The Functional f(R) Approximation

This chapter is a review of functional f(R) approximations in the asymptotic safety approach to quantum gravity. It mostly focuses 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 −∞ < R < ∞, 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.