Geometrical phases on hermitian symmetric spaces

TL;DR

This paper characterizes hermitian symmetric spaces as the unique homogeneous manifolds where the phase of scalar products of coherent states relates directly to symplectic areas, providing explicit calculations and connecting to known cocycles.

Contribution

It identifies hermitian symmetric spaces as the unique cases with this phase-area relationship and computes the multiplicative factor explicitly for complex Grassmannians.

Findings

The phase of scalar products equals twice the symplectic area on hermitian symmetric spaces.

Explicit calculations for complex Grassmann manifolds and their duals.

The multiplicative factor matches the two-cocycle from Lie group cohomology.

Abstract

For simple Lie groups, the only homogeneous manifolds $G/K$, where $K$ is maximal compact subgroup,for which the phase of the scalar product of two coherent state vectors is twice the symplectic area of a geodesic triangle are the hermitian symmetric spaces. An explicit calculation of the multiplicative factor on the complex Grassmann manifold and its noncompact dual is presented.It is shown that the multiplicative factor is identical with the two-cocycle considered by A. Guichardet and D. Wigner for simple Lie groups.