Representations of noncommutative quantum mechanics and symmetries

TL;DR

This paper develops a unified framework for representing quantum mechanics on noncommutative spaces, exploring phases, dualities, symmetries, and applications to specific physical systems like the harmonic oscillator and Landau problem.

Contribution

It introduces a general approach to noncommutative quantum mechanics representations, identifying phases, dualities, and symmetry classifications across dimensions.

Findings

Two distinct phases identified in noncommutative quantum systems.

Duality relations connect different phases in arbitrary dimensions.

Symmetry analysis applied to specific models like the harmonic oscillator and Landau problem.

Abstract

We present a unified approach to representations of quantum mechanics on noncommutative spaces with general constant commutators of phase-space variables. We find two phases and duality relations among them in arbitrary dimensions. Conditions for physical equivalence of different representations of a given system are analysed. Symmetries and classification of phase spaces are discussed. Specially, the dynamical symmetry of a physical system is investigated. Finally, we apply our analyses to the two-dimesional harmonic oscillator and the Landau problem.