Construction of a complete set of states in relativistic scattering theory

Abstract

The space of physical states in relativistic scattering theory is constructed, using a rigorous version of the Dirac formalism, where the Hilbert space structure is extended to a Gel’fand triple. This extension enables the construction of “a complete set of states,” the basic concept of the original Dirac formalism, also in the cases of unbounded operators and continuous spectra. We construct explicitly the Gel’fand triple and a complete set of “plane waves”—momentum eigenstates—using the group of space–time symmetries. This construction is used (in a separate article) to prove a generalization of the Coleman–Mandula theorem to higher dimension.