Relativistic invariance as gauge invariance and high-intensity Compton scattering

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Abstract
The problem of an electron interacting with the classical external field of a plane electromagnetic wave is reexamined. An explicit connection between symmetry properties of the free field and the behavior of an electron in the external field is exhibited. A new group of transformations leaving the electromagnetic field tensor unaltered is constructed and is used to derive the classical and quantum (semiclassical) evolution operators of an electron in the wave. The classical operator is shown to describe both the evolution of the electron and the evolution of its spin in the external field. The quantum operator is used to define a representation which seems well adapted to the calculation of scattering processes that occur in a laser beam. However, difficulties of uniqueness of this representation appear in the case of a monochromatic wave. The high-intensity Compton scattering amplitude is derived as a simple application of the new formalism.