Compton scattering S-matrix and cross section in strong magnetic field

Compton scattering of polarized radiation in a strong magnetic field is considered. The recipe for calculation of the scattering matrix elements, the differential and total cross sections based on quantum electrodynamic (QED) second order perturbation theory is presented for the case of arbitrary initial and final Landau level, electron momentum along the field and photon momentum. Photon polarization and electron spin state are taken into account. The correct dependence of natural Landau level width on the electron spin state is taken into account in general case of arbitrary initial photon momentum for the first time. A number of steps in calculations were simplified analytically making the presented recipe easy-to-use. The redistribution functions over the photon energy, momentum and polarization states are presented and discussed. The paper generalizes already known results and offers a basis for accurate calculation of radiation transfer in strong $B$-field, for example, in strongly magnetized neutron stars.