Asymptotic safety in quantum gravity

Asymptotic safety (sometimes also referred to as nonperturbative renormalizability) is a concept in quantum field theory which aims at finding a consistent and predictive quantum theory of the gravitational field. Its key ingredient is a nontrivial fixed point of the theory's renormalization group flow which controls the behavior of the coupling constants in the ultraviolet (UV) regime and renders physical quantities safe from divergences. Although originally proposed by Steven Weinberg to find a theory of quantum gravity, the idea of a nontrivial fixed point providing a possible UV completion can be applied also to other field theories, in particular to perturbatively nonrenormalizable ones. In this respect, it is similar to quantum triviality. Asymptotic safety (sometimes also referred to as nonperturbative renormalizability) is a concept in quantum field theory which aims at finding a consistent and predictive quantum theory of the gravitational field. Its key ingredient is a nontrivial fixed point of the theory's renormalization group flow which controls the behavior of the coupling constants in the ultraviolet (UV) regime and renders physical quantities safe from divergences. Although originally proposed by Steven Weinberg to find a theory of quantum gravity, the idea of a nontrivial fixed point providing a possible UV completion can be applied also to other field theories, in particular to perturbatively nonrenormalizable ones. In this respect, it is similar to quantum triviality. The essence of asymptotic safety is the observation that nontrivial renormalization group fixed points can be used to generalize the procedure of perturbative renormalization. In an asymptotically safe theory the couplings do not need to be small or tend to zero in the high energy limit but rather tend to finite values: they approach a nontrivial UV fixed point. The running of the coupling constants, i.e. their scale dependence described by the renormalization group (RG), is thus special in its UV limit in the sense that all their dimensionless combinations remain finite. This suffices to avoid unphysical divergences, e.g. in scattering amplitudes. The requirement of a UV fixed point restricts the form of the bare action and the values of the bare coupling constants, which become predictions of the asymptotic safety program rather than inputs. As for gravity, the standard procedure of perturbative renormalization fails since Newton's constant, the relevant expansion parameter, has negative mass dimension rendering general relativity perturbatively nonrenormalizable. This has driven the search for nonperturbative frameworks describing quantum gravity, including asymptotic safety which — in contrast to other approaches—is characterized by its use of quantum field theory methods, without depending on perturbative techniques, however. At the present time, there is accumulating evidence for a fixed point suitable for asymptotic safety, while a rigorous proof of its existence is still lacking. Gravity, at the classical level, is described by Einstein's field equations of general relativity, R μ ν − 1 2 g μ ν R + g μ ν Λ = 8 π G T μ ν {displaystyle extstyle R_{mu u }-{1 over 2}g_{mu u },R+g_{mu u }Lambda =8pi G,T_{mu u }} . These equations combine the spacetime geometry encoded in the metric g μ ν {displaystyle g_{mu u }} with the matter content comprised in the energy–momentum tensor T μ ν {displaystyle T_{mu u }} . The quantum nature of matter has been tested experimentally, for instance quantum electrodynamics is by now one of the most accurately confirmed theories in physics. For this reason quantization of gravity seems plausible, too. Unfortunately the quantization cannot be performed in the standard way (perturbative renormalization): Already a simple power-counting consideration signals the perturbative nonrenormalizability since the mass dimension of Newton's constant is − 2 {displaystyle -2} . The problem occurs as follows. According to the traditional point of view renormalization is implemented via the introduction of counterterms that should cancel divergent expressions appearing in loop integrals. Applying this method to gravity, however, the counterterms required to eliminate all divergences proliferate to an infinite number. As this inevitably leads to an infinite number of free parameters to be measured in experiments, the program is unlikely to have predictive power beyond its use as a low energy effective theory. It turns out that the first divergences in the quantization of general relativity which cannot be absorbed in counterterms consistently (i.e. without the necessity of introducing new parameters) appear already at one-loop level in the presence of matter fields. At two-loop level the problematic divergences arise even in pure gravity.In order to overcome this conceptual difficulty the development of nonperturbative techniques was required, providing various candidate theories of quantum gravity.For a long time the prevailing view has been that the very concept of quantum field theory — even though remarkably successful in the case of the other fundamental interactions — is doomed to failure for gravity. By way of contrast, the idea of asymptotic safety retains quantum fields as the theoretical arena and instead abandons only the traditional program of perturbative renormalization. After having realized the perturbative nonrenormalizability of gravity, physicists tried to employ alternative techniques to cure the divergence problem, for instance resummation or extended theories with suitable matter fields and symmetries, all of which come with their own drawbacks. In 1976, Steven Weinberg proposed a generalized version of the condition of renormalizability, based on a nontrivial fixed point of the underlying renormalization group (RG) flow for gravity.This was called asymptotic safety.The idea of a UV completion by means of a nontrivial fixed point of the renormalization groups had been proposed earlier by Kenneth G. Wilson and Giorgio Parisi in scalar field theory (see also Quantum triviality).The applicability to perturbatively nonrenormalizable theories was first demonstrated explicitly for the Non-linear sigma model and for a variant of the Gross-Neveu model. As for gravity, the first studies concerning this new concept were performed in d = 2 + ϵ {displaystyle d=2+epsilon } spacetime dimensions in the late seventies. In exactly two dimensions there is a theory of pure gravity that is renormalizable according to the old point of view. (In order to render the Einstein–Hilbert action 1 16 π G ∫ d 2 x g R {displaystyle extstyle {1 over 16pi G}int mathrm {d} ^{2}x{sqrt {g}},R} dimensionless, Newton's constant G {displaystyle G} must have mass dimension zero.) For small but finite ϵ {displaystyle epsilon } perturbation theory is still applicable, and one can expand the beta-function ( β {displaystyle eta } -function) describing the renormalization group running of Newton's constant as a power series in ϵ {displaystyle epsilon } . Indeed, in this spirit it was possible to prove that it displays a nontrivial fixed point. However, it was not clear how to do a continuation from d = 2 + ϵ {displaystyle d=2+epsilon } to d = 4 {displaystyle d=4} dimensions as the calculations relied on the smallness of the expansion parameter ϵ {displaystyle epsilon } . The computational methods for a nonperturbative treatment were not at hand by this time. For this reason the idea of asymptotic safety in quantum gravity was put aside for some years. Only in the early 90s, aspects of 2 + ϵ {displaystyle 2+epsilon } dimensional gravity have been revised in various works, but still not continuing the dimension to four. As for calculations beyond perturbation theory, the situation improved with the advent of new functional renormalization group methods, in particular the so-called effective average action (a scale dependent version of the effective action). Introduced in 1993 by Christof Wetterich and Tim R Morris for scalar theories, and by Martin Reuter and Christof Wetterich for general gauge theories (on flat Euclidean space), it is similar to a Wilsonian action (coarse grained free energy) and although it is argued to differ at a deeper level, it is in fact related by a Legendre transform. The cutoff scale dependence of this functional is governed by a functional flow equation which, in contrast to earlier attempts, can easily be applied in the presence of local gauge symmetries also.