Non-commutative quantum mechanics in three dimensions and rotational symmetry (Q2880362)

In this paper, the authors extend the formulation of non-commutative quantum mechanics to the three-dimensional non-commutative space. They begin with the definition of non-commutative Heisenberg algebra, and then they define the quantum Hilbert space \(\mathcal{H}^{(3)}_{q}\), whose elements represent the physical states, and on which the non-commutative Heisenberg algebra is represented. They demonstrate how the Voros star product can be defined in this odd-dimensional space \(\mathcal{R}^{3}\), and discuss how to construct the representation of the rotation group on this space and the deformation of the Leibniz rule, which implies that the action of the Schrödinger equation, in which the potential appears as a fixed background field, and Hamiltonian are no longer invariant under rotations.