  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.
