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On deformed quantum mechanical schemes and \(\star \)-value equations based on the space-space noncommutative Heisenberg-Weyl group. (English) Zbl 1264.81259

Summary: We investigate the Weyl-Wigner-Gröenewold-Moyal, the Stratonovich, and the Berezin group quantiza-tion schemes for the space-space noncommutative Heisenberg-Weyl group. We show that the \(\star\)-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 \(\star\)-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 \(star\)-valued equations for the 3 quanti-zation schemes considered and discuss the criteria that are also needed for them to exist.

MSC:

81R60 Noncommutative geometry in quantum theory
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
81Q99 General mathematical topics and methods in quantum theory
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