Background independent exact renormalization group for conformally reduced gravity

Abstract

Within the conformally reduced gravity model, where the metric is parametrised by a function f (φ) of the conformal factor φ, we keep dependence on both the background and fluctuation fields, to local potential approximation and O(∂ 2 ) respectively, making no other approximation. Explicit appearances of the background metric are then dictated by realising a remnant diffeomorphism invariance. The standard non-perturbative Renormalization Group (RG) scale k is inherently background dependent, which we show in general forbids the existence of RG fixed points with respect to k. By utilising transformations that follow from combining the flow equations with the modified split Ward identity, we uncover a unique background independent notion of RG scale,k. The corresponding RG flow equations are then not only explicitly background independent along the entire RG flow but also explicitly independent of the form of f . In general f (φ) is forced to be scale dependent and needs to be renormalised, but if this is avoided then k-fixed points are allowed and furthermore they coincide withk-fixed points.

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.…”

“…At this stage it is unclear whether all principles are strictly necessary for establishing a suitable fixed function. In particular, the list does not incorporate the recent observation [60,63] that the combination of a flow equation with the equation for the modified split Ward identities may lead to substantial simplifications of the PDEs prescribing the fixed functions of the theory.…”

Fate of nonpolynomial interactions in scalar field theory

“…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].…”

“…But this has to begin by building on a thorough understanding of what is legitimate in a well-understood situation such as that of scalar field theory. Moreover, recent work has shown that, despite the complications induced by diffeomorphism invariance and background independence [18,20,37], the overall scalar factor part of the metric (at least within the so-called conformal truncation) has RG properties that are just those of a scalar field theory -but crucially with wrong sign kinetic term [28,36]. Furthermore this one sign profoundly alters the RG properties (unless the field is continued to the imaginary axis) [36].…”

Solutions to the reconstruction problem in asymptotic safety

“…[2], there is a wealth of literature investigating asymptotic safety in this way. For reviews and introductions see [10][11][12][13][14], and for recent advances see for example [15][16][17][18][19][20][21][22][23][24]. In the vast majority of this work the RG flow equation takes the generic form [3]:…”