On the properties of the Volkov solutions of the Klein-Gordon equation

TL;DR

This paper provides a rigorous proof of the completeness and orthogonality of Volkov solutions to the Klein-Gordon equation in a plane wave electromagnetic field, supporting their use in laser-matter interaction calculations.

Contribution

It offers an elementary proof of the completeness of Volkov solutions and discusses constructing the Feynman propagator in this context, clarifying their role in laser physics.

Findings

Proof of completeness of Volkov solutions

Derivation of orthogonality relations

Justification of transition amplitude expressions

Abstract

We present an elementary proof based on a direct calculation of the property of completeness at constant time of the solutions of the Klein-Gordon equation for a charged particle in a plane wave electromagnetic field. We also review different forms of the orthogonality and completeness relations previously presented in the literature and we discuss the possibility to construct the Feynman propagator for the particle in a plane-wave laser pulse as an expansion in terms of Volkov solutions. We show that this leads to a rigorous justification for the expression of the transition amplitude, currently used in the literature, for a class of laser assisted or laser induced processes.