Renormalizable Quantum Gauge General Relativity

arXiv:gr-qc/0309041v18Sep2003RenormalizableQuantumGaugeGeneralRelativityNingWu∗InstituteofHighEnergyPhysics,P.O.Box918-1,Beijing100039,P.R.ChinaFebruary7,2008PACSNumbers:11.15.-q,04.60.-m,04.20.Cv,11.10.Gh.Keywords:generalrelativity,gaugefield,quantumgravity,renormalization.AbstractThequantumgaugegeneralrelativityisproposedintheframeworkofquantumgaugetheoryofgravity.Itisformulatedbasedongaugeprinciplewhichstatesthatthecorrectsymmetryforgravitationalinteractionsshouldbegravitationalgaugesymmetry.Thegravitationalgaugegroupisstudiedinthepaper.Thengravitationalgaugeinteractionsofpuregravitationalgaugefieldisstudied.ItisfoundthatthefieldequationofgravitationalgaugefieldisjusttheEinstein’sfieldequation.Afterthat,thegravitationalinteractionsofscalarfield,Diracfieldandvectorfieldsarestudied,andunificationsoffun-damentalinteractionsarediscussed.Pathintegralquantizationofthetheoryisstudiedinthepaper.Thequantumgaugegeneralrelativitydiscussedinthispaperisaperturbativelyrenormalizablequantumgravity,whichisoneofthemostimportantadvantageofthequantumgaugegeneralrelativityproposedinthispaper.Astrictproofontherenormalizabilityofthetheoryisalsogiveninthispaper.Anotherimportantadvantageofthequantumgaugegeneral∗emailaddress:[e-mail]1relativityisthatitcanexplainbothclassicaltestsofgravityandquantumeffectsofgravitationalinteractions,suchasgravitationalphaseeffectsfoundinCOWexperimentsandgravitationalshieldingeffectsfoundinPodklet-novexperiments.Forallclassicaleffectsofgravitationalinteractions,suchasclassicaltestsofgravityandcosmologicalmodel,quantumgaugegeneralrelativitygivesoutthesametheoreticalpredictionsasthatoftheEinstein’sgeneralrelaitvity.CONTENTS1.Introduction2.TheTranscendentalFoundations3.GravitationalGaugeGroup4.TwoPicturesofGravity5.PureGravitationalGaugefields6.GravitationalInteractionsofScalarFields7.GravitationalInteractionsofDiracField8.GravitationalInteractionsofVectorField9.GSU(N)UnificationModel10.UnificationofFundamentalInteractions11.ClassicalLimitofQuantumGaugegeneralRelativity12.PathIntegralQuantizationofGravitationalGaugeField13.Renormalization14.ClassicalTestsofQuantumGaugeGeneralRelativity15.QuantumEffectsofGravitationalInteractions16.Discussions21IntroductionIn1686,IsaacNewtonpublishedhisbookMATHEMATICALPRINCIPLESOFNATURALPHILOSOPHY.Inthisbook,throughstudyingthemotionofplanetinsolarsystem,hefoundthatgravityobeystheinversesquarelawandthemagnitudeofgravityisproportionaltothemassoftheobject[1].TheNewton’sclassicaltheoryofgravityiskeptunchangeduntil1916.Atthatyear,EinsteinproposedGeneralRelativity[2,3].Inthisgreatwork,hefoundedarelativistictheoryongravity,whichisbasedonprincipleofgeneralrelativityandequivalenceprinciple.Newton’sclas-sicaltheoryofgravityappearsasaclassicallimitofgeneralrelativity.Oneofthebiggestrevolutioninhumankindinthelastcenturyisthefoundationofquantumtheory.ThequantumhypothesiswasfirstintroducedintophysicsbyMaxPlankin1900.In1916,AlbertEinsteinpointsoutthatquantumeffectsmustleadtomodificationsinthetheoryofgeneralrelativity[4].Soonafterthefounda-tionofquantummechanics,physiciststrytofoundatheorythatcoulddescribethequantumbehaviorofthefullgravitationalfield.Inthe70yearsattempts,physi-cistshavefoundtwotheoriesbasedonquantummechanicsthatattempttounifygeneralrelativityandquantummechanics,oneiscanonicalquantumgravityandan-otherissuperstringtheory.Butforquantumfieldtheory,therearedifferentkindsofmathematicalinfinitiesthatnaturallyoccurinquantumdescriptionsoffields.Theseinfinitiesshouldberemovedbythetechniqueofperturbativerenormaliza-tion.However,theperturbativerenormalizationdoesnotworkforthequantizationofEinstein’stheoryofgravity,especiallyincanonicalquantumgravity.Insuper-stringtheory,inordertomakeperturbativerenormalizationtowork,physicistshavetointroducesixextradimensions.Butuptonow,noneoftheextradimensionshavebeenobserved.Tofoundaconsistenttheorythatcanunifygeneralrelativityandquantummechanicsisalongdreamforphysicists.The”relativityrevolution”andthe”quantumrevolution”areamongthegreatestsuccessesoftwentiethcenturyphysics,yettwotheoriesappearstobefundamentallyincompatible.Generalrelativityremainsapurelyclassicaltheorywhichdescribesthegeometryofspaceandtimeassmoothandcontinuous,onthecontrary,quan-tummechanicsdivideseverythingintodiscretequanta.Theunderlyingtheoreticalincompatibilitybetweentwotheoriesarisesfromthewaythattheytreatthegeom-etryofspaceandtime.ThissituationmakessomephysicistsstillwonderwhetherquantumtheoryisatrulyfundamentaltheoryofNature,orjustaconvenientde-scriptionofsomeaspectsofthemicroscopicworld.Somephysicistsevenconsiderthetwentiethcenturyasthecenturyoftheincompleterevolution.Tosetupacon-sistentquantumtheoryofgravityisconsideredtobethelastchallengeofquantumtheory[5,6].Inotherwords,combininggeneralrelativitywithquantummechanics3isconsideredtobethelasthurdletobeovercomeinthe”quantumrevolution”.In1921,H.Weylintroducedtheconceptofgaugetransformationintophysics[7,8],whichisoneofthemostimportantconceptsinmodernphysics,thoughhisoriginaltheoryisnotsuccessful.Later,V.Fock,H.WeylandW.Paulifoundtha