Covariant Quantization of the Superparticle Using

arXiv:hep-th/0105050v23Aug2001IFT-P.039/2001CovariantQuantizationoftheSuperparticleusingPureSpinorsNathanBerkovits1InstitutodeF´ısicaTe´orica,UniversidadeEstadualPaulistaRuaPamplona145,01405-900,S˜aoPaulo,SP,BrasilTheten-dimensionalsuperparticleiscovariantlyquantizedbyconstructingaBRSToperatorfromthefermionicGreen-Schwarzconstraintsandabosonicpurespinorvariable.Thissamemethodwasrecentlyusedforcovariantlyquantizingthesuperstring,anditishopedthatthesimplercaseofthesuperparticlewillbeusefulforthosewhowanttostudythisquantizationmethod.ItisinterestingthatquantizationofthesuperparticleactioncloselyresemblesquantizationoftheworldlineactionforChern-Simonstheory.May20011e-mail:[e-mail],theten-dimensionalsuperstringwascovariantlyquantizedbyconstructingaBRSToperatorfromthefermionicGreen-Schwarzconstraintsandabosonicpurespinorvariable[1].Althoughthismethodwassuccessfullyusedforcomputingtree-levelscat-teringamplitudes,theconstructionoftheBRSToperatorisnon-conventionalsoitisabitmysteriouswhythismethodworks.ThissameBRSTconstructioncanbeusedforcovariantlyquantizingtheten-dimensionalsuperparticleanditishopedthatbystudyingthissimplermodel,someofthemysterieswillbeeasiertounderstand.Sincethespectrumoftheten-dimensionalsuperparticlecontainsaspin-onefield,theconstraintsoftheworldlineactionshouldimplyspacetimegaugeinvariancesaswellasspacetimeequationsofmotion.ThisdiffersfromtheworldlineactionsfortheparticleorspinningparticlewheretheconstraintsimplytheKlein-GordonorDiracequationsofmotion,butdonotimplyspacetimegaugeinvariances.OneworldlineactionwhichdoesdescribeatheorywithspacetimegaugeinvarianceistheworldlineversionofWitten’sactionforChern-Simonstheory[2].ItwillturnoutthattheconstraintsandquantizationofthisChern-Simonsactioncloselyresembletheconstraintsandquantizationofthepurespinorversionofthesuperparticleaction2.Section2ofthispaperwillreviewtheproblemswithquantizingthestandardsu-perparticleaction.Insection3,theworldlineactionforChern-Simonswillbediscussed.Section4willreviewthesuperspacedescriptionoften-dimensionalsuper-Yang-Mills.Andinsection5,thepurespinorversionofthesuperparticleactionwillbequantizedinamannersimilartotheChern-Simonsaction.Theappendixwillcontainacomputationofthezero-momentumBRSTcohomologyofthesuperparticle.2.ReviewofStandardSuperparticleDescriptionThestandardactionfortheten-dimensionalsuperparticleis[3]S=Zdτ(ΠmPm+ePmPm)(2.1)whereΠm=˙xm−i2˙θαγmαβθβ,(2.2)2ThesimilarityofthetwoworldlineactionswasfirstpointedouttomebyWarrenSiegel.1Pmisthecanonicalmomentumforxm,andeistheLagrangemultiplierwhichenforcesthemass-shellcondition.Thegammamatricesγmαβandγαβmare16×16symmetricmatriceswhichsatisfyγ(mαβγn)βγ=2ηmnδγα.IntheWeylrepresentation,γmαβandγαβmaretheoff-diagonalblocksofthe32×32Γmmatrices.Theactionof(2.1)isspacetime-supersymmetricunderδθα=ǫα,xm=i2θγmǫ,δPm=δe=0,andisalsoinvariantunderthelocalκtransformations[4]δθα=Pm(γmκ)α,δxm=−i2θγmδθ,δPm=0,δe=i˙θβκβ.(2.3)Thecanonicalmomentumtoθα,whichwillbecalledpα,satisfiespα=δL/δ˙θα=−i2Pm(γmθ)α,socanonicalquantizationrequiresthatphysicalstatesareannihilatedbythefermionicDiracconstraintsdefinedbydα=pα+i2Pm(γmθ)α.(2.4)Since{pα,θβ}=−iδβα,theseconstraintssatisfythePoissonbrackets{dα,dβ}=Pmγmαβ,(2.5)andsincePmPm=0isalsoaconstraint,eightofthesixteenDiracconstraintsarefirst-classandeightaresecond-class.Onecaneasilycheckthattheeightfirst-classDiracconstraintsgeneratetheκtransformationsof(2.3),however,thereisnosimplewaytocovariantlyseparateoutthesecond-classconstraints.Nevertheless,onecaneasilyquantizethesuperparticleinanon-Lorentzcovariantmannerandobtainthephysicalspectrum.Assumingnon-zeroP+,thelocalfermionicκ-transformationscanbeusedtogauge-fix(γ+θ)α=0whereγ±=1√2(γ0±γ9).Inthisgauge,theactionof(2.1)simplifiestothequadraticaction[5]S=Zdτ(˙xmPm+i2P+(˙θγ−θ)+ePmPm)=Zdτ(˙xmPm+i4˙σaσa+ePmPm),(2.6)whereσa=√2P+(γ−θ)aanda=1to8isanSO(8)chiralspinorindex.2Canonicalquantizationof(2.6)impliesthat{σa,σb}=2δab.Soσaactslikea‘spinor’versionoftheSO(8)Paulimatricesσja˙bwhichsatisfyσja˙cσjb˙d+σjb˙cσja˙d=2δabδ˙c˙dwherejand˙bareSO(8)vectorandantichiralspinorindices.Onecanthereforedefinethequantum-mechanicalwavefunctionΨ(x)tocarryeitheranSO(8)vectorindex,Ψj(x),oranSO(8)antichiralspinorindex,Ψ˙a(x),andtheanticommutationrelationsofσaarereproducedbydefiningσaΨj(x)=σa˙bjΨ˙b(x),σaΨ˙b(x)=σja˙bΨj(x).(2.7)Furthermore,theconstraintPmPmimpliesthelinearizedequationsofmotion∂m∂mΨj=∂m∂mΨ˙b=0.SothephysicalstatesofthesuperparticlearedescribedbyamasslessSO(8)vec-torΨj(x)andamasslessSO(8)antichiralspinorΨ˙a(x)whicharethephysicalstatesofd=10super-Yang-Millstheory.However,thisdescriptionofsuper-Yang-MillstheoryonlymanifestlypreservesanSO(8)subgroupofthesuper-Poincar´egroup,andonewouldlikeamorecovariantmethodforquantizingthetheory.Covariantquantizationcanbeextremelyusefulifonewantstocomputemorethanjustthephysicalspectruminaflatbackground.Forexample,non-covariantmethodsareextremelyclumsyforcomputingscatteringamplitudesorforgeneralizingtocurvedbackgrounds.Sincethesuper-Yang-Millsspectrumcontainsamasslessvector,oneex