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1、PHYSICALREVIEWVOLUME160,NUMBER525AUGUST1967QuantumTheoryofGravity.I.TheCanonicalTheory*BRYCES.DEWITTInstitnteforAdvancedStldy,Prenceton,NewJerseyDepartmentofPhysics,UniversityofNorthCarolina,ChapelHill,NorthCarolenal'(Received25July1966;revisedmanuscriptreceived9January1967)Follo
2、winganhistoricalintroduction,theconventionalcanonicalformulationofgeneralrelativitytheoryispresented.ThecanonicalLagrangianisexpressedintermsoftheextrinsicandintrinsiccurvaturesofthehypersurfacex=constant,anditsrelationtotheasymptoticfieldenergyinaninfiniteworldisnoted.Thedistinc
3、tionbetweenfiniteandinfiniteworldsisemphasized.Inthequantumtheorytheprimaryandsecond-aryconstraintsbecomeconditionsonthestatevector,andinthecaseoffiniteworldstheseconditionsalonegovernthedynamics.Aresolutionofthefactor-orderingproblemisproposed,andtheconsistencyoftheconstraintsis
4、demonstrated.A6-dimensionalhyperbolicRiemannianmanifoldisintroducedwhichtakesforitsmetricthecoeKcientofthemomentaintheHamiltonianconstraint.Thegeodesicincompletabilityofthismanifold,owingtotheexistenceofafrontierofinfinitecurvature,isdemonstrated.Thepossibilityisexploredofrelatin
5、gthismanifoldtoaninfinite-dimensionalmanifoldof3-geometries,andofrelatingthestructureofthelattermanifoldinturntothedynamicalbehaviorofspace-time.TheproblemisapproachedthroughtheWEBapproximationandHamilton-Jacobitheory.Einsteinsequationsarerevealedasgeodesic"equationsinthemanifold
6、of3-geometries,modifiedbythepresenceofa"forceterm.Theclassicalphe-nomenonofgravitationalcollapseshowsthattheforcetermisnotpowerfulenoughtopreventthetrajectoryofspace-timefromrunningintothefrontier.Theas-yetunresolvedproblemofdeterminingwhenthecol-lapsephenomenonrepresentsarealbar
7、riertothequantum-statefunctionalisbrieflydiscussed,andaboundaryconditionatthebarrierisproposed.Thestatefunctionalofafiniteworldcandependonlyonthe3-geometryofthehypersurfacex=constant.Thelabelx'itselfisirrelevant,and"time"mustbedeter-minedintrinsically.Anaturaldefinitionfortheinne
8、rproductoftwosuchstatefunctionalsisintro
