Quantum Theory of Generation of Coherent X-Ray in a Wiggler

In this paper we present the nonlinear quantum theory of X-Ray FEL in a wiggler. We present the solution of the Dirac equation in a space periodic strong magnetic field, which describes the quantum dynamics of a single electron in a wiggler. With the help of obtained wave function of an electron the self-consistent set of the Maxwell and relativistic quantum kinetic equations is obtained. Then, the process of coherent X-ray radiation generation in nonlinear quantum regime, when the photon energy is larger than resonance widths due to energetic/angular spreads of an electron beam and resonance width caused by finite interaction length, is investigated.