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arXiv:hep-th/0009239v128Sep2000NYU-TH/00/09/10SNS-PH-00-14hep-th/0009239CovariantQuantizationoftheBrink-SchwarzSuperparticleP.A.Grassia,G.Policastroa,b,andM.Porratia(a)PhysicsDepartment,NewYorkUniversity,4WashingtonPlace,New-York,NY10003,USA(b)ScuolaNormaleSuperiore,PiazzadeiCavalieri7,Pisa,56100,ItalyABSTRACTThequantizationoftheBrink-Schwarz-Casalbuonisuperparticleisperformedinanexplicitlycovariantwayusingtheantibracketformalism.Sinceaninfi-nitenumberofghostfieldsarerequired,withinasuitableoff-shelltwistor-likeformalism,weareabletofixthegaugeofeachghostsectorwithoutmodi-fyingthephysicalcontentofthetheory.Thecomputationrevealsthattheantibracketcohomologycontainsonlythephysicaldegreesoffreedom.September20001IntroductionThecorrectquantizationofsystemswithaninfinitelyreduciblegaugesymmetryisalong-standingproblemand,recently,severalnewefforts(seee.g.[1])havebeenmadetowardtheconstructionofacovariantquantizationprocedurefortheGreen-Schwarzsuperstringmodel[2].Indeed,theGSsuperstringisthemostimportantandinterestingmodelen-joyingthefeatureofinfinitereducibility,butsincethemodelisverydifficulttohandleinitscomplexity,thestudyofsimplermodelsprovidesagoodtestforthequantizationtechniques.Thisisessentiallythereasonwhy,inthepastdecades,peopledevotedseveraleffortstryingtoquantizethesuperparticlemodelofBrink-Schwarz-Casalbuonitype[3].InthecaseofGSsuperstringandsuperparticle,therearefirst-andsecond-classcon-straints[4,5,6,7].Theoccurrenceofsecond-classconstraintsarisesfromthefactthattheGrassmanmomentaPαθ,conjugatetothefermionicvariablesθα,arenon-independentphase-spacevariables.If,asaformalprocedure,oneattemptstoconstructDiracbrackets,treatingallthefermionicconstraintsasiftheyweresecondclass,theresultingexpressionsaresingular.Analternativeprocedurewouldbeacarefulseparationoffirst-andsecond-classconstraints,but,inthatway,acovariantquantizationprocedureisimpossibletoachieve.InordertomaintainmanifestLorentzcovariance,onehastoexploittheκ-symmetry[2,8]ofthemodel,whichcancelshalfofthefermionicdegreesoffreedomrealizingthematchingbetweenthebosonicandthefermionicstates.Unfortunately,theκ-symmetryisareduciblelocalfermionicsymmetry.Thismeansthat,inreality,onlyfourdegreesoffreedomareeffectivelycanceledatthefirststage.Pursuingtheanalysis,itiseasytoshowthataninfinitetowerofghostsisnecessarytomatchthecorrectnumberofdegreesoffreedom.IntermsoftheHamiltonianformalism[6],thisisequivalenttothestatementthattheconstraintsareinfinitelyreducible:thereexistnotonlylinearvanishingcombinationsofconstraints,butalsozeromodesofthoserelations,andsoontoinfinitelymanylevels.Severalattemptstoasolutionoftheproblemcanbefoundintheliterature.Inpartic-ular,wewouldliketomentiontheideaofchangingtheclassicalconstraintsinorderthatallthesecond-classconstraintsaretransformedintofirst-classones[9,10].Thisessen-tiallyyieldsanextensionofthephase-spacewheretheκ-symmetryisgaugedbymeansofsuitablefermionicgaugefields.Otherapproachestothesuperparticlequantizationaretheharmonicsuperspace[11]–whichproducesanon-localsuperYang-Millsfieldtheory–andtheconstructionofmodelsbasedonagivenBRSToperatorwhichselectsthecorrectphysicalsubspace[12].Nevertheless,alloftheseapproachessharethecommonfeatureofinfinitenumberofclassicalandghostfields.Finally,thecomputationofthegauge-fixedBRSTcharacteristiccohomology[13,14]1forthesuperparticlemodelreceivedconsiderableattention[12,15,16,17].Ithasbeenshownin[17]that,duetotheinfinitelymanyinteractingfields,thecanonicaltransforma-tionsperformedtoimplementthegaugefixingturnouttobeill-defined.Moreover,asaconsequenceoffurthergaugesymmetriesofthegauge-fixedaction,atwo-stepgauge-fixingprocedureisrequiredanditmaycauseproblems,asarguedin[12].TheaimofourpaperistopresentanewsolutionoftheapplicationoftheBV-BRSTformalism([18,19];forareview,see[5,20,21])totheproblemofthesuperparticle.Inparticular,wetaketheadvantageoftheexistingliteraturetomakeessentialstepstowardsthecompletesolution.ThemainissuehereistheconstructionofaproceduretoquantizetheBrink-Schwarz-Casalbuoniclassicalaction,computingthecorrectantibracketmapandthecorrespondingcohomology.Thepresenttechniqueisbasedoncanonicaltransformationswhichimplementthegaugefixingofthenthorderghostswithoutaffectingthe(n−1)thorder.Thestructureofthesuperparticlemodelinvolvesword-linediffeomorphisms,on-shellclosureofthealgebraofsymmetriesandaninfinitetowerofghostfields.Totakeintoaccounttheopenalgebra,theBV-BRSTformalismisimplementedandthesolutionofthemasterequationscontainsquadratictermsintheantifields.Ontheotherhand,theκ-symmetryofthemodelentailsfield-dependenttransformationswhichproduceinteractionsamongtheghostfieldsofdifferentlevelsandthesuperparticlefields.Therefore,afterthegaugefixing,aninfinitenumberofghostsfieldsinteract,amongthemselvesandwiththephysicalfields.Inthissituation,twotypesofproblemsarise:i)therenormalizabilityofthemodelmaybelostduetoaninfinitenumberofparameters,ii)thecomputationofthegauge-fixedBRSTcohomologyandthedefinitionofthephysicalspectrumofthemodelareill-defined.Inordertosolvetheproblemofinteractionsamongtheghostfieldsofdifferentlevels,weapplytheideaofP.Town