The Universal Kepler Problem

TL;DR

This paper generalizes the classical Kepler problem using Euclidean Jordan algebras, introducing universal dynamical vectors within the TKK algebra framework, and explores classical and quantum realizations.

Contribution

It introduces universal Hamiltonian, angular momentum, and Laplace-Runge-Lenz vectors for all simple Euclidean Jordan algebras, extending Kepler problem symmetries.

Findings

Defined universal dynamical vectors in TKK algebra

Constructed classical generalized Kepler problems

Provided quantum realizations for the simplest case

Abstract

For each simple euclidean Jordan algebra $V$, we introduce the analogue of hamiltonian, angular momentum and Laplace-Runge-Lenz vector in the Kepler problem. Being referred to as the universal hamiltonian, universal angular momentum and universal Laplace-Runge-Lenz vector respectively, they are elements in (essentially) the TKK (Tits-Kantor-Koecher) algebra of $V$ and satisfy commutation relations similar to the ones for the hamiltonian, angular momentum and Laplace-Runge-Lenz vector in the Kepler problem. We also give some examples of Poisson realization of the TKK algebra, along with the resulting classical generalized Kepler problems. For the simplest simple euclidean Jordan algebra (i.e., $\mathbb R$), we give examples of operator realization for the TKK algebra, along with the resulting quantum generalized Kepler problems.