Investigating the Ultraviolet Properties of Gravity with a W(9)

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

expansionoftheform∞

Γk(φ,gi)=

whereg(n) g(n)(n)i(k)Oi(φ),(8)n=0i(n)

i(k)arecouplingconstantsandO iareallpossibleoperatorsconstructedwiththe eldφandnderivatives,whicharecompatiblewiththesymmetriesofthetheory.Theindexiisusedheretolabeldi erentoperatorswiththesamenumberofderivatives.Fromthede nitiongivenabove,itiseasytoshowthatthefunctionalΓksatis esthefollowing“ExactRenormalizationGroupEquation”(orERGE)[18,19]

kdΓk

2Tr Γ(2)+Rk 1kdRk

k

δφδφfortheinversepropagatorofthe eldφde nedbythefunctionalΓk.Ther.h.s.of(9)canberegardedasthe“betafunctional”ofthetheory,givingthek–dependenceofallthecouplingsofthetheory.Infact,takingthederivativeof(8)onegets

kdΓk

(n)dk=dgi