Quantum field theories with shadow states. I. Soluble models

Abstract

To construct a finite local relativistic quantum field theory we may introduce an indefinite-metric vector space, but then to avoid conflict with unitarity we must consider only a selected subset of states to be physical. The remaining states participate in the dynamics but are not among the complete set of physical states as far as probability interpretation is concerned. These states are called shadow states. The S matrix should be unitary when restricted to the physical states. In this paper we formulate and solve several simple models of field theories with shadow states and demonstrate the manner in which shadow states influence the dynamics and the structure of the scattering amplitude. The choice of a standing-wave boundary condition for the shadow states is shown to be completely consistent with the physical description of the scattering process in terms of wave packets. These methods are adapted to the study of low-energy pion-nucleon scattering in the following paper.