Dyson-Schwinger Equations, Renormalization Conditions, and the Hopf Algebra of Perturbative Quantum Field Theory

This book offers a systematic introduction to the Hopf algebra of renormalization in quantum field theory, with a special focus on physical motivation, the role of Dyson-Schwinger equations, and the renormalization group. All necessary physical and mathematical constructions are reviewed and motivated in a self-contained introduction. The main part of the book concerns the interplay between Dyson-Schwinger equations (DSEs) and renormalization conditions. The book is explicit and consistent about whether a statement is true in general or only in particular renormalization schemes or approximations and about the dependence of quantities on regularization parameters or coupling constants. With over 600 references, the original literature is cited whenever possible and the book contains numerous references to other works discussing further details, generalizations, or alternative approaches. There are explicit examples and remarks to make the connection from the scalar fields at hand toQED and QCD. The book is primarily targeted at the mathematically oriented physicist who seeks a systematic conceptual overview of renormalization, Hopf algebra, and DSEs. These may be graduate students entering the field as well as practitioners seeking a self-contained account of the Hopf algebra construction. Conversely, the book also benefits the mathematician who is interested in the physical background of the exciting interplay between Hopf algebra, combinatorics and physics that is renormalization theory today.

Paul-Hermann Balduf studied at Friedrich Schiller University Jena (Germany) and University of Lund (Sweden), receiving a Bachelor’s degree from the former. Having moved to Berlin (Germany), he received his Master’s degree from Humboldt-Universität zu Berlin in 2018. He did his doctorate in the group ＂Structure of Local Quantum Field Theories＂ at Humboldt-Universität under the supervision of Prof. Dr. Dirk Kreimer. Currently, he is a postdoctoral fellow at University of Waterloo (Canada). His research interest is in renormalization of quantum field theories. In particular, he is interested in the behavior of perturbative series at high order, in combinatorial properties of Feynman graphs, in numerical integration and in the physical background and interpretation of the Hopf algebra framework of renormalization.