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1、NuclearPhysicsB50(1972)318-353.North-HollandPublishingCompanyCOMBINATORICSOFGAUGEFIELDSG.'tHOOFTandM.VELTMANInstituteforTheoreticalPhysics,UniversityofUtrecht*Received31July1972Abstract:GaugefieldtheoriescanbedescribedbymanydifferentsetsofFeynmanrules
2、,de-pendingontheparticulargaugechosen.InthispaperaprescriptionforobtainingtheFeynmanrulesindifferentgaugesisgiven.Arigorouscombinatorialproofoftheinde-pendenceoftheS-matrixofthechosengaugeispresented.TheproofisgeneralandappliestoYang-MiUstypetheoriesa
3、swellastogravitation.ForrenormalizableYang-Millstypetheoriesitisshownthattherenormalizedtheoryisinvariantwithrespecttorenormalizedgaugetransformations.1.INTRODUCTIONAgaugefieldtheoryisdefinedtobeanyquantumfieldtheorywhichhasalocalgaugeinvariance[1].Th
4、elocalgaugegroupmaybeany(compactornoncompact)Liegroup.Examplesare:(i)MasslessYang-MiUsfields[2];(ii)MassiveYang-MiUsfieldsifthemassofthevectorbosonsisduetotheHiggs-Kibblemechanism[3,4,5];('di)Quantumtheoryofgravitation.Itappearsthatagaugefieldtheoryca
5、nbedescribedbymanydifferentsetsofFeynmanrulescorrespondingtodifferentgauges.Oneofthesesetswillonlycon-taininternallinesthatcorrespondtophysicalparticles(thatis,noghosts).Insuchaset,unitarityoftheS-matrixisrelativelysimpletoprove.Itwillbecalledthephy-s
6、icalgaugefromnowon.Ontheotherhand,somegaugefieldtheoriesturnouttohavealsoamanifestlyrenormalizablesetofFeynmanrules[4].Ifindeedthesedif-ferentsetsofFeynmanrulesdescribethesameS-matrix,andifthisremainstrueafterrenormalizationthenwehaveunitaryandrenorma
7、lizablevectorfieldtheories.ThefactthattheS-matrixisgaugeindependenthasonlybeenshowninaveryformalway,basedonpathintegrals(apartfromsomespecialcases[2,5]).This"proof"hastheshortcomingthatitignoresallinfinities,anditispreciselythoseinfinitieswhich*Postal
8、address:Maliesingel23,Utrecht,theNetherlands.G.'tHoofl,M.Veltman,Combinatoricsofgaugefields319cancausethewellknownanomalies.Theseanomaliescaninvalidatethewholetheory[8,9].Moreoverwehavenoinsightintherenormalizationprocedureandthepropertiesofth
