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

TL;DR

The paper tackles the problem of rigorously handling logarithmic divergences in quantum electrodynamics within Heisenberg's S-program framework. It introduces the deviation factor $W_0(t, eps)$ derived from the first approximation of the scattering operator and builds a regularized operator $S^R$, yielding a secondary generalized scattering operator $S^R(+inf,-inf,eps)$ without relying on an $eps$-expansion. The analysis addresses both infrared and ultraviolet divergences and applies to Coulomb-type problems, Friedrichs models, and concrete Feynman integrals, providing explicit formulas for deviation factors and higher-order terms. The results establish a mathematically rigorous realization of the S-program in the logarithmic-divergence regime and point toward extensions to other divergence types and self-energy analyses.

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?"