Generalized MICZ-Kepler Problems and Unitary Highest Weight Modules

TL;DR

This paper establishes a new dynamical symmetry for generalized MICZ-Kepler problems in odd dimensions, showing their bound states form specific unitary highest weight modules of a certain spin group, with a geometric realization.

Contribution

It introduces a novel $ ext{Spin}(2, 2n+2)$ symmetry for generalized MICZ-Kepler problems and characterizes their bound states as unitary highest weight modules.

Findings

Identifies $ ext{Spin}(2, 2n+2)$ as a dynamical symmetry group.

Shows bound states form a specific unitary highest weight module.

Provides a geometric realization of these modules.

Abstract

For each integer $n\ge 1$, we demonstrate that a $(2n+1)$-dimensional generalized MICZ-Kepler problem has an $\mr{Spin}(2, 2n+2)$ dynamical symmetry which extends the manifest $\mr{Spin}(2n+1)$ symmetry. The Hilbert space of bound states is shown to form a unitary highest weight $\mr{Spin}(2, 2n+2)$-module which occurs at the first reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. As a byproduct, we get a simple geometric realization for such a unitary highest weight $\mr{Spin}(2, 2n+2)$-module.