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.
Highlights
The exact solution of the Dirac equation in the presence of a background plane-wave electromagnetic field was found by Volkov well before the invention of the laser [1]
The corresponding one-particle electron states have been widely employed in order to describe quantum electrodynamical processes occurring in the presence of a strong laser field, starting from the pioneering papers by Reiss [2], by Gol’dman [3], by Brown and Kibble [4], and by Nikishov and Ritus [5]
The Green’s function so obtained is related to the exact Feynman propagator in the plane-wave field (Volkov propagator). By expressing the former function as an integral over the Volkov states with positive and negative energy, we show in a relatively straightforward way the completeness of the Volkov states themselves at a fixed time
Summary
Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany (Received 9 February 2018; published 30 March 2018). 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. 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 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. 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
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