Paul BaldufIn principle, the behaviour of high-energy quantum gravity can be computed with the methods of functional #renormalization group equations. In practice, a number of assumptions and approximations are required, for example choosing a suitable splitting of the full metric field into a background and quantum-fluctuations, and choosing an IR cutoff functional that leads to analytic simplifications. Of particular importance is the choice of truncation for the effective action: The effective action is the generating function of correlation functions, it contains all physical information. In gravity, these correlation functions can potentially depend on all possible tensor structures, and have an arbitrary dependence on momenta. The earliest truncation, used in the 1990s, was to assume that there are only two terms: One proportional to the cosmological constant, and one proportional to the curvature R. By now, many further terms have been included. The present review analyzes the case where an arbitrary function f(R) of the curvature is allowed. This includes arbitrary powers R^n, but also trans-monomials like exp(1/R).
https://arxiv.org/abs/2210.11356
The functional $f(R)$ approximation
This article is a review of functional $f(R)$ approximations in the asymptotic safety approach to quantum gravity. It mostly focusses on a formulation that uses a non-adaptive cutoff, resulting in a second order differential equation. This formulation is used as an example to give a detailed explanation for how asymptotic analysis and Sturm-Liouville analysis can be used to uncover some of its most important properties. In particular, if defined appropriately for all values $-\infty<R<\infty$, one can use these methods to establish that there are at most a discrete number of fixed points, that these support a finite number of relevant operators, and that the scaling dimension of high dimension operators is universal up to parametric dependence inherited from the single-metric approximation. Formulations using adaptive cutoffs, are also reviewed, and the main differences are highlighted.