Investigating the Ultraviolet Properties of Gravity with a W(7)

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

wemakeconnectionwitholdresultsfromperturbationtheory.InsectionVwerederivethe’tHooft–VeltmanoneloopdivergencefromtheERGEandweshowittobescheme–independent.WealsodiscusswhytheGoro –Sagnottitwoloopdivergencecannotbeseenwiththismethodandwediscussthesigni canceofthisfact.InsectionsVIandVIIweconsiderhigherderivativegravity.InsectionVIwederivetheexistenceoftheFPinthemostgeneraltruncationinvolvingfourderivativesatoneloop,andwehighlightthedi erencesbetweentheWilsonianprocedure[13]andearliercalculations.InsectionVIIweconsiderhigherpowersofcurvaturebutrestrictingourselvestopolynomialsinthescalarcurvature.Wegivemoredetailsofourrecentcalculations[14]andextendthemtopolynomialsofordereight.InsectionVIIIweassessthepresentstatusoftheasymptoticsafetyapproachtoquantumgravityanddiscussvariousopenproblems.

II.THEERGEANDITSAPPROXIMATIONS

ThecentrallessonofWilson’sanalysisofQFTisthatthe“e ective”(asin“e ective eldtheory”)actiondescribingphysicalphenomenaatamomentumscalekcanbethoughtofastheresultofhavingintegratedoutall uctuationsofthe eldwithmomentalargerthank

[15].Atthisgenerallevelofdiscussion,itisnotnecessarytospecifythephysicalmeaningofk:foreachapplicationofthetheoryonewillhavetoidentifythephysicallyrelevantvariableactingask4.Sincekcanberegardedasthelowerlimitofsomefunctionalintegration,wewillusuallyrefertoitastheinfraredcuto .Thedependenceofthe“e ective”actiononkistheWilsonianRG ow.

Thereareseveralwaysofimplementingthisideainpractice,resultinginseveralformsoftheRGequation.Inthespeci cimplementationthatweshalluse,insteadofintroducingasharpcuto inthefunctionalintegral,wesuppressthecontributionofthe eldmodeswithmomentalowerthank.Thisisobtainedbymodifyingthelowmomentumendofthepropagator,andleavingalltheinteractionsuna ected.Wedescribeherethisprocedureforascalar eld.WestartfromabareactionS[φ],andweaddtoitasuppressionterm Sk[φ]thatisquadraticinthe eld.In atspacethistermcanbewrittensimplyinmomentumspace.Inordertohaveaprocedurethatworksinanarbitrarycurvedspacetimewechoose