Investigating the Ultraviolet Properties of Gravity with a W(20)
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
Notethattheoverallsignofthesecondterminthesecondlineof(31)isnegativewhend>2.ThisisthefamousproblemoftheunboundednessoftheEuclideanEinstein–teron,wewillneedthetraces:
tr1=d(d+1)
RµνRµν+3
d 22d 5d2+8d+4;trW=d(d 1)
Onthed-dimensionalspherewecanwrite
U=1 νµ¯µ 2δµgC RννC.R Pd 2
dR .d(d 1)
Then,usingthesecondlineof(31),wehave
Γk=(2)Z
d(d 1)R d 2
dR .
(33)
Wewillnowdiscussseparatelyvarioustypesofcuto schemes.
A.Cuto oftypeIa
Thisistheschemethatwasusedoriginallyin[8].Itisde nedbythecuto term
1√ Sk[hµν]=ghµνRk( 2)µνρσhρσ dx
dZ
Z
22=ZK tRk( )+ηRk( ).
dt(37)
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