On calculation of cross sections in Lorentz violating theories

We develop a systematic approach to the calculation of scattering cross sections in theories with violation of the Lorentz invariance taking into account the whole information about the theory Lagrangian. As an illustration we derive the Feynman rules and formulas for sums over polarizations in spinor electrodynamics with Lorentz violating operators of dimensions four and six. These rules are applied to compute the probabilities of several astrophysically relevant processes. We calculate the rates of photon decay and vacuum Cherenkov radiation along with the cross sections of electron-positron pair production on background radiation and in the Coulomb field. The latter process is essential for detection of photon-induced air showers in the atmosphere.