Investigating the Ultraviolet Properties of Gravity with a W(6)

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

Inthee ective eldtheoryapproach(ind=4),Bjerrum-Bohr,DonoghueandHolsteinhaveproposedinterpreting

aclassofoneloopdiagramsasgivingthescaledependenceofNewton’sconstant[4].Theycalculate

G(r)=G01 167

r2 ,

whereristhedistancebetweentwogravitatingpointparticles.Ifweidentifyk=1/ar,withaaconstantoforderone,thiswouldcorrespondtoabetafunction

2167 βG=2G a

.ThiscalculationwasbasedonperturbativemethodsandsincetheFPoccurs

,itisnotclearthatonecantrusttheresult.WhatwecanatanotverysmallvalueofG167a2

saywithcon denceisthattheonsetoftherunningofGhastherightsignforasymptoticsafety.Clearlyinordertomakeprogressonthisissueweneeddi erenttools.

InthispaperwewilldiscusstheapplicationofWilsonianrenormalizationgroupmethodstotheUVbehaviorofgravity.InsectionIIwewillintroduceaparticularlyconvenienttool,calledthe“ExactRenormalizationGroupEquation”(ERGE)whichcanbeusedtocalculatethe“betafunctional”ofaQFT.Renormalizabilityisnotnecessaryandthetheorymayhavein nitelymanycouplings.InsectionIIIweillustratetheuseoftheERGEbycalculatingthecontributionofminimallycoupledmatter eldstothegravitationalbetafunctions.Inthissimplesetting,wewillreviewthetechniquesthatareusedtoextractfromthebetafunctionalthebetafunctionsofindividualcouplings,emphasizingthoseresultsthatare“schemeindependent”inthesensethattheyarethesameirrespectiveoftechnicaldetailsofthecalculation.InsectionIVweapplythesametechniquestothecalculationofthebetafunctionsforthecosmologicalconstantandNewton’sconstantinEinstein’stheoryinarbitrarydimensions,extendinginvariouswaystheresultsofearlierstudies[8,9,10,11,12].WealsoshowthattheFPthatisfoundinfour–dimensionalgravityisindeedthecontinuationfor →2oftheFPthatisfoundinthe2+ expansion.Wecomparevariouswaysofde ningtheWilsoniancuto and ndtheresultstobequalitativelystable.InsectionsVandVI