Coherent states on the circle

TL;DR

This paper constructs and analyzes coherent states on the unit circle, exploring their properties and demonstrating their resolution of unity, with implications for quantum systems with circular configuration spaces.

Contribution

It introduces a novel method for constructing coherent states on the circle using an analogy with finite periodic chains, expanding the understanding of quantum states in circular geometries.

Findings

Coherent states on the circle satisfy the resolution of unity.

The construction is analogous to finite periodic chains.

Properties of these states are systematically studied.

Abstract

We present a possible construction of coherent states on the unit circle as configuration space. In our approach the phase space is the product Z x S^1. Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction we use the analogy with our quantization over a finite periodic chain where the phase space was Z_M x Z_M. Properties of the coherent states constructed in this way are studied and the coherent states are shown to satisfy the resolution of unity.