Quantum field theory and renormalization \`a la St\"uckelberg-Petermann-Epstein-Glaser

TL;DR

This paper rigorously addresses the renormalization problem in perturbative quantum field theory by studying the extension of distributions representing Green functions, ensuring they satisfy physical axioms like causality.

Contribution

It provides necessary and sufficient conditions for extending distributions with ultraviolet divergences and explicitly constructs such extensions adhering to physical principles.

Findings

Established criteria for distribution extension in pQFT

Constructed extensions satisfying causality and other axioms

Clarified the mathematical structure underlying renormalization

Abstract

The problem of renormalization in perturbative quantum field theory (pQFT) can be described in a rigorous way through the theory of extension of distributions. In the framework of pQFT a certain type of distribution appears, given by products of Green functions which act by integration with a test function. They present ultraviolet divergences, whenever any pair of arguments coincide on one point of spacetime, and therefore, they are not defined everywhere. In this work we have studied the necessary and sufficient conditions for the extension (or regularization) of this type of distribution. Moreover, we have constructed such extensions explicitly, satisfying a series of physically relevant axioms, such as the axiom of causality.