Weak Dirac bracket construction and the superparti

arXiv:hep-th/9512036v16Dec1995WeakDiracbracketconstructionandthesuperparticlecovariantquantizationproblemA.A.Deriglazov∗,A.V.Galajinsky,S.L.LyakhovichDepartmentofTheoreticalPhysics,TomskStateUniversityTomsk634050,RussiaAbstractThegeneralprocedureofconstructingaconsistentcovariantDirac-typebracketformodelswithmixedfirstandsecondclassconstraintsispresented.Theproposedschemeessentiallyreliesuponexplicitseparationoftheinitialconstraintsintoin-finitelyreduciblefirstandsecondclassones(bymakinguseofsomeappropriatelyconstructedcovariantprojectors).Reducibilityofthesecondclassconstraintsin-volvedmanifestsitselfinweakeningsomepropertiesofthebracketascomparedtothestandardDiracone.Inparticular,acommutationofanyquantitywiththesecondclassconstraintsandtheJacobiidentitytakeplaceonthesecondclasscon-straintssurfaceonly.ThedevelopedprocedureisrealizedforN=1Brink–SchwarzsuperparticleinarbitrarydimensionandforN=1,D=9massivesuperparticlewithWess–Zuminoterm.Apossibilitytoapplythebracketforquantizingthesu-perparticleswithintheframeworkoftherecentunifiedalgebraapproachbyBatalinandTyutin[20–22]isexamined.Inparticular,itisshownthatforD=9massivesuperparticleitisimpossibletoconstructDirac-typebracketpossessing(strong)Jacobiidentityinafullphasespace.1IntroductionThesuperparticlecovariantquantizationproblemhaslongbeenrealized[1–4]toconsistinadequateextensionoftheinitialphasespace[5].Therewereanumberofattemptsinthisdirection.Themostsuccessfulapproachestodatearetwistor-likeformulations[3,6–10],harmonicsuperspacetechnique[11–15]andthenull-vectorsapproachofRef.2.Anothersighttotheproblemliesinthefactthat,insteadofquantizingtheoriginalBrink–Schwarztheory,itisconstructedthesuperparticlemodel[16,17andreferencestherein]whichwillleadafterquantizationtothecovariantSYM.Thekeyideaoftheharmonicsuperspaceapproach[13]wastointroduceadditionalharmonicvariables,whichplayedtheroleofabridgebetweentheSO(1,D−1)indicesandsomeinternalspaceindices,tosplittheinitialfermionicconstraintsintothefirstandsecondclassparts.Thentwistor-likevariablesmightbeused[3,13]toconvertthesecondclassconstraintsintothefirstclassones,whatbroughtthetheorytotheformadmittingconventionalcanonicalquantization.1However,theintroducedauxiliaryvariablesturnouttobevariousfordifferentdimensions[3,12].Theresultingconstraintsystemcanbeirreducibleorinfinitelyreducibledependingonthedimension[3].Inthelattercasetherearisesanadditionalseriousproblembeingconnectedwithconstructingafunctionalintegralforthemodel[2,3].Withintheframeworkofoperatorquantization,thewavefunctionsdependontwistororharmonicvariableswhatmakesunderstandingtheresultsintermsofordinary(super)fieldsdifficult(inthespecialcaseofacompactLorentz-harmonicsuperspace,however,theproblemcanbesolved[12]).Additionalsourceofdifficultiesliesinthegeneralstatusoftheconversionmethoditself.Actually,althoughtheapproachisknownforalongtime[27],itstillremainsunclearwhetherasystemafterconversionisalwaysphysicallyequivalenttotheoriginalone.Thegeneralformalismmaynowofferaproofoftheequivalencewhichisessentiallylocal[28].Suchaconsiderationisenoughforthecaseofaconventionalperturbativefieldtheory,butitislikelyunabletotakeintoaccounttheeffectofthereducedphasespaceglobalgeometrywhichmayhaveasignificantinfluenceonaphysicalspectrumoftheparticlemodel.Inviewoftheallmentionedproblematicpointsoftheconversionmethoditseemsinterestingtostudyanalternativeapproachdealingwiththesuperparticleinitsoriginalformulation[5].Inthisconnection,itisrelevanttomentionthe“unifiedalgebra”approachrecentlydeveloped[20–22]justforthesystemswithmixedfirst-andsecond-classconstraints.Inprinciple,theproposedconstructiondoesnotrequireanexplicitseparationofthefirstandsecondclassconstraints(whatisjustthebasicproblemofthesuperparticle,superstringmodels).Anapplicationoftheprocedureforconcretetheories,however,impliestheexistenceofsomeclassicalbracket(withalltherankandalgebraicpropertiesofthestandardDiracone)asaboundaryconditiontothebasicgeneratingequations[21].Althoughthegeneralconstructiondoesnotincludeanalgorithmofbuildingthisbracket,itisimpliedtobeknown“fromtheoutset”.Inthepresentpaperweproposethegeneralschemeofconstructingaconsistentco-variantDirac-typebracketformodelswithmixedfirstandsecondclassconstraints.ApossibilitytoapplythisbracketinthecontextofthequantizationmethoddevelopedinRefs.20–22isexaminedinthework.Therearetwonaturalwaysofbuildingthebracketwithneededproperties.Firstofthemconsistsinsplittingtheinitialconstraintsintoinfinitelyreduciblefirstandsecondclassparts(bymakinguseofsomecovariantprojectors)andsubsequentgeneralizingthestandardDiracbracketconstructiontothecaseofinfinitelyreduciblesecondclassconstraints.Thesecondlineistowritedownthemostgeneralansatzforthebracketand1WemostlydiscussN=1,D=10caseforwhichmanifestlycovariantquantizationistheprincipalproblem.thentorequireallneededrankandalgebraicpropertiesfortheconstruction(whatwillspecifythecoefficientfunctionsoftheansatz).Possibilitiestoconstructthebracketsofthefirstkindforthesuperparticle,superstringmodelswereexaminedinRefs.2,11,18,19,and25.ItseemssurprisingbuttheJacobi