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Journal of Mathematics
Volume 2013 (2013), Article ID 308392, 7 pages
http://dx.doi.org/10.1155/2013/308392
Research Article

Hilbert Space Representations of Generalized Canonical Commutation Relations

Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

Received 19 November 2012; Accepted 10 December 2012

Academic Editor: Stefan Siegmund

Copyright © 2013 Asao Arai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider Hilbert space representations of a generalization of canonical commutation relations , where 's are the elements of an algebra with identity , is the imaginary unit, and is a real number with antisymmetry . Some basic aspects on Hilbert space representations of the generalized CCR (GCCR) are discussed. We define a Schrödinger-type representation of the GCCR by an analogy with the usual Schrödinger representation of the CCR with degrees of freedom. Also, we introduce a Weyl-type representation of the GCCR. The main result of the present paper is a uniqueness theorem on Weyl representations of the GCCR.

1. Introduction

In this paper, we consider Hilbert space representations of a generalized canonical commutation relations (GCCRs) with degrees of freedom () of the following type: where ’s are elements of an algebra with identity , , is the imaginary unit, and (the set of real numbers) with antisymmetry such that, for some pair , . For convenience, we call (1) the -GCCR with degrees of freedom and the matrix the noncommutative factor for .

Note that, in the case where is equal to with being the unit matrix, (1) becomes the CCR with degrees of freedom. Namely, if we put in the present case, then we have where is the Kronecker delta. Thus, (1) is a natural generalization of the CCR with degrees of freedom.

The GCCR also includes some of non-commutative space times (e.g., [13]), non-commutative spaces (e.g., [4]), and non-commutative phase spaces (e.g., [511]). In fact, one of the motivations for the present work is to investigate general structures underlying those non-commutative objects. In this paper, however, we present only some fundamental aspects of Hilbert space representations of the GCCR. The main result is to establish a uniqueness theorem on Weyl type representations of the GCCR (for the definition, see Section 4).

In Section 2, we define Hilbert space representations of the GCCR and discuss some basic facts on them. It is shown that there exists a one-to-one correspondence between representations of the GCCR and the CCR with the same degrees of freedom. In Section 3, we introduce a Schrödinger-type representation of the GCCR, whose representation space is as in the case of the Schrödinger representation of the CCR with degrees of freedom. In Section 4, Weyl-type representations of the GCCR are defined by analogy with Weyl representations of CCR. In the last section, we prove the uniqueness theorem mentioned above. In Appendix, we present some basic properties of self-adjoint operators obeying generalized Weyl relations, which are used in the text.

2. Basic Facts on Hilbert Space Representations of the -GCCR

Let be a complex Hilbert space with inner product (antilinear in the first variable and linear in the second one) and norm . For a linear operator on , we denote its domain by . For linear operators on ,

Definition 1. Let be a dense subspace of and , , be symmetric (not necessarily essentially self-adjoint) operators on . Set . We say that the triple is a symmetric representation of the -GCCR with degrees of freedom if and (1) holds on .
If all the ’s () are self-adjoint, we say that is a self-adjoint representation of the GCCR.

Remark 2. The concept of self-adjoint representation defined above is different from the one used in representation theory of -algebra (e.g., [12, page 205]).

Remark 3. In each symmetric representation of the -GCCR, is infinite dimensional (if were finite dimensional, then, for such that , trace of, and hence one is led to a contradiction).

Remark 4. It follows from a well-known fact on commutation properties of linear operators (e.g., [13, Theorem 1.2.3]) that, for with , at least one of and is unbounded. Hence, one has to be careful about domains of ’s.

Remark 5. In the case of Hilbert space representations of CCR, symmetric representations, but nonself-adjoint ones, also play important roles. For example, such representations appear in mathematical theories of time operators [14] (see also [15, 16] for investigations from purely operator-theoretic points of view). Thus, it is expected that, in addition to self-adjoint representations of the -GCCR, non-self-adjoint symmetric representations of it may have any importance in applications to quantum physics.

Remark 6. In the context of quantum mechanics, for a symmetric operator and a unit vector , is called the uncertainty of in the vector state . Let be a symmetric representation of the -GCCR with degrees of freedom. Then, one has uncertainty relations of Robertson type [17]: for all unit vectors and ,

Let be a symmetric representation of the -GCCR as in Definition 1. We assume for simplicity the following:

Assumption 1. The noncommutative factor is regular (invertible).

Under this assumption, is a regular antisymmetric real matrix. Hence, by a well-known fact in the theory of linear algebra (e.g., [18, page 173, Problem 9]), the following fact holds.

Lemma 7. There exists a regular real matrix such that , where is the transposed matrix of and is defined by (3).

The matrix in Lemma 7 belongs to the set It is easy to see that for each , there exists a unique symplectic matrix (i.e., ) such that . Hence,

For a real matrix , we define We call the correspondence the -transform of  .

Let

Proposition 8. (i) For all , is a symmetric operator on .
(ii) For all , on .
(iii) For each and , on .

Proof. An easy exercise.

Proposition 8-(i) and (ii) show that is a symmetric representation of the -GCCR with degrees of freedom.

Proposition 8 (iii) implies the following.

Corollary 9. Let and Then, is a symmetric representation of the CCR with degrees of freedom.

Corollary 9 means that for each , the -transform of gives a correspondence from a symmetric representation of the -GCCR with degrees of freedom to a symmetric representation of the CCR with the same degrees of freedom.

One can easily see that (9) with implies that on . Thus, every symmetric representation of the -GCCR with degrees of freedom is constructed from a symmetric representation of the CCR with the same degrees of freedom via (14).

Conversely, if a symmetric representation of the CCR with degrees of freedom is given and let with and , then is a symmetric representation of the -GCCR and (9) holds with , and . Hence, every symmetric representation of the CCR with degrees of freedom is constructed from a symmetric representation of the -GCCR with the same degrees of freedom. Thus, for each , there exists a one-to-one correspondence between a symmetric representation of the -GCCR and a symmetric representation of the CCR with degrees of freedom.

3. Representations of Schrödinger Type

Let . By the fact on stated in the preceding section, we can define a class of representations of the -GCCR. Let be the Schrödinger representation of the CCR with degrees of freedom, that is, is the multiplication operator by the th component of and with being the generalized partial differential operator in , acting in . Let which is (15) with and . We denote the closure of by and set We call the triple the -Schrödinger representation of the -GCCR.

It is easy to see that for all , is essentially self-adjoint on (apply, e.g., the Nelson commutator theorem [19, Theorem X.37] with dominating operator ) (This can be proved also by applying Proposition 16). Hence is self-adjoint. Thus, we obtain the following.

Proposition 10. For each , the -Schrödinger representation is a self-adjoint representation of the -GCCR.

4. Representations of Weyl Type

Based on an analogy with Weyl representations of CCR, we introduce a concept of Weyl representation for -GCCR.

Definition 11. Let be a set of self-adjoint operators on a Hilbert space . We say that is a Weyl representation of the -GCCR with degrees of freedom if for all and , We call these relations the -Weyl relations.

For a linear operator on a Hilbert space, we denote its spectrum by .

Proposition 12. Let be a Weyl representation of the -GCCR on . Then, there is a dense subspace left invariant by each () such that is a self-adjoint representation of the -GCCR. Moreover, for every pair such that , and are purely absolutely continuous with

Proof. By (18), we can apply the results described in the Appendix of the present paper. In the present context, we need only to take, in the notation in the Appendix, , and . By Proposition A.4-(iii) and Corollary A.5, there exists a dense subspace left invariant by () and on . Thus, the first half of the proposition is derived. The second half follows from Proposition A.1.

Remark 13. As in the case of self-adjoint representations of CCR (e.g., [16, 20, 21]), the converse of Proposition 12 does not hold (i.e., a self-adjoint representation of the -GCCR is not necessarily a Weyl one).

We recall that a set of self-adjoint operators on is a Weyl representation of the CCR with degrees of freedom if for all and , the following Wey relations hold:

Remark 14. A set of self-adjoint operators on is a Weyl representation of the CCR with degrees of freedom if and only if with is a Weyl representation of the -GCCR, where is given by (3).

Let be arbitrarily fixed. The next proposition shows that the -transform of each Weyl representation of the -GCCR is a Weyl representation of the CCR with degrees of freedom.

Proposition 15. Let be a Weyl representation of the -GCCR on , and let be the -transform of . Then, each is essentially self-adjoint, and is a Weyl representation of the -GCCR.

Proof. The essential self-adjointness of follows from a simple application of Theorem A.6 in Appendix. Corollary A.7 in Appendix and the relation imply that satisfies the -Weyl relations.

In the same way as in the proof of Proposition 15, we can prove the following proposition:

Proposition 16. Let be a Weyl representation of the CCR with degrees of freedom on a Hilbert space . Let () be defined by (15). Then, each is essentially self-adjoint and is a Weyl representation of -GCCR with degrees of freedom.

This proposition shows that the converse of Proposition 15 holds too. Thus, for each , there exists a one-to-one correspondence between a Weyl representation of the CCR with degrees of freedom and that of the -GCCR with the same degrees of freedom.

It is well known [22] that the Schrödinger representation is a Weyl representation o the CCR with degrees of freedom. Hence, we obtain the following result.

Corollary 17. For each , the -Schrödinger representation is a Weyl representation of the -GCCR.

We say that a Weyl representation of the -GCCR on is irreducible if every closed subspace of which is invariant under the action of is or .

Proposition 18. Let . Then, the -Schrödinger representation as a Weyl representation of the -GCCR is irreducible.

Proof. Let be an invariant closed subspace of (). We have on . Hence, by an application of Theorem A.6 in Appendix, and () can be written, respectively, as a scalar multiple of . Hence, is invariant under the action of and (). It is well known that is irreducible. Thus, or .

5. Uniqueness Theorem on Weyl Representations of the -GCCR

In this section, we prove the main result of the present paper, that is, a uniqueness theorem on Weyl representations of the -GCCR, which may be regarded as a GCCR version of the celebrated von Neumann uniqueness theorem of Weyl representations of CCR ([13, Theorem 4.11.1], [22], [23, Theorem VIII.14]).

Theorem 19. Let be a Weyl representation of the -GCCR on a separable Hilbert space . Then, for each , there exist mutually orthogonal closed subspaces (; or ) such that the following (i)–(iii) hold.(i). (ii)For each , is reduced by each , . We denote by the reduced part of to . (iii)For each , there exists a unitary operator such that where is the -Schrödinger representation of the -GCCR.

Proof. Let , be the -transform of and , (). Then, by Proposition 15 and Remark 14, is a Weyl representation of the CCR with degrees of freedom. Hence, by the von Neumann uniqueness theorem mentioned above, there exist mutually orthogonal closed subspaces such that (i) given above and the following (a) and (b) hold.(a)For each and all , and leave each invariant (). (b)For each , there exists a unitary operator such that
By (14), we have on . Hence, . By Proposition 16, is self-adjoint. Hence, . Therefore, by Theorem A.6 in Appendix, we obtain Hence, each leaves invariant (). Therefore, is reduced by each . We denote the reduced part of to by . Then, we have by (23) Thus, (22) follows.

Theorem 19 tells us that every Weyl representation of the -GCCR on a separable Hilbert space is unitarily equivalent to a direct sum of the -Schrödinger representation of the -GCCR, where is arbitrary.

The next corollary immediately follows from Theorem 19.

Corollary 20. Let be an irreducible Weyl representation of the -GCCR on a separable Hilbert space . Then, for each , there exists a unitary operator such that

The following result shows that the arbitrariness of the choice of in the -Schrödinger representation of the -GCCR is implemented by unitary operators.

Corollary 21. Let . Then, there exists a unitary operator on such that

Proof. We need only to apply Corollary 20 to the case where .

Remark 22. As in the case of non-Weyl representations of CCR, for non-Weyl representations of the -GCCR, the conclusion of Theorem 19 does not hold in general. Examples of such representations of the -GCCR can be constructed from non-Weyl representations of CCR (e.g., [15, 16, 20, 21]). A detailed description of some examples is given in [5].

Appendix

Some Properties of Self-Adjoint Operators Satisfying Relations of Weyl Type

Let be an integer, and let () be self-adjoint operators on a Hilbert space satisfying relations of Weyl type: where ’s are real constants. It follows that is antisymmetric in :

The unitarity of and functional calculus imply that Hence, we have the operator equality

For a linear operator on a Hilbert space, we denote the spectrum of by .

Proposition A.1. Suppose that there exists a pair such that (hence, ). Then, Moreover, and are purely absolutely continuous.

Proof. By (A.4) and the unitary invariance of spectrum, we have for all . Since , this implies the second equation of (A.5). By (A.2), we have . Hence, by considering the case of replaced by , we obtain the first equation of (A.5).
Relation (A.4) means that is a weak Weyl representation of the CCR with one degree of freedom [14, 15, 24]. Hence is purely absolutely continuous [14, 15]. Similarly, we can show that is purely absolutely continuous.

Proposition A.2. Let and be fixed. Then, for all , is in and

Proof. An easy exercise (use (A.4)).

For each function and each vector , we define a vector by where and the integral on the right-hand side is taken in the strong sense. We introduce where denotes the subspace algebraically spanned by the vectors in the set . It is easy to see that is dense in .

For (the set of complex numbers), we set .

Lemma A.3. Let such that . Then, .

Proof. Since is unitary, we have . Thus, the desired result follows.

For each , we define a function on by

Proposition A.4. (i) For all and , leaves invariant.
(ii) For each , leaves invariant (i.e., ) and for all , where is defined by and is the times composition of with (identity).
(iii) For all ,

Proof. (i) Let be as above. Then, we have . By (A.1), we have Hence, with It is easy to see that is in . Hence, . Thus, leaves invariant.
(ii) By (A.14), we have for all ,  . It is easy to see that . Hence, by Lemma A.3, Therefore, is in and . Hence, (A.10) with holds. Then, one can prove (A.10) by induction.
(iii) This easily follows from (ii).

Propositions A.2 and A.4 immediately yield the following result.

Corollary A.5. For all , on .

Theorem A.6. For all , , is essentially self-adjoint on and where for a closable operator , denotes the closure of .

Proof. For each , we define an operator by By using (A.1), one can show that is a strongly continuous one-parameter unitary group. Hence, by the Stone theorem, there exists a unique self-adjoint operator on such that . By Proposition A.4, leaves invariant and strongly differentiable on with Hence, is a core of (e.g., [23, Theorem VIII.10]). Hence , . Thus, the desired result follows.

For all , , we set

Corollary A.7. For all and ,

Proof. By direct computations using (A.17) and (A.1).

Acknowledgment

This work is supported by the Grant-In-Aid no. 24540154 for Scientific Research from Japan Society for the Promotion of Science (JSPS).

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