Relation of the O(2, 1) Partial-Wave Expansion to the Regge Representation

The general two-particle scattering amplitude is expanded in terms of partial waves corresponding to the crossed channel little group, O(2, 1). Under the assumption of square integrability over the group manifold, the invariance of the S matrix under the complex Lorentz group, which follows from the Bargmann-Hall-Wightmann theorem, enables this expansion to be identified with the Regge representation in the crossed channel, whenever no dynamical singularities occur to the right of Re j = −½. The identification requires the assumption of the fixed t dispersion relation necessary for the definition of the Regge representation.