On the completeness and orthonormality of the Volkov states and the Volkov propagator in configuration space

TL;DR

This paper provides simplified proofs of the completeness and orthonormality of Volkov states in QED with intense plane-wave fields, and explores their analytical properties and gauge transformation behaviors in configuration space.

Contribution

It offers new, more straightforward proofs of fundamental properties of Volkov states and analyzes the Green's function and propagator in a plane wave context.

Findings

Proved completeness of Volkov states using Green's function properties.

Established orthonormality of Volkov states via geometric arguments.

Derived a closed-form expression for the Green's function in terms of special functions.

Abstract

Volkov states and Volkov propagator are the basic analytical tools to investigate QED processes occurring in the presence of an intense plane-wave electromagnetic field. In the present paper we provide alternative and relatively simple proofs of the completeness and of the orthonormality at a fixed time of the Volkov states. Concerning the completeness, we exploit some known properties of the Green's function of the Dirac operator in a plane wave, whereas the orthonormality of the Volkov states is proved, relying only on a geometric argument based on the Gauss theorem in four dimensions. In relation with the completeness of the Volkov states, we also study some analytical properties of the Green's function of the Dirac operator in a plane wave, which we explicitly prove to coincide with the Volkov propagator in configuration space. In particular, a closed-form expression in terms of modified Bessel functions and Hankel functions is derived by means of the operator technique in a plane wave and different asymptotic forms are determined. Finally, the transformation properties of the Volkov propagator under general gauge transformations and a general gauge-invariant expression of the so-called dressed mass in configuration space are presented.