Charged particles in an intense, plane electromagnetic wave: Limitations of the nonrelativistic theory

We discuss the classical and quantum theory of a charged particle moving in a plane wave electromagnetic field. In the nonrelativistic theory forbidden regions in the momentum spectrum of the particle occur, while no such regions exist in the relativistic theory. The gaps in the momentum spectrum are shown to be an artifact of the nonrelativistic approximation and are caused by the possibility of transforming to a Galilean frame of reference moving at the speed of light in which the particle moves in a time independent periodic potential. A quantitative estimate of the momentum gaps for optical frequencies shows that they are proportional to a very large power n≂105 of the ratio of particle velocity to the speed of light, and hence, beyond the limits of accuracy of the nonrelativistc theory. For this result to hold in the low frequency limit we have to employ a conjecture about the radius of convergence of the perturbation series expansion for eigenvalues of Mathieu’s equation. While no rigorous proof seems to exist at present, this conjecture is well supported both by numerical studies and by our WKB calculations.