Pullback Coherent States, Squeezed States and Quantization

Abstract

In this semi-expository paper, we define certain Rawnsley-type coherent and squeezed states on an integral Kähler manifold (after possibly removing a set of measure zero) and show that they satisfy some properties which are akin to the 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 ℂPⁿ. Any other submanifold for which the criterion of the identity theorem holds exhibits this type of Berezin quantization. Also, this type of quantization holds for totally real submanifolds of real dimension 𝑛 of a general homogeneous Kähler manifold of real dimension 2𝑛 for which Berezin quantization exists. In the appendix, we review the Rawnsley and generalized Perelomov coherent states on ℂPⁿ (which is a coadjoint orbit) and the fact that these two types of coherent states coincide.