$\mathcal {PT}$-symmetric non-commutative spaces with minimal volume uncertainty relations

Abstract

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 PTsymmetry 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.

Time-dependentq-deformed coherent states for generalized uncertainty relations

“…We demonstrated that all key requirements for coherent states are satisfied. A direct comparison with the results obtained in [5] is not possible as the analysis in there relates to a nontrivial limit q → 1, which is not directly obtainable from the setting presented here, see [1,4]. However, qualitatively we found a somewhat different behaviour with regard to the key question addressed in this manuscript.…”

“…Let us now investigate some properties of these states and in particular investigate to which kind of expectation values they lead for observables and compare with the results for the nontrivial q → 1 limit studied in [5]. In the latter case these states were found to be squeezed states up to first order in perturbation theory in τ when parameterizing the deformation parameter as q = e 2κ 2 6 τ , where κ 6 is explained in [4]. Most importantly we wish to investigate whether these states respect the generalized uncertainty relations.…”

“…In order to verify the uncertainty relations projected onto these states we commence by recalling [1,15,4] that the analogues of the canonical variables expressed in terms of the q-deformed oscillator algebra generators…”

“…The algebras satisfied by the canonical variables resulting from q-deformed oscillator algebras have been shown to be related to noncommutative spacetime structures leading to minimal lengths and minimal momenta as a result of a generalized version of Heisenberg's uncertainty relations [1,2,3,4]. An important question to address in this context is whether explicit states satisfying these relations actually exist and how they can be constructed.…”

Entangled squeezed states in noncommutative spaces with minimal length uncertainty relations

“…More interesting structures, leading for instance to minimal length and generalized versions of Heisenberg's uncertainty relations, are obtained when θ µν is taken to be a function of the momenta and coordinates, e.g. [14][15][16][17][18][19].…”