Generally covariant quantization and the Dirac fie

arXiv:gr-qc/0606033v45Apr2007GenerallycovariantquantizationandtheDiracfieldM.LeclercSectionofAstrophysicsandAstronomy,DepartmentofPhysics,UniversityofAthens,GreeceJune7,2006AbstractCanonicalHamiltonianfieldtheoryincurvedspacetimeisformulatedinamanifestlycovariantway.SecondquantizationisachievedinvokingacorrespondenceprinciplebetweenthePoissonbracketofclassicalfieldsandthecommutatorofthecorrespondingquantumoperators.TheDiractheoryisinvestigatedanditisshownthat,incontrasttothecaseofbosonicfields,incurvedspacetime,thefieldmomentumdoesnotcoincidewiththegener-atorsofspacetimetranslations.ThereasonistracedbacktothepresenceofsecondclassconstraintsoccurringinDiractheory.Further,itisshownthatthemodificationoftheDiracLagrangianbyasurfacetermleadstoamomentumtransferbetweentheDiracfieldandthegravitationalbackgroundfield,resultinginatheorythatisfreeofconstraints,butnotmanifestlyhermitian.1IntroductionQuantizationincurvedspacetimehasalonghistoryandthereexistsanequallylonglistofproblemsrelatedtothesubject.Itisnotourintentiontogiveareviewofthoseissues(see,e.g.,[1]foradiscussionofmanyoftherelatedproblemsaswellasforalistofrelevantreferences).Here,instead,wewish,inacertainsense,tostartfromzeroandinvestigatesomeoftheconsequencesofastraightforwardcanonicalquantizationperformedincurvedspacetimethatariseindependentlyofeventualadditionalproblemslikethosedescribedin[1]andmanyotherarticles(e.g.,theobserverindependentconceptofaparticle).Webaseourinvestigationontheprincipleofrelativity,whichstatesthat,locally,wecannotdistinguishbetweengravitationalandinertialfields.Therefore,ifwecanputagivenspecialrelativistictheoryintoamanifestlygenerallycovariantformandperform,alwaysinamanifestlycovariantway,thesecondquantization,thentheincorporationofgravitationalbackgroundfieldswillbetrivial,sinceallourrelationswillremainidenticalinform.(Thisholdsaslongaswesticktotheminimalcouplingprinciple,avoidingthusexplicitcurvaturecouplings.)Accordingtothisprocedure,thespecialrelativistictheorydictatestheformofthecorrespondingtheoryingravitationalbackgroundfields.Ifthe1resultingtheoryisnotfreeofproblems,thenthereareessentiallythreepossibilities:First,theproblemsmightbesolvedinsomeway.Thisisthepointofviewweadopthere.Inmanycases,thisinvolvesdifficultiesconcerningtheinterpretationofcertainresults,whichmightbestraightforwardinflatspacetime,butleadstoambiguitiesincurvedspacetime.Second,thecanonicalquantizationproceduremightnotbecompletelycorrect,andthereforeleadstoproblemswhich,bychance,donotmanifestthemselvesinthespecialrelativisticlimit.Wedonotinvestigatethispossibilityhere.Finally,thereisthepossibilitythatitissimplynotpossibletoperformsecondquantizationinamanifestlycovariantway,andthus,ultimately,inanobserverindependentway.Otherwisestated,insuchacase,thequantizationprocessdoesnotcommutewiththechangeofthecoordinatesystem,i.e.,wewillnecessarilyhavetofixfirstthecoordinatesystem,andthenperformthequantization.Obviously,thisisnotaveryattractiveoption,sinceitwouldessentiallymeanthatatthequantumlevel,theprincipleofrelativityceasestobevalid.Assumingthatthecanonicalquantizationprocedureisvalidandcanbeperformedinagen-erallycovariantway,weproceedstraightforwardlytotheanalysisofbothbosonicandfermionictheories.Westartbytheformulationofclassicalfieldtheory,whereweuseadirectgeneraliza-tionofthePoissonbracketusedbyOzaki[2]intheframeworkofspecialrelativistictheories,andperformthequantizationofthetheorybyinvokingthetraditionalcorrespondenceprinciplebetweenPoissonbracketsandcommutators(oranticommutators).Theresultingquantumthe-oryisessentiallyacurvedspacetimegeneralizationofSchwinger’smanifestlyLorentzcovariantformulationofspecialrelativisticfieldtheory[3].Ofparticularinterestistheresultthat,inDiractheory,thegeneratorsofspacetimetrans-lationsarenotgivenintermsofthefieldmomentumoperator.Thus,inparticular,thetimecomponentofthefieldmomentum,whichisconventionallyreferredtoasfieldHamiltonian,doesnotgeneratethetimeevolutionofthequantumfields,asisthecaseinflatspacetime.Instead,itturnsoutthatthefieldmomentumgeneratesakindofgeneralizedtranslationswhicharedirectlyrelatedtothehermitianmomentumoperators˜pkderivedinourpreviousarticle[4].ThereasonisfoundintheoccurrenceofsecondclassconstraintsinDiractheory.Further,analternativewaytoquantizetheDiracfieldispresented,wheretheLagrangianismodifiedbyasurfaceterm.Thesurfacetermisshowntoleadtoachangeinthefieldmomentum,thatcanbeinterpretedasamomentumtransferbetweentheDiracfieldandthegravitationalbackgroundfield.Theresultingtheoryisfreeofconstraints,andthegeneratorsofspacetimetranslationsarenowgivendirectlyintermsofthefieldmomentumoperator.However,thetheoryisnotmanifestlyhermitian,andthesymmetrybetweenthefieldvariablesψand¯ψisbroken,thelatterplayingmerelytheroleofaLagrangemultiplier.Mostinteresting,inthespecialrelativisticlimit,boththehermitianandthenon-hermitianformulationsturnouttobeequivalent,whichisthereasonwhytheissueofhowwedealconsistentlywiththeconstraintsinDiractheoryisusuallypassedoverintherelatedliterature.Thearticleisorganizedasfollows.Inthenextsection,weexplainournotationsa