Coherent Soft-Photon States and Infrared Divergences. III. Asymptotic States and Reduction Formulas
T. W. B. Kibble · Physical Review · 1968
In paper II of this series, the mass-shell singularities of the Green's functions of quantum electrodynamics were investigated. In this paper, the relationship between these singularities and the asymptotic states of the theory is studied by considering the nature of the intermediate states that can contribute to the corresponding discontinuity functions. The basic principle underlying this work is that the asymptotic states of the theory should not be specified a priori but should be determined from the structure of the Green's functions themselves. The pure soft-photon asymptotic states, which can be created from the vacuum by operators constructed from the soft-photon part of the electromagnetic field, are studied first. These states are defined by appropriate weak limits and are shown to span a space with the same structure as in the noninteracting case. Next, states containing a single particle (massive particle or hard photon), together with soft photons, are investigated. These states can appear as intermediate states in the two-point function. They are again defined by weak limits, and are shown to be stable in the absence of external currents. It is demonstrated that the near-mass-shell components of the field operator, acting on the vacuum or on a soft-photon coherent state, yield a state containing one particle and a soft-photon coherent state. Finally, the analysis is extended to two-particle and multiparticle states. The only essentially new feature here is the appearance of factors related to the "Coulomb phases." General reduction formulas are obtained that permit matrix elements between arbitrary asymptotic states to be extracted from the Green's functions. In effect, these matrix elements may be identified with the coefficients not of poles but of branch-point singularities.