Coherent states and geodesics: cut locus and conjugate locus

TL;DR

This paper explores the relationship between coherent states and geodesics on symmetric spaces, providing methods to determine cut and conjugate loci, with applications to complex Grassmann manifolds.

Contribution

It establishes the equivalence of the cut locus with orthogonal coherent vectors and introduces a simple method for calculating conjugate loci in Hermitian symmetric spaces.

Findings

Cut locus of point 0 equals orthogonal coherent vectors.

Method for calculating conjugate locus in Hermitian symmetric spaces.

Application demonstrated on complex Grassmann manifold.

Abstract

The intimate relationship between coherent states and geodesics is pointed out. For homogenous manifolds on which the exponential from the Lie algebra to the Lie group equals the geodesic exponential, and in particular for symmetric spaces, it is proved that the cut locus of the point $0$ is equal to the set of coherent vectors orthogonal to $\vert 0>$. A simple method to calculate the conjugate locus in Hermitian symmetric spaces with significance in the coherent state approach is presented. The results are illustrated on the complex Grassmann manifold.