  We present a nonlinear theory for relativistic X-ray free electron lasers in
the quantum regime, using a collective Klein-Gordon (KG) equation (for
relativistic electrons), which is coupled with the Maxwell-Poisson equations
for the electromagnetic and electrostatic fields. In our model, an intense
electromagnetic wave is used as a wiggler which interacts with a relativistic
electron beam to produce coherent tunable radiation. The KG-Maxwell-Poisson
model is used to derive a general nonlinear dispersion relation for parametric
instabilities in three-space-dimensions, including an arbitrarily large
amplitude electromagnetic wiggler field. The nonlinear dispersion relation
reveals the importance of quantum recoil effects and oblique scattering of the
radiation that can be tuned by varying the beam energy.
