  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.
