Investigating the Ultraviolet Properties of Gravity with a W(8)
时间:2025-07-10
时间:2025-07-10
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
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