Lorentz Group and Oriented MICZ-Kepler Orbits

TL;DR

This paper explores the geometric and algebraic structures of MICZ-Kepler orbits, revealing how they are interconnected through Lorentz group symmetries and a novel 4D parametrization, advancing understanding of these magnetized orbital systems.

Contribution

It introduces a new 4D Minkowski space parametrization of MICZ-Kepler orbits, linking different magnetic charges via Lorentz symmetries and extending the orbit classification.

Findings

Lorentz group acts transitively on elliptic and parabolic MICZ-Kepler orbits.

New Minkowski parametrization relates orbits of different magnetic charges.

Symmetry group extends to structure group of Euclidean Jordan algebra.

Abstract

The MICZ-Kepler orbits are the non-colliding orbits of the MICZ Kepler problems (the magnetized versions of the Kepler problem). The oriented MICZ-Kepler orbits can be parametrized by the canonical angular momentum $\mathbf L$ and the Lenz vector $\mathbf A$, with the parameter space consisting of the pairs of 3D vectors $(\mathbf A, \mathbf L)$ with ${\mathbf L}\cdot {\mathbf L} > (\mathbf L\cdot \mathbf A)^2$. The recent 4D perspective of the Kepler problem yields a new parametrization, with the parameter space consisting of the pairs of Minkowski vectors $(a,l)$ with $l\cdot l =-1$, $a\cdot l =0$, $a_0>0$. This new parametrization of orbits implies that the MICZ-Kepler orbits of different magnetic charges are related to each other by symmetries: \emph{${\mathrm {SO}}^+(1,3)\times {\mathbb R}_+$ acts transitively on both the set of oriented elliptic MICZ-Kepler orbits and the set of oriented parabolic MICZ-Kepler orbits}. This action extends to ${\mathrm {O}}^+(1,3)\times {\mathbb R}_+$, the \emph{structure group} for the rank-two Euclidean Jordan algebra whose underlying Lorentz space is the Minkowski space.