# Hilbert Space Representations of Generalized Canonical Commutation Relations

Authors: Asao Arai
Journal: Journal of Mathematics
 (2013)

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## Abstract

We consider Hilbert space representations of a generalization of canonical commutation
relations(CCRs):[Xj,Xk]:=XjXk−XkXj=iΘjkI(j,k=1,2,…,2n),
where Xj's are the elements of an algebra with identity I, i is the imaginary unit, and Θjk is a real number with antisymmetry Θjk=−Θkj(k,j=1,2,…,2n). 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 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.


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## Body

## 1. Introduction

In this paper, we consider Hilbert space representations of ageneralized canonical commutation relations (GCCRs) with n degrees of freedom (n∈ℕ∶={1,2,3,…}) of the following type:
(1)[Xj,Xk]=iΘjkI(j,k=1,…,2n),
where Xj’s are elements of an algebra with identity I, [Xj,Xk]∶=XjXk-XkXj, i is the imaginary unit, and Θjk∈ℝ (the set of real numbers) with antisymmetry Θjk=-Θkj(j,k=1,…,2n) such that, for some pair (j,k), Θjk≠0. For convenience, we call (1) the Θ-GCCR with n degrees of freedom and the 2n×2n matrix
(2)Θ∶=(Θjk)j,k=1,…,2n
the noncommutative factor for {Xj}j=12n.Note that, in the case whereΘ is equal to
(3)J∶=(0In-In0),
with In being the n×n unit matrix, (1) becomes the CCR with n degrees of freedom. Namely, if we put Qj∶=Xj,Pj∶=Xn+j(j=1,…,n) in the present case, then we have
(4)[Qj,Qk]=0,[Pj,Pk]=0,[Qj,Pk]=iδjkI(j,k=1,…,n),
where δjk is the Kronecker delta. Thus, (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., [1–3]), non-commutative spaces (e.g., [4]), and non-commutative phase spaces (e.g., [5–11]). 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 Section2, 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 L2(ℝn) 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.


## 2. Basic Facts on Hilbert Space Representations of theΘ-GCCR

Letℋ 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 A on ℋ, we denote its domain by D(A). For linear operators A1,…,Ap on ℋ,
(5)D(∑i=1p‍Ai)∶=∩i=1pD(Ai),D(A1A2)∶={Ψ∈D(A2)∣A2Ψ∈D(A1)},D(A1⋯Ap)∶=D((A1⋯Ap-1)Ap)(p≥3).Definition 1.
Let𝒟 be a dense subspace of ℋ and Xj, j=1,…,2n, be symmetric (not necessarily essentially self-adjoint) operators on ℋ. Set X∶=(X1,…,X2n). We say that the triple (ℋ,𝒟,X) is a symmetric representation of the Θ-GCCR with n degrees of freedom if 𝒟⊂∩j,k=12nD(XjXk) and (1) holds on 𝒟.
If all theXj’s (j=1,…,2n) are self-adjoint, we say that (ℋ,𝒟,X) 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(ℋ,𝒟,X) of the Θ-GCCR, ℋ is infinite dimensional (if ℋ were finite dimensional, then, for (j,k) such that Θjk≠0, 0=trace of[Xj,Xk]=iΘjkdimℋ≠0, 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 (j,k) with Θjk≠0, at least one of Xj and Xk is unbounded. Hence, one has to be careful about domains of Xj’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 Θ-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 operatorA and a unit vector ψ∈D(A), (ΔA)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 [17]: for all unit vectors ψ∈𝒟 and j,k=1,…,2n,
(6)(ΔXj)ψ(ΔXk)ψ≥12|〈ψ,Θjkψ〉|.Let(ℋ,𝒟,X) be a symmetric representation of the Θ-GCCR as in Definition 1. We assume for simplicity the following:Assumption 1.
The noncommutative factorΘ is regular (invertible).Under this assumption,Θ 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 regular2n×2n real matrix T0 such that tT0ΘT0=J, where tT0 is the transposed matrix of T0 and J is defined by (3).The matrixT0 in Lemma 7 belongs to the set
(7)MΘ∶={tTΘT=JT∣Tisa2n×2nrealmatrixsuchthattTΘT=J}.
It is easy to see that for each T∈MΘ, there exists a unique 2n×2n symplectic matrix W (i.e., tWJW=J) such that T=T0W. Hence,
(8)MΘ={T0W∣tWJW=J}.For a2n×2n real matrix L=(Ljk)j,k=1,…,2n, we define
(9)XjL∶=∑k=12n‍LkjXk,j=1,…,2n.
We call the correspondence X↦XL∶=(X1L,…,X2nL) the L-transform of  X.Let(10)ΘL∶=tLΘL.Proposition 8.
(i) For allj=1,…,2n, XjL is a symmetric operator on ℋ.
(ii) For allj,k=1,…,2n,
(11)[XjL,XkL]=i(ΘL)jk
on 𝒟.
(iii) For eachT∈MΘ and j,k=1,…,2n,
(12)[XjT,XkT]=iJjk
on 𝒟.Proof.
An easy exercise.Proposition8-(i) and (ii) show that (ℋ,𝒟,XL) is a symmetric representation of the ΘL-GCCR with n degrees of freedom.Proposition8 (iii) implies the following.Corollary 9.
LetT∈MΘ and
(13)Qj∶=XjT,Pj∶=Xn+jT(j=1,…,n).
Then, (ℋ,𝒟,{Qj,Pj}j=1n) is a symmetric representation of the CCR with n degrees of freedom.Corollary9 means that for each T∈MΘ, 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 (9) with L=T∈MΘ implies that
(14)Xj=∑k=12n‍(T-1)kjXkT
on ∩j=12nD(Xj). 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 (14).Conversely, if a symmetric representation(ℋ,𝒟,{Qj,Pj}j=1n) of the CCR with n degrees of freedom is given and let
(15)Xj(Q,P;T)∶=∑k=1n‍(T-1)kjQk+∑k=1n‍(T-1)(n+k)jPk
with Q∶=(Q1,…,Qn) and P∶=(P1,…,Pn), then (ℋ,𝒟,X(Q,P;T)) is a symmetric representation of the Θ-GCCR and (9) holds with L=T, XjT=Qj,Xn+jT=Pj(j=1,…,n) and Xj=Xj(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 eachT∈MΘ, there exists a one-to-one correspondence between a symmetric representation of theΘ-GCCR and a symmetric representation of the CCR withndegrees of freedom.


## 3. Representations of Schrödinger Type

LetT∈MΘ. 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 (L2(ℝn),C0∞(ℝn),{qj,pj}j=1n) be the Schrödinger representation of the CCR with n degrees of freedom, that is, qj is the multiplication operator by the jth component xj of x=(x1,…,xn)∈ℝn and pj∶=-iDj with Dj being the generalized partial differential operator in xj, acting in L2(ℝn). Let
(16)Xj(q,p;T)=∑k=1n‍(T-1)kjqk+∑k=1n‍(T-1)(n+k)jpk,
which is (15) with Q=q and P=p. We denote the closure of Xj(q,p;T) by X¯j(q,p;T) and set
(17)X¯(q,p;T)∶=(X¯1(q,p;T),…,X¯2n(q,p;T)).
We call the triple πST∶=(L2(ℝn),C0∞(ℝn),X¯(q,p;T)) the T-Schrödinger representation of theΘ-GCCR.It is easy to see that for allj=1,…,2n, Xj(q,p;T) is essentially self-adjoint on C0∞(ℝn) (apply, e.g., the Nelson commutator theorem [19, Theorem X.37] with dominating operator N=∑j=1n‍(qj2+pj2)+I) (This can be proved also by applying Proposition 16). Hence X¯j(q,p;T) is self-adjoint. Thus, we obtain the following.Proposition 10.
For eachT∈MΘ, the T-Schrödinger representation πST is a self-adjoint representation of the Θ-GCCR.


## 4. Representations of Weyl Type

Based on an analogy with Weyl representations of CCR, we introduce a concept of Weyl representation forΘ-GCCR.Definition 11.
Let{Xj}j=12n be a set of self-adjoint operators on a Hilbert space ℋ. We say that {Xj}j=12n is a Weyl representation of theΘ-GCCR with n degrees of freedom if for all s,t∈ℝ and j,k=1,…,2n,
(18)eitXjeisXk=e-istΘjkeisXkeitXj.
We call these relations the Θ-Weyl relations.For a linear operatorA on a Hilbert space, we denote its spectrum by σ(A).Proposition 12.
Let{Xj}j=12n be a Weyl representation of the Θ-GCCR on ℋ. Then, there is a dense subspace 𝒟0⊂ℋ left invariant by each Xj (j=1,…,2n) such that (ℋ,𝒟0,X) is a self-adjoint representation of the Θ-GCCR. Moreover, for every pair (Xj,Xk) such that Θjk≠0, Xj and Xk are purely absolutely continuous with
(19)σ(Xj)=σ(Xk)=ℝ,j=1,…,2n.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, N=2n, ajk=Θjk and Aj=Xj. By Proposition A.4-(iii) and Corollary A.5, there exists a dense subspace 𝒟0 left invariant by Xj (j=1,…,2n) and [Xj,Xk]=iΘjk on 𝒟0. 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 Θ-GCCR is not necessarily a Weyl one).We recall that a set{Qj,Pj}j=1n 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:
(20)eitQjeisPk=e-istδjkeisPkeitQj,eitQjeisQk=eisQkeitQj,eitPjeisPk=eisPkeitPj.Remark 14.
A set{Qj,Pj}j=1n of self-adjoint operators on ℋ is a Weyl representation of the CCR with n degrees of freedom if and only if {Xj}j=12n with Xj∶=Qj,Xn+j=Pj(j=1,…,n) is a Weyl representation of the J-GCCR, where J is given by (3).LetT∈MΘ 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 15.
Let{Xj}j=12n be a Weyl representation of the Θ-GCCR on ℋ, and let XT be the T-transform of X. Then, each XjT is essentially self-adjoint, and {XjT¯}j=12n is a Weyl representation of the J-GCCR.Proof.
The essential self-adjointness ofXjT follows from a simple application of Theorem A.6 in Appendix. Corollary A.7 in Appendix and the relation tTΘT=J imply that {XjT¯}j=12n satisfies the J-Weyl relations.In the same way as in the proof of Proposition15, we can prove the following proposition:Proposition 16.
Let{Qj,Pj}j=1n be a Weyl representation of the CCR with n degrees of freedom on a Hilbert space ℋ. Let Xj(Q,P;T) (j=1,…,2n) be defined by (15). Then, each Xj(Q,P;T) is essentially self-adjoint and {X¯j(Q,P;T)}j=12n is a Weyl representation of Θ-GCCR with n degrees of freedom.This proposition shows that the converse of Proposition15 holds too. Thus, for each T∈MΘ, 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 [22] that the Schrödinger representation {qj,pj}j=1n is a Weyl representation o the CCR with n degrees of freedom. Hence, we obtain the following result.Corollary 17.
For eachT∈MΘ, the T-Schrödinger representation {X¯j(q,p;T)}j=12n is a Weyl representation of the Θ-GCCR.We say that a Weyl representation{Xj}j=12n of the Θ-GCCR on ℋ is irreducible if every closed subspace ℳ of ℋ which is invariant under the action of eitXj(t∈ℝ,j=1,…,2n) is {0} or ℋ.Proposition 18.
LetT∈MΘ. Then, the T-Schrödinger representation {X¯j(q,p;T)}j=12n as a Weyl representation of the Θ-GCCR is irreducible.Proof.
Letℳ be an invariant closed subspace of eitX¯j(q,p;T) (t∈ℝ,j=1,…,2n). We have
(21)qj=∑k=12n‍TkjXk(q,p;T),pj=∑k=12n‍Tk(n+j)Xk(q,p;T)
on ∩j=12nD(Xj(q,p;T))=∩j=1nD(qj)∩D(pj). Hence, by an application of Theorem A.6 in Appendix, eitqj and eitpj (t∈ℝ) can be written, respectively, as a scalar multiple of eitX¯1(q,p;T)…eitX¯2n(q,p;T). Hence, ℳ is invariant under the action of eitqj and eitpj (t∈ℝ,j=1,…,n). It is well known that {eitqj,eitpj∣t∈ℝ,j=1,…,n} is irreducible. Thus, ℳ={0} or ℋ.


## 5. Uniqueness Theorem on Weyl Representations of theΘ-GCCR

In this section, we prove the main result of the present paper, that is, a uniqueness theorem on Weyl representations of theΘ-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{Xj}j=12n be a Weyl representation of the Θ-GCCR on a separable Hilbert space ℋ. Then, for each T∈MΘ, there exist mutually orthogonal closed subspaces ℋℓ (ℓ=1,…,N; N∈ℕ or ∞) such that the following (i)–(iii) hold.(i)
ℋ=⊕ℓ=1Nℋℓ.(ii)
For eachj=1,…,2n, Xj is reduced by each ℋℓ, ℓ=1,…,N. We denote by Xj(ℓ) the reduced part of Xj to ℋℓ.(iii)
For eachℓ, there exists a unitary operator Uℓ:ℋℓ→L2(ℝn) such that
(22)UℓXj(ℓ)Uℓ-1=X¯j(q,p;T),j=1,…,2n,
where {X¯j(q,p;T)}j=12n is the T-Schrödinger representation of the Θ-GCCR.Proof.
LetT∈MΘ, XT be the T-transform of X and Qj∶=XjT¯, Pj∶=Xn+jT¯ (j=1,…,n). Then, by Proposition 15 and Remark 14, {Qj,Pj}j=1n 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 ℋℓ such that (i) given above and the following (a) and (b) hold.(a)
For eachj=1,…,n and all t∈ℝ, eitQj and eitPj leave each ℋℓ invariant (ℓ=1,…,N).(b)
For eachℓ, there exists a unitary operator Uℓ:ℋℓ→L2(ℝn) such that
(23)UℓeitQjUℓ-1=eitqj,UℓeitPjUℓ-1=eitpj,t∈ℝ,j=1,…,n.
By (14), we have Xj=Xj(Q,P;T) on ∩j=12nD(Xj). Hence, Xj⊂X¯j(Q,P;T). By Proposition 16, X¯j(Q,P;T) is self-adjoint. Hence, Xj=X¯j(Q,P;T). Therefore, by Theorem A.6 in Appendix, we obtain
(24)eitXj=eit2∑k<m2nJkm(T-1)kj(T-1)mj/2eitQ1⋯eitQneitP1⋯eitPn,j=1,…,2n.
Hence, each eitXj leaves ℋℓ invariant (ℓ=1,…,N). Therefore, Xj is reduced by each ℋℓ. We denote the reduced part of Xj to ℋℓ by Xj(ℓ). Then, we have by (23)
(25)UℓeitXj(ℓ)Uℓ-1=eit2∑k<m2nJkm(T-1)kj(T-1)mj/2eitq1⋯eitqneitp1⋯eitpn=eitX¯j(q,p;T).
Thus, (22) follows.Theorem19 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∈MΘ is arbitrary.The next corollary immediately follows from Theorem19.Corollary 20.
Let{Xj}j=12n be an irreducible Weyl representation of the Θ-GCCR on a separable Hilbert space ℋ. Then, for each T∈MΘ, there exists a unitary operator U:ℋ→L2(ℝn) such that
(26)UXjU-1=X¯j(q,p;T),j=1,…,2n.The following result shows that the arbitrariness of the choice ofT in the T-Schrödinger representation of the Θ-GCCR is implemented by unitary operators.Corollary 21.
LetS,T∈MΘ. Then, there exists a unitary operator V on L2(ℝn) such that
(27)VX¯j(q,p;S)V-1=X¯j(q,p;T),j=1,…,2n.Proof.
We need only to apply Corollary20 to the case where Xj=X¯j(q,p;S).Remark 22.
As in the case of non-Weyl representations of CCR, for non-Weyl representations of theΘ-GCCR, the conclusion of Theorem 19 does not hold in general. Examples of such representations of the Θ-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].



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