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J,.--. .5- :_"_7. -'- ‘~~j P-It ' .-_'_r »|NtsnuAz|onAtE lbtiflflflflfliflvlflfillflflfi ¢§§§$§§@¥§§§flWE$TE$§iV? _, _ rn§@;k§~aa%n";y&a%:ao>i II i -1I 1'?-_:.|_-.-_l_ -_|.1.-L-m_A_| u_-.._._|.._»..._-__m_._-.__.i 1 1 ! .1 i Ii !i E I. i .1 1 ! t| 1 ._.._p.|.|n..-.._-.-....._......__i._..-_1._.-._---Wannat—are—" 01/ ISAS-INTERNATIONAL sc|-|oo|. 6,8a,,,_M* FORADVANCED STUDIES -r" !%}f:}____ 71-} AUG.31.1988 X/K '3=n;a1a%:§+ V1099#351154at *'f"7J-to"in ..‘T, '1-.__ __ -u-1-‘_.-.__-“H ANINTRODUCTION T0THESPINORML CHESSBOARD Paolo Budinich International School forAdvanced Studios -34014 Trieste, Italy and Andrzoj Trautman 1nstytutFizyki Tooretyoznej, Uniworsytct Warszawski -00681 Warszawa, Poland 1988 TRIESTE _ _ _ __ _..F — T _ _____ -— -n-T7 — ----_ 6/88/F.M ANINTRODUCTION TOTHE SPINORIAL CI-IESSBOARD Paolo Budinich International School forAdvanced Studies -34014 Trieste, Italy and And1zejTrautman Instytut Fizyki Teoretycznej, Uniwersytet Warszawski -00681 Warszawa, Poland Abstract, Thearticle contains abrief review ofthefirststage oftheauthors‘ research onspinors associated with higher-dimensional geometries and,inparticular, onthephysical relevance of Cartan‘s simple (pure) spinors. Historical remarks arefollowed byashort description ofthe relation between spinors andnullelements. General properties (grading, bilinear forms, charge conjugation) ofClifford algebras associated withrealvector spaces with scalar products are described andtheirdouble periodicity modulo 8isexhibited. Thelatter gives risetoachessboard arrangement ofthealgebras; itisshown howtherelevant properties ofthespinrepresentation of every realClifford algebra canbesimply obtained fromthose oftherepresentation ofanalgebra belonging tothechessboard. 1..Introduction Spinors --andstructures associated withthem—-areamong thegeometrical notions whose importance wasrecognized asaresult ofresearch inphysics. Foralongtime, theinterest of physicists inspinors wasrestricted tothree- andfour-dimensional spaces (Euclidean and Minkowski). Spinors associated with them have twoorfourcomponents. Recent work on fundamental interactions andtheirunification makes essential useofgeometries ofmore thanfour dimensions. Forthisreason, spinor structures inhigher dimensions and,inparticular, Elie Cartan's "simple" or"pure" spinors, havenowmore chance ofbecoming relevant tophysics than theyhadatthetimeoftheappearance ofthearticle byBrauer andWeyl (1935) andCartan's (1938) lectures. Thisarticle contains abriefreview ofthefirststage ofourresearch oriented towards physical applications ofspinors associated withhigher-dimensional geometries. Afuller account isbeing published under thetitleTheSpinorial Chessboard intheSpringer-Verlag series ofTrieste Notes inPhysics. Itisintended tobefollowed byanaccount ofthespinor groups andstructures, the geometry ofsimple spinors andtwistors, andoftheassociated differential equations. 2..Alittle ofhistory There isaprehistory ofspinors: theperiod oftime, before thediscovery ofthespinofthe electron, when mathematicians considered notions andideas closely related tothose ofspin representations (inthepresent dayterminology). Itbegins probably withLeonhard Euler (1770) andOlinde Rodrigues (1840) whodiscovered newrepresentations ofrotations inthree-dimensional space. Thelatter wrote anequation forarotation (x,y,z)-1(x',y‘,z‘)equivalent to x'=(1+,1-(m1+n2+p2)-luxui (1) 1+;-ip §(im+n) zX-iy u= ,x= , Q)where t gain-n) 1-gap x+iy-z andsimilarly forX‘.Therighthandsideof(1.1)isrational inthecomponents ofthevector (m,n, p)parallel totheaxisofrotation; theangle ofrotation isto=2arctg11-\im2+n2+p2 andtheunitary |- unimodular matrices iUcos§tocover therotation inquestion. Thismaybeinterpreted tomean thatEuler andRodrigues knew thatSpin(3) =SU(2). Formulae forrotationssimilar to(1)were alsoknown toCarlLudwig Gauss (cf.Cartan 1908). 1 Thediscovery ofquaternions byWilliam Rowan Hamilton (1844) ledtoamuch simpler, "spinorial" representation ofrotations: ifq=ix+jy+kzisa"pure" quaternion anduisaunit quatemion, then q-->uqu"1 isarotation andevery rotation canbesoobtained. Thisobservation, which canbeusedtoestablish theisomorphism Spin(3) =Sp(l), wasmade byArthur Cayley (1845) whomentioned, however, thattheresult hadbeenknown toHamilton. Cayley discovered alsoaquaternionic representation of rotations infourdimensions thatwasequivalent tothestatement Spin(4) =Sp(l) xSp(l) (Cayley 1855). Quaternions arenowanimportant partofthestructure ofrealClifford algebras. Inthis context, itisinstructive torecall theview ofLord Kelvin (quoted afterKline 1972): "Quatemions came from Hamilton afterhisreally good work hadbeen done; andthough beautifully ingenious, have been anunmixed eviltothose whohave touched them inanyway... Vector isauseless stuvival, or offshoot from quaternions, andhasnever been ofthe slightest usetoanycreature." TheHamilton-Cayley representation ofrotations in3and4dimensions byquaternions was generalized tohigher-dimensional spaces byRudolf O.Lipschitz (1886) whousedforthispurpose theassociativealgebras introduced byWilliam K.Clifford (1878). Thealgebras considered by Lihtford andLipschitz weregenerated bynanticommuting "units" eawithsquares equal to-1.In E.Cartan's "Nombres complexes: Expose, d'apres l'article allemand deE.Study (Bonn)" thereis adefinition andclassification ofrealClifford algebras ofarbitrary signature (Cartan 1908). Theroad tospinors initiated byEuler andessentially completed byClifford andLipschitz maybedescribed asbeing based ontheideaoftaking thesquare rootofaqundraticform. Indeed thematrix Xgiven by(1.2)islinear inx,y,zandhastheproperty xi=(x2+yz+Z2)1 (3) where Iistheunit2by2matrix; Clifford algebras provide auniversal method ofgeneralizing (3)to higher dimensions andarbitrary signatures. Spinors haveanother parentage, related tothestudy ofrepresentations ofLiegroups and algebras. TheLiealgebras oforthogonal groups have representations which donotlift 2 ("integrate") tolinear representations ofthegroups themselves. Forexample, theLiealgebra of SO(3) isisomorphic toEH3withthevector product playing theroleofthebracket, i=1.=2]==e3.etc. (4) Therepresentation of(4)given byea—->on/2i, where thePauli matrices are O1 O-' 10 1 .- does notlifttoarepresentation ofSO(3), butintegrates toarepresentation ofSU(2), the simply-connected double cover ofSO(3), or,inother words, toatwo-valued representation of SO(3). Cartan (1913) determined allirreducible representations oftheLiealgebras ofthegroups SO(n) andfound that,forevery n>2,thereareamong themrepresentations which donotliftto SO(n). Thisissobecause thegroups SO(n) arenotsimply-connected; thedouble valuedness comes from TF1 = fDI' H>2 andSpin(n) isthedouble cover ofSO(n) which issimply-connected forn>2.Cartan's approach wasinfinitesimal: heconsidered representations ofLiealgebras only. Brauer andWeyl (1935) found global, spinorial representations ofthegroups Spin(n) foralln.Thisroadtospinors maybe called topological: itisrelated, inanessential way,tothenon-triviality ofthefundamental groups 1:1ofthegroups ofrotations. Ithasthevirtue ofallowing ageneralization ofthenotion of spinorial representations togeneral linear groups (Ne'eman 1978). Asamanifold, thegroup GL"(n, IR)ofnbynrealmatrices withpositive determinant ishomeomorphic totheCartesian product ofmanifolds, SO(n)Xn"<"+1>/2. (1) Iltjrefore, forn>2,rt!(GL‘*(n, lR))=Z2andthegroup hasasimply-connected universal cover GL'*(n, FFl)horneomorphic to Spin(n) Xn"("+1>/1. (8) Thegroup GI.."'(n, IR),forn>2,hasnofinite-dimensional faithful representations. Inother words, spinors associated withthegeneral linear group haveaninfinity ofcomponents. They have thevirtue ofnotrequiring, fortheirdefinition, anyquadratic form orscalar product; theycanbe 3 contemplated ona"bare" differentiable manifold without metric tensor. Thetopological approach tospinors ismore general thantheonebased ontheideaoflinearization ofaquadratic form. Theimportance ofthetwo-valued representations oftherotation goup forphysics became clear afterthediscovery oftheintrinsic angular momentum ---spin--oftheelectron (Uhlenbeck andGoudsmit 1925) andthrough thework ofWolfgang Pauli (1927), PaulA.M. Dirac (1928) and many other physicists onwave equations describing thebehaviour offermions, i.e.particles with half-integer spin. According toB.L.vanderWaerden (1960), thename spinor isduetoPaul Ehrenfest. Hermann Weyl (1929) putforward arelativistic wave equation formassless particles described byatwo-component spinor function. Weyl's equation wascriticized byPauli (1933) on theground thatitwasnotinvariant under reflections. Ettore Majorana (1937) introduced another equation, closely related toWeyl's, based onareality condition equivalent totheidentification of theparticle anditsantiparticle. Two-component equations became accepted inelementary particle physics afterthediscovery ofparity violation inweak interactions. Atfirst,spinors baffled physicists who, under theinfluence ofrelativity theory anddespite Lord Kelvin‘s opinion, were becoming accustomed toscalars, vectors andtensors. Inthewords ofC.G.Darwin (1928): “Therelativity theory isbased onnothing buttheideaof invariance anddevelops fiont ittheconception oftensors asamatter ofnecessity; anditisrather disconcerting to find thatapparently something hasslipped through the net,sothatphysical quantities exist, which itwould be, tosaytheleast, very artificial andinconvenient to express astensors". What isspinor? Every physicist uses thisnotion frequently andknows itwell, but amazingly diverse definitions ofspinors aregiven intheliterature. Thedifferences among the definitions ofspinors aremore profound thanthose related tovectors andtensors; forspinors, there aredifferences inthesubstance andnotonlyintheform ofthedefinitions. Geometry andphysics require ascheme todealwithfields ofquantities suchasvectors, tensors andspinors. Tensors ofvarious types arefirstdefined interms ofvectors: forexample, theymaybedescribed asmultilinear maps onCartesian products ofvector spaces andtheirduals. This algebraic definition isthenextended todifferentiable manifolds bytaking thetangent bundle andapplying toitthe"functor“ corresponding tothetypeoftensors under study. Nosuch functorial ornatural construction canbegiven forspinors because therearetopological obstructions 4 totheirexistence onmanifolds. Moreover, the"obvious" algebraic definition ofaspinor space maybeextended ininequivalent ways tomanifolds (Trautman 1987). Thealgebraic definition, maybeformulated asfollows (Chevalley 1954): assume, forsimplicity, that_Visa2m-dimensional realvector space withascalar product go.Thespace of(Dirac) spinors of(V,go)isthecarrier space S0ofacomplex, faithful andirreducible representation oftheClifford algebra Cligo). Since thealgebra C£(g0) issimple, allsuchrepresentations areequivalent andthe2"‘-dimensional space S0isdetermined uptoisomorphism. There areatleast twoinequivalent extensions ofthealgebraic definition ofspinors to manifolds. Werecall them hereforthespecial caseofa2m-dimensional oriented manifold Mwith apositive-defmite Riemannian metric tensor g. (i) Thestandard definition (Haefliger 1956, Borel andHirzebruch 1958-60) ofaspinor structure onM:itisaspinprolongation Pofthebundle F3oforthonormal frames ofcoherent orientation onM.There arebundle maps Z2 l Spin (Zrn) ->P—->M l lu S0(2m) ->Fg—>M (see,forexample, Dabrowski andTrautman (1986) fordetails andreferences). Thebundle 2-—>M ofDirac spinors isassociated with P->Mbythestandard representation ofSpin (2m) inS0= =llI2.Theprolongation Pexists if,andonlyif,thesecond Stiefel-Whitney classofMvanishes. (ii) IfMadmits anorthogonal almost complex structure J,thenonecandefine a"Chevalley bundle" S=ANCA(III®TM) where Nisthetotally nullsubbundle of[E®TMconsisting ofallcomplex vectors oftheform u-iJ(u),where ueTM. Thebundle S-->MhasS0asitstypical fibreandthereisabundle map C£(g)xS—-)8 making thefibreofS—>MatxeMintothecarrier space ofarepresentation oftheClifford algebra Cllgx) associated with(TXM, gx),where gxistherestriction ofgtothetangent space TIM. 5 Thebundles ZandSareinequivalent: among even-dimensional spheres onlythose of dimension 2and6admit bothChevalley andDirac bundles. TheDirac bundles ofspheres areall trivial (Gutt 1986), buttheChevalley bundle ofS2isnot.Allcomplex manifolds admit Chevalley bundles defined bytheir complex structure. Inparticular, thisistrueoftheeven-dimensional complex projective spaces which havenoDirac bundles. Formostpurposes, oneassumes thestandard definition (i).Wehavementioned definition (ii)toemphasize acertain non-uniqueness inthenotion ofspinors onmanifolds. Thelatter definition isclosely related totheapproach tospinors through differential forms (Ivanenko and Landau .1928, Kahler 1960, Graf 1978) andtotherepresentations ofClifford bundles considered byKarrer (1973). 3. Null elements and simple spinors Theapproach tospinors exposed byElieCartan (1938) isbased ontheuseofnull1) (light-like, optical) geometrical elements: vectors with vanishing squares andlinear spaces containing non-zero vectors orthogonal tothespace. Theconnection between spinors andnull elements isoffundamental importance fortheapplications ofspinors inthetheory ofrelativity (Penrose 1960, Penrose andRindler 1984, 1986). Itisatthebasis oftheNewman-Penrose {.1962} ft.-rmaliszn develop-e.i .'...;...1ati_, andsolve Einstcit1's equations. Thediscovery oftnrstors by Penrose (1967) isclosely linked toobservations concerning aremarkable Robinson congruence of nulllines inll-linkotvsici space (Penrose 1987}. 'T‘=.vi:~*.tors have ledtodeep results, such asnew methods forsolving bothlinear andnon-linear equations (Penrose andMacCallum 1972, Ward 1977). Aconnection between spinors andnullvectors canbeillustrated ontheoldproblem of Pythagorean triples, i.e.triples x,y,zofpositive integers suchthat x2+yz-=22 (9) Equation (9)means thatthevector (x,y.z)isnullwithrespect toascalar product ofsignature (2.1). Itisequivalent tothestatement thatthesymmetric matrix 1)Inpuremathematics theadjective "isotropic" isusedtodenote vectors withvanishing square and alsovector spaces consisting ofsuchvectors (Porteous 1981). Physicists refertosuchobjects as "null". Theformer choice issomewhat misleading since theword "isotropy" isoften usedina different context: there istheisotropy subgroup defined bytheaction ofagroup inaspace. 6 z+y x _ X.-=,‘_-s( (10) x z-y isofrank1:detX=0andX¢0.There thusexists atwo-component real"spinor" (p,q)suchthat PX=()0»q) (11)q 01" X=21>q. y=r->2-q’. z=p2+q2- (12) Notonlydoes(12)giveasolution of(9),butevery Pythagorean triple ofrelatively prime integers (x,y,z)canberepresented asin(12)bychoosing asuitable couple ofrelatively prime integers p andq. Asanexample closer tophysics, consider thevectors EandBofanon-zero electromagnetic field, thecomplex vector F=E+iB=(F1,F2,F3), (13) andthesymmetric matrix F1+iF2 iF3 <l>= . (14) iF3 F1-iF2 Itsdeterminant, dfit (D= "|' + vanishes if,andonlyif,theelectromagnetic fieldissimple ornull,i.e..when E-B=0 andE2=B2.. (15) ¢1Ifthisisso,thenthereisacomplex two-component spinor ¢==(Jsuchthat ¢2 7 91 ¢= ()(¢1~ (192)- ¢2 Thespinor ¢E[E2isdetermined byFuptoasignandcanbealsoused toform theHermitean matrix t»,__\l!=()(¢'p¢g)- (16) 92 Equation (16)canbeabbreviated toread \|!=¢(blandthematrix 1|!represented asalinear combination ofthethree Pauli matrices andtheunitmatrix 00=I, 141=kl‘on (summation over p.=0,...,3) (17) Therealvectorrtota‘withC0mp0net'ltS gtvohby(11)isHullwithrespect totheMinkowski scalar product ofsignature (1,3). Moreover , k°=lEI=lBl and k°k=ExB, (18) where (kl,k2,k3)=-k.Simple electromagnetic fields characterized by(15)and(18)playamajor roleinthetheory ofshear freecongruences ofnullgeodesics inLorentzian manifolds; theygive risetoan"optical geometry" andaCauchy-Riemann structure onthespace ofnullgeodesics (Robinson 1961, Penrose 1983a, Trautman 1985, Robinson andTrautman 1986). Toputinperspective these examples, consider thecomplex vector space V=lllzmwitha scalar product gandafaithful irreducible representation _ 7:C£(2m) —->llI(2’“) (19) ofitsClifford algebra Ct(2m). Let¢eS=[Elmbeanon-zero Dirac spinor. Itsdirection dir¢ defines avector subspace ofV, N(dir¢|)=[ue Vl'y'(u)¢=0]. (20) From thebasic property oftherepresentation (19), 'r(u)vtv)+xv)'r(11)=2s(11.v). (21) 8 itfollows thatN=N(dir<|>)istotally null,i.e.every vector inNisnull. Thedimension ofNisnot larger thanm.Anecessary condition forNtobeofthemaximal dimension .misthat¢beaWeyl spinor, i.e.aneigenvector ofthehelicity operator F=imY1Y2...Yzm, where ya=i/(ea) andea(or=1,...,2m)arethevectors ofanorthonormal basis inVembedded in C[(2m). This condition isalsosufficient form--=1,2,and3:there isanatural, bijective correspondence between theprojective space ofWeyl spinors andthesetofmaximal, totally null planes ofthecorresponding helicity. Form24thecomplex dimension 2""!-1 oftheprojective space ofWeyl spinors islarger thanthedimension m(m-1)/2 ofthemanifold 30(2m)/U(m) (23) 1 ofmaximal totally nullplanes. ElieCartan callsaspinor simple (intheFrench edition, Cartan 1938; intheEnglish translation, theadjective pure isused) ifitdefines by(20)atotally nullplane ofmaximal dimension. Cartan shows thataWeyl spinor ¢issimple if,andonlyif, <Bo,ya!ya? yap¢:>=0 (24) forallsequences ofintegers ctsuchthat 1 1sot1<otz<...<0tp52m and Ospsm-1. (25) Here B:S->S*issuchthat“ya-=B‘yaB4anditisunderstood thatforp=0condition (24) reduces to <B¢,¢>=0. (26) Them-form withcomponents given by(24)forp=mcharacterizes them-dimensional totally null plane associated withthesimple spinor ¢. Ineight dimensions (m=4)equation (26)istheonlycondition for¢tobesimple. Herc simple spinors lieona"nullcone" intheeight-dimensional space ofWeyl spinors; aninteresting rrialiry, orsymmetry between thethreeeight-dimensional spaces (vector space andtwospaces of Weyl spinors), appears inthiscase(Study 1903, Car-tan 1925, Weiss 1933, Chevalley 1954, Tits 1959,Porteous 1931,Penrose andRindler 19s6). ‘ 9 Simple spinors canbedefined inasimilar manner forrealvector spaces withaneutral scalar product. Forother signatures, ifoneinsists onstaying within thedomain ofrealnumbers, the situation ismuch more complicated andsubtle. Forexample, ifthescalar product is positive-definite, thenthere arenonulldirections whatsoever andthegroup SO(n) ofrotations acts transitively ontheprojective space lHPn_1 ofvector directions. Forsufficiently highn,however, theaction ofSpin(n) ontheprojective spinor space isnottransitive. The"simplicity" ofaspinor canbemeasured bythedimension ofitsorbit under theaction ofthespingroup: thelower the dimension, thesimpler thespinor. Only partial results have been sofarobtained onthe classification oforbits ofSpin(k,0)andthegeometrical interpretation ofsimple spinors inthose cases (Porteous 1981, Igusa 1970, Popov 1977, Benn andTucker 1988, Budinich 1986b, Budinich andTrautman 1986). 4.. General properties ofClifford algebras Inthispaper, wedescribe inconsiderable detail thespinorial representations oftheClifford algebras associated withcomplex andrealvector spaces. Wegiveexplicit methods tofindthe representations forarbitrary dimension andsignature. Wealsopresent alltheessential information about theinvariant bilinear andHermitean forms onthecarrier spaces oftherepresentations. Special attention isdevoted totheappearance ofWeyl andMajorana spinors (oftwokinds), to charge conjugation andtothesymmetry andsignature oftheinvariant forms. Ourmaintoolisthe classical theorem about representations ofsimple algebras (§4.2). Toobtain anoverall picture oftherepresentations ofClifford algebras itisconvenient to divide thestudy intoseveral stepsinsuchawaythatateachstepanewstructure isintroduced. (i) Atfirst,oneforgets about theClifford algebra everything butitsstructure ofalgebra AFor anyalgebra o,wedenote by2'3thedirect sumQ-'3®(B,cf.§4.5. There aretwotypes of complex algebras, llI(2'“) and2llI(2""), andfivetypes ofrealalgebras, lH(2""), 2lFl(2"‘), H(2"‘), 2H(2"‘) andE(2"‘). Theinteger missimply related tothedimension oftheunderlying vector space. For example, considered asabstract algebras, thethree algebras C£(4). C£(4,1) andCi(2.3) areall 10 (ii) iii) (ivisomorphic tollI(4). Here andinthesequel Ctik, P)denotes therealClifford algebra associated withascalar product ofsignature (k,I2). Itseven subalgebra isdenoted by c@(1<,2). 1 IftheClifford algebra isconsidered together withitsZ2-grading given bythemain automorphism ct,thentherearestilltwotypesofcomplex algebras, butalready eightclasses ofrealalgebras, cf.Table I,"Therealclock". Thisprovides aclassification finerthanatthe previous step,butonecannot determine thesignature oftheunderlying vector space fromthe soleknowledge ofitsgraded Clifford algebra filo->fit.Forexample, thegraded algebra 2FR(8) --1»FFl(l6) isisomorphic toCQ)(8,0) -~>C£(8,0), CQ)(4,4) --1»C£(4,4) andCQ)(0,8) —->CtI0,8). Theclass oftherealalgebra Ctik,J?)depends on k-I?mod 8. (27) If 7:fll-—->End S (28) isafaithful irreducible representation ofasimple algebra fitwith aninvolutive antiautomorphism B,thenthecontragredient representation ' 3":fit->EndS"', where ’Y(a)=‘Y(B(a)), isequivalent to7andthere exists anisomorphism B:S—>8"‘intertwining 7and‘y.Iffitis central simple, thenBiseither symmetric orskew; itdefines aninner product onS.The symmetry ofBdepends onthedimension noftheunderlying vector space B forn=0, 1,2,7 modS ‘B= (29) -B forn=3,4,5,6 mod8 Thedouble periodicity mod8given by(27)and(29)gives risetoachessboard arrangement ofrealClifford algebras alluded tointhetitleofthiswork andpresented inTables II-V. There isagreat wealth ofstructure inaClifford algebra flltaken together withthevector spaceVthatgenerates it: 11 1.Thenatural linear isomorphisms .2=AVcAV* (30) allow aninterpretation ofelements oftheClifford algebra asmultivectors orforms. 2.Thegrading, .2=H0®£1,may beused todefine anassociated graded or"super" Lie algebra. Itsunderlying vector space coincides withfitandthegraded bracket is [a,b]=ab-(-1)P‘l ba, where ae befllq, andp,q=0orl.Ofparticular interest isthegraded Liesubalgebra L=KovoA2v. Ifu,vEV,then [u,v]=uv+vu=2g(u, v) (31) sothat [K,1.1=0,[v,v]cK,[v,A2v1cv and [A2v,A2v]cA2v. Thelastinclusion means thatA2Visan(ungraded) Liesubalgebra: itistheLiealgebra ofthe orthogonal andspingroups. These groups arealsosubmanifolds ofA;wedefer their detailed description tosubsequent work. 3.IfBisaminimal leftidealofasimple algebra withunity FLthen 7:fit->EndQ,where 7(a)b=ab, forevery aefitandbeB,isafaithful irreducible representation of3.Thisgives Chevalley‘s (1954) interpretation ofspinors aselements ofaminimal (left)idealofaClifford algebra. I 12 r AllClifford algebras are"supercentral" :numbers (scalars) aretheonlyelements which supercommute withallelements oftheClifford algebra (Wall1964). If(ea)isanorthonormal basisforascalar product ofsignature (k,I2),thenthesquare ofthevolume element T]'-=C1C2-..Cb“; is 112=(-1)0=-P) tk-P-1)/2_ Fork-I?a2or3mod4thesquare isnegative and1]belongs tothecentre ofJ10orA,respectively. Itmay, therefore, berepresented byitimes theunitendomorphism ofthespace ofWeyl orDirac spinors. There areatleasttwoother "independent" ways ofintroducing complex numbers inquantum theory. Thefirstcomes fromtheobservation thatenergy andmomentum arerelated totranslations. Infinitesimal translations arerepresented byfirst-order differential operators. Tomake them (formally) self-adjoint onehastomultiply thembyi.Arelated observation isthattheLaplacian on compact Riemannian spaces isanegative operator. Another reason forconsidering complex wave functions and,inparticular, spinor fields, has todowithelectromagnetic interactions. According tothegauge, or"minimal interaction" principle, wave equations forcharged particles contain thegradient operator dalways inthecombination d-ieA, where eisthecharge andAthepotential ofthe(external) electromagnetic field. Theicomes from thefactthattheLiealgebra ofthegroup U(l)—-thegauge group ofelectrodynamics —~ consists ofpureimaginary numbers. Itisnotatrivial orobvious matter thatthethreei's(spinorial, quantum-mechanical andelectromagnetic) areoneandthesame; buttheyareasindicated bythe successes oftheDirac equation. Similar remarks haverecently been made byChen Ning Yang (1987). (v) LetAdenote C£(1<,.0)orClb(l<, I?)depending onwhether k+9=2mor2m+1, repsectively. Thealgebra J-Iiscentral simple and,therefore, hasonlyone,uptoequivalence, irreducible faithful representation. Let(28)besucharepresentation inaspace Sofcomplex dimension 2"‘.Thecomplex conjugate representation Y:I-I->End§ isreal-equivalent to7.There thusexists alinear isomorphism C:S-->§intertwining yand ‘ti. _ 13 "'f(a)C=C'y(a), aefl. Itisdefined uptoacomplex factor which canbechosen sothat I fork-9 e0,l,2,7 mod8, E‘:c= -I fork-.0 a3,4,5,6 mod8. Depending onwhether CC=Ior-Itherepresentation 'yisrealorquaternionic. Ifitisreal, thenthere areMajorana spinors (ofthefirstkind) defined byC¢= Fork-1?E6mod8 onecandefine Majorana spinors (ofthesecond kind) aseigenvectors ofC'y(r|), where 'r1is thevolume element given by(32). There arenoMajorana spinors ofanykindfork-I?E3, 4,5mod8. Fork+.0=2m+1,thefullalgebra C£(k, 9)admits anirreducible representation 7ina complex 2"‘-dimensional space. This representation isfaithful when restricted totheeven subalgebra andcanbechosen sothat -Ym)=iV(\’-1)/2 1,. wherev alt-I?mod8 and0svS7. Therefore, there istheequivalence ofrepresentations, ii ‘Y for v=1 and5, ‘Y... "yea for v=3 and7, where otisthemain automorphism ofClilt,9). 14 5. Representations ofrealClifford algebras Inthissection wegiveashortsummary oftheproperties ofrepresentations ofClifford algebras ofrealvector spaces inalanguage familiar tophysicists. The2"‘-dimensional spinor space Sisidentified withfllzrlte endomorphsims ‘yaare2"‘by2'“matrices andthesymbols ‘A, ATandKdenote theusual transpose, Hermitean conjugate andcomplex conjugate ofthematrix A, respectively. Therefore AT=‘E. If(k,I?)isthesignattue, k+Q=2mor2m+1, thenthere arek+IJDirac matrices yaeE|I(2"‘) suchthat ya75+75‘ya=0forct¥=B,ctandB=1,.._,k+t1', (33a) 2ya=Iforkvaluesofctand1:=-1fort1valuesofot. (33b) Wedonotinsist herethatthefirstkvalues ofthelabelshould correspond toDirac matrices withpositive squares; onlythetotalnumbers ofpositive andnegative squares matter. Letk-9=8p+v, where pisaninteger and0Sv57.Thematrix 1"=i"’(""D/2 71 ‘yamanticommutes with‘ya, (34) and 1"1=1. (as) There existinvertible matrices A,B,C,D,Ee[|I(2"‘) suchthatforevery ct yin=Ay,,A-1, (sex) ‘ya=B7,,B'1, (36B) 1,=C'yuC‘1, (sec) ts They satisfyv;'.=-I>v,.1>"‘. (sen) “ya=-E'yuE'1. (365) IB=(-1)'"<t"-1)/2B (3713) lE=(-1)"1<"‘+1>/2E (375) Ir=(-1)"'nra-1 (arr) Thedefining properties (36)determine thematrices A,...,Euptocomplex factors. These factors canbechosen sothat Theremaining freedom isA->FLA,B->7t|.tB, C-->|.tC,D-1»RD,E—>7t.p.E, where Kisreal1*0 andttiscomplex ofunitmodulusEc=(-1)‘*<\’-2)/8 1 (as) A=‘tic=Al (39A) 1)=ac=oi (39a) E=tier (40) IfUisaninvertible matrix, UelE(2m), thenthematrices havetheproperties (33).Marking withprimes ontheleftthematrices associated by(36A-E) with thematrices ‘Ya,wehave‘ya=U'1y,,U (41) 16 at=UTAU, (42A) '3=ittau, (42B) 'c='fi'1cU. (420) andsimilar relations for'I‘,'Dand‘E. TheHermitean formsq>iA<pandaloe,wheretpert?,areneutral exceptinthefollowing CHSBSZ elseisdefinite forr=0,k>0, (43a) tpiDtp isdefinite fork=0,9>0. (43D) These forms restrict tonon-degenerate I-Iermitean forms onthespaces ofWeyl spinors if, andonlyif,kiseven. Foroddk,thematrices AandDchange thehelicity ofWeyl spinors. Letk-P=8p+v, where pisaninteger and1SvS7.Onecanchocsethematrices 71,...,'_Y2m+1 SOthat 71'y2m+1 =i"("'1)/2 I. (44) There existmatrices A0,BoandC0suchthat,forevery ct -{Ta=(-1)”A0yaA-1° (4SA) 17 ‘r,,=(~1)“" B,,r,,B",, <4sB> 7.,=<-1)*<""‘>” C01,,C‘, (450) and A0=B000=A01‘ (46) IBo=(_1)m(m+l)/2 Bo (4-7) Eoco=(-1)<*’'1)/B1 (43) TheHermitean form <piAotp isneutral except inthecasewhen either k=0or1?=0:itisthen definite. Letk-+0=2mandk-9=8p-+v, asbefore. TheZmt-l matrices 'Yl,...,'Y2m afid =F areDirac matrices foraspace withsignature (k+1,P) and AforI?even, A0 ‘-'-’ (SOA) DforJ?odd, Bformeven, BO = {E (SOB) formodd, Cforv=0or4, CD= . (50C+) ' CI"forv=2or6,- where thematrices 1",A,...,Eareasin§5.1. ' 18 Sirnilarly, the2m+1 matrices Y1,...,Yam and Y2"-H_1 = areDirac matrices foraspace withsignature (k,P+l). Theintertwining matrices A0andB0areasin (SOA) and(SOB). but C= Cforv=2or6, (soc) O Cl"forv=0or4. Asanexample, wegiveexplicitly allrelevant quantities foranextension fromsignature (k',9) to(k+1, 9+1). Wechoose anextension ofspecial kindthatallows asimultaneous treatment of even- andodd-dimensional spaces. Onecantake "Y'u=O'®'Ya((l=l,...,k+Q), 'Y'k+p+1=T®I flnd"Y'k+P+1 (i) Fork+J?=2mwehave l"'=o® I‘,C‘-=I®C A‘=':® 11>,D‘-=(-1)"ie®A(52)B'=1:®E,E'=(-1)’*1ie®B (ii) Fork+Q=2m+1 wehave {it-:®A0forPeven, A‘= (53A) ° 1:oA0for:2odd, -ie®Bo formeven,-B'o={ (SSB) 1:®Bo formodd, {I®Co forv=1or5, 63¢c'= )0 i0®CO forv===3or'7, 19 where k-Q=8p+v andthematrices No,B‘OandC‘°areinthesame relation to7‘uasthematrices A0,B0andCoareto‘ya,cf.§5.2. 6. The spinorial chessboard There areseveral "periodicity properties“ ofrealClifford algebras andtheirrepresentations. Thetypeofthealgebra depends onlyonk-Pmod8.Butthesymmetry properties oftheinvariant bilinear forms depend onk+1?mod8.There isa"double periodicity" inthesetofallrealClifford algebras: itisconvenient todescribe itbyreferring ittoachessboard. Wedefine thespinorial chessboard tobethesetof64realalgebras {C£(k,I.l) I05k,Q57] where itisunderstood thatCQ)(0,0) —>610,0) isthealgebra lFl-+lFl, i.e.C[1(0,0) =[0].ln addition tothechessboard —andrepresentations ofitselements ---weconsider thetwoeight-dimensional Euclidean algebras 018,0) andC£(0,8). According totheperiodicity property, ifk‘=k+8p and1?'= =P+8q, then C£(k‘,I?') =Cl(k,I?) ®|Fl(16P"'q). (54) Therefore, every Clifford algebra canberepresented asin(54),withC£(k,£l) onthechessboard. Thesi_anifica.*v-rr cfthisremark. goes beyond themere isomorphism ofalgebras (54): the representations ofC[(k‘,1?') andtheassociated bilinear andHennitean forms canbeeasily constructed from those ofC£(k,9). Adding eight dimensions makes larger theClifford algebra and theassociated spinor spaces, butpreserves theiressential properties suchasthesymmetry ofB, typeofC,etc. Tomake thelast statement more precise, consider avector space V=lR8 with a positive-definite scalar product. Thefaithful irreducible representation ofitsClifford algebra, CK8,0) —->EndS, (55) isrealsothatScanbetaken tobeareal, 16-dimensional space (ofMajorana spinors). Let (e1,...,e8) beanorthonormal basisinV.Thesetof23products oftheform ee...e ,wherelSt1<ct<...<otS8,_at‘ct: up 1 2 P 20 constitute abasisofthealgebra. Thisbasis isorthogonal forthescalar product hon018,0) defined by h(fl.b)=T1‘‘Y(B(fl)b)- Indeed, if a=ea! cupandb=ea! efiq, where 1stx1<...<otps8 and1s[3,<...<Bqs8, then B(a)b =1whenever p=qand0:1=B1,...,up=Bp, and Try(B(a)b) =0otherwise. Therefore, thescalar product hispositive-definite andthesymmetric bilinear form Bisalso positive-definite. Wechoose abasisinSsuchthatBisrepresented byaunitmatrix withrespect to thisbasis, andweusethebasis toidentify Swith Hi16sothattherepresentation (55)canbe described as _ 9:018,0) -->lH(l6) (56) and5==9,i.e.theDirac matrices 90,=9(ea), ct=1,...,8, aresymmetric, i6“=Ba. They maybechosen tobe 61=o®I®I®I, 92=e®c®I®I, 63=e®o'®e®I, 64=t-:®o'®o®o, (57) 95-=t-:®o®t®e, B6=e®1:®I®e. 97==e®t®e®o, 63=e®1:®e®1:, Their product 9=r®I®I®I 21 isalsosymmetric and(92=I.There isthedecomposition 6°==6+Q6_, where , 0,=cr,,(a,0) ->ms) (ss) aretheinequivalent Weyl representations oftheevenalgebra. Since 9anticommutes withtheDirac matrices, onecanconstruct afaithful irreducible representation oftheopposite algebra #6:ClI0,8) -—)FFi(16) (58*) byputting 4-96,=99“, (I.=1,...,8, (59) sothattheDirac matrices (59)areskew and he,=o.,t-1,,ca-1. (so) Let 7:Ctll-:,l?) -->EndS (61) the-arepresen tation oftheClifford algebra C£Ik,I?). Onecanextend ittorepresentations Y"t"!{l<r:»8,.t-1} --;>lF?{l6) ®EndS and 1'":CtIk,9+8) ->|¥i(16) ®EndS byputting 'Y‘q=®®Yq='Y"q (05:11 ---1k+p)s (623) 'Y'u+k+9 =9a®I (CL=1,...,8), (62b) and v"am,=oeast (<1=1,...,s). (62.-;) Marking withprimes ordouble primes thequantities corresponding totheextensions 7'or 7",respectively, weobtain fork+I?even 22 Adding 8"positive" or"negative" dimensions preserves thecharacter ofA,B,C orD,E,C, respectively. IfAorDisdefinite, thensoisA‘orD‘,respectively. There aresimilar results for k-I-Podd,namelyP=9eF=P' Q5123’:=I®A, A"=@®A =I®B, B"=®®B =I®C=C" D‘=9®D, D"=I®D E'=9®E, E"=IeE. Ag,(forreeven)andA"0(forrodd)=IoA0, B'o(formeven) andB"o(formodd)=1®B C'OandC"o(forv=1or5)=I®Co, A‘o(forJ?odd)andA"o(forI?even) =(E)®A B‘o(formodd) andB“o(formeven) =(E)®B C'oandC"0(forV=3or7)=-'-(9®Co. 23OT(63) (64-A) (6413) (54C) (65A) (65B) (65C) Table I IR/(7212 0 'R1\\$ la tiQ IH ‘Q 2/4 . Therealclock"‘ maybeusedtofindtheClifford algebra Ctlk,P)anditsevensubalgebra CQ)(k, I2):compute firstthe hour itsuchthat9-k=8p+|.t,where pisaninteger and05|.tS7.Theletters adjacent tothe hourdetermine thetypeofthealgebras. Thedimension ofthefullalgebra is2*"9.Forexample, £4,435) ->.ctt3,5)ist:(s)->H(8)because, inthiscase,u=2anddimH(8)=28. "‘Thecomplex clock ismuch simpler: ithasatwo-hour dial. 24 MPO_mD__SEhT kMM'___L__|_H___H_MHt__i1 'lI|IbH ________L________ I,‘-II‘Iiil__l__'_______|___|__I_il|l_IIII'III‘IIllIHM_8"InHH4__,__ _______“WW‘wa _____O_i ___8__C___ _idraO -%.%._%_.% WVW%.%- -%fl-%l%-%.%fl%E%u%u% ___atkb55hC “MQMHiwMMAMdM_dmHWEhS MWheRMWmmw_r£mam}mJc_m maX8rhmMMB‘Htf_0SmMW_Gm CsflwKrCwmflghohm“___6JPM £6 MD__ BgCmWW.“ Oqflrs_m_h fnOwGd MH1 w____mWfl atPO__SHH mmqmC__________s amm WRd__v_J___mm_m I_Wmo_m H_n_m WmI MrmCw PUWtEM”_nm_HGWhfl J26( W_nCS¢dS3$mWS -1 -1Table HI Thestructure ofthealgebras occurring onthechessboard maybedetermined from thefollowing data: ‘I ‘IC-1 -1 -1 -1 1 ZEEE;;i..l!i.illQRQ;i.k‘i!.N“NREE.iiRE-RBQS. ,0) (s,0 |1 1'-1 --1 1 1-1 ._1"v-av O)5-.9.L0).§§.€RNN..RN.RRNN.ERRENE.ERR\_\_Al -I IgnEC White andblack dotsreplace herethesquares ofthechessboard. Thefigures ontheleftandlower sidesarevalues ofthevolume element squared. Those ontherightandupper sidesdetermine the type(real if1,quatemionic if-1)ofthefull(fork+9even) oreven (fork+I?odd) Clifford algebra. 26 BTable IV Thebilinear forms andtheirSymmetries e 1-H-) (+'+) ("'."') ("3") ("W") (_'+)(_'+) (-'-) .“.n.m W Hii§'§.§'§.aa.n <-l»»mlII2I )04 ° ’ ° ' if . ‘".1.N(+~-l (+‘H’0,2 (+'_)0,1 tr=ao =52) anmuummu9%inn""I|-|I"""'-""' "-F‘+-.1‘III-nil"A "'Il-|l"""o1"‘-"H 1.3,pI-"Ila, i.g4!"-"M. 2'7.......I'*i,I.{I.Is~Z§§§_I' I - 1-_ I ' ' '-_ '_ ' _ -1d:7=-EvE‘!areeither symm I11_ . dEd finedbyw ::B'f Ban q, II. Thc‘sum:-,pms1?; Bautumn; oranticotgmute siiththehelicity operator I‘.These propeskewand eyel ¢I°° tA33,1431‘ .- -= -.Theyaredefined byB=B1indicated above byPal"('51-92)“'h°r° E1and£2+or t1-B;andsimilarly forE. Table V TheDirac (I-Iermitean) forms Ddefinite "‘-' Q0-00 '3‘--0~'-00-0 -kg. *3Q-QFO*O-—->3:.21>P /'\J(0,? (0.0)s (0.0) (0.4) (o,a (0.2)0 W’III(at-)=(0.0) nottam(so)Adeflnlte Theisomorphisms AandDaredefined by770,==A‘yaA'1andvia=-D-yaD4,Theyboth@315;fo, even dimensional spaces. Inanoddnumber ofdimensions, exactly oneofthetwoexists, depending ontheparity ofk;thisisindicated bytheletterAorDnexttothecorresponding white dot.TheHennitean forms A(¢,4))are(positive) definite forthealgebras Clil<,0); similarly, the Hermitean forms D(¢,¢)are(positive) definite forCKO,P).Otherwise theyareneutral. - 1I 28 7. Concluding remarks andoutlook Every physicist willagree thatspinors areanecessary andimportant toolinthedescription of fundamental interactions. Thesuccess oftheDiracequation isoneofthemostbeautiful chapters of theoretical physics. Spinors playamajor roleinessentially allrecent attempts atbuilding new models (mand unification, supersymmetry, strings andmembranes). They arealsoveryuseful in theclassical, relativistic theory ofgravitation (Penrose andRindler, 1986). Animpressive example oftheusefulness ofspinor analysis inanewdomain hasbeenprovided byEdward Witten (1981a) whoproved the"positive energy theorem" inEinstein's theory inamanner which ismore transparent thantheearlier proof duetoSchoen andYau. Thirring (I972) showed thatbyspinors inafive-dimensional space onecanobtain CPviolation inageometrical way. Recent renewal of interest ingeneralized Kaluza-Klein theories (cf.,forexample, thepapers byWitten (l98lb), Abdus Salam andJ.Strathdee (1982), andSteven Weinberg (1983)) hasledtoconsidering spinors inspaces ofdimension greater thanfour. Inasomewhat different context, oneofus(Budinich 1979, l98ob) proposed toconsider fields ofsimple (pure) spinors insuitable higher-dimensional spaces andtorelate them towave-functions ofphysical particles. There areindications thatinthis manner a"natural" wayofderiving interaction terms ofLagrangians ofparticles withinternal symmetry maybeobtained. Attempts havebeenmade towrite adifferential equation forsimple spinors, consistent withthequadratic constraints (24). Forexample, themethod ofLagrange multipliers, applied toavariational principle in7space-time dimensions, leads I08.Weyl equation forsimple spinors witha"mass term" induced bytheconstraint (26)(cf.Budinich andTrautman 1986 andthereferences given there). Aremark onthepossible physical relevance of‘simple spinors hasalsobeenmade byA.D.Helfer (1983). There aresome "unexpected" applications ofspinors: spinor connections onlow-dimensional spheres coincide with simple, topologically non-trivial gauge configurations (Budinich and Trautman 1986). Spinors provide afinetoolforthestudy oftopological properties ofmanifolds (Atiyah, BottandShapiro 1964, Atiyah andSinger 1968). There isaremarkable "spinorial" form oftheEnneper-Weierstrass formula forsolutions oftheequation forminimal surfaces andofits extension tostrings (Budinich 1986a, Budinich andRigoli 1987, andthereferences given there). Itisbased onarepresentation ofcomplex andrealnullvectors interms ofspinors, analogous to those described in§3. Considerations suchastheseconvince usthattheremaybesomething moretospinors than hasbeensaidandseensofar.Thisviewhasbeenputforward, quitealongtimeago,byRoger Peru-ose whopursued themost comprehensive andfarthest reaching programme ofapplying spinors -~andtheirclose relatives, twistors —-infundamental physics. Weshare hisview"that wehavestillnotyetseenthefullsignificance ofspinors —-particularly the2-component ones-—in thebasis structure ofphysical laws" (Penrose 1983b). Weareinclined, however, toextend the 29 belief inthesignificance ofspinors tothose associated withhigher-dimensional geometries and replace thephrase about the2-component spinors byonereferring tosimple spinors andthe homogeneous spaces mentioned in§s3.(Note that,infour-dimensions, simple spinors havetwo components. More generally, Weyl spinors aresimple inneutral spaces ofdimension S6.In particular, twistors aresimple). Ourwork isanattempt tofollow thisroad. Thepresent article isapreparation fora systematic study ofthespinandpingroups andoftheir representations inrelation tosimple spinors. Weintend tomake more precise theideathatthedimension oftheorbitisameasure ofthe simplicity ofspinors itcontains, useourmethods toderive thebiquadratic spinor identities (Case 1955), study (simple) spinor fields onhomogeneous spaces -suchastheones arising from conformal compactification --andconsider thepossibilities offered byvarious schemes of dimensional reduction. 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