## Matrix Elements of the Translation Operator:

Variations on the Jost-Hepp Theorem

### David N. Williams

Department of Physics and Astronomy

The University of Michigan

Ann Arbor, 1970
**Abstract:**
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.

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