Hopf Algebra Theory of Renormalization

The second chapter is a detailed account of renormalization theory. The goal is, in particular, to demonstrate that the modern Hopf-algebraic formulation is entirely natural and compatible with the more traditional approach. We start in Sect. 2.1 with a review of formal power series and Hopf algebras. While well-known to mathematicians, these topics are not usually taught in physics courses, and the section includes basic definitions and examples in order to be accessible to physicists. The core finding is that a Hopf algebra is a systematic framework that encodes, in one way or another, the insertion of combinatorial objects into each other. The canonical example for this is the insertion of power series, f(g(x)), which lies at the heart of all renormalization and can naturally be visualized in terms of rooted trees. We examine renormalization of Feynman integrals in Sect. 2.2. The original physical motivation for renormalization is that the predictions from quantum field theory contain unspecified “bare” parameters, such as coupling constants, and that one needs to systematically infer these values from a measurement in order to make concrete physical predictions.