Generalized Kepler Problems I: Without Magnetic Charges

TL;DR

This paper generalizes the classical Kepler problem to a broad family of dynamical systems associated with simple Euclidean Jordan algebras, introducing new quantum and classical models with rich symmetry properties and explicit spectral results.

Contribution

It introduces a unified framework for Kepler-type problems linked to Jordan algebras, providing new classical and quantum models, explicit spectra, and connections to representation theory.

Findings

Quantum bound state spectra are explicitly derived.

Hilbert spaces realize scalar-type unitary lowest weight representations.

Classical Lagrangian retains simple Kepler form on Jordan algebra spaces.

Abstract

For each simple euclidean Jordan algebra $V$ of rank $\rho$ and degree $\delta$, we introduce a family of classical dynamic problems. These dynamical problems all share the characteristic features of the Kepler problem for planetary motions, such as existence of Laplace-Runge-Lenz vector and hidden symmetry. After suitable quantizations, a family of quantum dynamic problems, parametrized by the nontrivial Wallach parameter $\nu$, is obtained. Here, $\nu\in{\mathcal W}(V):=\{k {\delta\over 2}\mid k=1, ..., (\rho-1)\}\cup((\rho-1){\delta\over 2}, \infty)$ and was introduced by N. Wallach to parametrize the set of nontrivial scalar-type unitary lowest weight representations of the conformal group of $V$. For the quantum dynamic problem labelled by $\nu$, the bound state spectra is $-{1/2\over (I+\nu{\rho\over 2})^2}$, I=0, 1, ... and its Hilbert space of bound states gives a new realization for the afore-mentioned representation labelled by $\nu$. A few results in the literature about these representations become more explicit and more refined. The Lagrangian for a classical Kepler-type dynamic problem introduced here is still of the simple form: ${1\over 2} ||\dot x||^2+{1\over r}$. Here, $\dot x$ is the velocity of a unit-mass particle moving on the space consisting of $V$'s semi-positive elements of a fixed rank, and $r$ is the inner product of $x$ with the identity element of $V$.