Two Dimensional Plane, Modified Symplectic Structure and Quantization

TL;DR

This paper explores noncommutative quantum mechanics on the plane by modifying the symplectic structure, leading to a deformed Heisenberg group and altered canonical commutation relations.

Contribution

It introduces a modified symplectic structure within Isham's quantization scheme to account for noncommutative coordinates, resulting in a nonabelian symmetry group.

Findings

Modified symplectic structure leads to a nonabelian symmetry group.

The canonical group becomes a deformed Heisenberg group.

Results align with standard noncommutative quantum mechanics.

Abstract

Noncommutative quantum mechanics on the plane has been widely studied in the literature. Here, we consider the problem using Isham's canonical group quantization scheme for which the primary object is the symmetry group that underlies the phase space. The noncommutativity of the configuration space coordinates requires us to introduce the noncommutative term in the symplectic structure of the system. This modified symplectic structure will modify the group acting on the configuration space from abelian $\mathbb{R}^2$ to a nonabelian one. As a result, the canonical group obtained is a deformed Heisenberg group and the canonical commutation relation (CCR) corresponds to what is usually found in noncommutative quantum mechanics.