2021/05/29 by Dey, Rukmini, Samuel, Joseph, Vidyarthi, Rithwik S. · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2105.14283
Do co-adjoint orbits of Lie groups support a Kähler 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, SU(2) and SU(1,1). In cases, where the orbits admit a Kähler structure, we show that coherent states give us a Kähler 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 SUT(2,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.