Deformation of Noncommutative Quantum Mechanics

TL;DR

This paper explores the deformation quantization of the symmetry group in noncommutative quantum mechanics, deriving new algebraic structures and star-products that generalize existing quantum group results.

Contribution

It introduces a three-parameter deformation of the symmetry group in NCQM and constructs corresponding quantum algebras and star-products, extending prior work on quantum Heisenberg groups.

Findings

Derived a three-parameter family of quantum groups for NCQM

Established agreement with known quantum Heisenberg group results in special cases

Obtained explicit star-product expressions for the deformed symmetry group

Abstract

In this paper, the Lie group $G_{NC}^{\alpha,\beta,\gamma}$, of which the kinematical symmetry group $G_{NC}$ of noncommutative quantum mechanics (NCQM) is a special case due to fixed nonzero $\alpha$, $\beta$ and $\gamma$, is three-parameter deformation quantized using the method suggested by Ballesteros and Musso in "Quantum Algebras as Quantizations of Dual Poisson-Lie Groups" [J. Phys. A: Math. Theor., 46 (2013), 195203]. A certain family of QUE algebras, corresponding to $G_{NC}^{\alpha,\beta,\gamma}$ with two of the deformation parameters approaching zero, is found to be in agreement with the existing results of the literature on quantum Heisenberg group. Finally, we dualize the underlying QUE algebra to obtain an expression for the underlying star-product between smooth functions on $G_{NC}^{\alpha,\beta,\gamma}$.