Heisenberg's S-matrix program and Feynman's divergence problem

TL;DR

This paper introduces a method to regularize divergent scattering operators in quantum electrodynamics using deviation factors, addressing longstanding divergence issues in Feynman's approach.

Contribution

It proposes a novel regularization technique based on deviation factors to handle divergences in scattering operators without expansion in small parameters.

Findings

Successfully regularizes divergence in scattering operators

Addresses infrared and ultraviolet divergence cases

Provides a rigorous approach to Feynman's divergence problem

Abstract

In the present article, we assume that the first approximation of the scattering operator is given and that it has the logarithmic divergence. This first approximation allows us to construct the so called deviation factor. Using the deviation factor, we regularize all terms of the scattering operator's approximations. The infrared and ultraviolet cases as well as concrete examples are considered. Thus, for a wide range of cases, we provide a positive answer to the well-known problem of J. R. Oppenheimer regarding scattering operators in QED: ``Can the procedure be freed of the expansion in $\varepsilon$ and carried out rigorously?"