Scattering of Infraparticles

Abstract

In the hope of getting some new insight into the infrared problem we study the space-time aspects of electron scattering events in quantum electrodynamics. Localization of electrons is provided by „counters”︁ described by a quasilocal field. An initial n-electron state, prepared in a specified bounded region of space-time, is obtained by applying n electron field operators, smeared over suitable test functions, to the vacuum. The probability that m counters localized in bounded four-dimensional neighbourhoods of the points x1, …, xm are triggered, is the vacuum expectation value of a product of two time-ordered products of n electron fields and m counter fields each. These vacuum expectation values are closely related to the Green's functions of the theory. For macroscopic counter separations xi–xi we can evaluate the triggering probability with the help of asymptotic methods, starting from the p-space singularities of the Green's functions. We assume the singularity structure of perturbation theory summed over soft photons. As typical examples we analyze one and two electron initial states. The results are those expected for classical particles. For a one-particle state with momentum p, prepared in a neighbourhood of the origin, the probability that counters in x1. …, xm are triggered is measurably non-zero only if x1, …,xm lie on a ray through O in direction p. The dependence of the probability on the distances ∥xi – xi∥ is as given by geometry. Similar results hold for two- particle states, in which case scattering events are seen. Only elastic scattering is considered. The dependence of the scattering probability on the momenta of the four particles involved is given by a factor |M(p, p2, q1, q2)|2. M replaces the S-matrix element of the familiar case without infrared problems. The definition of M is a generalization of the reduction formula expression of S in terms of Green's functions. The properties of M are not discussed. Processes involving photons as observed particles are not considered.