UV and IR behaviour in QFT and LCQFT with fields as Operator Valued Distributions:Epstein and Glaser revisited

TL;DR

This paper revisits Epstein-Glaser's operator valued distribution approach in quantum field theory, demonstrating how it rigorously handles UV and IR divergences without infinities, and highlighting its potential for non-perturbative methods.

Contribution

It introduces a mathematically rigorous formulation of QFT using operator valued distributions with test functions, effectively managing divergences and aiding non-perturbative analysis.

Findings

UV divergences are avoided through test function regularization.

IR behavior is effectively controlled in massive scalar field theory.

The approach simplifies the treatment of divergences in non-perturbative contexts.

Abstract

Following Epstein-Glaser's work we show how a QFT formulation based on operator valued distributions (OPVD) with adequate test functions treats original singularities of propagators on the diagonal in a mathematically rigourous way.Thereby UV and/or IR divergences are avoided at any stage, only a finite renormalization finally occurs at a point related to the arbitrary scale present in the test functions.Some well known UV cases are examplified.The power of the IR treatment is shown for the free massive scalar field theory developed in the (conventionally hopeless) mass perturbation expansion.It is argued that the approach should prove most useful for non pertubative methods where the usual determination of counterterms is elusive