Investigating the Ultraviolet Properties of Gravity with a W(14)

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

addupto

d/2 =0Q

=2

=2Bd( 2+E) tRkg( 1) trE b2i( 2)Ed/2b0( 2) dx√(d/2)!

Therefore,inadditiontobeingindependentoftheshapeofthecuto function,thesecoef- cientsarealsothesameusingtypeIortypeIIcuto s.

Asanexamplewewillnowspecializetofour-dimensionalgravitycoupledtonSscalar elds,nDDirac elds,nMgauge(Maxwell) elds,allmasslessandminimallycoupled:

√¯µ µψ+1 µφ µφ+ψγΓk(gµν,φ,ψ,Aµ)=d4x2

d;E(M)=Ricci;E(gh)=0.(20)

Here“Ricci”standsfortheRiccitensorregardedasalinearoperatoractingonvectors:Ricci(v)µ=Rµνvν.Forthegauge eldswehavechosentheLorentzgauge,and (gh)istheoperatoracting

onthescalarghost.(Itcanbeshownthattheresultsdonotdependonthechoiceofgauge[26].)

WithatypeIIcuto ,foreachtypeof eldwede nethemodi edinversepropagatorPk( (A))= (A)+Rk( (A)).Then,theERGEreducessimplyto

(S)dΓk tRk( ) tRk( (D))Tr(S)Tr(D)22 tRk( (M))Tr(M)2Pk( (gh))