Investigating the Ultraviolet Properties of Gravity with a W(8)

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

asuitabledi erentialoperatorOwhoseeigenfunctions n,de nedbyO n=λn n,canbetakenasabasisinthefunctionalspaceweintegrateover:

φ(x)= n n n(x),φ

naregeneralizedFouriercomponentsofthe eld.(Wewilluseanotationthatiswhereφ

suitableforanoperatorwithadiscretespectrum.)Then,theadditionaltermcanbewrittenineitherofthefollowingforms:

Sk[φ]=1

2 n 2Rk(λn).φn(7)

ThekernelRk(O)willalsobecalled“thecuto ”.Itisarbitrary,exceptforthegeneralrequirementsthatRk(z)shouldbeamonotonicallydecreasingfunctionbothinzandk,thatRk(z)→0forz kandRk(z)=0forz k.Theseconditionsareenoughto

ncorrespondingguaranteethatthecontributiontothefunctionalintegralof eldmodesφ

toeigenvaluesλn k2aresuppressed,whilethecontributionof eldmodescorrespondingak-dependentgeneratingfunctionalofconnectedGreenfunctionsby

e Wk[J]=Dφexp S[φ] Sk[φ] dxJφtoeigenvaluesλn k2areuna ected.Wewillfurther xRk(z)→k2fork→0.Wede ne

andamodi edk-dependentLegendretransform

Γk[φ]=Wk[J] dxJφ Sk[φ],

where Sk[φ]hasbeensubtracted.ThefunctionalΓkissometimescalledthe“averagee ectiveaction”,becauseitcanbeinterpretedasthee ectiveactionfor eldsthathavebeenaveragedovervolumesoforderk d(dbeingthedimensionofspacetime)[17].The“classical elds”δWk/δJaredenotedagainφfornotationalsimplicity.Inthelimitk→0thisfunctionaltendstotheusuale ectiveactionΓ[φ],thegeneratingfunctionalofone-particleirreducibleGreenfunctions.ItissimilarinspirittotheWilsoniane ectiveaction,butdi ersfromitinthedetailsoftheimplementation.

Theaveragee ectiveactionΓk[φ],usedattreelevel,givesanaccuratedescriptionofprocessesoccurringatmomentumscalesoforderk.Inthespiritofe ective eldtheories,weshallassumethatΓkexistsandisquasi–localinthesensethatitadmitsaderivative