Semiclassical method of calculating the characteristics of the spontaneous radiation of a charge moving in periodic structures

It is shown that in quantum electrodynamics with an external field the intensity of spontaneous radiation of a charge (in the first order in the radiation field) can be presented in a form of a functional of classical trajectory. The proof is given for the case of spinless particles moving in periodic structures, using quasiclassical solutions of the Klein–Gordon equation. As an example, the explicit calculation is performed for a flat magnetic ondulator including the first order correction.