Coherent Soft-Photon States and Infrared Divergences. II. Mass-Shell Singularities of Green's Functions

This is the second in a series of four papers in which a new, field-theoretic approach to the problem of the infrared divergences of quantum electrodynamics is presented. The primary aim of the present paper is the study of the mass-shell singularities of the Green's functions, which are branch points rather than simple poles. This is an essential preliminary to the discussion of asymptotic states and scattering matrix elements in subsequent papers. A conventional separation is introduced between hard- and soft-photon regions of momentum space. It is shown that the soft-photon contribution to an arbitrary Green's function may be isolated in a single function of the external momenta, independent of the spins of the particles involved. An explicit expression for this function is obtained and its mass-shell singularities are studied in detail. In particular, it is shown that, in contrast to the case where there are no massless particles, the mass-shell singularities in different momenta are not independent, but depend on the order in which the various momenta are allowed to approach their mass shells.