General quantum theory of Thomson scattering

It is well known that the interaction of radiation with matter can be described using classical or quantum physics in relation to three systems: incoming radiation, matter, and scattered radiation. Currently, there is no general non-perturbative theory of electromagnetic wave scattering by free electrons, which describes the scattering process using quantum physics. In this paper, such a non-relativistic theory is presented, where statistics and the number of incoming photons is taken into account, and in scattering, statistics with the number of scattered photons is obtained. This theory is completely analytical, considering an arbitrary number of electrons in the system and, in a particular case, goes over into the previously known theory of scattering as the number of incident photons tends to infinity. It is shown that this theory can differ greatly from the previously known theory of Thomson scattering in the non-perturbative case and at relatively small numbers of incident photons. In addition, this theory is applicable to the scattering of ultrashort pulses by free electrons. This theory has good prospects for application in quantum optics, since it fully takes into account the quantum nature of the incident and scattered radiation when interacting with an arbitrary number of free electrons, including in the non-perturbative case.