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