Investigating the Ultraviolet Properties of Gravity with a W(19)

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

ofthetype:

SGF(g(B),h)=

whereZg(B)χµg(B)µνχν,1+ρ(29)χν= µhµν

2

includingthegauge xingterm,canbewrittenintheform

1

containingtheminimaloperator:

Γk

where8(2)µνρσ 1,whichleadstoconsiderablesimpli cation.Theinversepropagatorofhµν,ghµνΓk(2)µνρσhρσ µν µν=ZKρσ( 2 2Λ)+Uρσ,

;µνδρσ=(30)µνKρσ=1

2

1µνPρσ 1dgµνgρσ;µνµνUρσ=RKρσ+

gh.dµνUsingthatK=1

2

equation(30)ineitherofthefollowingforms:

Γk(2) P,ifd=2wecanrewrite=ZK( 2 2Λ1+W)Z=2 P 2 2Λ1 4

2(d 2)(Rρσgµν+gρσRµν Rgρσgµν).