Compton scattering in external magnetic fields: Spin-zero charged particles

Methods recently developed by Schwinger are applied to derive a general expression for the total spin-0 unpolarized Compton cross section in the presence of an arbitrarily strong homogeneous external magnetic field. This expression is correct to order ${\alpha }^2$ in radiative effects, but treats the external field exactly. The motion of the incident photon is restricted to lie along the field direction. Special limiting cases corresponding to classical and weakfield limits are discussed.