## Macroscopic Causality and Permanence of

Smoothness for Two-Particle Scattering

### David N. Williams

The Institute for Advanced Study

Princeton, New Jersey

August, 1967
**Abstract:**
Recent developments in the formulation of causality restrictions
on the *S* matrix are reviewed, with attention focused on
the behavior of matrix elements of the translation operator
between suitably localized in and out states. Rapid decrease
for large translations outside the timelike velocity cone of the
center of momentum follows from Poincaré invariance and
boundedness of *S*, as a result of a generalization of a
theorem of Jost and Hepp. At present, rapid decrease can be
proved in the Haag-Ruelle scattering theory, when the in state
is translated to large positive times, but not for the remaining
timelike directions, where thresholds of intermediate particles
play a role. In the case of two-particle reactions, we show
that rapid decrease for timelike directions is equivalent to
permanence of smoothness of the *p*-pace wave function, as
an application of rapid convergence properties of the angular
momentum expansion.

Published in *Lectures in Theoretical Physics *, vol. XB,
*High Energy Physics and Fundamental Particles*, (Gordon and
Breach, New York, 1968), pp. 357–376.

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