The Epstein-Glaser approach to pQFT: graphs and Hopf algebras

TL;DR

This paper explores the Epstein-Glaser approach to perturbative quantum field theory, emphasizing its formulation through graphs and Hopf algebras, and clarifies the mathematical structure underlying renormalization and causality.

Contribution

It introduces a Hopf algebra framework for Epstein-Glaser pQFT, linking algebraic structures with physical concepts like causality and renormalization.

Findings

Identifies multiple Hopf algebras associated with physical concepts.

Provides a mathematical reasoning for ultraviolet divergences.

Models renormalization scheme changes via a Hopf algebra similar to Kreimer's.

Abstract

The paper aims at investigating perturbative quantum field theory (pQFT) in the approach of Epstein and Glaser (EG) and, in particular, its formulation in the language of graphs and Hopf algebras (HAs). Various HAs are encountered, each one associated with a special combination of physical concepts such as normalization, localization, pseudo-unitarity, causality and an associated regularization, and renormalization. The algebraic structures, representing the perturbative expansion of the S-matrix, are imposed on the operator-valued distributions which are equipped with appropriate graph indices. Translation invariance ensures the algebras to be analytically well-defined and graded total symmetry allows to formulate bialgebras. The algebraic results are given embedded in the physical framework, which covers the two recent EG versions by Fredenhagen and Scharf that differ with respect to the concrete recursive implementation of causality. Besides, the ultraviolet divergences occuring in Feynman's representation are mathematically reasoned. As a final result, the change of the renormalization scheme in the EG framework is modeled via a HA which can be seen as the EG-analog of Kreimer's HA.