Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 028, 14 pages      arXiv:2108.08082
Contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quantum in honor of Leon Takhtajan

Pullback Coherent States, Squeezed States and Quantization

Rukmini Dey and Kohinoor Ghosh
International Center for Theoretical Sciences, Sivakote, Bangalore, 560089, India

Received December 07, 2021, in final form March 30, 2022; Published online April 09, 2022

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 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 Kähler 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.

Key words: coherent states; squeezed states; geometric quantization; Berezin quantization.

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