The combinatorics of Bogoliubov's recursion in renormalization

TL;DR

This paper explores the combinatorial structures underlying Bogoliubov's recursion in renormalization, linking it to the Magnus expansion and Hopf algebra frameworks, providing new insights into the algebraic foundations of quantum field theory.

Contribution

It establishes a novel connection between Bogoliubov's recursion and the pre-Lie Magnus expansion within the context of Hopf algebras, broadening the mathematical understanding of renormalization.

Findings

Identifies Bogoliubov's preparation map with the pre-Lie Magnus expansion.

Applies combinatorial methods to any connected filtered Hopf algebra.

Highlights the role of pro-nilpotency in the Lie algebra of infinitesimal characters.

Abstract

We describe various combinatorial aspects of the Birkhoff-Connes-Kreimer factorization in perturbative renormalisation. The analog of Bogoliubov's preparation map on the Lie algebra of Feynman graphs is identified with the pre-Lie Magnus expansion. Our results apply to any connected filtered Hopf algebra, based on the pro-nilpotency of the Lie algebra of infinitesimal characters.