The Conformal Group SU(2,2) and Integrable Systems on a Lorentzian Hyperboloid

TL;DR

This paper classifies eleven maximally superintegrable Hamiltonian systems on a Lorentzian hyperboloid, derived from the conformal group SU(2,2), and demonstrates their solvability via separation of variables.

Contribution

It introduces a new classification of superintegrable systems on a Lorentzian hyperboloid using the conformal group SU(2,2) and explores their integrals of motion and solvability.

Findings

Eleven superintegrable Hamiltonian systems identified.

All systems derive from a free Hamiltonian on a homogeneous space.

Classical and quantum equations solvable by separation of variables.

Abstract

Eleven different types of "maximally superintegrable" Hamiltonian systems on the real hyperboloid $(s^0)^2-(s^1)^2+(s^2)^2-(s^3)^2=1$ are obtained. All of them correspond to a free Hamiltonian system on the homogeneous space $SU(2,2)/U(2,1)$, but to reductions by different maximal abelian subgroups of $SU(2,2)$. Each of the obtained systems allows 5 functionally independent integrals of motion, from which it is possible to form two or more triplets in involution (each of them includes the hamiltonian). The corresponding classical and quantum equations of motion can be solved by separation of variables on the $O(2,2)$ space.