Pole_Factorization Theorem in Quantum Electrodynamics
  In quantum electrodynamics a classical part of the S-matrix is normally factored out in order to obtain a quantum remainder that can be treated perturbatively without the occurrence of infrared divergences. However, this separation, as usually performed, introduces spurious large-distance effects that produce an apparent breakdown of the important correspondence between stable particles and poles of the S-matrix, and, consequently, lead to apparent violations of the correspondence principle and to incorrect results for computations in the mesoscopic domain lying between the atomic and classical regimes. An improved computational technique is described that allows valid results to be obtained in this domain, and that leads, for the quantum remainder, in the cases studied, to a physical-region singularity structure that, as regards the most singular parts, is the same as the normal physical-region analytic structure in theories in which all particles have non-zero mass. The key innovations are to define the classical part in coordinate space, rather than in momentum space, and to define there a separation of the photon-electron coupling into its classical and quantum parts that has the following properties: 1) The contributions from the terms containing only classical couplings can be summed to all orders to give a unitary operator that generates the coherent state that corresponds to the appropriate classical process, and 2) The quantum remainder can be rigorously shown to exhibit, as regards its most singular parts, the normal analytic structure. 
Henry P. Stapp