Investigating the Ultraviolet Properties of Gravity with a W(13)
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
reducessimplyto:
∞
Tr tRk( )
(4π)d/2 QdB2i( )(16)
i=0Pk
whereB2i( )aretheheatkernelcoe cientsoftheoperator andtheQ-functionals,de nedin(A14,A15)aretheanalogsofmomentumintegralsinthiscurvedspacetimesetting.Wehavewritten tRktodenotethederivativewithrespecttotheexplicitdependenceofRkonk;whentheargumentofRkdoesnotcontaincouplingsthiscoincideswiththetotalderivatived
Pk( 2)+E.SinceEislinear
incurvature,inthelimitwhenthecomponentsofthecurvaturetensorareuniformlymuchsmallerthank2,wecanexpand
tRk
P +1.
k
Eachoneofthetermsonther.h.s.canthenbeevaluatedinawayanalogousto(A10),sointhiscasewegetadoubleseries:
Tr tRk( 2)
(4π)d/ ∞2 ∞Qd
i=0 =0Pk +1 dx√
Pkm
appearingin(16)and(18)willbeequaltok2(n m+1)timesanumberdependingonthepro le function.AsdiscussedinAppendixA,theintegralswithm=n+1areindependentoftheshapeofRk.Thus,ineven-dimensionalspacetimeswithacuto oftype
thecoe cientoftheterminthesum(16)withi=d
Pk II,andusing(A19),Bd( )=2Bd( ).On
theotherhandwithatypeIcuto ,using(A18),(A19)and(A5)thetermswith =d
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