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arXiv:hep-th/9502023v13Feb1995February1,2008TheQuantizationofAnomalousGaugeFieldTheoryM.Martellini⋆I.N.F.N.,SezionediRoma”LaSapienza”,Roma,ItalyM.SpreaficoDipartimentodiMatematica,Universit`adiMilano,MilanoandI.N.F.N.,SezionediMilano,ItalyK.YoshidaDipartimentodiFisica,Universit`adiRoma,RomaandI.N.F.N.,SezionediRoma,ItalyABSTRACTWediscussthesocalledgaugeinvariantquantizationofanomalousgaugefieldtheory,originallyduetoFaddeevandShatashvili.ItispointedoutthatthefurthernoninvarianceofrelevantpathintegralmeasuresposesaproblemwhenonetriestotranslateittoBRSTformalism.Themethodbywhichweproposetogetaroundofthisproblemintroducescertainarbitrarinessinthemodel.Wespeculateonthepossibilityofusingsuchanarbitrarinesstobuildseriesofnonequivalentmodelsoftwodimensionalinducedgravity.⋆OnleaveofabsencefromDipartimentodiFisica,Universit´adiMilano,Milano,ItalyandI.N.F.N.,SezionediPavia,Italy1.IntroductionTheconsistentquantizationof(classical)gaugeinvariantfieldtheoryrequiresthecompletecancellationofanomalies[1][2].Here,”consistent”meansthatwewantnotonlytorequirerenormalizability(perturbativefiniteness),butalsouni-tarityofS-matrix,non-violationofLorentzinvarianceetc.Moreover,inphysical4dworld,anomalycancellationconditionitselfoftenleadstothephysicalpredic-tions.ThewellknownexampleistheequalityofnumbersofquarksandleptonsintheStandardModelofWeinbergandSalam.Onlowerdimensional(eg.d=2)fieldtheory,thecancellationofanomaliesisstillthecrucialingredientforthemodelbuilding.Thecriticalstringdimensiond=26isoftenquoted[3]astheconsequenceofanomalyfreeconditionforbosonicstring(althoughinthisexamplethecancellationofanomalydoesnotguaranteefullconsistencyofthemodelinabovesense,duetothepresenceoftachyons).Inthecaseoflowerdimensionalfieldtheory(d4),oneoftentriestoquantizeagaugefieldtheorywhenthereisnowayofcancellingitsanomaly.TheclassicalexampleofthissituationistheattempttothequantizationofchiralSchwingermodelbyJackiwandRajaraman[4][5].Theyhaveshownthatthemodelcanbeconsistentlyquantized(=freefieldtheory)evenwhenthegaugeinvarianceisbrokenthroughanomaly.Ingeneralthereseemtobetwowaysforattemptingthequantizationofanoma-lousgaugefieldtheory:1)GaugenoninvariantmethodOneignoresthebreakingofgaugesymmetryandtrytoshowthatthetheorycanbequantizedevenwithoutthegaugeinvariance.TheexampleofthisapproachistheaboveJackiw-RajaramanquantizationofthechiralSchwingermodel.Theproblemhereisthatitisnoteasytodevelopthegeneraltechnicscoveringwide1classofphysicallyrelevantmodelswithanomaly.2)GaugeinvariantmethodInthiscase,onefirsttriestorecovergaugeinvariancebyintroducingnewdegreesoffreedom.Thetheoryisanomalouswhenonecannotfindlocalcountertermtocancelthegaugenoninvarianceduetotheoneloop”matter”integralsinpresenceofgaugefields,bymakinguseexclusivelyofthedegreesoffreedom(fields)alreadypresentintheclassicalaction.Inref.[6],FaddeevandShatashvili(FS)havetriedtojustifytheintroductionofnewdegreesoffreedomwhicharenecessarytoconstructtheanomalycancellingcounterterm.Theirargumentisbasedontheideaofprojectiverepresentationofgaugegroup.Theyobservethattheappearanceofanomalydoesnotmeanthesimplebreakdownof(classical)gaugesymmetry,butitrathersignalsthatthesymmetryisrealisedprojectively(thisisrelatedtotheappearanceofanomalouscommutatorsofrelevantcurrents).Sucharealization,throughprojectiverepresen-tations,necessitatestheenlargementofphysicalHilbertspace.Thustheyarguedthattheintroductionofnewfieldsinthemodelisnotanadhoc(andlargelyarbitrary)construction.Independentlyoftheir”philosophy”,theFSmethodgivesthegaugeinvariantactionatthepriceofintroducingtheextradegreesoffreedom(generallyphysical).Theseriousproblemofthismethodis,however,thatthegaugeinvariancethus”forced”uponthetheory,doesnotautomaticallyguaranteetheconsistencyofthetheory.Thisisincontrastwithourexperiencewithsome4dmodelssuchastheStandardModel.Forexample,onemayapplytheFSmethodtothecelebratedcaseofchiralSchwingermodel[4][5][5A].Inthiscase,wehavetheclassicalactionS0=Zdz∧d¯z2ih¯ψRγ¯z(¯∂+R)ψR+¯ψLγz∂ψL+14TrF2i2whereψR/L=1±γ52ψR/L=A1±iA2,F=¯∂L−∂R+[R,L](weareusingtheeuclidiannotation).SoisinvariantunderthegaugetransformationψR→ψgR=S(g)ψRψL→ψLAμ=gAμg−1+g∂μg−1foranyg(z,¯z)∈G.Thetheoryisanomalousbecausetheoneloopintegrale−WR(R)=ZDψRD¯ψRexp−Z¯ψRγ¯z(¯∂+R)ψRisnotgaugeinvariantunderR→gRg−1+g¯∂g−1(foranychoiceoftheregularization).FollowingFS’technic(seenextsection)however,onecanintroducethelocalcountertermΛ(R,L;g),(g(z,¯z)∈G)sothatthegaugevariationofΛcancelsthenoninvarianceofWR(R).ThereiscertainarbitrarinessinthechoiceofΛbuttheconvenientoneisΛ(R,L;g)=−αL(L,g)+14πZTr(RL)3whereαL(L,g)=14πh−Zdz∧d¯z2iTr(g−1¯∂g,L)+12Zdz∧d¯z2iTr(g∂g−1,g¯∂g−1)−121Z0dtZdz∧d¯z2iTr(g′∂tg′−1,[g′∂g′−1,g′¯∂g′−1])ig′(0,z,¯z)=1,g′(1,z,¯z)=g(z,¯z)istheWess-Zumino-Novikov-WittenactioncorrespondingtotheanomalyofleftfermionψL,¯ψL(αLisnotgloballyalocalactionbutitissofor”small”g≃1+iξ).Thatis,onecanwriteαL(L,g)=WL(Lg)−WL(L)wheree−WL(L)=ZDψ′LD¯ψ′Lexp−Z¯ψ′Lγz(∂+L)ψ′L(notethatψ′L,¯ψ′LhavenothingtodowithψL,¯ψLinS0).Withthischoiceofcounterterm,onecanshowthatthetheoryisequivalenttoa)freedecoupledfermionψL,¯ψLa