Further Representations of the Canonical Commutation Relations

TL;DR

This paper introduces a new class of representations for the canonical commutation relations by perturbing the Fock representation, providing a comprehensive characterization of their properties and relationships to existing representations.

Contribution

It generalizes known classes of CCR representations by perturbing the infinitesimal generator with square-integrable functions, and characterizes conditions for irreducibility and equivalence.

Findings

New class of CCR representations constructed

Conditions for irreducibility established

Criteria for unitary and quasi-equivalence provided

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.