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Journal of Mathematical Physics

Direct-Product Representations of the Canonical Commutation Relations

John R. Klauder, J. McKenna, E. J. Woods

Published May 1, 1966• 55 citations
Publisher

Abstract

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.

DOI
10.1063/1.1931213

Related papers

Partial-Tensor-Product Representations of the Canonical Commutation Relations: Classification

G. Reents

Partial-tensor-product representations of the canonical commutation relations are a generalization of direct- (or tensor-) product representations as discussed by Klauder, McKenna, and Woods. A classification is given of those representations up to unitary equivalence. The equivalence or inequivalence, respectively, of two representations is expressed in terms of the corresponding reference vectors.

Journal of Mathematical Physics•Jun 1, 1970•2 citations

Kernel Integral Formulas for the Canonical Commutation Relations of Quantum Fields. II. Irreducible Representations

Gerhard C. Hegerfeldt

Continuing our investigation of the kernel or group integral for the canonical commutation relations introduced by Klauder and McKenna, we prove that a representation fulfilling any sort of kernel integral formula is irreducible. This has been conjectured by Klauder and McKenna. After collecting some auxiliary results, a complete classification of all representations is given for which a kernel integral formula in the form of a limit superior holds. It is shown that these are just the partial tensor-product representations and that the limit superior can be replaced by an ordinary limit over a fixed subsequence, thus allowing the transition from norms to scalar products. Then the basis-independent kernel integral in the form of a sup lim¯ is investigated. Here the supremum is taken over all bases of the test-function space. Under a not-very-strong irreducibility assumption, we show that this can be reduced to the vacuum functional and that there exists a fixed sequence of subspaces of the test-function space such that the sup lim¯ can be replaced by an ordinary limit which again allows a transition to scalar products. The results are strikingly similar to the case of cyclic field. This tempts us to conjecture that a representation fulfilling a kernel integral formula is both irreducible and cyclic with respect to the field just as in the case of finitely many degrees of freedom.

Journal of Mathematical Physics•Jan 1, 1970•3 citations

Further Representations of the Canonical Commutation Relations

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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.

arXiv:math-ph/0006011•Proceedings of the London Mathematical Society•Mar 1, 2000•13 citations

On the Representation of the Canonical Commutation Relation of Bose Fields

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A presentation of the canonical commutation relation of Bose fields is given in a way which is independent of the choice of the bases of the test functions and covariant with respect to the Euclidean transformation of the coordinate system. It is shown that the representation is characterized by an integral on the conjugate space L* of the space L of the test functions and a real function on Σ⊗L* where Σ is the group of the transformations f→u-1f+φ; fεL*, φεL and u is a Euclidean transformation of L*. The conditions for the irreducibility of a representation and the unitary equivalence of the representations and the existence of unique vacuum state are given. An example of the inequivalent Euclidean covariant irreducible representations containing unique vacuum state is given.

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Kernel Integral Formulas for the Canonical Commutation Relations of Quantum Fields. I. Representations with Cyclic Field

Gerhard C. Hegerfeldt

We investigate the kernel or group integral for the canonical commutation relations introduced by Klauder and McKenna and its generalizations. For the finite case the kernel integral formula has been proven by means of the Schrödinger representation. Motivated by the close similarity of the Schrödinger representation to the form of a general representation with cyclic field, we examine these representations with respect to kernel integral formulas. A general criterion is derived in which the dimensionality of the test function space does not enter, i.e., it is independent of the number of degrees of freedom. In this way the finite and infinite case can be treated on equal footing. The criterion contains as special cases the kernel integral formulas of Klauder and McKenna for finitely many degrees of freedom and for direct-(or tensor-) product representations of fields. For partial tensor-product representations we obtain a somewhat modified formula. After these applications, a considerably sharpened form of the criterion is derived in which only the vacuum expectation functional enters. Under a certain cyclicity assumption it is shown that the validity of a kernel integral for just some cyclic vector implies its validity for all vectors. It is further shown that the basis-independent integral defined by a supremum over all bases of the test function space U can be replaced by an ordinary limit over a kind of diagonal sequence through finite-dimensional subspaces of U. In the last section a representation is constructed which possesses a cyclic field but does not fulfil a kernel integral formula; this is an instructive illustration of a general theorem to be proved in II of this series of papers.

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L2(E*, μ)-WEYL REPRESENTATIONS

Xiaoshan Hu, Zhiyuan Huang, Xiangjun Wang

For the canonical commutation relations in infinite dimensions, we offer an explicit direct construction of Weyl representations Wϕ generated from the Fock representation by any ϕ ∈ L2(E*, μ, R) over the Q-space (E*, μ). Moreover, we obtain that, for any ϕ, ψ ∈ L2(E*, μ,R), Wϕ+ψ and Wϕ are unitarily equivalent, proving a conjecture posed by Robinson in Ref. 2. Our construction employs Wiener–Itô decomposition of the space L2(E*, μ, R) (respectively L2(E*, μ, C)).

Infinite Dimensional Analysis Quantum Probability and Related Topics•Dec 1, 2002•0 citations
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.
Continuing our investigation of the kernel or group integral for the canonical commutation relations introduced by Klauder and McKenna, we prove that a representation fulfilling any sort of kernel integral formula is irreducible. This has been conjectured by Klauder and McKenna. After collecting some auxiliary results, a complete classification of all representations is given for which a kernel integral formula in the form of a limit superior holds. It is shown that these are just the partial tensor-product representations and that the limit superior can be replaced by an ordinary limit over a fixed subsequence, thus allowing the transition from norms to scalar products. Then the basis-independent kernel integral in the form of a sup lim¯ is investigated. Here the supremum is taken over all bases of the test-function space. Under a not-very-strong irreducibility assumption, we show that this can be reduced to the vacuum functional and that there exists a fixed sequence of subspaces of the test-function space such that the sup lim¯ can be replaced by an ordinary limit which again allows a transition to scalar products. The results are strikingly similar to the case of cyclic field. This tempts us to conjecture that a representation fulfilling a kernel integral formula is both irreducible and cyclic with respect to the field just as in the case of finitely many degrees of freedom.
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.
A presentation of the canonical commutation relation of Bose fields is given in a way which is independent of the choice of the bases of the test functions and covariant with respect to the Euclidean transformation of the coordinate system. It is shown that the representation is characterized by an integral on the conjugate space L* of the space L of the test functions and a real function on Σ⊗L* where Σ is the group of the transformations f→u-1f+φ; fεL*, φεL and u is a Euclidean transformation of L*. The conditions for the irreducibility of a representation and the unitary equivalence of the representations and the existence of unique vacuum state are given. An example of the inequivalent Euclidean covariant irreducible representations containing unique vacuum state is given.
We investigate the kernel or group integral for the canonical commutation relations introduced by Klauder and McKenna and its generalizations. For the finite case the kernel integral formula has been proven by means of the Schrödinger representation. Motivated by the close similarity of the Schrödinger representation to the form of a general representation with cyclic field, we examine these representations with respect to kernel integral formulas. A general criterion is derived in which the dimensionality of the test function space does not enter, i.e., it is independent of the number of degrees of freedom. In this way the finite and infinite case can be treated on equal footing. The criterion contains as special cases the kernel integral formulas of Klauder and McKenna for finitely many degrees of freedom and for direct-(or tensor-) product representations of fields. For partial tensor-product representations we obtain a somewhat modified formula. After these applications, a considerably sharpened form of the criterion is derived in which only the vacuum expectation functional enters. Under a certain cyclicity assumption it is shown that the validity of a kernel integral for just some cyclic vector implies its validity for all vectors. It is further shown that the basis-independent integral defined by a supremum over all bases of the test function space U can be replaced by an ordinary limit over a kind of diagonal sequence through finite-dimensional subspaces of U. In the last section a representation is constructed which possesses a cyclic field but does not fulfil a kernel integral formula; this is an instructive illustration of a general theorem to be proved in II of this series of papers.