Investigating the Ultraviolet Properties of Gravity with a W(21)

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

Thecalculationin[8]proceededasfollows.ThebackgroundmetricischosentobethatofEuclideandeSitterspace.Themodi edinversepropagatorisobtainedfrom(33)justreplacing 2byPk( 2).Usingthepropertiesoftheprojectors,itsinversionistrivial:

Γ(2)+R 121

kk=P2

k 2Λ+d 3d+4d 2P (38)

dR

Decomposinginthesamewaythetermd

tRk+ηRk

dt=1

Pδ tRk

k 2Λ+d2 3d+42TrxLPν.

dR TrxLµ

d

Onecannowexpandto rstorderinR,usethetraces(32)andformula(A10)toobtain:

dΓk

(4π)d/2 dx√

Pk

d(d+ 2Λ dQd+1)tRk+ηRk

2 16Qd

d(d 1)

2 4Qd tRk+ηRk2 Pk Pk tRk