Quantum Description of Angles in the Plane

TL;DR

This paper explores the quantum formalism in the real plane, illustrating concepts like covariant integral quantization, quantum measurement of light polarization, entanglement, Bell inequality violations, and spin-1/2 coherent states as entangled angles.

Contribution

It provides a quantum framework for understanding angles in the plane, connecting geometric and quantum concepts with novel interpretations.

Findings

Demonstrates covariant integral quantization in the plane

Shows how light polarization can be viewed as a quantum measurement

Explains entanglement and Bell inequality violations in this context

Abstract

The real plane with its set of orientations or angles in $[0,\pi)$ is the simplest non trivial example of a (projective) Hilbert space and provides nice illustrations of quantum formalism. We present some of them, namely covariant integral quantization, linear polarisation of light as a quantum measurement, interpretation of entanglement leading to the violation of Bell inequalities, and spin one-half coherent states viewed as two entangled angles.