Non-commutative Quantum Mechanics in Three Dimensions and Rotational Symmetry

TL;DR

This paper extends non-commutative quantum mechanics to three dimensions, focusing on the representation of rotation symmetry, its deformation, and the resulting symmetry breaking in the Hamiltonian and equations of motion.

Contribution

It introduces a formulation of three-dimensional non-commutative quantum mechanics, detailing the representation of rotation group and the deformation of symmetry operations.

Findings

Rotation symmetry is deformed in 3D non-commutative space.

Rotational invariance is broken at the level of the Hamiltonian and Schrödinger equation.

Deformation causes the commutator of Hamiltonian and angular momentum to be non-zero.

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, the construction of 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 Schroedinger 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.