On the Plethora of Representations Arising in Noncommutative Quantum Mechanics and An Explicit Construction of Noncommutative 4-tori

TL;DR

This paper constructs explicit representations and gauge structures in noncommutative quantum mechanics, introduces noncommutative 4-tori, and develops associated modules and connections, enriching the mathematical framework of the field.

Contribution

It provides an explicit construction of noncommutative 4-tori, detailed gauge families, and projective modules, advancing the mathematical understanding of noncommutative quantum systems.

Findings

Unified gauge potentials in noncommutative quantum mechanics

Explicit construction of noncommutative 4-tori and star products

Development of projective modules with constant curvature connections

Abstract

We construct a 2-parameter family of unitarily equivalent irreducible representations of the triply extended group $\g$ of translations of $\mathbb{R}^{4}$ associated with a family of its 4-dimensional coadjoint orbits and show how a continuous 2-parameter family of gauge potentials emerges from these unitarly equivalent representations. We show that the Landau and the symmetric gauges of noncommutative quantum mechanics, widely used in the literature, in fact, belong to this 2-parameter family of gauges. We also provide an explicit construction of noncommutative 4-tori and compute the associated star products using the unitary dual of the group $\g$ that was studied at length in an earlier paper (\cite{ncqmjpa}). Finally, we construct projective modules over such noncommutative 4-tori and compute constant curvature connections on them using Rieffel's method.