Lippmann-Schwinger equation and the connection between the scattering operator and the scattering amplitude in the relativistic case

TL;DR

This paper explores the relationship between stationary and dynamical scattering problems in relativistic quantum mechanics, establishing a connection between the scattering amplitude and the scattering operator using the Lippmann-Schwinger equation.

Contribution

It introduces a relativistic version of the Lippmann-Schwinger equation and links stationary and dynamical scattering concepts, extending classical ergodic formulas to quantum mechanics.

Findings

Established the connection between scattering operator and amplitude in relativistic case.

Extended classical ergodic formulas to quantum scattering theory.

Provided a framework for analyzing relativistic scattering problems.

Abstract

In this paper, we consider two types of the scattering problems (relativistic case), namely, the stationary scattering problem, where the distance $r$ tends to infinity, and the dynamical scattering problem, where the time $t$ tends to infinity. Using our results on Lippmann-Schwinger equation in the relativistic case, we found the connection between the stationary scattering problem (the scattering amplitude) and the dynamical scattering problem (the scattering operator). This result is the quantum mechanical analog of the ergodic formulas in the classical mechanics.