Coadjoint orbits and K\"ahler structure: examples from coherent states

TL;DR

This paper investigates when co-adjoint orbits of Lie groups have a K"ahler structure, using coherent states to embed these orbits into projective Hilbert space, and explores geometric quantization of certain orbits.

Contribution

It provides examples and insights into the K"ahler structure of co-adjoint orbits via coherent states, including embeddings and quantization, with a semi-expository approach.

Findings

Co-adjoint orbits of certain Lie groups support K"ahler structures.

Coherent states enable K"ahler embeddings of these orbits.

Squeezed states provide symplectic, but not K"ahler, embeddings.

Abstract

Do co-adjoint orbits of Lie groups support a K\"{a}hler structure? We study this question from a point of view derived from coherent states. We examine three examples of Lie groups: the Weyl-Heisenberg group, $\mathrm{SU(2)}$ and $\mathrm{SU(1,1)}$. In cases, where the orbits admit a K\"{a}hler structure, we show that coherent states give us a K\"{a}hler embedding of the orbit into projective Hilbert space. In contrast, squeezed states, (which like coherent states, also saturate the uncertainty bound) only give us a symplectic embedding. We also study geometric quantisation of the co-adjoint orbits of the group $\mathrm{SUT(2,\mathbb{R})}$ of real, special, upper triangular matrices in two dimensions. We glean some general insights from these examples. Our presentation is semi-expository and accessible to physicists.