Locally constant and crossed field approximation for description of the photon emission in a strong oscillating electric field

The process of photon emission by a scalar particle (spinless analogue of an electron) in the presence of a strong time-dependent and homogeneous electric field is considered. Provided that the emitting particle is ultra-relativistic and the field component transverse to its propagation direction is large enough, in the calculation of the process amplitude, it usually is enough to apply the locally constant field approximation. Using the example of the linearly polarised oscillating electric field, we analyse the validity limits of this approximation and test it against the numerical calculation of the process amplitude, derived using the semi-classical solution of the Klein–Gordon equation in a time-dependent field.