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1、VOLUME57,NUMSER18PHYSICALREVIEWLETTERS3NovEMsER1986NewVariablesforClassicalandQuantumGravityAbhayAshtekarPhysicsDepartmentS,yracuseUniversity,Syracuse,IVewYork13244,andInstituteforTheoreticalPhysics,UniversityofCalifornia,SantaBarbara,SantaBarbara,California93106
2、(Received18Ibm:mber1985;revisedmanuscriptreceived29August1986)AHamiltonianformulationofgeneralrelativitybasedoncertainspinorialvariablesisintroduced.Thesevariablessimplifytheconstraintsofgeneralrelativityconsiderablyandenableonetoimbedtheconstraintsurfaceinthepha
3、sespaceofEinstein'stheoryintothatofYang-Millstheory.Theimbeddingsuggestsne~waysofattackinganumberofproblemsinbothclassicalandquantumgravity.Someillus-trativeapplicationsarediscussed.PACSnumbers:04.60.+n,04.20.FyAttemptsatconstructingperturbativequantumgravitytang
4、entspaceofZandmakesthemSU(2)spinorindices.havebeenunsuccessful.ItisnowgenerallybelievedthatFurthermore,eachsolderingform,ct,isa"squareroot"theproblemliesinthebasicassumptionoftheperturba-ofa(positivedefinite)metricq,bonZ,tiontheorythatthetruespace-timestructureca
5、nbewellABMNgab.a&bgMBNTr&a&b,approximatedbyaclassicalbackgroundgeometryevenbelowthePlanckscale.Fromthisstandpoint,thereislit-andsinglesoutauniquetorsion-freeconnectionDonten-tlehopeinretainingthegeneralperturbativeframeworksorandspinorfieldsonZsatisfyingandsimply
6、changing,e.g.,theformoftheEinsteinLagrangeanbyaddinghigher-derivativetermsorsuper-Dega0,Dcttt0.(2)symmetricmatter.AmorepromisingdirectionistofaceTheconfigurationspace8forgeneralrelativityistotheproblemnonperturbatively.For,ashasbeenem-bethespaceofall(suitably)reg
7、ularand,ifZisnoncom-phasizedby3.Klauderovertheyears,quantumgravitypact,asymptoticallywell-behavedsolderingformsa'"tt.maywellexistasanexacttheoryinspiteofperturba-ThephasespaceIisthecotangentbundleover8.Thus,tivenonrenormalizability.Thecanonicalquantizationapointo
8、fIconsistsofapair(a'"tt,M~tv),whereM,aschemeprovidesanaturalavenueinthisdirectionsinceitdensityofweight1,isthemomentumcanonicallyconju-doesnotrequirethefixatio
