-symmetric noncommutative spaces with minimal volume uncertainty relations
We provide a systematic procedure to relate a three dimensional q-deformed oscillator algebra to the corresponding algebra satisfied by canonical variables describing noncommutative spaces. The large number of possible free parameters in these calculations is reduced to a manageable amount by imposing various different versions of -symmetry on the underlying spaces, which are dictated by the specific physical problem under consideration. The representations for the corresponding operators are in general non-Hermitian with regard to standard inner products and obey algebras whose uncertainty relations lead to minimal length, areas or volumes in phase space. We analyze in particular one three dimensional solution which may be decomposed to a two dimensional noncommutative space plus one commuting space component and also into a one dimensional noncommutative space plus two commuting space components. We study some explicit models on these type of noncommutative spaces.
The simplest and most commonly studied version of noncommutative spaces replaces the standard set of commuting coordinates by new ones obeying , with being a constant antisymmetric tensor. However, even in the very first proposals on noncommutative spaces  the tensor was taken to be a function of the position coordinates, i.e. . Further possibilities arise when one breaks the Lorentz invariance of the tensor and allows for a general dependence of position and momenta [2, 3, 4, 5]. It is known for some time that such a scenario leads to the interesting versions of a generalized version of Heisenberg’s uncertainty relations [2, 3]. In particular when the commutation relations are modified in such a way that their structure constants involve higher powers of the momenta or coordinates one encounters minimal lengths or momenta, respectively. As a consequence of these type of relations lead to more radical changes in the interpretation of possible measurements than the conclusions usually drawn from the standard relations. Whereas the conventional relations, which simply have Planck’s constant as structure constant, only prevent that two quantities commuting in this manner, e.g. and , can be known simultaneously, the modified versions prohibit that the observables can be known at all below a certain scale, the minimal length or minimal momentum. This scale is usually identified to be of the order of the Planck scale. Combining some of these minimal lengths in two dimension leads to minimal areas and in three dimensions to minimal volumes. The need for such type of noncommutative space structures has arisen in many contexts, such as for instance in certain string theories  and models investigating gravitational stability . For a general review on noncommutative quantum mechanics see for instance  and for a review on noncommutative quantum field theories see [9, 10].
By now many studies on the structure of such type of generalized canonical relations have been carried out [2, 3, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], albeit mostly in dimensions less than three. Besides leading to different physical results, a furher crucial difference between the Snyder type noncommutative spaces and those with broken Lorentz invariance is the way they are constructed. Whereas the former spaces can be thought of as arising naturally from deformations based on general twists [26, 27], the construction of the latter is less systematic and is usually based on the deformation of oscillator algebras [28, 11, 3]. In [16, 18, 19] it was shown in one and two dimensions, respectively, how to map q-deformed oscillator algebras onto canonical variables. The main purpose of this manuscript is to extend these considerations to the full three dimensional space. This approach has the advantage that it allows for the explicit construction of the entire Fock space [12, 13, 14].
Our manuscript is organized as follows: As an introduction we explain in section 2 how one can systematically construct canonical variables on a flat noncommutative space starting from standard generators in Fock space by exploiting -symmetry to reduce the number of free parameters. In section 3 we extend these considerations to q-deformed oscillator algebras and present in particular one solution in more detail for which we construct in a nontrivial limit a non-Hermitian representation. In section 4 we demonstrate that, depending on the dimension, this solution leads to minimal length, areas or volumes. In section 5 we study the non-Hermitian -symmetric versions of the harmonic oscillator on these spaces and in section 6 we state our conclusions.
2 Deformed oscillator algebras and noncommutative spaces
In the limit we denote and recover the standard Fock space commutation relations
We assume further that the relations (2) are linearly related to the standard three dimensional flat noncommutative space characterized by the relations
where the , having dimensions of length or momentum for or respectively. The commutation relations obeyed by the canonical variables , , , , , associated to the deformed algebra (1) are yet unknown and are subject to construction. The algebra they satisfy may be related to (1) by similar relations as (4), but since the constants and are in general complex, this amounts to finding real parameters. To reduce this number to a manageable quantity one can utilize -symmetry.
2.1 The role of -symmetry
Whereas the momenta and coordinates in (3) are Hermitian operators acting on a Hilbert space with standard inner product, this is no longer true for the variables associated to the deformed algebra (1) as they become in general non-Hermitian with regard to these inner products. Thus a quantum mechanical or quantum field theoretical models on these spaces will, in general, not be Hermitian in that space. However, it is by now well accepted that one may consider complex symmetric non-Hermitian systems as self-consistent descriptions of physical systems [29, 30]. Guided by these results one may try to identify this symmetry for the noncommutative space relations (3). In  the authors argue that this would not be possible and one is therefore forced to take the noncommutative constants to be complex. We reason here that this is incorrect and even the standard noncommutative space relations are in fact symmetric under many different versions of -symmetry. All one requires to formulate a consistent quantum description is an antilinear involutory map  that leaves the relations (3) invariant. We identify here the several possibilities:
Taking for instance , the algebra (3) remains invariant under the following antilinear transformations
We may also attempt to keep different from zero, in which case we have to transform the as well in order to achieve the invariance of (3)
A further option, which also allows to keep different from zero, would be to introduce permutations amongst the different directions
Clearly all of these maps are involutions . In fact, there might in fact be more options. The occurrence of various possibilities to implement the antilinear symmetry is a known feature previously observed for many examples [33, 34, 35] in dimensions larger than one. More restrictions and the explicit choice of symmetry result from the specific physical situation one wishes to describe. For instance might be appropriate when one deals with a problem in which one direction is singled out, requires the noncommutative constant to appear as a parameter in the model and suggest a symmetry along the line . For the creation and annihilation operators this symmetry could manifest itself in different ways, for instance as , or by the permutation of indices , when they label for instance particles in different potentials, see e.g. . Once again the underlying physics will dictate which version one should select. The general reason for the occurrence of these different possibilities are just manifestations of the ambiguities in defining a metric to which the -operator is directly related. What needs to be kept in mind is that we only require the symmetry of some antilinear involution  in order to obtain a meaningful quantum mechanical description.
2.2 Oscillator algebras of flat noncommutative spaces
Let us first see how to represent a three dimensional oscillator algebra in terms of the canonical variables in three dimensional flat noncommutative space. For definiteness we seek at first a description which is invariant under . The most generic linear Ansatz for the creation and annihilation operators to achieve this is
with dimensional real constants . We note that we have for . The nonsequential ordering of the constants in (8)-(10) is chosen to perform the limit to the two dimensional case in a convenient way. For we recover equation (2.4) in . It is useful to invoke this limit at various stages of the calculation as a consistency check. We then compute that the operators (8)-(10), expressed on the three dimensional flat noncommutative space (3), satisfy the standard Fock space commutation relations (2) provided that the following nine constraints hold
where the matrices have entries
These expressions satisfy the commutation relations(3) when we invoke the constraints (11)-(2.2) and the standard Fock space relation (2). In that case we also have the simple relation . By changing the Ansatz (8)-(10) appropriately one may also obtain or -invariant solutions.
3 Noncommutative space-time from q-deformed creation and annihilation operators
Next we construct the commutation relations for the deformed noncommutative space satisfied by the canonical variables , , , , , , which we express linearly in terms of the creation and annihilation operators obeying the deformed algebra (1). Guided by the fact that in the limit we should recover the relations (23)-(28) of the previous subsection. We therefore make the Ansatz
with for having the dimension of a length and for possessing the dimension of a momentum. The constants for are therefore dimensionless. We deliberately keep here all dimensional constants different from . With the help of the -deformed oscillator algebra (1) we compute
Inverting now the relations (30)-(35) we find that it is indeed possible to eliminate entirely the creations and annihilation operators from these relations. However, this leads to very lengthy expressions, which we will not present here. Instead we report some special, albeit still quite general, solutions obtained by setting some of the constants to zero and imposing further constraints.
3.1 A particular -symmetric solution
We now make the assumption that . This choice still guarantees that none of the canonical variables become mutually identical. The consistency with the direct limit in which we want to recover (3) enforces the constraints
The only non-vanishing commutators we obtain in this case are
3.1.1 Reduced three dimensional solution for
The solution (47)-(51) possesses a non-trivial limit leading to an even simpler set of commutation relations. For this purpose we impose some additional constraints by setting first , , and subsequently we take the limit . The relations (47)-(51) then reduce to
where has the dimension of an inverse squared length, has the dimension of an inverse squared momentum and is dimensionless. We find a concrete representation for this algebra in terms of the generators of the standard three dimensional flat noncommutative space (3)
Evidently the quantities , and are non-Hermitian in the space in which the , , , , , are Hermitian. In order to study concrete models it is very convenient to carry out a subsequent Bopp-shift of the form , , , , , and express the generators in (54) in terms of the standard canonical variables. Since there is no explicit occurrence of , the representation (54) is trivially invariant under as well as . Taking, however, the representation (54) and in addition this evidently changes, as by direct computation one of the commutation relations is altered to . Setting furthermore the representation in (54) is also invariant under the -symmetry stated in (7).
3.1.2 Reduction into a decoupled two dimensional plus a one dimensional space
The algebra (47)-(51) provides a larger three dimensional setting for a noncommutative two dimensional space decoupled from a standard one dimensional space. This is achieved by parameterizing , setting , and subsequently taking the limit (, ) reduces the algebra to a two noncommutative dimensional space in the ,-direction
decoupled from a standard one dimensional space in the -direction
which are of course still non-Hermitian with regard to the standard inner product.
3.1.3 Reduction into three decoupled one dimensional spaces
We conclude this section by noting that all three directions in the algebra (47)-(51) can be decoupled, of which one becomes a one dimensional noncommutative space previously investigated by many authors, e.g. [2, 16]. It is easy to verify that this scenario is obtained from (47)-(51) when parameterizing and subsequently taking the limit , , . The remaining non-vanishing commutators are then
Thus all three space directions are decoupled from each other. It is known that the choices , or , constitute representations for the commutation relations (58) in the -direction.
3.2 A particular -symmetric solution
Instead of solving the complicated relations (36)-(45) one may also start by making directly an Ansatz of a similar form as in (54) without elaborating on the relation to the -deformed oscillator algebra. Proceeding in this manner with an Ansatz respecting the -symmetry we find for instance the representation
yielding the closed algebra