Scattering of a High-Intensity, Low-Frequency Electromagnetic Wave by an Unbound Electron

"Thomson" scattering of a high-intensity, low-frequency, circularly-polarized electromagnetic wave by a free electron is considered. We find that by neglecting radiative corrections and pair effects, the Feynman-Dyson perturbation expansion is summable, and the sum can be analytically continued in the form of a sum of continued fractions. By imposing the boundary conditions that at $t=\pm \infty$ the photons and target electron propagate as free particles, we obtain results which differ from those reported by Brown and Kibble and by Goldman. In particular our results differ in two aspects. The first difference is in the kinematics; namely, we find no intensity-dependent frequency shift in the scattered photon. The second difference is in the dynamics; that is, we obtain a different expression for the scattering amplitude. Both of these changes originate in the choice of boundary conditions. Instead of treating the asymptotic radiation field classically, we choose our states as linear combinations of occupation-number states. Finally, contact is made with the results of Brown and Kibble and of Goldman using a mixed set of classical and quantum boundary values.