The Symmetry Groups of Noncommutative Quantum Mechanics and Coherent State Quantization

TL;DR

This paper investigates the group theoretical structure of noncommutative quantum mechanics in two dimensions, constructing representations and coherent states that reproduce the standard noncommutative commutation relations.

Contribution

It identifies the relevant symmetry groups, extends them, and develops coherent state quantization consistent with noncommutative quantum mechanics.

Findings

Identified the two-fold central extension of the Galilei group as fundamental.

Constructed unitary irreducible representations and coherent states.

Derived the same noncommutative commutation relations via coherent state quantization.

Abstract

We explore the group theoretical underpinning of noncommutative quantum mechanics for a system moving on the two-dimensional plane. We show that the pertinent groups for the system are the two-fold central extension of the Galilei group in $(2+1)$-space-time dimensions and the two-fold extension of the group of translations of $\mathbb R^4$. This latter group is just the standard Weyl-Heisenberg group of standard quantum mechanics with an additional central extension. We also look at a further extension of this group and discuss its significance to noncommutative quantum mechanics. We build unitary irreducible representations of these various groups and construct the associated families of coherent states. A coherent state quantization of the underlying phase space is then carried out, which is shown to lead to exactly the same commutation relations as usually postulated for this model of noncommutative quantum mechanics.