Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT

TL;DR

This paper links Rota-Baxter algebras with the algebraic Birkhoff decomposition in perturbative quantum field theory, revealing a non-commutative generalization of Spitzer's identity through Hopf algebra characters.

Contribution

It demonstrates that solutions to Birkhoff factorization in Rota-Baxter algebras naturally extend Spitzer's identity to a non-commutative setting in pQFT.

Findings

Establishes a connection between Rota-Baxter algebras and Birkhoff decomposition.

Provides a non-commutative generalization of Spitzer's identity.

Analyzes the algebraic structure using complete filtered Rota-Baxter algebras.

Abstract

In this article we continue to explore the notion of Rota-Baxter algebras in the context of the Hopf algebraic approach to renormalization theory in perturbative quantum field theory. We show in very simple algebraic terms that the solutions of the recursively defined formulae for the Birkhoff factorization of regularized Hopf algebra characters, i.e. Feynman rules, naturally give a non-commutative generalization of the well-known Spitzer's identity. The underlying abstract algebraic structure is analyzed in terms of complete filtered Rota-Baxter algebras.