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

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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 properties 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 framework 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.

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