Alternative actions for quantum gravity and the in

arXiv:hep-th/9606140v121Jun1996NBI-HE-96-22June1996AlternativeactionsforquantumgravityandtheintrinsicrigidityofthespacetimeJ.AmbjørnandG.K.Savvidy1TheNielsBohrInstituteBlegdamsvej17,DK-2100CopenhagenØ,DenmarkK.G.SavvidyPrincetonUniversity,DepartmentofPhysicsP.O.Box708,Princeton,NewJersey08544,USAAbstractUsingtheSteiner-Weylexpansionformulaforparallelmanifoldsandthesocalledgonihedricprinciplewefindalargeclassofdiscreteintegralinvariantswhicharede-finedonsimplicialmanifoldsofvariousdimensions.TheseintegralinvariantsincludethediscreteversionoftheHilbert-EinsteinactionfoundbyReggeandalternativeactionswhicharelinearwithrespecttothesizeofthemanifold.Inadditiontheconceptofgeneralizeddeficitanglesappearinanaturalwayandisrelatedtohigherordercurvatureterms.Theseanglesmaybeusedtointroducevariousaspectsofrigidityinsimplicialquantumgravity.Keywords:Simplicialgravity,integralinvariance,rigidityofspacetimePACSno:02.10.Rn,02.40.Ky,04.20.Cv,04.60.+n1Permanentaddress:YerevanPhysicsInstitute,375036Yerevan,Armenia11IntroductionStringtheoryandtheoriesbeyondstringtheoryhaveincreasedtheinterestforquan-tumgravityandphysicsatthePlanckscale.Oneprobleminthesetheoriesisthelackofanon-perturbativedefinition.Althoughstringdualityseemstoopentantaliz-ingpossibilitiesforextractingnon-perturbativephysicsfromdifferentperturbativesectorsofthetheory,itisstillnotclearthatwewillbeabletoaddressthefullphysicalcontentatthePlanckscalewithoutanon-perturbativedefinitionofstringtheoryoritsgeneralization.Inparticular,itisnotclearhowmuchwewillbeabletosayabouttheaspectsrelatedtothequantumgravitysectorofthetheory.Itisthereforepossiblethatquantumgravity,oratleastimportantaspectsofquantumgravity,maybedescribedentirelywithintheframeworkofanon-perturbativefieldtheory.Afirststepinthisdirectionisanon-perturbativedef-initionofthefieldtheorywewilldenotequantumgravity.Thisisanon-trivialtaskbecausethecontinuumtheoryhastobeinvariantunderreparametrizations.Latticegaugetheoryisanexampleofasuccessfulnon-perturbativeregularizationofaquan-tumfieldtheorywithacontinuousinternalsymmetry.TheregularizationbreaksEuclideaninvariance(whichisrestoredinthescalinglimit),butmaintainsfromacertainpointofviewtheconceptoflocalinternalinvariance.Inthecaseofgravitythesituationismoredifficultsincewedealwithlocalsymmetriesofspace–timeitself,andanylatticeregularizationwillbreakthissymmetry.InclassicalgravityaverynaturaldiscretizationwassuggestedbyRegge[1]bytherestrictiontopiece-wiselinearmanifolds,andheshowedhowtheEinstein–Hilbertactionhadanaturalgeometricrepresentationontheclassofpiecewiselinearmanifoldsandcouldbeexpressedentirelyintermsofintrinsicinvariantsofthepiecewiselinearmanifolds.Inthiswayoneachievedacoordinatefreedescriptionofthisclassofmanifoldsandtheiractions,wherethedynamicalvariableswerethelinklength.TheuseofReggecalculusasaprescriptionforquantumgravityislessstraightforward,sincethereisnotaone-to-onecorrespondencebetweenthepiecewiselinearmetricandthelengthassignedtothelinks(foradiscussion,see[2]andalso[3]andreferencestherein).However,itisveryencouragingthatavariantofReggecalculus,knownasdynam-icaltriangulation[4,5,6],worksperfectintwo-dimensionalquantumgravity.Inthisformalismonefixesthelink–lengthofthepiecewiselinearmanifold,andtheassignmentofametricdependsonlyontheconnectivityofthetriangulation.Thesummationovertriangulationswithdifferentconnectivitytakestheroleofintegra-tionoverinequivalentmetricsandtheactionassignedtothemanifoldiscalculatedbyRegge’sprescriptionforapiecewiselinearmanifold.Inthisformalismthescalinglimitagreeswiththeknowncontinuumresultsoftwo-dimensionalquantumgravity[7,8],i.e.onegetsareparametrizationinvarianttheoryinthescalinglimit.Whileitiseasytogeneralizethedefinitionofthetwo-dimensionaldiscretemodeltoboththree[9]andfourdimensions[10](seealso[11]foranearlierslightlydifferentformu-lation),themodelscanpresentlyonlybeanalyzedbynumericalmethods[12]andwehavenocontinuumtheoryofEuclideangravityindimensionslargerthantwowithwhichwecancompare.WhetheroneusestheformalismofdynamicaltriangulationsortheoriginalformalismofReggewithfixedconnectivityandvariablelinklength,2itmightwellbethatthesimplestversionsofdiscretizedEinstein–Hilbertactionwhichhavebeenusedsofaratthediscretizedlevelareinsufficientinproducinganinterestingcontinuumlimitindimensionslargerthantwo.Ontheotherhanditisveryappealingtousesomeclassofpiecewiselinearmanifoldsasthenaturalchoiceofdiscretizationinquantumgravitysincethereisaone-to-onecorrespondencebe-tweenpiecewiselinearstructuresandsmoothstructuresformanifoldsofdimensionslessthanseven.Thismotivatesthesearchfor”natural”integralinvariantswhicharedefinedonpiecewiselinearmanifolds.Inthisarticlewewouldliketoadvocateageometricalwaytoconstructintegralinvariantsonasimplicialmanifold.ItcontainsRegge’sresultasaspecialcaseandallowsustoconstructalargeclassofnewintegralinvariantswhichmightbehelpfulintheattemptstodefinearegularizedpathintegralinquantumgravitywhichpossessesaninterestingcontinuumlimit.ThemethodisbasedonSteiner’sideaofaparallelmanifold[15,16,17].LetMn−1beasmooth(n–1)-dimensionalm