On the classical limit and the infrared problem for non relativistic fermions interacting with the electromagnetic field

Abstract

The classical limit and the infrared divergence problem for a non-relativistic charged quantum particle interacting with the quantized electromagnetic field are analized. Overall three momentum conservation is taken into account. A unitary transformation associated to the coherent state corresponding to a particle surrounded by a cloud of soft photons is performed upon the hamiltonian and the particle-field states. The transformed state represnting a moving dressed quantum particle and its energy are given by the Brillouin-Wigner perturbation theory. It is shown formally that the quantum energy approaches the classical electromagnetic field at the Plank constant h goes to zero. Moreover,all Feynman diagrams contributing to this quantum energy are infrared finite, without needing add diagrams of the same order in the electric charge to obtain the infrared finiteness. Those properties justify the usefulness of the unitary transformation. The Compton effect in the forward direction is studied using dressed charged particle states after the unitary transformation has been performed. The quantum cross section approaches the classical limit (Thomson´s formula) as h ͢͢͢͢͢ 0, and the Feynman diagrams are free of infrared divergences.