New exact solutions of the Dirac equation of a charged particle interacting with an electromagnetic plane wave in a medium

Exact solutions are presented of the Dirac equation of a charged particle moving in a classical monochromatic electromagnetic plane wave in a medium of index of refraction nm < 1. The found solutions are expressed in terms of new complex trigonometric polynomials, which form a doubly infinite set labelled by two integer quantum numbers. These quantum numbers represent quantized spectra of the energy–momentum components of the charged particle along the polarization vector and along the propagation direction of the applied electromagnetic plane wave field (which is considered as a laser field of arbitrarily high intensity propagating in an underdense plasma). The found solutions may serve as a basis for the description of possible quantum features of mechanisms for the acceleration of electrons by high-intensity laser fields. (letter)