On Locality in Quantum General Relativity and Quan

gr-qc/960304425Mar1996OnLocalityinQuantumGeneralRelativityandQuantumGravityEduardPrugovec˘kiDepartmentofMathematicsUniversityofTorontoToronto,CanadaM5S1A1Abstract.Thephysicalconceptoflocalityisfirstanalyzedinthespecialrelativisticquantumregime,andcomparedwiththatofmicrocausalityandthelocalcommutativityofquantumfields.Itsextrapolationtoquantumgeneralrelativityonquantumbundlesovercurvedspacetimeisthendescribed.Itisshownthattheresultingformulationofquantum-geometriclocalitybasedontheconceptoflocalquantumframeincorporatingafundamentallengthembodiesthekeygeometricandtopologicalaspectsofthisconcept.Takeninconjunctionwiththestrongequivalenceprincipleandthepath-integralformulationofquantumpropagation,quantum-geometriclocalityleadsinanaturalmannertotheformulationofquantum-geometricpropagationincurvedspacetime.Itsextrapolationtogeometricquantumgravityformulatedoverquantumspacetimeisdescribedandanalyzed.1.INTRODUCTIONThephysicalconceptoflocalityinquantumphysicshasundergoneanumberofrevisionsduringcourseofthiscentury,asitbecameincreasinglyintertwinedwiththatofcausality.Itsoriginsare,however,clearlyrootedinthegeometricconceptualizationof“point,”andofthetopologicalconceptof“neighborhood.”Inarecentlyadvancedformulationofquantumgeneralrelativity(1)ithasbeenshownthataformulationofthisconcept,whichdistinguishesbetweenitsvariousgeometricandtopologicalnuances,canprovidethebasisforaconsistentunificationofgeneralrelativityandquantumtheory.Intheaxiomaticapproachtogeometry,whichculminatedinHilbert'sworkonthesubjecttowardstheendofthelastcentury,theconceptof“point,”togetherwithotherkeygeometricconcepts,isnotdefined.Thus,HilberthimselfstartshisGrundlagenderGeometriebyrequiringthereadertoimagine“threekindsofthings...calledpoints...lines...planes,”andthenenumeratestheaxiomstobefulfilledbythese“things,”andbytherelationshipsbetweenthem.Ontheotherhand,Riemann'sattitudetogeometry,whichinfluencedEinsteininhisformulationofgeneralrelativity,isempirical.Thus,Riemannassertsthat“thetheoremsofgeometrycannotbededucedfromgeneralnotionsofquantity,butthatthosepropertieswhichdistinguishSpacefromotherconceivablytriplyextendedquantitiescanonlybededucedfromexperience.”(Ref.2,p.135.)Whensuchanattitudeisadopted,geometrybecomesofdirectrelevancetophysics,andcanbeusedasananalytictoolinphysicaltheories.InSec.2wepresentananalysismeanttodisentanglefromeachotherthecontemporaryconceptsoflocality,microcausalityandlocalcommutativityofquantumfields.Withtheessentialaspectsoftheconceptoflocalitythusillucidated,weturninSec.3totheformulationofthisconceptinspecialrelativisticquantumtheoryinMinkowskispace.Weexplainwhyinthatcontexttheconventionalformulationofthisconceptisdeficient,andhowitcanbereplacedbyaconsistentformulationincorporatingafundamentallength.WethenextrapolateinSec.4thismodifiedconceptoflocalitytoquantumtheoryincurvedspacetime,andexplainthedistinctionbetweenitsgeometricaspectsanditstopologicalaspects.InSec.5weturntotheformulationofthequantumbundlesandofthelocalquantumframesrequiredforthemathematicalimplementationoftheseideas,anddiscusstheoperationalinterpretationoftheresultingfibretheoreticalframeworkforquantumgeneralrelativity.Finally,inSec.6wedescribe,inprimarilynontechnicalterms,howtheensuingconceptoflocalitythatincorporatesafundamentallength(whoseexistencehasbeenpredictedbymeasurement-theoreticalarguments)canbeappliedtotheformulationofageometricframeworkforquantumgravitywhichavoidsthefoundationaldifficultiesencounteredbyvariousexistingconventionalframeworksforquantumgravity.22.LOCALITY,MICROCAUSALITYANDLOCALCOMMUTATIVITYAsiswell-known,inEuclideangeometrya“point”isdefinedastheintersectionoftwolines.Withtheconceptoflineapriorigiven,thisdefinitionthrowsnolightonthephysicalnatureoftheobjectsthatcanberepresentedbypoints.Nevertheless,thenotionof“materialpoint”playedacrucialroleinEinstein'sformulationofbothspecial(3)andgeneral(4)relativity.Clearly,inadoptingthisnotionEinsteinreliedonthe“intuitive”graspofthisnotionthatoneacquireswithexposuretoclassicalmechanics.There,however,thisconceptisviewedasanidealization.Inthisidealization,undercertaincircumstancestheactualmotionofmacroscopicbodiesisreplacedwiththatoftheircenterofgravity,withtheirspatialextensionbeingtherebyneglected.When,however,theconceptofmaterialpointplaysafundamentalroleintheformulationoftheveryconceptofspacetime,thenquestionsariseastothephysicallegitimacyofthisconcept.Hence,inthecollection(5)ofessayshonoringEinstein'sseventiethbirthday,H.Margenauvoicedthecriticismthat“particlesmaynotberegardedaspointsbutasstructuresoffinitesize...,whichthreatensthevalidityof[relativistic]causaldescription”(Ref.5,p.259).Thisobservationbecomesextremelypertinentatthemicrolevel,whenonetriestocombinequantumtheoreticalnotionswithrelativisticones.Inparticular,itaffectsevenQFT(i.e.quantumfieldtheory)since,asemphasizedbyBohrandRosenfeld,(6,7)thequestionoflocalizabilityofquantumfieldsisinextricablyintertwinedwiththatoftestbodiesrequiredfortheirmeasurement.InconventionalspecialrelativisticQFTithasbecomecustomarytoidentifythenotion