Investigating the Ultraviolet Properties of Gravity with a W(4)

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

InordertoaddressthesecondproblemwehavetoinvestigatethesetofRGtrajectoriesthathavethisgoodbehaviour.WewanttousetheconditionofhavingagoodUVlimitasacriterionforselectingaQFTofgravity.IfalltrajectorieswereattractedtotheFPintheUVlimit,wewouldencounteravariantofthesecondproblem:theinitialconditionsfortheRG owwouldbearbitrary,sodeterminingtheRGtrajectoryoftherealworldwouldrequireinprincipleanin nitenumberofexperimentsandthetheorywouldlosepredictivity.Attheotherextreme,thetheorywouldhavemaximalpredictivepoweriftherewasasingletrajectoryendingattheFPintheUV.However,thismaybetoomuchtoask.AnacceptableintermediatesituationoccurswhenthetrajectoriesendingattheFPintheUVareparametrizedbya nitenumberofparameters.Atheorywiththesepropertiesissaidtobe“asymptoticallysafe”[5].

Tobetterunderstandthisproperty,imagine,inthespiritofe ective eldtheories,ageneralQFTwithallpossibletermsintheactionwhichareallowedbythesymmetries.Wecanparametrizethe(generallyin nitedimensional)“spaceofalltheories”,Q,bythedimensionlesscouplingsg i.Weassumethatredundanciesinthedescriptionofphysicsduetothefreedomtoperform eldrede nitionshavebeeneliminated,i.e.allcouplingsare“essential”(suchcouplingscanbede nede.g.intermsofcrosssectionsinscatteringexperiments).WethenconsidertheRenormalizationGroup(RG) owinthisspace;itisgivenbythebetafunctions

βi=kdg i

dk

where=Mij( gj g j ),

Mij= βi(3)

2RGtransformationsleadtowardslowerenergies,andthetrajectorieslyinginCarerepelledbytheFPunderthesetransformations.Forthisreason,Cisalsocalledthe“unstablemanifold”.SinceweareinterestedinstudyingtheUVlimit,itismoreconvenienttostudythe owforincreasingk.