Smatrix for finite quantum electrodynamics in the Heisenberg representation

Abstract

In this paper we present a covariant formulation of the $S$ matrix in the Heisenberg representation for finite quantum electrodynamics. The theory involves the use of an indefinite metric implemented with the idea of shadow states. The indefinite metric is used to make the theory finite, and the idea of shadow states is introduced to make the $S$ matrix for physical scattering processes unitary. The equations of motion for the field operators are solved with the boundary conditions fixed in accordance with the idea of shadow states. The consequences of this choice of boundary conditions are exhibited in the problem of vacuum polarization. It turns out that the analytic properties of the $S$ matrix element in this formalism are quite different from those in conventional theory. Following the methods of Källén and of Yang and Feldman, we proceed to construct the $S$ matrix for physical scattering processes. The explicit forms for the $S$ matrix in the perturbation expansion up to second order in $e$ are given. Some unfamiliar features of this formalism, such as (1) the presence of noncausal effects, (2) piecewise analyticity of the scattering amplitudes, and (3) the reinterpretation of crossing symmetry are also discussed.