How to construct a coordinate representation of a Hamiltonian operator on a torus

TL;DR

This paper explores the quantization of a particle on a torus using Cartesian and toric coordinates, highlighting the challenges and solutions in representing the Hamiltonian operator in different coordinate systems.

Contribution

It provides a method to construct a coordinate representation of the Hamiltonian operator on a torus, addressing the transformation issues between Cartesian and toric coordinates after quantization.

Findings

Toric coordinates simplify commutation relations and their solutions.

Two quantum Hamiltonians emerge in the toric coordinate system.

Transforming from Cartesian to toric coordinates introduces specific problems in the quantum framework.

Abstract

The dynamical system of a point particle constrained on a torus is quantized \`a la Dirac with two kinds of coordinate systems respectively; the Cartesian and toric coordinate systems. In the Cartesian coordinate system, it is difficult to express momentum operators in coordinate representation owing to the complication in structure of the commutation relations between canonical variables. In the toric coordinate system, the commutation relations have a simple form and their solutions in coordinate representation are easily obtained with, furthermore, two quantum Hamiltonians turning up. A problem comes out when the coordinate system is transformed, after quantization, from the Cartesian to the toric coordinate system.