Electron in a quantized plane-wave field and in the classical field of a longitudinal electric wave

The Dirac equation is solved for an electron moving in a quantized plane-wave field in the classical field of a longitudinal traveling electric wave propagating in one direction. Through a canonical transformation of the photon creation and annihilation operators the problem is reduced to a quasiparticle problem; the quasiparticle energy depends on the time and the coordinates.