Complexifier Coherent States for Quantum General R

arXiv:gr-qc/0206037v113Jun2002ComplexifierCoherentStatesforQuantumGeneralRelativityT.Thiemann∗MPIf.Gravitationsphysik,Albert-Einstein-Institut,AmM¨uhlenberg1,14476GolmnearPotsdam,GermanyPreprintAEI-2002-045AbstractRecently,substantialamountofactivityinQuantumGeneralRelativity(QGR)hasfocussedonthesemiclassicalanalysisofthetheory.Inthispaperwewanttocommentontwosuchdevelopments:1)Polymer-likestatesforMaxwelltheoryandlinearizedgravityconstructedbyVaradarajanwhichusemuchoftheHilbertspacemachinerythathasprovedusefulinQGRand2)coherentstatesforQGR,basedonthegeneralcomplexifiermethod,withbuilt–insemiclassicalproperties.Weshowthefollowing:A)Varadarajan’sstatesarecomplexifiercoherentstates.Thisunifiesallstatesconstructedsofarunderthegeneralcomplexifierprinciple.B)AshtekarandLewandowskisuggestedanon-AbeleangeneralizationofVaradarajan’sstatestoQGRwhich,however,arenolongerofthecomplexifiertype.Weconstructanewclassofnon-AbeleancomplexifierswhichcomeclosetotheoneunderlyingVaradarajan’sconstruction.C)Non-AbeleancomplexifiersclosetoVaradarajan’sinducenewtypesofHilbertspaceswhichdonotsupporttheoperatoralgebraofQGR.TheanalysissuggeststhatifonestickstothepresentkinematicalframeworkofQGRandifkinematicalcoherentstatesareatalluseful,thennormalizable,graphdependentstatesmustbeusedwhichareproducedbythecomplexifiermethodaswell.D)Presentproposalsforstateswithmildenedgraphdependence,obtainedbyperformingagraphaverage,donotapproximatewellcoordinatedependentobservables.However,graphdependentstates,whetheraveragedornot,seemtobewellsuitedforthesemiclassicalanalysisofQGRwithrespecttocoordinateindependentoperators.1IntroductionAmathematicallywell-definedcandidateWheeler-DeWitt(orHamiltonianconstraint)operatorforcanonicalQuantumGeneralRelativity(QGR)hasbeenproposedin[1,2]onthekinematicalHilbertspaceH0definedbyAshtekar,IshamandLewandowski[3].ThisHilbertspacepresentlyunderliesliterallyalltheconstructionswithinQGR.(See[4]foranup-dated,detailedintroductiontoQGRand[5,6]fornon-technicaloverviews).Thus,thereismodesthopethatH0indeedsupportsa(dual)representationofallconstraintoperatorsandDiracobservablesofthetheoryandthatthereexistsawell-definedquantumfieldtheoryforLorentzianmetrics(plus,possiblysupersymmetric,matter)infourspacetimedimensions.Itispossibletoexplicitlystate(andtosomeextentevensolve)the,∗[e-mail]1necessarilydiscrete,quantumtimeevolution.Openproblemsaretheconstructionofaninnerproductonthespaceofsolutionsandofasuitablesetofgaugeinvariantobservables.However,beforetacklingtheseproblems,themorecrucialquestionis,whetherthatquantumfieldtheorycanpossiblyhavegeneralrelativityasitsclassicallimit.Theanswertothatquestionisfarfromobviousbecausethetheoryisbackgroundindependent,thatis,non-perturbativelydefined:Usualquantizationprocedures,ofwhichweknowthattheyguaranteethecorrectclassicallimitsuchasforMaxwelltheory,alwaysmakecruciallyuseofthe(Minkowski)backgroundmetric.However,generalrelativitydoesnotdistinguishanybackground,itisabackgoundindependenttheory,inotherwords,thebackgroundmetricbecomesdynamical,aquantumfieldoperator.Thereforetheseusualquantizationtechniques,usuallybasedonFockspaces(noticethattheMinkowskimetricexplicitlyslipsintothedefinitionofaFockspacethrough,e.g.thed’Alembertoperator,thepositiveandnegativefrequencyone-particlewavefunctionsetc.)cannotbeused.Thisfactmanifestsitselfintheregularizationandrenormalizationoftheoperatorin[1]onthebackgroundindependentHilbertspaceH0whichemployscompletelynewmathematicaltechniqueswhichwehavelittleexperiencewithanditisthereforeindeednotmanifestwhetherthequantumEinsteinequationsproposedreducetotheclassicalonesintheclassicallimit.Tosettlethisquestiononehasseveralpossibilities.Asaminimumtestoneshouldverifywhetherthenewquantizationtechniqueworksinsituationswherethetheorycanbesolvedexplicitly,sayindimensionallyorKillingreducedmodelsofgeneralrelativity.Thishasbeendonein[2]for2+1gravityandin[7]forBianchicosmologies,inbothcaseswithsuccess.Theanalysisin[7]isbasedonaquantumsymmetryreduction,thatis,oneworksontheHilbertspaceofthefulltheoryandimposestheKillingconstraintsthere,incontrasttotheusualprocedureofimposingthesymmetrybeforequantization.Inthatsensethecalculationsperformedin[7]arealmostfully3+1dimensional,justthatoneneglectstheexcitationsofallbutafinitenumberofdegreesoffreedom,whichiswhythisprovidesareallyfirmtestoftheproposalmadein[1].Theagreementofthequantumtheorywiththeclassicaltheorydemonstratedin[7]isindeedspectacularlysharp,thediscretespectrumofquantummetricoperatorsconstructedin[1]approachestheclassicalcurvemuchfaster(forverylowquantumnumbers)thanonecouldhavehopedfor(correspondenceprinciple).Whatismore,theoperatorproposedin[1]certainlysuffersfromquantizationambiguitiesbutin[7]thesewereshowntobeirrelevantfortheclassicallimit.Finally,forthesamereasonthatthequantizationtechniqueinin[1]madetheWheeler-DeWittoperatorfinite,thedeeperreasonbeingbackgroundindependence,itavoidscertainclassicalcurvaturesingularitieswhichcouldbecheckedexplicitlyin[7]whereitwasshownthattheclassicalbigbangsingularityisavoidedinthequantu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