Symbol calculus for the Kepler problem

TL;DR

This paper develops a quantum-classical correspondence for the Kepler problem using generalized coherent states on a specific homogeneous domain, linking quantum observables to classical phase space functions.

Contribution

It constructs a system of coherent states for the quantum Kepler problem on a homogeneous domain and demonstrates the classical limit via a momentum map and observable averages.

Findings

Quantum observables' averages tend to classical functions.

Commutators scaled by -i/h approach Poisson brackets.

The quantum system's structure aligns with classical Kepler dynamics.

Abstract

We construct the system of generalized coherent states for the quantum Kepler problem corresponds to the homogeneous domain $SU(2,2)/S(U(2)\times U(2))$. We show that the SU(2,2)-equivariant momentum map for this domain yields the momentum map for the classical Kepler problem via appropriate limiting passage. We also show that under this passage the average values of quantum observables in this system of coherent states pass into the functions on classical phase space and $-i/h$ times commutator pass into the Poisson bracket.