Low-frequency photons in processes involving particles with spin

The method of $S$-matrix electrodynamics is used to analyze processes in which spin particles participate. A number of effects associated with soft phonons (real and virtual) are calculated nonperturbatively:
1) The terms of order $k^{-1}$, $\ln k$, $k^0$, $k\ln^2k$, in the matrix element for emission of a soft photon ($k$ is the photon energy) are found and expressed in terms of the matrix element of the process without emission and the static electromagnetic characteristics of the particles (charge and magnetic moment);
2) a threshold theorem is obtained for Compton scattering on a spin $1/2$ particle up to terms $k^2\ln k$;
3) for the scattering amplitude of charged spin $1/2$ particles two terms $O(t^{-1})+O(t^{-1/2})$ in the expansion in the square of the momentum transfer $t$ are obtained.
The second term gives the correction to the Rutherford formula for the scattering cross section. The obtained results, which hold in all orders in the coupling constant and can in principle be verified experimentally, are essentially nontrivial consequences of gauge invariance.