Relativistic multiphoton bremsstrahlung

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Abstract
It was shown some time ago by Brown and Goble [Phys. Rev. 173, 1505 (1968)] that the low-frequency approximation for single-photon bremsstrahlung derived originally by Low [Phys. Rev. 110, 974 (1958)] could be extended to the two-photon case. A further extension to the case of N-photon bremsstrahlung is obtained here. For definiteness the scattering system is taken to be that of a Dirac electron in the presence of a static potential, but greater generality is possible, since, as in the earlier treatments, the argument is based on the generally valid gauge-invariance requirement. The first two terms in the low-frequency expansion may be constructed from a knowledge of the on-shell field-free scattering amplitude. If, in addition, the physical single-photon bremsstrahlung amplitude is known to first order in the frequency, the first four terms in the expansion of the N-photon amplitude, for N>1, may be determined. Some remarks are included concerning the possible utility of the theorem in external-field problems and in the derivation of infrared radiative corrections to scattering processes.