Ellipsoids of U(3) model

TL;DR

This paper applies the Cartan model to analyze the structure of coadjoint orbits in u^*(3), introduces a reduced Poisson algebra, and explores canonical parametrizations and quantization methods for these systems.

Contribution

It presents a novel reduction of the u^*(3) algebra using the Cartan model, along with new canonical coordinates and a schematic Hamiltonian for analyzing coadjoint orbits.

Findings

Derived the seven-dimensional Poisson algebra u_SO(3) from u^*(3).

Identified canonical parametrization of u^*(3) orbits [p_1,p_2,p_3].

Developed a reduced 4D system of equations for volume-conserving Hamiltonians.

Abstract

The Cartan model of SO(3)/SO(2) matrices is applied to reduce of rotational degrees of freedom on coadjoint orbits of u^*(3) Poisson algebra. The seven--dimensional Poisson algebra u_SO(3) obtained by SO(3) reduction of u^*(3) algebra is found and canonical parametrization of u^*(3) orbits [p_1,p_2,p_3] is studied. The structure of bands formed by so--called families of S and P ellipsoids obtained by searching extremes of many--body SO(3) invariant Hamiltonians is investigated. The reduced four--dimensional system of equations of motion describing the simple schematic Hamiltonian based on the volume conservation is presented. A new set of canonical coordinates regarding the separation of motion for independent modes is found with the help of the Jacobi approach. Bohr Somerfield's quantization of new momentum space is studied.