Quantum Theory of Gravity. II. The Manifestly Covariant Theory文稿

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1、I'HVSICAIREVIEWVOLUME162,NUMBER52&OCTOBER1967QuantumTheoryofGravity.II.TheManifestlyCovariantTheory*BRVCES.DEWITTInstituteforAdvancedStudy,Princeton,NewJerseyCwdDepartmentofPhysics,UniversityofNorthCarolina,Chape/Hill,NorthCaroiinat(Received25July1966;revisedmanuscriptreceived9January1967)Contrary

2、tothesituationwhichholdsforthecanonicaltheorydescribedinthefirstpaperofthisseries,thereexistsatpresentnotractablepureoperatorlanguageonwhichtobaseamanifestlycovariantquantumtheoryofgravity.Onemustconstructthetheorybyanalogywithconventional5-matrixtheory,usingthec-numberlanguageofFeynmanamplitudesw

3、hennothingelseisavailable.Thepresentpaperundertakesthisconstruction.Itbeginsatanelementarylevelwithatreatmentofthepropagationofsmalldisturbancesonaclassicalbackground.Theclassicalbackgroundplaysafundamentalrolethroughout,bothasatechnicalinstrumentforprobingthevacuum(i.e.,analyzingvirtualprocesses)

4、andasanarbitraryfiducialpointforthequantumfluctuations.Theproblemofthequantizedlightconeisdiscussedinapreliminaryway,andtheformalstructureoftheinvariancegroupisdisplayed.AcondensednotationisadoptedwhichpermitstheYang-Millsfieldtobestudiedsimultaneouslywiththegravitationalfield.GenerallycovariantGr

5、eensfunctionsareintroducedthroughtheimpositionofcovariantsupplementaryconditionsonsmalldis-turbances.ThetransitionfromtheclassicaltothequantumtheoryismadeviathePoissonbracketofPeierls.Commutationrelationsfortheasymptoticfieldsareobtainedandusedtodefinetheincomingandoutgoingstates.Becauseofthenon-A

6、beliancharacterofthecoordinatetransformationgroup,theseparationofpropagateddisturbancesintophysicalandnonphysicalcomponentsrequiresmuchgreatercarethaninelectrodynamics.Withtheaidofacanonicalformforthecommutatorfunction,twodistinctFeynmanpropagatorsrelativetoanarbitrarybackgroundaredefined.Oneofthe

7、seismanifestlyco-variant,butpropagatesnonphysicalaswellasphysicalquanta;theotherpropagatesphysicalquantaonly,butlacksmanifestcovariance.Thelatterisusedtodefineexternal-linewavefunctionsandnon-radiatively-correcte