Structures in Feynman Graphs -Hopf Algebras and Symmetries

TL;DR

This paper reviews the combinatorial structures of Feynman graphs in quantum field theory, focusing on their decomposition into primitive graphs and the implications for Dyson--Schwinger equations.

Contribution

It introduces a detailed analysis of graph decomposition and explores the algebraic and symmetry properties related to Hopf algebras in quantum field theory.

Findings

Primitive graph decomposition elucidates the structure of perturbative expansions.

Unique factorization of Dyson--Schwinger equations into Euler products is established.

Highlights the role of Hopf algebras in understanding symmetries in Feynman graphs.

Abstract

We review the combinatorial structure of perturbative quantum field theory with emphasis given to the decomposition of graphs into primitive ones. The consequences in terms of unique factorization of Dyson--Schwinger equations into Euler products are discussed.