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