The Asymptotic Safety Scenario in Quantum Gravity -- An Introduction
M. Niedermaier
TL;DR
The paper investigates whether gravity can be renormalizable and unitary within the asymptotic safety framework, characterized by a non-Gaussian fixed point where essential couplings ${\mathrm g}_i(\mu)$ tend to finite limits ${\mathrm g}_i^*$ as $\mu\to\infty$. It surveys multiple lines of evidence—$2+\varepsilon$ expansion, perturbative higher-derivative gravity with large-$N$ matter, 2-Killing vector reductions, and truncated flow equations of the effective average action—arguing for a coherent UV completion. A central claim is that UV fixed-point dynamics induce a dimensional reduction, with self-interactions becoming effectively two-dimensional and anomalous dimensions like $\eta_N = 2-d$ at the fixed point, consistent with a two-dimensional effective description in the extreme UV. The paper discusses unitarity considerations, the role of coarse graining, and outlines concrete avenues to establish a genuine continuum limit and characterize observables, highlighting four future research directions to consolidate the fixed point and understand the microstructure of quantum geometries.
Abstract
The asymptotic safety scenario in quantum gravity is reviewed, according to which a renormalizable quantum theory of the gravitational field is feasible which reconciles asymptotically safe couplings with unitarity. All presently known evidence is surveyed: (a) from the 2+\eps expansion, (b) from the perturbation theory of higher derivative gravity theories and a `large N' expansion in the number of matter fields, (c) from the 2-Killing vector reduction, and (d) from truncated flow equations for the effective average action. Special emphasis is given to the role of perturbation theory as a guide to `asymptotic safety'. Further it is argued that as a consequence of the scenario the selfinteractions appear two-dimensional in the extreme ultraviolet. Two appendices discuss the distinct roles of the ultraviolet renormalization in perturbation theory and in the flow equation formalism.