Quantizations on the circle and coherent states

TL;DR

This paper constructs coherent states on the unit circle using Borel quantization and group-theoretic methods, exploring their properties and relation to canonical coherent states, with applications to quantum phase space and periodic chains.

Contribution

It introduces a novel construction of coherent states on the circle via Borel quantization and group methods, extending the concept to infinite periodic chains.

Findings

Coherent states satisfy resolution of unity.

Properties show similarities and differences with canonical coherent states.

Framework applicable to quantum phase space and periodic chains.

Abstract

We present a possible construction of coherent states on the unit circle as configuration space. Our approach is based on Borel quantizations on S^1 including the Aharonov-Bohm type quantum description. The coherent states are constructed by Perelomov's method as group related coherent states generated by Weyl operators on the quantum phase space 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 and coherent states over a finite periodic chain where the quantum phase space was Z_M x Z_M. The coherent states constructed in this work are shown to satisfy the resolution of unity. To compare them with canonical coherent states, also some of their further properties are studied demonstrating similarities as well as substantial differences.