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

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.