Reflections on the renormalization procedure for gauge theories

TL;DR

This paper reflects on the development of renormalization procedures in gauge theories, highlighting key insights, the role of symmetries like BRST, and personal recollections related to the mathematical foundations of quantum field theories.

Contribution

It provides a historical and conceptual overview of the formulation and renormalization of gauge theories, emphasizing the importance of symmetries and the mathematical understanding of quantum field theories.

Findings

Understanding multiple derivations of Feynman rules

Role of gauge constraints in consistent calculations

Significance of BRST symmetry in renormalization

Abstract

Various pieces of insight were needed to formulate the rules for working with gauge theories of the electro-magnetic, weak and strong forces. First, it was needed to understand how to formulate the Feynman rules. We had to learn that there are many different ways to derive them, and it was needed to know how different formulations of the gauge constraint lead to the same final results: the calculated values of the scattering amplitudes. The rules for dealing with the infinities that had to be subtracted were a big challenge, culminating in the discovery of the Becchi-Rouet-Stora-Tyutin symmetry. Fond recollections of the numerous discussions the author had with Raymond Stora on this topic are memorised here. We end with some reflections on the mathematical status of quantum field theories, and a letter sent by Stora to the author