Feynman's linear divergence problem

TL;DR

This paper addresses the problem of linear divergence in QED by constructing secondary generalized scattering operators, providing a rigorous approach to scattering theory without expansion in small parameters.

Contribution

It introduces a method to handle linear divergence in QED through secondary generalized scattering operators, advancing the mathematical rigor of scattering theory.

Findings

Derived modifications of commutation relations for scattering operators.

Constructed secondary generalized scattering operators for linear divergence in QED.

Provided a positive answer to Oppenheimer's problem on rigorous scattering procedures.

Abstract

First, we consider generalized wave and scattering operators and derive modifications of commutation relations (between scattering operators and unperturbed operators) when the corresponding deviation factors behave as $\exp\{i t {\mathcal C}_{\pm}\}$ for $t\to \pm \infty$. Then, we construct so called secondary generalized scattering operators for the related case of linear divergence in QED, which gives a positive answer (in that case) 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?"