文档介绍：Dyson Schwinger Equations: From Hopf algebras to
Number Theory
Dirk KREIMER
Institut des Hautes Etudes´ Scientifiques
35, route de Chartres
91440 – Bures-sur-Yvette (France)
Septembre 2006
IHES/P/06/45
Fields munications
Volume 00, 0000
Dyson Schwinger Equations: From Hopf algebras to Number
Theory
Dirk Kreimer
IHES´
Le Bois Marie, 91440 Bures sur Yvette, France
******@
Abstract. We consider the structure of renormalizable quantum field
theories from the viewpoint of their underlying Hopf algebra structure.
We review how to use this Hopf algebra and the ensuing Hochschild co-
homology to derive non-perturbative results for the short-distance sin-
gular sector of a renormalizable quantum field theory. We focus on the
short-distance behaviour and thus discuss renormalized Green functions
2 2
GR(α, L) which depend on a single scale L = ln q /µ .
1 Introduction
The crucial notion of locality, the structure of Dyson–Schwinger equations and
the appearance of mixed motives in the evaluation of Feynman graphs are intimately
related. We want to exhibit how these e together in quantum field
theory. We emphasize the role of Dyson Schwinger equations in this interplay.
Renormalization theory is a time-tested subject put to daily use in many
branches of physics. We have seen many of its facets illuminated here at the Fields
Institute. In this paper, we focus on its applications in quantum field theory, where
a standard perturbative approach is provided through an expansion in Feynman
diagrams. In perturbation theory it is mainly binatorial problem: determine
the needed correction to parameters in the Lagrangian such that putation
allows for finite results in the desired order of perturbation. Whilst the resulting
combinatorics of the Bogoliubov recursion, solved by suitable forest formulas, has
been known for a long time, the subject regained interest on the conceptual side
with the discovery of an underlying Hopf al