Investigating the Ultraviolet Properties of Gravity with a W(17)
时间:2025-07-10
时间:2025-07-10
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.
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