Investigating the Ultraviolet Properties of Gravity with a W(12)

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

III.MATTERFIELDSANDCUTOFFSCHEMES

Inthissectionweillustratethemethodthatisusedtocomputethetraceinther.h.s.of

(9)inagravitationalsettingandtoevaluatethebetafunctionsofthegravitationalcouplings.Quitegenerally,wewillconsiderthecontributionof eldswhoseinversepropagatorΓ(2)isadi erentialoperatoroftheform = 2+E,where isacovariantderivative,bothwithrespecttothegravitational eldandpossiblyalsowithrespecttoothergaugeconnectionscoupledtotheinternaldegreesoffreedomofthe eld,andEisalinearmapactingonthequantum eld.Ingeneral,Ecouldcontainmasstermsortermslinearincurvature.Forexample,inthecaseofanonminimallycoupledscalar,E=ξR,whereξisacoupling.Apriori,nothingwillbeassumedaboutthegravitationalactionandalsothespacetimedimensiondcanbeleftarbitraryatthisstage.

InordertowritetheERGEwehavetode nethecuto .Fortheoperatortobeusedinthede nitionof(7),severalpossiblechoicessuggestthemselves.LetussplitE=E1+E2,whereE1doesnotcontainanycouplingsandE2consistsonlyoftermscontainingthecouplings.Wecallacuto oftypeI,ifRkisafunctionofthe“bareLaplacian” 2,oftypeIIifitisafunctionof 2+E1andoftypeIIIifitisafunctionofthefullkineticoperator = 2+E.Thesubstantialdi erencebetweenthe rsttwotypesandthethirdisthatinthelattercase,duetotherunningofthecouplings,thespectrumchangesalongthe ow.Forthisreasonthesecuto saresaidtobe“spectrallyadjusted”[25].6

LetusnowrestrictourselvestothecasewhenE2=0,i.e.thekineticoperatordoesnotdependonthecouplings;thenthereisonlyachoicebetweencuto softypeIandII.ThederivationofthebetafunctionsistechnicallysimplerwithatypeIIcuto .InthiscasewechoosearealfunctionRkwiththepropertieslistedinsectionIIandde neamodi edinversepropagator

Pk( )= +Rk( ).(15)

IftheoperatorEdoesnotcontaincouplings,using(A10)thetraceinther.h.s.oftheERGE