Three paths toward the quantum angle operator

TL;DR

This paper explores three different mathematical methods to construct a quantum angle operator linked to a number operator, advancing the theoretical understanding of phase in quantum mechanics.

Contribution

It introduces three novel approaches for defining quantum angle operators, including operator theory and integral quantization methods, expanding the mathematical framework for quantum phase.

Findings

Three distinct methods for quantum angle operator construction.

Connection between angle operator and phase space quantization.

Generalization of Berezin-Klauder approaches to quantum phase.

Abstract

We examine mathematical questions around angle (or phase) operator associated with a number operator through a short list of basic requirements. We implement three methods of construction of quantum angle. The first one is based on operator theory and parallels the definition of angle for the upper half-circle through its cosine and completed by a sign inversion. The two other methods are integral quantization generalizing in a certain sense the Berezin-Klauder approaches. One method pertains to Weyl-Heisenberg integral quantization of the plane viewed as the phase space of the motion on the line. It depends on a family of "weight" functions on the plane. The third method rests upon coherent state quantization of the cylinder viewed as the phase space of the motion on the circle. The construction of these coherent states depends on a family of probability distributions on the line.