Non-commutative quantum mechanics in three dimensions and rotational symmetry

Abstract

We generalize the formulation of non-commutative quantum mechanics to three-dimensional non-commutative space. Particular attention is paid to the identification of the quantum Hilbert space in which the physical states of the system are to be represented. We then demonstrate how the Voros star product (like the Moyal) can be defined in this odd-dimensional space by identifying the appropriate basis in which the state has to be represented. Then, we discuss how to construct the representation of the rotation group on this space, the deformation of the Leibnitz rule accompanying this representation and the implied necessity of deforming the co-product to restore the rotation symmetry automorphism. This also implies the breaking of rotational invariance on the level of the Schrödinger action and equation, as well as the Hamiltonian, even for rotational-invariant potentials. For rotational-invariant potentials, the symmetry breaking results purely from the deformation in the sense that the commutator of the Hamiltonian and angular momentum is proportional to the deformation.