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On the deformability of Heisenberg algebras

Based on the vanishing of the second Hochschild cohomology group of the enveloping algebra of the Heisenberg algebra it is shown that differential algebras coming from quantum groups do not provide a non-trivial deformation of quantum mechanics. For the case of a q-oscillator there exists a deforming map to the classical algebra. It is shown that the differential calculus on quantum planes with involution, i.e. if one works in position-momentum realization, can be mapped on a q-difference calculus on a commutative real space. Although this calculus leads to an interesting discretization it is proved that it can be realized by generators of the undeformed algebra and does not posess a proper group of global transformations.


Introduction
It is known that the deformation of an algebra, either of Lie or associative type, is connected to its (Chevalley or Hochschild) cohomology [13]. More precisely, for an algebra g the second cohomology group H 2 (g, g) contains the information if a non-trivial deformation of it exists or not. In particular, if H 2 (g, g) = 0 then there exits no non-trivial deformation of g .
This result can readily be applied to the case of quantum groups [7,15]. Here one takes for example a finite-dimensional semisimple Lie algebra g and addresses the question of existence of deformations of its enveloping algebra U(g). It is well known that we have non-trivial deformations denoted by U h (g) 1 as long one considers U h (g) as being a Hopf algebra or at least a bialgebra. The non-triviality of this deformation comes from the fact that H 2 (U(g), U(g)) bialgebra ≃ Λ 2 (g) = 0 [8], [16] ch.18, where Λ(g) denotes the exterior algebra. If one in contrast would consider only the algebra part of U(g) the classical Whitehead lemma applies in this case. That lemma states that for a finite dimensional semisimple Lie algebra g and and finite-dimensional left-g-module M it holds that: This result gives rise to the following: Theorem 1.1 [8] There exists a unique morphism of topological algebras, such that α ≡ id mod h.
Practically speaking this result means that there exist deforming maps in the sense of [6] which connect the generators of the deformed to the ones of the undeformed algebra respectively.
It has been shown in the fundamental paper [2] that quantum mechanics itself can be understood as a deformation of classical mechanics. Already in that work the question of stability of quantum mechanics with respect to further deformations has been addressed. This question was answered affirmatively some years ago [9] in showing that the second cohomology group of the Weyl algebra regarded as a bimodule over itself is zero. This means that (at least in the reasonable sense of star-products) quantum mechanics cannot be further deformed.
However, in recent years the possibility of q-deformations of the Heisenberg algebra has been studied extensively within the context of quantum groups, see e.g. [1,3,10,12,22,23,24] and references therein. The characteristic relation arising from these studies is of the type: In general one has two possiblities of considering these algebras according to the inequivalent antilinear involutions one can choose on the algebra (3). The first choice is to take a Bargmann-Fock type conjugation, i.e.x = −ip. In this case (3) becomes a q-oscillator algebra. Here the equivalence with the undeformed case is known, see e.g. [27]. There exists, however, a second possiblity which has been addressed in [24]. In this case one wants to interpret the generators of (3) as being momentum and position operators respectively. The aim of this paper is to show that due to the mentioned rigidity theorem for the Weyl algebra even the second approach (in any dimensions) does not provide a true deformation of the Heisenberg algebra.
The plan of the paper is as follows. In section 2 Heisenberg and Weyl algebras are defined and the Hochschild cohomology is calculated. The immediate consequences for qoscillator algebras are explained. We study the algebra (3) with position and momentum operators in section 3. It turns out that this algebra does not have a proper group of global transformations containing the Weyl group of ordinary quantum mechanics as it should in order to have a quantum mechanical interpretation. In section 4 q-difference algebras on almost commutative spaces with involution on both the coordinates and the q-difference operators will be considered. The existence of this involution is necessary for having a position-momentum interpretation of the generators. A uniqueness result for the calculus will be obtained. The more general case of a q-differential calculus on the real SO q (N) quantum plane is addressed in section 5. It will be proved that there exists a deforming map which provides an isomorphsim of this q-differential calculus to the q-difference calculus on a commutative space. Finally we summarize and comment on prospects in section 6.

The cohomology of the Weyl algebra
We denote by h n the Heisenberg algebra which is generated by the set of generators p 1 , q 1 ; p 2 , q 2 ; . . . ; p n , q n and the unity together with the relations: The enveloping algebra of h n is denoted by A n and will be called Weyl algebra. We note that by minor changes in the definitions one can also consider the Weyl algebra A n (k) over an arbitrary field k of characteristic zero. It is obvious that elements of the form ( . . , i n , j n ∈ N 0 generate A n as a vector space. One naturally has a realization of the generators of A n by differential operators: We can now state the following theorem on the cohomology H ⋆ (A n (k), A n (k)).
Theorem 2.1 [9]: It holds that: Due to the importance of this result we sketch the proof. Proof: One first proves a more general result. Let a be a finite-dimensional nilpotent Lie algebra over k and B its enveloping algebra. We denote by b an ideal of a and by λ a character of b. For ξ ∈ b the set of elements in B of the form ξ − λ(ξ) is denoted by b λ . Obviously b λ is a sub-vector space of B stable under the adjoint action of a. Moreover we define B λ := B/Bb λ . Using the inverse process of homological algebra (see e.g. [4]) one can show that for a B λ -bimodule X the following statement is true: We can now pass to the special case of the Weyl algebra. In this case a is the Heisenberg algebra h n and b = z its center which is of course trivial. Obviously it holds that B λ = A n (k) with 2n + 1 being the dimension of a. In order to prove the theorem we set X = A n (k). By (7) we are enabled to take H ⋆ (A n (k), A n (k)) = H ⋆ (h n /kz, A n (k)).
Using the realization of the generators of h n in terms of differential operators (5) the problem of calculating H ⋆ (h n /kz, A n (k)) is mapped to a problem in de Rham cohomology. The observation that all the derivations in A n (k) are interior ones completes the proof of the theorem. ✷ The essence of this theorem is that there does not exist a non-trivial deformation of the Heisenberg algebra within the category of algebras.
That q-deformations of Heisenberg algebras are trivial from the general deformation theory point of view is evident if one considers q-deformed oscillator algebras. In one dimension this algebra takes the form: Studying its representations one finds the same Fock space as in the undeformed case. The only difference between the classical and the q-case are the norms of the operators. The undeformed oscillator algebra is given by: If one chooses a certain completion of these algebras in the h-adic topology it is possible to introduce an element of the form exp(hN) = q N . In the sense of theorem 1.1 we then get the following deforming maps: We note that deforming maps are not unique in general. This deforming map can be applied to higher dimensional q-boson algebras as well. For instance it has been shown in [17] that the q-differential calculus on quantum planes belonging to quantum groups SL q (N) and SO q (N) (see section 5 for more details) can be mapped into a tensor product of mutually commuting algebras of the form (8). However, this result applies only to cases without any reality conditions neither on the quantum plane nor on the differential operators. This means that the mentioned map in [17] is not at all compatible with the natural involution (real form) coming from the quantum group.
3 Remarks on "q-deformed quantum mechanics" As mentioned in the introduction there exists another approach to the deformation of Heisenberg algebras which comes from the Wess-Zumino differential calculus [26]. In this approach one considers the quantum plane which is a certain comodule of a quantum group and the q-differential calculus on it. The differential relations are interpreted as q-Heisenberg relations.
In contrast to the q-boson algebras the involution in this approach is not of Bargmann-Fock type. As mentioned above it has been shown in [17] that with an involution of Bargmann-Fock type the full q-differential algebra in the case of SL q (N) and SO q (N) can be transformed into a tensor product of mutually commuting algebras similar to (8).
The authors of [24] investigate the following algebra: Like in ordinary quantum mechanics the generators p and x ought to be interpreted as momentum and position operators on some Hilbert space. Since q is taken to be real one cannot find an antilinear involution which allows for taking both generators to be real under involution. However, one can takep = p and introduces an additional generatorx together with the obvious relations: For convenience an additional object is introduced by using usual commutators: The aim is to get a q-Heisenberg algebra with formal real objects. If p is interpreted as real momentum an obvious choice for a real position would beξ := x +x. This definition results in the algebra: The problem with this algebra is that even though r q-commutes with p it has rather involved relations withξ. To circumvent this problem one uses the observation that r and r can be decomposed, however non-uniquely, into a formal real and and a quasi unitary object.
Denoting T := rr one is allowed to writer := √ quT 1/2 and r := √ qT 1/2ū which of course implies uū = q −1 =ūu. Applying these decompositions to the algebra (14) and redefining the position to be yields the following algebra: Although this algebra suggests to be interpreted as a deformation of the Heiseberg algebra it turns out that all relations can be entirely realized within a certain completion of the enveloping algebra of the usual Heisenberg algebra [x c , p c ] = i. One has the freedom to interpret the momentum p to be the usual momentum generator p c . This gives: The following realization follows directly from (16): It is shown in [24,25] that the spectra of p and ξ are discrete. In momentum representation the Hilbert space states are given by vectors |n π 0 where n ∈ Z and the continuous parameter π 0 ∈ [1, q) labels the the different irreducible representations of the algebra (16). One then has : By a proper Fourier transformation [25] it is possible to show that one can also construct irreducible representations in which ξ is diagonal. Its spectrum then is similar to the one of p in (19). Actually the momentum eigenstates |n π 0 can be realized by ordinary functions. Up to normalization we have : The operator p = p c then acts on these states by ordinary Schrödinger representation although the application of x c does lead out of the irreducible representations of the algebra (16).
Although it seems quite interesting to interpret the algebra (16) to be a q-deformation of the Heisenberg algebra which provides a discretization quite similar to the one arising from ordinary lattice quantum mechanics we have the following Lemma 3.1: The algebra (16) does not allow for a group of global transformations which contains the Weyl group of usual quantum mechanics consistent with the irreducible representations of that algebra.
Proof: We have seen that all generators of the algebra (16) can be realized in terms of the generators of the classical Heisenberg algebra. Strictly speaking (16) is not generated by the classical Heisenberg algebra itself which is nilpotent but by the minimal solvable extension of it. This means that on the algebra level we have an additional generator equivalent to p c x c . The enveloping algebras of these algebras are identical since the additional generator lies in the vector space spanned by elements of the form (p c ) i (x c ) j with i, j ∈ N.
If we denote the classical nilpotent Heisenberg algebra by H 1 the corresponding group is the Weyl group W 1 . Obviously H 1 is a subalgebra and moreover a Lie bi-ideal of its minimal solvable extension. The usual uniform lattice discretization corresponds to considering the Weyl group W 1 not over the real numbers but over the integers Z.
The group corresponding to the minimal solvable extension of H 1 is some product of W 1 with a group of dilatations generated by p c x c which we will call D. We denote this fact by W D 1 = W 1 • D.
The spectra of (16) would correspond -by the conformal invariance of the real lineto fix the group parameters in the pure D part of W D 1 to the set hZ. For consistency the space of group parameters for the full group W D 1 has to be hZ. That the global transformations corresponding to this set is inconsistent with the spectrum of (16) is evident. The only thing one has to do is to apply a translation of the form exp(αp c ) to the Hilbert space of the algebra (16) with α/h ∈ Z. This completes our proof. ✷ This result can be applied to higher dimensional cases. In [14] for example the case of SO q (N) covariant quantum mechanics has been investigated. It turned out that the total Hilbert space of the theory is a tensor product of two Hilbert spaces. One of them corresponds to U q (su2) and the other is the one coming from (16). In the sense of theorem 1.1 and of lemma 3.1 that models do not provide a true deformation of quantum mechanics.

q-derivatives on almost commutative spaces
In this section q-difference calculi on commutative or almost commutative spaces with involution are considered. By almost commutative (sometimes also referred to as qcommutative) we mean two objects A and B having a commutation relation of the form AB = q r BA where r might be any number different from zero. Having the background of quantum groups we require that for q = 1 we obtain the usual continuous calculus.
Let us first assume that the configuration space is a finite-dimensional commutative algebra generated by objects x α . We will talk about these generators freely as coordinates.
An (anti-linear) involution on this space is introduced by the rule x α := x −α . This is just the standard involution on an euclidean space in a lightcone like basis. We remark that our results are of course basis-independent. Since we later want to make contact to the quantum group case the chosen basis is convenient. Let D α be q-partial derivatives acting on x α . For the application we have in mind and hence in analogy to (16) we require these partial derivatives to have the standard conjugation property D α = −D −α . As has been mentioned in the introduction the existence of an involution of this kind is essential for our considerations because in the sense of the previous section we have in mind a position-momentum interpretation of the differential algebra. For the remainder of this paper no summation over repeated indices is assumed. By Diff q k(α) (x α ) we denote the algebra generated by x α , D α , u α and u −1 α where k(α) is some number not equal to zero belonging to the index α. The ideal of relations in Diff q k(α) (x α ) is generated by: Proposition 4.1 Under the previous assumptions together with k(α) = −k(−α) and α ∈ I it holds that: Every linear q-difference calculus on a commutative space, obeying the formal hermiticity The tensor product here implies that all the off-diagonal relations are commutative. Before proving the proposition we state the following Corollary 4.2 The usual continuous partial derivatives on the coordinates are given by The formal hermiticity of the relations (21) restricts u α to be of the form u α = exp (h x α ∂ α ). Moreover we have u α = q −1 u −1 −α . Proof: If we assume that the proposition is true, i.e. relations (21) hold for all α ∈ I, the corollary is a direct consequence of these relations. Since we require that for q → 1 the derivatives D should turn into the continuous derivatives it must hold for all α maybe up to some normalization that D ≡ ∂ mod h. Together with the linearity of the calculus and the formal hermiticity of the q-derivatives this requires that in the realization of D α some linear combination of the u k(α) α and (u k(α) α ) −1 must appear. Hence, a realization of any D α is of the form: The product appearing in this expression is taken over some u's such that the hermiticity of iD is not spoiled. λ is a number tending to zero as q → 1. I is a polynomial in x's,D's, and maybe classical ∂'s and is required to have the dimension of a partial derivative. It is easy to show that the q-derivative in the above expression can be shifted and rescaled. This means that according to the assumptions of the proposition D α has the following realization: Now it is easy to conclude. The only thing which has to be done is to calculate the commutation relations of the so realized D α with all other generators. The relation with x α is just (21) while it commutes with D β and x β for α = β. ✷ We note that the considerations of section 3 apply to each component of the tensor product (22).
Remark 1: It can easily be seen that if the coordinate algebra is almost commutative the corresponding q-difference calculus can be transformed into the form described in proposition 4.1.
To illustrate this we use a simple three-dimensional example which, however, is generic. We take commutative coordinates x −1 , x 0 and x 1 subject to the above mentioned conjugation rule. We can make these coordintes q-commuting by using the q-shift operators u α . In the simplest case we set: By corollary 4.2 this unitary transformation preserves the real structure of the coordinates and leads to the commutation relations: The q-derivative with respect to x 0 has to be rescaled as well by D q 0 := (u −1 (u 1 ) −1 ) −1 D 0 while the other ones remain the same. The outcome is a q-difference calculus in which the diagonal relations are almost the same as (21) while the off-diagonal ones are qcommutative. Although the Hilbert spaces of the commutative and of the almost commutative case are not identical the algebras can simply be related. These almost commutative calculi appear for example by reduction of a GL q -quantum group to some lower dimensional orthogonal quantum group (see e.g. [5]).

Remark 2:
It is a well known problem in the study of inhomogeneous quantum groups (e.g. [19]) that it is difficult to find a coproduct which preserves the formal antihermiticity of q-derivatives. It can be read off the algebra (21) that the comultiplication of the formal antihermitian q-difference operator necessarily involves either the quantity u or u −1 . Due to the conjugation property stated in corollary 4.2 it is clear that the comultiplication of a q-difference operator can hardly preseve the formal anti-hermiticity.

Remark 3:
Another problem with the differential calculus on quantum spaces is the nonlinear conjugation rule of the derivatives [18]. Although we will treat this case in the next section in some detail, already at this stage some comments are necessary. The q-partial derivatives as they turn out of the Wess-Zumino calculus [26] are unsymmetric in the sense that they produce q-shifts in only one direction in contrast to the case considered in (19). It can be shown that the nonlinear conjugation rule occurs already in simpler cases. Let us use the commutative three dimensional space introduced in remark 1, and introduce unsymmetric but commuting q-difference operators by: The action of these derivatives on the coordinates looks almost like the ones coming from the SO q (3) covariant calculus. The diagonal relations are for instance of the form ∂ 0 x 0 = u −1 u 1 + qx 0 ∂ 0 . The off diagonal relations are almost commutative. If one introduces a formal antilinear involution, denoted as above by a bar, on the derivatives we get e.g.
The relation between∂ 0 and ∂ 0 is established introducing the quantity Λ := u 2 −1 u 2 0 u 2 1 . We get: Analogous relations hold also for the other q-partial derivatives. Thus, the aim of this remark is that the nonlinear conjugation rule is not surprising when considering unsymmetric q-difference operators in the sense of (27).

q-differential algebras coming from orthogonal quantum groups
In this section we want to extend the results obtained in the previous section to the case of differential calculi on quantum planes coming from orthogonal quantum groups. These quantum planes fit well in our treatment since the compact form of an orthogonal quantum group SO q (N) naturally induces a real structure on the corresponding quantum plane.
In order to present the main result of this section we define the ingredients and fix the notation. For further details see [18] and references therein.
TheR-matrix for the orthogonal quantum group in N dimensions, SO q (N), posesses a decomposition into the following projection operators: the q-analogs of the symmetrizer P + , the antisymmetrizer P − , and the trace projector P 0 . The latter defines the q-analogue of the metric tensor g ij by P 0 ij kl = cg ij g kl , where c is some constant. We then have: The quantum plane corresponding to SO q (N) is the algebra generated by N generators x i with i running through the index sets I 2N +1 or I 2N which have been defined at the beginning of the previous section. The ideal of relations in this algebra is generated by k,l P − ij kl x k x l = 0. The so defined algebra will be denoted by V q (N). By definition V q (N) is a SO q (N)-comodule. The metric defines a SO q (N)-invariant object (by the comodule mapping) L : Due to the real form of the quantum group there exists an antilinear involution on V q (N) given by: It has been shown in [26] that it is possible to construct an algebra of q-partial derivatives on V q (N). We denote these derivatives by ∂ i with i ∈ I. The set of ∂ i spans a SO q (N)comodule algebra (just another quantum plane) with relations dual to the ones in V q (N), namely: The element ∆ := (1 + q N −2 ) −1 ( i,j g ij ∂ i ∂ j ) is central in the algebra of the q-derivatives and invariant under SO q (N)-coaction. The action of the q-partial derivatives on the generators of V q (N) is given by: This action is unsymmetric by its definition in the sense of remark 3 in the previous section.
The involution on V q (N) as defined in (30) cannot be extended to the q-differential algebra. Another copy of partial derivatives has to be introduced. Using the notation of (30) we have∂ i := −q N k,l g ik g kl ∂ l . Although the algebra of these derivatives is generated by relations identical to (31) the action of the conjugated derivatives is given by: For consistency the q-derivatives have to satisfy the relation ∂∂ = q −1R−1∂ ∂.
The both kinds of q-derivatives can be connected using an element which is similar to the one introduced in remark 3 of the previous section. Using the classical analogue of V q (N) with commutative generators x i c and the usual continuous partial derivatives ∂ c i (i ∈ I) we define a classical Euler element E c := i∈I x i c ∂ c i . This definition gives rise to introduce: This element can be expressed in terms of the generators of V q (N) and the partial derivatives [18]. A straightforward calculation gives the following almost commutative relations: The notation (c) means that these commutation relations hold for both the generators coming from the quantum groups and for their classical analogs. (35) shows that the element Λ is well defined in the q-algebra and in the classical algebra as well. Its coproduct is grouplike but not consistent with the involution on the quantum group [19]. Using this element the original and the conjugated derivatives can be related by the formula:∂ This equation should be compared with (28) in the previous section. We denote the full algebra generated by {x i , ∂ i ,∂ i , Λ, Λ −1 |i ∈ I} together with their algebraic relations by Diff c SOq (N). In the spirit of section 3 and 4 the task is to construct a differential calculus on V q (N) consisting of formal anti-hermitian q-derivatives. The immediate guess for an object posessing this property is (cf. [11]): The relations of these newly introduced objects D i are identical to (31). D i is a linear combination of q-derivatives acting either viaR (32) or viaR −1 (33) on the generators of V q (N). Therefore one has two possibilities for writing the diagonal actions.
This relation is valid for any i ∈ I except for x 0 in the odd dimensional case due to the properties of theR-matrix. x 0 requires a q rather than a q 2 in the diagonal relation. This should be noted for all the remaining formulae in this section. The objects r i andr i can be calculated without any difficulties using (32) and (33) respectively. The off-diagonal commutation relations are involved. Moreover the algebra of r i andr i with all generators of Diff c SOq (N) is quite complicated but their explicit form is not needed for our purposes. Due to the properties of the involution (30) and (36) one finds: We now proceed in analogy to the one-dimensional case in section 3 and split the objects r i . For the decomposition we have in mind the quantities u i as they have been obtained in corollary 4.2 are required. Of course, the so defined u i is not supposed to obey any managable commutation relation with the generators of Diff c SOq (N). Introducing for any i ∈ I elements ρ i we make the ansatz: The second expression in (40) is a consequence of (39). It is clear that this decompostion is not unique. However, its existence is guaranteed. Note that in the odd dimensional case it holds for the zero component r 0 = u −1 0 ρ 0 . This ansatz can now be inserted in (38) yielding: Now we can apply proposition 4.1 and equation (21). The statement there is that the quantities u i , u −1 i , or any power of them are defined by by q-difference relations on commutative spaces. Thus the factors ρ −1 i and ρ −i −1 in (41) provide deforming maps and we have the following Theorem 5.1 The q-differential algebra Diff c SOq (N) is isomorphic to the q-difference calculus of the same dimension α∈I Diff q k(i) (x i ) with k(i) = 2 for i > 0, k(i) = −2 for i < 0, and k(0) = 1 as outlined in proposition 4.1. ✷ We note that this result is not covered by the treatment in [17]. As mentioned above in the differential calculus on a commutative space obtained in that paper is not consistent with an involution on the coordinate algebra. In contrast that calculus is only consistent with a Bargmann-Fock type conjugation rule on the full differential algebra.

Conclusions
We have pointed out that coming from the differential calculus on quantum groups one is led to two a priori non-equivalent approaches towards a q-deformation of quantum mechanics. The first one is based on q-oscillator algebras. The second approach arises from the q-differential calculus on involutive quantum planes and focusses on positionmomentum interpretation of the generators of the q-differential algebra. Although both approaches might have interesting applications, e.g. either in statistical mechanics or in the theory of generalized hypergeometric series, it has been shown that due to the rigidity theorem 2.1 for the Heisenberg algebra both approaches do not yield a true deformation of quantum mechanics. Moreover it has been shown that it is hardly possible to find a hermiticity perserving comultiplication on the generators of both the q-difference calculus and of the q-differential calculus. For these reasons one has to think carefully if quantum planes, although they have interesting features [20,21], provide a reasonable base space for quantum field theories.