Introduction to the Functional RG and Applications to Gauge Theories

This lecture course is intended to fill the gap between graduate courses on quantum field theory and specialized reviews or forefront-research articles on functional renormalization group approaches to quantum field theory and gauge theories. These lecture notes are meant for advanced students who want to get acquainted with modern renormalization group (RG) methods as well as functional approaches to quantum gauge theories. In the first lecture, the functional renormalization group is introduced with a focus on the flow equation for the effective average action. The second lecture is devoted to a discussion of flow equations and symmetries in general, and flow equations and gauge symmetries in particular. The third lecture deals with the flow equation in the background formalism which is particularly convenient for analytical computations of truncated flows. The fourth lecture concentrates on the transition from microscopic to macroscopic degrees of freedom; even though this is discussed here in the language and the context of QCD, the developed formalism is much more general and will be useful also for other systems. Sections which have an asterisk ∗ in the section title contain more advanced material and may be skipped during a first reading. This is not a review. I apologize for many omissions of further interesting and important aspects of this field (and their corresponding references). General reviews and more complete reference lists can be found in [1, 2, 3, 4, 5, 6, 7, 8]. A guide to more specialized literature is given in the Further-Reading subsections at the end of some sections.