Investigating the Ultraviolet Properties of Gravity with a W(18)

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

G

FIG.1:Thegenericformofthe owinducedbymatter elds.

IV.EINSTEIN’STHEORY

Asa rststeptowardstheinclusionofquantumgravitationale ects,wediscussinthissectiontheRG owforEinstein’sgravity,withorwithoutcosmologicalconstant.Thistruncationhasbeenextensivelydiscussedbefore[8,10].Herewewillextendthoseresultsinvariousdirections.Sincethedependenceoftheresultsonthechoiceofgaugeandpro lefunctionRkhasalreadybeendiscussedin[10,11]hereweshall xourattentiononaparticulargaugeandpro lefunction,andanalyzeinsteadthedependenceoftheresultsondi erentwaysofimplementingthecuto procedure.Thesimplicityofthetruncationwillallowustocomparetheresultsofdi erentapproximationsandcuto schemes,aluxurythatisprogressivelyreducedgoingtomorecomplicatedtruncations.

ThetheoryisparametrizedbythecosmologicalconstantΛandNewton’sconstantG=1/(16πZ),sothatwesetg(0)=2ΛZandg(2)= Zinequation(8).Allhighercouplingsareneglected.Thenthetruncationtakestheform

√Γk=dx