Springer Theses- 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.

This book offers a systematic introduction to the Hopf algebra theory of renormalization in quantum field theory. Special emphasis is put on physical motivation for mathematical constructions, the role of Dyson–Schwinger equations (DSEs), renormalization conditions, and the renormalization group. The bulk of the book deals with the similarities and differences between two popular renormalization conditions, kinematic renormalization (MOM) and Minimal Subtraction (MS). MOM is a physical global boundary condition for Green functions. DSEs can then be solved in terms of power series which only involve finite renormalized quantities. Conversely, MS is defined order-by-order based on divergences of the unrenormalized Green function. We show that MS is equivalent to MOM with coupling-dependent renormalization point. We determine the large-order growth of series coefficients in different renormalization schemes and derive a novel analytic formula for the all-order solution of linear DSEs in MS. Finally, we derive the changes in off-shell Green functions and counterterms under nonlinear redefinition of field variables for self-interacting scalar fields. The book is aimed at mathematically oriented physicists and physically interested mathematicians who seek a systematic overview of the Hopf algebra theory of renormalization and DSEs.

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.