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arXiv:math-ph/0006011 · Proceedings of the London Mathematical Society

Further Representations of the Canonical Commutation Relations

Martin Florig, Stephen J. Summers

Published Mar 1, 2000• 13 citations

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 (in other words, the free quantum field) by a function of the field which is square-integrable with respect to the associated Gaussian measure. We characterize perturbations which 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, or just quasi-equivalent, with Fock, coherent or quasifree representations. 1991 Mathematics Subject Classification: 81S05, 46L60, 81T05.

DOI: 10.1112/s0024611500012259

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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 (in other words, the free quantum field) by a function of the field which is square-integrable with respect to the associated Gaussian measure. We characterize perturbations which 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, or just quasi-equivalent, with Fock, coherent or quasifree representations. 1991 Mathematics Subject Classification: 81S05, 46L60, 81T05.

This paper studies a class of representations (called quadratic) of the canonical commutation relations over symplectic spaces of arbitrary dimension, which naturally generalizes coherent and symplectic (i.e. quasifree) representations and which has previously been heuristically employed in the special case of finite degrees of freedom in the physics literature. An explicit characterization of canonical quadratic transformations in terms of a 'standard form' is given, and it is shown that they can be exponentiated to give representations of the Weyl algebra. Necessary and sufficient conditions are presented for the unitary equivalence of these representations with the Fock representation. Possible applications to quantum optics and quantum field theory are briefly indicated.

A class of representations of the canonical commutation relations is studied, each of which is characterized by an expectation functional that is the exponential of a Euclidean-invariant quadratic form of the test functions. The underlying field operators are realized as the direct product of two Fock representations and the consequences of this realization are analyzed. Compatible Hamiltonians are constructed and an extensive study of the most general quadratic Hamiltonians is presented. In order to include thermodynamic examples, the analysis includes indefinite Hamiltonian spectra as well as the usual definite spectra. Finally, conditions are given for a theory to be local in the sense that all time derivatives of the field operator commute with one another at equal times but unequal spatial arguments.

We consider direct-product representations of the canonical commutation relations. An irreducible representation is defined on each of the incomplete direct-product spaces (IDPS) of von Neumann. We prove that two such representations are unitarily equivalent if and only if the corresponding IDPS are weakly equivalent, for which simple analytic tests exist. The matrix elements of these representations, coupled with a Friedrichs-Shapiro type of integral, fulfill group orthogonality relations. This classification into unitary equivalence classes also applies to direct-product representations of the canonical anticommutation relations.

In models of a quantum harmonic oscillator coupled to a quantum field with a quadratic interaction, embedded eigenvalues of the unperturbed system may be unstable under the perturbation given by the interaction of the oscillator with the quantum field.A general mathematical structure underlying this phenomenon is clarified in terms of a class of Fock space representations of the * -algebra of the canonical commutation relations over a Hilbert space.It is also shown that each of the representations is given as a composition of a proper Bogolyubov (canonical) transformation and a partial isometry on the Fock space of the representation.