Equivalence of Faddeev-Jackiw and Dirac approaches

arXiv:hep-th/9610067v19Oct1996EquivalenceofFaddeev-JackiwandDiracapproachesforgaugetheoriesJ.AntonioGarc´ia∗andJosepM.PonsDepartamentd’EstructuraiConstituentsdelaMat`eriaUniversitatdeBarcelonaAv.Diagonal64708028BarcelonaCatalonia,SpainAbstractTheequivalencebetweentheDiracmethodandFaddeev-Jackiwanalysisforgaugetheoriesisproved.Inparticularwetraceout,inastagebystageprocedure,thestandardclassificationoffirstandsecondclassconstraintsofDirac’smethodintheF-Japproach.WealsofindthattheDarbouxtransformationimpliedintheF-JreductionprocesscanbeviewedasacanonicaltransformationinDiracapproach.UnlikeDirac’smethodtheF-Janalysisisaclassicalreductionprocedure,thenthequantizationcanbeachievedonlyintheframeworkofreduceandthenquantizeapproachwithalltheknowproblemsthatthistypeofprocedurespresents.Finallyweillustratetheequivalencebymeansofaparticularexample.UB-ECM-PF95/15hep-th/9610067∗OnleavefromInstitutodeCienciasNucleares,UniversidadNacionalAut´onomadeM´exico,ApartadoPostal70-543,04510M´exico,D.F.E-mail:antonio@teorica0.ifisicacu.unam.mx1J.A.Garc´ia,J.M.Pons,‘EquivalenceofF-JandDirac...’21IntroductionTheclassicaltreatmentofthedynamicsofgaugetheories(includingthosewithrepara-metrizationinvariance)describedbyavariationalprinciplewasfirstsolvedbyDirac[1]andBergmann[2]intheearly50’s.DiracwasmainlyinterestedintheHamiltonianapproachtogeneralrelativity,andhisleadingeffortsinthisfieldwerecompletedinthe60’swiththeADMformalism[3].TheapplicationofDirac’sworktogaugetheorieslikeEMorYMcamelaterinthe70’s(forinstance,thebasiccanonicalcommutationsfortheelectromagneticpotentialinCoulombgauge,firstobtainedthroughheuristicmethods,werethenunderstood[4]asadirectapplicationoftheDiracbracket).Alsointhe70’sthegeometrizationofDirac’smethodwassuccessfullyaddressed[5]andinthe80’sthereappearedgeneralproofsoftheequivalenceofthe(classical)HamiltonianandLagrangiantreatmentforgaugetheories[6].Withregardtothequantizationpro-gram,Dirac’sapproachopenedthewaytothesocalledDiracmethodofquantization,whereconstraintsareimplementedasoperatorsinHilbertspace.Theequivalenceofthismethodwiththesocalledreducedquantization(i.e.,whentheclassicaldegreesoffreedomareeliminatedbeforequantization)isstillasubjectofcontroversy[7].Letusmentionalsothediscoveryinthe70’softheBRSTsymmetry[8],whichappearedasareminiscenceoftheclassicalgaugesymmetryafterfixingthegaugethroughFaddeev-Popovprocedure[9].Inthissense,theclassicalapproachbyDirachaditsquantumcontinuationthroughtheBRSTmethods.Thesemethodshaveprovidedthemostpow-erfultool,thefield-antifieldformalism,toquantizeanykind(reduciblegaugealgebra,softalgebra,openalgebra,etc.)oflocalgaugefieldtheory[10].ThemainfeaturesofDiracapproach,eitherinLagrangianorHamiltonianformalismarea)thepossibilitytokeepallthevariablesinphasespaceorvelocityspace,b)theconstructionofanalgorithmtodetermine,throughastepbystepprocedure,thefinalconstraintsurfacewherethemotiontakesplace,c)theelucidationofthetruedegreesoffreedomofthetheory,separatedfromthegauge–andhenceunphysical–ones.Ingeneral,quotientingoutthegaugedegreesoffreedomcanbedoneby“fixingthegauge”,whichamountstotheintroductionofanewsetofconstraintsfromtheoutset,theGaugeFixingconstraints.Morerecently,anewmethodtoclassicallydealwithgaugetheorieswasdevisedthroughtheworkbyFaddeevandJackiw[11].UnlikeDirac’s,intheF-Jmethod,thevariablesarereducedtothephysicalonesand,atleastformally,theprocedurelookssimpler.WehaveconsideredthatitisworthtoexploreF-Jmethodtoseetowhatextentitdiffers–ifitdoes–fromDirac’s,andwhataretheadvantagesofeachone.Letusimmediatelystateourconclusion:thoughtechnicallydifferent,bothmethodsareequivalent,asitshouldbe.ThemaindifferenceinprocedureisthatF-Jmethodisamethodofreductiontothephysicaldegreesoffreedom–anditsvariationalprinciple.ItistrueneverthelessthatDirac’smethodalsoincludesthepossibilitytoeliminatevariables.Infact,theadoptionoftheDirac’sbracket(whichisthebracketassociatedtothesymplecticformthatonecandefineinthesecondclassconstraintsurfaceastheprojectionofthesymplecticforminphasespace)allowsfortheeliminationofavariableforeverysecondclassconstraint.Instead,intheF-Jprocedure,eveninthecasewhenJ.A.Garc´ia,J.M.Pons,‘EquivalenceofF-JandDirac...’3somefirstclassconstraintsarepresent,thereductioncanbestillperformed,althoughtheprojectedsymplecticformbecomesdegenerate.Thisimpliesthatsomedynamicalvariablesanditsequationsofmotionmaybelost.AkeyfeatureofF-Jprocedureisthatthisfactdoesnotaffectthephysicalcontentofthetheory.Aswewillseeinthenextsection,F-Jprocedurefocusesexclusivelygettingtheequationsofmotion(or,equivalently,itsassociatedvariationalprinciple)forthesetofphysicalvariables,discardinganythingelse.SomeofthevariableswhichinDirac’sapproacharerelatedtothephysicalonesthroughconstraintsarequicklyeliminatedinF-Jmethodtogetherwithothervariableswhicharegoingtobecomegaugedegreesoffreedom.ThisfactexplainstheefficiencyofF-Jmethod:Itdoesnotproducessuperfluousinformationthatisgoingtobediscardedlateron,contrarytowhathappensinDirac’s,wherewecankeepthissuperfluousinformationtilltheend.Obviously,thisefficiencymustpaysomeprice.Apartfromthetechnic