Investigating the Ultraviolet Properties of Gravity with a W(11)

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

makestheintegrationconvergent.So,wecanregardthederivationgivenaboveasmerelyformalmanipulationsthatmotivatetheformoftheERGE,butthentheERGEitselfisperfectlywellde ned,withouttheneedofintroducinganUVregulator.IfweassumethatatagivenscalekphysicsisdescribedbyarenormalizedactionΓk,theERGEgivesusawayofstudyingthedependenceofthisfunctionalonk,andthebehaviorofthetheoryathighenergycanbestudiedbytakingthelimitofΓkfork→∞(whichneednotcoincidewiththebareactionS).

Inmostcasesitisimpossibletofollowthe owofin nitelymanycouplingsandacommonprocedureistoconsideratruncationofthetheory,namelytoretainonlya nitesubsetoftermsinthee ectiveactionΓk.Forexampleonecouldconsiderthederivativeexpansion(8)andretainalltermsuptosomegivenordern.Whateverthechoice,onecalculatesthecoe cientsoftheretainedoperatorsinther.h.s.of(9)andinthiswaythecorrespondingbetafunctionsarecomputed.IngeneralthesetofcouplingsthatonechoosesinthiswaywillnotbeclosedunderRGevolution,sooneisneglectingthepotentiale ectoftheex-cludedcouplingsontheonesthatareretained.Still,inthiswayonecanobtaingenuinenonperturbativeinformation,andthisprocedurehasbeenappliedtoavarietyofphysicalproblems,sometimeswithgoodquantitativeresults.Forreviews,see[22,23,24].

Ifwetruncatethee ectiveactioninthisway,thereisusuallynosmallparametertoallowustoestimatetheerrorwearemaking.Oneindirectwaytoestimatethequalityofatruncationreliesonananalysisofthecuto schemedependence.Thee ectiveactionΓkobviouslydependsonthechoiceofthecuto functionRk.Thisdependenceissimilartotheschemedependenceoftherenormalizede ectiveactioninperturbativeQFT;onlyphysicallyobservablequantitiesderivedfromΓkmustbeindependentofRk.Thisprovidesanindirectcheckonthequalityofthetruncation.Forexample,thecriticalexponentsshouldbeuniversalquantitiesandthereforecuto –independent.Inconcretecalculations,usuallyinvolvingatruncationoftheaction,criticalexponentsdodependonthecuto scheme,andtheobserveddependencecanbetakenasaquantitativemeasureofthequalityoftheapproximation.Ultimately,thereisnosubstituteforperformingcalculationswithtruncationsthatcontainmoreterms.Notethatagoodtruncationisnotnecessarilyoneforwhichthenewtermsaresmall,butoneforwhichthee ectofthenewtermsontheoldonesissmall.Inotherwords,insearchofanontrivialFP,wewanttheadditionofnewtermsnottoa ecttoomuchtheFPvalueofthe“old”couplings,northe“old”criticalexponents.