Linear Hamiltonians on homogeneous K\"ahler manifolds of coherent states

TL;DR

This paper develops explicit differential operator representations of Lie algebra generators on coherent state manifolds, deriving equations of motion for linear Hamiltonians, including Riccati equations for symmetric cases and polynomial equations for non-symmetric spaces.

Contribution

It provides explicit formulas for the differential action of Lie algebra generators on coherent state manifolds, enabling the derivation of equations of motion for linear Hamiltonians in these settings.

Findings

Equations of motion are first order differential equations with polynomial coefficients.

For hermitian symmetric manifolds, equations reduce to matrix Riccati equations.

Non-symmetric space example has polynomial equations of degree up to 3.

Abstract

Representations of coherent state Lie algebras on coherent state manifolds as first order differential operators are presented. The explicit expressions of the differential action of the generators of semisimple Lie groups determine for linear Hamiltonians in the generators of the groups first order differential equations of motion with holomorphic polynomials coefficients. For hermitian symmetric manifolds the equations of motion are matrix Riccati equations. It is presented the simplest example of the non-symmetric space $SU(3)/S(U(1)\times U(1)\times U(1))$ where the polynomials describing the equations of motion have the maximum degree 3.