Elements of the Continuous Renormalization Group

Abstract

This lecture covers some of the advances that underpin recent progress in deriving continuum solutions from the exact renormalization group. We concentrate on concepts and on exact non-perturbative statements, but in the process will describe how real non-perturbative calculations can be done, particularly within derivative expansion approximations. An effort has been made to keep the lecture pedagogical and self-contained. Topics covered are the derivation of the flow equations, their equivalence, continuum limits, perturbation theory, truncations, derivative expansions, identification of fixed points and eigenoperators, and the rôle of reparametrization invariance. Some new material is included, in particular a demonstration of non-perturbative renormalizability, and a discussion of ultraviolet renormalons.