Hopf algebras and Dyson–Schwinger equations

Abstract

In these lectures I discuss Hopf algebras and Dyson-Schwinger equations. The lectures start with an introduction to Hopf algebras, followed by a review of the contribution and application of Hopf algebras to particle physics. The final part of these lectures is devoted to the relation between Hopf algebras and DysonSchwinger equations.

Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction

“…For a pedagogical introduction to some of these concepts, see Refs. [17,[32][33][34][35][36]. In the context of Eq.…”

Graphons and renormalization of large Feynman diagrams

“…Each term X n is produced on the basis of the terms with the lower degrees and furthermore, terms X n s play the role of generators for a Hopf sub-algebra associated to the equation DSE [1,18,19,25,26,40,41]. This presentation of the solution X, which belongs to the completion of H [[α]] with respect to the n-adic topology, has been applied as a starting point to build some new mathematical structures which are capable to encode non-perturbative parameters [33,35,36].…”

“…The completion of the Hopf algebra H with respect to this topology is the extended Hopf algebra H = n≥0 H n which has elements of the form n≥0 Γ n such that Γ n ∈ H n . It is shown that solutions of combinatorial DSEs belong to this completed Hopf algebra [1,19,25,26,40,41]. On the other side, for each n, X n is a finite Feynman diagram which guarantees the finiteness of the terms Y n s. Each Y n could be constructed from Y n−1 by a growing (not generally uniform) attachment graph sequence.…”

“…Our best chance in this situation is to make stronger the mathematical foundations of non-perturbative QFT. In this direction, search for some new advanced mathematical structures originated from Dyson-Schwinger equations could be helpful for the better understanding of the phenomenology of non-perturbative parameters [33,35,36,40,41].…”

Combinatorial Dyson–Schwinger equations and inductive data types

“…1.1 The Connes-Kreimer Hopf algebra of (rooted) trees. For discussions of the notions of bialgebra and Hopf algebra, see the contribution of Weinzierl [28] in the present volume. The Connes-Kreimer Hopf algebra of (rooted) trees (also called the Butcher-Connes-Kreimer Hopf algebra) is the free algebra H CK on the set of isomorphism classes of combinatorial trees, such as , ,…”