Hilbert Space Representations of Generalized Canonical Commutation Relations

We consider Hilbert space representations of a generalization of canonical commutation relations (CCR): [ X j , X k ] := X j X k ¡ X k X j = i Θ jk I ( j, k = 1 , 2 , . . . , 2 n ), where X j ’s are elements of an algebra with identity I , i is the imaginary unit, and Θ jk is a real number with Θ jk = ¡ Θ kj ( j, k = 1 , . . . , 2 n ). Some basic aspects on Hilbert space representations of the generalized CCR (GCCR) are discussed. We deﬁne a Schr¨odinger type representation of the GCCR by analogy with the usual Schr¨odinger representation of the CCR with n 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.


Introduction
In this paper, we consider Hilbert space representations of a generalized canonical commutation relations (GCCR) with n degrees of freedom ( n ∈ N := {1, 2, 3, . . .}) of the following type: [X j , X k ] = iΘ jk I (j, k = 1, . . . , 2n), (1.1) where X j 's are elements of an algebra with identity I, [X j , X k ] := X j X k − X k X j , i is the imaginary unit, and Θ jk ∈ R (the set of real numbers) with anti-symmetry Θ jk = −Θ kj (j, k = 1, . . . , 2n) such that, for some pair (j, k), Θ jk = 0. For convenience, we call (1.1) the Θ-GCCR with n degrees of freedom and the 2n × 2n matrix Θ := (Θ jk ) j,k=1,...,2n (1.2) the non-commutative factor for {X j } 2n j=1 . Note that, in the case where Θ is equal to with I n being the n × n unit matrix, (1.1) becomes the CCR with n degrees of freedom. Namely, if we put Q j := X j , P j := X n+j (j = 1, . . . , n) in the present case, then we have [Q j , Q k ] = 0, [P j , P k ] = 0, [Q j , P k ] = iδ jk I (j, k = 1, . . . , n), where δ jk is the Kronecker delta. Thus (1.1) is a natural generalization of the CCR with n degrees of freedom. The GCCR also includes some of non-commutative space-times (e.g., [10,11,14,20]), non-commutative spaces (e.g., [16,17]) and non-commutative phase spaces (e.g., [5,6,15,18,19,25,32]). 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 L 2 (R n ) as in the case of the Schrödinger representation of the CCR with n 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.

Basic Facts on Hilbert Space Representations of the Θ-GCCR
Let H be a complex Hilbert space with inner product · , · (anti-linear in the first variable and linear in the second one) and norm · . For a linear operator A on H, we denote its domain by D(A). For linear operators A 1 , . . . , A p on H, Definition 2.1 Let D be a dense subspace of H and X j , j = 1, . . . , 2n, be symmetric (not necessarily essentially self-adjoint) operators on H. We set X := (X 1 , . . . , X 2n ). We say that the triple (H, D, X) is a symmetric representation of the Θ-GCCR with n degrees of freedom if D ⊂ ∩ 2n j,k=1 D(X j X k ) and (1.1) holds on D. If all the X j 's (j = 1, . . . , 2n) are self-adjoint, we say that (H, D, X) is a self-adjoint representation of the GCCR.

Remark 2.2
In each symmetric representation (H, D, X) of the Θ-GCCR, H is infinite dimensional (if H were finite dimensional, then, for (j, k) such that Θ jk = 0, 0 = trace of [X j , X k ] = iΘ jk dim H = 0 and hence one is led to a contradiction).

Remark 2.3
It follows from a well known fact on commutation properties of linear operators (e.g., [22,Theorem 1.2.3]) that, for (j, k) with Θ jk = 0, at least one of X j and X k is unbounded. Hence one has to be careful about domains of X j 's.

Remark 2.4
In the case of Hilbert space representations of CCR, symmetric representations, but non-self-adjoint ones, also play important roles. For example, such representations appear in mathematical theories of time operators [2,3,4,8,9,21] (see also [29,30] for investigations from purely operator-theoretic points of view). Thus it is expected that, in addition to self-adjoint representations of the Θ-GCCR, non-self-adjoint symmetric representations of it may have any importance in applications to quantum physics.

Remark 2.5
In the context of quantum mechanics, for a symmetric operator A and a unit vector ψ ∈ D(A), (∆A) ψ := (A − ψ, Aψ )ψ is called the uncertainty of A in the vector state ψ. Let (H, D, X) be a symmetric representation of the Θ-GCCR with n degrees of freedom. Then one has uncertainty relations of Robertson type [26]: for all unit vectors ψ ∈ D and j, k = 1, . . . , 2n, Let (H, D, X) be a symmetric representation of the Θ-GCCR as in Definition 2.1. We assume for simplicity the following: Assumption: The non-commutative factor Θ is regular (invertible).
Under this assumption, Θ is a regular anti-symmetric real matrix. Hence, by a well known fact in the theory of linear algebra (e.g., [27,p.173,Problem 9]), the following fact holds: Lemma 2.6 There exists a regular 2n × 2n real matrix T 0 such that t T 0 ΘT 0 = J, where t T 0 is the transposed matrix of T 0 and J is defined by (1.3).
The matrix T 0 in Lemma 2.6 belongs to the set It is easy to see that, for each T ∈ M Θ , there exists a unique 2n × 2n symplectic matrix W (i.e., t W JW = J) such that T = T 0 W . Hence For a 2n × 2n real matrix L = (L jk ) j,k=1,...,2n , we define We call the correspondence X → X L :

Proposition 2.7
(i) For all j = 1, . . . , 2n, X L j is a symmetric operator on H.
Proof. An easy exercise.
Proposition 2.7-(i) and (ii) show that (H, D, X L ) is a symmetric representation of the Θ L -GCCR with n degrees of freedom. Proposition 2.7-(iii) implies the following: .
is a symmetric representation of the CCR with n degrees of freedom.
Corollary 2.8 means that, for each T ∈ M Θ , the T -transform of X gives a correspondence from a symmetric representation of the Θ-GCCR with n degrees of freedom to a symmetric representation of the CCR with the same degrees of freedom.
One can easily see that on ∩ 2n j=1 D(X j ). Thus every symmetric representation of the Θ-GCCR with n degrees of freedom is constructed from a symmetric representation of the CCR with the same degrees of freedom via (2.7).
Conversely, if a symmetric representation (H, D, {Q j , P j } n j=1 ) of the CCR with n degrees of freedom is given and let with Q := (Q 1 , . . . , Q n ) and P := (P 1 , . . . , P n ). Then (H, D, X(Q, P; T )) is a symmetric representation of the Θ-GCCR and (2.3) holds with L = T , X T j = Q j , X T n+j = P j (j = 1, . . . , n) and X j = X j (Q, P; T ) (j = 1, . . . , 2n). Hence every symmetric representation of the CCR with n degrees of freedom is constructed from a symmetric representation of the Θ-GCCR with the same degrees of freedom. Thus, for each T ∈ M Θ , there exists a one-to-one correspondence between a symmetric representation of the Θ-GCCR and a symmetric representation of the CCR with n degrees of freedom.

Representations of Schrödinger Type
Let T ∈ M Θ . By the fact on X(Q, P; T ) stated in the preceding section, we can define a class of representations of the Θ-GCCR. Let (L 2 (R n ), C ∞ 0 (R n ), {q j , p j } n j=1 ) be the Schrödinger representation of the CCR with n degrees of freedom, i.e., q j is the multiplication operator by the jth component x j of x = (x 1 , . . . , x n ) ∈ R n and p j := −iD j with D j being the generalized partial differential operator in x j , acting in L 2 (R n ). Let which is (2.8) with Q = q and P = p. We denote the closure of X j (q, p; T ) by X j (q, p; T ) and set X(q, p; T ) := (X 1 (q, p; T ), . . . , X 2n (q, p; T )).
We call the triple π T It is easy to see that, for all j = 1, . . . , 2n, X j (q, p; T ) is essentially self-adjoint on C ∞ 0 (R n ) ( apply, e.g., the Nelson commutator theorem [24, Theorem X.37] with dominating operator N = n j=1 (q 2 j + p 2 j ) + I) 1 . Hence X j (q, p; T ) is self-adjoint. Thus we obtain the following: S is a self-adjoint representation of the Θ-GCCR.

Representations of Weyl Type
Based on an analogy with Weyl representations of CCR, we introduce a concept of Weyl representation for Θ-GCCR.
j=1 be a set of self-adjoint operators on a Hilbert space H. We say that {X j } 2n j=1 is a Weyl representation of the Θ-GCCR with n degrees of freedom if, for all s, t ∈ R and j, k = 1, . . . , 2n, e itX j e isX k = e −istΘ jk e isX k e itX j . (4.1) We call these relations the Θ-Weyl relations.
For a linear operator A on a Hilbert space, we denote its spectrum by σ(A).  (j = 1, . . . , 2n) such that (H, D 0 , X) is a self-adjoint representation of the Θ-GCCR. Moreover, for every pair (X j , X k ) such that Θ jk = 0, X j and X k are purely absolutely continuous with Proof. By (4.1), we can apply the results described in Appendix A of the present paper. In the present context, we need only to take, in the notation in Appendix A, N = 2n, a jk = Θ jk and A j = X j . By Proposition A.4-(iii) and Corollary A.5, there exists a dense subspace D 0 left invariant by X j (j = 1, . . . , 2n) and [X j , X k ] = iΘ jk on D 0 . Thus the first half of the proposition is derived. The second half follows from Proposition A.1.

Remark 4.3
As in the case of self-adjoint representations of CCR (e.g., [1,13,30]), the converse of Proposition 4.2 does not hold (i.e., a self-adjoint representation of the Θ-GCCR is not necessarily a Weyl one).
We recall that a set {Q j , P j } n j=1 of self-adjoint operators on H is a Weyl representation of the CCR with n degrees of freedom if, for all s, t ∈ R and j, k = 1, . . . , n, the following Wey relations hold: Let T ∈ M Θ be arbitrarily fixed. The next proposition shows that the T -transform of each Weyl representation of the Θ-GCCR is a Weyl representation of the CCR with n degrees of freedom:

Proposition 4.5 Let {X j } 2n j=1 be a Weyl representation of the Θ-GCCR on H and let X T be the T -transform of X. Then each X T j is essentially self-adjoint and {X T j } 2n j=1 is a Weyl representation of the J-GCCR.
Proof. The essential self-adjointness of X T j follows from a simple application of Theorem A.6 in Appendix. Corollary A.7 in Appendix and the relation t T ΘT = J imply that {X T j } 2n j=1 satisfies the J-Weyl relations. In the same way as in the proof of Proposition 4.5, we can prove the following proposition: Proposition 4.6 Let {Q j , P j } n j=1 be a Weyl representation of the CCR with n degrees of freedom on a Hilbert space H. Let X j (Q, P; T ) (j = 1, . . . , 2n) be defined by (2.8). Then each X j (Q, P; T ) is essentially self-adjoint and {X j (Q, P; T )} 2n j=1 is a Weyl representation of Θ-GCCR with n degrees of freedom.
This proposition shows that the converse of Proposition 4.5 holds too. Thus, for each T ∈ M Θ , there exists a one-to-one correspondence between a Weyl representation of the CCR with n degrees of freedom and that of the Θ-GCCR with the same degrees of freedom.
It is well known [31] that the Schrödinger representation {q j , p j } n j=1 is a Weyl representation o the CCR with n degrees of freedom. Hence we obtain the following result:   R, j = 1, . . . , 2n). We have Hence, by an application of Theorem A.6 in Appendix, e itq j and e itp j (t ∈ R) can be written respectively as a scalar multiple of e itX 1 (q,p;T ) . . . e itX 2n (q,p;T ) . Hence M is invariant under the action of e itq j and e itp j (t ∈ R, j = 1, . . . , n). It is well known that {e itq j , e itp j |t ∈ R, j = 1, . . . , n} is irreducible. Thus M = {0} or H.

Uniqueness Theorem on Weyl Representations of the Θ-GCCR
In this section, we prove the main result of the present paper, i.e., a uniqueness theorem on Weyl representations of the Θ-GCCR, which may be regarded as a GCCR version of the celebrated von Neumann uniqueness theorem of Weyl representations of CCR ( [22,Theorem 4.11.1], [23, Theorem VIII.14], [31]).  H ( = 1, . . . , N ; N ∈ N or ∞) such that the following (i)-(iii) hold: Proof. Let T ∈ M Θ , X T be the T -transform of X and Q j := X T j , P j := X T n+j (j = 1, . . . , n). Then, by Proposition 4.5 and Remark 4.4, {Q j , P j } n j=1 is a Weyl representation of the CCR with n degrees of freedom. Hence, by the von Neumann uniqueness theorem mentioned above, there exist mutually orthogonal closed subspaces H such that (i) given above and the following (a) and (b) hold: (a) For each j = 1, . . . , n and all t ∈ R, e itQ j and e itP j leave each H invariant ( = 1, . . . , N ).
(b) For each , there exists a unitary operator U : By (2.7), we have X j = X j (Q, P; T ) on ∩ 2n j=1 D(X j ). Hence X j ⊂ X j (Q, P; T ). By Proposition 4.6, X j (Q, P; T ) is self-adjoint. Hence X j = X j (Q, P; T ). Therefore, by Theorem A.6 in Appendix, we obtain Hence each e itX j leaves H invariant ( = 1, . . . , N ). Therefore X j is reduced by each H . We denote the reduced part of X j to H by X ( ) j . Then, we have by (5.2) U e itX ( ) j U −1 = e it 2 P 2n k<m J km (T −1 ) kj (T −1 ) mj /2 e itq 1 · · · e itqn e itp 1 · · · e itpn = e itX j (q,p;T ) , Thus (5.1) follows.
Theorem 5.1 tells us that every Weyl representation of the Θ-GCCR on a separable Hilbert space is unitarily equivalent to a direct sum of the T -Schrödinger representation of the Θ-GCCR, where T ∈ M Θ is arbitrary.
The next corollary immediately follows from Theorem 5.1: be an irreducible Weyl representation of the Θ-GCCR on a separable Hilbert space H. Then, for each T ∈ M Θ , there exists a unitary operator 3) The following result shows that the arbitrariness of the choice of T in the T -Schrödinger representation of the Θ-GCCR is implemented by unitary operators. Proof. We need only to apply Corollary 5.2 to the case where X j = X j (q, p; S).

Remark 5.4
As in the case of non-Weyl representations of CCR, for non-Weyl representations of the Θ-GCCR, the conclusion of Theorem 5.1 does not hold in general. Examples of such representations of the Θ-GCCR can be constructed from non-Weyl representations of CCR (e.g., [1,13,29,30]). A detailed description of some examples is given in [5].

A Some Properties of Self-Adjoint Operators Satisfying Relations of Weyl Type
Let N ≥ 2 be an integer and A j (j = 1, . . . , N ) be self-adjoint operators on a Hilbert space H satisfying relations of Weyl type: where a jk 's are real constants. It follows that a jk is anti-symmetric in (j, k): The unitarity of e itA j and functional calculus imply that Hence we have the operator equality For a linear operator A on a Hilbert space, we denote the spectrum of A by σ(A).
Proposition A.1 Suppose that there exists a pair (j, k) such that a jk = 0 (hence j = k). Then Moreover, A j and A k are purely absolutely continuous.
Proof. By (A.3) and the unitary invariance of spectrum, we have σ(A k ) = σ(A k −ta jk ) for all t ∈ R. Since a jk = 0, this implies the second equation of (A.4). By (A.2), we have a kj = 0. Hence, by considering the case of (j, k) replaced by (k, j), we obtain the first equation of (A.4).
Relation (A.3) means that (A k , A j ) is a weak Weyl representation of the CCR with one degree of freedom ([2]- [4], [29]). Hence A j is purely absolutely continuous [2,21,29]. Similarly we can show that A k is purely absolutely continuous.

Proposition A.2 Let j and k be fixed. Then, for all
Proof. An easy exercise (use (A.3)).
For each function f ∈ C ∞ 0 (R N ) and each vector ψ ∈ H, we define a vector ψ f by where t = (t 1 , . . . , t N ) ∈ R N and the integral on the right hand side is taken in the strong sense. We introduce where Span{· · ·} denotes the subspace algebraically spanned by the vectors in the set {· · ·}. It is easy to see that D 0 is dense in H. For f : R N → C (the set of complex numbers), we set f 1 := R N |f (t)|dt.
Proof. Since e it j A j is unitary, we have ψ fn − ψ f ≤ f n − f 1 ψ . Thus the desired result follows.
and F j is the times composition of F j with F 0 j := I (identity).
(iii) For all 1 , . . . , N ∈ N ∪ {0}, Proof. (i) Let ψ f be as above. Then we have e itA j ψ f = R N f (t)e itA j e it 1 A 1 · · · e it N A N ψdt. By (A.1), we have Hence . . . , t j−1 , t j − t, t j+1 , . . . , t N )e −itg j (t) . It is easy to see that f Therefore ψ f is in D(A j ) and iA j ψ f = ψ F j (f ) . Hence (A.9) with = 1 holds. Then one can prove (A.9) by induction. where, for a closable operator C, C denotes the closure of C.
Proof. For each t ∈ R, we define an operator U (t) by U (t) := e it 2 P N j<k a jk c j c k /2 e itc 1 A 1 e itc 2 A 2 · · · e itc N A N .
By using (A.1), one can show that {U (t)} t∈R is a strongly continuous one-parameter unitary group. Hence, by the Stone theorem, there exists a unique self-adjoint operator A on H such that U (t) = e itA , t ∈ R. By Proposition A.4, U (t) leaves D 0 invariant and strongly differentiable on D 0 with Hence D 0 is a core of A (e.g., [23,Theorem VIII.10]). Hence Aψ = N j=1 c j A j ψ, ψ ∈ D 0 . Thus the desired result follows.