Further Representations of the Canonical Commutation Relations

Abstract

. We construct a new class of representations of the canonical commutation relations, which generalizes previously known classes. We perturb the infinitesimal generator of the initial Fock representation (i.e. the free quantum field) by a function of the field which is square-integrable with respect to the associated Gaussian measure. We characterize which such perturbations lead to representations of the canonical commutation relations. We provide conditions entailing the irreducibility of such representations, show explicitly that our class of representations subsumes previously studied classes, and give necessary and sufficient conditions for our representations to be unitarily equivalent, resp. quasi-equivalent, with Fock, coherent or quasifree representations. Typeset by A M S-T E X 2 MARTIN FLORIG AND STEPHEN J. SUMMERS I. Introduction The canonical commutation relations (henceforth the CCR) were initially introduced in 1927 by Dirac as generalizations of Heisenberg's commuta..