Abstract
Based on the vanishing of the second Hochschild cohomology group of the Weyl 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 aq-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 aq-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 possess a proper group of global transformations.
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Pillin, M. On the deformability of Heisenberg algebras. Commun.Math. Phys. 180, 23–38 (1996). https://doi.org/10.1007/BF02101181
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DOI: https://doi.org/10.1007/BF02101181