Combinatorics of renormalization as matrix calculus

TL;DR

This paper presents a matrix calculus approach to the combinatorics of renormalization in quantum field theory, based on the Birkhoff decomposition in Hopf algebras, offering a potentially useful computational method.

Contribution

It introduces a simple matrix-based framework for understanding the combinatorics of renormalization derived from Hopf algebra principles.

Findings

Provides a matrix calculus formulation of renormalization combinatorics.

Derives the method from first principles using Birkhoff decomposition.

Offers a potentially practical computational tool for quantum field theory.

Abstract

We give a simple presentation of the combinatorics of renormalization in perturbative quantum field theory in terms of triangular matrices. The prescription, that may be of calculational value, is derived from first principles, to wit, the ``Birkhoff decomposition'' in the Hopf-algebraic description of renormalization by Connes and Kreimer.