Pullback Coherent States, Squeezed States and Quantization

TL;DR

This paper generalizes properties of coherent and squeezed states on Kähler and smooth manifolds, establishing a Berezin-type quantization framework for certain submanifolds, with implications for geometric quantization.

Contribution

It introduces Rawnsley-type coherent and squeezed states on Kähler and smooth manifolds, demonstrating their properties and establishing a Berezin-type quantization for specific submanifolds.

Findings

Coherent and squeezed states satisfy maximal likelihood and resolution of identity properties.

A Berezin-type quantization is constructed for totally real submanifolds of complex projective space.

The quantization framework extends to submanifolds in homogeneous Kähler manifolds.

Abstract

In this semi-expository paper, we define certain Rawnsley-type coherent and squeezed states on an integral K\"ahler manifold (after possibly removing a set of measure zero) and show that they satisfy some properties which are akin to maximal likelihood property, reproducing kernel property, generalised resolution of identity property and overcompleteness. This is a generalization of a result by Spera. Next we define the Rawnsley-type pullback coherent and squeezed states on a smooth compact manifold (after possibly removing a set of measure zero) and show that they satisfy similar properties. Finally we show a Berezin-type quantization involving certain operators acting on a Hilbert space on a compact smooth totally real embedded submanifold of $U$ of real dimension $n$, where $U$ is an open set in ${\mathbb C}{\rm P}^n$. Any other submanifold for which the criterion of the identity theorem holds exhibit this type of Berezin quantization. Also this type of quantization holds for totally real submanifolds of real dimension $n$ of a general homogeneous K\"ahler manifold of real dimension $2n$ for which Berezin quantization exists. In the appendix we review the Rawnsley and generalized Perelomov coherent states on ${\mathbb C}{\rm P}^n$ (which is a coadjoint orbit) and the fact that these two types of coherent states coincide.