RG flows of Quantum Einstein Gravity on maximally symmetric spaces

Abstract

Abstract:We use the Wetterich-equation to study the renormalization group flow of f (R)-gravity in a three-dimensional, conformally reduced setting. Building on the exact heat kernel for maximally symmetric spaces, we obtain a partial differential equation which captures the scale-dependence of f (R) for positive and, for the first time, negative scalar curvature. The effects of different background topologies are studied in detail and it is shown that they affect the gravitational RG flow in a way that is not visible in finite-dimensional truncations. Thus, while featuring local background independence, the functional renormalization group equation is sensitive to the topological properties of the background. The detailed analytical and numerical analysis of the partial differential equation reveals two globally well-defined fixed functionals with at most a finite number of relevant deformations. Their properties are remarkably similar to two of the fixed points identified within the R 2 -truncation of full Quantum Einstein Gravity. As a byproduct, we obtain a nice illustration of how the functional renormalization group realizes the "integrating out" of fluctuation modes on the three-sphere.

A proper fixed functional for four-dimensional Quantum Einstein Gravity

“…Starting from the flow equation derived in [24], the existence of fixed functions f * (R) have been investigated by various groups in d = 3 [53][54][55] and d = 4 [56][57][58][59][60] spacetime dimensions. Quite unsettling, the verification of a suitable NGFP at the level of fixed functions turned out to be extremely challenging: while the finite-dimensional computations always produced a suitable UV fixed point regardless of the computational details and setting, up to now the fixed function completing the four-dimensional NGFP in the ansatz (1.2) is still elusive.…”

“…of JHEP08 (2015)113 the flow, encoding the quantum effects, becomes trivial and the flow equation reduces to the classical scaling relation. This was the strategy implemented in [55], which constructed fixed functions on the compact interval 0 ≤ r ≤ r (0),term = 6. While this compactification simplifies the numerical analysis, it will not be implemented here and we will consider fixed functions which are well-defined on the entire positive half-axis 0 ≤ r < ∞.…”

Fate of nonpolynomial interactions in scalar field theory

“…We see that for the quantised interactions the t-dependence of their renormalised coupling can at large z be instead attributed to mean-field evolution as in (25). (This is just the RG argument for accepting these as renormalised couplings [2,38,39], run in reverse.)…”

“…At first sight this picture seems deeply at variance with the fact that the large field behaviour is fixed to be mean-field, viz. (25), meaning that in physical variables the large field dependence (23) remains that of the original non-polynomial perturbation (24) and does not actually depend on t at all.…”

“…However this is also an area where there is little guidance from current experimental observation or other techniques, and therefore one must place particular reliance on a rigorous understanding of the mathematical structure that the exact RG exposes, in so far as this is possible. This is especially so with recent work on "functional truncations" [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].…”