  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.
