Unified Field Theory From Enlarged Transformation

arXiv:gr-qc/0401090v121Jan2004UNIFIEDFIELDTHEORYFROMENLARGEDTRANSFORMATIONGROUP.THECONSISTENTHAMILTONIANDavePandres,Jr.andEdwardL.Green1(Pandres,D.andGreen,E.,Unifiedfieldtheoryfromenlargedtransformationgroup.TheconsistentHamiltonian.,InternationalJournalofTheoreticalPhysics,42,1849-1873.)1DepartmentofMathematicsandComputerScience,NorthGeorgiaCollegeandStateUniversity,Dahlonega,Georgia30597,[phone],[e-mail]reviouslyinwhichthegeometricalstructureofarealfour-dimensionalspacetimemanifoldisexpressedbyarealorthonormaltetrad,andthegroupofdiffeomorphismsisreplacedbyalargergroup.Thegroupenlargementwasaccomplishedbyincludingthosetransformationstoanholonomiccoordinatesunderwhichconservationlawsarecovariantstatements.Fieldequationshavebeenobtainedfromavariationalprinciplewhichisinvariantunderthelargergroup.ThesefieldequationsimplythevalidityoftheEinsteinequationsofgeneralrela-tivitywithastress-energytensorthatisjustwhatoneexpectsfortheelectroweakfieldandassociatedcurrents.Inthispaper,asafirststeptowardquantization,aconsistentHamiltonianforthetheoryisobtained.Someconcludingremarksaregivenconcerningtheneedforfurtherdevelopmentofthetheory.Theseremarksincludediscussionofapossiblemethodforextendingthetheorytoincludethestronginteraction.1.INTRODUCTION.InSections1and2,wedescribeatheoryinwhichtheclassical(unquantized)gravitationalandelectroweakfieldsappearasmanifesta-tionsofgeometricalstructureinarealfour-dimensionalspace-timemanifold.InSection3,weobtaintheHamiltonianforthetheoryasafirststeptowardquantizingthetheory.InSection4,wemakesomeconcludingremarksconcerningthefurtherdevelopmentofthetheory.Oneoftheseremarkssuggestsamethodforextendingthetheorytoincludethestronginteraction.[NOTE:Inseveralpriorpapers,oneofus(Pandres,1981,1984A,1984B,1995,1998,1999),hasbasedthetheory,notonamanifold,butonaspaceinwhichpaths,ratherthanpointsaretheprimaryelements.Inthispaper,however,weshowthatthetheorycanbebasedentirelyonamanifold.TypesetbyAMS-TEXItiswellknownthatanygeneralrelativisticmetricgμνmaybeexpressedintermsofanorthonormaltetradofvectorshiμ.Theexpressionisgμν=gijhiμhjν(1)wheregij=gij=diag(−1,1,1,1),andthesummationconventionhasbeenadopted.Indicestakethevalues0,1,2,3,andgμνisdefinedbygμνgνα=δμα,whereδμαistheKroneckerdelta.Latin(tetrad)indicesareraisedandloweredbyusinggijandgij,justasGreek(spacetime)indicesareraisedandloweredbyusinggμνandgμν.Partialdifferentiationisdenotedbyacomma.CovariantdifferentiationwithrespecttotheChristoffelsymbolΓαμν=12gασ(gσμ,ν+gσν,μ−gμν,σ)isdenotedbyasemicolon.1.1.Motivation.Werecall(Pandres1962,1999)anargumentwhichisagen-eralizationofthe“elevator”argumentthatledEinsteinfromspecialrelativitytogeneralrelativity.Thespecialrelativisticequationofmotionforafreeparticleisd2xids2=0,(2)where−ds2=gijdxidxj.Considertheimage-equationofthisfree-particleequationunderthetransformationdxi=hiμdxμ(3)wherethecurlfiμν=hiν,μ−hiμ,νisnotzero.Eq.(3)establishesaone-to-onecorrespondencebetweencoordinateincrementsdxianddxμ.Sincehiν,μ−hiμ,νisnotzero,wecannotintegrateEq.(3)togetaone-to-onecorrespondencebetweencoordinatesxiandxμ.However,itfollowsfromEq.(3)thatdxids=hiμdxμds.Upondifferentiatingthiswithrespecttos,usingthechainrule,andmultiplyingbyhiα,weseethatEq.(2)maybewrittend2xαds2+hiαhiμ,νdxμdsdxνds=0.(4)WefollowEisenhart(1925)indefiningRiccirotationcoefficientsbyγiμν=hiμ;ν=hiμ,ν−hiσΓσμν.Multiplicationbyhiαgiveshiαhiμ,ν=Γαμν+γαμν,anduponusingthisinEq.(4)wehaved2xαds2+Γαμνdxμdsdxνds=−γαμνdxμdsdxνds.(5)Therelationγμνi=hjμγjναhiαillustratesourgeneralmethodforconvertingbe-tweenGreekandLatinindices.Now,theaffineconnectionforspiningeneralrelativityisexpressedintermsoftheRiccirotationcoefficientsbyΓμ=18γijμγiγj−γjγi
+aμI,wheretheγiaretheDiracmatricesofspecialrelativity,Iistheidentitymatrix,andaμisanarbitraryvector.Itiswellknownthatthespinconnectioncontainscompleteinfor-mationabouttheelectromagneticfield,andthatonehalfofMaxwell’sequationsareidenticallysatisfiedonaccountoftheexistenceofthespinconnection.Further-more,themannerinwhichtheelectromagneticfieldentersthespinconnectionis2inagreementwiththeprincipleofminimalelectromagneticcoupling.Anunder-standingofthespinorcalculusinRiemannspace,andtheroleplayedbythespinconnection,wasgainedthroughtheworkofmanyinvestigatorsduringthedecadeafterDirac’sdiscoveryoftherelativistictheoryoftheelectron;see,e.g.,BadeandJehle(1953)forageneralreview.Manyoftheseinvestigatorsrecognizedthede-scriptionoftheelectromagneticfieldaspartofthespinconnection.AnespeciallyluciddiscussionofthishasbeengivenbyLoos(1963).Thesubsequentunifica-tionoftheelectromagneticandweakfieldsbyWeinberg(1967),andSalam(1968)causesustoexpectthatthespinconnectionmightalsocontainadescriptionoftheweakfield.Wenowrecall(Pandres,1995)calculationsthatsuggestthattheelectroweakfieldisdescribedbyMμνi,the“mixedsymmetry”partofγμνiunderthepermutationgrouponthreesymbols.Onemayobjecttousingγμνitodescribetheelectoweakfieldsinceγijμisusedinthespinconnection.However,thesegeometricobjectscannotbeconsideredtobethesamesincethemethodofconvertingfromonetot