Matrix Elements of the Translation Operator: Variations on the Jost-Hepp Theorem

We consider matrix elements of the translation operator in any continuous, unitary representation U(a, A) of the covering group iSL(2,C) of the Poincaré group on a Hilbert space, between C∞ and analytic vectors of the restriction U(0, A) of the representation to the homogeneous subgroup. By applying the Jost-Hepp technique, we extend the directions of rapid decrease (assuming any vacuum is excluded) to include not only spacelike ones, but all directions outside the center of momentum velocity cone generated by the four-momentum support. When the vectors are analytic vectors of U(0, A), and have only physical momentum support contained in V+ (again excluding any vacuum), we show that the decrease is exponential for large, spacelike translations. Then the p-space measure corresponding to the matrix element is analytic in a strip in the spatial components p of the mass-momentum variables, analogous to the Jost-Hepp result that the measure is C∞ in p, for C∞ vectors. ∗March 17, 2008: This LTEX version has only cosmetic changes from the original.