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Construction of a Complete Set of States in Relativistic Scattering Theory

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.