Scattering Matrix in the Heisenberg Representation for a System with Bound States

Abstract

A definition is formulated of the scattering matrix for a closed physical system with bound states which makes use throughout only of the assumed observable p~operties of the system. A direct product space, X, is defined in which the ingredient factor space comprises the steady states-vacuum, one­ particle, and bound states-of the physical system. It is argued that the boundary conditions for a scattering experiment are suitably expressed in terms of vectors in X and that these stand in unitary correspondence, U, to the Heisenberg states. Indeed, one defines two operators U to express outgoing and ingoing wave boundary conditions, and the scattering matrix is constructed from these in the usual way. A suitable Yang-Feldman formalism is then developed in which the operators in the remote past and future also describe the bound states of the system. A representation of the frame­ work thus constructed in terms of field operators for individual fields results in the well-known formulas for S-matrix elements in terms of covariant amplitudes.