Dyson Schwinger Equations: From Hopf algebras to Number Theory

TL;DR

This paper explores the algebraic structures underlying renormalizable quantum field theories, specifically Hopf algebras and Hochschild cohomology, to derive non-perturbative insights into their short-distance behavior.

Contribution

It introduces a framework connecting Hopf algebra structures to non-perturbative analysis of quantum field theories' short-distance singularities.

Findings

Hopf algebra structures encode renormalization properties.

Hochschild cohomology aids in deriving non-perturbative results.

Analysis of Green functions reveals insights into short-distance behavior.

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 cohomology to derive non-perturbative results for the short-distance singular sector of a renormalizable quantum field theory. We focus on the short-distance behaviour and thus discuss renormalized Green functions $G_R(\alpha,L)$ which depend on a single scale $L=\ln q^2/\mu^2$.