Quantization on a torus without position operators

TL;DR

This paper develops a formulation of quantum mechanics on a two-dimensional torus that relies solely on momentum and shift operators, avoiding position operators, and establishes a foundational link to noncommutative geometry.

Contribution

It introduces an algebraic framework for quantum mechanics on a torus without position operators and proves the equivalence of its irreducible representations, supporting noncommutative torus theory.

Findings

Constructed an algebra with momentum and shift operators for the torus

Demonstrated the algebra's irreducible representations are unitarily equivalent

Provided a foundation for noncommutative torus quantum mechanics

Abstract

We formulate quantum mechanics in the two-dimensional torus without using position operators. We define an algebra with only momentum operators and shift operators and construct irreducible representation of the algebra. We show that it realizes quantum mechanics of a charged particle in a uniform magnetic field. We prove that any irreducible representation of the algebra is unitary equivalent to each other. This work provides a firm foundation for the noncommutative torus theory.