The Dirac Equation of the Compton Effect

We are looking at a Dirac electron in the electromagnetic field of a plane monochrome polarized X-ray. It will be attempted to link the terms of a certain (joint) asymptotic expansion of the Heisenberg propagations of momentum- and total-energy- observables with collision of the electron with 1, 2, 3, photons. Our asymptotic expansion neglects terms small for states of very high frequencies, and might be natural, in many respects. A special focus will be given to the single collision term. We attempt a description of this term as a (distribution) integral operator, to analyze directionally different oscillations of the electron, to be compared with those of the Compton model. We obtain agreement of scattering frequencies, but not of scattering directions. It might be recalled, in that respect, that the Dirac particle is not a simple mass point - it has a dual nature as electron-positron; it has a spin who possibly could rotate after the collision, then absorbing energy-momentum, etc.