The Functional Renormalization Group in Quantum Gravity

Abstract

The gravitational asymptotic safety program envisions a high-energy completion of the gravitational interactions by an interacting renormalization group fixed point, the Reuter fixed point. The primary tool for investigating this scenario are functional renormalization group equations, foremost the Wetterich equation. This equation implements the idea of the Wilsonian renormalization group by integrating out quantum fluctuations shell-by-shell in momentum space and gives access to the theory's renormalization group flow beyond the realm of perturbation theory. This chapter gives a pedagogical introduction to the gravitational asymptotic safety program with a specific focus on clarifying conceptual points which led to confusion in the past. We provide a step-by-step introduction to the Wetterich equation and its most commonly used non-perturbative approximations. This exposition also introduces recent developments including the minimal essential scheme and N-type cutoffs. The use of the Wetterich equation in explicit computations is illustrated within the Einstein-Hilbert truncation which constitutes the simplest non-perturbative approximation of the gravitational renormalization group flow. We conclude with a brief summary and comments on recent developments originating from other quantum gravity programs.

Foliated asymptotically safe gravity in the fluctuation approach

“…The critical exponents agree with the ones obtained from canonical power counting, in agreement with the definition of a GFP advocated in [9,10]. The GFP is a saddle point in the g-λ-plane.…”

“…Within the gravitational asymptotic safety program, reviewed e.g. in [8][9][10][11][12][13][14][15][16][17], the construction of renormalization group (RG) flows based on ADM-variables has been developed in [18][19][20][21][22][23], also see [24] for an implementation of a foliation structure based on a gauge-fixing construction and [25,26] for further discussions. 1 Conceptually,…”

Weak-gravity bound in asymptotically safe gravity-gauge systems

“…Systematic extensions of this set of operators are then necessary to i) estimate the systematic uncertainties in a given truncation, and ii) to obtain qualitatively reliable results of physical observables. For detailed reviews on the FRG, and its applications in gravity see, e.g., [36,37,105,143]. In the present work we approximate the gravitational dynamics by the Einstein-Hilbert action, see Eq.…”

“…Following a proposal by Weinberg [4] and early evidence in 2+ϵ dimensions [6][7][8], see also [9][10][11] for recent studies, a breakthrough came with Reuter's adaptation of functional Renormalization Group techniques [12][13][14] to quantum gravity [5]. With these techniques, compelling evidence for the requisite interacting RG fixed point has been collected, both in pure gravity and in gravity-matter theories, see [15][16][17][18][19][20][21][22][23] for early evidence and [24][25][26][27][28][29][30][31][32] and references therein for the most recent evidence, as well as [33][34][35][36][37][38][39][40][41] for recent reviews. There is a critical open question, namely whether the theory is physically viable in that it respects causality and unitarity.…”