Quantum Mechanics on the Noncommutative Torus

TL;DR

This paper studies the algebraic structure of a charged particle on a noncommutative torus under a magnetic field, revealing a critical magnetic field where the algebra simplifies and connecting it to noncommutative bundles.

Contribution

It introduces a set of algebra generators analogous to the commutative case, identifies a critical magnetic field value, and relates the algebra's representations to noncommutative geometry.

Findings

Existence of a critical magnetic field where the algebra reduces

Explicit irreducible representations of the algebra

Connection between algebra representations and noncommutative bundles

Abstract

We analyze the algebra of observables of a charged particle on a noncommutative torus in a constant magnetic field. We present a set of generators of this algebra which coincide with the generators for a commutative torus but at a different value of the magnetic field, and demonstrate the existence of a critical value of the magnetic field for which the algebra reduces. We then obtain the irreducible representations of the algebra and relate them to noncommutative bundles. Finally we comment on Landau levels, density of states and the critical case.