Investigating the Ultraviolet Properties of Gravity with a W(13)

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

reducessimplyto:

∞

Tr tRk( )

(4π)d/2 QdB2i( )(16)

i=0Pk

whereB2i( )aretheheatkernelcoe cientsoftheoperator andtheQ-functionals,de nedin(A14,A15)aretheanalogsofmomentumintegralsinthiscurvedspacetimesetting.Wehavewritten tRktodenotethederivativewithrespecttotheexplicitdependenceofRkonk;whentheargumentofRkdoesnotcontaincouplingsthiscoincideswiththetotalderivatived

Pk( 2)+E.SinceEislinear

incurvature,inthelimitwhenthecomponentsofthecurvaturetensorareuniformlymuchsmallerthank2,wecanexpand

tRk

P +1.

k

Eachoneofthetermsonther.h.s.canthenbeevaluatedinawayanalogousto(A10),sointhiscasewegetadoubleseries:

Tr tRk( 2)

(4π)d/ ∞2 ∞Qd

i=0 =0Pk +1 dx√

Pkm

appearingin(16)and(18)willbeequaltok2(n m+1)timesanumberdependingonthepro le function.AsdiscussedinAppendixA,theintegralswithm=n+1areindependentoftheshapeofRk.Thus,ineven-dimensionalspacetimeswithacuto oftype

thecoe cientoftheterminthesum(16)withi=d

Pk II,andusing(A19),Bd( )=2Bd( ).On

theotherhandwithatypeIcuto ,using(A18),(A19)and(A5)thetermswith =d