Some combinatorial interpretations in perturbative quantum field theory

TL;DR

This paper explores how combinatorial methods can elucidate the algebraic structure of perturbative quantum field theory, focusing on Dyson-Schwinger equations and Feynman graph periods, with specific examples of geometric series reduction and denominator reduction techniques.

Contribution

It introduces novel combinatorial interpretations that improve understanding and simplification of Dyson-Schwinger equations and Feynman graph calculations in quantum field theory.

Findings

Enhanced reduction techniques for Dyson-Schwinger equations

Identification of subgraphs aiding denominator reductions

Explanation of Brown and Schnetz's trick for graph analysis

Abstract

This paper will describe how combinatorial interpretations can help us understand the algebraic structure of two aspects of perturbative quantum field theory, namely analytic Dyson-Schwinger equations and periods of scalar Feynman graphs. The particular examples which will be looked at are, a better reduction to geometric series for Dyson-Schwinger equations, a subgraph which yields extra denominator reductions in scalar Feynman integrals, and an explanation of a trick of Brown and Schnetz to get one extra step in the denominator reduction of an important particular graph.