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A relativistic quantum mechanical model to describe the quantum FEL dynamicshas been developed. Neglecting the spin of electrons in the impacting beam,this model is based on the Klein-Gordon equation coupled to the Poissonequation for the space-charge potential and the wave equation for thetransverse components of the radiation field. Furthermore, a system of couplednonlinear envelope equations for the slowly varying amplitudes of the electronbeam distribution and the radiation field has been derived.  The fundamental system of basic equations have been cast into a suitablehydrodynamic formulation. In the framework of the hydrodynamic representation,a new dispersion relation has been derived and analyzed in both the quantum andthe quasi-classical regimes, where the space-charge oscillations of theelectron beam are taken into account.

Quantum Description of Free Electron Laser Radiation and Nonlinear Amplitude Equations