained innbh(x, 6).Ingeneral theproximity function onIR”need notcoincide with themetric onIR” regarded asavector space. fAboundary element xofasetS’contained inthesetSwith distance LlI1C'[l.OI1 d1Sanelement such that nbh(x, 5),forsome positive 6ElR, contains both elements inS’and elements notinS’.The setofall boundary points ofS’iscalled theboundary ofS’.Inparticular if SEnbh(x,_i5) CSthen S’does notcontain itsboundary andiscalled anopen setin(S,d).Ifanyboundary points arenotinthesetthen itis anopen set.Ifall. boundary points areinthesetitisclosed. Ingeneral it1Spossible tofind different distance functions that iiletermine thesame class ofcontinuous functions. Avaluable genera- lsation then. istoconcentrate ontheopen sets themselves asthe primitive notions and reformulate ‘nearness’ directly interms ofthem Ilatqei" than interms ofanyparticular proximity function. The immediate CSeuness ofopen sets isareformulation ofthe definition ofa f0I1t1nuous. function f:U—>W. iscontinuous atpEUifandonly if, coranyoneighbourhood Wcontaining f(p)there isaneighbourhood U’ Oolltalning pwhose image f(U)CW’. Such anotion ofcontinuity relies Htheopen setstructure ofthespaces related byfand notona particular choice ofproximity function used inspecifying these open 1Sets. Consequently one attempts tobypass anymention ofaproximity unction and establish amore general definition ofopen sets onany Space. The declaration ofwhich subsets ofaspace aretobeconsidered 3Sopen isocalled adefinition ofitstopology provided such afamily of subsets satisfy thefollowing axioms.— Mi“ It ‘Q 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,<5)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) Enbh(p,<5)and thespace issaid tohave ametric topology (which isalways Hausdorff). One ofthecommonest metric topologies is associated with IR”and d(x, y)E|x——y|,x,yEIR". With theabove d(x, y)onIR”theopen sets may bevisualised asallpossible open hypercubes inIR”. Itisuseful tohave such examples ofanatural metric topology inIR" since they can beused toinduce topologies onsubsets ofIR”. 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 IP13provide some ofthesimplest visualisable models oftopological spaces. Thus the sphere S2isthesubset ofIR3defined bylxlE1,xEPR3with atopology induced from themetric topology ofIR3.Itistopologically equivalent (homeomorphic) totheellipsoid (azxz +bzyz +c222 =1,a,b,cEIR) with thetopology induced from that ofIR3;that isonecanestablish a homeomorphism between them. Neither ishomeomorphic tothe2- torus, S1XS1.However allthese examples (and indeed any two- surface) have points with neighbourhoods homeomorphic totheopen disc {x||x|<1,xEIRZ}. Such spaces aresaid tobelocally homeomor- phic. The fact that they need not behomeomorphic issometimes phrased bysaying thatthey have different global topologies. Ifoneexploits thevector space structure ofIR3onecanproject any sufficiently small region ofatwo-surface onto asuitable two-plane inIR3i IitTOPOLOGICAL MANIFOLDS 127 toobtain aneighbourhood inIR2andabijective 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 setofIR”.Acollection ofopen setsisabasis foratopology ifevery neighbourhood canbeexpressed astheunion of members inthebasis. Theelements ofatopological manifold areoften referred toaspoints. Itisclear from theexamples above that onecannot ingeneral find a homeomorphism from thewhole topological space toanopen setofIR“. Theabove definition ofatopological manifold issufficiently general that notalltopological two-manifolds aresubsets ofIP13. 4.2Derivatives ofFunctions IR”->IR” Ourdiscussion ofcontinuity culminated inthedefinition ofatopological manifold asbeing locally homeomorphic toIR”.This local correspond- ence with IR”canbeused toestablish acriterion fordifferentiability of maps onmanifolds. We first brieﬂy review the differentiation of vector-valued functions onlR”'. Ifisafunction from lR’"toIR”then thederivative offatpElR’”in thedirection ofVEIR’”isgiven by D./f(p)=lgg(”"””,?‘f”’)) (4.21) where hEIR.(Other commonly used notations forDvf( p)aredf(p)V, df,,(V) andf’,,V.) Whereas thediscussion ofthecontinuity offonly involved thetopology oflR”‘ and IR”, 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'"toIR”. Df(p):VI——>Dvf(p), determining thelinear part ofanapproximation 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’ inIR“, With itstailatpandtipatp+V,toan‘arrow’ inIR",with f(p) astail and f(p)+Dvf( p)astip. We may formalise this bydefining the tangent space toIR"‘atp,T,,lR’", tobethesetofpairs (p,V)forall VElR"’. These pairs (tangent vectors) form avector space, isomorphic toIR“, with therule—----—---iyi 128 MANiFoLDs JL(p, V)+u(p, U)E(p,/IV+uU) /I,ttElR. (4.2.2) Wemay now define thederivative offatp,ortangent map, f,.,,: f*P’TP1Hm ’”“'> Tf(r>)IBn (P,V)E>(f(p), Dvf(P))- (4-2-3) Since Df(p)isalinear transformation onlR”"itfollows that fipisa linear map onT,,IR"‘. Thetangent space oflR'"atpisjustasubspace of thedirect sum ofIR“with itself, andsothere isanatural wayofadding tangent vectors lying indifferent tangent spaces. This feature willnot carry over tothefollowing 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. IR“ la.V)/'/ 2 T E’ . f p+V flpl+UyflPl’*P, lf(pl.DVf(pl) IR, P fl/Jl V I Figure 4.1Thetangent map off:IR'" —>IR". If{e,-} and {ej-} arethenatural bases forlR"‘ and IR”then the component functions off,fl:lR’">—>IRiE1,...,n,aregiven by f(p)=§11r@:-. (4.2-4) Thedirectional derivatives ofthese component functions along thebasis vectors for]R"‘arecalled thepartial derivatives, andaspecial notation iscustomary: D,,f’(p) E(5f’/8)C’)(p). (4.2.5) ForanyVElR"‘ Dam»)=2Dvf’(P)@i =v1D.,r'e:i=1 ,tE1iE1 bythelinearity ofDf‘(p). Thus thematrix ofthelinear transformation Df(p) isformed bythe partial derivatives. The nXmmatrix [(8f"/8xl)(p)], with iflabelling therows, iscalledthe Jacobian andwe" DERivATivEs OFFUNCTIONS lR"‘->IR" 129 have Dr/f<1>>=§il(@f’/er")(p)lV"@I-- (42.6) Thepartial derivatives may beregarded asrealfunctions ofthepoint pandhence higher partial derivatives may beformed. Amap between subsets ofIR”and IR”forwhich allpartial derivatives uptoorder k exist andarecontinuous issaidtobeaC"map. Ahomeomorphism that isaCkmap with aCkinverse iscalled aC’dijfeomorphism Weshall beprimarily concerned with C°°maps, which willbecalled smooth. Example 4.1 Letf=B’*->IB’.a E(X1.xi)*—+f(a) =((x‘)2. xlxz+1x2)TakingV=(v1,U2)in(4.21) gives DVf(p):(2U1X1,X1U2 +xzvl v2)E(v1v2)(2)61 x20) ’ ’ 0x11 where theentries inthematrix arerecognised asthepartial derivatives ofthefunction f. 4.3Differentiable Manifolds With thenotion ofsmooth maps between IR’"andIR"established we proceed now todefine adifferentiable manifold. Atopological manifold islocally homeomorphic toIR”.Bysetting upaSystem ofcharts that map neighbourhoods ofthemanifold onto neighbourhoods ofIR”wecan usethedifferential structure onIR"todefine thedifferential structure ontopological manifolds. ﬁrgl egglceisgto t?ot1vate thedefinition ofadifferentiable manifold letus _ eproblem ofcoordinating apatch ofatopological pltanifold by_return1ng totheexample ofS2asasubset ofIR3.Suppose issubset isconstructed from thin perspex and theboundary ofa region 1Smarked outbypainting aclosed curve ontheperspex surface. ﬁllitlzlegpnczrte paint afizhnet ofcurves within andonthis‘boundary so boundar imelurves intleneltintersect only once andeach intersects the C y geonce aso. magine alight isshone through thisnetof urves andexamine theimage shadow onanytwo-plane placed conve- niently tocollect_the shadow. Ifeach intersection inthenetofpainted eﬁrves casts aunique shadow onthetwo-plane then theneighbourhood :h:Sen onthesphere yields a.proper coordinate patch with respect to label[l)iI'I:)]6CI10Il scheme. Each intersection canbeuniquely labelled by gallthecurvilinear lineshadows uniquely. Ifalens ofsuitable 130 MANIFOLDS 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 setofIP12formed bytheshadow. Toeach point peU weassign tworeal coordinates <p(p) =(cp1(p), <p2(p)) eIB2. 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 <;0isahomeomorphism forthen andonly then 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 the relations 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 setofIR”. Ifwelabel one such homeomorphism cp,:U,—><p,(U,) then thepair (U,, <p,)iscalled a coordinate chart forU,(with thechart domain U,). The image <p,(p) forpeU,assigns tothepoint pthenreal coordinates ((p,1,(p), (pf;(p), ..., <pZ(p)). For each chart labelled byathereal-valued function (p{,:U,->IR,(j=1,...,n)iscalled thejthcoordinate function andis projected from tp,bythej-projection map trl H1113" —~—>IR,<Pt(P) *—>Fl0<Pt(P) E<P’.}(P) (4-3-1) forallpeU,.When wework inaprescribed chart weoften drop the chart label ‘a’on andacommon notation forthesetofnnumbers {<P"(P)} is{X’(P)}-One ofthemost important hurdles toovercome when first working with general coordinates istoresist theinstinct toinfer anymetric or distance properties ofthemanifold from theuseofthesymbol xl. Whereas thecoordinates {xl(p)} ofpareelements of1B”,regarded asa Euclidean vector space, themetric onIR”need notdefine anymetric or distance function onthemanifold. Forexample, x‘andx2could bethe ‘usual’ polar coordinates 6,tpforaneighbourhood ofthetwo-sphere. Although theEuclidean metric isused on(6(p), <p(p)) todifferentiate functions onthesphere thisisnotnecessarily related toanymetric on thesphere, certainly nottothestandard metric. Acollection ofcharts (U,, (p,)a=1,2,...becomes anatlas forMeM DIFFERENTIABLE MANIFOLDS 131 provided theunion ofalltheU,isMitself. Two charts (U,, (pa)and (Ub, 99,)such that U,HU,abQ1give risetoahomeomorphism between neighbourhoods of1B”. IfU,F)U,EU,, then we define (see figure 4.2) hab E(Pb0(Pt? :(pa(Uab) W‘) (pb(Uab)' an*9, j ‘pt: . KP(Ul he-:==\°t=-O\PE.—1 i bb waiuablav.~”"'_._ '1 *'—r ¢_ W I . i r._. I_ _. Figure 4.2Thechart maps forU,OU,CM. Then (p,(p) =h,,,<>q0,(p) expresses thencoordinates (p§,(p) ofpin the‘b’chart interms ofncontinuous functions hf,ofthecoordinates q9{,.(p)ofpinthe‘a’chart, thatisacoordinate transformation expresses thecoordinates ofpinone chart interms ofthecoordinates ofthe same point inanother overlapping chart. Ifasisoften done wewrite X‘E<;0,(p) andy’E<p§,(p) then xi=hf,,(y1, yz,...,y”)i=1,...,n. Similarly h,j,1isahomeomorphism from <p,(U,,,) to<p,(U,,) andgives theinverse mapping between thecoordinates. The maps [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 lib,0h,,=h,,then /1,5,1=11,,andsotheinverse chart transformations aredifferentiable; hence they arediffeomorphisms onIR”. New charts (U, cp)can beadded totheatlas [(U,, <p,)] provided <p<>q0,‘ and Q0,O<p'1aredifferentiable foralla,inwhich case (U,rp)iscompatible with theatlas. Ifevery member ofone atlas iscompatible with every member ofanother atlas then the two atlases are compatible. A differentiable structure onatopological manifold isspecified bygiving a differentiable atlas from theclass ofallcompatible differentiable atlases forM.Ifatopological manifold canbeprovided with twodifferentiable atlases that areincompatible then thetopological manifold issaid to admit two different differentiable structures. An n-dimensional C°° SE1 132 MANIFOLDS DIFFERENTIABLE MANIFOLDS 133 manifold (orsmooth manifold) isdefined asann-dimensional topologic- Simply afunction onM.Iffisdefined onanopen setWofM almanifold together with aC°°differentiable structure. f:W—> IR,then inalocal chart (U,(p)itdefines afunction Asanexample ofhow thetopological space 1B(the realline) canbe assigned different C°°structures consider theatlas with single chart f<Pllp(U lelW)_" B (4-34) (IR,rp)with (p:IR+—>IR,x—>x.Consider another atlas for1Rwith chart bytherulefQ,=f0Q9-1, thatis ,. (lB,[-3)where ,8:IB +—>IR,x—> x3.Then (cp<>j6*l)(x) =xl’3which isnot . __ differentiable atx=0.Hence (IR,(p)and(IR,B)arenotcompatible and *n f(p) _(fa°(PXP).: -r each atlas defines adifferent C°°structure onthesame underlying a =f(p(q;1(p), (f(p), ___,(pr=(p)) Vpepj/_ topological manifold. Inwhat follows weshall always assume that our i manifolds have been given aparticular differentiable structure. ’ Wedeﬁne 90*bytherule¢. 1 Ifthemanifold admits acovering bycharts such that each h,,is1 (¢*f(p) =fq,Qq9_ (4_3_5)1- ‘i:' § orientation preserving (that isthedeterminant oftheJacobian ofthe t, map (hab)* iseverywhere ofthesame sign foralla,b)then the - Writing f:f¢>° (P:(p*f¢>i themap falsSale tobePulled back from manifold issaid toadmit anorientation. Every oriented differential rp(U W)toUFlW.‘ _ o.This notion generalises toanydiffeomorphism 1/1between themani-manifold admits twoorientations corresponding tothetwosigns ofthe Jacobian determinant. The ribbon with one twist (Mobius band) isan folds MandN‘Forf 2N_)lewedeﬁne _ A=¢¢mmm¢=A~u--"mm-\OOIexample ofatwo-dimensional differential manifold that isnon- ; q,*f; M__>13 Pi_;.(1j,*f)(p) =f(q,(p)) (4_3_6) orientable. Ifitisregarded asbeing asubset ofEuclidean three- . dimensional space onenotices that itisnotpossible toassign unambi- 1 andSaythatthereahlalued function fonNhasbeen pulled baek tothe guously asmooth field ofever)/Where normal unit vectors tosuch a if mahlalued function lllefonM(See ﬁgure 4'3)" ltlellews immediatelySurface_ thatunder acomposition ofdiffeomorphisms: Having used thedifferentiability offunctions onIR”toestablish the A (fpO¢)*=1j)*0q;*_ (4.3.7) notion ofasmooth manifold wecannow similarly define differentiable I maps between smooth manifolds. Amap ffrom asmooth manifold M1 41 toasmooth manifold M2issaid tobedifferentiable atpeM1if,for M'i"""' '"N some charts (U1, cpl)forM1and(U2, Q92)forM2, themap (pgofotpfl isdifferentiable at(p1(p). Since achange ofchart isadifferentiableI operation thedifferentiability offdoes notdepend onthechart used to represent it.Ahomeomorphism between smooth manifolds isadiffeo- ell f morphism ifboth itanditsinverse aredifferentiable. Amap fsuch thatji P2:f(Pi) P2EM2>Pi€Mi HQ mayberepresented inlocal coordinates bywriting Figure 43Thepun_back map (P2(P2) =(P2°f(Pi) =(P2°f° €9i_1°‘i91(Pi) Zfzi°€01(P1) Where Suppose fisasmooth map from amanifold Mtoamanifold N.If fzlE(paOf, gait d1l'ﬂ(fi(T,M))‘= rthen fissaid tohave rank ratpeM.The tangent " map f,.,,issaid tobeinjective atpifr=dimM (dimM sdimN),If IfWeWrite xl(P2) E(Pi(P2) ="'i((P2(P2)) l:1»---,Fl,f0fTheC00fdiI1- F=dimN then f*pissaidtobesurjective. The mapping fforwhich fip alesofP2111(U2» (P2)and}’l(Pi) =(Pli(P1) =7Tl((P1(Pi)) J=1»--.,m, isinjective forallpeMIScalled animmersion andMisanimmersed f0fI116C001'diI1ﬂll@S OfP1in(U1,(Pi), then submanifold ofN.When theimmersion fisinjective itisreferred toas ,- ,- ,, i; 'bdd' d ' ' ' x(P2):f2l(y 1(p1)’y2(p1)’''"y(P1))' (433) I Zpleclilfilecti otlilegrwzige byldsulglmezleifglllllbiﬁgéfliﬁiil filfgariliagdfiiiilbeeliieldlsullirlfilzzs Ifwetake M2tobe1Bandwrite M1=Mthen fisusually called ifold. Inthiscase coordinate systems forNexist around f(p) endowing 134 MANIFOLDS f(M) with asmooth manifold structure. Asanexample consider themap f:S‘——>1B2where theimage point traverses thefigure 0once without stopping. Then fisaninjective immersion since both fand f,.are injective, and f(S‘) isaone- dimensional imbedded submanifold of1B2. Ifthe map uniformly traverses theimage setmore than once itbecomes animmersion, with f nolonger injective. Similarly iftheimage f(Sl)isthefigure 8traversed uniformly once themap isanimmersion, since although again ftis injective fisnot. The map fr[—1, 1]—>]B, xi->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 (4.2.1) 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 carve Conamanifold Misamap from anopen interval IC1BtoM.Ifpisanypoint ontheimage ofCand(U,cp)isa chart fortheneighbourhood ofpthen Cmay bespecified inthis neighbourhood bynreal-valued functions 1rlip[C(t)] EgalQC(t) teI. (4.4.1) Thus denoting (pl<>CbyClwewrite inalocal chart therepresentation ofC f(p)=C"(t). (4.42) Different parametrised curves canhave thesame image onM.Ifh maps theopen interval JCIBinto IC1Bthen C’:J+——>Missaid tobe areparametrisation ofC:I—>MifC’=C0h(seefigure 4.4). Where- asreparametrised curves have thesame image, ifwethink ofthe parameter asatime, achange ofparameter affects therate atwhich thatimage evolves.PARAMETRISED CURVES 135 CM I J:4 -..*—X/v Figure 4.4Different parametrised curves with thesame image. Iffisasmooth function defined intheneighbourhood ofPO=C(t0), 'hC '' - 3/:1;ined:?ggth atto,then thederivative offalong Catpg,Vgu(f)is Vfiiri=§(reCm). (4.43) (The reason foradopting thenotation Vf,U(f)willbeclear later.) Since foCisamap from ItoIB,smooth atto,thederivative in(4.4.3) needs I10further explanation. If(U, ip)isachart foraneighbourhood of P0=C(t0) then thechart map cpcanbeused toexpress V,‘,,.,(f)interms ofthedirectional derivative offq,=f0qr1_ Wemay write foCasthe composition ofmaps from 1to1B”and1B”to1B: f°C=(f°<P")<>(qv<>C). Thechain ruleofdifferentiation then gives ;,‘1,<r@oat)=(art/@x*><<io@>> (it). (44.4) If“I/"isthevector ‘inIR”with components dC"(t0)/dt then (4.4.4) expresses thederivative offalong Casthedirectional derivative off,, Eiancvee thishrelation golds forallfunctions fwehave acorrespondence tomien (ecurve ,with image containing P0,andthetangent vector .,q9(p0), V). Acurve C,with C1(/lo) =p0will be(jallgd equivalent toCatp0ifV,e,§(f)=Vf,U(f)forallfunctions f.Thus, for some choice ofchart map, equivalent curves atp0correspond tothe Same tangent vector inT,,L,,,,)IlFi”. Bytaking allcurves passing through p0 Weobtain aone-to-one correspondence between equivalence classes of Curves andvectors inT,(,,)lF1" (seefigure 4.5). 136 MANIFOLDS llIR I f I *0 wtfLpof ‘P Figure 4.5This diagram illustrates therelation between real func tions onMandcurves. 4.5Tangent Vectors Inview oftheprevious section wecould define atangent vector tothe manifold Matthepoint p0tobeanequivalence class ofcurves passing through p0.Such aclass ofcurves defines adirection atthepoint p0 andenables functions tobedifferentiated. Further, foranychart map wecanputthisclass ofcurves intocorrespondence with atangent vector inIR”,thishaving been previously defined. Itismost convenient (and usual) toadopt anequivalent definition oftangent vectors, modelled on theabstraction ofdifferentiating along acurve. Atangent vector atp0 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,e,(f)to denote thederivative offalong Catp0:with thedefinition that we shall give I/inwillbeidentified with atangent vector, thetangent tothe curve CatP0,whose action onfisgiven by(4.4.3). Similarly the definition ofthetangent vector toIR”,based ontheintuitive idea ofa directed linesegment, isequivalent tothemore abstract definition of being aderivation into IBonfunctions. Given thetangent vector (p,V)eTpllei” 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.‘ .i. IR” £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 Mto atp.The collection ofallsuch equivalence classes iscalled thecollection ofgerms ofC°°maps atp.Clearly elements in [fM]yield thesame image forp.P Forexample consider thegerms ofC°°maps C:IR-+ Nat;_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 [CR]rather than toberelated toaparticular curve inthisclass. IffIM-—>Nand g:N—>Pareany representatives ofthegerms IfM,,I and I8iv,] Then thecomposition [gN]0[fM] isthegerm obtained bycomposing representatives :namely glof.Similarly wedefine the pull-back ofgerms interms ofanyrepresentitives [fl*[el =[fie]=Is°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=f(i?) +g(p) and(fg)(p) =f(p)g(,n). Byidentifying theconstantunctions with thereal numbers thering @(M) may beregarded asa rial vector space, and hence analgebra. Aderivation into IRon IJ"(M)p] ISEllinear map XI[@(M),] ->IBthatobeys theLeibnitz rule X(fif2) =X(fi)f2(I-'7) +fi(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) =0and hence, bylinearity, Xannihilates any element of The vector space ofderivations oftheabove germs at pEMisdefined asthetangent space T,,M ofthesmooth manifold atp. d_We introduced earlier the pull-back map flassociated with the llfeemmphlsm M-> N,P =f(p). The tangent map atp associated with fisdenoted f.,.,,andisdefined interms off*by fip:T,,M ——> T,N Xi—->f,pX =Xf'=_ (4_5_3) Thus (see figure 4.6) f*,,X isaderivation onelements ge[%(N)f(p)] obtained bypulling back gwith fl‘andthen acting with X,thatis 138 MANIFOLDS (f*,.X)(8) =X(f*(8)) =X(3°f)- (4-5-4) From thispoint onweshall alsoapply thedefinition ofatangent vector being aderivation into IBonfunctions, totangent vectors tolB“‘. We must therefore show theequivalence with theprevious definition ofa tangent vector being anordered pairofelements from lB"‘. LetXE(p, V)eT,lB"‘. Ifhisareal-valued function onlB""then wedefine Xto map htoIRbytaking thedirectional derivative, thatis X(/1)=Dvh(P)- With thisrule thetangent vector Xisaderivation onfunctions inthe neighbourhood ofp.Italso ensures theconsistency ofthedefinition of thetangent map given in(4.5.3) with theearlier definition (4.2.3), as willbeexplicitly demonstrated inamoment. IR 90>‘ 9 f_,_p \\ / XETMP IRf,,,,XET,N M f N Figure 4.6Thetangent mapf,.,,:T,,M ->T,N. Wenow construct alocal basis forT,_.,M interms ofalocal chart germ atp,[cpp] :M—> IB”,that assigns thepoint peMtotheorigin inIB". As usual letx°", v=1,...,ndenote the coordinate maps <p":UM->IR.Then qp*isamap from function germs inIR”tofunction germs inMand(pi,maps tangent vectors from T,,M toTOIB". Ofall thederivations onreal-valued functions onIR”wedenote byX,,eTOIB", thepartial derivative: Xt=I@(1B”)nI —"->13 [fl%>I(5f/9X"’)(0)l- (4-5-5) Suppose a"X,, =0forsome nrealnumbers a",then since (X,,(x*“))(0) =at,acting onx“gives at“=0.Thus theX,,arelinearly independent andthentangent vectors {X,,}form alocal basis forthen-dimensional vector space TOIB . We may express any tangent vector XeT,,M interms ofii '|u\I'i'nh¢';Il;A6§:"me-w..TANGENT VECTORS 139 tp,,Xe TOIB”. Iffe @(M) then bywriting f=q9*f,, wehave X(f)=X(<P*fn) =(<P*pX)fn- (4-5-6) Inanatural basis associated with thechart (UM,Q9) cp,,X=Za"(8/E9x"’), (4.51)v=1 where itistobeunderstood thatthederivative actsatx"=0,thisgives X(f) =[(£1a"(8/8x"))f,, (x’(p), ...,x”(p)). (4.5.8) Often forcomputations, real-valued maps fonMarespecified locally interms oftheir local representatives f,=f0tp'lonIB"andthedetails ofthechart tparesuppressed. However itmay beimportant when dealing with global properties ofmanifolds toremember thedistinction between fandf,,since forageneral manifold itisnotpossible tofind anatlas consisting ofasingle chart. Just asthecharts areoften suppressed when discussing real-valued maps, inasimilar way the representative ipof<><p'lofamap between manifolds isoften written with thegharts ipandtpomitted. Inthefollowing weshall denote such a map byf.Wemay specify anyXeT,,M bygiving q0*,,X, asin(4.5.7). Itiscommon nottodistinguish cp*,X from X,identifying (8/ex V)with a tangent vector toM.Having pointed out the distinction weshall nevertheless employ thisabuse ofnotation inthefollowing sections. Consider theexpression forthe tangent map f.,.,, where fisa representative ofagerm atpfrom some n-dimensional manifold Mto some m-dimensional manifold N.Suppose (xl, ...,x")arelocal chart functions that assign topeMtheorigin ofIB"and (yl, ...,y”‘) are local chart functions that assign tof(p) eNtheorigin ofIR”. Thus f may bespecified interms ofthemreal-valued functions (fl, ...,1”") andwerepresent itbythemap t:v<n")--+i=r~. (X1, ...,x")i—-—> (yl=fl(xl, ...,x”), ...,ym=f”'(xl, ...,x”)). Werecall that {X.,} ={(8/8x ")}isabasis forTOIB” inthischart. Ifgis anyelement of[@(lB”’)0] then (f*n(@/@X”))8 =(3/@X")(f*3) =(9/@X")(e °ll =;(38/9)/”)(9)(@f“/@X”)(0) ormore simply f,.0(8/Bx”) =i(8f”(0)/8x")(8/By“). (4.5.9) 140 MANIFOLDS The action offinonanarbitrary vector inTOIB“ now follows directly since froislinear: f*0(a"’(E3/8x")) =a"'f,.0(8/8x”) =a"’(8f“/8x")(0)(8/Syi“). (45.10) The Jacobian matrix gives arepresentation ofthelinear map between T,,M andTm,)N. ‘_ Equation (4.5.10) expresses the chain rule ofdifferentiation and establishes theequivalence ofdefinitions (4.2.3) and (4.5.3) forthe tangent map onTPIB”. IfAeTOIB” isregarded asanordered pair, A=(0,a)with a=Eﬁzlale, inthenatural basis forIFi"~then Ais equivalent tothederivation a"(8/8x "')|0. The effect off*Qonthis 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 {(2,} isthenatural basis forIBT‘. From (4.2.6) werecognise thisas(f(0), D,f(0)), which istheform off*0A given in(4.2.3). Figure 4.7summarises therelationship between ipandfandthemaps thatthey induce. Letusnext observe thatif1/1:U1(lB") —>U2(IB”) xi"i>x'it =1j,#(x1, ,__,x") (4.5.11) wemay infer from theabove that ip*0(E:9/E-Bx”) =(at/Ir/axr)(0)(a/aw). (4.512) Thetangent vector XatpeUM, thatwasrepresented inthechart (UM, go)bycp..,,X =a"(8/59x"), will have adifferent representation inthe chart (UM, ip0(p),since (1))0q;),pX =1/1,,0(p,,,X =tp*0(a"(E9/6x”)) =a"’(&)ipf’/8x”)(0)(8/8x’P) (from (4.5.12)) Ea'»"(8/<'Jx"°) where a’f’=(61/1*’/£9x'“)(O)a‘“. - iii T,p{pjIRn filflpl /ﬁg V W 1,0,, Ulmpi 7-M faip THPIN P at - - (pip) (qJof1tpl=(fotp)lp) IR-lri T 7 if 7WeWOT74? IWe ejeiee" % ﬂ P i f(p) M TIWe zeeeﬁnwweif N Figure 4.7Relations between tpandfandthemaps they induce.TANGENT VECTORS 141 This representation ofthesame tangent vector XeTPM atpina different chart should bedistinguished from the tangent vector f,,,X eTf(,)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 {(6/8x'P)} atptothe natural components {av} ofXinthebasis {(8/E9x")} ofadifferent chart about p,may berecognised asaGl(n, IR)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(t0) =p0then thetangent vector toCatp0is Vs,EC.,,,(8/St) (4.5.13) soforfe%(M) V5.(f)=<o..<@/@t>><r> =(8/@t)(f° oat).Thus thetangent vector toCatP0maps functions totheir derivative along thecurve atp0,aswas anticipated bythechoice ofnotation in (4.4.3) (c..),,(a/at) =((E3Cl/8t))(t0)(8/E9x)l GT,,,M. (4.5.14) Asanillustration consider C:(0,1) —>IB2given by Cl(t) =asin bt C2(t) =acosbt a,beIB. If{(8/Sxl), (8/8x2)} isanatural basis forTmlli-I2 then c..,,(a/at) =(act/ai)(i,)(a/av‘). From theabove wehave (<3/8t)Cl(t0) ECl(t0) =abcosbto=bC2(t0) (SC2/&9t)(t0) EC2(t0) =—-absinbi,=—bC1(t0). 4.6Vector Fields Sofartangent vectors have been associated with points onthemanifold. Bysmoothly assigning atangent vector toeach point wedefine avector 142 IVIANIFOLDS 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(1f+tie)=/lX(f) +i4X(s) 4,ve1B;f.seWM) X(fs) =X(f)e +fX(s)- (4-6-1) (Intheprevious 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 Tl(M). Given anXeTl(M) wemay define avector X,eT,M, forany peM,by (Xf)(p) ZX,f. (4.6.2) Itisclear from thederivation properties ofXandX,,that thisdoes indeed define atangent vector. Since vector fields map functions to functions wemay define aproduct inanobvious way. For X, YeTl(M) XY: @(M) ——> @(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)(fs) =(XY)(f)e +f(XY)(s) +X(f)Y(s) +Y(f)X(s)- From thisitisclear that wecanobtain anew vector field from the commutator oftwovector fields [X,Y]=XY-YX. (4-6-4) Being thecommutator ofanassociative product thisbracket operation onvector fields isantisymmetric andsatisfies theJacobi identity [{x,Y],2]+[[Y,z],X]+[[2,x],Y]=0. (46.5) Smooth vector fields form amodule (see Appendix A)over @(M), and hence avector space over IBidentified with theconstant functions. The commutator then turns thevector fields into an(infinite-dimensional) Liealgebra. Thecommutator isalsocalled theLiebracket.4 i ifVECTOR FIELDS 143 Iff:M—>N isasmooth map between manifolds then, forany peM, thetangent map ft,sends T,,M toTfL,)N. IfXand Yare smooth vector fields onMandNrespectively, with X,,andY,given by (4.6.2), such that 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,_.,,X Ef,.,X. Iffisnotonto then smooth vector fields onNthat are f-related toXeTl(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 XeTl(M) there is aunique YeTl(N) such that Aswenoted intheprevious section itiscommon nottodistinguish XPeT,M from itscoordinate representation (p..,,X. Thus if(U,cp)isa chart fortheneighbourhood ofp,with coordinate functions {xl}, one identifies {(8/8x")],} with abasis forT,M. IfXeTl(M) then inthe neighbourhood ofpwecan express XasX=X"(El/Sxl), where Xle@(M) arenotdistinguished from their representations inthischart. The elements (8/ex’) form abasis forTl(U), the9?-module ofsmooth vector fields onU.They form thenatural local basis orlocal coordinate basis. Since thering ofsmooth functions isnotadivision ring there is noreason why the@-module Tl(M) should have abasis, andingeneral itwillnothave. 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 Zn-dimensional manifold TM bycollecting together allthetangent spaces T,,M from allpoints ofM: TM=UT,M. (4.7.1)P Anelement ofTMisatangent vector XP,labelled bythepoint pand 144 MANIFOLDS itscomponents insome basis forT,,M. Moreover theconstruction of TM must satisfy certain smoothness criteria with respect tothese assignments. Ifatangent vector X,,eT,,M isrepresentedin alocal chart ((1,,W)withcoordinate mapstx’).by<i4,X,, =We/E14’ thenWedefine (f(p), yf(p)) eIRE”asthecoordinates ofapoint inTM. That is, thechart (UM, cpM) forMinduces achart (U-,-M, cpTM) forTMby (e0TM)(Xp) E(f(p), f(p)) where (pM(p) Exl(p)e,- and((pM')=l<pXp I)’i(P)(a/ax’) i(‘PMI(PI’ lei} belllg thenatural basis forIR”.Aswehave remarked earlier atangent vector toIR”isequivalent toanelement ofIP12”: thederivative inthedirection Vatpbeing equivalent to(p,V).Thus (p7~M assigns toX,,theelement ofIP12”equivalent to(tpM),,X, eT,,(,)IFi”.. _ . . Since Misadifferentiable manifold itispossible togive a.topoIogy anddifferentiable manifold structure toTM. If(UTM,rpTM) isalocal chart forTM, induced by(UM, goM),then themap specifying achange ofcoordinates inTM: (cpTM), 0(tp}M), :IBZ”—>I32”, 13glveﬂ 111terms of themap specifying achange ofcoordinates onM (<vn)2O(<t>n‘)t11B”—>IR” f(p)E-—>x’l(P)- Thetangent map is (((pM)2 °(<PU))t*t<nnttp> IT(§0.n)t(P)lRn _’T<tn>21B”THE TANGENT BUNDLE 145 From itsconstruction UTMisdiffeomorphic toUMXIR”, butglobally TMneed notbeaproduct manifold. Aproduct manifold M><Nis formed from ordered pairs ofelements from themanifolds MandN.If {(U,. <p,)} and{(V,, ip,)} areatlases forMandNrespectively then an atlas forM><Nisdefined bythecollection ofcharts (pa Xlpb ZUa XVb We lBdimM+dimN (pt l‘__> ((pa(p)> Such acollection ofmaps satisfies thecriteria forbeing anatlas. The local product structure ofTM allows the definition ofanatural projection map II:TM———->M,X, I——>p (4.7.3) which identifies thepoint onMtowhich thetangent vector inTMis attached. Itisconvenient topicture UTM,with itslocal product structure exposed, asaspace over UM(see figure 4.9). Allthetangent vectors at paredrawn asthespace T,,M associated bytheprojection IItoapoint pofM.The local coordinate representative ofHisusually given the same name, II:I82”—>IR”,(xl,yl)I—>xl.(The inverse image setT,,M is sometimes denoted lT'l(p) and UTM denoted HCl(U M)although this notation should notbeconfused with thenotion ofaninverse mapl). r,/*4 U k k tt k t5 ii TM IRZD y/<5)/ax t-_->y((61: /ax))a/ax (where weareusing summation convention) sothat ((q),.,,), O((p},i,,)1)(xlt y")(q) =(f(p), y"(P)(@X"/@x")(P))- (47-2) These maps define (seefigure 4.8)adiffeomorphism (cpTM),2 of‘Pm p,X) -4 -44> --4 1 *-—--2-..-It-4.-in5-< ((PrM)i((UrM)i O(UTMI2) UM [PM ‘Rn onto ((pTM)2((UTM)'l Q(UTA/1)2)~ at’€lermli llpriili llp7Ml12 Figure 4.8---—-—o-i-— Z>—— -—-0---P X Figure 4.9Thelocal product structure ofthetangent bundle. Theexistence ofaprojection map makes TMinto afibred space, the elements related topbyHbeing 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); indifferent ways. This islikethedifference between a cylindrical ribbon with atwist andonewithout atwist. Inboth cases the twist can beeliminated from any neighbourhood butisanessential characteristic distinguishing oneribbon from theother. 146 MANIFOLDS Asmooth section ofTMisaC°°map 0":M_>TM (4.7.4) such that H00 =(id)M. Thus o(p)e T,M forallpeM. Itmay be represented inlocal charts (UTM,Q9rM),(UM= (PM) by 5(x) =(xi,yiE()'l(x)) (4.'/.5) where the{ol} arereal functions onUM» (see figure 4.10). Thus o smoothly assigns atangent vector toeach point peM.Wemay identify asmooth section owith asmooth vector field Xby (Xf)(P) =0(P)f Vfe@(M)- (4-7-6) Inthisway every smooth vector field onMisequivalent toasmooth section ofTM. IfFTM isthespace ofsmooth sections ofTMwewill henceforth usetheabove toidentify Tl(M) with FTM. Ur/4 \ltJ.Xl rpm (x.y) ___________ __ll_________ #_ / 4 U ‘CI L UM kppf IR!) p X Figure 4.10 Alocal section anditsrepresentation. 4.8Differential 1-Forms The smooth 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 toTl(M); thatis,an@-valued 9-linear map onvector fields. A1-form issmooth ifitmaps smooth vectors tosmoothDIFFERENTIAL 1-FORMS 147 functions. Asmooth 1-form onMwill also becalled adifferential 1-form. The space ofsmooth 1-forms onMisdenoted T,(M). If XeTl(M) assigns X,,e TPM tothepoint pthen forweT1(M) we define co,by :wp(Xp)- (4-8-1) Clearly 0),,isalinear map from T,,M toIB,that is,anelement ofthe dual space T",‘,M. Elements ofT“;,M arecalled co-vectors or1-forms atp.Thus wsmoothly assigns anelement ofT’j,M toevery point pof M.Inanalogy totheconstruction ofTMwemay collect together allthe cotangent spaces andform anewspace T*M=L5JT°‘;,M (48.2) Like TMthespace T*M inherits amanifold structure from that ofM, with anatural projection from T*M toM.With this structure T*M becomes thecotangent bundle. Wemay identify asmooth 1-form onM with asmooth section ofT*M. SoifFT*M isthespace ofsmooth sections wehave anatural equivalence between elements ofFT*M and T,(M). Forevery fe§(M) wemay associate anelement dfeT1(M) bythe rule X(f) =(df)(X) VXe Tl(M). (4.8.3) That is,dfeFT*M assigns (df), eT’f.,M tothepoint pwith X,.<t)=<4r),,<X..>- (48.4)Theelement (df), which maps T,,M toIBisrelated tof,,,which maps T,,M toTf(,,)lB :infact they are naturally isomorphic. Ifgisa real-valued function onIB,AI-——>g(/I), then from (4.5.4) = Of)- Bythechain rule dsXi.-(8 °f)=Xp(f)?1_Z(f(p))' Thus f,t,X,, eTf(,)lB isequivalent totheordered pair (f(p),-‘Q-f) =(f(p)1(df)p(Xp))‘ Theexistence andlinearity offt,ensures that (4.8.3) really does define a1-form. Despite thisnatural isomorphism weshall distinguish themaps (df), andf.,,,. _ Ifx’isoneofthecoordinate functions and(8/Bx’) isavector from thenatural local basis then (4.8.3) gives dxl(8/Sxi) =(Sxl/Eixl) = (4.8.5) 148 MANIFOLDS Thus {dx’} isalocal basis forT1(M) naturally dual tothebasis {(69/E9xl)}. Insome coordinate neighbourhood, foranyfe@(M), dfcan beexpanded inalocal basis df= df(E9/8x")dx",. giving theclassical expression df=(Sf/8x")dx" (4.8.6) from (4.8.3). Itisworth emphasising that inthisexpression thedxlare not‘infinitesimal increments ofthecoordinates’ butlinear mappings on thetangent vectors. Byevaluating thisexpression onavector tangent to some curve weobtain thederivative offalong thecurve: inthisway df encodes theway inwhich thevalue offchanges asthepoint inM begins tomove. The components ofa1-form with respect tothenatural basis {dxl}, associated with thechart (UM, (,0),areused tocoordinate thebundle T*M. Ifa/6Tl(M) with a=oz”dx”, a/He@(M), then ais associated with thesmooth section p p:M——-—> T*M represented inalocal chart by X”(P) *—>(X”(P)» %(P))- Iff:M—>Nisasmooth map then wehave already defined the pull-back map f*that takes asmooth function gonNtoasmooth function f*gonM,fkgIgOf.Thus f*gisevaluated atpbyusing fto send pfrom MtoNwhere itisevaluated with g.Inthesame spirit we candefine thepull-back ofa1-form coonNtoa1-form f*w onM.If Xp6TPM wedefine (f"‘w),,X,, =wﬂ,,)(f,.,,Xp). (4.8.7) Weneed tocheck that forasmooth assignment ofXptoTPM anda smooth wonNthisrule assigns (f*w)p smoothly toT”:,M. This canbe seen from thelocal coordinate expression for(4.8.7). Firstly wenote thatforg6@(M) (f*(8w))pXp =(gw)f(P)(.f*pXp) =(8Of)(p)wf(Pl(f*pXp) =(f*8)(P)(f*w)pXp thus f*(8w) =(f*8)(f*w)- (4-8-8) If{xl} i=1,...,mand{yf}j=1,...,rtarelocal coordinates forM and Nsuch that the coordinate representation offisgiven by yl=f7(x‘), then ifX=X’(8/fix’) f*pXp =X’(P)(@f’l/@X’)(1>)(@/@yl)lap)DIFFERENTIAL 1-FoRMs 149 andfordxlE'I"’}U,)N dxj(f*pXp) =Xi(P)(5f"/5X’l)(P) =(9f’l/5X")(P) dX‘(Xp)- Itfollows from (4.8.8) thatifw=50,-dxl flu) =((0,Of)(8fl/Bx‘) dxi (4.8.9) andthesmoothness offand thecomponent functions wjensure that f“cuissmooth. IfweT";,M then wecanuseanychart fortheneighbourhood ofpto represent co,using thenatural local basis. Given twodifferent charts we cancompare therepresentations ofcobyusing thepull-back ofthemap that relates thecharts. Suppose (UM, cp)isachart fortheneighbour- hood ofpwith q0(UM) =U1.Given adiffeomorphism 1/1:U1(IR”)-> U2(lB") wehave anewchart (UM, tp0rp).If1/1isspecified by 1//IU1(1B") _—>U203”) x”l———>x’*‘ =1/)(x1, ...,x") then wehave theinverse map 1//"IUz(1B”)——> U105”) x'”I——> x”=1/1“1”(x", ...,x’”). in§4.5 weshowed that ifXeTpM isrepresented inthe(UM, (p)chart Y (p*pX : 1,)l(p(p) then therepresentative in(UM, 1/10cp)is (111°<P)*pX =X"(@1/1*“/9X")(<P(P))(9/5X'”)ln,.».@)@)- When representing a1-form wehave toremember that thepull-back map acts intheopposite direction tothemap itself. Since chart maps areinvertible weT’j,M isrepresented in(UM, qra)by<;0;§,§“w, with ‘19¥<i=»T“’ =‘Wdxvlw-> say. Inthe chart (UM, 1/1Orp) the representation of0)is (1/10q0)(',,,1;)(,,) co,where (‘P°‘P)<1i$><p)w =1/’<1l$><p)‘l’@?tl»’i 6°=w(;'1:;)(P)wvdxv =wv(51P_1v/3x!’u)((W °(P)(P)) dx'”|(¢.¢>)(p) ‘ULdx'”|(w@<v)(p) from (4.8.9). Thus whereas thecomponents ofthetangent vector Xare transformed with theJacobian matrix representing tp,thecomponents of the1-form totransform with theinverse matrix since 150 MANIFOLDS (E91/1“/Eix “‘)(E3q1_1“'/Eix’ ")=<5-Z. That is,thecomponents oftotransform contragradiently tothose ofX. The behaviour ofthechange inthecomponents oftoinduced by changing thecoordinate basis isthehistorical characterisation ofa covariant vector (seefigure 4.11). UM r>>1ll||>U1 \P \l1o\P U2 \ ->gg‘.77 q, ‘F = i L131:w(_-6 D V bx’V 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 covariantdegree randcontravariant degree s. Itispurely forconvenience that wehave 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 theother wayaround. Clearly wehave T§(T“§,M) =T§(T,,M). Whereas thecotangent space atanypoint isarealvector space theset of1-form fields forms an@-module. Inthesame wayasweconstructed thetensor product ofvector spaces wemay construct thetensor product ofthe@-module of1-form fields with itself and thedual module ofTENSOR FIELDS 151 5mOOth vector fields. Elements ofthetensor product module, Tf,(M), arecalled tensor fields ofcovariant degree randcontravariant degree s. Weidentify T3(M) with @(M). Thus anelement ofTf,(M) smoothly assigns anelement ofT§(T’f,M) toeach point pinM.Asforthecase of vector and1-form fields thisway ofregarding tensor fields isformalised interms ofafibre bundle, thebundle ofmixed tensors TfM. Thus TQM =UpTi(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 Ti(M) with thespace ofsmooth sections l"T;‘.M. If(U,(,0)isachart forsome neighbourhood ofM,with chart maps {x‘}, then {(8/<'Jx")} and {dxf} arebases forT1(U) and T1(U) respec- tively. Thus locally anytensor field TeT§(M) canbewritten as T=Tili} ‘J.rit.l2- --lidxli ® dxiz ®...Q)arr®(E23/Z-Bxll) ®(a/axh) ®...®(8/Eixli). (4.9.1) This isjust aformula from §1.5 rewritten with dx"1 replacing elland (8/Eixll) replacing XI-1.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. 'The submodule ofTi(M) formed byalltotally antisymmetric covar- ianttensor fields forms theexterior algebra ofdifferential forms, A(M), under theexterior product of(1.2.2). Weshall identify A0(M) with WM). Thus asmooth differential form isassociated with asmooth section oftheexterior bundle AM =UpA(T*:,M), Whereas anglemgnt oftheexterior algebra ofanarbitrary vector space iscalled anexterior gorm, theterm differential form isreserved foranelement ofthe JP-module /\(M). If[3eF/\,M, section ofthebundle ofexterior r-forms Wemay usealocal coordinate basis towrite ﬁ= Zli,,,,,,___,,,(dx"i),\(dx"2)A...,\(dxi“*) (49.2)#1$~tJ2$- --Hr equivalently 1 E Z kjﬂﬂlﬂg. . A A '‘°A Where thesummation convention isused. These formulae aretrans- cribed from §1.2with thesubstitution ofdx*“*fore”*‘. 152 MANIFOLDS Given asmooth map fbetween two manifolds theinduced maps on thetangent andcotangent spaces can, tosome extent, beextended to tensor fields. Iff:M->Nthen weextend themap f.,.,,toanIR-linear map oncontravariant tensors atp f*pITS(T°;M)i> TS(T’}(p)N) X1®X2®...®X, i——>f*pX1®f.i,,X2 ®...®f.,.,,X, X,-G TPM. (4.9.3) 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 f*which enables smooth 1-form fields onNtobepulled back tosmooth 1-forms onM. This pull-back map may beextended toanB-linear map onsmooth covariant tensors onN FI W w‘®w2®...®w’>—> Fm‘ ®f*w2 ®...®fire" rule T1(N). (4.9.4) Forsuch adefinition tomake sense itisimportant thatwehave (4.8.8), thatis f”“(ew) =(f*e)(f"‘w) sE@(N), weT1(N)- ForﬁeT,(N) and{Xi} eTPM i=1,...,rwehave (f*ﬁ)p(X1¢ X2: ''-1Xr) 2ﬁf(p)(f*pX1>f*pX2> '''>f*pXr)' Ingeneral thesmooth map f:M—>Ndoes notinduce amap on smooth contravariant tensor fields onM;noronmixed tensor fields, the maps f.,.,,andf”‘:,,acting inopposite directions. Forthespecial case ofa 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. Ifrp:M—>Nisadiffeomorphism then wedefine (pby $1Ti(M) —>Ti(N) q’5(w1®a)2®...®co"®X1®...®X,) =<,‘0‘1*cu1® tp“‘*w2 ®...®cp“1*w’ ®<p.,.X1®... ®<p*X, ofeT1(M), X,eT1(M). (4.9.5) Again werequire (4.8.8) forconsistency. Equivalently. ii ,. 2 -=I M-:_. Pl‘- ,.. fr»,- .-;_. --}- ‘iTENSOR FIELDS 153 gator®a,2®...®(,,»®X,®...®Xi~)(Yi,Y2..... Y,,a/‘,...,a*‘) :o)1@w2®...®o)’®X1®___ @Xi<<i*1*Yt<i~1tYt. H-t<P**Y,.<P*a/1. ...,Q9*6l’s). (49.6)Example 4.2 Forthesmooth map ‘l"lB2“"lB2 P'““"<P(P) W»1’)'*“’(<P1(P)> <P2(P)) =(acost+bsint, boost -asin,-) with taconstant, theinverse isgiven by ‘P'11B2 Hr152 Pr-—><r‘1(p) la’bl"T ((‘/’h1)1(P)» ((P_1)2(P)) =(acost —bsin t,bcost +asin t). Foravector field Y,<;’5Y=.q9,,Y_ Taking Y=x2(@/ax) _|_xy(a/8),), with xandythestandard coordinates on1B2,gives <P*pYp =x2(P){(3<P1/5X)(P)(5/5X)lW) +(a<p2/ax)(p)(a/ay)|,,(,,} +x(P)J/(P){(3(P1/3y)(P)(3/5x)|W) +(a<p2/ay)(p)(a/ay)|,M,,} (‘P*Y)¢>o> = (xzcost +xysint)(p)(E9/E9x)[,,,(,,) +(xycost —xzsin t)(p)(8/8y)[q,(p) Wemayuse <p‘1toexpress thecoordinates ofpinterms ofthose of fP(p), giving _ ((l9*Y)¢(p) =@2905‘ “X)’SinT)(<P(P))(3/3x)l¢»(p) 3 So 2] <P*Y =(xzcost —xysint)(8/ox) +(xycost -yzsin t)(E9/8y). .9 Weconsider now a1-form cr=x2dx+xydy ((l5a’)w(t>) =(<P_1*a’)¢@) =Q9_1*a/1, 1 ='”2(1’)l<a(‘P_1)I/a")(‘P(P))dxlup) +(5(<P_1)1/3y)(<P(P))dyl<p(p)}+(x)’)(P)i(5(§9%1)2/ax)((P(P))d-Ylwt) +(3(q9*1)2/8y)(<p(p))dy|,,,(p)} ={)t2(p) cost +(xy)(p) sint}dxlq,(p) +{(xy)(p)<>0Sr —r2(p)Sinr}dy|.,M,, ==(xicost "X)’Si"l)(<P(P))dx|¢»(p) +(xycost —yzsin t)(<p(p))dy|,,,(p)+(xyCOS‘ *Yzsln l)(‘P(P))(3/3y)l¢@)- ll 154 MANIFOLDS asintheprevious example, so (pct=(xzcost —xysint)dx +(xycost —yzsin t)dy. Wehave (4%/)(<’t7’Y)(P) =(xzcost —xysint)2(p) +(xycost —y2sint)2(p) ={(xcost ——ysin t)4+(xcost —ysin t)2(y cost +xsin t)2}(p) =(X4+X2)’2)(<P"1(P)) =(¢Y(Y))(<P_1(P)) =(¢Y(Y) °<P"'1)(P) so (<i5¢Y)(<i5Y) =<'t3(<1<(Y))- 4.10 Exterior Derivatives In§4.8 weassociated with every fe97*(M) anelement dfEPT*M. Thus wehave anoperator mapping functions to1-forms. Wemay extend this operator toanIR-linear map onPAM: d:PAPM :> PAPHM (4.10.1) with theproperties: df(X) =Xf XePAM, fe‘¥M (4.10.2a) d(a/Aﬁ) =da,(ﬂ +(-—1)Pa./,(dB a/ePA,,M, ﬁePAM (4.10.2b) ddEd2=0. (4.10.2c) The operator discalled theexterior derivative. Itsexistence and uniqueness aremost easily demonstrated using alocal chart and the properties oftheexterior algebra. Inanycoordinate neighbourhood of Manelement ofPAM canbeexpressed inalocal natural basis. Since d isIR-linear itissufficient toconsider itseffect onanelement oftheform cu=gdx‘1,(...,(dx‘* ge@(M). From properties (4.10.2b) and(4.10.2c) dw=dg/(dx"* ,\...,(dx‘A* with dggiven byproperty (4.10.2a). Sofortheassumed form oftowe have theunique form fordw.Thedefining properties ofdenable do)toii 3:3.‘ I.";j%s€ IF?-.'-F4.' 2%EXTERIOR DERIVATIVES 155 beevaluated onasetofvector fields, foranywePAM, WeC()1'15idgf firsta1-form, itbeing sufficient toassume w=gdx x,ge@(M) thus d“’(X1=X2)= (d8/\dI)(X1,X2) =i{d8(X1)dx(X2) _'dg(X2)dx(X1)} from thedefinition oftheexterior product. From property (4.10.2a) 1dw<X1»X2)=X1(g)dim)-mg)dx(X1) =X1(edX(X2)) *gX1(dx(X2)) -X2(gdx(X1)) +gX2(dx(X1)) :X1(w(X2)) TX2(w(X1)) +8'lX2, X110‘)- Using thisproperty once again inthelastterm gives 2dw(X1i X2)=Xi(¢°(X2))"' X2(¢°(X1)) ""0J([/Y1, X2]). Itfollows thatforanyavePAIM (dw)(X, Y)=(1/2){X(a/(Y)) -Y(a/(X)) -a([X,Y])}.(4.103) Similarly ifaePAZM (d<r)(X> Y,Z)=(1/3){X(w(Y, Z))+Y(a/(Z, X))+Z(a(X, Y)) 6<r(lX,Y],Z)~a’([Y=Z],X)-a/([2.X],Y)} VX, Y,ZePTM. (4_1()_4) Forthegeneral caseofaePA,M 1’ _ A <d<r><X@» X1»---.X.)=;;—,~ (“1)lX;(¢1’(X0, ....X,-,....X») 1 . A A+ r+10<j;}r(<(__1)]+ka/([Xj‘7 Xk]7 X0! '''9Xja **'9Xk? '''9Xr) VXUJ X1: '''aXrErTM where X,-means omit thisterm from theargument list. Animportant property ofdisthat itcommutes with thepull-back map ff:PAN —>PAM induced from adiffeomorphism f:M->N_First observe thatifge@(M), XePTM, then (f*ds)(X) =ds(f*X) =(f*X)(g) (by4.10.2a) =X(f*s) __ =d(j°“g)(X) (using property (4.10.2a) again) giving f*ds=d(f*e) (410.6) 156 MANIFOLDS Now consider d{f*(gdx*'* /\dxlz/\~--/\dxikll =d{f*(8dxi‘) /\f*(dx"2) /\---/\f*(dxl")} =difklgdxnl /\dlfkxlzl /\---/\dlfkxlulll from above =d(f*(sdr‘*)) /\d<r*i~>/\-../\d(f*r‘*) as<12=0 =d((f*s)(f“dX“)) /\f*(dX"2 /\---/\dxl‘) =d(f*s)xf*dx‘1/\f*(dX"/\- ~~/\did) =rdgmdxi/\r*(dx'>iv-»/\dc) =r*d<gdi~ /\we/\-..,\dwi- Itfollows since dandffareIB-linear maps that fwdZdf>I< (4.107) onarbitrary elements ofPAM.asd(f"dx’) =dd(f*Xi) :0 4.11 One-Parameter Diffeomorphisms andIntegral Curves .. . - - ' nedwith situationsInmany situations intheoretical physics one1:illgglgggld, This technical that canbedescribed Inwnfns ofﬂowtli Ozim lestcase tovisualise, the term isborrowed from What lsperhaps th6rfage The motion ofaﬂuidlaminar ﬂow ofafluid around asmoo sillofE-Iﬂow Ifeach element around avortex 1Sanother familiar fpxanip Followed in-time ittraces out - '' aowi _ofthemediupi 6Xp€£:3I1(i1II;%]E;lCfOr aSmooth ﬂow one canestabhsh a theimage oacur . . h ' 1lvector field. T6correspondence between local ﬂuid ﬂow_and auoca H_ _n notion ofaﬂow intime isnaturally associated with abilectlve m3PP1 g=. - 'tonamanifold MI0theﬂow taking anelglibourhood (Pgoftanpgllqﬁqime ForSomg ﬁxed aneighbourhood U(p)insome fixe ine - interval twedescribe such anevolution by (pt:U(p)—>U(P')- (4.11.1) — 'frI,where 1,,CBisanopen interval about '0.TO 65¢ arbitrary time interval wedefine cpinterms Of‘Pibl’ (P,WC(1XM)_>.M,(1.p)e—><P(1>P)=WP) (4-11-2) - - h‘ f som6 Where’ for each IEI’(prlsa106? dlﬁ-":iOml(llp)lSCmM Ifhldie isanU(p) CMtoU(p’) CM.Conversely, 01'6Y PIii-ONE-PARAMETER DIFFEOMORPHISMS AND INTEGRAL CURVES 157 1,,CIsuch that (p,isadiffeomorphism from U(p)toU(p’) for;E{P, Motivated bytheexample ofﬂuid ﬂows, inwhich theconfiguration of ﬂuid elements atanytime canbeobtained from thesuccessive compo- sition ofevolution maps, wedemand that (pg0(p,.l=<p,,,,2 Vtl,:2eIsuchthat:1+:2eI andthat Q99(p) =p VpeM. (4.11.3) Inparticular, tpfl=$1,. Families ofdiffeomorphisms ofthistype are called local one-parameter diffeomorphisms onM.Aone-parameter family oflocal diffeomorphisms gives risetoavector field onM.For every peMthemap (pdefines acurve <p(p)starting atp ___>M I*——>¢>,(.v)- Tosaythat thecurve starts atpmeans that g00(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 <p,n(p) = <p,;](p’) forsome (to,p)and(t{,,p’), then(4.11.4) €0i(P') =<P(t-t;,)+t(.(P’) =(Q9:-15° ‘Pi;,)(P') by(4-11-3) =vi-it(vti(P’)) =(Pi-i.;(<t>i.(P)) =(<29:-it °‘Pt(,)(P) =(Pt-t@+i,(P) by(4.11.3) again. Thus ifthecurves <p(p’) andtp(p) have image points iiicommon then g0(p') isareparametrisation of<p(p). Theparametrisa- tions merely differ bythe addition ofaconstant and so<p(') Pit =‘P(P)*(i_,,;+,,), and thetangent vectors agree where theimage points co' 'd ' ' inci e.Hence theone-parameter family oflocal diffeomorphisms defines atangent vector ateach point ofM;thesmoothness ofcp, 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 oflocald. . . ._iffeomorphisms 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,ti—>p(t), iscalled anintegral curve ofXifX lsC-related to(8/St). That is,ifCisspecified byC:tl——>)C’“ =)v“(t), 158 MANiEOLDs giving C...(8/St) =/'l“(t)(E9/E9x”)|M,), and inlocal coordinates X=f*"(8/ 8x”), then Cisanintegral curve ofXif /l*“(t)=f~(i1(i), ...,/l"(t)) (4.11.5) forju=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). The smoothness oftheft‘ ensures that such solutions arenotonly smooth functions oft,fortin some interval ICIB,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(E9/8y) —y(8/8x) ePTIBZ. Let C:I—>IB2, ti—> (A1(t),l2(t)) beanintegral curve ofXthat starts atthepoint (a, b)GIB2. Solving Ill=-12, Z2=/llsubject tothis condition gives: h1(t) =acost bsin t,)l.2(t) =bcost +asin t.Here we may take I=1B,themaximal integral curve mapping thewhole reallineinto the circle, thecurve being periodic with period 211. Avector field whose maximal integral curves starting atparedefined onallof1B,forevery peM,iscalled complete. Ingeneral thiswillnot bethecase, thedomain ofthemaximal integral curves depending on which point they start at.Introducing asuggestive notation wedenote by(p(p)themaximal integral curve ofXEPTM starting atp v(P)11,.»—>M t*> WP)- IftoE1,,with (p,U(p) =qthen setting h:Jq———>Ip t|—>t+t0 gives acurve 1p(q) =(p(p)0h.The images ofip(q) and<p(p)coincide, asdotheir tangent vectors since thereparametrisation merely involves theaddition ofaconstant. Thus 1p(q) iscertainly anintegral curve ofX, starting atq.IfIp=(a,b)then Jq=(a—t0, b—to)and since a<0<bwehave —t0elq, giving 1/)_,0(q) =p.Ifip(q) were not maximal, with JQCIQ,then reversing theargument would contradict I,, 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 thatifcp,isdefined by tp,:pii>q_o,(p) Vpwitht6IpONE-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.IfIpCIVpthen WC(I><M),andtpisdefined by (PIW~—> M,(I.P)meWP)- Each ip,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 p*1-> <p,(p)isrepresented by X”(P)im¢>*“(13x1(P), ---=x"(P)) M=1,---,rt where <P”(0»x1(P)» ~--tr"(P)) =X*“(1>) and (l0”(tl +I21x1(p)v '''>xn(p)) :(plultb (P101: x1(p)v '''9xn(p))7 (l92(t1= x1(p)> ''"vxn(p))i' --a q0”(t1, x1(p), ...,x"(p))}. (4.11.6) Wemay usethesmoothness ofthefunctions co”inthevariable tto obtain alinear approximation ofgo”forsmall t <t>‘“(t=x‘(1>), ---,X"(P)) =<P”(0»x1(P)» ~~‘1xn(p)) +tcp“(0, x1(p), ...,x”(p)) +...(4.11.7) where cpl“denotes thederivative with respect tot.Since <p(p) isan integral curve ofX,starting atp,ifinlocal coordinates X=f”(E9/8x”) wehave W0,f(p)» ~~-tx"(p)) =x”(P) and <P”(9,X1(P), ---iX"(P)) =f”(P)~ (4-11-8) Thus fortsufficiently small (4.11.6) may beapproximated by x”(p) ii> x”(p) +tf*“(p) +.... (4.11.9) Example 4.4 Ifxcoordinates 1Bthen asmooth vector field onBisX=x2(8/Bx). If q9(t,p)isthemaximal integral curve starting atpwerequire 160 MANIFOLDS ¢>(t,p)=<t>(t»P)2 v(0.p)=P- The gglution isq;(r, p)=p/(1 ~tp). Ifp>0wemust have te(—<><>.p'1). ifp-0.re(—e-it @@)wh11SIfOrr <0»IE(P»°°)~Thedomain W=UPIPistheregion ofIB2bounded byhyperbolae inthe bottom-left andupper-right quadrants. This isshown infigure 4.12. We canverify thatindeed (pgO(pm=cp,,+,1. P/(1—tip) P(l0l3(q9l1p) _ 1__([1 —(pIi+!g(p)- Wehave shown infigure 4.12 theeffect ofone ofthelocal diffeo- morphisms cp,. / T %\\ \ti.av:0'0VI’CQWOO asit¢'¢‘o'¢‘6'o'o‘¢‘¢'¢9o‘~o'¢'§‘§‘0‘ §\0'90‘f4____ _____ ‘l W ﬁr/‘ / Figure 4.12 This diagram illustrates theeffect ofalocal diffeomorphism cp,.£ -i -Tél. l()NE-PARAMETER DIFFEOMORPHISMS AND INTEGRAL CURVES Ifwemodify theabove example byrestricting Xtothemanifold M consisting oftheopen interval (1,O0),then themaximal integral curve starting atphasdomain I},=(p" -1,p'1). Sointhis case co,is defined somewhere ifte(—1, 1).The domain W’=Up]; Vp_eM is shown infigure 4.12. Kllﬂlkpj/;(pJ1 ' Y X X R0,/;lp) A’ .5‘ l Y i \ ix,ri,, P lfX:1 I Figure 4.13 The geometrical interpretation ofthecommutator [X,Y]of thevector fields XandY. Exercise 4.1 LetXandYbevector fields with cp(p)andtp(p)therespective integral curves starting atp,andtp,and"mp,theassociated local diffeomorphisms (see figure 4.13). Fortsufficiently small andpositive aone-parameter family oflocal diffeomorphisms isgiven byC,=1/1_(/, <><p_.(/, <>1/IV, <><;0(/,, with C(p) :ti—>C,(p) asmooth curve starting atp.IfC0(p) is thetangent vector toC(p)atthepoint pshow that C0(p)=[X,Y]P. Hint: For fe@(M)fo tp,=f+tXf+t2/2X2f +O(t3), where X2f=X(Xf). 4.12LieDerivatives 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 weshowed inthat section how theaction ofanyvector field onafunction isthederivative ofthatfunction along acurve; namely the integral curve ofthevector field. That is,ifQ7isanintegral curve ofX, starting atp <Xf>=g(re¢>><0> weam/1) =limI-1{mpom—f(p)}. <4-12-1) r—>0 I Since Xisasmooth vector field then associated with thecurve (p(p) starting atp,isthelocal diffeomorphism cp,ontheneighbourhood ofp. Itisinstructive torewrite theabove interms ofthepull-back (pi (Xf)(P) =13311-1 {(<P‘§f)(P) -f(P)}- (4-12-2) This form ofthederivative, Xonf,suggest ageneralisation toa derivative onanarbitrary tensor field Tel"T§M. Wemay usethemap r’;5_,, associated with thevector field X,tomap Twp) back tothepoint pwhere itcanbecompared with Tp.Ifthelimit asttends tozero of thedifference between thetwo tensors divided bytheparameter t exists, itiscalled theLiederivative atpofTwith respect toX,denoted by(§£XT)(p) (seefigure 4.14) ($T)(p) =limF1{((;b T)p—Tp} VpeM. (4.12.3)X r~—>0 ‘ht Ywt1‘P Xqlﬂpl ‘Prlpl X»?.___ LIEDERIVATIVES 163 The definition of<’;5,_, isgiven in(4.9.5). Intheparticular case of fEWM) ®—rf =(p:i*f =(plif andwehave ffxf=Xf VfeWM)» (4.124) Itfollows from (4.9.5) that SEXisaderivation onthealgebra oftensor fields 55X(S®T)=EXS®T+s®§£XT (4.125) inparticular iffe@(M) §£X(fT) =(Xf)T +f~5£XT- (4.12.6) From (4.9.6) wemay deduce thatSEXcommutes with contractions: $X(T((1/1’ ""a/7" X1’ '"t =(CEEXY-l)(a/19 "'1a/ra Xla ..-;Xs) ‘l’kg T(Cl’1, ...,$XCYk, ...,a’,, X1, ...,XS) :1 +T(a/1,...,a/,,X1,...,sexxk, ...,X,). (4.127) ‘Applying thegeneral definition ofaLiederivative toavector field Y gives (§£XY)(p) =1ri_13;%{(<P-,*Y),, -Yp}' (4.12.s) Foranyfe@(M) (§£XY)pf: t—1{[((p*’)*(Y<P,(r>))lf '_Ypfl‘ =1393:-1{YW>(¢*_,f> -Yin. Now from (412.2) <P’if= f+IXf+ O02) lLp'l*Yl’° hence (§£XY) f=1i_51r”1 {Yq,,(p)[;‘ -tXf+o(¢2)] -Ypf}Y0 . P I =-Yp(Xf) +li1_)r[‘}11:-1{rm}; ~Ypf} Figure 4.14 This diagram illustrates thevectors used inthedefinition of theLiederivative ofavector field Ywith respect tothevector field X=—Yp<Xf> +13;;I-1{<Yr><<m>>) ~<Yr>(p>}. (tangemto Someimegralcm-v¢)_ pp Since Yfe@(M) weseefrom (4.12.1) that thelastterm isXp(Yf), 164 MANIFOLDS therefore (§EXY)pf= Xp(Yf) —Yp(Xf) ((§£’XY)f)(P)= (X(Yf) -Y(Xf))(P) VP or §gXy= [X,Y] (4.129) where [X,Y]EXY—YZisthecommutator ofthetwovector fields. From thisexample wenote that £8”vif§£X where fe@(M). Wesay that 51?Xisnot @-linear inX.This reﬂects thefact that the Lie derivative along acurve depends ontheparametrisation ofthecurve andnotjustonitsimage. When using acoordinate basis toevaluate §EX ontensor fields with contravariant components itisuseful tonote from (4.12.9) that §E(a/ax‘)(a/axl) :[(3/axi)» (3/axl)l =0- The properties of.§£Xthat 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 [ggxa ggy]=5/3lX_Y]. (4.12.10) Thecommutator 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 (4.12.3) conveys thegeometrical significance: §EXT =0ifandonly ifthetensor field Tisinvariant under thelocal diffeomorphisms generated byX. Example 4.5 We shall evaluate .S£XY forX,YEFTIB2 given byX=x(8(6y) — y(8/Bx), Y=x2(6/8x) +x)/(8/By). First weshall apply thedefinition (4.12.8) directly. Inthefirst example of§4.11 wefound theintegral curves ofXstarting atp=(a,b).This gives thediffeomorphism Q9, (Q,b)i---> (p,(a, b)=(acost —bsint, bcost +asint). Wehave already computed <p_,,.Y, inexample 4.2(the map Q9there being called <p_,here). So(4.12.8) becomesi @l. LIEDERIVATIVES 165 SEXY = F1[(x2cost —xysint —x2)(€-3/8x) +(xycost -—yzsint —xy)(8/8y)] =li_r.nr‘1(cost —1)[x2(8/8x) +xy(E~J/8y)] —ltlllélF1sint[xy(8/8x) +y2(8/8x)] =—xy(8/8x) —y2(8/8y). Wenow evaluate §EXY more practically, using thederived properties of 55X §£XY =X(x2)(8/8x) +x2§£X(E9/Bx) +X(xy)(8/By) +xy§EX(E9/8y) =X‘[X2)(3/ax) “x2§8(8/6x)X +X(Xy)(3/3}’) *x}’$(a/ay)X =X(x2)(6/8x) —x2(8/8y) +X(xy)(8/6y) +xy(8/Bx) since §.£(a,a,,)(6'/6y) =0 =—xy(8/6x) —~y2(E9/8y). 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 cp* foranysmooth map go,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 (1Xw)p =1Xpwp. Thus thegraded derivation iXis@-linear inX.Since 55Xcommutes with contractions itfollows that When acting ondifferential forms theLiederivative canbeexpressed in terms oftheexterior andinterior derivatives The equality ofthese expressions ismost readily seen bynoting that 166 MANIFOLDS both arederivations ontheexterior algebra, commuting with dand agreeing onfunctions. Foriffe@(M) ((11,.+1,a)f=1,df=df(X)=Xf=i’Xf~ IfaeI“/\,,M andBePAM then (dix+ixd)(¢1’/\/3) =dlixa’/\/3 +('1)pa’/\iX /3] +ixlda’/\ﬁ ‘l’(_1)pa’/\ dﬁl =diXcv,\;6 +(—1)P“iXa* A<15+(—1)" <14/\ix/3+w/\Clix/3 +iXdaA5+(—1)P*1da/ AiX[J’+(-1)PiX<r /\db’ +a<,\iXdﬁ =(dix +ixdia’/\l6 +fl’/\(dix ‘l’ixd)/1 Itisstraightforward toseethat (diX +iXd)commutes with dsince dz=0.The existence ofalocal coordinate basis forMensures that the above properties aresufficient toestablish (4.12.13). Example 4.6 ForXEFTIR2 and a/el"T*lB2 weshall evaluate §£Xa/, first from the definition then, aswillalways bedone inpractice, from theestablished properties ofLPX. Wetake X=x(8/8y) —y(8/8x), or=x2dx+xydy. Aswasnoted intheprevious example wemay useearlier examples to proceed from thedefinition Sfxar =limF1[(x2cost —xysint —x2)dx +(xycost —y2sint —xy)dy]r—>O =—xy dx—yzdy. alternatively, §£Xa =X(x2)dx +x2§EXdx +X(xy) dy+xy.§8Xdy =X(x2)dx +xid(Xx) +X(xy)dy +xyd(Xy) =—-2xydx+X2d(—y) +(x2~yl)dy+xyd(x) =—xy dx—yzdy. Forthevector field Yoftheprevious example (§f3X¢Y)(Y) =—x3>’ -W3 a/(§£XY) =—x3y —xy3 whereas a'(Y) =x4+xzyz.ILIEDERIVATIVES 167 These areindeed related by =€£X(a(Y)) =(=(£Xa)(Y) +a’(§£XY)- 4.13 Integration OnManifolds The differential forms derive acertain prominence amongst thetensor fields onamanifold from thefact that they give risetoatheory of integration, generalising theRiemann integral in1B’.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 inIR’bymeans ofiterated integrals. The classical notation foraRiemann integral suggests a natural definition fortheintegral ofanr-form on1R’over anoriented domain. With such adefinition amapping from 1B’toann-dimensional manifold Menables anr-form onMtobeintegrated: weusethemap topull itback toIR’where theintegration isdefined. Properly formulated theabove idea gives thetheory ofintegration ofdifferential forms over oriented chains. Let[0,1]’bethesetofpoints pelB’ that satisfy Os0"(p) =5;1, k=1,...,rinanynatural chart {of} forIR’.Thus [0,1]’istheunit cube inIB’.Introduce Q’forthenatural ‘volume’ r-form dol,\do'2 A Ado’ which serves toorient [0,1]’. Anoriented r-cube onan n-dimensional manifold Misthepair (C’, Q’)where C’isaC°°map C’:[0,l]’—> M.(Tosaythat C’isC”ontheclosed setmeans that there isaC°°map <6’between open sets containing thedomain and image ofC’such that C’isobtained from <6’byrestriction.) Inalocal chart (U,x)wemay represent themap C’:pe[0,1]’+—>qeMbyits components, (/11,...,A’) x‘(q) =)t'(01(p), ...,0"(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"1(p) =e,where s=0,1.Denoting the(r-—1)- faces byC[,f,1) :[0,1]"1 ->Mwehave then CEE§>(0'1(P), ---,<I*"“(P)» v*‘“(P), -~-,(f(p)) =C’(U1(P)= -~-»Ul_1(P)»5»0l+1(P)» ---,v'(P)) j=1,...,r;s=O, 1 (4.13.2) 168 MANIFOLDS Each (r-1)-face may begiven aunique orientation Qffgn, induced from theorientation ofC’: _Z(“1)£+1l(5/aUJ')Qr :1,...,I”;8:0,].Cf“CL~ from which itfollows that thefaces labelled by:5=O,1have opposite induced orientations. Anoriented 1-cube hastwo oppositely oriented 0-faces (itsend 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 allo’(p)equal tozero orone. Forbfe1Bthefinite sum Ejb,-C]-', that maps some set{C}-', 9]} 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’, Q’)which isdefined as 8(C’, Q’)=22(Cfjj), §2f,f,1)). (4.13.4) i=1 £=0,1 Theboundary operator 8extends naturally toallr-chains: a(Zb,(c;,§2,’))=Zz>,a(c;, 9;). Itfollows directly from thedefinition ofC"2 that 88=0since the (r—2)-faces cancel pairwise. From anr-form er,defined ontheimage ofC’,wecanusethemap C’to‘pull back’ orto[0,1]’.The r-form (C’)*a hastherepresentation /2ClOi‘ ,\d0’1,\. ..,(do’*, he¢f(IB’). The orientation Q’ofC’isnow used todefine e,=i1by Qr =5,.dO'il /\dO'l-2/\ .../\dO'i'. Wedefine theintegral ofC’*a over [0,1]’interms oftheRiemann integral ofh ] C’*cr =5,] hdo“ ...do".[9~1l’ [0»1]’ This may beevaluated astheiterated integral 1 ii h(cr1,02,...,0’) do1)do2 ...do’. Wemay now define theintegral ofanr-form onMover anoriented r-cube La=][_],(c*)*a/. (4.13.5)01 This definition isextended toinclude 0-forms bydefining theintegralINTEGRATION ONMANIFOLDS 169 ofa0-form over a0-cube tobethedifference between thevalues ofthe 0-form taken atthetwoendpoints. Ifcp:[0,1]’+—>[0,1]’isasmooth reparametrisation thatpreserves orientations andC”=C’0gothen F‘ Ior= ac" .t(c*.e) Zl1](CrO('9)azlli¢*(Cr*a/) JO, ’ 0,1’ : Cr*a :J. Cr*a .1q:[0,1]’ [0,1]’ since thelastequality follows from achange ofvariable 0|—>0’=(p(o") intheiterated integral. Hence (.4).CF 6.?‘ andwesaythat theoriented r-cubes C’and C"areequivalent. The integral oftrover ther-chain C=E,-bl,-C] isdefined tobe [Ca =%b,¢]C;c1/. The culmination ofthistreatment ofr-form integration over oriented r-chains istheelegant generalisation ofStokes’s theorem afforded by thisformalism. Foranysmooth r——1form Bdefined intherange ofthe r-chain C(r21)wehave ]Can=L65. (413.6) 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. IfereFA,M, )6el"A,,M then dfa’/x5) =do’/M3 ""(_1)’ a’/\d5- Consequently forsome (r+q+1)-chain C ICd(Of/\ﬂ) =J’CdG.’/\ﬁ "‘l' JCCCY/\Clﬁ :L661/Aﬁ byStokes’s theorem. IfSC=0oraA)8=0onSCwehave thesimple result 170 MANIFOLDS ]Cda'Aﬁ =(—1)’+1]CarAdﬁ. (4.13.8) Example 4.7 Weconsider thechain C: C:[0,1]2——>lB3 (t,0)+—> (sinatcos2110, sinatsin2110, cos1rr). If(r,6,cp)arethestandard polar coordinates for1B3then thismap sends (i.',0) tothepoint ontheunitsphere with polar coordinates (1,rrr, 2110). The spherical polar coordinates (6,rp)donotcover thesphere, there are coordinate singularities at6=0,1rand Q0=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 00=r3sin6d6Adipover C.Note first that toissmooth on thewhole ofIP13.This canbeseen bychanging toCartesian coordinates thatcover allofIB3,giving w=xdyAdz+ydzAdx+2dxAdy.We have C*d6 =rrdt, C*dq0 =21rd0 giving C*w =2112sin(1rt)drA d0and 11 ]2C*w =2112] sin(1rt)dr)d0 =411.[0,1] 00 e=o 0 \P=0 T Figure 4.15 Thetwo-sphere asatwo-chain.F .. INTEGRATION oNMANIFOLDS 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 beput into 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 thatoftheimage. That is,if{(8/80")} isan oriented basis forthecube then {C,,.,,(8/80“)} 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 iftoisanr-form onNwhich isparametrised byboth C’andC’’then fcrw =1Ora). Itistherefore meaningful todefine l,~»=l,~»where C’parametrises S.Although weshall notprove theabove we observe that itiscertainly reasonable. Ontheinterior oftheir domains C’and C" areinvertible, and hence (C’)_1 OC’’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=E,C’, 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 one canparametrise 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 p6Mitprovides ametric 172 MANIFOLDS tensor onthespace T,,M. Ifgisasymmetric positive-definite non- degenerate metric tensor field themanifold issaid tobeaRiemannian manifold. Ifgisasymmetric but indefinite 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,go,-,4). If{dx”} isalocal basis for1forms forT"§,M we may write thetensor field gas g=gmdx” ®dx’ (4.14.1) where then(n+1)/2real-valued functions g,,,,=g(8/8x”, E9/8x’) satisfy gm=g,,,,(it,v=1,...,n).Ag-orthonormal basis {Xa} ofT,,M isone thatsatisfies g(X,, Xb) =17,),=i1 a,b=1,...,n. (4.14.2) Anordered basis oflocal vector fields defines alocal frame onMand anordered basis of1-forms alocal co-frame. The components 17,),ofg inag-orthonormal co-frame arerealconstants andwemay write a=nae”®6’ where {e"} EFT*M isag-orthonormal co-frame satisfying e“(X,,_) =53 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 ereFT*M then EreFTM isdefined by g(c'i', X)=a/(X) VXE FTM. (4.14.4) Thecontravariant (pseudo-Riemannian) metric g*isatensor field onM that when restricted toapoint peMprovides ametric onthevector space T’j,M, defined by g*(cr, /3)=g(c'i%, fr’) Va/, ﬁeFT*M. (4.14.5) Inalocal chart wemay write g*=g”"’8/8x“ ®E9/Bx’ =17“’bX,, ®X), where g"»“=gm’E@(M) and gluvgvp Z n"’r1a =<5?- The Gl(n, 1B)elements effrelating natural andg-orthonormal co-frame fields,‘$22-'x>'+_—. pg ii‘METRIC TENsoR FIELDS 173 e“=ejdxt“ (4.146) arenow functions onM.Some authors refer totheco-frame {ea} asan n-bein, others reserve theterm n-beins forthen2functions eje§?(M). Itshould benoticed that, unlike thenatural co-basis, ingeneral de“=#0, a=1,...,rt. Ifthe 1-form wiswritten locally asw= wpdx" =wae“ then the metric dual is(D=co*“8/8x“ =co"X,, where wt‘=g'“’w,, and (pa=17""w,,. Similarly, if locally X=-F8/8x" =EX“, than X=.§,,dx” =§,e“ where §,,=g,,,,<’§" andE,=17,,-Eb (see Appendix A). The index notation isdoing double duty here, theGreek andRoman alphabets indicating that thecomponents arewith respect toanatural and orthonormal basis respectively. The symbols wt‘=w(dx“) and 0)“=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 cu=w(8/8x*“)dx*“ =a)(Xa)e" X=dx”(X)E9/Eix” =e“(X)X,. Wediscussed inChapter lhow 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 toTPM forallpeM. Equivalently, if(U,,, (pa)and(Ub, 09),)areanyoverlapping charts inthis atlas, with coordinate functions {xf} are{y'“} respectively, then the real-valued function fonU,F)U),,defined bydxlAdx2A...dx”= fdy1Ady2A 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 isz=e1Ae2A...Ae” where {e"} isag-orthonormal moving co-frame. Wemay now extend the construction oftheHodge map given earlier toMwith *1= 2.This enables thedomain oftheHodge map tobegeneralised tosections of AM. Ifcp:Ml—>Nisasmooth diffeomorphism between (pseudo)- Riemannian manifolds MandNsuch thatthemetric tensor fields gMon MandgNonNarerelated by an=<e*g~ then tpissaid tobeasmooth isometry. Asaspecial case ifM=N then goisasmooth isometry ofM.If{<p,-} isasetofsuch maps onM 174 MANIEoLDs 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,-,-"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 in1R”may be 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’, q0U: U+—>IR",(pv:V+—>1B"provided cpu<>qof)issmooth and1:1 with anon-zero Jacobian onUOV.This manifold has anatural Riemannian structure. Inaglobal chart {x’,18”} themetric tensor field takes theform g=E,-=1, __,,dx’®dx". 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 asthePoincare 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 CTJandPoston T1977 Tensor Geometry (London: Pitman) Hawking SandEllis G1973 TheLarge Scale Structure ofSpace-Time (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)T"-:"----Illilq Applications inPhysics 5.1Galilean Spacetimes Since thetime ofAristotle 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 and theprojection 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 that lieonthesame curve canbe regarded ashaving thesame spatial position with respect tothisfamily. Each such family defines acoordinate system. The Galilean bundle is provided with apreferred 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 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 the Galilean 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). —--1%UJ IR“ _._____.---T21 Hp)ill I-I--I IIC_ —'_. -|—~—__t___._ ~_J-ii rllpl-—---—--—-——i---0--i----i-..._i Figure 5.1The Galilean bundle with aEuclidean three-space assigned anarbitrary time coordinate byprojection. Theexistence 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 that isrobust and whose behaviour inexternal fields offorce caninprinciple becompen- sated for.) Itisfurther assumed that allgood 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 INPHYSICS ofapparatus such asstandard metres, telescopes, oscilloscopes etcand theNewtonian description ofphenomena relies fundamentally onsuch a framework. However if,asEinstein did, one builds 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 anynew setofmeasurement 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 theideal clocks ofa non-Galilean description measure theelapsed time between events in1B4 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 dimensionsMAxwELL’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,x1,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. The numerical values chosen forsuch dimensioned parameters establish thechoice ofunits for thesystem. Ifonewants towork with coordinates having thestandard dimensions oftime andlength, sayQ,£1,g_c_2,_.v_g_3), onemay introduce three standard dimensioned units such asc,astandard speed, ha standard unit ofaction and areference mass mo. The restoration of physical units follows from thesimple chart transformations t=(mocz/h); x"=(moc/h)gt_“ lc=1,2,3. (5.2.l) Adimensionless tensor field will have components with dimensions when referred toabasis induced from alocal chart with dimensioned coordinates. Itisafundamental property ofmatter that itcanexert along-range inﬂuence onother matter byboth theeffect ofitsmass (thegravitation- alinteraction) anditselectrical charge (theelectromagnetic 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 was originally conceived toconsist ofapair ofvector fields (E,B)onEuclidean IB3parametrised byauniversal time t.Ifwe denote bythe(time-dependent) function p:IB3—>IRtheelectrical charge density inCm73 and byjthe(time-dependent) vector field onIB3 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/so curlE =~83/81 SEdivB =O curlB =uUj+ (5.2.2)(;31‘ Weareassuming thatthesources (p,j)exist inafreespace or‘vacuum’ environment. IfE=_}El,-(E3/81’) eI"TlB3 then by(SE/E91) one means (SE,/8t)(8/8;’) where, inthechart (£‘,£2,£3) forIB3,theEuclidean 180 APPLICATIONS INPHYSICS metric tensor field hastherepresentation _g‘_=Z;7’:1d£‘®d£’. The con- stants £0,in,and cE(eO1.t0)_1’2 ensure that theequations aredimen- sionally coherent. They areassigned dimensions asfollows [tn]= I41=[1292i ML3 The functions (E,-, _li,-):lB3—> IR,each depending onthetime para- meter t,willbecalled theMKSCartesian components oftheelectric and magnetic field respectively. The Cartesian components oftheelectric field have dimensions [ML/TQQ], with MKS units ofNC'1, 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 lB“endowed with a particular metric structure. Instead ofassociating theCartesian compo- nents ofE,Bwith vector fields onIB3,they areused toconstruct a 2-form Eon1B‘.Using alocal chart Q,£1,£2,£3)wedefine E=Q‘l’(ll/\_1-‘Z (5-2-3) where Q=Qidlz /\(12.3+lizdll /\dill+Qsdll /\912 E=EAE+Em?#L@§ Inasimilar way weunify thecomponents ofthecurrent and charge density toconstruct the3-form Cg: 2=Cruel/\ dL+(P/Cgoldill /\dig/\9&3 (5-2-4) where L=iAﬁA@?+p@%w£+1ﬁ£A®f The {j,-} arethecomponents ofthevector current j.The choice of dimensioned coefficients ensures that cgandZhave thesame dimen- sions, namely [h/Q]. Note that, foranyform f,dfandfhave thesame physical dimensions: theexterior derivative does notchange thephysical dimensions oftheform onwhich itacts. Themetric tensor field adopted onIB‘isgiven inthischart by g=—e2d1®d1 +g. (5.2.5) Hence (cdf, dil‘) isanorthonormal co-frame with respect tothisg.In terms oftheHodge map j_associated with thisLorentzian-signatured metric Maxwell’s equations may beexpressed elegantly astheexterior equationssMAxwELL’s EQUATIONS AND MINKOWSKI SPACETIME 181 di£=2 626 d£=u wan 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: l=(30/702 where e0istheelementary charge ontheelectron. Ingeneral, equations involving theHodge map make reference toaspecific metric. The equations (5.2.6) and(5.2.7), however, remain unchanged ifwereplace gby2/lgwhere /lisanypositive-definite real-valued function onIR‘. This follows since Eisa2-form infour dimensions. Itisconvenient for ustoexploit thisfreedom here torescale gbyanyconstant with the dimensions of[L]3’~ and useadimensionless metric tensor field g= L'2g. Weshall denote theHodge map associated with gassimply *and rewrite theMaxwell equations: mF=j Baa dF=o 61% One isofcourse free touseeither dimensioned ordimensionless coordinates inextracting component equations from thisset.Wehave spelt outindetail thestraightforward manner inwhich onecanmake contact with theconventional 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 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,x1,x2,x3)inwhich themetric tensor field isgiven by 3 g=—dt®dt +2dx’®dx’. (5.2.10)i=1 Weobserve that thevector field (8/St) hasanegative norm whilst (E9/Bx’) hasapositive norm fori=1,2,3 8/St ,8/St )=-1 ‘M ( ) (5.2.11) g(E9/Bx’, 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 To=(E9/3!), Tk=(3/axk) k=1,Z,3 K3=x1(E9/8x2) -x2(&)/8x1) K2=x3(8/8x1) —x1(B/8x3) (5.2.12) K1=x2(8/8x3) —x3(<3/8x2) Bk=[(3/axk) +x"(8/St) k=1,2,3. The isometry group ofMinkowski space iscalled thePoincare’ group. Thevectors T,,,it=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 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 5.3Observer 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- tion oftheparametrisation ofC:[0,1]—>M.If(t,xk)arelocal chart maps for Mwe represent Cparametrically bythe equations t(p)=C°(r), x"(p) =C"(r) and werestrict ourselves tomonotonic functions ofrthatmake Cafuture timelike curve: g(C.,.8.,, C.,.8,) <0. (5.3.1) Thelength ofCisdefined tobetherealnumber 1 S=]01g(c.a,, c.a,)11‘2Cl'l.'. (5.32) Under achange ofparametrisation rl—->r'(r) mapping [0,1]—>[0, 1] with (81%/Sr) >0Vtthen Cid, l—>(8,r’)(C§.E9,») and drl—> (817/8r')d1:', soweseethat theintegral isinvariant under such areparametrisation. Aparameter 1:issaidtoprovide aproper-time parametrisation forCif g(C..a,, c..a,)=-1. (5.3.3) 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- tion along 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. This time 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 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 VEk(8, +Zvldxi) (5.3.4) 1:1 inalocal chart (t,xi)inwhich theMinkowski metric gtakes theform (5.2.10). The field islabelled byreal constants k,0‘,02,03.Visa velocity vector if k2[1_(v1)2 _(U2)2 _(U3)2]-1/2_ (5_3_5) What observer curve Chasatangent vector that coincides with Vat each ponitsimage? Forthiswerequire 3 c..a,1, =(at/at)a,], +2(6xl/8r)8,1],, =V1,1':1 that is((8t/81:), (Sxl/é)i:)) Ek(1, vi). These equations fixtheparamet- risation ofCuptoanadditive constant forr.For Clabelled bythe triplet v=(01,02,o3)elB3 afamily ofobserver curves through the origin ofthe (t,xk) chart has the representation t(p) =kt, xl(p) =kulr orxl(p) =vlt,j=1,2,3.For arbitrary constant vthe vector field VisaKilling field. Wedefine astationary observer tobea proper-time parametrised integral curve ofatimelike Killing vector. Thus thevector field V,with arbitrary constant v,yields athree- parameter family ofstationary observers inMinkowski space. Any global Minkowski space chart inwhich themetric takes theform (5.2.10) isoften referred toasaninertial chart. Thechart 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 {8,,8,1} associated with theinertial chart isanexample ofaninertial frame along theintegral curves of8,.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 3 stationary one. Since wenow contemplate arbitrary observers we concentrate onT,,M rather than thewhole history ofthearbitraryP- ‘I. -,1 3;._-.3..,-Z;" .:"t"l;="='“.=?:*.-A f5-;> 2“ 1 : .3‘:-1Bl?- Sn“?-. -.'-I_: :‘-‘;" - 3"?. I: ,-.é1t-1|'‘er-rj-. tn--‘j/1'41 ,--_,_,-=2 .3‘.-3' :='g;'.I.-I:-''44‘ "rsL.. J ,1‘ >‘&_,:. -14../{E';é+5.'I- ..-..-21 age- ._.o “ad.- Kl .1'1».._i;¢_'.'. .3.,,. -1.1:‘._<\.. ?’<"f' ililii sz=- Y;-1‘=I?<-. .1,, .,,i\.i.|:s'~'".,...OESERvER CURvEs 135 observer world line. Apoint peM together with afuture-pointing timelike vector with norm -1willbecalled aninstantaneous observer atp._ LetCbesuch aninstantaneous observer associated with thegeneral observer Cand letAbeany timelike future-pointing 1-chain (not necessarily another observer) with tangent vector Aatp.Wewish to define the Newtonian velocity ofAobserved byCatp.Since A,CeT,,M wehave aunique orthogonal decomposition i=P+aC 63o where $6113 .and _g(P, C)=O. This latter condition implies =—g(/1. C)s1nce g(C, C)=-1,hence A=P—s(1.~C)C'- (5.37) The Newtonian velocity ofAobserved byCatpisnow defined with respect tothisorthogonal decomposition asv=P/%, or v=—P/s(A1. C‘) (5.3.s) showing thoatov depends onboth AandC.The vector Pisspacelike and issaidtolieInaninstantaneous three-space ofCatp.This isdefined as theorthogonal complement ofCinT,,M. Wenext consider thecase ofanull 1-chain Fobserved byC.The condition g(1", F)=0inserted intoF=P—g(P, C)C gives, with theaid Ofs(P.P)=s(l".P). F=PK?—N) (5.39) where %E—g(F, C)andNE(g(F, C)/g(P, P))P. %iscalled theenergy that Cobserves forFatpwhilst Nisthespatial direction observed for F.Note thatNisspacelike with g(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 null curves may model theﬂow ofelectro- magnetic radiation, orphotons. The images oftimelike future-pointing curves aremodels foreither massive point particles orthestreamlines ofmass—energy ﬂows. Apoint particle ofmass mismodelled byafuture-pointing curve pwith 8(l3’»19)=—m2. Then p=P+BCimplies g(P, P)+m2=%2;P=%'\ therefore implies g(P, P)=%2g(v, v)=%2~m2. (5.3.10) Hence %andPmay beexpressed interms ofvas 772 %[1-_gmUH”, (5.3.11) 186 APPLICATIONS INPHYSICS moP [1_g(v, Uni/2. (5.3.12) Clearly ifmE0,g(v, v)=1—(m/E)’ <1:that is,massive particles areobserved tohave bounded Newtonian velocities. If,forexample, C=8,],, andA=1tl(r)E),,1],, +t(r)E),|,, inaninertial Minkowski chart then g(C, A)=—i and A=P+i(r)E),],, gives P=.tl(r)E),,1. Hence inthischart v=(Jtl(r)/i(t))E9,1[,, istheNewtonian velocity ofAobserved byCatp. Iftheprojection onto theinstantaneous three-space orthogonal toC atpiseﬂfected byfhe projection operator H,,:T,,M —>(C)j, Hp=(1—{C(C)}"‘1C® C),,, then theNewtonian length ofanyspace- likevector VeTPM observed byCisdefined as(g(II,, V,HpV))”2. If Wisasecond spacelike vector inTPM then theNewtonian angle between VandWobserved byCisgiven by cos6E g(HpV’Hp W) e . (5.3.13)[g(H,,V, II,,V)g(II,,W, II,,W)]1’2 The presence oftheprojector 11,,inthese formulae, defined bythe observer curve, means that theNewtonian length andangles specified in this way depend ontheobserver aswell asonthevectors being observed. For thegeneral future-pointing vector A=P+BCwesee that VhasNewtonian length (g(v, v))1’2 =[g(P,P)]1’2/E. IfAisnull, g(P, P)=E2andhence allnullvectors arealways observed tohave unit length Newtonian velocities. Wehave already noted that g(v, v)<1if vistheNewtonian velocity ofaparticle with mE0.Ifg(v, v)<<1we may expand (4.3.11), (4.3.12) using thebinomial expansion %=m+§mg(v, v)+... (5.3.14) P=mv+.... (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 has had aprofound 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, +tE),.1 (5.3.16) passing through apoint pgwith coordinates, . .P 1 OBSERVER CURVES 187 (l(P0.)» x1(P0)= x2(P0)-.» x3(P0)) inaninertial chart. Weshall take p0tolieoutside the‘light cone of (0,0,O,0)’,defined asthesetLofpoints psatisfying §1(xt(p)r -(t(p>>2=0. This ensuresthat atp0Xistimelike. Fordefiniteness weshall assume t(p@) >0,x’(p0) >0j='1,2,3.The integral curve isgiven parametri- Cally ast(p) =/lO(T), X’(P) =A’(I) jE1,2,3,where thefunctions A(“:[(), 00)—->IR,it=0,1, 2,3satisfy dA1/di: =1°,Cl/1.0/£11,‘ =11 dA2/d1.‘ =U,dA3/d't' =0. Thus thecurve isgiven bythesolution A1(r) =A1(0) coshr +A°(0)sinh1: A°(r) EA°(0) coshr +A1(0) sinh17 (5.317)/12(1)=11(0) A3(t) =/13(0). Eliminating 1:between A1(r) andA°(r) gives part ofahyperbola through P0andp (r(1>r-(i°(r>r=(r(0>r-(/v(o>>2. (5.313)Ifwerelabel thefunctions Al“with coordinate names (with p0specified byr=O),equations (5.3.17) may berewritten as 1 ‘((1) ""”7xll=+.l‘>tl.§"’ (5-3-19> t(p)-‘(Pll';':)f,€°) (5.3.20) where coshr =1/(1—02)” andsinhr =v/(1~02)” >0.These famil- iarequations relate the point p0tothe point plabelled bythe parameter v=tanh ralong theboost orbit (5.3.18). Forafixed vwehave adiffeomorphism, generated byX,thatmay be used torelate twoobserver fields. Define themap 12..IM—>M.P—>P’ t(P’)=(t(P)+vx‘(P))/(1 —v’)”’ X‘(P’) =(r‘(P) +vt(P))/(1 ~v’)"’ x"(P’) =r"(P) k=2.3 ' “H 188 APPLICATIONS INPHYSICS then ((pv)*:atlp *—>(91+Uax)lp’/(1 —v2)”- Thus thefixed parameter 0canbeidentified astheNewtonian velocity of(q0,,)..S,|,,1 asmeasured byS,|,,» forallp’.Note that forall1:, 0=tanhr< 1.Itisofinterest tonote that since two Successive diffeomorphisms oftheabove type parametrised byr,and 1'2respec- tively produce adiffeomorphism parametrised by1:,+1'2: (ptj O¢TQ Z ¢lT1+ T2 weobtain asNewtonian velocity parameter 012corresponding to1:,+‘C2 U12 :‘tanh (T1 ‘l’T2) tanh T1+tanh T2 01+02 1+tanhr1tanh1.'2 1+0102' Forall01,02<1,012=02,<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. Iforisap-form onU,UCM,satisfying theequation dot=0itis saidtobeclosed onU.Then there exist some region WCUforwhich cr=db’,forBa(p—1)-form onW.The p-form aisthen said tobe exact onW.Itisanimportant result that theglobal topology ofU determines whether ornotallclosed forms areexact onU.Forour 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 Ae9'-*(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,xk)wemay write 3 A=EAkdx" +(pdt k=1 and hence relate thereal-valued functions Ak,cptosome electro-ji 1-eiatggg-.1,-.-'-.-g"|,_-,-_<.-,."--1;;.-1-;.:=ELECTROMAGNETISM 139 dynamic ‘vector’ and‘Scalar’ potentials. Introducing alocal potential A means that (5.2.9) issatisfied identically andtheother equation (5.2.8) becomes d*dA=1". (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 OIFAPM-—)FAp_1M by 6=*“d*11. (5.4.2) (Recall from (1.1.2) that if(pisap-form ncpErp"=(-1)P(p_) Since on p-forms *4=(_1)P(~*P)____detg ]detg| E77"_1detg (5.4.3) Idetg] Itfollows Immediately that (5hastheproperty 56=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 (tr,16)E]M<rA*1? ct.tier/\,ivI. (5.4.4) An‘integration byparts’ gives, with Stokes’s theorem andthecompact- ness ofM, (<12.<11/1)=((5%it») feel"/\,.M. wef/\,. _1M. That is,6isthe adjoint ofdwith respect tothis product. The Laplace—Beltrami operator Aisdefined by AE-(<15+6d). (5.4.5) Note that since d((5) 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): (<22.Ate)=—(<P.d<5¢)~(<12.(5d<P) =—(<5<P. (510)~(9%d<P)-M7314 l ‘A1 190 APPLICATIONS INPHYSICS Thepositivity oftheRiemannian metric ensures thattheright-hand Side isnegative-definite, thus soareanyeigenvalues. Theequation (5.4.1) canbewritten interms ofAas (A+(16)/I=-*1. (5.4.6) Itispossible toselect arepresentative potential from theclass of co-homologous 1-forms such that (SA=O.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 SA’E(5dA=0also ifAischosen tobe harmonic, thatissatisfy AA=O.) Letusexamine some solutions toMaxwell’s equations inaregion of Minkowski spacetime free ofsources. Suppose weseek asolution to (5.4.1) oftheform AEfdt, fe‘Ji(M) using apolar chart (t,r,6,cp)in which g=—dt®dt +dr®dr +r2dl9®dl9 +r2sin28drp®d(p. Weshall look forastatic ‘spherically symmetric’ solution Satisfying the symmetry condition SEK,FE0where thetimelike Killing vector is K0=(S/St) andtherotational Killing vectors take theform K1: sin(pS,, +cot(:lcos(;0S,, K2=—cos(,t>S9 +cot6sin(pS,,, (5.4.7) K3=S,,. This canbeachieved ifthefunction finvolves only thecoordinate map r.Aconvenient orthonormal co-frame is{dr,dr,rd6l, rsinddqo}. Then dA=S,fdrAdt =S,fe1 Ae0, so if *1Ee1Ae2A e3Ae0 then *dA =(S,)fe3 Ae2=(S,)fr2 sin6d(pAd6. Thus d*dA =S,(S,fr2)drA sin6d(p Adi9. This iszero iff=--k/rforsome constant k.The solution A=kdt/r yields theelectric 2-form F=dA=—(k/r2)dr Adt. This is the Coulomb solution. The frame-dependent electric field 1-form EE ia,F= (k/r2)dr gives theelectric field vector E=(k/r2)S,, the integral curves ofwhich give the familiar radial Coulomb pattern associated with astationary charge inthisframe. Forageneral Fwedefine jC*F astheelectric charge Qcontained in theinterior oftheSphere which istheimage ofC.(Ifthecharge is non-zero then thisS2cannot betheboundary ofasource-free region!) (Restoring dimensioned variables, [$.15=(A./~@)1'2_a515:-- ELECTROMAGNETISM 191 determines acharge QinCoulombs.) Aclass of2-chains willdetermine thesame electric charge. Wedefine anequivalence relation on2-chains asfollows: C1='C2 iffC1=C2+SE,where 2isanySource-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 thp 2-chain inMinkowski spacetime whose image isthe Sphere t-constant, r=constant. Then fortheCoulomb solution ]C*F=kfcsin (9d(9A661 11/2 27; =Zkfo Sin6ld6l]0 dtp=411k, Kdwemay deduce ijliat ifF_St'If‘anhaL16denvatlwls commute with 3_>fOrm jthen cgFsatisﬁes thaIsIesltlqlMaxwell equations with Source anunderlying isbmetr emWlt t'6Smlrce gm. The-gxlst-ellce 9f _ group ofspacetime isoften used implicitly In constructing new solutions ofMaxwell’s equations from simpler ones. If :iErI§Ei31('il]1lrihi:3e)(;ltfil)f(1)I:)1li(lOHlOflihB Liederivative, andcompare it.with the Solution 3);alimit OfCElvJ3glEg[1OHl usegl toconstruct theelectric dipole indegd expect thefonowin qua ‘an opposite Coulomb solutions, we gpotential toprovide asource-free solution l _H /< 14 1A —k_;C_)‘$(S/S)t‘)_r_dt =—;5(=(£(a/8xl)l’)dt : T ' 1 - . .. , hevector field (S./Sx )represents aMinkowski space Killing vector in aninertial chart. Since theLiederivative commutes with d k F=7%-=E£(S/Sx')(_,:2_drA dl) lsthefield ofastatic electric dipole with moment it.Ingeneral for P951t1V@ Integers P,Q,F3‘P.(5,t"—type electric multipole solution fol- lows from Poincare covariance as F={5'5(@/@x*’>}“{55(@/Ari}"{5£(6/6P1},l%d’/\ dtl' F l!_}!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 556s=Ace (54.8) 192 APPLICATIONS INPHYSICS forsome scale function /‘LC.For any n-dimensional space (neven) it then follows thatifFe1“/\,,,2M then .§E(;(*F) =*(.EECF). (5.4.9) Hence inaspacetime (n=4)with ametric g,ifsuch aCexists andthe Maxwell 2-form Fsolves Maxwell’s equations with Source jthen .§£CF willbeasolution, inthesame metric, with source Sfcj. 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*“(6/Sx“) AD=2 KM=g(D, D)(E9/ax“) —2x,,D AK”=—-4x,, (5.-4.10) it=1,2,3,0. These vector fields along with the10Killing vectors generating the Poincare 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 ll/Uandwrite anyFuniquely as F=EAI7+B (5.411)P\-I where Bisa2-_f_o_r_m satisfying iVB=0and Ea1-form satisfying ivg =O.(Note: 8/E91‘ =—dt.) One refers toBe1"/\2M asthemagnetic 2-form associated with T7and F,and Eel"/\1M astheassociated electric 1-form. Theelectric field observed bythisclass ofobservers is all---.._..u E—iVF. (5.4.l2) The magnetic vector field observed bythisclass canberelated toFas¢-‘~‘-‘I'.':..115':T ELECTROMAGNETISM 193 follows. We use thevelocity vector todefine ametric gonthe instantaneous three spaces s=~V®V+§» wan) Wemay factor thevolume four-form as *1=T7/i‘*1. (5.-4.14) Any p-form wcanbe‘3+1decomposed’ with respect tothevelocity vector V:'-"ad ww+VA5 (5.-4.15) with ivoz=iv/3=0.IfiiistheHodge map associated with gthen *w=—(‘*=‘a/)/\ T7—‘*‘ﬁ- (5.416) Applying thisresult to(5.4.11) gives *F (‘*‘B)/\ V+QB. (54.17) Butiv(”'?Ev_) =_0$0it/*1_'7 =—§?B. Wedefine thevector field B=53as themagnetic field associated with V;hence interms ofF a-—--.__,a I i B——1v*F- (5.418) If{Ya} isaframe ontheinstantaneous three-space, orthonormal with respect to.§,then theelectric andmagnetic field components insuch a basis aregiven interms ofFas i?'(Y..)=(iVF)(Ya) =2F(v,Y.) (i*“B)(Ya) =_(iV*F)(YQ) =~2*F(V, Ya)- Asanexample consider theCoulomb solution: F=—q5drAatr r2=x2+y2+z2 with observer curves tangent toV=(8/8:) and W=1/((8/Z-9:) + v(8/8x1)), V=(1=v2)'”2. With respect toV: E=—:-2‘l(a/er) B=0. Ontheother hand, since rdr=xldxl +x2dx2 +x3dx3, Wobserves _ K 1 12'=-’§i’l\(a/er) +i":—(a/an) B’=—-%y—(x (8/8x3) —x3(8/8x2)) instead ofEandBatp. 194 APPLICATIONS iNPHYSICS Itisworth stressing that although observers inMinkowski space experiencing arbitrary motion donot have 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 oftheroleplayed 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 Applying theexterior derivative and using Maxwell’s equations forF produces dry=%{diVF,( *F—iVF,(j —div*F,( (5.4.20) Recall theidentity .5-EX=diX+iXdVX: hence divF =.58‘/F (5.4.21) asdF=0.Similarly div*F =§EV*F —ivj.Inserting thisin(5.4.20) gives dr-,/= %{.§£’VF,(*F—§EV*F,\F—iVF,\j+iVj,(F}. (5.4.22) IfCisaconformal Killing vector then §BC*F,\F =*§E¢F,(F = FA*§£CF =.§£CF A*F.Hence specialising tothecase ofaconformal Killing vector dTc=_iicF/\]. ‘l’iicj/\F- Since iC(j,( F)=iCj,( F—j,(iCF and, being a5-form, j,(Fiszero we have dTC : Foreach conformal Killing vector these equations describe a‘local conservation equation’ inasource free region (j=0).The identificationI '-ELECTROMAGNETISM 195 ofaclosed 3-form 3with alocal conservation lawisappropriate inan arbitrary spacetime. Forconsider aregion described bysome 4-chain U whose boundary may bewritten 51/=21+E2+H (5.4.24) with theimage ofeach 2,aspacelike hypersurface (each tangent vector toE,being spacelike). ForCiclosed LU?=lUd.§P=0 (5.4.25) byStokes’s theorem, thus Ll?=L22?—(Hi. (54.26) IEcases where Umay bechosen sothat IRE =0onerecognises that 2S)uxofC?through Z1equals theﬂux ofgéthrough 22(see figure itS?. \A /. sI / g \\ '<\ ,2 Fl ‘ ‘Ks ‘\ig I/\ \ . \"unnu- I /f|-—- (I\\""In-- \"In_""'q- $iti~ \ Y’ //>\\\ J-__ \\ I ,2/1.4I\(§§'‘- Figure 5.2This diagram illustrates theequation EBU=E1+22+II. Suppose that wehave afield system describing asimply connected Eziéiﬁi-I€I;.¢i;r_ Iiﬁglgll UofMinkowski space. If istheproper time of chart {I Iapserver palssing through thisregion then inanadapted Some: CO=n(;t;£ =0Ifvililil taleJ.tolieinthehypersurface r(p)=cj,for distances from beeectromagnetic field vanishes atlarge spatial boundar ofU ehohseryler then wemay take II‘tocomplete the inthiscgse theSflllC fattheelectromagnetic field vanishes onII.Thus inde d uxo9,13trough theinstantaneous three-space istime pen ent. Ifwewrite ffinterms ofa2-form current gfﬁand an 196 APPLICATIONS INPHYSICS associated 3-form density 6,C?=,9Adi: +6with i(;,,a,)§ =Oand i;@,@.,)6 =0,then clearly 2*} =6and Lg=La. (54.27) Itistempting toreinterpret theconse/rvation ofC?-ﬂux associated with Uinterms ofalocal flow ofcurrent ,9andanassociated variation of density 6.Certainly the3-form equation dc?=Oimplies alocal contin- uityequation. Intheabove chart wemay write dwhen acting onEas d=Q+drA§sP(a,@,) where _distheexterior derivative associated with theinstantaneous three-space. Hence (asQ6=O) Q3‘-.s£(,,,,,,a =0. (5.4.2s) Ifweexpress and6inabasis adapted toE: $5=.9140’/\dp3+$240’Ade’+$340’/\<1P2 (5=P991/\ (102/\(1193 (5.4.28) isequivalent to (a5,/apt) -(Sp/61) =0. (5.4.29) The interpretation ofthislocal continuity equation must, however, be treated with caution. If,9isaclosed 3-form onUthen sois ,9’=5'9+dfliwhere THisanysmooth 2-form. Iff7{ischosen such that fagflf =0then §éandQ9’both have theSame fluxthrough Z,although C?’ 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: F‘--.4 P‘-..u I"-.4 1 P"--I I"-_v f'\v F‘--4 TV:_EABAV+§(EA/’7?E+BA”EB). The physically dimensioned components ofthevector obtained by taking themetric dual ofthe2-form FfABwith respect tog‘was identified byPoynting asthelocal field energy transmitted ‘normally’ across unit area persecond (that isthelocal field energy current). Similarly theg‘-dual ofthe3-form §(B’A SE+BAQB) may, after restoring physical dimensions, beinterpreted asalocal field energy density. Since, forexample BAQB=g’(E,E)?'?1, thesignature of§ ensures that this density ispositive-definite. This interpretation has persisted although with thecaveats above wewould prefer toidentify theoriented integral fzrv, inasource-free region ofspacetime, asthe.,--'‘.441-'-- ELECTROMAGNETISM 197 field energyassociated with thespacelike 3-chain Eand fszivdrv asa power ﬂuxacross anoriented spacelike 2-chain S2. Suppose weconsider aspacelike Killing vector field Xgenerating spacelike translations along open integral curves and decompose rX according to IX=HX/\V+‘ﬁr (54.31) With _ii/MX =ii/‘BX =0.The Maxwell stress 2-form ;.iXmay beused to identify mechanical Newtonian forces produced bya‘ﬂow’ ofaNewto- nian field momentum density 3-form ‘QX. 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 {X}a=0123in. , (I 7 3 3 spacetime, with natural dual co-frame {eh}, wemay obtain 16real functions Tabdefined by*rXh =Tbcec orTab=(>l<'[Xa)(Xb)_ These may beused todefine asecond-rank tensor T=Tib@“®e” (54.32) which isreferred toasthestress tensor. Exercise 5.2 Show that ifTX,/\@z> =TX,,Ae,, then Ta),=TM: the stress tensor 15Symmetrm Show thatifTX,/\6”=0then Tb”=0:thestress tensor iszraceless. d_Thqse properties aresatisfied fortheMaxwell stress tensor asfollows llI'6CI yfrom thedefinition. Weshall meet these properties again ata aterstage inthecontext ofaClifford representation forthistensor. Exercise 5.3 IfF=§F,,,,e“ AebShow that Tab :“%<gabFCdFcd _FacFCb- Exercise 5.4 USethethree angular momentum 3-forms rK,toevaluate thetorque on anelectric dipole inauniform static electric field. (Hint' calculate the Coltal electromagnetic 2form andusethisin(5.4.19) where theKilling rrents arecomputed with theaidoftherotational Killing vectors.) Bibliography Misner C,Thorne Kand Wheeler A1973 Gravitation (San Francisco: WH Freeman) 193 APPLICATIONS INPHYSICS Sachs RKandWuH1977 General Relativity forMathematicians (New York: Springer) Schutz BF1985 AFirst Course inGeneral Relativity (Cambridge: Cambridge University Press) I pt \ .6 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 inthefactthat.SEXT isnot@-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 one totheother 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 butnotonhow bumpy 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 6.1Linear Connections Alinear connection onamanifold Misamap V:FTM ><FTM -—>PTM thatsatisfies thefollowing, Vf,g e@(M), VX, Y,ZeFTM: VfX+gYZ =fVXZ +gVyZ (6.1.1) VX(fY +gZ) =X(f)Y +fVXY +X(g)Z +gVXZ. (6.1.2) Thus VXisalinear mapping onvector fields which isalso@-linear inX: itiscalled covariant differentiation with respect toX. From these properties itfollows that wecanspecify Vbygiving the components ofthevector VXGXbinanyconvenient basis {XG}: VXGXE) Z rabCXC. The n3functions Fab“, where n=dimM,areknown astheconnection components, orconnection coefficients inthisbasis. These coefficients canbeused todefine asetof1-forms, theconnection 1-forms, (gab =Fcbaef (6.104) where {ea} istheco-frame dual to{Xa}. Thus wecanwrite (6.1.3) equivalently as VXuXb :CUCb(Xa)XC. If{Ya} isanew basis, related to{Xb}byageneral linear transform- ation Ya=AabX,,, then VyaYb :AapVXp(AbCXC) =A,PA,@r,,,tX, +A,PX,,(A,,C)X,_.. Iftheinverse transformation isgiven byA‘1p'“A,," =62,then the connection coefficients l"',,),’ inthebasis {Ya} aregiven by 11,,’ =A,PAbCTp,_.‘¥A'1q’ +AaPXp(A),“)A ‘lc’. (6.1.6) Equivalently theconnection 1-forms inthisbasis aregiven by w’ab=A,,‘1tu’qA"”1," +A‘1q“dA,,‘>’. (6.1.7) The ‘inhomogeneous’ term inthistransformation represents adeparture from thetransformation ofthecomponents ofatensor, reﬂecting the factthatthemap X,Y_-> VXY isnot@-linear inY. Asanticipated thecovariant derivative ofavector field with respect toX,evaluated atthepoint p,depends only onthevalue ofXatp. ForifVisanyvector field and{XQ}isabasis intheneighbourhood of pthen (Vi/Z)|,, =V"(p)(VXaZ)|,,. SoifVvanishes atptheniii‘-- LINEAR CONNECTIONS 201 (VvZ)|p =0VZ. Thus ifXandYarevector fields Such thatXI),=Y|,, then (VXZ)lp =(VyZ)l,, VZ. Hence forany XpeTPM wehave a covariant derivative inthedirection ofXp,_VXp:FTM-—>TPM. LetCbeacurve with tangent vector C.Then ifYisavector field wemay covariantly differentiate Yinthedirection ofthetangent vector atanypoint C(t) onthecurve. Anassignment ofavector Ya.) toevery TQAM isa(smooth) vector field along Cifthemap t+~—>Ycmf isa smooth function oftVfe@(M). Thus ifCisasmooth curve V('-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 V¢Y by DY/dt where tparametrises C.) Avector field Yalong acurve Cissaid tobeparallel along Cifit satisfies theequations Vt-Y=0. (6.1.s) Ifwe expand Y=Yl61-inalocal coordinate chart inwhich xl(p) =Cl(t) represents C,j=1,...,nthen C=C..(6/St) =C"(t)(6/ Eixk). Hence V¢(Yl(8/6xl)) =(CYl)(6/8x1’) +Y/'V¢(6/Eixl). But CYI =[C..(8/6t)]Y/' =C"(t)(6Yl/6x") =d(Yl=>C)/dt and V¢(6/6x7) =Ck(t)V(a,6,i)(6/6x1‘) =C"(t)l",(,-"'(6/E9x’”). Thus (6.1.8) gives thefol- lowing differential equations forthecomponents Y1<>C ofYonC: §;(Y’"oC) +(Yi'-c)c'<(i)(r,.,~%c) =0. (61.9) Forgiven functions C"(t)andconnection components Pk,-"‘(C(t)) these equations areknown tohave aunique solution Y’"(C(t)) specified by thechoice ofinitial components Y’"(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(-(O) isany vector inTC(0)M and Yistheunique vector field along Csuch that VCY =0then Ycm iscalled theparallel translate ofYg-(0) along C. LetYbeanysmooth vector field along Cwith Yam a‘=0,andfthe smooth function such that (f<=C)(t) =t.For tsufficiently small Zisa smooth vector field onC Z5Y+Z1§i)=g-‘)_}-/ (6.1.10) where (V¢)2Y =V(-;(V@ Y)etc.Wehave VCZIWY in(“f)”;l'1(Vc)"Y+ i(—f)"(:f)"* ‘Ygince Cf21) 202 CONNECTIONS T gn(—f)n ’;!l(VC)nY +i1(__f)tt(7l!C._)n+1)/ im-/"""\m +1)(_f)m(VC)m+1Y 0°(_f)n(VC_)n+lY ='= ’(m+1)m! J"; ‘in!’ '0' SoZisparallel along Cwith ZQO) =Yqo), thus Zcm must bethe parallel translate ofYcw) toC(t). Note thatanyvector field Ysatisfying Yqo) =Acanbetaken in(6.1.10) toevaluate theparallel transport of AETc(0)M along C(seefigure 6.1). T10)/fE01:l/[{t}'fvfl/la?" Yuri f rm is Yrioi ((01 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, x2)byF111 =--x1 andF222 =—x2 with all other connection components zero inthischart. Prove that fora,beIB thevector field Y=aexp[(x1)2/2](6/6x1) +bexp[—(x1)2/2](6/6x2) isparallel along thecurve C:[0,1]Ii> (x1(p) =sint, x2(p) =cost). Alinear connection enables ustodefine a‘straight line’, generalising oneoftheintuitive properties ofstraight lines inEuclidean space. A curve Cisanautoparallel (ofV)ifitstangent vector field isparallelLINEAR CONNECTIONS 203 along C.Such curves aregiven assolutions totheequation v¢C=0. (6.1.11) (Autoparallels aremore frequently called geodesics although weprefer toreserve thisterminology fortheautoparallels ofapseudo-Riemannian connection which willbediscussed later. Students everywhere willbe 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 byxr"(p)=Cl(t)then theClmust satisfy thesystem ofdifferential equations %c:~1+(F;q~m<>C)C“(t)Cl =6 OI‘ cm+(r,.,'"-C)C’<Cf =0. (6.1.i2) Itisimportant tonote that thesolution 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 Coh isalso anautoparallel ifand only if h=at+ bfora,be]B. For anarbitrary curve Cwedefine the acceleration tobethevector field VCC onC.(Thus theacceleration Each autoparallel isfixed uniquely byspecifying (Cl, Cl)forsome initial t.That is,forevery XPeTPM there isaunique maximal autoparallel starting atpinthedirection ofXP.Let 7/Xpbethis autoparallel. The exponential mapping atp,Expp, maps asubset of TpM into M:ExppXp =1/Xp(1). Clearly Exp,,isdefined onthose XPfor which 1/XPisdefined on[0,1].Since forit6IBy,(Xp(t) =yXp(lt), ifExpP isdefined onXPthen itisalso defined on./IXP forite[0,1].Itinfact follows from thenature ofthedifferential equations (6.1.12) that for every pEMthere isaneighbourhood oftheorigin inT,,M, N0,such thattheexponential mapping isadiffeomorphism onto aneighbourhood ofp,Np. Ifsuch anN0isstar shaped then itiscalled anormal neighbourhood. (TosaythatN0isstarshaped means thatifoeN0then AvEND V/le [0,1].)Anormal neighbourhood ofpistheimage ofa normal neighbourhood inT,,M under theexponential mapping. For every qinanormal neighbourhood ofp,NP,there isoneandonly one QETPM such that q=Exp,,Q. Thus if{X,-} isanybasis forTpM with Q=Z§‘=1Q’X,- themapping q+~—->{Q"} provides acoordinate system for Np(seefigure 6.2). Such coordinates arecalled normal coordinates atp. 204 CONNECTIONS r,M_ 1 _ _ _ , 1 ~ _ — _----'—" X‘‘\ -.._~\xx‘\\-ekx __‘\‘ta I \-‘K I -i-___‘~_in“"'-S 1--.-,-___-.- q-inqjiiji4" 4%?’ --I I " \ P1 Figure 6.2This diagram illustrates theexponential map andnormal coordin- ates. Exercise 6.2 9 _ If1",-1" aretheconnection coefficients with respect toanormal co- ordinate basis atpShow that rm)+rm)=@~Hint: Show that 1/f,(t) =viwhere v=vlXt- 6.2Examples andNewtonian Force Togain some insight into covariant derivatives weturn toI84. This manifold hasanabsolute parallelism: theparallel-transport map ispath independent. If{xl} arestandard coordinates and XP=Z,-c"E9,-l,, then theparallel translate ofXpatqisXq=E;C‘3ilq- Thus 1“5“Ch_ 3 standard chart theconnection isdefined byV;,».161-=0.Such 8COHHECIIOI1 isreferred toasthestandard connection onIR”. _ Itisofinterest tocompute thestandard connection for1B2inapolar chart (r,6)related tothestandard oneby x1=rcos6 0<r<<><> 621 (--) x2=rsin6 0<6-==§2ti. This induces acoordinate frame transformation: gr=(xl/,»)g), +(x2/r)a, (6.22)\ \ .. \ \ 2%. \ \\\ ’9~ExAivIPLEs 205 and 69=—x2E31 +1:162. (6.2.3) Since 61and62areparallel wehave V413,) =la.-(X1/t)]@i +[@,(-Y2/t)]@2 =0 V6130) ='"l9r(x2)l51 +l5r(x1)l32 =(1/'")ae (62.4) Va9(8Q) =-73, V@6(6,) = Writing Var(E9 9)asl",9’6, +F,9"66, etcwemay read offthecomponents oftheconnection inthepolar chart; 119,9 =Fm‘? =(1/r), F99’ =—r with allothers zero. Ifacurve Cisgiven inthenatural chart asx1(p) =Cl(t) then V¢C =Jc'1'(t)E9]-. Let usevaluate the acceleration ofthe curve C:[0,1]—>1R2given intheabove polar chart by (F°C)(I)=10(1),(9°(7)0)=9(1) forsmooth realfunctions pand(9oft.Thetangent vector toCmay be written hence V¢C=pa,+($36.,+pvca, +(-1~')V@a,,. But l V66, =pV;,r8, +QVGBG, =((9/p)69 and V6--6., =pvaa, +®Va,a,, =(P/(9)36 "065%- Hence thenatural IB2acceleration ofCis V@C =(,6—p@2)6, + +Zp(§'-))(1/p)69. (6.2.5) With respect tothestandard Euclidean metric onIE2 g=a,®a2 +a2®a2 =a,®a, +(1/r2)69®66 (6.2.6) 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. I 2;! 206 CONNECTIONS Exercise 6.3 Usethestandard connection inIB3tocompute theorthonormal compo- nents oftheNewtonian acceleration ofthecurve C:[0,1]->1B3given by(F0C)(I)=R0)» (90C)(t)=90)» (trOC)(I)=¢>(I)Where themaps (r,8,go)arestandard polar coordinates in1B3. The above examples inI82and IB3suggest that the Newtonian 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 ‘JonC given by a=V¢(mC). (6-27) Inmany problems inphysics 5arises asarestriction toCofavector field onIB3determined 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 (I)=GM/r inapolar chart, where GistheNewtonian gravitational constant, to obtain a=-m<i'E'I'5 =(GMm/r2)E9,. (6.28) Foraparticle with electric charge qtheNewtonian Lorentz force is Q=q{E +i?B}. Instandard coordinates {x’}, E=E,-dx’ and B=Bldxz Adx3 +B2dx3 Adxl +B3dx1 Adxz are 1-and 2-forms re- spectively onIB3,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 with 108ms”. 6.3Covariant Differentiation ofTensors Wehave introduced thecovariant derivative VXasamap onvector fields. Toextend the definition toitsaction onsmooth 1-formsICOVARIANT DIFFERENTIATION OFTENSORS /36I"/\1M wedefine VXB by (VXB)(Y) =—)8(VXY) +X(B(Y)) X,YEFTM. (6.3.1) Iffe§(M) itfollows from thisthat VX(fﬁ) =fvxﬁ +(Xf)/>7 (6-3-2) If{Xa}, {eb} aredual bases itfollows from e"(X,,) =62thatifcob,are defined by(6.1.5) then VXae“ =—-coC,,(X,,)e". (6.3.3) Ifforfe@(M) VXf EX(f) (6.3.4) wenote that(6.3.1) isequivalent toadopting therule VX(l3(Y)) =(VXi5)( Y)+/3(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:FTf.M —-—> FT§M VXT(X1, ...,X,,e1,...,es) =—T(VXX1,..., X,,e1,...,es)—... -T(X1, ...,X,,e1,...,VXe~‘) +VX(T(X1, ...,X,,e1,...,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~-~“',,],_ _“bxinanybasis itisconventional todenote the components ofVXkT inthesame basis byT“1~~--=“r,,1,__q,,s,.,(. For any TeFTf.M thetensor field VTe I"Tf+1M defined by (VT)(X, X1,...,X,,e1,...,es)=(VXT)(X1, ...,X,,e1,...,es) vx,X,EFTM, e“6FT*M (6.3.s) 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 6.4Curvature 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 f€@(M): VX(fY) =(Xf)Y +fVXY and -93’X(fY) EIX,fY] =(Xf)Y +flX,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)is@-linear inboth arguments. Consequently associated with Tisatype (2,1)tensor field Tknown asthetorsion tensor ofV: T(X, Y,/5)=[a’(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{ea} isanyco-frame, inwhich theconnection 1-forms are{w",,}, then thetorsion 2-forms aregiven by 4 T“=de“+w“,,Aeb. (6.4.5) This iscalled thefirst structure equation. Itmay beproved bycontract- ingonapairofarbitrary vectors. Using (4.10.3) have 2(de" +co“),Ae")(X, Y) =X(@“(Y)) —Y(@“(X)) "@“(IX.» Yl)+fv“6(X)@b(Y) Tw”i(Y)@b(X) =X(@“(Y)) —l7.Y@“(Y) TY(@"(X)) +Vi/<?“(X) —@“(IX» Yl) by(6.3.3). The right-hand Side may besimplified byusing (6.3.1), producing (de“' +cu“),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 ofCURVATURE AND TORSION TENSORS OFV 209 Vthatforanytensor field U VXVWU =fVXVyU +(Xf)VyU VA/VXU =fV,/VXU VX, YEFTM. Ifwedefine R(X, Y)U =VXVYU —VyVXU —V[X_y]U (6.4.6) VU, X,Y,then again wehave @-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[VX, Vy]—V[X,Y]isa type-preserving derivation onthealgebra oftensor fields R(X,Y)(U®W) =R(X,Y)U®W +U®R(X, Y)W (64.9) 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,6)=B(R(X, Y)Z). (6.4.10) Since R(X, Y)=—R(Y, X)wemay introduce asetofcurvature 2-forms Rd,by R=2R“'c®e‘®Xd. (6.4.11) Interms oftheconnection forms 6)“),with respect toanyco-frame {ea}: Rab=dw“), +co“,Awcb. (6.4.12) This isthesecond structure equation. Forverification wecontract onan arbitrary pairofvectors: 2(d(t)“b +w".,Aro°,,)(X, Y) =X(wa6(Y)) _Y(wab(X)) _wa6(IX= Yl)+wa<.~(X)wC6(Y) _wac(Y)wCb(X) =X(@a(VYX6)) —Y(@a(VxX6)) “@a(V[x, Y]Xb) ‘VXea(Xc)eC(VYXb) 1 +VYea(Xc)eC(VXXb) =X(@a(VYX6)) _Y(@a(VxX6)) ‘"@a(V[X. Y]Xb) TVX@a(VrXb) +VYQQWXX6) =e“(R(X, Y)X,,) 210 CONNECTIONS =R(X,Y,X,,,e”) :2Rab(X, Bycontracting the(3,1)curvature tensor weobtain a(2,0)tensor: theRicci tensor. That is, Ric(X, Y)=R(X,,X,Y,e“) (6.413) where thearbitrary bases {XQ}and {e“} aredual. For ageneral connection ‘Ric’ hasnoparticular symmetry properties. Itissometimes more convenient towork with thesetofRicci 1-forms {P,}, elements ofwhich aredefined by Pb : IXaRab hence P,=Ric(X,,, X,,)eb. (6.4.15) Because oftheir @-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 qeN,, liesonone and only one (uptoalinear reparametrisation) geodesic radiating from p.Wedefine XqeTqM byXq=rqpXp where rqpistheparallel translation map along theautoparallel. This assign- ment ofatangent vector toevery qeN,, issmooth: wedenote the resulting vector field byX.Wesimilarly extend Yptoavector field Y. We have constructed Xsuch that VZPX=0VZP eTPM, thus T(Y, X)]p =[X,Y],,. From exercise 4.1attheendof§4.11 weseethat T(Y,,, Xp)isthetangent atptothecurve formed from theintegral curves ofXandY(seefigure 6.3). Inconsidering thecurvature weextend XPand Ypdifferently: this time tocommuting vector fields Xand Y.We could, forexample, choose normal coordinates {xi} with O 3 6x1P:Xp and 6x2P:Yp with X=—a-— and Y= 6x1 6x- If<p(p) and 1/t(p) aretheintegral curves ofXand Yrespectively, starting atp,then weform thequadrilateral shown infigure 6.4. Wedenote theparallel translation map from T,,M toTQM, along the.\ ':' '.-U I-, I I‘-':'-’:;‘- '=-“:3 I .-._-‘,CuRvATuRE AND TORSION TENSORS OFV 211 "rXan _./ PpYp / P t(>j,,x,,i Figure 6.3Geometrical interpretation ofthetorsion tensor. TFQTQPZP Xi~=uu,o~p,iipi Y TstTmT4/JZ/J Teezti X -inlp) S y 9-r Z, I p I Tﬂ5T$-'"T=’G‘T-'-IPZP Figure 6.4Geometrical interpretation ofthecurvature tensor. curve shown, byrqp.IfZpisanyvector inT,,M then wecalculate the parallel translate around thefigure byusing (6.1.10), dropping terms of order greater than t2: rqpZ,, ={Z—tVXZ +:2/2VX2Z}q +O(t3) r,,t-,,z, ={Z-t(VXZ +vyz)+:2/2(vX2z +V}/22+2vYvXz)}, +O(t3). 212 CONNECTIONS Proceeding around theloop wecompare r,,,r,,r,qrq,,Zp with Zp: _r,r,,r,r Z—Z ‘ii’ ’—([vY’VX]Z)P =R<Y.,X..>Z.~ since [X,Y]=0.This expression shows thatthecurvature 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(1,1)tensor field (VXR)(Y, Z)isdefined by(VXR)(Y, Z)(W, B) =(VXR)(Y, Z,W,/3).ForanyX,Y,ZEFTM consider thevector °-‘J={(vXR)(Y1 Z)+(VYR)(Z> X)+(VzR)(X, Y)}(V) -,;;,t(vXR>(Y. Z)}(V)~ Here 3X,y_Zdenotes thecyclic sum ofX,Y,Z.Now (VXR)(Y, Z)=VX(R(Y, Z))—-R(VXY, Z)—R(Y, VXZ), sowemay write O3=X_5£Z{AxYz —BXYZ}(V) where AX)/Z(V) =V.Y(R(Y» Z))(V) =VX(R(Y, Z)(V)) —R(Y,Z)(VXV) BXYZ(V) =(R(VXY» Z)+R(Y.VXZ))(V)- Wemay express AXYZinterms ofthecurvature operator AX‘/Z(V) =VX(R(Y> Z)(V)) —R(Y,Z)(VXV) :IVx- IVY» Vzl_V(Y.z]IV- Foranyoperators P,Q,Rwehave the(Jacobi) identity ,,g,iP.iQ. Rll=0 hence Aiz/4 XYZ(V) I;._E,f_ZIVx- V[Y.Z]lV' E.-5;.- -'.§-.\s.' 3‘ﬂ“J:' limBIANCHI INDENTITIES Writing outBXYZinterms ofV BxYz(V) :(VVXYVZ _VZVVXY —V[vXY, Z]+VYVVXZ —VVXZVY —V[Y,vXz])V, , \ X‘%’ZBXYZ(V) =X?.)Z(VvYzVx '_VXVV}/Z —V[v,.z, X]+VXVVZY “VVZYVX +V[VZY, X])V :XﬁZ([V[Y, 2]»Vxl_V[[Y,z].x] +Ivrtr. Z)»VXl_V[T(Y, z),x])V- Using theJacobi identity again gives X‘E£'ZBXYZ(V) I*X_<<{,0_Z{IVX» V|Y.Z]I"R(T(X, Y)»Z)}V- Thus 9.9=—X‘§£‘Z{R(T(X, Y),Z)}V andsince thisisvalid forarbitrary V: 9’{(VR)(Y, Z)+R(T(X, Y),Z)}=0 (6.5.1)X,Y,Z X VX, Y,ZeFTM. This isknown asBianchi’s second identity. Inasimilar way weobtain anidentity bycovariantly differentiating thedefining relation forthetorsion tensor. Weleave itasanexercise to prove Bianchi’s first identity: X§]jZ{R(x, Y)(Z)-r(r(x, Y),z)-(VXT)(Y, 2)}=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 do)“, byR“,—(oakAoak,then thesecond Bianchi identity isexpressed as dRab + (l)a(. ARC); * Rac. /\(UCb :3 0. Similarly byapplying dtothefirst structure equation and expressing div“, back interms ofR“,anddebback interms ofTbgives thefirst Bianchi identity as d7-la +(Dab ATb :Rab A€b.213 , 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 that thisparallel-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))E0.Since forfe@(M) C(f) EVcf, andVcommutes with contractions ('?(s(Y» Y))=Vc(s(Y, Y)) EV¢g(Y, Y)+2g(V@Y, Y). IfYisparallel along Cthen thesecond term iszero. Requiring that the length ofallparallel vectors along Cbeconstant gives V5,-g =0.Fora metric-compatible connection this holds forallC,soVismetric compatible ifandonly if VgE0. (6.6.1) If{X,-}isanylocal basis then covariantly differentiating thefunctions gt;=g(Xi> X1)gives X(8t)') =VX€(Xt» Xi) ‘I'8(wkt(X)X1<.- Xi) +3(Xt» wkj(X)Xk) =VX8(Xr» X1)+wkt(X)81q' +wkj(X)gik- If{X,-}isorthonormal then thefunctions g,-1.areconstant. Soinan orthonormal frame theconnection forms ofametric-compatible connec- tionsatisfy theantisymmetry condition 63,,+63,,=0 (6.6.2) where to,-,Eg_,-,.cuI‘,-. 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. (6.6.3) If{ea} isanorthonormal co-frame such that *1= e1Ae2A ...Ae”'1}-I‘ I_1'I_ ‘ :‘I:I _'_e-i I -: I Ti'.".E: _->2:-|'.\.'- " "Ii--ii..,,.-- ,.,._>~_.METRIC-COMPATIBLE CONNECTIONS 215 then VX*1= —w1,,(X)e" Ae2A ...Ae” +...+(—1)"e‘A ...A a)”,,(X)e“. Now w1,,(X)e" Ae2A ...Ae” =co11(X)e1Ae2A ... Ae” and so(6.6.3) follows from (6.6.2). Itcan now beSeen from the definition (1.4.5) thatifVismetric compatible 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(s(V, W))+V(s(W, U))"W(s(Ui Y)) =s(VUV. W)+g(V,VUW) +s(Vi/W, U)+g(W,VI/U) —s(VwU. V)—e(U.VWV): Thedefinition ofthetorsion operator enables thistoberewritten as 2s(Vi/V, W)=U(s(V, W))+I/(s(W. U))EW(s(U, Y)) —s(U, [ViW1)+s(Vi[WtU1)+g(W,[UrV1) —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 [Xw Xb] :Cab£IXC. Ifthree different basis vectors areinserted in(6.6.5) then wecansolve fortheconnection coefficients: rabp :%gCp{Xa(gbc) +Xb(gca) —Xc(gab) "FCbcdgad +Ccadgbd +Cabdgcd “HT(Xb> Xcv Ya) +T(Xcr Xai Yb) +T(X,,X,,,}Z,)}. (6.67) Here gq’istheinverse matrix togab, gpcg“? E<53.If{ea} isthedual basis then XQ=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 260(1), : €“iXaiXb((l€d "— Td) -I‘ IXb(£l€a _ Ta) — iXa(d€j, "_ 216 CONNECTIONS This formula isofgreat computational utility. From now onwewillonly consider metric-compatible connections. 6.7 TheCovariant Exterior Derivative Itisoften convenient towork with sets ofdifferential 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 {X1-}with dual {el}. Suppose that Sissuch atensor oftype (r+q,p).Wedefine aset ofr-forms SI""P,-1 J,-Qby Sii.,.ipj1___jq(X1, -..,Xr) I S(X1, ...,Xr, Xjl, ...,Xjq, eil, ..-,elp). (6.71) Wedefine thecovariant exterior derivative DoftheS"1':-‘P,-]___,-Q in terms ofaconnection Vby (F+i)I>st1---i»,~1,,,,~,(x,,, ...,X,) =2(-i)tvX,s(x(,, ...,)'Z,,...,x,,X,-1,...,x,-,, e‘1,...,e"/>),i=0 -Z(-i)1+'<s(T(X,-, Xk),X0,...,22,,...,2%,,..,(JE,i<kE..r Xr, Xjl, ...,Xjq, 6‘), ...,€‘P). The‘hat’ above asymbol indicates that that term isomitted from the sequence. Tisthetorsion operator ofV.Itfollows from theabove rather cumbersome expression that I 1...! DS1 pjl-'-lq =dSl'1__-ipj1.I.fq +wt,l_$ASi,...t,J_lmjq +___+wlipis ASi,..-,_5j]'|'jq —(L}j’J,']ASi1"'ipjS‘__}-Q — ...—Cl}]I’jqASi1"'ipj1_._j-S. This canbeverified byusing (4.10.5). For thespecial case inwhich pEq=0thecovariant exterior derivative reduces totheordinary exterior derivative. Wecanthen infer from (6.7.2) that 6“ AVXQ : d— Ta AIXQ.THE COVARIANT EXTERIOR 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, 2t...l _ I i§...t'_ _ ti‘ i...i,_ _ DSI pj]___jq—RIiYAS pJ1.__]q+...+Rp]SAS1 hlmjq ti/\ is---it — AS£]"'ipj'l Itfollows from (6.7.l) that under achange ofbasis thesetofforms S‘II'""IP,-I jqtransform according totheclassical tensor transformation rules. The 9?-linearity inthearguments oftheright-hand side of(6.7.2) ensures that theDSI1---‘E,-l___,q transform liketheS‘)---‘Ir,-1_,_,-q under a change ofbasis. IfSIandTlaresetsofr-forms ands-forms respective- ly,labelled bythemulti-indices Iand Jthen, asmay beseen from (6.7.3), I>(s1ATl)=DS’AT1+(-1)’S’ ADTJ. (6.76) 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 LXHEDix“ +IXGD (6.7.7) then LX8maps asetofp-forms into asetofp-forms indexed bythe extra index a.Itactsasaderivation onexterior products LXa(SI AT1) =LXQSI AT] ‘I’S1ALXQTJ. First weconsider asetof1-forms, A"1:--Ir,-1___,q. For Aany 1-form (6.7.4) gives iXadA : VXRA " €biXaVXbA ‘I’ IXaTbIXbA. Using thisin(6.7.3) gives iXaDAi1... ip__| H_r J I.‘ ‘ = i1...i_ __ 6: i1...i_ I -6- i...i_ IVXQA p]l___jq €1XaVXbA PJl'_'jq+1XaT1XbAI Ph___Jq - i,_t...}’,t,...t_ __ r,_- t...’i,t,...t_ _':'I'1X..w1,AI pll---iq w61X./11 pll---lq —' lr. I1---I. '.\. . it.‘ 1!---3. *>. . "' 1X..w JISA P11-~ ]S.fr"'.lq +wti1X.A P11-~ .lS.lr~-'11]. Now ifAisany1-form VXbA = VXbiXcA€C — IXcA(I)Cp(Xb)ep 218 CONNECTIONS so iXaVXbA =VXbiXaA —o)fa(Xb)iXcA and e"’iXaVXbA =diXaA ——w‘,,iXCA (by(6.7.4) again) so IXADA I1IiIIpli...Iii :VXaAi,...ip]_]H_jq _diXaAi1...i,,]_lH_jq +wcaiX€Ai1...i,,j1H‘jq +iXaTbiXbAi]...ipI1"-}_q +iXawi,l_rAt1... 'i‘,i,... ipj]Hijq _wi,iriXaAi1... ?,i,...i,,J_]m]_q ___ _iX.‘”I’i.A“""’ii__- LIA.--ti +wI’i.iX.A“"'I’ii-.- i.i>---ti» Recognising theright-hand side ascontaining DiXQAII---‘P,-I ___J,-Qenables thistobewritten as LXaAl1...lp)i].--}_q =(VXQ _|_iXaTb AiXh)Al']...lpjl".]_q +wi,ir(Xa)Ai1... i',i,,...i,,j_]_Hjq _ j,_ i...i_ A__ _ w;,(Xa)A1 p)1...j,],...)q' Wehave obtained thisexpression fortheA‘1'~-‘r,-1___,q 1-forms; butLX“, VX0and iXaT” AiXb areallderivations onexterior products ofmulti- indexed p-forms soitisconsequently valid onarbitrary p-forms: LXaSi,...i,,]_lmjq =(Vx, ‘I’iX,Tb /\iX,,)SII '''I”),...1‘,+¢I)i‘t,.(Xa)SI‘ '"U"''II’),...1, ...—o)l',-$(X,,)SI'--"Ir,-1_,_ ;S,r___,q. (6.7.9) IfSI1"--‘P,-1_H,-q ES(eI1, ...,eh,X,-1,...,X,-q)then thiscanbewritten as LXaSi;...lp]_1...jq =VA-aS(eII, ...,ele,X,-1,...,X,-Q) +IXQTI’ AiXbS’II--:IP,]___,q. (6.7.10) Forthespecial case oftp,any@-valued p-form, thisreduces to iXad<p +Dixatp =VXaq0 +iA-GTI’ Aixbgo. (6.7.11) The definition (6.7.2) canbeapplied to0-forms where there isthe simplification thatthetorsion terms donotenter. Since gab=g(X,,, Xb) wehave Dg,,j,(X) =VXg(X_,,, X,,).Thus forametric-compatible connec- tion Dgab =..6'1.- . .;. %,,,_ ‘-._;I.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=(n!)‘1e,l___,ne"1A ...Ae‘?= then e,-,___,~n =n!*1(X,-1, ..., X,-H). So forametric-compatible connection D851 ___in= Asanticipated theBianchi identities (6.5.3) and (6.5.4) cannow be written as DR“,=0 (6.7.14) DT“=R“,AeI’. (6.7.i5) 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 (6.7.15) canbecontracted toobtain various other identities. Weleave it asanexercise toprove thefollowing contracted Bianchi identities: St’iXiXiXDT =2(iXiXR —iXiXR )(6.7.16)p,q,r’a p q r a a qpr p raq IXpIXqiXaDTa= IXQPP _ IXPPQ IXPIXQIXJYD Tag“ +p%rIXpiXqDTr= %S§riXqRpr iXaDT“= P,AeI’. (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 9%isgiven by QR=Ric(X_,, X“) (6.8.1) Where asusual X"=g“I’Xb. Interms oftheRicci 1-forms Pa, gi, = IXHPH. 220 CONNECTIONS Inn-dimensions theEinstein (n—1)-forrns G,aredefined byTHE CURVATURE SCALAR AND EINSTEIN TENSOR 221 Bianchi identity. Inn-dimensions wecanexpand theHodge dual as = 'abGC Rab /\1Xt*e ' *6’Ilf2i3 —i—-————1 Q“ ‘5 I0 I ' =t<515213 These may berelated totheRicci forms; wehave Gc=R66/\iX,_.iX"*ea =Rba/\iXI’iX,.*ea =iXb(Rba /\ix,*@a) “Pa/\ix,*@aE(n_3), /\5'/\ /\@”1X,,, 1X,51x,, 9 Now iX,”...iXmiX,4*e‘1‘?"3 isproportional to*1contracted onnvectors, thus itscovariant exterior derivative iszero, so D*eI1I2I3 Ei—(T‘*A e"-SA ...AeI~ —eI*A TPA ...Ae"~ =IX"(R4b /\iX“iX."1) _P4/\iX."@“ I (”T3)! =iX,,{iX,(Rab /\iXC>I<1) _Pb/\iXc=i<1} _pa/\iXc.i.e4_ I +...+(—1)" ""e"~*A ...ATI~)iX,n ...iX,5iX,4*e*i*2I3 Now RabAIX*1E0,since itisan(n+1)-form, so 1 i i i‘ ‘ iii=fyli/(65/\ A€"lXl_n ...1X’,5*€‘23,-4 C Q 2, (n 4)! G,E—9R*eC +PbAiXtiX,*1— P,,AiX,*e" E—97t*eC —2PAea, and Pa/\*6“=iX"’Paeb /\*9“ =ix”Pa{—iX,(@b /\*@a) +8b<,~*@a) =ix”Pa(*86a*@¢ ‘Igi>¢"ea) E—iXaP"*eC +iX(P“’*e,,. The contracted Bianchi identity (6.7.17) gives theantisymmetric part oftheRicci tensor interms ofthetorsion, so P“A*e,,C E—9t*eC +iX~PC*e,, +iXiiX(iXbDT”*e,, E—9h*e, +*PC+*iX,iXhDT" thus G,E9t*e,, —2*P,. ——2*iX,_iXbDTI’ or *"1G,. Egtef —2P).—2iX€iXhDTI’. (6.8.4) The setofEinstein forms areequivalent toa(2,0)tensor. The Einstein tensor Gisdefined by G=-*1o,®@t. (6.s.5) The antisymmetric part oftheEinstein tensor isdetermined bythe torsion. Using (6.7.17) once again gives IXb*#IlG(_. — iX(_*_‘1Gb I —2IXbIX(iXIIDTa. The covariant exterior derivative oftheEinstein forms canalso be related to the torsion. Writing G"ER,,,,A*e“I"' we have DG‘ EDRQ), A*e“”‘ +RabAD*e"I". The first term iszero bythefirst=T14A>l<e‘li2i3I_4 thus DGC :Rab ATp A*€abCp. Equivalently thisrelation canbewritten interms ofthe(2,0)Einstein tensor. The divergence ofG,V.G, isa1-form defined by (V.G)(Y) EVXaG(X”, Y) (6.8.8) thus V.G E(iX~*‘1VXaG,, -—co‘,,(X,,)iX~*'IGC)eP E*“(e“ AVXQGP —wcpAGC)eP. Wemay now use(6.7.4) togive V.G E*‘1(DG,, —T“AiXaG,,)eP and(6.8.7) then gives V.G E—*‘1(T‘? AiXqR,,,, A*e“I’,,)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 E1",,,P. Forthisreason atorsion-free connection isoften called ‘symmetric’. The connection coefficients ofthepseudo-Riemannian connection expressed 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 irt2(n —1)independent connection coefficients. Foracoordinate basis thezero-torsion condition cuts down thenumber ofconnection coeffi- Ii='.' cients to§n2(n +1).Thus inanorthonormal basis there aren2fewer connection coefficients. ‘ Because oftheBianchi identities thecurvature tensor ofatorsion-free connection has extra symmetries. Equation (6.7.16) reduces toan expression ofthe ‘pairwise interchange’ symmetry ofthe Riemann2-»1'=-. tensor. Equation (6.7.17) shows thatforzero torsion theRicci tensor is-3-If_I symmetric. Forzero torsion theEinstein tensor issymmetric, by(6.8.6), anddivergenceless by(6.8.9). . Wecanuse(6.7.4) towrite theexterior derivative interms ofany I torsion-free connection. Since anymetric-compatible connection satisfies-I'__;;'t.T (6.6.4) weobtain auseful relation between thepseudo-Riemannian .=.:1-6Egg I-71-ii‘!=..r connection andtheco-derivative 6which wasintroduced in(5.4.2). Iftp isadifferential p-form then iXQVX000 iscertainly a(p—1)-form. Q';;-. Introducing theHodge map anditsinverse: - -:its. I iX“VX..(i7=iX"**_1VX..(p =iX"*VX.,*_1(p by<6-64> -*<vi.*—1<pAe) (by(1-47)) .'.*E.J'.-rtt.. =*(@“/\Vx,l7* T197) ]] where T]isdefined in(1.1.2). Wenow use(6.7.4): II ix"VX,,(P Z*d77*_1(P- The inverse oftheHodge map isgiven in(5.4.3). Byconsidering ther-\ II. cases ofeven andodddimensions separately itcanbeseen thatthiscan berewritten asiXiVXﬂcp E—*'Id*r)09, thatis iX.vX,¢ =-66». (6.91) I-A From now on,unless wespecify tothecontrary, weshall restrict ourselves tothe pseudo-Riemannian connection. For most ofwhat follows itwill beessential that theconnection ismetric compatible, whereas inmost places torsion merely contributes extra terms. Exercise 6.4 -.. AnEinstein space isoneforwhich RicEcgforsome constant c.Show thatif,inthree ormore dimensions, RicEfgforfe§(M) then: (i)fE91/n (ii)G.=*—T(”QZIQYW. :r. .- (iii)d9RE0.THE PSEUDO-RIEMANNIAN CONNECTION 223 Example 6.1 Let gbethe metric tensor ofafour-dimensional spacetime: 3E—e°®e° +Ei:1e“®e“. Inalocal chart with coordinates (t(p), r(p), 6(p), g0(p)) aclass ofspherically symmetric metrics may be parametrised byfunctions H0,H1,H2ofr(p)andafunction /Ioft(p), bychoosing alocal orthonormal co-frame as 60 =Hgdi e‘Ee"H1dr e2Ee’IH2d6 e3Ee’IH2 sinBdqo. Asanexample ofusing (6.6.8) verify thattheconnection 1-forms 01,), ofthepseudo-Riemannian connection aregiven inthisbasis bytable 6.1.Hence construct table 6.2forVXaebwhere X,,isadual orthonormal frame: eb(X,,) E62. 6.10 Sectional Curvature Atwo-dimensional subspace SofT,,M willbecalled atangent plane to Matp.If{X,Y}isanybasis forSand Q(X, Y)=s’(X,X)s(Y. Y)—(s(X, Y))2 (6-19-1) then Q(X, Y)E0ifandonly ifginduces adegenerate metric onS. Such atangent plane iscalled degenerate. IfSisanynon-degenerate tangent plane atpthen thesectional curvature ofMatp,along the plane section S,isK(S): s(R(X. Y)X.Y)1<(s)-EQ(X,Y)». (610.2) 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,,(J\f0 F)S)isatwo-dimensional Riemannian submanifold ofM.Let 973(r) beanopen ball ofradius rcentred about theorigin inN0OS, with rsufficiently small that Expp isadiffeomorphism onto B(r), an open ballcentred about p.Lets4(r) bethearea of9B(r) andA(r) beiI 616 Iii forms60,),E—o),,,,for freeorthonormaconnectoniv-Ii -—| OTSOl'l-II—I I The Table6.1Q .—()I/H)e-I H3)€_'l€I —(cot6/H2)e-IIe~‘Z »--~—- ~._.-' I~ themetricofexampe62 (/1/H)e2HH)e"’:2 AC. ._@ \ “*(H2/ v—t —(A/H)e.--\'6 nq-4 HH)e-'Ie0 - ,_._ ~I_1 /—-.4-4 —(H’/ CID C.'>~—~('\1c')dl/dtHEdH,,/dr“Q 0- A54 -I-..~I-.. a-\-I-4-._ fyngeI’(X_,)E<51? II-II basessatSIi cientsspecifiedbyVxuebntheduaII—1 6u—| oncoeff\I-ll ofLev-CevtaconnectI? aI—I Pi Table6.2Associatedtabe4 l'\l Q.) I? Q I Crpm M./‘-‘\ 0 —(H/HOH)e"'Ie°|-4 a—\s. , ‘_| H/HOH)e"Ieq—| A8. , -< VX°=€>< >-<l>I> DQCD 0H)e°-Q&-r n,_i —(/I/H)e——(H3/HH3)e-'Ie-—(cot6/H2)e-'Ie3,_., -Ii. 1-». —¢ ¢—-I-.1 0 .—i )e‘*e/163 \ (/l/(H2/HH2cot6/H2)e-'—1 (_ —(/I/H0)e° H2)€_A€241.1 I-Ii. -1 — -(ii/H.,)e2(HQ/HH2)e-'Ie3u—I1 —(/I/H)@~7¢—--v 0 (‘NI 1"“; IIEat/atH;EdH,/dr0*T 1! 5r2 - "I '5.SECTIONAL CURVATURE 225 thearea ofB(r). Thus s5i(r) isdetermined bytheEuclidean geometry of TPM whilst A(r) isdetermined bythe Riemannian geometry of Exp,,(J\(U F)S).The sectional curvature isdetermined byacomparison ofthese twoareas: K(S)=121312 . (610.3) The proof ofthese assertions canbefound in,forexample, Helgason (1978). Exercise 6.6 Take Mtobethetwo-sphere with thestandard metric induced from B3 (see figure 6.5). Calculate thesectional curvature using (6.10.2). Verify that (6.10.3) gives thesame result. (Note that B(r) isaspherical cap with geodesic radius r(figure 6.5).) alt‘) iiﬂ Figure 6.5 Amanifold issaid tohave constant curvature ifitssectional curvature is constant. Exercise 6.7 Show thatMhasconstant curvature cifandonly if RabEce"". (6.10.4) 6.11 TheConformal Tensor Two metric tensor fields gand gsuch that gEexp(2).)g forsome function 2.aresaid tobeconformally related. Whereas aconformal rescaling ofthemetric will change thecurvature itispossible to construct atensor outoftheRiemann tensor that isinvariant under 226 CONNECTIONS such scalings. Let{ea} beag-orthonormal co-flame, with dual {Xa}, and{e"} a§-orthonormal co-frame, with dual {XQ},where Q=6Xp(i)@== 2?;=exp(—/1)Xa. (6.11.1) If9isthepseudo-Riemannian connection ofgwith connection forms/‘\/""'-.. a ma),with respect to{e}then from (6.6.8) (mg)=6.1+X1<»1)e“—X.</rm. (6.112) Similarly thecurvature forms RC),of9inthe{e/5} basis are (Rig)=Ra),+vXb62.,( ea-vXa62.,( 6,,+X,,(2t)@a Ad/I —X..'(/U66 /\<1/I—XC(l)X‘(1)@66- (611-3) Wehave used DXa(/1) —Vxad/1, which follows from (6.7%11).Contract- ingwith )?‘gives theRicci forms andcurvature scalar ofV: /\. @Xp(/i)Pb -—Pb + ““'n)VXbdl +(I1 +(2-n)XC()L)X‘(}t)eb -lXaVXﬂdAeb (611.4) 6xp(2i)@?‘r =at-2(n-1)1X1vXb62t +(1-n)(n-2)XC()t)XC(/1). (611.5) The conformal 2-forms Cab aredefined (inmore than two dimen- sions) interms ofthecurvature 2-forms andtheir contractions by C=R ———;(P e—P e)+ ~16 6eagle /(6),. ab ab n____2 a/\ b b/\ n (n__2)(n_1) a (6.11.6) These 2-forms have theimportant property ofbeing invariant under conformal scalings ofthemetric. That is,ifCab aretheconformal 2-forms of§with respect to{ea} then cj;=cab. (6.117) Ifthe(3,1)conformer! tensor (orWeyl tensor) Cisdefined by C=2C“b®eb®Xa (6.11.8) then equivalently @=c mum From their definition the conformal 2-forms Ca), are manifestly antisymmetric under interchange ofaandb.They alsosatisfy (forzero torsion) analogous identities tothose forthecurvature 2-forms, namely Cab A6"=U (611.10) iXaiXhCpq : iXpiXqCab.THE CONFORMAL TENSOR 227 Inaddition there istheidentity iX6Ca,, =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 :égleab. Since there isonly onetangent plane wewrite thesectional curvature simply asK.This isrelated tothecurvature scalar by K=gar. (612.2) 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=2Rab®e"b. Both factors inthetensor product are2-forms, anditisoften convenient tohave anotation forthetensor obtained by taking theHodge dual ofeither factor. Wewrite *R=2*Ra),®e"" (6.12.3) and R*=2R,,b®*e“b (6.12.4) Inthree dimensions thedual ofa2-form isa1-form, so R>=<=2Ra),®e‘ ,(iXC*e“" =2RabiX(*e“b®e‘. The first factor now involves theEinstein forms, which were given in (6.83). Soif ‘QE2GC®e‘ (6.l2.5) wehave R*=‘Q,or R=‘§*". (6.12.6) 228 CONNECTIONS Now <§*—1 =_GCiXaiXb*_1e“®e""b =(—9t*eC +2*PC)iXaiXb*“e‘®e“b by(6.8.4). Tosimplify thefirstterm write _1 i 6 6 6 ___1 i 6 . 6 _l I I _1 1X..1X6* @C‘?<~ —1X“1X..1X1.* 196—1X"(1X..1X6* 1/\er)+3‘X..1X1.* 1 . . . _* . _l _1 : ]X<.~{1Xa(lXb* 11/\ QC) —1Xb* lgac} ‘i’ 3* 6),“ II =iX"iX,,.(ec /\F166) “iX,,iX,,*—11 ‘l’3*”666 =gbciX‘iX,,,*—l1 +2*_16’66 =*_1@6a- Inexactly thesame wayweobtain P,.iXaiXh*“1e“ =iXbP(.*“1e,f —iXuPC*_'e)," +€’7t*_‘e,,,,. Using thesymmetry oftheRicci tensor, (6.7.17), gives PCiXuiXb**1e‘ =**1(e,, AP),—e),AP, +§F?.e),,,). sowehave ‘§*_1 =2(§@teb,, +PaAeh—P),Ae,,)®e“". Thus (6.l2.6) shows thatinthree dimensions Ra),=%97?.e),,, +PaAe), —PbAea. (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 thatifinn dimensions YaEDP, —[2(n. —1)]“d97i Aea then Ya=exp(—l) ><[Ya+(rz.—2)Xb(/l)C,,,,]. (ii)Show that YaAeb—Y),Aea=(2—n)DC,,,, andYaAe“=O. Inthree dimensions Ca),E0andsointhiscase thetensor Y,,®e“' is conformally invariant. Thus thevanishing ofY,®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 R1“E%(Ri>1=-1R*) (6.l2.8) wehave R‘=(PpAeq —PqAe,, —§9tepq)®eP‘? (6.12.9)I i .3;-i ';r;-.5 _gr».- aé _>--IJ, Q;-.--...\. -.-1 .:-.-4'_.-.11.,-=.-:->:;SOME CURVATURE RELATIONS INLOW DIMENSIONS 229 and R+"-C+197te ®eP‘1 (6.12.10)_ Epq where C=2Cpq®eP‘*‘. These relations can beverified inexactly the same way astheir three-dimensional analogues. 6.13Killing’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 are several 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 Itimmediately follows that AXisaderivation ontensor fields that commutes with contractions, also satisfying AXf= 0Vfe@(M). In particular AX(g(Y. Z))=0=(/1Xg)(Y. Z)+g(/IXY. Z)+g(Y.AXZ)- ForVmetric compatible AXg=.SEXg sotheabove becomes §(/IXY, Z)+6'(Y.»AXZ) =-()i’Xs)(Y, Z)» Since foranyvector field Ywehave AXY =[X,Y]—VXY, ifVis torsion freethen AXY=-VYX, hence g(VYX, Z)+g(VZX, Y)=($Xg)(Y, Z). (6.13.2) Ififisthe1-form related bythemetric toXthen itisoften convenient torewrite theabove intheequivalent form izvyir +1YvZ3{' =(§BXg)(Y, 2). (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) 5: 230 CONNECTIONS K1LuNo’s EQUATION 231 v,,6i>‘ =v,,6@,(vX,?' +ea,(vVvX,"Y' =-e“(VVX),)eb ,(vX,'17 +6“,(v,,vX,f>' :_eh/\VVI/Xbi? 'i'€a/\VVVXapYi =9“/\(V1/VX, Vvvxﬂl Y =ea/\(R(V, X6) +VX,,VV +V[V,X,,] —Vv.,X,,)? =e“A(R(V, Xa)+VXQVV —VVXHV) 17since Vistorsion-free. =e“,(R(v, X,)i>' +av,/Y‘ -e“,(vWfY. (613.5) Now 1X6?=iXe“VXal7 -e“Aixvx, "Y" =vXi>'-e“A(1XvX, +1X,vX)? +vXi>' sothat VXP=§iXdI7 +geeA(iXvX,? +1X,vX'Y). Using (6.13.3) wehave VXP=§iXdi7 +g.se,,g(x, X,,)e“. (6.13.6) This gives @~AvW,"Y" =§e"AiVXavdI7 +;;eYg(vX,v, X,)@@b =re/\(vX..<i./cl?) -1./vX.dY'> +tirygtvrv. Xbleab =§di,,dP +timeAVXad '17)-gvvai“/' +iggl/g(VX,,V: Xbleab =;61,,6i7 -§v,,6”Y‘ +;.2.P,,g(vX,v, X,,)e“” (613.7) since d2=0.Using (6.13.6) once again aw? =garter +g6(.seYg(v, X,,)e“') =;6i,,6i7 +gv,,,,._<e,,g(v, X,,)eb“ +;6>,,g(vX,v, X,,)e"“. (613.3) Returning now to(6.13.5) with (6.13.7) and(6.13.8) produces VVdP =26“,(R(v, X,.)i>' +vX,§£Yg(v, X,,)eb". This canbeexpressed interms ofthecurvature 2-forms as vvai/‘ =2YevbR,,,. +vX,§eYg(v, X,,)e”“ (613.9) where Y“=e“(Y) etc.Operating onthiswith theinterior product gives anexpression with theRicci forms: : —2YaPa + VXt_£Yg(XC, Xa)€“ _" VXb$Yg(Xa, Xa)€bi ._.,. =5;.E51-? :--=-'1... .'.1.-'-= 3351.» I-=:--i::l.. '516,-6'.. ,5or,by(6.9.1) 661/_2Y“P,, vX,.se,,g(XC, X,)@~ +vX,.&eyg(X“, X,,)@b. (6.13.10) Obviously such expressions are particularly useful forvectors that generate symmetries. Exercise 6.9 Avector field Kiscalled aconformal Killing vector if§£Kg EZltgfor some function A.Show that Ksatisfies (1)61?=621 (613.11) (11)66K_21<@P, +2(n1)6/1. (6.13.12) Exercise 6.10 Forsome calculations oneneeds tobeable tocommute aLiederivative past acovariant derivative. If DX(Y) E[.251/, VX] —VWIX] (6.13.13) show that (i)DX(Y) isatensor derivation thatcommutes with contractions (ii)D)x(Y) =fDX(Y) forf6WM) (iii)DX(Y)fS =fDX(Y)S foranytensor field S (iv)DX(Y)Z =DZ(Y)X (since Vistorsion-free) IfDXa(Y)X), EM,,b"(Y)X,. show that Mabphyl =igcp(VX,,*§£Yg(/Ye: X6) _VX,.§~PY8(X6» X6) +VXh.§£yg(Xa, XC)). (6.13.14) Hint: starting from DXﬂ(Y)(g(X b,X6))=Ofollow 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) Gravitation 7.1Lorentzian Connections Aswenoted in§6.2 thespace IR”hasanatural connection. This is defined such that anatural coordinate basis isparallel. Wehave already seen how Newtonian dynamics may bedescribed with thenatural connection onIB3.InChapter 5Minkowski spacetime wasmodelled on IR‘, the natural coordinate basis being declared orthonormal with respect toaLorentzian metric. Such afield ofglobal orthonormal frames isparallel with respect tothenatural 1R4connection, andthus we may now recognise theclass ofinertial frames asconsisting ofallframes that areparallel 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 VC-~C may beattributed toafour-force @:@=V@(itC). Forexample, ifCdescribes aparticle ofelectric charge qmoving ina background electromagnetic field described by__the 2-form Fthen the force isgiven bythe Lorentz rule @=qi¢F. Hence Cmay be determined bysolving theequation V('j([,lC) =qi'§"I~"'. (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=EAdt+B.Similarly weexpress thetrajectory four-velocity Cin terms oftheNewtonian velocity 0",k=1,2,3,with respect tothe same inertial observer, asC=7/(8,+v"87,), where 7/'1=(1-—v"u,,)1’2. Itisstraightforward tocalculate Vc(uC) =C(ur)@i +C(m'v")@i. (7.12) and ii=-1/E,v»'e, —)/Eta,+)/61i';,i'3. (7.i.3) Wehave__w_ritten E=E,-dxl andused i(;dt =7/,(ii=—8,, dY1'=E),-_If wewrite ia,B =—ek’"‘B78,,, (where ck)", istotally antisymmetric k,l, m=1,2,3and £123=1) then inaninertial chart forMinkowski spacetime (7.1.1)becomes C(m/vm) =—q)/(Em +v"8i.i,..B’) C(,u7/) =—q)/E,,,v"‘. Since C=C*8,, C(t) Edt/dr E7/relates theinertial time variable tto theproper time "ratpoints onthecurve. Similarly C(x") =dxk/dr =7/0"=(dt/dr)o", hence 0"=(dxk/dt). Setting pk=it)/0", %=try gives theequations intheform d $(Pm) Z_¢l(Em "'vk8klmBl) Cl . 325% = —qEiUi. V)’/(e seethat theNewtonian equations ofmotion arerecovered for 00,,<<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 (E,77,y’,2’). t=Esinh 17,x=‘gcosh 17,y’=y, 2’=ztoexpress theMinkowski metric tensor intheform gE—§2d77®di7 +d§®d.§ +dy'®dy’ +dz'®dz' onapatch defined byOEE,77,y’,2'<00.Verify that theonly non-vanishing connection components inthis chart are given by V558,, =(1/§)8,, =Van8~;- and Vaq8r7 =E85. Show that C=<98", ‘ﬁegt, solves (7.1.1) foraconstant electric field expressed intheinertial chart asF=Eﬂdx Adt if‘Q=—qE0/m. Hence derive thehyperbolic orbit (5=<9“, 17=‘Qt,y’=0,z’=0)and show that thisasymptotes toa light cone. Note ‘Qisthenorm oftheconstant four-acceleration ofthe particle: g(V¢C, VCC) =(Q2. 234 GRAVITATION 7.2Fermi—Walker Transport IfCisanygeodesic ofanarbitrary spacetime (V7-C E0)then ifg(X, C)E0atanypoint onthecurve then Xremains orthogonal toCatall points ifV¢X E0.Butiftheacceleration field ACEV@C along Cis notzero then thisproperty islost. However, onanygiven Cwemay usefully define anew connection Vinterms ofVandthemetric tensor field g.Acting onanyvector field Xrestricted toC 'vCXEVCX+g(C,X)A(; -g(AC, X)C. (7.2.1) 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,vCY)+goCX,Y) vx,Y66c(7.2.2) so iscompatible with themetric tensor g.IfACisanobserver curve (g(C, C)E-1)then g(AC, C)E0andhence V(";C E0:soavelocity vector isalso F-parallel. For any vector /field YonC,C(g(Y, C)) Eg(VCY, C), soifYisF-parallel (VCYE0)then the metric projection ofYonC(ortheangle between Yand C)ispreserved along C.Inparticular ag-orthonormal frame {Xa} atonepoint ofC, with atimelike basis vector X0EC,will remain orthonormal with XAECatallpoints 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 that inspacetime anF-parallel spacelike vector Ssatisfying theorthogonality condition g(S, C)E0along atimelike curve Cmodels thebehaviour ofanideal gyroscope (one that experi- ences nonon-gravitational torques) onC.Three such mutually ortho- gonal gyroscopes (g(S,-, S,—)E5,-J,-) together with Cthen define a non-rotating frame along C.Itisinteresting tonote thatthisconcept 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 ofTHE EINSTEIN 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. Theidea 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 ga:M—>qaMadiffeomorphism, (p*gwillsolve thediffeomorphic image ofEinstein’s equations oncpM. 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 that each 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 ofthespacetime manifold M.If{G6} arethe Einstein 3-forms associated with theLorentzian connection, given in (6.8.3), then Einstein’s equations forgare KG.+r.(e.)=0 6=0,1,2,3 (7.3.i) where {t,,(g, )} isasetofstress 3-forms determined inthisco-frame bysome choice ofmatter fields, denoted generically here by(D,andKis some (positive) coupling constant. (The notation indicates that re depends ongandrather than being contracted onthese fields.) We shall supplement these equations with asetofmatter field equations denoted collectively by <6(g,)=0. (7.3.2) Wecannot choose thestress forms arbitrarily, since forzero torsion (6.8.6) and(6.8.7) reduce to DG, E0 (7.3.3) and Ga /\€b =G7, /\Ca. The matter stress forms defined with respect to{ea} determine the stress energy tensor field 27=*-16,,®@@. (73.5) Any matter model forEinstein’s equations must therefore give risetoa symmetric second-rank stress tensor 9'E?T,,,,e“®eb that isdivergence- less: V.?TE0.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.l), that inndimensions Einstein’s equations canbewritten as\§::'='=‘--. _-5-*--=.-?- A.»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 §(V, V)E0 foralltimelike V.This condition ismotivated byassuming that anobserver whose curve istangent toVwould interpret g(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 7:;E—*7:V forrvEt,,e“(V). Alternatively one canimpose conditions onthestress tensor byrequiring that thecorresponding (viaEinstein’s equations) Einstein tensor hascertain properties, resulting ingravity being, insome sense, attractive. The condition onthestress tensor such that Ric(V, V)E0foralltimelike Viscalled the strong energy condition. Details ofthese energy conditions canbefound inHawking andEllis. 7.4 Conservation Laws InNewtonian dynamics thetotal energy and momentum 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 may be 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. LetETbeasymmetric (2,0)tensor whose metric related (O,2)tensor hascomponents FF"insome orthonormal frame {X6}. Foranyvector field Vwehave §£vg(X,,, Xb)+g(.§EVX,,., X,,)+g(X,,, $vX,,) E0since .§£V[g(X,,, X,,)]E0forany orthonormal frame {Xa}. Hence since gabE9'5”andVistorsion free: i§£Vg(Xa, Xb)gab*1 : _'2g(i$i/X0, Xb)gab*1 = _“2g(VVXa _“ Vxav, Xb)9_ab*1 2P_.,._1,( iX"*"%,.(__ =-—{g(VvX,,, x,)+g(x,,VVX7,)}§""*1 +2g(vX,v, x,)a*~b*1. K __ ._.-_ C n-2 1-r.-.- -1 ' =i.. .-‘.2-~:*._-,i,...__ '4.., F 238 GRAVITATION Now g(VVXQ, X,,) +g(X,,, VVX),) EOsince V{g(X,, X7,)} E0andso %§£Vg(Xa, Xb)gab*i = g(VXaV, Xb)gab*1 :VXu{g(V, Xb)gab}*1_ VXuXb)gab*1_ Xb)VX“g'ab*1_ ii iii‘ini»Now forany(n—1)-form Jwemay write d-7=5“/\VX..J =Vx,(@“ A1) —Vxﬁ” AJ- Sointroducing j“Ee“A]wehave d]EVXuj“ —(VXae“)(X7,)jb. Thus wehave A i§£v8(Xw X6)gab*1 IVX,i8(V» X6)=O7ab*1i _(VX,6’a)(X1>)8(V» Xc)gbC*1 +(VX.@“)(X6)s(V, XC)9_bC*1 Eg(V, VXaXb)€J'“b*1 —g(V, X7,)VXﬂ“"*1 Ed{V,ﬂ‘”’*e,} —{e“(VX“X,,)VC9"*’c +g(V, VXaX,,)9"" +V,,VXa€T“"}*1 =d{Vb?T,,),*e"} —{VXie"(X,,)9"7,Ce‘ +VXaeb§,b +X”(E’T,,b)eb}(V)*1. Wemay write thisinterms ofthe(n—1)-form JVEV"9T,,,,*e" as §(§£vg)(X,,, Xb)€’T“"*1 EdJv—(V-€J')(V)*1. (7.4.1) From this relation weconclude that ifthe spacetime admits a conformal Killing vector field C,icg E2/lg, then /1a",@*1 =61¢-(v-sr)(c)*1. Hence aclosed (n—1)-form may beconstructed outofadivergenceless traceless stress tensor inaspacetime with conformal isometries. Ifthe vector field KisKilling (§BKg E0)then irrespective ofthetrace of9' 611,,=0. IfP0,(P,-) areKilling vector fields onfour-dimensional spacetime generating open timelike (spacelike) integral curves then theintegrals of thecorresponding 3-forms over aspacelike 3-chain 2define theenergy (momentum) contributed byETtoE.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 L important property that onemay define thetotal charge contained ina '6CONSERVATION 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 the stress tensor vanishes thespacetime isRicci ﬂat. Soifthespacetime admits aKilling vector Kthen, from (4.l3.12) 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—Gordon system Themassive realscalar field cpePAOM istaken tosatisfy d*d<p Etlz“-‘Q9 +U'(cp)*1 (7.5.1) where ,uissome realparameter and Uisapolynomial ingo.The stress forms inthelocal frame {Xa}aregiven by It=%(i..d<aA *<1</1+dfﬂ/\i6*d(P) E(iazirz +U)*@@ (75-2) where i,EiX“.The stress associated with aconstant Uissometimes attributed toa‘cosmological term’. Aswehave remarked, inorder tobeconsistently equated tothe Einstein tensor, thestress forms should satisfy Dr, E0.Taking the expression in(7.5.2) gives Dru=ii(DiX,,d(p/\ *dq>+iX,d<a/\<1*<I1<aEd<PADiX,,*d¢) -(itztp +U’)dqoA *e,. (7.5.3) Now wemay use(6.7.11) (forzero torsion): D76 =i(VX,,d(P/\ *9‘?+iX..d(l-9/\ d*d(P _99‘?/\ Vx,*d99 +999/\ix,,d*d€0) —iX..<1¢>(a2<t> +U’)*1~ Since V is metric-compatible dcpAVXfdrp E dqaA*VXddqa E VXudcpA *dipandsotheterms involving Vcancel. Since d(pA iXad*d<p E —-iXu(dcpA d*dq0) +iXﬂd(pA d*dqo, and d(pA d*d(p isa5-form infour dimensions Dr.=X..(<r)(<1*d<r —ii2*<P—U'*1)~ Thus whenever thefield equations (7.5.1) hold Dr, EO. Z40 GRAVITATION Exercise 7.2 Show that§(X(,, X(,)*1 Er0Ae“andthatfor(7.5.2) 13 7 7 ‘J To/\@U =(§2(X.i(§0))" +iH‘<P" TU)*1-=6 S thatis,forasuitable potential Utheweak energy condition issatisfied. TheEinstein—Proca system The‘massive’ reall-form field Aistaken tosatisfy drdﬁ=—a2*A (75.4) with itsome realnon-zero constant. Theassociated stress forms are 6,=§(i,6.i A>162-i,,*dAA62) +;ii1(i,,,li A*2+21‘Ai,*,li). (7.5.5) Itmay benoted that anintegrability condition follows byapplying rdto (7.5.4): all=0. (7.5.6) TheEinstein-Maxwell system For theelectromagnetic field 2-form Fwehave thecurved space Maxwell equations d*F E0 (7.5.7) dFE0 (7.5.8) with associated stresses r,E§(i,FA *F—i,*FA F). (7.5.9) TheEinstein Yang—Mills system LetAEA,»T‘ beaLie-algebra-valued 1-form, A,-el"/\1M and {Ti} a basis forsome Liealgebra, with Liebracket [T", Tl].The Yang—Mills field strength istheLie-algebra-valued 2-form FEdA+§[A,A]EF,-T‘ where the bracket between aLie-algebra-valued p-form Hand a Lie-algebra-valued q-form Bis [H, Z : _ (_1)pqBAH anddAEdA,-T’. Itisuseful todefine anexterior covariant derivative ontheLie-algebra-valued p-forms H: on=611+[A,H]. (7.5.10)i-‘-5,: 1-;.- I ii; .-.1?SoiviE MATTER FIELDS 241 From thedefinition ofFwehave theBianchi identity DFEO. (7.5.11) Thefield equation analagous to(7.5.7) is D*F E0. (7.5.12) Thesystem iscoupled toEinsteinian gravity with thestress forms Ta : _ia*Fi/\Fi)' TheEinstein—Maxwell-charged scalar system Inthiscase anelectrically charged complex scalar field (I1couples to both gravity and electromagnetism. The Maxwell equations now have electric current sources j[g,<11]: d*F Ej (7.5.14) dFEO (7.5.15) where thecurrent 3-form is jEIm(*@*) (7.5.16) andtheU(l) exterior covariant derivative isdefined by 93¢)Ed+iA interms ofthe 1-form Asatisfying FEdA. Under the maps A+—>A—dit,(I)i—>ei" foritanyrealfunction onM,§D|—>ei’l€Z>. All electrically charged tensors andtheir U(1) covariant derivatives belong tosome representation ofthegroup U(1). The Maxwell stress forms are now supplemented by r,,[g, A,<D]E§Re(i,,§ZJA *9.D* +§DAi,,*91)*) —i(M2l¢’|2 +U(|‘1>l2))*@t- (7-5-17) TheU(l) covariant field equation for(Dis €:D*E€D E,u2* +U’*1 (7.5.18) with U’EdU/d||2. Exercise 7.3 Show that thetotal stress tensor, thesum ofthose in(7.5.9) and (7.5.17), satisfies Dr,E0when thecoupled Maxwell—Klein-Gordon equations, (7.5.14), (7.5.16) and(7.5.18), aresatisifed. 242 GRAvITATioN Ideal-ﬂuid stress Inastrophysical problems oneoften models massive ﬂuids onatimelike vector field. IfVisalocal vector field with g(V, V)E-1each integral curve isconsidered todescribe theworld lineofamassive fluid element. Iftheﬂuid hasmass density specified bytheO-form p,the3-form mass current is I-6.:6 7-p*V (7.5.19) andthemass inaspacelike 3-surface EisI;j.Ifthenumber ofparticles inthefluid remains constant then djE0.Weexamine thesymmetric tensor field I-My 5?!”-pV® V. (7.5.20) Since vX,a"'=(X,,p)V®V +,6vX,v®v +pv®vX,v then (VX,?7)(@“. )=(X..t>)V“V +P(VX.V)(@“)V +aV“VX,V- Thesymmetric tensor field Thasdivergence v.3""'=v(,6)v +,6v.vv +pVvV but(VXa(pV))(e") EV(p) +pV.V, hence v.3’=V.(pV)V +[JVVV =-—5(pV)V +pvvv E-(*dj)V +pVVV. Thus for 5:tobedivergenceless the acceleration ofVmust be proportional toV.ButifVistimelike with constant norm itsaccelera- tionisorthogonal toitself. Sothedivergence ofTiszero ifandonly if djEOandVisageodesic vector field, VVV E0. Electrically charged ﬂuid stress Suppose that each integral curve ofVmodels theworld line ofan electrically charged fluid element. Letthecharge density p,ofthefluid be(e/rn)p. Thus each world linemay betaken tocorrespond toapoint particle with electric charge eand mass rn.The gravitational field equations aretheMaxwell—Einstein equations where theMaxwell equa- tions have as3-form current source 1"‘-61 J,--(p,v). (7.5.2i) Thesymmetric stress tensor forthesystem ofelectromagnetic fields and fluid is-':».. '..i:'r- ~"._., "Ti-£1;-.i=--- Q33?’ _..L;‘r¢;. =w_"'."‘3"i -Av.-LL-6;._:=-,-.- _....-'.6ma¢e-.r'=i--.--.-.-.-SOME MATTER FIELDS 243 I"'\-Ir--66.1 where 57M) istheMaxwell stress tensor. Ifd*F EJ,then from (7.5.9) Drum, EFAiXaJ,. Since 9Tenjoys similar properties totheEinstein tensor <3,anargument analagous tothat leading to(4.8.9) shows that VET E*“Dr,,e". SoforanyX,V.?T(M)(X) E*'1(FAiXJ,). Repeatedly using (1.4.7) with **E—i7: FAIXJ, =FAiX**J,. =FA*(*J,A 2'?) I(*Je/\‘Y)/\*F‘: *F/\*]@/\}Z- sothat *(F/\ iXJe) =*(*F/\ *]@/\-Y)=iX*(*F/\ *]@) EiXi,j;.,**F E—iXi;;,_F and V.3(M)(X) EiXi;;,,F Ei;;.eF(X). Thus VFW) Ei;3,F E(ep/m)i(/F, by(7.5.21), so i'7"."a‘=-%i"{;i'r +pV(/V-(*dj)V. (7.5.22) Aswenoted before ifVisofconstant norm then itsacceleration is orthogonal toitself,la_i1d iF;F(V) EiViVF E0.Thus byequating tozero thecomponents ofV8” parallel andorthogonal toVweseethat 3is divergenceless ifandonly iftheparticle number isconserved, djE0 and EI-‘I--11 vvv iVF. (7.5.23)m Werecognise thisastheLorentz force lawequation forcharged world lines. 7.6TheReissner—Nordstr6m 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 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 gE——H(,(r)2dt®dt +H1(r)2dr®dr +r2d6®dt9 +r2sin26d<p®d<p. (7.6.1) The chart isspecified by{OE.6<77,0Ecp<27t,0<t<00}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,,@,,g EOand(8/8t) is orthogonal tothehypersurfaces with tEconstant. Aswepointed outin Chapter 6itisconvenient tochoose anorthonormal co-frame inwhich tocompute theconnection forms. Choosing thelocal co-frame: {e0EHodt, e1EH1dr,e2 Erd6l, e3Ersinﬁldrp} onecomputes thenon-vanishing connection forms HO’ 0atE—6o0E———-e 61 1 HOH1 __ “L25912 W21 rH3i 1 3(H13=""6031 =-7,"?1 cott9 , (923=-0932 =_‘T¢"- (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 ,R23 I "" i)eh3 r" H7 R(H5 1101_HHOH1_:L. .ci--:.-‘J. =-,3’; M .-.>;~ -;i=-"--- '=f::' . ;=_: .-,1__. ':=-_':'.;';- -1".="=-.-2!THE REISSNER—NORDSTROM SOLUTION 245 =m(_1_)',..31rH,H1 H9 602R.=-E0”rHiH0 R _ F612 12T rH1 H1 H’-R0} =—"%0—€03. C FHTHO Taking *1Eeom theEinstein forms arecalculated: G0 I—2R72,\e3 _ZRQ3/\€l "'ZR31/\e2 [2(1)' 1 1-_2 A +~-,erH1 H1 r3 r2H7 I ( H0 1 1 )()0; G1E22 - +--,e"*rHiH0 r2 r2H7 G2 =_2](r)e013 G3 I2J(r)e012 I --.-1-) (r) H H()H| FHIH(] rHl HI Thevaccuum equations G“E0arenow allsatisfied bywhere (H[,)' 1 Hf) 1 1’ l 1 l “)1/2 H1:F:1”71 I forsome constant it.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 AEf(r)dt, ensuring that §£KJF E0forFEdAand K,any Killing vector ofthe spherically symmetric static metric. The Maxwell 2-form is FEL(r)e‘ Ae“where L(r) Ef’/(H0H,). Integrating thedifferential equations d*FEOgives LrzEqforsome constant q.From (7.5.9) the Maxwell stress forms follow simply 2 q2 q2 qzA 33_ ___ 612 To:L613‘ TlI__:ei)23, I2=E4301 ,1;_-46 _ Zr‘ Zr 2!’ 2?” 246 GRAviTATIoN Thepresence ofthestress modifies theequations above to K2(l)’ 1+ 1]+q3_0 rHiH1 r2r2H§ 4r‘ H’ 1 1 2i<(2E°-—+——,)+"—-=0 I_ 41,4rHiH0 r2 VH1 I<](r)-i-2=0.4r4 These equations areallsatisfied by H1_ M qz 1/2 H0-‘1T1—(1+7+I—2) . (7.62)Kl’ The electromagnetic 2-form field isFE(q/r2)e1Ae°, sowemay interpret this solution asthegravitational field ofaspherically sym- metric static electrically charged source. Itisknown astheReissnerE Nordstrom solution. Intheabove solution wehave twoarbitrary constants itandq.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 (60=F616’,ek=F‘1dx" t<=1,2,3} with Fafunction ofthethree spatial coordinates. Show that the geodesic C:I —>M,r)——> (x°(r), x"(r)) isdetermined by 15°+2x°F")‘C(F) =0 12"+[(x°)2F3 +xix,-F7116,-F E2x"F‘1C(F) =0. For Ctimelike choose aproper-time parametrisation toreplace these with g(C,C)=—1 +ia,-F2 +2F-1(x1‘x,-8, -x*x»'6,-)F E0.THE REISSNER—-NORDSTROM SOLUTION 247 Ifnow <<1andF2E1—hwith h<<1then these approximate to Xi : Bycomparing with Newton’s law ofmotion foraslowly moving particle inaNewtonian gravitational potential (I),make theweak field identification <1)E—h/2. Exercise 7.5 Inthe above metric (7.6.1), setqE0and make the coordinate transformation HM2—R +E" 216R towrite itintheisotropic form 4R+,u)2--E 6 g (4R_M dt®t 4 +(1-%)(dR®dR +R2di9®d6 +R2616Ze6¢®6q>). Inaregion where ti<<4Rthisisofthetype considered inexercise 7.4, (change from standard R3polar toR3Cartesian coordinates.) Recall that forapoint source ofNewtonian gravity duetoamass M, thepotential (I)E—GM/r where GistheNewtonian gravitational coupling constant. Hence from hEGM/r identify theconstant inthe Schwarzschild solution; itE—2GM. Exercise 7.6 Inthe metric inexercise 7.4 above verify that for h<<1, G0E—2(8k8"h)e1 Ae2Ae3.Foranideal ﬂuid ofdensity pshow that r0Epe‘Ae2Ae3intheframe {Xa} inwhich itsvelocity VEX0. Hence usetheNewtonian Poisson equation Vztp E4rtGp torelate ourK totheNewtonian coupling Gby K__1_T1660' Exercise 7.7 Usetheresult ofexercise 7.1torewrite Einstein’s equations intheform PC~—geﬂt E87rG*'1rc. Intheabsence oftheelectromagnetic field (qE0)theReissner— Nordstrom metric reduces totheSchwarzschild metric. That is,wehave avacuum spacetime with metric 248 GRAVITATION 2M 2M ‘lgE (1 )dt®dt +(1—- dr®dr +r2(dt9®d6 +sin26dqa®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(8/8t) +q(8/8r) has norm squared (1—2M/r)'*q2 —(1—2M/r)p2 andistherefore timelike if (i(<1~2/IE.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 torE2M inthe (r,t)chart. Ontheother hand, ifone calculates thescalar curvature near rE2Mitappears well behaved, suggesting that theSchwarzchild coordinates may cover only part ofsome Lorentzian manifold. Ifwe introduce theEddington—Finkelstein coordinates (T,r’,6,<79)where TEt+r+2Mlog (r——2M) and r’Erthen itisstraightforward to compute dTinterms ofdtanddrandwrite theabove metric inthese coordinates as ZMg (1 )6T®6T +dT®dr’ +dr'®dT +r'2(dt9®d6r +sin26dqa®dq0). (7.6.4) The region ofspacetime covered byre(2M, O0)te(-00, O0)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’E0.Thus there exists ahorizon at r’E2M, 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’E0where thecurvature tensor becomes unbounded. Such events donotbelong toaLorentzian manifold and-K 46‘-'- " 5 Q2531’. -_,_gr..- 2.‘.-’-L-I._>".=. ' L .r.i- [ '‘.'_r,~;§.- -6 ..__u_ .11 1':l 1, i ..i' ﬁg T‘ >-.1,‘ -is-I.'- .iii. ii?-.-,;:_. .1-'.]I .- all :.=-,.. .waTHE 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 isrttimes 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 that theconnection 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 ofthe connection 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 ercoupled togravity according tothefield equations [14] in/2G“ E—r“[a/] +Act/4*e“ (7.7.1) 66-66/2 =2/1C1’3*1 (7.7.2) 250 GRAvITATioN dcrTa=8“A-57 (7.7.3) with r“E§c(i“da/A *da +dotAi“*da). (7.7.4) Thenon-vanishing realparameters Aandcarecoupling constants. (For /1E0thismodel isequivalent toatheory ofgravitation proposed by Brans andDicke REF[15].) The equation (7.7.3) involving thetorsion may besolved fortheconnection forms (6.6.8): fibdar iadoz(Uab :gab ‘i’ 8,, _ T Eb interms ofthetorsion-free connection forms Q,,,,. 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. inPare Mathematics volXXVII, p425 i “ Clifford Calculus onManifolds The first three chapters ofthis book 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 ofthe exterior 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 the terms 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 than exterior algebra. Just asone candevelop 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 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 asetof7/-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, andresults, of Chapter 2.Inparticular weshall juxtapose differential forms todenote their Clifford product. 8.1Covariant Differentiation ofClifford Products If(I)isanarbitrary inhomogeneous differential form andAanarbitrary 1-form onapseudo-Riemannian manifold Mthen (2.1.7) gives A(I>EAA(I)+i,;(I>. IfVisthe pseudo-Riemannian connection then VX(i,.;) E i6X,;(I> +iAVX(I>, since VXcommutes with contractions, and VXAEVXAsince Vismetric compatible. Hence anditfollows that VXisaderivation onClifford products. (This does notrequire zero torsion.) Adding andsubtracting equations (2.1.7) and (2.1.8) gives usrelations that permit AA(I)andi,.((I> tobeexpressed in terms ofClifford products: 1 A(I> +(I>’lA E2AA(I> (8.1.2) ACD —(I>"A E2i,.;(I>. (8.1.3) For{ea} alocal orthonormal co-frame wedenote e“Aebbyeab.Then (8.1.3) gives [ebci ea] :2(nac€b _nabec) where theleft-hand side isaClifford commutator and 77"”arethe orthonormal components ofthemetric. Soifweusetheconnection 1-forms tointroduce the2-form UX E¢l1wbc(X)ebC :iVXea /\ea wecanwrite (6.3.3) as VXe“ E[oX, ea]. (8.1.6) Ifweintroduce anorthonormal multibasis {e1} forPAM then, since anCovARIANT DIFFERENTIATIQN orcLiFFoRD PRODUCTS 253 exterior product ofmutually orthogonal 1-forms isthe same asa Clifford product _ VXEI =[UX, 31] - Ifweexpand anarbitrary differential form as(DE(D161 then IfSisanyinvertible element oftheClifford algebra andE“ESe“S '1 then itfollows from (8.1.6) that VXE“ E[EX, Ea] with EXE SoXS'1 +VXSS'1. Ifs6,1“ then {e“’Ese"s'1} isanother orthonor- malframe. If0}denotes theexpression in(8.1.5) computed with the connection forms inthisnew basis then 01,7EsoXs“ +VXss‘1. (8.1.9) Certainly thetwosides ofthisexpression canonly differ byanelement ofthecentre. Since oxisa2-form andse,1“ then soXs'1 isa2-form andweneed only check that VXss'1 isa2-form. Forse,1“ wecan write sExlxz ...x" where thexiare1-forms such that_(x’)2 Eil, then VXss'1 E(VXx‘x2 ...x"+x‘VXx2 ...x"+... +x1...x”‘1VXx")[(x”)“ ...(x2)_1(x1)‘1)] EVXx1(x1)(1 +x1[VXx2(x2)_1](x1)_1 +... +xl...x”“[VXx"(x”)"](x1...x"“)"1 . Since (x‘)2 isaconstant VXx’anticommutes with x‘and hence with (rd)-1-r‘/(r")‘*- sov.,x"<x*):1 =i<vXr<x*')"1 ~(x*)~1vXr) =VXx‘ A(x’)_1. Itfollows thatVXss_1 isa2-form. If{e1}isanorthonormal multibasis forFAM then differentiating (8.1.7) expresses thecurvature operator asR(X, Y)e" E[9tXY, e1]for gixir =VXUY "VYUX "lax» Uri “U[X,Y]- (8-1-10) Since thecurvature operator is@-linear then forany(I)ePAM R(X, Y)(I> E[9'tXy, (I>]. (8.1.11) Itcanbeverified that (lim, isunchanged ifoxi—-—> SoXS'1 +VXSS_1 forany invertible S.The forms gin are certainly related tothe curvature 2-forms Rab; We now establish the exact relationship. Differentiating (8.1.5) and using (8.1.7) gives VXoY E§X(wbC(Y))e”" +[oX, oy], andhence QRXY :-iiX(wb¢(Y)) _Y(wb6(X)) "w66(iXi Ylliebc +lax» Uri - Referring to(4.10.3) wecan simplify thefirst three terms: gin E §d6o,,,.(X, Y)e"” +[oX, cry]. Torecognise thelastterm wewillusethe 254 CLIFFORD CALCULUS 0NMANIFOLDS following useful relation: [e66, 666]:2,766,966 _2,766,966 +2,766,966 _2,76<.-e66_ (8002) Wecanusethisandtheantisymmetry oftheconnection forms, towrite lvx.Uri=i(w66(X)w"’6(Y) —w66(Y)w’t(X))@“‘ =%(w66/\w“C)(X.» Y)("‘- Sowehave gin E§R7,C(X, Y)e"" E—§iXiyR,,Ceb" . (8.1.13) This canberewritten, using the‘pairwise symmetric’ Bianchi identity for zero torsion (6.7.16), as 9?.Xy E§e“(X)e"(Y)R,,,, . (8.1.14) Exercise 8.1 Use(2.1.7) and(2.1.8) toshow that (forzero torsion): Rr,6b=P“ (61.15) 11,6“=at (61.16) R,,6b@ =at. (61.17) 8.2Theoperator ¢l Many equations inphysics canbeelegantly formulated interms ofthe exterior derivative dandtheco-derivative <5.InChapter 6weshowed how these operators could beexpressed interms ofthe pseudo- Riemannian connection. Wenow define anoperator 91onPAM by dEe“VXa . (8.2.1) Thus from (6.7.4) and(6.9.1) wehave (clEd—6 (8.2.2) with 6defined in(5.4.2). The operator dissometimes called theHodge de-Rham operator. Unlike dand6separately, (tiisnotahomogeneous operator ondifferential forms; whereas dincreases thedegree ofaform byone, 6decreases thedegree byone. The square ofgzlishomogeneous forsince dand6arenilpotent (112EA (8.2.3) where AistheLaplace—Beltrami operator of(5.4.5).'5,:__-,. 'HinTHE oPERATOR(zl 255 We can trivially rewrite the pair of Maxwell equations d*FE],dFE0as 61FEj (8.2.4) where jE—*_1J. Asanexample ofmanipulating Clifford expressions wenow re-express the Maxwell stress tensor interms ofClifford products andevaluate itsdivergence. The stress tensor isrelated tothe stress forms by‘JE*‘1r,, ®e“-**1r,,(X,,)e" (>9e“. For afour- dimensional Lorentzian spacetime **E—17, andthestress tensor com- ponents are9'0,Ei0*r,,.From (7.5.9) 26,=i,,FA*F-i,,*FAF. First weuse (8.1.2) toexchange theexterior products forClifford products: 46,=1,11*F+*Fi,F-i,*FF-Fi,,*F. Now weuse(8.1.3) 8r,E(e,,F —Fe,,)*F +*F(e,,F -—Fed) —(e,,*F —*Fe,,)F -F(e,,*F —*Fe,,). Finally weuse(2.1.19) towrite theHodge dual interms ofthevolume 4-form 2: ___1r,-§Fe,,Fz . Wehave used F5E—Fsince Fisa2-form and2(1)E(D42. Once again weuse(8.1.3) toobtain thestress tensor components 9],,E§(Fe,,Fe), +e7,Fe,,F) . (8.2.5) When covariantly differentiating thestress tensor thederivatives ofthe co-frames intheabove components will cancel thederivatives ofthe tensor basis, hence (V-3),, E§(VXcFe,,Fe‘ +Fe,,VX(Fe‘ +e‘VX,Fe,,F +e"Fe,,VXcF) . Wewant tousetheMaxwell equations (8.2.4) tosimplify this, butthe terms VX,Fande“donotalloccur intheright order towrite them as The above expression iscertainly a0-form, sobyapplying the homogeneous projector (cf(2.1.12)) 9’0wedonothing. Under this projector, factors intheClifford product can becyclically permuted (2.1.17). (We cannot, ofcourse, then remove theprojector.) Sowe have (V-815),, E%9’0(¢iFe,,F +VX,Fe"’Fe,,) . Since F5’E—-F, then VX,Fe‘ E—(¢iF)“-*7. Wecaninsert thisintheabove andthen use9’0(I) E.9’0<I)9‘ toobtain (V-97),, E3’0((zlFe,,F). Wecannow 256 CLIFFORD CALCULUS ONMANIFOLDS usetheMaxwell equations (8.2.4): 7-7"=§n(ne.)e =56(6)using. (2.1.18). Since jisa1-form and Fisa2-form then F]E]AF—i,-Fand sofinally ’ V-F=-i,-F. (62.6) (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 (I)onapseudo-Riemannian manifold: 66>=66>-iA(I). (8.3.1) The term involving Adescribes theelectromagnetic coupling tothe Maxwell field FEdA. Hewas apparently motivated todevelop a ‘calculus ofinfinitesimals’ inwhich relations oftheform dx“Adx” E0 and dx”Vdx”+dx"’Vdx“E2gt"’ could co-exist on apseudo- Riemannian manifold. Kahler recovered Dirac’s solution describing the wave mechanics ofarelativistic electron ofmass 7ainahydrogen atom when heanalysed (8.3.1) inﬂatMinkowski spacetime. Itwas adesire tofind afirst-order equation, such that thecompo- nents satisfied thesecond-order Klein—Gordon equation, that motivated Dirac toformulate hiscelebrated equation in1928 [19]. Because of (8.2.3), and since theLaplace—Beltrami operator ishomogeneous, the p-form components 9’,,((I)) ofanarbitrary solution to(8.3.1), in theabsence ofanelectromagnetic field, satisfy Asr,() =62s>,() . (3.32) However, anarbitrary complex differential form onspacetime has sixteen complex components; whereas aspinor ofthecomplexifiedTHE KAHLER EQUATION 257 Clifford algebra has four complex components. Thus anarbitrary solution to(8.3.1) 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 {xa} inwhich e“Edx“, aE0,1,2,3consti- tute anorthonormal basis. Wecanconstruct asetofglobally defined primitive idempotents {P,} outofthisparallel co-frame. The resulting idempotents will also beparallel, VXP,E0VXGPTM. Thus if <72,E(I)P,- then go,isinaminimal leftideal. If(I)satisfies (8.3.1) then multiplying (8.3.1) ontheright byP,gives dip,Eago,—iAqa,- iE1,2,3,4 (8.3.2) 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 oftheKahler 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 field could 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 ofthisprinciple itisdifficult toreconcile our current understanding ofthequantum mechanics ofelectrons with the 258 CLIFFORD cALcULUs ONMANIFOLDS four copies ofthePauli—SchrOdinger equation obtained from (8.3.1). In anarbitrary curved spacetime (gravitational field) theKahler equation willnotdecouple into four minimal leftideas (there willnotbeglobally defined parallel primitives). Although theexperimental significance of thisisfarfrom clear thefact that thedegeneracy oftheMinkowski space system can bebroken would seem tolead tointerpretational problems forthequantum theory. Exercise 8.2 Define intheusual Minkowski spacetime polar chart (t,r,6,cp)the local 1-forms 55."=r"‘¢1(r"YZ”(6. <7)))=/<YZ”(9. <10)+H11’? k=0,1,2... —kEmE.k interms ofstandard spherical harmonics satisfying ;zl2(r"Yj,") EO.Verify that 1—k=LT)" andthatforanyinhomogeneous differential form Rindependent ofdt: . 1—k§Zi(RSZ‘) =(§+ilR +R'7§"—7-— d7‘)SZ‘ . Verify that asolution ofKahler’s equation with aCoulomb 1-form potential AE(e/r)dt inthisspacetime may bewritten k W=222Rim(r.6.<v)Tt(t)€Ei itmE—k where Rim,E{ff,(r) +g‘,§(r)dr}S§" and TE(t) Eexp(iw€t)(1 +iedt) and foreach e,kthe0-forms fand gsatisfy theordinary differential equations: r+(‘j"r ‘,2g+(w—6>6=0 s’+-(L:—iQe+§,,if—(w+t))f=0- Exercise 8.3 The 1-form harmonics SZ’may also beused toanalyse Maxwell’s equations (AFE0.First observe that the1-forms aimEZ§(Ar)S Eobey gii2a'E —/lza and the 2-forms 76'f,,,,EZf,()tr)drSZ’ obey ¢l276E —h2f3, where Z‘iklabel theindependent Bessel solutions oftheequationF I THE KAHLER EQUATION 259 Writing FEEdt+Bwith EEicoE and EEicoB write theharmonic component Maxwell equations asthecomplex pair: 615=-1666 66=-16115 and seek solutions oftheform EEp,,(r)S§,” forsome 0-forms pk. Hence construct themultipole expansions: E’=29’1(Aim1Ti6)@Xi)(iw‘() B’=$1415’e.k,m B”=2i9’z(Bi6/3i6)@XI)(iw‘I) E”=$68”€,k,m where Him EZf,(6or)81drS’,§’ toE0 ﬁjm,EZ§,(tor)drS’,f toE0 81EemandAim, Bi”,areanycomplex constants. Exercise 8.4 _ Thestress tensor fortheEinstein—Kahler coupled system (with AE0) is T=%3)0((I)5"6aVX€€‘6b "1'<D5’l€bVXc(I>€c€,,)€“ ® eb - Verify that V-TE0. Hint. Since theco-frames with contracted indices willnotcontribute to thedivergence concentrate ontheterms 4(V~T),, ESf0(VXa(I)5”e"VXC(I)e"e,, +(I)‘-5’le“VX,VX,(I)e‘e0 +(I)“5"e“VXC(I)VXﬁe‘e0 +(P‘”(?6@6Vx,VX,‘PeCea +(I)‘5~""e),VX,(I)VXae‘e“). Note6r.(vX.%.vi..<1>e¢e“) =0since@661“-=')=ivforanyfv.Using(8.3.1) anditsiterate, Ad)E7.t2(I), theabove terms cancel with theaid oftherelations 6@=-(6)%7 6~‘r-f=(6)~f='i 61=—(d(I))” 66>":—(6(I))" 2 0,6__k) v,,6@= —(d(I)+66>)". i()”(") +;P’(”) +(12T""*rT")P(F) =9 1E9-‘Ariix‘-1': _-,._,,;?=;;f-'T6==.“-1j_:_'-<. l =et.'ii1.%f=-Thelastrelation follows from (8.1.3). 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.,-a<1>_=no (8./4.1) where (DiE§(li17)(I>. This corresponds totheDuffin—-Kemmer—Petiau equation. Writing thisinterms ofClifford products, e"’VXa +VXﬂe“ =2,u weseethat thesecond term prevents thedecoupling oftheequation into minimal leftideals inMinkowski space. Since dand 5map even (odd) forms toodd(even) ones (8.4.1) isequivalent to d+ =;,t_ 5_ =,u+. Asaconsequence 6(1), =0andd_ =0soanysolution to(8.4.1) will alsosatisfy theKahler equation for'11). Inthemassless case (8.4.l) exhibits thegeneralised gauge symmetry +rZ>+ +dpgs _|—--><D_ +(§;(+ anddescribes what inthephysics literature areoften called antisym- metric tensor gauge fields. Bibliography Chisholm JSRandCommon AK(ed) 1986 NATO ASISeries 183'5 ...<;.. l_.‘.i';:'. .»-¢..-:.- ll '-la’;-2%‘..-.r- 5-i7l§:' l, if F if .;| . -,1.--'-.-.;-=_;_.~iif -t-.:- ZSpinor Fields In§2.5 spinors (orsemi-spinors) were defined ascarrying irreducible representations oftheClifford algebra. Any such 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 Mis6*/611 dimensional sothat thereal Clifford algebra iscentral simple. ThuS 262 SPINQR FIELDS C(T”§,M,g) =Jl/t,(lB) ®D(]B), where A/t,(lB) 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 that Clifford multiplication is D-linear. Let.S5(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 §(M) and5°’(M) areequivalent ifandonly ifthere isa diffeomorphism relating them such that fibres of.9>(M) above pare mapped intofibres of9’(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 .§l(M) aspinor bundle. Fibres oftheClifford bundle areisomorphic tothealgebra ofD-valued matrices. If{e"(“)} isalocal orthonormal co-frame defined ontheopen neighbourhood U0,ofMthen anisomorphism between C(T”j,M, g)and D-valued matrices may begiven ateach peU,,.interms ofthe generators {e"("‘)| p}andtheconstant matrices {ya} satisfying ya‘;/b +yby“ =2g“”1. (9.1.1) For agiven choice ofD-valued y-matrices wemay correlate alocal orthonormal co-frame with alocal basis ofsections of5P(M). OnU,,. there isalocal basis forspinor fields {b,-ml} such that e“(“)b§“) =bialyjj-. (9.1.2) (Note that wejuxtapose symbols todenote theClifford action of sections ofC(M) onsections of9(M).) (Thus thebasis {b§"’)} 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§“)’} also satisfies (9.1.2) then f(“’)isanon-zero function onU,,.such that.3;-‘-'3'--is'1|I_'.I'3-SPINOR BUNDLES 263 bf-“"’=f<e>b$e>. (9.13) OnUaﬁEU,,U U5there must besome local section ofC(M), sllial, such that big‘=s(5“'lb(“) But eafﬁlbim =bill”)/f =s‘l’“lb(“ly‘-’- = 1 I' 1 t ,1 ,1: S(lB¢Yiea(a’)bEal_ SO ea(B)s(B5Y)b$a) :S(l3'51’)ea(a’)bg‘7) and earn=5(l3¢Y)ee(aJS(ﬁa)"_ (9_1_4) Thus certainly sw“) isintheClifford group F.Ifthespinor bases are changed asin(9.l.3) then swal’ =f(5)s(5“lfl“)I1. Itturns outthatwecan, infact, always choose thelocal bases in(9.1.2) such that theClifford elements sill“) relating them onoverlaps areinii“.(Itisastandard result that anyFbundle isreducible toaifbundle since I“/il" =lB“‘, seeforexample Kobayashi and Nomizu [23].) On triple overlaps U0,UU5UU,EU,5,theClifford elements relating spinor bases satis- fythecoherence condition S(al3lS(i3l/) Z S(a’l/)_ 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.) IfMisanypseudo-Riemannian manifold then wecanchoose local orthonormal frames, related onoverlaps byanorthogonal transform- ation, A(5“l say.Wecanchoose answ“) eifsuch that;((s(l’°‘)) =AW). Ontriple overlaps wemust have slaﬁlswll =ism). Ingeneral, we cannot choose the{s(“5)} 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. This condition 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 _%(1i2),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 aSpinc 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 =D.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)J" ._.,,,-, .=‘>“: ._,.i=- Fin25% |....:_,_,6:, ,;;,.>' .._ Q. .r-..__.,,-,,1‘ ‘..g-;.., _. r’-at - Le-,. R11; ‘-3;-’*{- "'3*=;;‘,..|'u- ,-r:|".1’ <--.--.,.-.-5+-1-1-.-"II '.-'»._;_-;--.;.,,. \ ‘5-t€:=. .>,‘;,;.-_..,._u.. _,,5".“.32.- xiii: ,.-.;>-.' 5%?"- .‘3.§-<' I -I=3'_1..:"-'='-1.)-... .?~€"f- fr ;1;.-; r'_; *.'-4 .1A‘‘,-I: '.'.*. '.5- ...\' <;;5 :";_ .-t .11‘--'\.=€*4:,ji§:=‘---=71»<-; .ﬁg.:?..?_ .-at"“- _'I....=,_ _2. ,1;=-1-1.;== ;.-1 V. E. .3 |I.PRODUCTS ONSPINOR FIELDS 265 Clifford algebra isisomorphic tothealgebra ofall(real orcomplex) matrices, with theinvolution $17similar totransposition. Inthiscase for matrices asin(9.l.1) there isamatrix C,symmetric orskew, such that Cy“TC‘1 =—y“'. (9.2.1) If{b§“’l} isastandard spinor frame, satisfying (9.l.2), then abilinear product onlocal spinor fields isspecified bydefining (bin/‘ii, bjia/))(a,) = The product hasbeen labelled with thesubscript (a/)since inprinciple wehave adifferent product foreach U0..Wewant toshow thatonU,5 theproducts (,)(,,)and(,)(,3)coincide, forthen wehave awell defined product onspinor fields. First weshow thatthese local products arespin invariant. For any such local product then (suppressing the (at)- labelling) (bi: 9%)) =(bi: b/<l’i;") =CQIYZ; Z(C_1l’a)i;' I_(l’aTC_l)r; by(9.2.1), so (bi: eabj) I_Yaii<(/‘Q1 =_l/iijctfji :_'l/Zi(bk1 :—(eabir Thus foranyspinor fields and meFC(M) (go,mt/2)(,,) =(m5"<p, 1,p)(a,) andhence these local products arespin invariant, having Enasadjoint involution. On Uaﬁthe standard spinor frames are related by by”=s(5“lb§‘“’) forsw“) 6+1“. SoonUM, (ban, b§ﬁ>)(a) =(5(ﬂa)bta), S(ﬁw)b§e))(a) =(S(Ba/)i”S(Ba)b§e)’ b§,<r))(a) =(bi-0‘), bi-a))(a) I(big), b,iﬁ))(5)- Thus foranylocal spinor fields (cp,1/1)(,,.) =(go,1/1)(5). 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 the standard local spinor frames arenot unique. Ifthelocal orthonormal co-frames arerelated byAW) then the s(""B) e+1“ such that X(s(“5l) =Alaﬁl isdetermined uptoasign. Soif {b§“l’} isalso astandard spinor frame with b§“)’ =f(“’lb§°’) foralocal function f(“)then onoverlaps wemust have fits)=if“). Soanon-zero function fisdefined onMbyfp=(sgnf(“))f(“), with b§“’l’ =iﬂ)§“l. Soif(b§“’)’, b§“")' =(b§“l, bi”), then foranyspinor fields f2((p, 1/1)’= (Q0,tp).Itiseasily seen that thechoices oforthonormal co-frames, 1/-matrices andmatrix Ccannot affect thespinor product bymore than aconformal scaling. Thus thisprescription determines aclass ofconfor- mally related spin-invariant products. 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, 5*or§n*. If,forMeven dimensional, 5°(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"‘*=l'l1_1']/“I'll, with m*=imhl. (9.2.3) OnUQ,theoperator c(a/) isdefined by lllcw) :(bia)1lli)c(a) =bia)mji1l)j*- (9-2-4) If#(a) isthelocal operation onspinor fields that complex conjugates thecomponents inthebf“)basis then weusethesame symbol todenote theautomorphism ofthecomplexified Clifford algebra defined by (,,.,j,)#<a> =a#<a>,j,#<a> _ (9_g_5) Thus ifab§"l =bj“’)a,-,- then a"*“*')b§"‘) =bj"'la,-,~*. (Care isneeded with the notation. Bya,-,-*wemean thecomplex conjugate ofthecomponents of a,whereas a*1,,arethecomponents oftheClifford element a*.The difference between these isthedifference between *and#(a).) Ifm(°‘) isthelocal Clifford form such that m(“’)b§“’) =bj“’)m,,- then itfollows from (9.2.3) that ' a#("‘l =m(“’)“]a*m(“’l (9.2.6) so (aw)c(a) 2' Z Z :n'j(a)a#(a’)bEa’)'q)i* :rn(5Y)a#(a’)n»l(a')_1r¢C(a’) :a*'lpC(a’)_ Ifweexpand 1/1as1/1=bimt/1*“ then 1116(5) =bjmm,-,1/;i*, but tp= s"3“)bj“)1/1" so we=,<e>*br>m,.~.r =br>m,,,**since s(B“)* =s95“) fors93“) e+1”. Hence theoperations c(a/) and c(j6’) agree onU,5andwehave awell defined operation ofcharge conjuga- tion, denoted c.IfMisodd dimensional thecomplexified Clifford algebra issemi-simple. Inthiscase either *or17*isaconjugate-linear involuntary automorphism ofthesimple component algebras. Inthe-,_.-,..:..;.-_._,,,-. -_ "-".j"_i%¥j1 ' 31;§.;"..I ‘ -i‘*E"' ‘If‘;' -;';s3§=--;.-" "£1!-i"Pnooucrs onSPINOR FIELDS 267 latter case wecandefine charge conjugation using 17*instead of*. Ineven dimensions wehave spin-invariant products ontherealspinor bundle with adjoint involutions 5andE17.The automorphism 17isinner with a"=zazil forzthevolume form. Using asubscript tolabel the product byitsadjoint involution wehave (111,<r);=,,=(w,Z€0)»;- (9-27) Inthecomplexified case wehave similarly (Ill,(ﬁle=(W,Q9): (9-2-3) and (1/1,<P)r,,~==(W,Z(P)r~ (9-2-9) For asemi-simple real Clifford algebra there isaproduct onthe semi-spinors associated with either 5orE17.When thereal Clifford algebra isisomorphic tocomplex matrices then either EorEnis associated with acomplex bilinear product, theother being associated with aconjugate—linear product; theproducts being related by‘charge conjugation’ defined using T].Forthebundle ofcomplex semi-spinors in odddimensions then either Eor$17isassociated with acomplex bilinear product; either §*or§n*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. Formost 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.l.5) and (8.l.6), wehave VXe“(“’) =[o_§,5“l,e“(“'l] where o§§‘)=,1;co§,‘§)(X)e"‘(“’). We 268 SPINOR FIELDS canusethis local orthonormal co-frame todefine astandard spinor frame satisfying (8.l.2). Wecanintroduce acovariant derivative S$3’)of local spinor fields bydefining Sf@')b§“) =of,§"lb§"’). (9.3.1) IfUzi“)isanarbitrary local spinor field then SE19’)isdefined by ssi>te>=sse<1>r~>,r> =55?"bi“’1/1‘ +br>X<,r>. <9-3.2)The components 1/1‘areD-valued functions andtheabove requires that weknow how todifferentiate these. Quaternionic orcomplex-valued functions aredifferentiated asordered quadruples orpairs ofreal functions; that is,thealgebra Dhasaparallel basis. Ofcourse, wewill want toshow that iftpisalocal spinor field defined onUaj, then S§§”lIp=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 1/19*’)alocal spinor field then $€r’(Aw‘“>) =S£r’(A,e“bE“>w‘) =5E,é”(A,b}“’r;’,1/1‘) =X(A,)b§“’n-w‘ +A,vS?‘)b§“’r,f%1/1‘ +Aabrwz-X(w") =X(A,)e“w<“> +o£?’Aw<“> +AbE“>X(w”) =vxxwe+Ao§,‘3‘l1p("‘) +Ab§°‘>X(1/1‘) by(8.1.8), so Sgg/)(Aw(a/)) =VXA1j)(a/) +ASgg)1jj(a»)_ Since thisistrue foralllocal 11%“)andthel-forms generate theClifford algebra wehave S§§‘)(a1p(“‘l) =Vxatp-(‘Yl +aS§§")t/fa’) (9.3.3) foranyClifford form a.Ifnow 1/Jisanyspinor field then onUM,we have 59¢ =5,"Z»”(bi‘”1l1‘) =03% +bi’3‘X(1l1‘)- ButonUaﬁwehave b§’6l=s(5“)b§“) fors(5"’) e+1"*,so 5529111=53?)(S“"‘)bi“W1‘i) IVXs(,8a*)b(a’)wi +S(Bcr)O-(€)b(a)wi +S(ﬂa)b(a)X(wi) =vXS<ra>S(ﬁa)*,j, +_g(t3a/)Ugg)S(t5a)"w +b§rr>X(,j,,)_ Hence from (8.l.9) weseethat Sf{-‘)1/1=SE91/1 andwehave acovariantCOVARIANT DIFFERENTIATION oFSPINOR FIELDS 269 derivative SXdefined onarbitrary spinor fields such that SXtpial = S(e)w(a')_ X/We 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§“’l, bj"‘)) =C,]‘forsome constant matrix C.Then if1/Jand cparearbitrary spinor fields with w=bﬁmw’onU, (Silt/, fr)+(W,5X<P) _ _ =(vS?"1/1 +b§""X(1/1’), <0)+(111,vS?"(/1+b5"’X(<P‘))- Since of?)isareal 2-form 035.“ =-05,?) forany9that istheadjoint involution ofaspin-invariant product. Hence (Sir/1, av)+(1/1,Sw)=(bE“’X(1/1‘), fr)+(1/1,bE“’X(<t>’)) =(X(w‘))*C;.‘<p* +(1//)’C.7.1X(<t>") where the product isDl-linear inthe first variable. Since (X(t/)"))1' =X((1/1")l) andthematrix C”isconstant (SXI/1, (P)+(1/1,5X90)=X(1/1,tr)- (9-3-4) 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, SX'Lf)C z 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 @-linearity inX Sfx=fSX (9.3.6) the‘Leibnitz’ rule SX(at/1) =VX4111; +aSX1p VaeFC(M), Vt/1el"55(M) (9.3.7) andcompatibility with some spin-invariant product (SXw= Q0)+(111,5x09)=X011,<P)- (9-3-8) 270 SPINOR FIELDs Given anSthat satisfies these axioms, isitunique? Suppose that S}, alsosatisfied theaxioms above. Then ifLXES§(—SXwehave L,IF9(M) —>F9(M) (93.9) La=fLX (93.10) LX(9w) =aL,»w (9.311) (99,LX111)+(LX919, 1/1)=0, (93.12) Equation (9.3.ll) says that LXcommutes with Clifford multiplication, soLX1/1=tppx forsome D-valued function pX.Putting thisin(9.3.12) g1ves (99,11119,»)+(9919,. 9»)=0, Iftheproduct isD1‘-linear inthefirstvariable then, since pXeD. (99,1/»)19X+9i',»(<9,1/»)=0- (9.313) The D-linearity in1/1ensures that rp,1/J-> (rp,1/1)maps l".S°(M) ><l"9(M) onto 1),sowecanchoose (,0and 1})such that (cp,1/1)=l.This shows that p’X=~pX, andifthisissubstituted into (9.313) then weseethat pXmust beinthecentre ofD.IfDisoneofthecentral algebras Ror ‘I-Ithen wemust have pX=0.Similarly ifD=Cwith jtheidentity Involution. However, fortheremaining case ofD=Candjcomplex conjugation then pXcanbeanyimaginary function. Since themapping X——>pXisrequired tobe@-linear (by(9.3.6)) then ifSXsatisfies (9.3.6)—-(93.8) then sodoes SX,with S’X1/1 =SXt/1 +iA(X)1/1 (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 thatSXb§“') =o"f{,")b§‘”l. .In§2.6 weused aD-valued spin-invariant product tomap aspinor 1ntotheD-linear dual space. Wewillusethedefinition andnotation ofCOVARIANT DIFFERENTIATION OFSPINOR FIELDs 271 (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 cpanddual spinor 1])wehave (pi/V1intheClifford algebra. Although thenotation ofsimply juxtaposing thespinors isaslight liberty when thespinor fields arenot intheClifford algebra westillhave amapping taking aspinor and a dual spinor totheClifford algebra; tp,ij]I——->(pi/H)where (q9il3)9 =99099) E<9(1/9,19) V1991"~5((M) -(9-3,15) Iftheadjoint spinor 17;isdefined with respect toaproduct with which SXiscompatible then wehave theuseful relation VX(f9"?t3) =5x991? +q9§}7/»- (9-3-19) This follows bydifferentiating (9.3.l5); using theLeibnitz property on theleft-hand sideandthemetric compatibility ontheright-hand side. Thecurvature operator ofSisdefined intheobvious way, There isalways alocal basis inwhich SXb,-=oXb,-, and hence S(X,Y)b,- =97?.Xyb,- where gtxy isdefined in(8.l.10). Since thecurva- ture operator is9-linear then foranyspinor field S(X,Y)1p =9RX,,ip. (9.318) Using (8.l.13) and(forzero torsion) (8.1.l4) wecanwrite thisinterms ofthecurvature 2-forms giving Z —§iXlyRa),€“b1}) OT S(X,Y)1/1 =§_,@9(x)@9(Y)R,,q,. (93.20) 9.4LieDerivatives ofSpinor Fields Because theClifford product involves themetric then unless thevector field VisKilling theLiederivative S8,,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 onspinor fields from that ontensor fields byusing thesymbol .%€X. 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 that commutator ensures the‘Leibnitz’ property. In(6.l3.l) weintroduced theOperator AvEéffv -—Vv, satisfying Av(frp) =fAvrp foranyfunction fanddifferential form Q9.Since Avisaderivation on theexterior algebra wehave Avq0=Ave”AiX“cp Vrpel“/\M. We can use (8.l.2) and (8.1.3) towrite the interior and exterior products interms ofClifford products, producing At/(P :iii/ivea /\ea:‘Pl+iiX,,AI/(3699 “i(Av@a‘Pnea +@a~(P”Av@a)- The Clifford commutator isaClifford derivation. The2-form Ave“ Aea can bewritten interms oftheexterior derivative ofI7.Since Av commutes with contractions andAvf =0forfe@(M), if{e“} and{Xa} are dual bases and AvX,, =m,,"X,, for some matrix ma” then Aveb =—m,”e“. Then Ave“ Aea=—mb“e” Aea=mabeb Ae“, using theantisymmetry oftheexterior product, soAve“ Ae, =ma Ae”. Now AvXa E[V,X,,] —VvXa, soifVistorsion free AvX, =—VXuV, thus Ave“ Aea=e“AVXHV =e“AVXQV=dV (by(4.7.4)). Theremaining terms intheexpression forAvingeneral prevent itfrom being, aClifford derivation. Ifwritten interms ofthematrix mab then only thesymmetric part enters: iiX,,Av@a(t9 —;i(Av5’a99“@a +¢’a(Pi‘Av@a) =-%m,‘";9 +%(m,,+m,,)(9’<9”9“ +9”99”9") using theusual index-lowering convention. Since g(AvXa, X,,)=mm, andAvcommutes with contractions mab +mba :—AVg(Xar Themetric compatibility ofVenables ustowrite Avg =§£vg and A1/99=[idl7,99] +i§Bve(X,, X“)99 —§.§£vg(X,,, X,,)(e"’cp’le" +e“rp”e"). (9.4.1) Thus, asexpected, Av, and hence Sfv, isaClifford derivation ifand only ifVisaKilling vector._51:' -'5, ._i_.:_- if - ..I|'".-‘it: ,~=.r.. iw ‘i __,>.1», .... . ‘,1 . _.-_._ ;.__. _,.,.-,-ii-ﬁgLIEDERIVATIVES OFSPINOR FIELDs 273 IfKisaKilling vector then theabove simplifies to Sfvqo =Vvcp +[-,‘,dK, Q9]. (9.4.2) Soif{ea} isanorthonormal co-frame wehave, from (8.1.6) §£Ke" =[UK+§dK, e“] (9.4.3) where UK=iivwpqepq. 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 .§EKb,~ =(UK +§dK)b,-. If1/1=b,1/1" andgxl/) =b,K(1p‘) +éEKb,-1})‘ then, recalling thedefini- tionofthecovariant derivative, wehave s9,,,v=s,<v+gain. (94.4) Such adefinition can (and will) betakenfor theLiederivative on spinors with respect toanarbitrary vector, butonly inthecase of Killing vectors isthere aclear geometrical interpretation with .52having useful properties. When KisaKilling vector then, like SK,SQKsatisfies a‘Leibnitz’ prOpe[IyI <%K(a'|1ll) : §£K6Z1/J + a,%K'l.p. F'-.4 This follows from (9.4.2) and (9.3.7). If1})isthespinor adjoint toip, with respect toanyspin-invariant product, then forKKilling §£1<(§01ﬂll) I§£K(P1ll +(P§£I<1ll- (9-4-6) IftheLiederivative iswritten using (9.4.2) then thisfollows from the analogous property ofSX,(9.3.16). Equations (6.l3.l3) and (6.13.l4) give thecommutator ofaLie derivative with acovariant derivative. We now obtain theanalogous expression forthespinor operators. This willbeuseful forexamining thecovariances ofspinor equations inthenext chapter. Straight from thedefinition wehave [C-£gK,Sj/J ""S[K,j/] : _ The curvature ofSisgiven in(9.3.20), andVvdK canbeexpressed as in(6.l3.9) togive [$1081/]— S[K‘V] Z —§VXb§EKg(V,Xa)€b“. 274 SPINOR FIELDS Forthespecial case ofKaconformal Killing vector with ££Kg =2kg ite@(M) (9.-4.8) theabove simplifies to i=%)K,Svl _S[1<,v] =—%dA/\ (9.4.9) Wecanusethecommutator oftheLiederivative with acovariant derivative toevaluate thecommutator oftwoLiederivatives, [5-PX, 551/ill! -§£[x,v]1l/ 2i=%)x,5Yl1ll _S[X,I/]‘l/ +i-1q'3X(d Hill’) _idYgxill _ From (9.4.1) 99,(9'Y'v») -<1179,9=99,9‘Y9-i§£Xe(X.,. x~)t1"rt, +§§£Xg(X,,, Xb)(e"d Ye“ +e"dKe")1/1 andsince ebdl/He“ +@9t1I769=2g99t1"Y -2(e"A1,,,,t1i? +t-9A1X..t1I7) then gseXg(X,, x,,)(t,9<1 17,,“+e“d'Y‘e9) =‘i‘S£Xg(Xa’ Xald? _%‘E£Xg(Xar Xb)ea /\iX"di7- Itfollows fromthedefinition of"Ythat :9,Y’=$7,}?+seXg(Y, x,)@9. Since theLieandexterior derivatives ondifferential forms commute .§£XdY d[X,Y]+d(§£Xg(Y, X,,)e“) h =dIXT'Y1+v,,a,g<r.x,)@e +99,g(v,,Y.x,)e9IIUS 99,(dY9)—digiv —d[9'?.""Y]v :Vx,-C€X8(Y>Xa)@balP +§BXg(VXbY>Xa)ebaw —2§£Xg(XasXb)ea /\ Returning now tothecommutator oftheLiederivatives weuse(9.4.7) toobtain ~ igx, gr] "§g[X, Y]=i§£X8(VX,,Y, X,,)€"“ —§§£Xg(X,,, Xb)e" AiX+,d Y. The right-hand side may besimplified soastoexhibit explicitly the antisymmetry inXandY: e“AiX,dF =e“AVX1,FY' —ix,VX6173“ .:_.l l-:-QLIEDERIVATIVES OFsP1NOR FIELDs 275 so 2§£’Xg(VXhY, X,,)e"" —.§EXg(X,,, X,,)e“ AiX1,dﬂl7 : —,$Xg(Xa, Xb)iXb VA/6 T/H866 _§£Xg(Xa, 176'“. UseofKilling’s equation, (6.l3.3), produces thefinal result [gXa -351/l -=3-Q[X,Y] =—ri§£x8(Xa, Xi-)§£Y8(Xb, Xt)eaC- (9-4-10) Ifeither XorYisconformal Killing then theright-hand sidevanishes. Exercise 9.2 Show thatif{K,-}isanalgebra ofKilling vectors inﬂatspace then ...,I,..._1..____.. IidK,,,d1<.-1-,dlK,-,1<.1_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 IR”with aflat pseudo-Riemannian metric then wehave aspinor sub-bundle ofthe Clifford bundle. Let {.99} beaglobal parallel orthonormal co-frame. Then forsome choice ofconstant y-matrices there isaglobal matrix basis {ev} forClifford forms such that é“=vf,-e,,~. Elements ofthis matrix basis can bewritten asClifford polynomials oftheparallel co-frames with constant coefficients, andsoareparallel. Then 5i(M) isa spinor sub-bundle ofC(M) ifthefibres of.S9(M) aretheminimal left ideals spanned by{e,1}. Sections of.5l9(M) (spinor fields) are in- homogeneous differential forms. The pseudo-Riemannian connection V induces aconnection on.99(M). Infact this iseasily seen tobethe spinor covariant derivative, generally denoted S,forthis particular spinor bundle. Wecanofcourse always choose non-parallel co-frames, saye“=$.99,-1 forsE+1”, with VXe“ =[oX,e“] forOX=VXss“1. The corresponding standard spinor basis is{b,-=se,-1} satisfying VXb,- =o"Xb,-. IfTdenotes theinvolution oftransposition inthematrix basis {e,-,-} and CistheClifford element such that ail?=CaTC'1 then aspin- invariant product onsections of9(M) isgiven by (<9,111)=5”t,(C77'<P*i”1/1)- (9-5-1) Notice that the0-form projector 500gives aproduct with values inthe 276 SPINOR FIELDs real numbers rather than theisomorphic algebra with e11asidentity, Forthespecial spinor bundle here thisproduct accords with thegeneral prescription of§9.2. Although forthisparticular spinor bundle theconnections Sand V coincide there isstillaneed todistinguish 5-£14from iv. ForKaKilling vector these areseen, using (9.4.2), toberelated by 9510/1=iﬁxv+iwdli (95.2) The Liederivative SEXdoes notinduce anoperator onthesub-bundle .¢(M): itdoes notpreserve theminimal leftideals. The addition ofthe second term ensures thativy; EI‘.$(M) forall1/1el“£P(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_0(lPt). Show that there isa unique vector asuch that ipa=1/1,V1/1eI.Hint: Take anorthonormal frame {e1,e2} andconstruct amatrix basis using P,=%(1iel).Then If10.= C2,0(lFi)P+ then I=[OSforsome invertible S.Expand Sinthe prev1ously constructed matrix basis and explicitly construct theasuch thatP,Sa =P,S. (ii)Argue that thereal Clifford bundle ofatwo-dimensional sphere does notcontain asp1nor sub-bundle ofminimal leftideals (since there ISnonon-vanishing vector field onasphere). The sphere does, however, admlt aspinor structure. Wehave emphasised that wecannot ingeneral find aspinor sub- pupddle oftheClifford bundle, andthus cannot ingeneral identify spinor d1es.w1th certain differential forms. However, wecan1fwew1sh always othrslocally. Foreach open neighbourhood U,ofMwecanchoose a local basis forthe‘Clifford algebra {e§f‘lQ),“l}. The local matrix frame {e,-,9}.commutes with thebasis {Qifl} forthed1v1s1on algebra. OnUm,» there 1salocalCl1fford form S(“(3)such thateifl=S(("")e§,")(S i/5“))'1 and Q?)=S((3")Q§<"”(S(/i‘*’))”1. IfF“)istheminimal leftideal spanned bythe first column ofe§-if")and Disthe‘standard’ division algebra with basis {av} then Ii“)isaright D-module with theruleejflqk Ee§j")Q§,.“'l. Ifwe canchoose theSW) coherently, that isS(‘*’l5)S("31’) =SW’ onU,,.,,,,, then wecandefine anequivalence relation between Ii“)andIi/5‘onUaj, to form paspinor bundle. Thus theS(“(7canbechosen coherently ifand only IfMisaspin manifold. Ifthisisthecase then for1/1;,“e1},”and (Pi,-ii‘EIE?‘.p,qeUM,wedefine theequivalence relation by 1Pi,‘”""(Pi?) iffp=qandcpl?’=1//j,“"(S<(5°'>)*‘. (9.5.3)REPREsENTINO SPINOR FIELDs WITH DIFFERENTIAL FORMS 277 The resulting equivalence classes ofdifferential forms form abundle. OnU_,.wemay represent asection ofthisbundle byadifferential form lying intheminimal leftideal I‘"2,onU5wemay choose arepresenta- tiveform in1(5), these being related onU0,3bytheabove relation. Ifa isanarbitrary Clifford form and qEDthen forcpiﬁl~1/1"’) wehave acpwlq ~at//(“lq so,indeed, thisbundle isaspinor bundle, carrying an irreducible representation oftheClifford bundle with aD-linear struc- ture. Although sections ofthisbundle 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 thespinor 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, and theproperties ofthese donotdepend onhow we choose torepresent spinor fields. Bibliography Geroch RP1967 J.Math.Phys. 8782 -— 1968 J.Math.Phys. 91739 11970 J.Math.Phys. ll11 Greub WandPetry HR1978 Lecture Notes onMathematics vol675(Heidel- berg: Springer) Isham C1978 Spinor fields in4-dimensional space—times Pr0c.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 HR1984 Spin Structures onLorentz Manifolds, Trieste ISAS-44/84 Pressley AandSegal G1987 Loop Groups (Oxford: Oxford University Press) 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 isSEe"SXa. The co-frame {e“} isdual tothe arbitrary tangent frame {Xa}. 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. TheDirac equation foracomplex spinor field 1/1is 2 $11»=mp (10.1.1) where itisacomplex constant. The nature ofthemanifold may restrict theeigenvalue ittocertain real orimaginary values. Inother cases we may only beinterested inreal orimaginary eigenvalues forphysical reasons. IfSQ)isthestandard spinor covariant derivative of§9.3.1 and AisaU(1) connection 1-form then aU(l)-covariant spinor derivative is given by S§§')tp =S5?)tp +qiA(X)t/1 (10.1.2) where qisthe‘charge’ coupling constant. The original equation of Dirac Involved such aU(1)-charged covariant derivativeTHE DIRAC OPERATOR 279 Exercise 10.1 Show thatS(‘l)(X, Y)1p ES(‘))(X, Y)1p +iqiXivF1p 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 1/1isreduced into ‘Weyl’ spinors 1/Fcarrying irreducible representations oftheeven subalgebra by t/F=§(1i§)tp. (10.1.3) The projectors %(1i E)anticommute with members oftheco-frame {ea} andareparallel. Soif111satisfies amassless (it=0)Dirac equation then sodotheWeyl spinors 1/F.Such massless equations fortheWeyl spinors areknown inphysics asWeyl equations. Spinors ofthereal Clifford algebras can also besubjected tothe Dirac equation (10.1.1) (with itreal). For signature (p,q)satisfying p—qE0,2 mod8 thereal Clifford algebra isatotal real matrix algebra andthespinors areknown inphysics asMajorana spinors. In this case theDirac equation may beknown asaMajorana-Dirac equation. (Although theeigenvalue uin(10.1.1) canbetaken tobeany real constant such anequation cannotbeobtained from avariational principle. Without recourse to‘anticommuting’ parameters avariational principle will only give aMajorana—Dirac equation with zero eigen- value.) Asweremarked atthebeginning of§9.5, forthespecial case ofaﬂat parallelisable manifold theClifford bundle contains aspinor sub-bundle ofminimal leftideals. Thepseudo-Riemannian connection Vinduces the spinor covariant derivative onthis sub-bundle. Thus inthiscase the operator (:1,restricted tosections ofthisspinor sub-bundle, isaDirac operator onspinor fields. One ofDirac’s requirements forhisequation fortheelectron wasthat thecomponents ofthefield should satisfy aKlein—Gordon equation. As wehave just noted above the operator (:1,which squares tothe Laplace—Beltrami operator, induces aDirac operator onspinor fields in ﬂat space. Sothis Dirac operator squares totheLaplace—Beltrami operator, acting ondifferential forms inthespinor sub-bundle. More generally, thesquare oftheDirac operator isknown asthespinor Laplacian. Wehave (W1/9=9“SX.(9”SX.v) =gz1e”SXb1/1 +,_%(e“e” +e"e“)SX,__SXh1/1 +§(e“eb —e“’e“)SXaSXb1p =fleasxﬂll +Sx,,SX9‘l) 1'ieabiSX,,, SX,,l1l) =r19“S,,v +SX,S,-v +%9“bS(X,, X,)v+%9“"Sv,.,_ ,-,.,w- 280 SPINOR FIELD EQUATIONS Now [X,,, Xb]=iXﬂiXbde‘XC, andso %€“bS[XmXh]t,I) : —(Ii€CSX(_1ﬂ. gives 3°99=i,,-v,<,9”S,-.1/1 +$X.5,-9 +%9‘"’S(X,, X,)v, Using (9.3.20) thecurvature operator ofScanbewritten interms of thecurvature 2-forms togive %9"bs<X,, X,)v=iR,,999v- From (8.1.17) wehave, forzero torsion, Rode“) =~9R, thecurvature scalar, andso 521/1=(5X“+iX,VX9@")5x..1/1 T$9711/L (10-1-4) Exercise 10.2 Analogously express theLaplace-Beltrami operator as ¢2¢ = (VXB +1XbVXr>€“)VXa(I) — — §RCd(I)€Cd. 10.2 Covariances oftheDirac Equation andConserved Currents Generally weexpect equations formulated onpseudo-Riemannian mani- folds tohave acovariance corresponding toanyisometries. Forexam- ple,in§5.4 weshowed how theLie.derivative with respect toaKilling vector maps solutions toMaxwell’s equations into new solutions. Inthe same way wemay usetheLiederivative onspinors toobtain new solutions totheDirac equation inspaces with isometries. Foravector field Kwehave 21(3) : (VKQQ + €“])SXa + €“.§€KSXa. Ifnow Kisaconformal Killing vector, with L’Kg =2/lg, then forAany 1-form SZKA =VKA +§[dK, A]+/1A. This follows from (9.4.1) and theobservation thatforXPap-form e,,X,,e" ==(n—2p)Xj§ (10.2.1) so §€K,$= §EKe“'SXa —/1,5‘+e“&’KSXﬂ E§EKe“SXa —2,8+SEEK +e“’S]K_XQ] —§e“(d/1A ea) by(4.4.9). Since $K(e“(X,,,)) =0then $Ke"SXa +e“S[K,Xa] =0,and e“(d/1A ea)=e“A(d/1A ea)+i”(d/1A ea)EX_,,(/1)e" —ndit=(1—n)d/1 so[é-€K, ,8]=—/1,8 ——§(1—n)d/1. Since ,S'(Mp) Edill/1+21,81]: thismay beCOvARIANcEs OFTHEDIRAC EQUATION 281 written as [99,+;(,,-1)/1,,t]=—/1,5". (10.2.2) IfKisaKilling vector (2=0)then .€€Kcommutes with theDirac operator and ifipsatisfies theDirac equation (10.1.1) then sodoes £8K1/1.Forthemassless case (ti=0)wealso have acovariance forKa conformal Killing vector: ifSip=0then ,$’[$K +§(n—1)/1]1/1 =0. Out ofanytwo solutions totheDirac equation wemay construct a closed (rt—1)-form. Fordefiniteness wetake (,)tobeaHermitian- symmetric product oncomplex spinors with 517*asadjoint involution. Then Re(,)isareal-valued symmetric product. Ifweexpress an (n—1)-form 59asC?=j,e“z, with 2thevolume n-form, then 9.592eh/\VX,,(]-tteazl =eh/\(VX,,jaeaZ J"lttVx,@a(Xc)eCZ)- Now e"A(e”z) =ebAiX,z =—iX,(e" A2)+g“"z =gabz, so d}=(VX,j,, +iXt,VXbe“j,,)z. (10.2.3) Taking §=R6(’l/J, e,,<p)e“z (10.2.4) gives <1?=R9($X,v, 9‘9>)->1+R90/9,,$c9)Z=—R9($1/»,99)Z +R90/9,,$99)Z where thecovariant derivative SXiscompatible with thespinor product. (This covariant derivative could contain aU(1) coupling.) Thus if Sip=mp, foritreal, andsimilarly fortp,then dc?=0.Inthisway we obtain aconserved current (aclosed (n—1)-form) from anypair of solutions tothefield equations. (Had wetaken aspinor product with §* asadjoint involution then theform ,9would beclosed for~spinors satisfying theDirac equation foranimaginary eigenvalue.) IfI/1isthe adjoint to1/)with respect totheyHermitian-symmetric product then (Ill,@099)?“ :9)0(1l)@@(P)ea =9i0(9-')‘l”3a)ea =9i1(‘Pil))- SO the ('1—1)‘ form in(10.2.4) canbewritten as rm,‘ a-‘,4 § Re9°1(q0tp)z *Re6"1(tp1/1). (10.2.5) Inparticular, taking cp=it/1in(10.2.4) gives theU(1) current ,9?=(1/1,ieat/J)e“z. (10.2.6) This current would provide asource fortheequation (such asMaxwell’s equation) foranyU(1) field entering into thespinor covariant deriva- tive. Wenow only consider theDirac equation without aU(1) coupling. The presence ofisometries, generated byaKilling vector K,ensures that if1/1isasolution tothefield equations then soisSQK1/1.Wethus have theassociated closed currents 282 SPINOR FIELD EQUATIONS $5,;=Re(tp, e,,.§€;,-t}1)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."ii 'i;=.;t.=s-* .I13--; -THE DIRAC EQUATION INSPACETIME 283 Although itisbeyond thescope ofthisbook toenter intotherealms 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 andthesimple 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 onamanifold willcease tobeimportant. The Dirac equation foracomplex spinor (aDirac spinor with unit charge) iponspacetime is ﬂip+i/11/9=mtp (10.3.1) where wehave explicitly exhibited the U(1) interaction with the electromagnetic 1-form potential A.The real eigenvalue mwill be interpreted asamass. The spinor field provides anelectromagnetic current 1-form j, 1'=5(1(i1//175) (10-3-Z) where i/7isthespinor adjoint of1pwith respect tothepseudo-Hermitian product whose adjoint involution is<§1j". The Maxwell 2-form FEdA satisfies 6F=j @033 with 6theco-derivative of(5.4.2). The electromagnetic current 1-form jisfuture-pointing andtimelike foranyspinor 1/1.The argument that thisissoisalgebraic. Wefirst consider thecharge density p=(1/1,ie‘)1p). Ifwetook thespinor adjoint asin(2.8.13) then thepositivity ofpwould follow immediately. Thefact that thespinor adjoint canbecastinthisform follows ultimately from thepositivity ofthemetric onthethree-dimensional spacelike sub- spaces. Itisinstructive toargue thepositivity ofpdirectly from properties ofthevarious spinor products. Let{e“} bealocal orthonor- malco-frame and 2Eiem such that 22=1.Letu,beaspinor such that 2u,,=eu,, with .1;=i1.Then u,carries asemi-spinor representa- tion ofthesubalgebra generated by{e1,e2,e3}. Let (,)g, bethe pseudo-Hermitian product associated with §*then (us, ue')§* :(Egan: us‘)-fa-1* Z'9(ue=- 2S*ur')§* 28£!(uev us’)??- Ifafour-dimensional spinor 1/1isdecomposed as1/1=u,+u_then i (ll),ll/lei =(“+, H,-).§* +(*4-, ”-),=*- Weknow from §2.7 that 5*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 on11+and anegative-definite product onu_. Forthree-dimensional semi-spinors wehave -0 _ -As,.0 _ -0/\ __ -0(M8,1eus»)—s(u,,1z~’l"e u,.,)-s(u,,,1e zu,») -ae'(u,,,1e u,.,). Sothecharge density pisdiagonal inthethree-dimensional semi- spinors: pE(u+, ie‘)u+) +(u_, ieUu_).THE DIRAC EQUATION INSPACETIME 285 Now (u_,,ie°u,) =e(u,,ie°ie123u,,) =c(u,, zug). Thevolume 4-form 2relates theproducts associated with 4317*and5*so thatwehave (u,,ie°u,,) =e(u,., u,)§, and P=(9,,H+).:,-* —(91,,u_)i.,=~=- (19-3-4) Thus pispositive-definite orzero since thefirst product ispositive- definite andthesecond negative-definite. Toshow thatthecharge density ispositive-definite above wesplit the four-dimensional spinor into semi-spinors ofthethree-dimensional sub- algebra. This argument implies thatg(j,V)islessthan orequal tozero forallfuture pointing timelike vectors Vandconsequently that jmust 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. Let1/FE%(1:1:iz)1/1, thatis121/Ii =itvi, then (1/rs,at//8') =se'(iz1/J8, aizipfi) =as’(1/18, zazip’) =—ee'(1p", a"Ip“). Sothecomponents ofjarediagonal in01*and1/F: j“=(tp*, ie"1p+) +(1/1', ieaijf) Ejf.+j‘1. (10.3.5) Thenorm ofj,isgiven by —ti=(1/»€9“v£)(v£,9,v8) =1't7£9“vrii‘9,wE- Using (10.2.1) 4 991/rt/7*i9, =§’,}(4—2p)(-1)P§P,(vtv7E)~P Now (v91'17r)9=izv91"/Friz --iztinr --1P"1t7"andsoonly oddpenter intothesum. Wehave v,(v£v70 =fr,(v£17Eiz)iz =—9P,(v£i?i70iz =—99”,(vrii9)izthus -1'i=—2v7r9r,(v@17t)vE —29v7E§P,(v£17£)izvE =-41'/799”,(Wl7£)vE =—4ff,(vrii£9,)1i3“9"v8 =-4(v:’, 9,vE)(w£, 9%//8)=41%, Soj+andj_areboth null and, since pE0,future pointing. Since the sum oftwofuture-pointing null vectors liesinorontheforward light cone thecurrent jisfuture 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 EQUAT1ONs 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 also allow adiscussion ofthediscrete orientation-changing transforma- tions. Let{xa} beglobal inertial coordinates forMinkowski space, such that {dxa} 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‘1. (10.3.6) Ifaisanarbitrary differential form then aEa,dx’ with themulti-index Ilabelling aparallel basis fortheexterior (orClifford) algebra. Then rr*(s)a =(a,9rr(s))sdx’s‘1 (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 if1pE1p"b,-, wecan deﬁne .rr(s)-1p E(1p"9:r(s))sb,-. (10.3.8) Since dxlb, —l“j-',b,- for1"],constants, itfollows from (10.3.7) that (10.3.8) satisfies rt(s)-(atp) E(rt*(s)a)(rt(s)-ip) VaeFC(M). (10.3.9) IfXisanarbitrary vector field wealsohave .7T(S)'SX"l]) ES,,;-1 (5)X(7T(S)'1/J). (10.3.10) This follows from (10.3.8) since VxsE0and X(1p‘) 9:rt(s) E (rr.,T1(s)X)(ip‘9rt(s)). Since rt(s) isanisometry, thepullback ofthe Clifford product oftwoforms istheproduct ofthepulled-back forms. If {e“} and {Xa} aredual bases then soare{rr*(s)e“} and {.rt,,'1(s)X,,}, thus n'(s)-,8 E,5)-rr(s). (10.3.11) Itimmediately follows that iftpandAsatisfy (10.3.1) then sodort(s)-ip and rt*(s)A forn'(s) any Lorentz transformation. The pullback map 5-,.11"t_a;"- .:i-I "jt '-1,.3?2.‘- l1 ‘Ir!-'. ':~a¥';. ii! l'.f:I.":- ,-:-(.1 '_.j-E-{.1-1 3'iE'.'="' ..-='.:‘..#-THE DIRAC EQUATION INSPACETIME 287 commutes with theexterior derivative and, inthecase ofanorientation- preserving isometry, with theHodge map andhence theco-derivative 6. 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 sodorr*(s)F andrt*(s)j. Butjisafunctional ofthe spinor field 1p—to symbolise thiswewillhere write j(1p) forthe1-form determined by(10.3.2). Isitthe case that j(n(s)-1p) Err*(s)j(1p)? Equation (10.3.2) involves thespinor adjoint with respect toaproduct whose invariance group does notcontain thewhole Clifford group, but only *1“.(This isthesubgroup defined with thenorm u,sosi"Es“)for sE+17.) The image under the vector representation of*1“isthe orthochronous Lorentz group. SoifTeil"issuch that ;g(T)isa reflection changing thetime orientation, then rr(T)-1p andrr*(T)A will notsatisfy thecoupled Maxwell—Dirac equations given that1pandAdo. 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,scpsip E)((s)-(quip) if(and oply if)sisintheinvariance group ofthespinor product used todefine 1p.Aswewillsee,ifinstead sisintherealsubalgebra such that (sip, sip)E(tp,1p)*foraproduct on complex spinors then stpsip EX(s)~(tp1p)*. 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, (,)g,,. Inanappropriate basis, charge conjugation simply complex conjugates thespinor components andwehave (1981/1‘);-,, =(<19,1t»)”‘f;,=,,~ (10.3.12) Ifwenow define (Q91 E(“P61 7~l))§r] then (,)certainly has511*asadjoint involution. The factor ofiensures thattheproduct isHermitian symmetric: (99,111) =(i<9‘,1/1);-1 =—(i99,1/1‘)’%,, =(1/)‘»i9°)’i,t =(11/)‘=‘P)‘i91 = (19,99)*- Since charge conjugation isinvolutory andthecomplex bilinear product in(10.3.13) isskew symmetric wehave 288 SPINOR FIELD EQUATIONS ((0%11/‘)=—(<P»1P)* (103-14) If istheadjoint of1,0with respect to(,)then foranythree spinors (W/7)*p =((<M7)P‘)‘ =((1/1,PW)‘ =(111,p‘*)*<P“ =(1/»"lp‘*)*<P‘ =—(1/1‘)PW by(10.3.14). So(tp1/7)*p =-—(q9‘ ti-/~1“*)p and <¢iiI)*=~¢>w7‘: (106.15) Consider now theelement Teil"with ;((T)atime-orientation-changing reﬂection. Then T5'”* =T5”=—T”, so T<P‘77/7?‘ =-T<P"1/7“T* =T(<Pi/7)*T"‘ by(10.3.15). Wenow define ?T.tp Err(T).q1‘ (10.3.16) andthen have 3‘.cp9.t/1 =rr*(T)(<;01p)*. (10.3.17) The operation 9isknown asWigner time reversal onspinors. It obviously satisfies a".(a¢) =rr*(T)a*(9".i/J). (10.3.18) Wenow examine thecovariances oftheMaxwell—Dirac system under thisoperation. IfAand 1/1satisfy (10.3.1) then sodo~.rr*(T)A and 9”-tp. Itfollows from (10.3.17) that j(9*-1,0) =—:r*(T)j(1,/1) and hence —Jr*(T)A and9"-1/J also satisfy (10.3.3). (Notice that whereas —1r*(T)A andJr(T)-1/1 satisfy (10.3.3) they donotsatisfy (10.3.1).) Plane-wave solutions play animportant part inthephysical interpre- tation ofthefree (A=0)Dirac equation, andtothese wenow turn. If bisaparallel spinor then welook forasolution to(10.3.1), forA(= 0, oftheform 1/1=exp(if)b forfsome realfunction. Then $1/1=idfl/1 and werequire idfip =mi/J. Itfollows that the1-form dfmust betimelike, with (df)2 =—m2. (10.3.19) Wecanwrite thealgebraic condition onbas §_(1+idf/m)b =b. (10.3.20) Ifsisaunit spacelike l-form orthogonal todfand zisthevolume 4-form then (zs)2 =—sz2s =52=1and zsdf =—zdfs =dfzs. So §(l+zs)isanidempotent orthogonal to§(1+idf/m) sothat %(1+ idf/m)§(l +zs)isprimitive. With sand 0taking thevalues ila F-E‘;-l‘-'i":'.“-§*""I."___A >5:542.1.THE DIRAC EQUATION INSPACETIME 289 complete setofpairwise orthogonal primitive idempotents isgiven by {Pm =§(l+eidf/m)§(1 —ozs)}. (10.3.21) Wecanchoose abasis ofspinors such thateach isaneigenspinor ofone ofthese primitive idempotents. Aninertial observer would useinertial coordinates {t,x,y,z}tointerpret df(8/Gt) asanenergy anddf(8/8x) asacomponent ofmomentum along thex-axis. Ifweassume that dfand sareparallel then wecanchoose inertial coordinates {t,x,y,2}such thatf=mtands=dx.Then wecanlabel plane-wave solutions byeand0, 111,1,=exp(ismt)b,(, (10.3.22) where bw=Pwb 8.,for PH,=§(1+iedt)§(1 +odydzdt) (10.3.23) and wehave" chosen z=dxdydzdt. Ifwechoose some parallel b++ then wecanbuild uptherest ofthespinor basis bytaking Clifford products. Forexample, wehave dxPw =P_,_,,dx anddyP,., =P__.,,,dy andhence dxdyP,,, =P,_(,dxdy. Sowemay choose thebasis as {b++, b__=dxb+,, b_+=dyb++, b,_=dxdyb++}. (10.3.24) If(,)has§n*asadjoint then wemay usethealgebraic properties of thisbasis towork outthenon-vanishing products, wehave =i= (baa: bs'0') :(peabeor Pe'0’bs’o') :(b€U= Pfvgn P£'<J'bE'0') :(beoa P—eaP£’c:’bs’0’) =6—E€'500'(b80> bE'0')' Thus the only non-zero independent products are (b++, b_,) and (b+2,b__). Ifwechoose thebasis asin(10.3.24) then these arerelated for (b+_,b__)=(dxdyb++, dxb++) =(b...dyb++) =(b...b_+)~ Sobysuitably scaling b++wehave (b++. b_+)=(b+_. b_-)=1- (10-3-25) Thus thetwo-dimensional subspaces with fixed 2areisotropic, whilst those with fixed 0areunitary subspaces ofmaximal index. The.9-label of1/1,0specifies theeigenvalue ofthespinor Liederivative inthe8/St direction, gs/aiwea =i5m‘Pw- (10326) Similarly 0may beused tolabel theeigenvalue oftheLie.derivative with respect tothevector ya/E92 ——28/8y that generates rotations about thex-axis, wehave 290 SPINOR FIELD sou/arrows g(y8/82 -26/8y)wso : Z8/ay)w£0 =édydzt/18,, =éidydzdtidtipﬁ, =éiedydzdttpw ‘§£(y8/82 —28/8y)we0 :2i8Uw£0' Theeigenvalues ofi"§ilead 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 ‘§n*asadjoint involution, then Re(,)isreal valued and symmetric. Let 9-ab=Re(1P=@aSX,,"J’) +R@(1//»@z>Sx,,lP)- (10-41) IfSiscompatible with thespinor product then Xa(gab) =R@(5x~1/1»@.i5x,,1/1) +Re(‘/1» VX"@a5x,,1l’) +Re(1/7» 5’aSX“SX,..‘/J) +Re/(S)c*i1//» ebSX,,1/4) +Re(1/1, VXn€bSX“1/J) +Re(ip, e),SX~SX“1p).(l0.4.2) Changing theorder ofthecovariant derivatives @aSX"SX,,W Z€aS(Xa= Xb)1/J +ea-SX,,SX“w +@aS[X,,,X,,]1/1 =6’aS(Xa.» X1011’ +SX,,>$1P _VX,.¢’“SX,,W +eC(VX,,Xb _VX,.Xa)@aSX,,.W ifVistorsion free. Now e‘(VXhX,,)e“ =—VX,e“(Xa)e“ =—-Vxhe‘ ande‘(VXaX,,) =—VXﬂe“(X,,). From (9.3.20) e“S(X,,, X,,)i/1 =§ePR,,b1p, andforzero torsion thiscanbewritten interms oftheRicci forms, e(IS(Xar :2Pb1/)9 SOTHE STRESS TENSOR 291 @a5X..5Xbq; =e“S(X,,, X,,)1p +SXb,$ip -VXae‘(X,,)e“SX,1p. (10.4.3) From (10.1.4) wehave SX..SXaip =$21/J—VXae‘(X,,)SX(1p +;‘,9iip. (10.4.4) We can rewrite VXie,, as(VX~e,,)(XC)eC =e‘(VX,,X“)€C = —VXﬂe‘(X")eC, andsimilarly VXae@, IVXieb(XC)e‘ =—eb(VX~XC)e‘ = —VX~e“(X,,)eC, collecting terms, X“(97ab) =R@(5x~1//i @a5x,.1//) +R@(5x~1/A @b5X.,1/1) "Vx,,@C(X“)R@(1/1» @C5x,,‘P) -Vx..6’C(X“)Re(l/1» ebsxﬁl/) ‘"VX,eC(Xb)R@(l/J» QQSXJP) _VX..@C(Xb)R@(1//» QCSXW’) +%Re(w.Pi1/1) +Rem».SX,,>Sw) +R@(1//»@b$21P) +iR@(1P» @iﬁ7W1)- (10-4-5) If9=§abe” ®ebthen vi..‘J<X“. Xb)=X.<wi> +vX.@“<X“>91»» +VX.@‘(Xb>?T“@ =Re(SX~i/1, e,,SX,w) +R<->($m/», @..$X.1/1) +RemSim»)+R60’)?ebﬂiv) +%R@(w.Pbw)+iRe(1/»»61%/1)» Since the spinor product issymmetric with 5537* asadjoint then Re(rp, A1/1) =—Re(1p, Acp) forAanyreal1-form and vX,2r(X“, Xi)=—-R<->($w» 5X.1/1) +R60/1, SX.$w) +Retwi @b$2w)- Itfollows thatV9 =0if$1/1=mi/1with mreal. The above isseen togothrough unaltered forreal spinors with a spinor product whose adjoint involution is$17.Had wetaken areal- valued skew-symmetric spinor product oncomplex spinors with 5*as adjoint then 9would bedivergenceless for$1};=imip. Forrealspinors andaskew-symmetric product with 5asadjoint thestress tensor would bedivergenceless if$1};=0. Exercise 10.4 ‘ Show that theMaxwell—Dirac stress tensor 1Sdivergenceless when the coupled equations aresatisfied. For the stress tensor of(10.4.1) the trace isgiven by 3,,“=2Re(ip, $1/1). When theDirac equation issatisfied wehave §]'a" =Zm(1p, (10.4.6) 292 SPINOR FIELD EQUATIONS-=;.::_",_ =-.":-=.I.-*= -1;.._,1l.,_ Certainly formzero thetrace iszero. Ingeneral thespinor product will bepseudo-Hermitian andsoformat0thetrace canstillvanish. S.l We have already noted in§7.4 that wecan construct aclosed (n—1)-form from the stress tensor and aKilling vector, namely JK=9al,K"e“z where 9,,l, arethecomponents ofthestress tensor 9, given by(10.4.1), which isdivergenceless when thefield equations l $1/J=mt/1 areimposed. In§10.2 weobtained byinspection aclosed (n—1)-form 9Kforeach Killing vector K.These twoforms, JKand .95K,infactdiffer byanexact form modulo thefield equations, aswe now demonstrate. We aregoing tohave torecognise theexterior derivative ofan(rz—2)-form when weseeone, sofirstwenote that if H=H,,l,e“”z then dHI2{Xb(Hba) “VX,.@c(Xb)Hba “VX’*@a(XC)H1>c}@aZ E(dH)a@aZ- l (10.4.7) ‘ Afairly tedious calculation produces I Retw,I'€SX.1/1) =%R@(1/1. eadli-$11/) —%(<r1H)@ +Raw.e..8iw) —%R@(w.@aKd>$1/1) +%R<~>(i/».I?e.$w) (1048) I where Hl,,,=Re(1/1, el,Xe, 1//)-—Re(t/1, el,,,X1/1). Aswell asfrequently using thedefining anticommutation relation oftheClifford algebra the calculation uses thefactthatsince theSpinor product has$171‘asadjoint l then Re(_t0, At/1) =0for ANany real 1-form. Thus for example Re(1/1, eaKebtp) =-Re(i/1, e”Ke,,I/1), asisnecessary forHal,=~—Hl,,,. (Although itistedious werecommend thatthereader verify (10.4.8), as itdoes help develop thecalculational proficiency that unfortunately is sometimes required.) Itfollows from (10.4.8) that 1"‘-,1 R901)» QGSKUJ) +R"3(1//» KSX,,w) =2R@(wi @..%l<w) —%(dH).. +R601»,(1?/\e..)$w)~ Ifweusethefield equations, ,$1p=mtp, thenI Re(‘/J» (Z/\@@)>$1P) ="”R@(1V»(1? /\@a)1P) I0 since areal2-form changes sign under §i7*, theadjoint involution ofthe i spinor product. Thus JK=2.9%,; modulo anexact form, modulo thefield equations. Exercise 10.5 Repeat theanalysis with askew product whose adjoint is5*with field lg* u < equations $111=1mIp. I,::"I==.1I'§-THE STRESS TENSOR 293 Example 10.1 Gravitational andNeutrino Waves Consider aspacetime inwhich themetric takes theform g=2(du®dv+dv®du—2Hdu®du+dz®dz*+dz*®dz) incoordinates (u,u,x‘,x2)with zExl+ixzandHarealfunction of M,zand2*.Itishere most convenient toadopt anullbasis. Wechoose thenull co-frame {na} a=1,2,3,4where n‘=du,n2=dv—Hdu, n3=dz,n4=dz* andtheduals areXl=8/Su +H8/80, X2=8/E90, X3=8/82, X4=E3/82*. The non-vanishing components ofthemetric are812=821=834=843=2»01'8127'8211'834=843= Since the components ofthemetric areconstant inthisbasis wecanevaluate the connection forms by(6.6.8), thenon-vanishing ones being (Um : —(1)13 : ZHZHI (U41: ""(Ul4 : 2HZ*ﬂl. Theonly non-zero Ricci form isPl=2H,»i,n1. Wenow adapt aspinor frame tothis null co-frame. Letblbea spinor such thatnlbl =n3bl =0.Wethen form thespinor frame {bl, b2=nzbl, b3=n4bl, b4=n3n"'bl}. (We canrepresent thespinor blbythedifferential form n‘/13,thislying inaminimal leftideal ofthecomplexified Clifford algebra. The other spinors {bl} arethen seen tocomplete thebasis fortheminimal left ideal.) If{X0} istheframe dual to{n"’} then, foroxdefined in(8.1.5), wehave OX]=(H_,n3 +H,in4)n1 with allother OXJ, zero. Itfollows thatthespinors blandb3areparallel. Hence ifhlandhparearbitrary complex functions ofuand tp=hl(u)bl +l13(u)b3 then ,S'i/1=0.To obtain Einstein’s equations wenow need toevaluate thespinor stress tensor. If(,) istheHermitian-symmetric spinor product with §r7*as adjoint then wecanusethealgebraic properties ofthespinor frame to evaluate theproducts. Forexample, (bu b3):(b1~ ”4b1):(”4b1» b1)* :""(bI~ ”3bI)* :0 since n3bl =0.Also b3=nzbl andson'b3 =nlnzbl =(1-n2rz‘)bl =blandhence (bl. bl)=(rtlbg, nlbg) =-(bl, nlnlbz) =0.Inthis way wecanshow that thenon-vanishing products arespecified bythe imaginary components (bl, b3)=(b3, bl). Bysuitably normalising bl wehave (bI~ b2):(bsi b-1):i- Theonly non-zero component ofthestress tensor of(10.4.1) isthen 9ll=4Re(ih*lh’l +ih"§h§). 294 SPINOR FIELD EQUATIONS Since forzero mass thespinor stress tensor istraceless wecanwrite the Einstein equations asZKPC =*”‘rC, andsothecoupled system reduces totheequation KH,*, =Re(ih*lh’l +ih‘*§h§). 10.5 Tensor Spinors Starting with thespinor representation ofthespin group 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 Vwith thespinor space of C(V, g).Such objects canbethought ofasspinor-valued tensors. Asanexample weconsider aspinor-valued 1-form ll’onspacetime. Then wecanwrite thisinanyco-frame {ea} as LII=1/»,®e“ (10.5.1) where each 1/1,,isaspinor. Wecanthink ofll!asamapping from vector tospinor fields: \I1(X)=ip,,e@(X) vx6FTM. (10.5.2) Equivalently if{bl} isanystandard spinor frame with 1/1,,=ii/§,bl then wecanwrite Was 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 the1,11,,areWeyl spinors, satisfying izipa =1/1,.Then the 111,,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). S0wemay regard aspinor-valued 1-form asa degree-three tensor onthespinor space. Ifuand vareanytwoWeyl spinors, lying inthesame semi-spinor space asthe1/1,,then wedefine.l,. _E3-; _I; -TENSOR SPINORS 295 ‘I’(H»v)E(H»1P.i)@“v- (19-5-4) Thebrackets ontheleft-hand side signify that LPisevaluated onMand 0.whereas thebrackets ontheright-hand side arethespinor product of Lland 111,,where theproduct has25asadjoint involution. (The skew- symmetric complex bilinear product onDirac spinors induces anon- degenerate product oneach ofthetwo spaces ofWeyl spinors. If u=izuthen thespinor product (u,1,116,)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 inuand0.In order todothiswewillneed thefollowing: (u,v)w ~——(w,U)Lt=4(u, W)v (10-5-5) foru,vand wanythree Weyl spinors. Toseethisletozbeanother Weyl spinor and consider theexpression (ti,u)(w, or).Using '17to denote theadjoint spinor wecanwrite this asuvwoz. Now wecan expand vWasin(2.l.l8) togive (H,v)(w, a)='iJ'9’ll(uﬁ7ejZ)e’*a’ =5‘ll(We"-fjlv) 'il'e"‘a =(w,e§l0)(u, eAa'). Now foruand0Weyl spinors andaanyClifford form (0,au)=(izu, aizu) =(0,zaz*1u) =(0,a’lu) so(v,au)=0foraodd. Inaddition (0,au)=(4250, ti)=~(u, div) $0f()1'Q3=—a(a5 =a)then (v,au)issymmetric (skew) inuand0.So (u,u)(w, er)—(u,w)(v, er)=(w,e§lv)(ii, e/‘a') —-(W<—>v) andthefirst bracket ontheright-hand side willonly contain those ejl that areeven under 17and under E.These arethe0-forms and the 4-forms, thus (u,u)(w, oz)—(u,w)(i), a/)=2(w, v)(u, er)—2(w, zv)(u, za/). Since vand crsatisfy zv=—iv and zaz=—ia theterms onthe right-hand side addup.Wecanusetheskew symmetry oftheproduct torewrite theleft-hand side, producing (w,u)(v, a’)—(U,u)(W, £1’)=4(W» U104» 9’)- Since thisistrueforallcrandthespinor product isnon-degenerate (W,M)v—(v,M)W=4(W»v)“- 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, U)—\I1(v, Lt)=4(Ll,U)€”1/)0. Thus thespinor-valued 1-form isanirreducible spin tensor ifitis ‘traceless’: en/2,=0. (10.5.6) Exercise 10.6 Usethecorrespondence between 1-forms and(1,1)spintensors given at theendof§2.8 tolabel thecomponents ofaspinor-valued 1-form with one ‘dotted’ and two ‘undotted’ indices. Show that the‘tracelessness’ condition isequivalent tosymmetry inthetwolikeindices. The spinor covariant derivative SXandthecovariant derivative VX can beextended bytheLeibniz rule toacovariant derivative, also denoted SX,onspinor-valued 1-forms. Intheobvious way SX\P=SX1/1,,®ea+10,,®VXe“. (10.5.7) (Ifanyconfusion islikely between thecovariant derivative onspinor- valued 1-forms andthat onspinors wecanwrite theformer asS92.)A representation oftheClifford algebra onspinor-valued 1-forms canbe defined by a\PE(a1pl,)® eh (10.5.8) sothatwehave aDirac-like equation $111=mlll. (10.5.9) The pair ofequations (10.5.6) and (10.5.9) aretheRarita—Schwinger equations forspin 3/2[25]. Exercise 10.7 Show that (10.5.6) and (10.5.9) imply the ‘Lorenz’ condition (Sx,,qJ)(Xa) =0- InMinkowski space wecanpick aparallel co-frame such that (10.5.9) reduces tofour Dirac equations. Wecanthen find plane-wave solutions asin§10.3. If{bw} isthespinor basis of(10.3.24) then wehave Dirac solutions asin(10.3.22) with theSign ofthefrequency correlated with the:3labelling thebasis spinors. Bytensoring onfour independent 1-forms tothetwo basis spinors with (say) s=+1wecanform eight linearly independent spinor-valued 1-forms. Wecanchoose four ofthese satisfying thetracelessness condition (10.5.6). The eight spinor-valued 1-forms canbechosen aseigenstates oftheLiederivatives with respect tovectors generating time translations and rotations about thex-axis. The 1-form basis can bechosen tohave eigenvalues of{i,—-i,0,0}2 '1' ... .'5; _li'1-Zi-ii'.' =§‘;I1' ;-i..i;4 G .:§".‘.:i .....=_-ir §': i.-52 .:-_» _._=:_ l._ l- \,-4-. -.1 -"A.1'I lI 1 1ilTENSOR SPINORS 297 under theLiederivative with respect totherotation, whereas thespinor basis haseigenvalues {§i,—§i}. Thefour traceless spinor-valued 1-forms arethen seen tohave eigenvalues {§i,ii,—§i, —§i}. For thebasis of (10.3.24) dxbﬂ, =b1,_l,, idtbw =eb,.,,, dybﬂ, =obOw,dzbw =ieb,,,,(10.5.10) soabasis forpositive-frequency solutions to(10.5.6) and(10.5.9) is {b,, ®(dz—-idy), b+_ ®(dz—idy) —2ib++ ®dx, bl, ®(dz+idy) —2ib,._ ®dx,b+_ ®(dz+idy)}. (10.5.11) These areeigenstates of§8',.l,,a, __,_.l,ll,,, 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 notrely onVbeing 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{Ta} arethetorsion 2-forms ofthe connection Vthen acovariant derivative ondifferential forms isdefined by ii,EVX+;IXT“,(1X,. (10.5.12) From (6.7.4) weseethatVisjustsuch that eaA6,,=<1. (10.5.13) IfSXisthespinor covariant derivative associated with Vthen a covariant derivative SXonspinor-valued p-forms isdefined by SXWESX11»,®e1+1;»,®VXQI (10.5.14) where efisap-form basis. ForLPaspinor-valued p-form wemay adopt theconvention thatforaanyq-form GAlp -EIf/I ® G/\BI. Thespinor covariant exterior derivative Dmaps spinor-valued p-forms to spinor-valued (p+1)-forms: D\I1E6“ASxgil. (10.5.16) If{bl} isastandard spinor frame associated with some orthonormal co-frame then wemay expand ll!as\I1=bl®1/1”where the1/1"areaset ofp-forms. Then wecanequivalently write thespinor covariant exterior derivative as'| IIE 1I I 11I 5 298 SPINOR FIELD EQUATIONS DIP=bl®dip‘+gepqbl ®(0,,A1p". (10.5.17) The Hodge dual ofaspinor-valued p-form isdefined intheobvious way, inanalogy to(10.5.15). IfNisaClifford-valued q-form, N==nA®eAfor11""arbitrary Clifford forms andeAabasis forq-forms then wechoose todefine N111 =nA1,//l ®e,.lAe’. (10.5.18) Having adopted these conventions weconsider theequation e*D\P =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 thatifII!isaspinor-valued 1-form then *(e*D\P) =SX,(@,Ip@) ®e“-,i¢\I1. Hint: youwillneed *(e"A*e“b) =g"‘e“ —g“‘e”. (ii)Show thatifrpisaspinor field then e*D2cp IebS(Xl,, X,,)(p ®*e". Hence show that iftheRicci andtorsion forms arezero (10.5.19) has the‘gauge’ symmetry ‘I1I—->111+Dip. Exercise 10.9 Consider thefollowing equation foraspinor iponspacetime: SX1/J-427510 =0 VXEFTM. Note that this isequivalent toequating tozero a‘traceless’ spinor- valued 1-form made from thecovariant derivatives of1//.Since Xand,$ both anticommute with thevolume 4-form thisequation decouples into twoequations forWeyl spinors. (i)IfKisaconformal Killing vector with .EEl<g =2)Igshow that if1/J satisfies theabove equation then sodoes .§€K1p —§/11/1. This can be shown inthesame way asfortheanalogous (but different!) result for themassless Dirac equation. (ii)Bydifferentiating theequation obtain theintegrability condition Rad/1 _i(@aSx,, _€’bSX,,))$W =0-! I'.:_if@''-_-i1»- _ --;-.§_- - [ ..I5' -2'1?! ':.':-I II §' I.I:. -14.3;-.TENSOR SPINORS 299 Clifford multiply toobtain thecontracted conditions Pm»+$4.584 -I-Siﬁw=0and 9711;;+3,8211» =0. Hence obtain theintegrability condition C4111/1 Z0- (Note that PaAel,—Pl,Aea=e,,Pl, —el,P,, forzero torsion.) (iii)If1/1=u+dfv, forsome function fandparallel Weyl spinors Ll andv,show that1/1solves theabove equation ifVXdf=X.Hence show that thisequation hasa‘twistor’ [9]solution with f=§i7,,l,x”x”, where {xa} 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 5*istheadjoint ofazero index Hermitian-symmetric product onDirac spinors, (,). Byintegrating over Mweintroduce another Hermitian product <1».<12)E[,,,(1/Afr)-Z where zisthevolume n-form ofM.The Dirac operator isanti-self- adjoint with respect tothisproduct. Toseethisweneed torecognise an exact form when weseeone. Tothisendwewrite an(rt—1)-form Jas J=j,,e"z and, for Vtorsion free, d]=(VX~j,, +iX4VXhe"j,,)z by (10.2.3). Since (,)hasE‘asadjoint involution with e”?=e“. <99»$11’) =(@0994 SX,.,1P> =[,,{v.i.,<e4>.1I») —vX.,4“<Xl.><4*’<4. 1»)-<54».11)};-'~ 1II _-=-.-._. 5 I F1 Eli [I‘. IL III 300 SPINOR FIELD EQUATIONS Now VXae“(Xl,) =—e"(VXaXl,) =—iXﬂVX4el,, and sowemay recognise anexact form intheintegrand. ByStokes’s theorem theintegral ofan exact form over acompact manifold iszero, thus (fa$11»)=—()$fP= 11>» (10-6-1) Since itisanti-self-adjoint with respect toaHermitian product theDirac operator onacompact Riemannian manifold hasimaginary eigenvalues. Asaspecial case oftheabove wehave <31».$4»)=—<>52IP» 11>. Since isa(zero-index) Hermitian product theleft-hand side is positive-semidefinite. Thus $211)=0<=>$1/J=0.Using (10.1.4) toexpand thespinor Laplacian gives ($14.$10=_<(Sx" +iX»Vn@“)Sn1I». 41>+%<%I».14>»Since <(SX,, +ix“5X,,@a)SX,1/% 1/1)=]MiVx“(Sx.,1P,1P) “(Sx"1P» Sxﬂ/1) +iX"VXl,ea(SX,,]1Uv 1/1)}-Z =_<SX“wv SX.w>wehave ($1/1» $1/1) I<SX“1/1» SX,,1//> +i<1/J» git/1) (10-6-2) If91B0then allthree terms are positive-semidefinite. If9%>0 then there arenozero eigenvalues oftheDirac operator: if9%=0then ,$1p=0<:>SX1lIi=0VX. When 9%isconstant, such asforthestandard metric onasphere, then weobtain alower bound fortheeigenvalues oftheDirac operator. If$1/1=imip, with rnreal, then (m2 _ <1/1»91>I(SXM/1» SX,W> andso m3> The above arguments canberepeated with real spinors. From table 2.15 weseethat theinvolution 5;’ofthereal Clifford algebra isthe adjoint involution ofazero-index product; theproduct being either R-symmetric, C*-symmetric orH-symmetric. 10.7 Killing Spinors Because oftheimportance ofaknowledge ofthegeodesics ona manifold aninteresting problem ingeneral relativity isthedetermination'. .1! .=iii!ii"-.-"E5"'- :r_ ._ 21"-!'.I-1 I .11‘:1-TIE‘I F "rip ;‘=_r*l?.-' '.--;:1.-"»- E /-YE-.-. I I I I iI i II _’_._.|_..__:§€__.___:_._:.-I-i=4---‘F?‘*4;-= .._¢_.-_=-..=--=.-"...;KILLING 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 A,satisfies sxw=1331;» (10.7.1) forallvector fields X,issaid tobeaKilling spinor. The name arises from thefactthatsuch spinor fields canbeused toconstruct conformal Killing vectors. Animmediate consequence of(10.7.1) isthat aKilling spinor isaneigenspinor oftheDirac operator, $1);=n/ii,/1. We have already noted inthesection above that onacompact Riemannian manifold, Amust bepure imaginary. Excluding thecase inwhich the signature ofthemetric onMis(p,q)with peven andqoddthen there isanHermitian symmetric product oncoiriplex spinor (orsemi-spinor) fields with 5*asadjoint involution. Let l[)~i)€ theadjoint spinor with respect tothisproduct. Then areal1-form Kisgiven by p-...,’ 1'"-.4 1(_5“l(q;1p)_ (10.7.2) Wecanexpand thisinabasis {e"} as Rd:y(I(]"iU]Fea)ea :y0(1ﬁ/;‘?¢i‘P)@a :(‘pi 3411090 SO 1'?*=(1/4,@..w)*@" =(4.11,4»)-4“=(1/4ewe“ and'1?isindeed real.Bydifferentiating (10.7.2) vii?=r.<sXI~I7 +1»§;.TI»)= rI<»I1'?1»I7 +4%)) =((11%8../U711») +(/13¢» @..1I»))@“ =((111./Ie..If1I») +(111.1I*[email protected]»))@“ =2Re</I)(w, 11))?+3i1m(/l)(1I1»(@n /\3f)w)@“ so (VXIZXY) +(V1/1’5)(X) =4R<‘>(l)(1I/» 1I1)s(Xi Y)- Using Killing’s equation, (6.13.3), wehave §£1<8=4R@(l)(iI»» w)a- (19-7-3) 302 SPINOR FIELD EQUATIONS Ifwetook aHermitian-symmetric product (,)with 537*asadjoint (the signature does nothave poddandqeven) then ifI?istheadjoint with respect tothisproduct then I"-_r K-sPl(1Il){l)) (10.7.4) isareal1-form. This satisfies are=-41m0)<4. 1/1)a~ (10.7.5) Exercise 10.11 Show.that ifIll)isaKilling spinor andKsome Killing vector field then £141/J isalsoaKilling Spinor with thesame eigenvalue PI. The existence ofKilling spinors onaRiemannian (asopposed toa pseudo-Riemannian) manifold necessitates interesting integrability con- ditions. Wefirstnote thatthesetoffirst-order differential equations for thecomponents ofIpgiven 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,Y)1/1=—iI2[X. i"]ll) vx,YEFTM. This canbewritten interms ofthecurvature 2-forms, using (10.3.20), as %e“(X)eb(Y)R,,l,1li =—/I2e“(X)eb(Y)[e,, ,el,]ip or Rabi}! =-4/l2e,,l,i)1. (10.7.6) Clifford multiplying bye“produces theRicci forms ontheleft-hand side: Pl,I/7 =~—4)I2(n —1)el,i/1. Now ifAisareal 1-form such that All)=0then certainly A2111 =0. ButA2=g(A, A)andsoforapositive-definite metric wemust have A=0for1;’)non-zero. Thus theabove integrability condition isthat Pb : -4120’! — 1)€b andthemanifold must beanEinstein space with curvature scalar given by 91=—4n(n —1)/12. (10.7.8) So/Imust 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,= Ral,+4A2e,,l, andhence (10.7.6) becomes'.‘- 45:?’‘HI. - -=.>:'_=.'."-" L51*-:-=.[-"" ._ 1'--"5.1 ;'L". I I-.e-,ieIIi.- I-':'-‘?%?}1‘”1 , II5 ':1 i ‘I17 _._-ir .- -I...- . :'I:’-ii ' .-Ia‘_;--lh? 'L':"'n" | "j..=-_5lr'-F’ KILLING SPINORS 303 Togofurther 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 Vxcab :CCbCUca(X) +C(lcCUCb(X) Differentiating (10.7.9) andusing (10.7.1) and(10.7.10) gives {Cpbwpa(Xc) +Capwpb(Xc)}w +Acpbeclfj I The first two terms vanish by(10.7.9), and sofor/1ab0wehave C,,l,el.I/1 =0.From (10.7.9) wehave el.C,,l,q) =0andsosubtracting these gives i,.C,,l,I/1 =0andhence (3,,=0. (10.7.11) Together (10.7.7), (10.7.8) and(10.7.11) show that Ral,=—4)I2e,,l,, that is,Mhasaconstant sectional curvature of—4/I2. Hence theonly locally symmetric Riemannian manifolds such that (10.7.1) hasasolution for /I940 arethestandard sphere, inwhich case /Iisimaginary, ora hyperbolic space with /1real, oraquotient ofthese spaces byadiscrete group. 10.8 Parallel Spinors Aspinor field I/1isparallel if SXIp =0 VXE PTM. (10.8.1) Thus aparallel spinor isaspecial case ()I=0)ofaKilling spinor. Not surprisingly Mmust betightly constrained ifitistoadmit aparallel spinor. Adiscussion ofparallel spinors necessitates abrief mention of Kahler manifolds. Atensor field Jel"T[M isanalmost complex structure onMif ﬁxE](J(X)) =—X vxePTM. (10.s.2) ARiemannian manifold (M,g)with analmost complex structure J thatisanisometry. g(]X,JY)=g(X.Y) vx,Yerrivi (10.8.3) andisparallel VXJ =0 VXe FTM (10.8.4) 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 E;ofthecomplexified cotan- gent space by x1lI1,, =0 forxET’j,MC iffxejg. (10.8.6) Wehave T";_.,M“3 =E;®,9;where x*E}; ifandonly ifxegg. Soa non-vanishing pure spinor field assigns amaximal isotropic subspace to thecomplexified cotangent space ofevery point. Let59+and§"bethe spaces ofcomplex differential 1-forms such that xe}*ifand only if xlpeﬁg. Given thesubspaces 9+and ,9", determined bythepure spinor, wecandefine analmost complex structure Jby Jx=ix Vxe$1’ (10.8.7) Jy= ~iy Vyeﬁ‘. (Note that wehere think ofJasanendomorphism ofthecotangent (rather than thetangent) space.) Since ithaseigenvalues iithen Jis certainly analmost complex structure, and since complex conjugation interchanges 3+and,9‘itisareal tensor field. Since Jpreserves the isotropic subspaces 3+and35‘, then tocheck that Jisanisometry we need only consider themetric evaluated onanelement of§*and of §l_.Letx69+ andy6}" then g(Jx, Jy)=g(ix, —iy) =g(x,y)andso Jsatisfies (10.8.3). Since 1,11isparallel then thesubspace $1(and hence E‘)ispreserved under covariant differentiation. Forifxi/1=0andipis parallel then VXxIp =0and hence Vxxe$1Vxe§l+, VXE PTM. Since covariant differentiation commutes with complex conjugation then italso preserves gli Now ifxe_§l+wehave Jx=ixand hence (VXJ)x +J(VXx) =iVXx. Since VXx e,§l+ wehave VXJx =0and VXJ.r* =0,hence VXJ =0.Thus wehave established (10.8.5). Wecanusethemetric toconstruct a2-form outofanalmost complex structure satisfying (10.8.3). IfJ=Jabe” ®Xl,then theusual index--.-I2\- . J!--'._,.‘-.:_. I ‘$1 *4 ;_._ _=,.... .-.'I.== -. ..-----'1=.\.=._-.-;.*- ,1?--_-:\g S‘--ét ..._-,_-4..-___¢_,-.__,- jPARALLEL SPINORS 305 lowering rulegives J,,l,=g(JX,,, Xl,). IfJsatisfies (10.8.3) then g(./xl, x,,)=—g(JX,,, J3Xl,) =—g(X... JX4)=—s(JXn XG) andJ,,l,=—Jl,,,. The 2-form QE%_]abg”b (10.8.8) iscalled theKahler 2-form. Ifxisany1-form then Jx : I gl(Qx)_ We showed above that aneven-dimensional Riemannian manifold admitting aparallel pure spinor isaKahler manifold. IILthiscase the Kahler 2-form canbeconstructed outofthespinor. If1/)denotes the adjoint spinor with respect totheHermitian spinor product whose adjoint involution is5*then areal2-form Fisgiven by F=g/t2(ll)) {j])_ (10.8.10) Forany1-form x 50111175) =yliilllﬁx _3’0(i1lllll)x} =3’0(i1ll1llX@n)@a "'500(i1llT)x and 8dm@ha)=amen%nwV)+¥%wMNMa—@aD %50()(i]~l}'2;-xea) :8(/Y, @n)5P0(i1llT) ‘i500(illllHll@nX) =go.e.)rn<i1»iI‘I) —i5°iI(iX1I”l7@4)- Ifnow 1})isapure spinor andxe§l+, asdetermined by(10.8.6), then thelastterm intheabove vanishes. Thus forx6§+ %wn=8awVn=Knwn~ Since (Ill,Ip)>0for1,11#50theKahler 2-form Qrelated tothealmost complex structure Jof(10.8.7) isgiven by Q=M, (10.8.11) (1/1,11/) Byonly considering parallel pure spinors wehave been able tousea basically algebraic argument toseedirectly that Mmust beaKahler manifold. IfMiseven dimensional andorientable, with dim Q6then ifMadmits aparallel spinor then itadmits aparallel pure spinor. IfM isorientable with ll)parallel then theWeyl spinors §(1iz’)1p arealso parallel where Ifisproportional tothevolume form onMsuch that ‘Z2=1.ButfordimMQ.6allWeyl 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 fortheexistence ofparallel pure spinor fields. The existence 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 that foraRicci-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 onthis sub- 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. -.:'1‘-3Appendix 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) [3].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 that ab=c. (ii)Theproduct isassociative, (ab)c =a(bc). (iii)There exists anidentity (orunit element), denoted 1,such that a1=la=a VaeG. (iv)Every element ahasaninverse a”, aa"1 =0710 =1. When agroup consists ofafinite number ofelements then this number iscalled theorder ofthegroup. Ingeneral thegroup product isII 308 APPENDIX A notcommutative, ab#5ba.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 ifuvEH Vu, vEH,u"EHVuEHand 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~bifb=ahforsome hEH. The equivalence class ofaiscalled theleftcoset ofG,relative toH, generated bya.Inanobvious way wedefine right cosets. Foraspecial type ofsubgroup thecosets inherit agroup structure. Asubgroup His called normal (orinvariant) ifghg"1 EHVgE G,VhEH. The nota- tion H<:Gdenotes that Hisanormal subgroup ofG.Itfollows that theleftandright cosets relative toanormal subgroup areequal. These cosets form agroup under theproduct defined by[a][b] =[ab]. Since [a]=[ah] forhEHthisdefinition 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,=Z/H. Ifmisanyinteger then m=qn+r,where 0ér<n,andsoevery element ofZisequivalent toapositive integer lessthan rt.The class ofthesum oftwosuch integers isrepresented by their sum modulo amultiple ofn.Forexample, Z3hastwoelements, [0]and[1],and[1]+[1]=[2]==[0].(The notation Z,willbeused to denote anygroup isomorphic tothese quotients. Forexample, theset {1,—1}forms agroup under multiplication, isomorphic toZ3.) Roughly speaking ahomomorphism isamapping between groups that preserves thestructure. Let rpbeamapping from GtoG’, then cpisa homomorphism ifcp(ab) =go(a)rp(b). The product ontheleft-hand side isthat ofGwhilst theproduct ontheright-hand side isthat ofG’.If every element ofG’istheimage ofsome element ofGunder go,then qt?]‘ipe j F l. .I_..id._ _.,_-_..- iI IF APPENDIX A 309 iscalled surjective (oronto). Ifnotwoelements ofGgetmapped into thesame element then cpiscalled injective (orone-to-one). Amapping that isboth injective andsurjective iscalled bijective. Groups that are related byabijective homomorphism arecalled isomorphic, and we write G’=G.Ingeneral ahomorphism cpisnotinjective, andtheset ofelements inGmapped onto theidentity ofG’iscalled thekernel of cp(kercp).Thekernel ofrpisanormal subgroup ofG,andwehave rp(G) =G/ker go. (A1) (This isknown asthefirstisomorphism theorem.) This may beproved byintroducing amap (D. (D:G/ker go——> qa(G) Ial*""">‘1’(IﬂI) =(P01)- Theproof consists ofshowing that notonly does such adefinition make sense, but<1)isabijection. Thefollowing Isusually known asthesecond isomorphism theorem. IfN<1Gand A¢GSuch that N<I A(IG then G/N-—-=G/A. (A2)A/N Theconditions onthesubgroups arejustsuch asarerequired forthisto make sense. Theequivalence class ofainGgiven byNiswritten [a]N; [a]Abeing similarly defined. The proof of(A2) isestablished by introducing amap cp, cp:G/N i> G/A Ialiv P‘) (P(IQIN) :Ia]A- Notonly issuch amap well defined butitisasurjective homomorphism with kernel 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, k3)aretwosuch pairs then we may define their product by(hl, lCl)(h3~ k2)=(l’lIl’l2» klk2)' IfG denotes thegroup formed bysuch pairs then Gisthedirect product of HandK,written G=H><K.Anisomorphism from agroup toItself iscalled anautomorphism. Ifgoand ill)areautomorphisms ofGthen their product may bedefined by(rpi/1)(a) =rp(t/1(a)). Under thisproduct thesetofallautomorphisms ofGforms agroup, AutG. IftIsany element ofGthen wehave aTintheautomorphism group given by r(a)=tat“. Such anautomorphism iscalled aninner automorphism. 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 for(Ul002EQ al,a2EGwedefine (al, o)l)(a2, 0.13)=(alwl(a2), (1)1602). With such a Pmducl Wehave (61,60)” =(a)“(a“1), of‘). The ordered pairs under thisproduct form thesemidirect product ofGand Q,Ksay written K=GQQ. Aring hastwo binary operations, addition, denoted +,and multi- plication, denoted byjuxtaposing elements. Under addition aringforms anAbelian group, theadditive identity being called thezero element. l\/Iultiplication isassociative (unless specifically stated otherwise) and distributive over addition, (a+b)c=ac+bc c(a+b)=ca+cb. Acommutative ring isoneinwhich multiplication iscommutative. The setofelements that commute with allother elements under multiplica- tion iscalled 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'1, that isaa‘1 =a“a =1. Aring in which every non-zero element isregular iscalled adivision ring. We havealready noted that theintegers, Z,form anAbelian group under addition; with multiplication they form aring. Similarly with multiplica- tionbeing defined modulo nthegroup Zl,forms aring. Afield isacommutative division ring. (Sometimes anon- commutative division ring iscalled askew field.) Familiar examples of fields aretherational numbers Q,therealnumbers IBandthecomplex numbers C.Forpaprime number then anexample ofafield with a finite number ofelements isZl,.Afield Fissaid tobeofcharacteristic pifthere isaprime number psuch that a+a+a...+a=0 VaEF. pterms. Inthiscase Fcontains Zl,asasubfield. Ifthere isnosuch pthen Fis said tobeofcharacteristic zero, andinthiscase itcontains therational numbers asasubfield. Weshall really only beconcerned with thezero characteristic fields IRandC.The complex numbers have theproperty ofbeing algebraically closed, which results intheproperty thatweshall 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-ii:. F- -*\ni3I;+- ' .aw:-''"-‘Z"ti .-r-.<l- _.._ -\..-- sr “F {J 144- =.-..;,=_. 1--.' F-‘>.‘J,7.2' -.->i..-- ',iia"IIa‘i1IlaI~_=f‘= -I 2*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: (i)(Mr=M744)(ii)(JI+u)x=/Ix+ux /I(x+y)=/Ix+)Iy V7I,,uEF,x,yEV. (iii)1x=x,where 1istheunitelement ofF. If{xl} isasetofvectors such thatx=El/llxl forAlEFthen xissaidto bealinear combination ofthexl.Asetofvectors iscalled linearly dependent ifanyonevector canbewritten asalinear combination of theothers. Conversely theset{xl} islinearly independent ifZ,-Jllxl =0 implies that allA’arezero. Asetofvectors {xl} 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, n el. 2 lI'=l n fl?" Z EB,-'6,-. I1 Substituting either expression intotheother gives 2 Z éjk i=1 H 2AI.kBkt =51.7" k=1 where theKronecker 6,-ltakes thevalue zero unless i=jwhen itsvalue isone. Thus the coefficients relating thechange ofbasis can be displayed asanon-singular n><nmatrix, 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 we1; W -1' _.1 312 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, UF)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 x=u+w,ueUweW. Ingeneral, such a decomposition ofxinto elements ofUand Wisnotunique. Itis, however, when UOW=0.Inthiscase thesum issaid tobedirect, written U(-9W.(Later weshall reserve thisnotation forthedirect sum ofalgebras, allvector space sums being direct unless stated otherwise.) Foranysubspace Uthere isasubspace Wsuch that V=U(BW;W being called the complement of_U in V. Obviously dimV=dimU+dimW.Any subspace Uisanormal subgroup under addition. The quotient group V/Ucanbegiven alinear structure by defining /l[x] =[/ix], where thebracket denotes theequivalence class of x,with x~yifx=y+uforsome ueU.With thisstructure V/Uis 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, goisalinear map from VtoWif (PW+My)=/l<P(-Y) +may) Vx.yEV,/1»#6F- Itfollows that every linear map sends thezero element ofVtothat in W.Alinear map may becompletely determined byspecifying itseffect onsome basis for V.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, .S€(V,W). Since any such linear map may bespecified bya dimVXdimWmatrix wehave dim.§E(V,W)=dimVdim W.Alineari':'=TI3":-‘-~:1‘ if :=‘ . 1-5-ii.'_:_-if:_- - .',:-J}-.'..i.=.L,.,__,,§.- ':<,,._'. niﬁlla--.-,.i.;'._ lAPPENDIX A 313 map from VtoVwillbecalled alinear transformation, orendomorph- ism, andwewillalso write EndVfor§E(V, V).Such linear transforma- tions can bemultiplied bycomposing maps, (cp1p)x =cp(1/1(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*=dimV.Foreach basis {e,-} ofVwe may establish anatural dual basis {e*l} ofV*such that e*"(e),-) =6‘,- Vi, j.(Note the conventional positioning ofindices.) Ifarbitrary elements bandBinVand V*respectively areexpanded indual bases as b=b"e,- B=B,-e*" (summation convention) then B(b) =B,-b". Inparticular, e*’(x) =xiexpresses 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{V,-} isasetofnon- intersecting subspaces such that V=E,-V,» and kinjectively assigns an element k(i) ofGtoeach V,-.Giscalled thegroup ofdegrees. Elements ofV,arecalled homogeneous ofdegree k(i), denoted degx =k(i) VxeV,~. Since thezero vector liesinevery subspace itishomogeneous ofevery degree. Paticularly when G=Zwewill 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 {V,-} and {W1} then alinear map cpiscalled homogeneous ofdegree kifthere isanelement keGsuch that q0(V,-) CW,-H, Vi6G.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 oAbilinear 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)=g(y, x)Vx, yEVandnon-degenerate ifg(x, y) =0Vyimplies that x=0.Weshall beprimarily concerned with the case ofF=IRwith 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, {ey}, inwhich g(e,-, ej)=17,-jwhere n,-I=:51if i=jorzero 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)) = 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 x6Vanelement Ii‘EV*bytherulethat ﬂy)=g(x,y) VyEV- Weshall refer tosuch anitasthemetric dual oradjoint ofx(with respect tog).Ifthecomponents ofginthebasis {e,-} aregiven by g,-j_=g(e,-, _e!-)and itisexpressed inthedual basis asIE=2,-6*" then X‘,-yl =g,~,-x'yl. Since thismust hold forallylitimplies that 55)=g,-1,-x". Frequently alowering convention isadopted forindices inwhich xi,Eg,-1-xi, such thatifx=x"e,-then if=x,-e*". The metric g:V ><V—-> 1B naturally induces ametric g*:V*><V*—>IBbytherule s*(X3Y‘)=s(r,Y) Vt.yEV~ If“the components ofg*inthe basis {e*‘} are the numbers g*'l=g*(e*", e*l) then g,,-g*l" E5",-. Thus thecomponents ofg*form theinverse ofthematrix ofcomponents ofg.The map '“from VtoV* isinvertible and wedenote itsinverse byD.Thus ifBeV*with B=B,~e*‘ then BE=Ble,» 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.ItR ..~r=;>.<-->",'i's'$~' -:' -;I-'-It.'..7;',.'-1;.=_=l.'-.'-"'_?_‘¢'.|;!7;';;;.---.-¢_;-,_1.-._.._:;>i\‘’-*.:'.e'~'*;==-5;;:.;;5;i-q','¢'.-1::P-';_.-_.-/_-' .3iii-‘I! '. l APPENDIX A 315 Aswell asrealvector spaces weshall beinterested invector spaces over thecomplex field. Invarious ways thesame Abelian group canbe endowed with both an1B-linear structure andaC-linear structure. When speaking ofthedimension ofsuch avector space itisimportant to distinguish between thetwolinear structures, andwhen there ispossibil- ityforconfusion weusedimm and dimg todenote thedimensions associated with thedifferent linear structures. Similarly wespeak of 1B-linear and C-linear transformations when there ispossibility ofconfusion. IfVisarealvector space then anendomorphism Jsuch thatJ2E-1,where Iistheidentity map, iscalled acomplex structure onV.Such aJcanonly exist ifVisofeven dimension. Acomplex structure can beused todefine multiplication ofelements inVby complex numbers. For/I+ineC,A,iteIR,wedefine ()L+iu)x=/lx+uIx VxeV. w Such aC-linear structure turns Vinto acomplex vector space V,the complex vector space associated with V(and J).We clearly have dimCV =%dim]BV. 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)"l"(352, Y2):(X1+X2»Y1+Y2) /l(x,y)=(/ix,/iy) /lelPi. With thisstructure theordered pairs form theexternal direct sum ofV with itself, VG)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°3E(VC-3 V),anddimCVC =dimlg V.Anelement ofVCisanordered pair ofelements from V.But since (x,y)=(x,0)+i(y,0)weshall write x+iyinstead of(x,y).Ifthen /1+iueCthisgives, asonewould expect, (1+iu)(x +iy)=/ix-uy+i(/ly +ux). Ifnow westart with acomplex vector space Ethen weautomatically have anassociated realvector space, EP‘,since IRisasubfield ofC.This real vector space comes equipped with anatural complex structure, multiplication byiinE.With thiscomplex structure E=(ER)? Agroup homomorphism rpbetween complex vector spaces iscalled conjugate linear ifq0(/Ix) =/'t*tp(x) for/1ECand /1*denoting the complex conjugate. Inparticular, if(pisanR-linear map onareal V thathascomplex structure Jsuch that, cplE—-Jcp then cpisaconjugate linear map onV.l 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- t1on.‘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 rpandprespectively then these are termed equivalent ifthere isanisomorphism S,mapping VtoW,such thatthefollowing diagram commutes forallg6G,x6V: (p(s) X"-——> tP(s)X Si J,S i.e.S<;0(g)S_1 Ep(g). p(s) StE P(s)$r Analgebra over thefield F,.vsl(F), consists ofavector space over F together with analgebra product, called multiplication, which satisfies a()tb +uc)E/lab+uac Va, b,ceat, VA,ueF andsimilarly formultiplication ontheright. Weshall callthedimension ofthe‘vector space the dimension ofthe algebra. The algebra is associative ifitsproduct satisifes a(bc) E(ab)c. Thus equivalently an associative algebra sﬂ(F) isaring atthat isavector space forwhich t1”(ab) Ea(0rb) E(a/a)b Va, bEﬂ, Va/6 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 atafinite-dimensional associative algebra over F,some arbitrary field; although inthisbook weshall only beconcerned with therealorcomplex field. Iftheunderlying vector space ofanalgebra .961isthedirect sum oftwo subspaces %then wewillwrite stE +%.These subspaces need notbesubalgebras, bywhich wemean avector subspace that isclosed under thealgebra product. The centre isanexample ofasubalgebra.:iiii:-5*? -E1F‘:-1;..,_'--..»,-.<s=--.V-1..;. .<_-i. - Ii:-it-ia..""_{.f§,=_|;_E.‘APPENDIX A 317 Forsubspaces 93,%wedefine theproduct 93%tobethevector space spanned byallproducts ofthebases for93and%.If%issome subspace such that atE%%...%then %issaid togenerate .d.Abasis for%will betermed asetofgenerators foroi.Ingeneral thedimension of9’>% willbelessthattheproduct ofthose of%and%.Infactwehave If {c,-} iE1,...,sis abasis for % then dim93%Edim93dim% iffEfzld,-c,» EOford,-693implies all d,-arezero. (A3) Forif{by} jE1,...,risabasis for95then 93%isspanned bythe setofallproducts bl,-c,-. Sodim95%Ersifand only ifthese areall linearly independent, thatis,if t:0 EM;"’>,"t-"-::___U"'<5 forA,-I»EFimplies all/1,,EO. N Ford,~E2;,/1,-I-b 1-thisisjustthestatement oftheresult. Asaspecial case wehave, forsome non-zero aesi,asﬂ.Eallifandonly ifthere is nonon-zero bsuch that abE0.The above result enables ustomake thefollowing simple observation, towhich wewilllater refer. Ifthere isanelement bsuch thatabE1then bistheunique inverse ofa. (A4) Itisobvious that ifahad aninverse then itwould beunique. Given abE1wehave abstl Es4.But absd Casdsowemust have asﬂEail, that is,from (A3), there isnonon-zero dsuch that adEO.Suppose there were acsuch that bacEc,thatisbac~—-cEdwhere dEO.This implies that abac —acEad.If,however, abE1then 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 atisthedirect sum of algebras %and%,sitE%®%, ifwehave avector space direct sum and 93%E%93E0.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 thatséllsil CI.Obviously ideals are subalgebras. Thus thecomponents ofareducible algebra areideals. Suppose oiE%+%,then wedefine anequivalence relation indby a~bifaEb+cwhere ce%. Wedenote theequivalance class ofa 318 APPENDIX A by[a].The elements of.94form anAbelian group under theoperation ofaddition; thisgroup may bequotiented bydefining [a]+[b]E[a+b]. Theequivalence classes aremade intoavector space bydefining 7L[a] E[ha] foritinF. Theobvious waytotryandmake theequivalence classes intoanalgebra isbydefining lallbl=[abl- If,however, c,de%then lallbl=la+Cllb+dl andsoforconsistency wewould need [ab] E[ab+ad+cb+cd] that is,(ad+cb+cd)e%. This will betrue foralla,beat and c, d6%if,andonly if,%isanideal. When thisisthecase then what we have described isthequotient algebra ofsailmodulo %,denoted all/%. If sdISaG-graded algebra with anideal Iwhich isaG-graded subspace then IwillinfactbeaG-graded algebra. Asavector space .94/Iinherits anatural G-gradation such that, ifaishomogeneous, deg[a]Edega. This makes all/IaG-graded algebra since dﬁsilallbll =deslabl Edegab Edega +degb Edeg[a]+deg[b].' 4 Analgebra homomorphism isalinear transformation from analgebra attoanalgebra %such that themultiplicative structure ispreserved. That is,ifcpisalinear transformation from atonto 93then (,0isan algebra homomorphism if<p(ab) Ecp(a)rp(b). When thelinear trans- formation isavector space isomorphism then wehave analgebra isomorphism, two isomorphic algebras also being called equivalent, denoted atE93.Anisomorphism from analgebra toitself iscalled an automorphism. Ifanalgebra hasaunitelement then foranyinvertible s themapping at—>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,“> II .-=.--;==»"v<:\,/- 'r--,;-1;.-_.0.»...;.._-.i&_.-'5,'-gag.-.~.-5 .-;3APPENDIX A 319 tp(a) EO,then <p(bac) E(p(b)(p(a)<p(c) EOforallb,candsothekernel isanideal. Inthesame way astheanalogous result forgroups is proved, wemay show that q0(sd) Eat/ker tp. (A5) Adifferent correspondence between algebras may bedefined as follows. Ifuisavector space isomorphism between atandst’ u:sﬂ—~——>d' aIi>a“ such that (ab)” Eb“a“ then sdandail’aretermed opposite algebras and weshall uses4°Ptodenote theopposite toat.Ingeneral s4°P79sd.For thecase when theopposite algebra isisomorphic toatthen st’may be replaced with allinthedefinition 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. If93and%arealgebras 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. Ifsﬂ,93,%arealgebras over Fwith dimensions mn, mandnrespectively such that93hasabasis {b,-} iE1, ...,mwith multiplication table I bib; :2Bt,='t<bt<k %hasabasis {op} pE1,...,nwith multiplication cacq ZZcaqtct then ailisthetensor product of93and%,atE93®%, ifitadmits abasis {a,-P} iE1,...,m;pE1,...,nwith multiplication given by (1,-pa)-q Z k2B,jkCpq,dk,. _.r This criterion for$4tobethetensor product of93and %involves particular bases for93and%,thus there isnow anonus toshow that it isinfactindependent ofthebases chosen. Ifwehave bases asdefined above then wecandefine abilinear map ®:93 ><%Z>s-=1 bi, CplWb;®Cp =£2,-P. Ifnow {bj-}, {c;,} areanybases for93,%then thebilinearity ensuresi.I1 I1 l ll l 320 APPENDIX A'.=:.'?=‘i€'-ii.\i.r---k.:|'..I1J-'-\$|1_.'1|=. 15.‘. W‘ll‘.-: i.-5.;-.:..'jn'»= .=..=..:_¥£é-3‘ 3 ;E..|§{Eh '.>;_'1, -=1:-:> :.- --..,.:'=...r.=-'1; a;'' APPENDIX A 321 thatthesetof{bj-®c;_,} arelinearly independent, andhence abasis for ENE KL1-(eU.®fpq)(e k,®frS) si.Further, if t>;t>;-=2Bi;"1<bi<k and ("'3"t:‘\n('3Qw-_ I I T2CptircrI‘ then wehave (b;<>;<>c;,)(b;®c*,) =23;,-,,c;,,,,.b;,<>@¢;.kr Soindeed thedefinition ofthetensor product isindependent ofthe bases for93and%.Itshould bestressed that thedefinition wehave given forthetensor product algebra defines itonly uptoequivalence. This willbeconvenient later when weshall make useoftheobservation that if93and %are mutually commuting subalgebras ofatwith dimsi Edim93dim% then atE93®%. Intheparticular case that allhas aunitelement itwillalsobetheunitelement of93and%. Afamiliar example ofann2-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 M”. Abasis formatrices oforder nisobviously provided byalltheelements with aunitintheithrowandjthcolumn andzeroes elsewhere. Weformalise thisbydefining anordinary matrix basis tobe {eff} i,j=l,...,I’l 9,-J,-Gk; : 0 k egejk : 9,-k. The identity isthesum ofthediagonal elements, that isIEeU+ ... +2le ,,,,.Matrix algebras have the following simple but important property. :A/lmn(F)' Theproof willconsist ofspotting how tolabel thebasis. If{e,-I-} and {fpq} areordinary bases for../I/tmandJ!/L,,then ifweset efj®fpq : EU where IE(i—1)n+p, JE(j—1)n+qabasis for./1/l,,,®Jl/l,, is{EU} 1,JE1,...,mn.IfKE(k—1)n+r,LE(l—1)n+sthen-".1"'-'rit-=+,._ _. 1-_'=;.§j:=;i'$ -.-=.-.-a.1¢;-"-'?':.*=-=I1r....-.§ '-<'.'.‘J.=.II3"i'I-‘M =5.‘-r.!..:'=-j.=.j.'g.§;: ' ' '-E,-_-';‘-=;L;-I'i.;=,5'i---'_-.';,-=j.;;;=,:.;§. .--:1,-.... .._\=__; 1.5’?_''~'.'->.=.a.!»'*;=:";'='=1-1;. - ‘"..-.=-:éjkéqre il®fps :6jl<6qrE IL 6(,i —l)n+q.(k—l)n+rEIL :5rI<E1r- One reason forthe importance 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 a6Vthen 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 asM:a->aM.Inthiscase wewill usetheconvention that aMN E(aM)N. Normally thislatter notation will bereserved forinvolutions. Addition oflinear transformations isdefined intheobvious way, anditisclear that EndVisatotal matrix algebra. Arepresentation ofanalgebra atisahomomorphism into EndV,for some V.Representations ofalgebras aretermed faithful, irreducible or equivalent using theobvious analogue tothecase ofgroup representa- tions. Ifoiisanalgebra then itiscertainly avector space andthus the algebra Endsél isassociated with it.Wemay putelements of.94into correspondence with certain elements ofEndsil asfollows. Foraeat, L(a) eEndsd isdefined by L(a)d Ead Vdesiil. Itfollows that Lisalinear map from siinto Endsﬂ such that L(a)L(b) EL(ab). Thus Lisahomomorphism, called the regular representation. Ifailhasaunit element then theregular representation is faithful. ForifL(a) EL(b) then L(a—b)dE0foralld,and taking dE1gives aEb.So,inthiscase, theset{L(a)} forallaeatforms an algebra, L(sll), equivalent tosd.Inanobvious fashion wedefine the mapping Rsuch that R(a)d Eda.Then R(a)R(b) ER(ba) andsofor analgebra with unit element R(s5l) Estl°P. Thus L(ﬂ) and R(.<2i) are subalgebras ofEndsi which arealsomutually commuting, for L(a)R(b)d EL(a)(db) Eadb ER(b)L(a)d since oiisassociative. What ismore, ifsdhasaunit element and SeEndsii commutes with allelements ofL(s.il) then Smust beinR(s§l). 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, .94isann-dimensional algebra with identity then Endsd isan l’l2-(llII1€I"lSlOI12ll algebra with L(s4) andR(sd) asn-dimensional commut- ingsubalgebras. Ifthedimension ofL(.sd)R(s4) were n2then Endsd would bethetensor product ofL(s4) and R(sﬂ). Although ingeneral thiswillnotbethecase itisinthefollowing situation. If9)isacentral division algebra then L(91)®R(9)) EEnd9J. (A7) If‘£3isn-dimensional weneed toshow that dim{L(9J)R(9J)} En2.The proof willrequire thefollowing Lemma L(9))R(9J) EL(€£D)u1 +L(9))u2 +...+L(§D)u, where ul,...,usareinR(€D) and thesums aredirect vector space sums. Since the identities ofL(93) and R(€:D) coincide we have R(€ZD) CL(9J)R(9)). Wepick anon-zero element ofR(2D), ulsay, and form L(9))u1. Then either L(9))u1 ER(99) orwecanpick aM2inR(9J) that isnotinL(9J)u1, giving L(@)u2 OL(2D)u1 E0.ForifaulEﬂu; Where a’,ﬁeL(9J) then forf-EEO U2E(/3‘1a/)u1, which contradicts u2eL(9))u1. Proceeding inthis manner completes theproof ofthe lemma. Inthemanner ofthelemma wewrite L(€:D)R(€D) EL(@)u1 +...+L(QZJ)u,. Since foruregular dim{L(€D)u} Enwehave dim(L(§b)R(g_1;)) =ns, with sEn.Ifs<nthen wemay extend theset{u1, H2,...,us}toa basis forR(9J) bychoosing u,+1, ...,u,,.Since, aswestated inthe lemma, R(€D) CL(9))R(§D) ii,,1=Ea/,~u, withii,€L(@).if t -“T31':I -_:-=1:1- -é--: . . _=- :2YT...5 , -'"'?i-. it-*-.1'" -.-....§ __'§_'a -\5-‘tit.:-'11" H >‘i.§_'_?:-_d_,__£APPENDIX A 323 However, since L(9)) andR(9)) arecommuting subalgebras [u,-,)8]E0 Vj5eL(€D) where thebracket denotes thecommutator. Inparticular [u.S'~+-1'15] =0giving 3 Ziaii ﬁlui : Since thesum isdirect, inthevector space sense, wemust have [a,~,f3]E0 Vf5’eL(9J)iE1,...,s. That is,theat,areinthecentre ofL(9J). But L(9J) EQD which is central, sothea/,-must allbemultiples oftheidentity bythebase field F.The expansion ofu,.+1 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 atisasubspace £8such that areC.58.Right ideals aredefined intheobvious way. Itfollows thatsingle-sided ideals aresubalgebras, andsowemay talkofnilpotent single-sided ideals. Thesumoftwonilpotent leftideals isanilpotent leftideal. (A8) Let93and%benilpotent leftideals ofindex aand[3respectively. Then 93+%iscertainly aleftideal. Ifanyelement of93+%israised tothe power kthen itwill bealinear combination ofterms oftheform aEQ1612 ...akwhere thea,-arein93or%.Suppose that pterms in thisproduct arein93,andthatjisthelargest integer such that aIe93. Then ifa),-_1e% weseta,-"la, Ea},where aj-E93, since 93isaleft ideal. Proceeding inthismanner wecanwrite aEbl...bprwith the b,-in93andrin%.Similarly, wehave aEcl...cqswhere thec,-are in% ands isin%. Herep+qEk. Soifk-a'+[J’—1thenp<cr gives qE)6’,whereas q<[3gives pEorandsowemust have aE0. That is,93+%isnilpotent with index nogreater than onelessthan the sum ofthose of93and%.Obviously thisresult isjustasvalid forright ideals.| I l j.I ;l ii-. | I ll I l l I P 324 APPENDIX A If.58isanilpotent leftideal then .4E£8+£8.14isanilpotent two-sided ideal. (A9) Firstly note that .4isindeed anideal; for34.38C.58since £8isaleft ideal andsos4.S8s4 C£8.94, thus .4isaleftideal. Similarly s4s4Cs4and so4&4C£8.94C.4,making 4aright ideal. Aswell asbeing aleftideal §8s4isnilpotent. ForifxG£8.94, xElaforle.58,aEAandx"Ell""‘* Ia where l’Eal,isin£8.So§8s4isnilpotent with index lessthan orequal tothatof.58.Since .4isthesum oftwonilpotent leftideals, theprevious theorem shows that.4isnilpotent. 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 £8isanilpotent leftideal then theabove result and(A9) combine togive £8C(.58+.%sz4) CN.Similarly forright ideals. ‘Before making theanticipated 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 istheonly idempotent inadivision algebra. (A11) Suppose P2EPand Pisnot zero. Then Pisinvertible and P*1P2 EP‘1P, 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 that ifanalgebra does notcontain anidempotent then infactit must benilpotent. Suppose that s4contains anasuch that siak Es14a"_1 forsome power k.Then if93Esi4a"_1,93isaleftideal ofs4,and hence analgebra, satisfying 93aE93and hence 93bE93 where be93isgiven bybEal‘.Sothere must besome Pe93such that PbEb,giving (P2—P)bE0.But 93bE93means that there isno non-zero xwith xbE0andso93,andthus .94,contains anidempotent. So if.94does not contain an idempotent we must have .-<. __-.:-;_; 2 '..I mﬂAPPENDIX A 325 dim(s4a"‘) <dim (s4a"“) forallpowers kofallaes4. The finite dimensionality ofs4means that there must beafinite atsuch that s4a°' E0;inparticular a°’+1 E0.Since this istrue foralla,Nis nilpotent. The existence ofanidempotent enables analgebra tobewritten asa direct vector space sum ofsubalgebras. LetPbeanidempotent in.94, then .S8(P) isdefined tobetheleftideal consisting ofaEs4such that aPE0.Similarly theright ideal 93(P) isdefined toconsist ofallae.94 such that PaE0,andwedefine 9(P) E§8(P) F)93(P). The following theorem gives thetwo-sided Peirce decomposition of.94. IfPisidempotent ins4then s4EPs4P +P.§8(P) +9t(P)P +5P(P). (A13) Alltheterms inthissum arealgebras. Pcd(P) consists ofallae94such that PaEaPEa;P§8(P) consists ofallaes4with PaEa,aPE0; 93(P)P consists ofallaeo4with PaE0,aPEaand ifae.4>(P) PaEaPE0.Soobviously these algebras arenon-intersecting andwhat weneed toshow isthatthey span s4.Toseethiswewrite aEPaP+P(a—aP)+(a—Pa)P+(a-—Pa—-aP+PaP) where each term inthesum liesinoneofthesubalgebras contained in thePeirce decomposition. Elements of5>(P) aresaid tobe(algebraically) orthogonal toP.An idempotent iscalled principal ifthere isnoidempotent orthogonal toit. Wecannow goonestepfurther from (A12) with Every non-nilpotent algebra contains aprincipal idempotent. (A14) Ifs4isnon-nilpotent then itcertainly contains anidempotent. Ifuisa non-principal idempotent then there exists anidempotent vsuch that uoEvuEO.That is,0e.4(u). IfPEu+othen Pisidempotent with PuEuPEuand PvEvPEo. SoifxPEOthenxuExPE0,andif PxE0then uxEuPxE0,that is,.Sl»(P) C5>(u). Infact§(P) must be strictly contained in.Si(u) forve.4(u) butnotin$‘>(P). IfPisnot principal then wesetP’EP+wwhere we9(P). Since 9(P) C.4(u) if thisprocess iscontinued itwilleventually produce aprincipal idempo- tentsince S3(u) isfinite dimensional. Offundamental importance are the primitive idempotents. An idempotent isprimitive ifitcan not bewritten asasum oftwo orthogonal idempotents. Thefollowing could have formed analternative definition ofaprimitive idempotent. Pistheonly idempotent ofPs4P iffPisprimitive. (A15) IfPwere not primitive then PEu+vwith uoEouEO.So 326 APPENDIX A PuEuPEuand PvEvPEvand thus both uand vareinPNP, 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. Every non-primitive idempotent isthe sum ofasetof pairwise orthogonal primitive idempotents. (A16) IfPisnotprimitive then PEu+u,where uand uareorthogonal idempotents. Suppose that 0isnotprimitive, then vEw+xwith w and xorthogonal. Now owEwvEwand uxExvExand so uwEuvw EO,wuEwou EO.Similarly uxExuEOandso{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 N‘EPNP +93 where 93EP.§8(P) +9t(P)P +.¢(P). 93isspanned by.S8(P) and93(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 .Sl(P), which contains allelements orthogonal toP,can contain noidempotent and thus must benilpotent. Since 93(P) and .%(P) consist ofallelements annihilated byleftandright multiplication byPrespectively 93(P)§8(P) C5>(P). SoiflE§8(P) andrE93(P) then (rl)°‘ E0where cristheindex of.9°(P). Since (lr)“"1 El(rl)"r E0then theideal §:8(P)93(P) isnilpotent ofindex lessthat orequal toa+1. Now .§8(P)N =..‘8(P){PNP +P.S8(P) +9t(P)P +r(p)} =NmmmP+NmNm. Since 9t(P) is aright ideal 93(P)P C9t(P) and since 5P(P) E§8(P) O obviously .¢(P) C93(P) and so .§8(P)N §8(P)9t(P). Inparticular, §8(P)§8(P) C§8(P)93(P). Since .§8(P)93(P) isI7 ;.a:_': '2 -1.-F 5 ,. ,.- . ,5--xv..',-1%,;5".._-;§.if=c.55.-_=-:-:';i.=;j.;I-.=,.j;-'-'._.-"t.-.:;.,.=-*;;‘-'='-2.",-:.. . _:.--=~:=.>-.-1 >=-.::;;.='1*'t‘Er.é.?%i,._;§=-n§.£..;—''-1If2+:-itsé-;'=-=?1i.i-’-"''1.‘-=':I'-E'=i.EI-;!5'¢::_--.7“=~="-"-:"--"-'N, APPENDIX A 327 nilpotent ofindex Ea+1 ifxE.§8(P) (x2)““1 E0,and so.S8(P) is anilpotent leftideal, contained intheradical. Inexactly thesame way weshow that93(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 (ay)““ =ta/P(aPr)"‘ Eay(PaPy)“’. ButPaPEPNP andsoPaPy CN0.Thus ifaistheindex ofN0,Nyis nilpotent ofindex Etr+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 iftheonly -ideals arethezero ideal 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.E .. ...“- ':.§,-:11'--=a:.-:s..=31‘;-=_. ;"'"§’.==I:?='+"=".-".1..--:i; = i15;-'=.;;_=,-,-,5Fl’,.. i 328 APPENDIX A I APPENDIX A 329 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. Let93beanideal ofNand 19;,betheunit in93.The Peirce decomposition ofNis at=i,,.a1,,, +1,,,.%e(i,,,) +9i(1%)1g,;, +4>(1%). which isatwo-sided ideal ofN.SoifbE93wehave bEbl+b2with blE5"(1gl) andblE5l(1lll). Then blglEblsince b2isorthogonal to1,1,, but blgll,Ebsoinfact wemust have 3’(1l.;l) E93.Since 4>(1%) is orthogonal to19;,itisorthogonal to93andsoNE93®.¢(19l). Lemma 2 Anon-zero ideal ofasemi-simple algebra issemi-simple. Suppose that 93isanideal inasemi-simple N,and that Nisthe radical of93.Then 93N93 CNsince Nisanideal of93andN93C93 since 93isanideal ofN.SoN(93N93)N C93N93which isthus anideal inN;further itisnilpotent since itiscontained intheradical of93. Since Nissemi-simple 93N93E0.Now (NNN)3 C(NNN)N(NNN)and NNNC93so(NNN)3 C93N93, which wehave shown iszero. That is, NNNisanilpotent ideal inasemi-simple NsoNNNE0.Since Nhasa unitelement thisgives NE0and93issemi-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. The components arecertainly ideals andsosemi-simple, and wemay proceed toreduce them. IfNisfinite then wemust 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) LetNE93l(-B ...(-D93, with the93,-simple. The identity ofNcanbe written asasum oftheidentities inthe93,-,1Ee,-(-3 ...€r)e,. Suppose NE%l(-B®%_,then%l,E%lel+%,,e2+ ...%,,e,VkE1, ...,s. If%llE%l,e,- then %,,,-CNe,E93,-andtheabove summust bedirect:';--='5|‘=. 1:;‘4';:.- '§!;‘."-.>'‘-5F'-“'....._,._ .\ -...-.2aala =Z.<a<a,,,a where walla cas,i=1 so%l,isanideal ifandonly ifallthe%l,,-areideals of93,-.Butthe93,- aresimple, so%l,,-E93,-or%,,,-E0.Ifthe%,,areirreducible then fora given knotmore than one%l,,-canbenon-zero anditfollows that the %l,arejustthe93,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 on.9 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 ofEnd V,thepth column say, carries anequivalent representation tothat carried byV. ForifVisdisplayed asa‘column vector’, with abasis {bk}consisting of zeroes except foraoneinthekthrow, then abasis forEnd V,{e,-,-},is formed bythearrays whose only non-zero element isaone inthe intersection oftheithrow and thejthcolumn. Elementary rules of matrix multiplication then give el,-bl, E6,-lb,-. Abasis forthepth column is{el,,,}where kranges over theorder ofthematrices, and e,-,-el,,,E<5,-ke,-,,. Sothepthcolumn, foranypcarries arepresentation equivalent tothatcarried byV. Weintroduce alinear transformation Sthat maps theminimal left ideal, .9“,ofNintothepthcolumn ofEndV: ss»=p(.i)e,,,,. Since 4carries anirreducible representation ofNthen p(.9>) carries an irreducible representation ofp(N) andsop(5’)e,,,, certainly transforms irreducibly under p(N). Butthisisasubspace ofthepthcolumn which transforms irreducibly, soeither Sisavector space isomorphism or p(5i)e,,,, E0.There must besome pforwhich this isnon-zero, for otherwise wewould have p(5>) E0,which cannot besince Nissimple. Soatleast forsome choice ofp,Sisavector space isomorphism between theminimal leftideal .9andthepthcolumn ofEnd V.IffE9 then thefollowing diagram shows theequivalence oftherepresentation carried bythepthcolumn (and hence V)andthatcarried by9: 330 APPENDIX A 1-(Q) f E >11)‘ S1 1S 0(0)p(f)eppi_"_i>p(a)p(f)epp =p(af)epp' Thus anyirreducible representation ofasimple algebra isequivalent to thatinduced onanyminimal leftideal bytheregular representation. Wearenow inaposition 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 the same kernel are infact equivalent. Asemi- simple algebra isthedirect sum ofsimple 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 iffNE9J®./I/1 where 9)isadivision algebra andA/1atotal matrix algebra. (A23) First wedotheeasy bitand assume NE9J®Jl/1. Then Nhasan identity. Letbbeanon-zero element ofanideal .55,then bE2,-,,-b,-,-e,-,- withat least one(bpl,say) non-vanishing coefficient in9D.But bpq = ,-pbe q,- andso 2bpq"18,pbeq,- = That is1CNbN C9,giving NC51*andthus Nissimple. Ifnow Nissimple ithasaunit element 1EEj‘=lP,- where the{P,} arepairwise orthogonal primitive idempotents. IfNl,EP,-NP, then the N,-J,arecertainly subspaces, andareinfactalgebras since they areclosed under multiplication. Multiplying twodifferent algebras gives dtrﬁpk E“tip1Ppdpk =~84ti'$4tk5ip . 1._ 1 , O3APPENDIX A 331 Now NP,-N isatwo-sided ideal, which isnotzero since itcontains P,-, andsothesimplicity ofNgives NP,-N ENandhence -Sig-ljgg-pk 1' Pl'~ﬂ»Pk(l5,'p 2 $4-ikéjp. Inparticular NllENl,-N,-l foranyj.Since PlENll there must be elements e,-l,el,inN,-landNl,-, respectively, such that el,-e,-l EPl.If W6 HOW Cl€flI16 G =9,-161, Il‘l€I1 Pkeij =elféik el,-Pl, =e,,6,,,. This gives etieaq :etiPiPPePq =eiieiqéia =°t1°1i°i1°1q‘5rp Ee,-lPlel,,6,-,, =°t1°1q5ip =e,l,<5,-,,. Inparticular thee,-,-areidempotent. Bute,-,-CP,-NP, with Plprimitive, soP,-NP, contains only oneidempotent, namely P,-,sowemust have ellEPl.Sothee,-,-span atotal matrix algebra Jlttwhose identity is 2ett=2iPt:1 theidentity ofN. i Since foreach kPkisprimitive, Pl,NPl, isadivision algebra with Pl, asidentity. Each Nkl,isanisomorphic copy ofNll, say.Forifa/(1)ENll wedefine cw“)EN,,,, byor“)Eel,la/lllelk. Then fora/(1), B11)ENll (,ll,(1i[,><1>)u<> =ekla/(1)ﬁ(1)e,k =ek1a(1)P1ﬁ(1)e1k since Plistheidentity inNll =ek1a(1)e1kek1ﬂ(1)e1k since thee,-,-areamatrix basis and so(ct/l1)l("3(1))(")E a*("lﬁ(")- This mapping from NlltoNllisobviously invertible andsoindeed wehave anisomorphism. Bytaking thedirect sum ofallelements inNllwith their isomorphic images inallNll,weobtain another copy ofNll, 9) say.That is,ifoil‘)ENllwedefine aE9.3tobe 332 APPENDIX A C1’Ea'(1)(-Ba”) ...®cr(”). Itisstraightforward tosee that 9)ENll; further, elements of‘.-ED commute with alltheelements ofJ1/1.ForifatEQB : if= U‘= G,-la’(1)e l,-6,-J, k =eila/(1)911. =eijejla/(1)911. =eija/(1') =2e,j.a/(kl k Forevery aENseta,-,-1")Eel,,-ae,-,,. Then aijlk) =ekleliaejlelk =ek1aij(1)e1k soifa,-,-EEka,-,-("l then al,E9).Further gage if:2661,} U.=gal). U ‘J bl» 1-1 = ,-,-618,-,-6 1]‘= lltle,-J; : Z Q. i,j i,j i,j Since thisistrue forevery aENwehave NE€D®/1/1. where 9)andJ1/1 areasconstructed intheproof. Theexpression ofasimple NasNE§Z1®Jl/l. cannot beunique. Forif e,-,-isamatrix basis then soise},Ese,1,-s‘1 where sisany regular element ofN.Then aEEll’,-aj-J,-ej-J, with _. _ -1 -1._ -1 -iall—29),,-aejk —Zse l,-sase,-ks —s(s as),-,~s k k that is,aj-,Es9Ds‘1. Itturns outthough that thechoice of9)and./I/1.is unique uptoaninner automorphism like this. Note that if NE9J®./1/1. E9J'®Jl/1 then wemust have 9)’EQB.Forif0tE9D' wecan write aEEl’,-a,-,-e,-J, with thea/,-,-E9),andifaistocommute with JI/1then £1’Eall(ell +e22+...+e,,,,) Eallsince theidentities inNandJill coincide. SoifNE9J®Jl/I. E9J'®J|/I.’ where Jl/1’ES./l/[S-1 then wecer- tainly have 9)’Es§.Ds“1. IfNissimple such that NE€D®Jl/1 and NE§.D’®JlA.' then there isansENsuch that./1/1’Esrl/Ls”, 9)’Es§.Ds‘1. (A24) Inview oftheabove comments itissufficient toprove thatJ1/1'EsJl/Ls'1- Let{e,-,-} i,jE1,...,nbeabasis forJ1/1and {e;,,,} p,qE1,...,m beabasis forA/1'. Without loss ofgenerality weassume mEn.We write Ell ': 26,,-6,, Cl‘),'E@ i.jEl ,-,1;-5;==“-'~'.-=§5.'e.-=--..-l;a__\- r-.=.1,.._,_-»_:I'=i'.l.I'.i<1',f_¢T_<:;_-=.l"\‘_"APPENDIX A 333 with atleast one(c,,,,say) ofthecl,notzero. Ifweset Q=Ci}.-1°1p°i1 (ii) and b=elleql (iii) then aEellNell, bEe’llNell with db = lpejleql H =cilie1112ciieiie in by(i)i,jE1 ...—-I __Cpqcaqell _ell thatis til) : ell. Also (ba)2 Eb(ab)a =belia Eba sobaisanidempotent ine’llNe'll E§D’e’ll; further itisnotzero since a(ba)b E(ab)2 =911, by(iv). Buttheidentity istheonly idempotent in 9)’sowemust have btl : (V) Ifwenow introduce h: 2:8,-ldell i=1 and 8'=Zeiibe 1;" (Vii);'=i then hg : 29,-16261,-6;-ll79l, = Zelldellbell i.jEl iEl n EZe,-label, since aENell i'El H :291191191; bY(iv)i=1 =29I, 31 ii£.ai§§£' -.5.iEl 334 APPENDIX A thatis hg=1. (viii) So/1must betheinverse ofg((A4)) andgh=1.But n n gh :2:6;-1b91J,-9,-16181,-= 26’;-1b811a9'1,~ r,;'=1 1:1 n n =Eeiibaeir =Zea. by(V)-i=l i=1 Since 2*”e’-~—1wemust have m—n,andhence A/t’~./I/L. Infact z=l nT" _- IT 89118-1 :2ei:-lbelpeijeqlaelq P.q=1 =eilbellaelj =ei"; by This completes theproof. Aconsequence ofthis theorem isthe following which wewill frequently use. IfPisanidempotent inasimple .94then P=Z§=1P,- where theP,arepairwise orthogonal primitives, and theuniquely determined riscalled therank ofP.Two idempotents in$4 aresimilar iffthey have thesame rank. (A25) Any idempotent cancertainly bewritten asasum ofpairwise orthogon- alprimitives, thisis(A16). Togofurther weshall use Lemma IfPisidempotent inasimple atthen P.v€lP issimple. Let9]?»beanon-zero ideal in-P.sﬂP. Since 93isanideal inP.<2§lP the left-hand side iscontained in93.But91939.4 isanideal inthesimple .14, andsotheright-hand sidegives P.v.4P. Thus 973=P.s>§iP. Ifatissimple then Ps§lP issimple with identity P.IfP=Zf:1P,- with theP,primitive then PQQP canbewritten asatensor product ofsome division algebra and atotal matrix algebra with theP,asdiagonal elements. The order ofthematrices willthen ber,which wasshown in (A24) tobeuniquely determined. Itwas also shown in(A24) that all matrix bases aresimilar, andsoasacorollary allprimitives aresimilar. If{P,} arepairwise orthogonal primitives then soare{sP,-s“1}, thus similarity preserves therank ofanidempotent. Toseethat having the same rank issufficient foridempotents tobesimilar note thatif P: : i=1 i=1 with {P,-} and {Q,-} being different sets ofpairwise orthogonal primi- tives then wecanchoose matrix bases with either the{P,} orthe{Q,-}M'.-i.I. 1;-?-it.1"J.\.'i:.1~_'?lF_- ..;__.3 an‘. 1:.‘-1 . --/=53!-.._--_-=. _\..APPENDIX A 335 asdiagonals, and(A24) then ensures theexistence ofans:Q, =sP,-s'1 Vi. Thetheorem above applies tosimple algebras. However thefirst part may beseen toapply tothesemi-simple case. ForifPisanelement of asemi-simple atthen P=Q1(-3Q2® ...(-BQ, 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) Ifadiscentral simple then $4=§D®Jl/In where Q11isacentral division algebra, and a4°P=2D°P®./I/L,,°P. The existence ofthe involution of transposition onmatrices shows that Jl/t,,°P =1!/tn, and so.sﬂ®.9.§l°P == €D®QD°P®./I/Lnz, by(A6). Wearenow inaposition, atlast, tomake use of(A7), giving .s24®.sz4°P =EHd@®./Ulnz, that iss&®s$i°P =A/Lmwhere m isthedimension of52¢,andwehave again used (A6). Weextend any automorphism, t,on.94tooneon.9.§l®.s&°P, T,bydefining (ab)T =a’b Vaesﬁ, be.sﬂ°P. Inthe‘uniqueness theorem’, (A24), weessentially proved that allautomorphisms ofatotal matrix algebra areinner. Thus forevery xe.<2ﬂ®.9Q°P, xT=sxs‘1 where ses"<4.®a§i°P, that isa‘=sas'1 forae.94andb=sbs‘1 forbe34°F. Thus smust commute with every element ofs.4°P. Since 521°?iscentral simple smust bein51¢,andsotis innen Sofarwehave assumed that allalgebras areover some field, F, which hasnotwarranted much attention; indeed wehave usually simply referred toanalgebra asailrather than asallover F.Inamoment we shall assume arestriction onthechoice ofF.The situation forthe simple algebras isalsosuch thatwemay regard asimple algebra over F asanalgebra over certain other fields. Ifatover Fissimple then the centre <6isacommutative division algebra, that is,afield. Inan obvious way.94isanalgebra over ‘{%,making adover <6central simple. In thefollowing section wewillexamine involutions ofasimple algebra .94 over Fwhere Fisassumed nottobeofcharacteristic two. (Asstated in theintroduction forthepurposes ofthisbook Fcanbetaken tobeone ofthezero characteristic fields IBorC.) Ifailover Fhas aninvolution Tthen thesetofT—symmetric 336 APPENDIX A quantities forms asubspace SPT. That is,aG9}ifandonly ifaT=a. Similarly wedefine 9;tobethesetofT-skew quantities, andthen we have .94=S";,~+gr. Forifae54, a=§(a+aT)+%(a—aT). The sum isdirect since ifa=aTand a=-—aT then a+a=0which (for characteristic nottwo) gives a=O.What ismore, ifthecentre contains aT-skew qthen s4=SPT+q9’T. Ifqisanon-zero element ofthe centre (ofasimple algebra) then ithasaninverse which isalso T-skew. Ifae9}then a=qq‘1a, and (q"1a)T =aTq'1T =aq“‘ =q‘1a. The T-symmetric quantities inthecentre willform asubfield of<6,6say. Wewillrefer toaninvolution asbeing aninvolution over 6,say, when 6isthesubfield ofthecentre <6leftinvariant bytheinvolution. Ifs4over Fissimple with Jand Tinvolutions over 6then TJisanautomorphism of.94over <6. (A27) IfTandJareinvolutions then TJiscertainly anautomorphism of94 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- ism on<6.Let qbeanon-zero J-skew element of<6then qqT is manifestly T-symmetric, and hence J-symmetric. But (qqT)~' =—qqTJ, which since qisinvertible, gives q”=—qT. So (q+qT)J = —(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 if94over Fissimple then .94over <6iscentral simple gives (A26) awider range ofapplicability than might atfirstsight besupposed. Inparticular, itenables ustoprove thefollowing. If.94over Fissimple and Tisaninvolution over 6then J:a|—>aJ isaninvolution over 6iffthere exists answith s=isTsuch thataj=saTs*1. (A28) First the easy bit. Ifa"=saTs“ then Jza+—>a’isananti- automorphism. Furthermore a”=s(saTs“)Ts“ =s(sT)‘1asTs'*, soif sT=is,Jisaninvolution. Inner automorphisms leave allelements of thecentre invariant. SoifTisaninvolution over 6then soisJ. Conversely letJbeaninvolution over 6,then JTisanautomorphism over <6((A27)). (A26) then ensures theexistence ofagsuch that an:gwlag 6\APPENDIX A 337 H’=(g“as)T =3TaT(s’T)"~ Since Jisaninvolution a=-<1”=aT(sTaT(g’)“)’(g’)“ =s’a"as(s"")“~ Since thisistrue forallawemust have gTg" =/le<6. If/1=—1then there isnothing lefttodo,ifnotthen sets=g+gT=g(1+/I)ands willhave thedesired property. Obviously thechoice ofsuch ansis determined only uptomultiplication byanelement ofthecentre. Afamiliar example ofaninvolution isprovided bytransposition of matrices. Insome ordinary matrix basis wedefine Tsuch thate,-E=e1-,. For some other basis {eﬂ-J-} wedefine Jbye},-J=ej-,-. Tand Jare examples ofwhat weshall callequivalent involutions. Two involutions, VandJ,willbecalled equivalent ifthere issome automorphism Ssuch thataj=asvswl E((aS)V) Sui.Ifaninner Srelates equivalent involutions JandV,related tosome ‘standard’ involution Tby av=vaTv“1 J._ T- a ‘Ia.’ 19 then j=)tsvsT 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 .94such as.94=€D®Jl/t astheWedderburn decomposition of.94.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 sum ofitsradical 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 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.94beaone-dimensional algebra over IR.Then abasis isprovided byuwhere 1.42=Au.If/I=0then .94isnilpotent ofindex two. Ifitab0 then itisinvertible andifP=/Flu, Pisanidempotent. Foranyae.94 wehave a=nP,heIRandI:a|—>itclearly establishes anisomorphism between s4andIR. The real algebra C(18) isatwo-dimensional algebra generated byi where i2=——1.This realcommutative algebra isnotcentral. Ithasthe well known involution ofcomplex conjugation *:ir—>—i. The real quaternion algebra H(IR) hasabasis {1,i,j,k}whose multiplication table isgiven intable A1.Thealgebra isgenerated bythe subspace spanned by{i,j},say. (We note here that theother four- dimensional real simple algebra A/L2(]B) isgenerated by{a,B}where a2=1, /32=1and ab’=-50./. For example, at=e12+em, /3’=e12—e21.) The quaternions arenotcommutative butthealgebra is central. Inthegiven basis, {i,j,k}span the subspace ofvector quaternions, whilst theidentity spans thescalar quaternions. Theinvolu- tion ofquaternion conjugation, q|—>Q,isdefined tochange thesign of thevector part ofevery quaternion. Then qr}isself-conjugate and hence inthecentre. Byinspection qr}isseen tobestrictly positive for non-zero q,sayqq'=8/I2. Then q"1=A'2q andindeed Hisadivision algebra. Suppose that Tissome other involution, then (A28) ensures thatqT=tcjt‘1 where f=it.Since theonly self—conjugate quaternions areinthecentre, togetaninvolution distinct from conjugation wemust have f=—t.Inparticular wedefine E1‘=kqk“ 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 isthatthecommutator [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,bytG=k,kc=t.This auto- morphism must beinner since Hisacentral division algebra andhence central simple. That is,t=gkg“ forsome g,andgil=l‘2§ forsomeAPPENDIX A 339 JteIR.Soifs=/l‘1g then t=sks, which isthecriterion forTtobe equivalent toreversion. Table A1Thequaternion algebra 1 i j k 7;‘!--i|—1p—i 77"-'*-'+-*'-u-doIliav-... ,_|. idW‘" " -1 k q " " —k -1 i Just asitisimportant toknow that anypositive real number 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=qr}. Reversion isrelated toconjugation byif=k‘1@’k, and forany qwe have q”=if/Iqjz, soify=)2then _1<“’yl< .Y1=*-"T (1)M Writing 1+q as 1+q=q"q +q=(1+ q'1)q gives q=(1+q"1)"(1 +q),foranyq.Inparticular, ifyoisaunit-norm reversion symmetric quaternion then r@==(1-t>m‘3l“(1i-yo) _k"1(1+k"y0k)k(1 +ya) by(D U+yH’1+ 2 =(--—---iii) sincek2=—-1. (ii)l1+Y0 l For any qwehave |k‘1qkj2 =k"’qkkqk"1 =k“q£jk =qq= from (i) l1+ Yrillz =|k_1(1+ )’0)k|2 :l1+ )’0l2- Thus (ii)gives yo=x2,forthereversion symmetric xgiven by 1+yX_1..._<?_ l1+Yuli Then forareversion symmetric yofarbitrary norm wecan write y=|yjy0 ={]y|1’2x}2, since anypositive realnumber hasarealsquare root. 340 APPENDIX A Itwillbeuseful tobeable toidentify thetensor products ofthese division algebras. Obviously ]B®IFi =IB,IFi®C =CandlR®H =H. Thealgebra 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(-D(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=ceac. (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 en=%(1+zi)and 922=5l.(1~zi)then ell, e22 are orthogonal idempotents with 1=en+e22. Ifwechoose e21-jell =ezzj and e12=—je22 =—e11j then thee,-,-form anordinary basis for./t/t2(lB), so C(1R)®H(B) =<l3(1B)®Jl/'~z(1B)- (A30) Wedonothave todoanywork todetermine thestructure ofH®H. Thequaternion algebra isacentral division algebra and, since ithasthe involution ofconjugation, H=HOP. Sofrom Theorem 4wehave Han®Hun=rnmn (An) 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 M><R——-->M:(x, q)I——>xq such that x(q1q2):(xql)q2 X(q1+(T2)=rel+Xe; (ii) (X+y)q=xq+rq xl=x (iii) where 1istheidentity inR. __,..._.iI'“.._’_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, .94,then Iisanexample ofa left6-module. If6issimple with .6=93J®Jl/L then Iisalso aright Q2)-module, formultiplication ontheright by9Dwillpreserve theI.In thiscase Iissimultaneously aleft.6-module andaright QD-module, with the.94action being right 2-D-linear, andtheQ2)action being left6-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 6=H®./I/t, then dimHI=r,whereas dimRI=4r. Bibliography Albert A1941 Introduction toAlgebraic Theories (Chicago: Chicago University Press) l 1961 Structure ofAlgebras (Am. Math. Soc. Coll. Publ. vol24) Greub W1978 Multilinear Algebra 2ndedn(Berlin: Springer) Appendix B Vector Calculus onIB3 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 {xi} i=1,2,3yield the familiar Cartesian coordinates x"(p)forpe1B3.Insuch aglobal chart theEuclidean metric tensor isexpressed as 3 g=2dx‘®dx" x’(p) eIR. i=1 The orthonormal frames {X,»}={S/8x1, 3/8x2, 8/8x3} and co-frames {e‘}={dx1, dxz, dx3} areinthiscase naturally dual toeachother. Observe also that df’=8/8x‘. For some problems other non-global charts areuseful. The familiar ‘spherical polar’ chart with coordinate functions (r,6,cp)hasco-domain 8 O<r(p)<0o 0<q9(p)5.27T O<6(p)<rr. The polar chart isrelated totheCartesian chart ontheoverlap bythe transformation ofcoordinates r:[(x1)2 +(x2)2 _|_(x3)2]1/2 __1 1)2+( 2)2]1/2 6Z5"‘[(x‘)2x+ <-»erx+(W11/2 Q9:COS-1[(x1)2 +8(x§)2 +(x3)2]1_/2' Ifwetried tocover thewhole surface r=constant (abO),with a single coordinate chart there would arise anambiguity inassigning I36.»-mwﬂAPPENDIX B 343 coordinates tothepoles ofthesphere. Such ambiguities cangive riseto ‘singularities’ insubsequent calculations, these pathologies reﬂecting only animproper useofcoordinates. Inapolar chart wemay write 3 V . . . . . ‘8x’ 8x‘ Eix‘ )(E-Bx’ Eix’ 3x’ )=-—a +—ae +—-<1 (>9-—a ‘<10 -—agElm ran seq’ Sr“Lee +3(p(p or,since x1= rsinblcoscp x2=rsin6sinq:> x3=rcosél g=dr®dr +r2d6®d6 +r2sin26drp®dq2. Similarly 3 e*=E(@.,®@x,)i=1 3 =Z[(ar/axqa, +(6)6/6)x‘)86 +(ea/axi)a,.]®[(ar/axi)a,i=1 +(86/E~Jx")89 +(@Q0/@X")@q,] 8 8 18 8 1 E9 6) "8r®E9r +,»1ae® ae+,~2Sin2@a<p®a<p' Hence anorthonormal co-frame inthis chart is{El} ={dr, rd6l, rsin6ld<p} with dual (orthonormal) frame E)18 1 8 {Y.-}—{..I SrrS6 rs1nt9 Srp Themetric duals ofdr,d6,diparethelocal vector fields .... 8 ~ 18 ~ 1 8 d-—-—,d6=—-—,d -e . r at r259 (P r2sin26 59¢}? (Observe thatpoints pwith r(p) =0,6(p) =0areoutside ourworking chart.) Ontheoverlap UofaCartesian chart and our polar chart, for fe@(U)wemay write df=(Sf/8x")dx" =(Sf/8r)dr +(Sf/E96)dt9 +(Sf/8(p)d(p. The metric dual ofdfiscalled thegradient off,sometimes written gradf.OnU 4-..,,, _ ¢--...‘, a-.._; a-..._4 gradf —df—(Sf/E9x’)8/8x‘ —(Sf/8r)dr +(Sf/66)dt9 +(Sf/E9(p)d<p ...(1)1,..1_(.§_f.)i ,-.1jar)8'\8rSr,1aeaer2sir136\3(p sup" 344 APPENDIX B Interms oftheorthonormal basis {Y,-} _8f)1(1)gradfn (Gr 1/1+ r86Y2+rsint9 EirpY3’ IfZisavector field onUwemay write Z=$8/8x‘ =§’E9/Sr +$98/E36 +$68/Scp where E’,E’,E9,E66F(U). The ‘rate ofchange off’inthedirection specified bythevector Z,orthedirectional derivative offinthe direction Z,isdefined asZ(f).Interms ofthevector field gradf Z(f)Edf(Z) =s(Z»ﬁll‘)=s(Z,gfﬂdfl Inthree-dimensional Euclidean space itiscustomary touse adot notation forthemetric evaluated ontwo vectors, namely g(X, Y)E X-Y.This casts theexpression forthedirectional derivative into the form Z(f) =gradf-Z. Letusexplicitly compute the*map associated with theEuclidean metric. If{El} isanyorthonormal co-frame with respect tothisgthen, with *1=ElAE2AE3,wefind *E1: EZAE3, *E2 =E3/\E1, *E3 =El/\E2 *(E1/\E2) =E3»*(E2/\E3) =E1»*(E3/\El)= E2 *(E1AE2AE3) =I. Consequently, inthis case, **=1onallforms. The *map for Euclidean 1B3establishes arelation between 2-forms andl-forms. The metric dual, ~,maps 1-forms tovector fields. Thus there isacorres- pondence given bytheEuclidean metric tensor between 2-forms and vector fields onIB3.Given twovector fields inanyg-orthonormal frame, X=§"Y,-, Y=§lY,~, wehave XV/\ 5;=(5162 *§2§1)i71 /\Y2+(5263 *6366372 /\Y3 +(6361 -<§l§3)ﬁYd3/\ T71- Butsince {"Y,-}isanorthonormal co-frame *0?/\Y)=(€1C2— s1c1)'Y'3+<r2t3 -§3C2)S/V1 +(e3§1— &1:3)'Y'i erT*1R3_-I-'—"1-—-—-11-1-P" .-v no Hence the orthonormal components ofthevector field *(XAY) correspond tothecomponents ofthecross orvector product oftwo vectors with orthonormal components (El), (Q4) respectively. Such a correspondence alsoenables ustomake contact with theoperation curl.om APPENDIX B 345 Foravector field VonUe1B3wedefine curlV= Forexample, inaCartesian chart with V=VIG,-: V=Vldxl di7=(alt/2 -S2V1)dx‘ Adxz+(azt/3 -E33V2)dx2Adx3 +(E33V‘ —81V3)dx3Adx1 *di7=(an/2-a,v1)<:n3 +(OZV3-e,v2)<Ix1 +(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=vra,+V986+via, =v1Y,+VQY2+WY, where V1=V’,V2=rV9, V3=rsin6V‘?", then I7=v1E1+V2152+V3E3 =V’dr +r2V9d6 +r2sin26V‘l’d(;0 where E1=dr,E2=rdt9, E3=rsin6dq9. Hence di7=E96V’d6Adr +a,,v*<1<pAar +8,(r2V")drAd6 +8,,,(r2V9)d<pA d6+8,(r2 sin26V*")dr Adtp +6,9(r2 sin26V¢’)d6lAdq2 1=[E9,(r2V9) ~89V’];E1 AE2+[86(r2sin26V‘l’) 1 —a¢(r2V6)] E2A E3 -l"[8,,V’ ~8,(r2sinZevr)];-£5-553 AE1 so *d\7=[Z-3,(r2V"') -a.,v'](1/r)E3 +[a,,v' —8,(r2 sin26V‘l’)]1/(rsin 6)E2 +[89(r2 sin2l9V‘*’) —8Q.,(r2V9)]1/(r2 sin6)E1. 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 and0-forms on1B3.The0-form divVassociated with avector field Vis defined by (divv)=*a*i7. InaCartesian chart *V=Vldxz Adx3+V2dx3 Adxl +V3dx1 Adxz d*l7=(alvl+azvz+83V3)dx1Adx2 Adx3- Butinthiscase *1=dx‘AdxzAdx3so *d*l7=an/1+ an/2+83V3. Exercise Bl Compute divVinthepolar chart above. Thus the operations ofgrad, curl and div inIB3are 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(f/1)=(gradf)/1 +f(g1'a<1 /1) curl(fv) =(gradf) ><v+f(curlv) div(fv) =g(gradf, 0)+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=0. Bycomposing theoperator *dwith itself oneobtains ahigher-order differential operator onforms. Iffe@(lB3) then inaCartesian chart *df=81fdx2Adx3 +E92fdx3Adx1+ 83fdx1Adx2 *(;l*(lf =—'(3%+8%+ 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 d2=0foreach natural basis form, but torevert totheorthonormal co-frame prior totaking aHodge dual. For example, inanypolar chart af=E9,fdr+a,_,fde+a,,,fa¢ =8,fE‘+(1/»~)a,,fE1 +1/(rsin 6)8,,,fE3APPENDIX B 347 *af=8,fE2A153+(1/r)86fE3 AE‘+1/(rsin 6l)E5>,,,fE~1 AE2. l Or,reverting toanatural basis, *df=E9,fr2sin 6d6A dqo+sin686fdcp Adr+(1/si1I6)E9,,,fdrA d6. Now apply dtaking notice ofthefactthatd6Ad8=Oetc: . . 1d*df =(E9,(r2 s1n6l8,f) +89(s1nt989f) +gi-;]—~é(8§,f))drA d6A (lQ9. But *(drA d6Adcp) =1/(r2 sin6)*(E1 AE2AE3)=1/rzsin 6. Thus ﬁnaﬂy 1 1 . 1*Cl*(lf =75@,,(I’25,_f) + 89( S1111986f) + @if. The notion ofaLaplacian can begeneralised toanoperator on p-forms, inwhich case itisusually called more generally theLaplace—- Beltrami operator. Ifarel“Ap(U) then Aa/E I“/\p(U) isdefined in Euclidean 3-space by Aer=(—1)P*1(d*d* ~*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)on1B3subject totheequation d*d =0or,after applying * A=O. Solutions tothis equation define avector field X=dd)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 laws ofmotion. Surprisingly wemust wait until Chapter 6before thenotion ofparticle acceleration isdefined. Suffice to sayhere that amassive particle isendowed with aparameter m,its inertial mass, such thatitexperiences theNewtonian gravitational ‘force’ mdé. 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 =p*1. (NB Both sides ofthisequation eFA3(]R3).) 348 APPENDIX B s Exercise B2 Obtain intheIB3cylindrical polar chart with coordinates (r,cp,2)and orthonormal co-frames e1=dr,e2=rdcp, e3=dzthecomponent equa- tionfortheNewtonian potential <1), (1/r)8,(r8,) +(1/r2)a§, +ago=p.-u»- 1-: REFERENCES 349 References Albert A196l:Structure ofAlgebras (Am. Math. Soc. Coll. Publ. vol24 Dickson L1960 Linear Algebras (Cambridge Tracts) (Cambridge: Cambridge University Press) :3:Kochendorffer R1981 Introduction toAlgebra (Groningen: Wolters- Noordhoff) Jauch JMandRohrlich F1959 TheTheory ofPhotons andElectrons (New York: Addison-Wesley) :5:Cartan E1966 TheTheory ofSpinors (Cambridge, MA: MIT Press) Chevalley C1954 TheAlgebraic Theory ofSpinors (New York: Columbia University Press) Budinich PandDabrowski L1985 Math. Phys. 10L7 Budinich PandTrautman A1986 Lett. Math. Phys. ll315 :8:vanNieuwenhuizen P1983 Anintroduction tosimple supergravity andthe Kaluza—Klein program, inRelativity and Topology I1(Les Houches) I983 (Amsterdam: North-Holland) pp825-932 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. 113934 :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. 110B 206 :15:Brans CandDicke RH1961 Phys. Rev. 124925 Dicke RH1962 Phys. Rev. 1252163 Darwin CG1928 Proc. R.Soc. 118654 Ivenko DandObukhov Y1985 Ann. Phys., Lpz 4259 Kahler E1962 Rend. Mat. 21425 _:Dirac PAM1928 Proc. R.Soc. 117610, 118341 :20: Benn IMandTucker RW1983 Fermions without spinors Commun. Math. Phys. 89341 :21: Duffin RJ1938 Phys. Rev. 541114 Kemmer N1939 Proc. R.Soc. A17391 :22] Milnor JW1963 Enseignernent Math. 9198 (23: Kobayashi SandNomizu K1963 Principles ofDifferential Geometry (New York: Interscience) 1:24] Benn IMandTucker RW1986 inGeometry andSpinors, Trieste Conf. I986, 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 Duff MJ,Nilsson Band Pope CN1986 Phys. Rep. 130 1-142 Nilsson B1986 Class. Quantum Grav. 3141-5‘WIF141F1‘AlI-*3l-5l-I‘P"-*\OO0‘--1O'\-IP;--4P_—A__l I 350 lREFERENCES 9:1Cahen M,Gott A,Lemaire Land Spindel P1986 Killing spinors, in Geometry andPhysics, Trieste Conf. I986 tions. inGeometry andPhysics, Trieste Conf. I986 1:Hitchin N1974 Adv. Math. 141-55 2:Yau T1978 Commun. Pure Appl. Math. 31339-411 ' 3:Shanahan PTheAtiyah—Singer Index Theorem. AnIntroduction (Springer0:Lichnerowicz A1986 Killing spinors according toOHijazi, andapplica- j Lecture Notes inMathematics vol638) Abelian, 308 Chain rule, 135 Acceleration, 203 Characteristic Adjoint involutions, 67,71 field, 310 Algebra, 307, 316 zero, 310 Almost complex structure, 303 Charge Alt,alternating map, 5 conjugate spinor, 96 Angular momentum, 197 conjugation (Dirac spinor), 287 Anti-automorphism (algebra), 319 conjugation (ofspinor fields), 266 Anticommuting spinors, 103 electric, 190 Antisymmetric, 4 Charged scalar field, 241 tensor gauge fields, 260 Chart transformations, 131 Atiyah—Singer index, 306 Chiral spinor, 97 Atlas, 130 Cl‘map, 129 Automorphism, 3 Christoffel symbols, 222 group, 119,308, 313 Clifford Autoparallel, 202 2-forms, 252, 253 algebra, 23 Basis (vector space), 311 ,algebra, (complexified), 60,80 Bianchi’s firstidentity, 213 commutator, 50,107 Bianchi’s second identity, 213 group, 42 Bijective, 309, 312 group (Lie algebra of),51 Bilinear covariants, 93 product (relation toexterior Bilinear form. 314 product), 24 Bispinor, 100 sub-bundles, 276, 306 Boost, 186 subgroups, 46,71 orbit, 187 Clock, 183 Boundary, 125, 168 Closed forms, 188 Brans—Dicke theory, 250 Closed sets, 125 Co-derivative, 189 Calabi—Yau, 306 Coherence (onoverlaps), 263 Central algebra, 316 Co-homologous, 188 Centre (ring), 310 Commutative, 308 Centre, 308 ring, 310 352 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 2 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, I30 Coset, 308INDEX Cotangent 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, 312Directional 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(ti)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, 5INDEX 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 with torsion, 249 Gravitational mass, 206 Gravitational waves (with neutrinos) 293 Group, 307 representation, 316 Gyroscopes, 234 H-module, 60 Harmonic, 190 Hausdorff, 126 354 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 also 5),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 Left ideal, 323INDEX INDEX Left R-module, 341-2 Minimal leftideal, 55 Length (ofacurve), 183 Minkowski spacetime, 181-2 Levi-Civita antisymmetric symbol, MiX6d IBHSOT, 16 15 , Module 340 Lichnerowicz theorem, 299 MOIHBIIIHII1, 186 Liealgebra ofClifford group, 51 Mlllti-i11d6X, 9,27 Lie-algebra-valued p-forms, 240 Mllltililleﬁf, 2,16 Liebracket Multipolei 191 Liederivative onspinors, 271 mform: 10 ontensors, 161 Natural Light-cone, 181 Linear connection, 200basis, 143 dual basis, 313 local basis, d°P°"d@"°“’- 311 Neighbourhood, 124 frame’ 311 Neutrino waves (with gravity), 293 map’ 312 Newtonian 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, 126acceleration, 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, 14,132355 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, 223Semi-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‘ ofspin(3, 1),53 Spin-invariant products, 62 Spin manifold, 262 Spinor bundle, 261 covariant exterior derivative 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, 1,84 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 9’,(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 35s :: Twisted vector rep1... ,31;, ;,_. _vcondition, 237 Twistor, 299 ~"' ‘:.i‘" 5-:Istructure theorems), equation, 298 Two-component formalism, I .us,279 Weyl U(l) spinor, 97,100, 108 covariant derivative, 241 tensor, 226 ofspinor, 270 Wigner time reversal, 94,288 exterior covariant derivative, 241 Witt basis, 107 Unit element (ring), 310 Witt index, 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 ~----.. . - ..__-_ -- =.L..-.. -=- '. - ---I‘-.--E. |-=-="r=--'=. '.|-r-'.:'--"I='-=-%---r- .5: -.~‘I.§: ...'-‘..-1:.. =.=-..-=.='- -.....;=--':--.1.--='--.. -=1-.'...-.- -\- ._;‘_ -- --. .»=~» ._,.w1 1!’: 1* ii?’