Combinatorics of (perturbative) quantum field theory

TL;DR

This paper reviews the algebraic structures, such as Hopf and Lie algebras, underlying perturbative quantum field theory, highlighting their roles in renormalization, operadic graph insertions, and connections to the Riemann-Hilbert problem.

Contribution

It elucidates the algebraic frameworks in perturbative QFT, emphasizing the role of Hopf and Lie algebras and their relation to renormalization and the Riemann-Hilbert problem.

Findings

Hopf algebra structures underpin renormalization via forest formulas.

Lie algebra arises from operadic graph insertions.

Connections established between algebraic structures and Feynman diagram numbers.

Abstract

We review the structures imposed on perturbative QFT by the fact that its Feynman diagrams provide Hopf and Lie algebras. We emphasize the role which the Hopf algebra plays in renormalization by providing the forest formulas. We exhibit how the associated Lie algebra originates from an operadic operation of graph insertions. Particular emphasis is given to the connection with the Riemann--Hilbert problem. Finally, we outline how these structures relate to the numbers which we see in Feynman diagrams.