A Proof of Haag-Swieca's Compactness Property for Elastic Scattering States

Abstract

It is proved that in any massive relativistic quantum field theory satisfying two-particle asymptotic completeness, all the bounded energy components in the elastic two-particle range of all subsets of states which are excitations of the vacuum state by uniformly bounded observables localized in a given finite region of spacetime are compact in the Hilbert space of states. This result, which is in agreement with Haag-Swieca's conjecture, is also given a more precise form in terms of the rate of decrease of the ``N–dimensional thickness'' (or approximation number) of such sets of states when N tends to infinity. A similar computation, valid at arbitrarily high energies, is also given for the massive free-field case.