Quantum localisation on the circle

TL;DR

This paper develops a covariant integral quantisation method for a particle on a circle using coherent states related to the Euclidean group E(2), addressing the quantum angle operator and its properties.

Contribution

It introduces a new covariant integral quantisation approach for the circle, providing a detailed analysis of the angle operator and its quantum properties.

Findings

Constructed the quantum angle operator and analyzed its spectrum.

Derived the commutator between angle and angular momentum.

Compared the new approach with existing methods for quantum angle.

Abstract

Covariant integral quantisation using coherent states for semidirect product groups is studied and applied to the motion of a particle on the circle. In the present case the group is the Euclidean group E$(2)$. We implement the quantisation of the basic classical observables, particularly the $2\pi$-periodic discontinuous angle function and the angular momentum, and compute their corresponding lower symbols. An important part of our study is devoted to the angle operator given by our procedure, its spectrum and lower symbol, its commutator with the quantum angular momentum, and the resulting Heisenberg inequality. Comparison with other approaches to the long-standing question of the quantum angle is discussed.