Renormalization and periods in perturbative Algebraic Quantum Field Theory

TL;DR

This paper explores the mathematical structures in perturbative algebraic quantum field theory (pAQFT) for massless scalar fields, linking renormalization, Feynman graphs, and Kontsevich-Zagier periods to deepen understanding of quantum field computations.

Contribution

It introduces a reformulation of the pAQFT renormalization group flow using Feynman graphs and connects periods to the pAQFT β-function, offering new insights into the mathematical underpinnings.

Findings

Relation between Kontsevich-Zagier periods and pAQFT computations

Reformulation of renormalization group flow in terms of Feynman graphs

Identification of periods in the calculation of the pAQFT β-function

Abstract

In this paper I give an overview of mathematical structures appearing in perturbative algebraic quantum field theory (pAQFT) in the case of the massless scalar field on Minkowski spacetime. I also show how these relate to Kontsevich-Zagier periods. Next, I review the pAQFT version of the renormalization group flow and reformulate it in terms of Feynman graphs. This allows me to relate Kontsevich-Zagier periods to numbers appearing in computing the pAQFT $\beta$-function.