On deformed quantum mechanical schemes and ★-value equations based on the space-space noncommutative Heisenberg-Weyl group

Abstract

We investigate the Weyl-Wigner-Gröenewold-Moyal, the Stratonovich and the Berezin group quantization schemes for the space-space noncommutative Heisenberg-Weyl group. We show that the ⋆-product for the deformed algebra of Weyl functions for the first scheme is different than that for the other two, even though their respective quantum mechanics' are equivalent as far as expectation values are concerned, provided that some additional criteria are imposed on the implementation of this process. We also show that it is the ⋆-product associated with the Stratonovich and the Berezin formalisms that correctly gives the Weyl symbol of a product of operators in terms of the deformed product of their corresponding Weyl symbols. To conclude, we derive the stronger ⋆-valued equations for the 3 quantization schemes considered and discuss the criteria that are also needed for them to exist.

Noncommutative Coherent States and Quantum Cosmology

“…In a more mathematical context they constitute the appropriate language to study harmonic analysis in Lie groups and, as it was shown in [2], they bridge different deformation quantization schemes. As for the latter it was also demonstrated in [3] that such bridge remains true even for the case of Noncommutative Quantum Mechanics understood as the quantum theory associated to the Lie algebra h θ 2n+1 of commutators:…”

“…Therefore, it is tempting to implement the full coherent-state formalism in the case of a Noncommutative Quantum Cosmology, where the configuration observables will be described by the algebra (1.1), in the hope that at an effective level the dynamical properties of noncommutativity suffice to generate solutions which are free of singularities. Overcomplete sets of coherent states have been constructed elsewhere for the case of noncommutative theories [3,8,9,10], where it must be emphasized that the representations used in all of these references are not equivalent and consequently the results obtained from their implementation in a given problem should differ. In the present work, however, we will opt for the use of the coherent system developed in [3].…”

“…Overcomplete sets of coherent states have been constructed elsewhere for the case of noncommutative theories [3,8,9,10], where it must be emphasized that the representations used in all of these references are not equivalent and consequently the results obtained from their implementation in a given problem should differ. In the present work, however, we will opt for the use of the coherent system developed in [3]. Thus, to investigate the consequences of introducing a noncommutative structure at the level of a Quantum Cosmology we organized this work in the following manner: In §2 and §3 we review the basics of Perelomov's definition of Generalized Coherent States and reproducing kernels [11,12], with their application to Feynman integrals of constraint theories by means of projectors, according to Klauder's procedure of coherent state quantization [13].…”

TwistedC⋆-algebra formulation of quantum cosmology with application to the Bianchi I model

“…This, from the viewpoint of Deformation Quantization where the Moyal ⋆-product arises as a deformation of the algebra product of the Weyl symbols of quantum operator observables, has no conceptual support. Moreover, as we have shown in [14] (and references therein) a more logical noncommutative replacement for the Schrödinger equation is the ⋆-value equation involving the deformed Moyal ⋆-product of the Weyl symbol of the quantum Hamiltonian operator and the Wigner function. It may be meaningful to notice here also that in a previous work [15] of the type mentioned above, the region close to the singularity has not been explored and the wave functions have branch points which imply an undetermined behavior near the singularity, which could very well be attributed to the authors use of this unsubstantiated Moyal product in the Wheeler-de Witt equation.…”

“…Alternatively, the C ⋆ -algebra A, on which our approach is based, is in particular a good example of the strategy of Noncommutative Geometry, and a motivational argument for basing our approach on this formalism hinges, on a nut shell, on the theoretical observations that since physically meaningful quantities should be independent of the choice of a gauge, the concepts of gauge potentials or connections had to be incorporated into the formulation of Action base topological M space in Classical Bianchi I Cosmology is an R 3 , for which translations are isometries, whereas physical space at the Noncommutative Geometry level is described as a sort of a subjacent discrete noncommutative cellular structure (posets), we let A be the algebra of the noncommutative extended Heisenberg-Weyl group [14], G be the discrete topological group of translations in R 3 , (α, σ) the twisted action of G on A, with α denoting the map α : G → Aut(A) and σ : G × G → T (A) is a normalized 2-cocycle on G with values in the multiplicative group T of all complex numbers of unit modules, such that σ(x 1 , x 2 )σ(x 1 + x 2 , x 3 ) = σ(x 2 , x 3 )σ(x 1 , x 2 + x 3 ), x 1 , x 2 , x 3 ∈ G σ(x, 0) = σ(0, x) = 1.…”