Investigating the Ultraviolet Properties of Gravity with a W(17)

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

kn 4gi(n)are

dg i(n)

4 n,

1(25)inparticularwritingg(0)=2ZΛandg(2)= Z=

dt dG +8πa(0)G +16πa(2)G Λ ,= 2Λ

4nS 4nD+2nM

dependingonwhethertherearemoreNotethattheFPoccursforpositiveornegativeΛ

,ontheotherhand,willbebosonicoffermionicdegreesoffreedom.TheFPvalueofG

positiveprovidedtherearenottoomanyscalar elds.Forn=4,(24)givesalogarithmicrunning

gi(k)=gi(k0)+ailn(k/k0),

implyingasymptoticfreedomforthecouplings1/gi.ThisisthesamebehaviorthatisobservedinYang–Millstheoriesandisinaccordancewithearlierperturbativecalculations

[29,30].Asnoted,itfollowsfrom(A24)thatwiththeoptimizedcuto ,forn>4,g i =0.ThecriticalexponentsatthenontrivialFPareequaltothecanonicaldimensionsoftheg(n)’s,soΛandGareUV–relevant(attractive),1/gi(4)(n)(4)(4)(4)(4) nS+2nD+4nM.(27)aremarginalandallthehighertermsareUV–irrelevant.NotethatinperturbationtheoryGisirrelevant.AtthenontrivialFPthequantumcorrectionsconspirewiththeclassicaldimensionsofΛandGtoreconstructthedimensionsofg(0)andg(2).Thismusthappenbecausethecriticalexponentsforg(0)andg(2)areequaltotheircanonicaldimensionsandthecriticalexponentsareinvariantunderregularcoordinatetransformationsinthespaceofallcouplings;thetrasformationbetween andgG (2)isregularatthenontrivialFP,butitissingularattheGaußianFP.

Thissimple owisexactinthelimitN→∞,butisalsoaroughapproximationwhengravitone ectsaretakenintoaccount,asweshalldiscussinsectionsIV-GandVI.Itisshownin gure1.