The classical dynamic symmetry for the $\mathrm{U}(1)$-Kepler problems

Abstract

For the Jordan algebra of hermitian matrices of order $n\ge 2$, we let $X$ be its submanifold consisting of rank-one semi-positive definite elements. The composition of the cotangent bundle map ${\pi }_X$: $T^{\ast }X\to X$ with the canonical map $X\to \mathbb{C}{P}^{n-1}$ (i.e., the map that sends a hermitian matrix to its column space), pulls back the Kähler form of the Fubini-Study metric on $\mathbb{C}{P}^{n-1}$ to a real closed differential two-form ${\omega }_K$ on $T^{\ast }X$. Let ${\omega }_X$ be the canonical symplectic form on $T^{\ast }X$ and $\mu$ be a real number. A standard fact says that ${\omega }_{\mu }:={\omega }_X+2\mu {\omega }_K$ turns $T^{\ast }X$ into a symplectic manifold, hence a Poisson manifold with Poisson bracket $\{,{\}}_{\mu }$. In this article we exhibit a Poisson realization of the simple real Lie algebra $\mathfrak{su}(n,n)$ on the Poisson manifold $(T^{\ast }X,\{,{\}}_{\mu })$, i.e., a Lie algebra homomorphism from $\mathfrak{su}(n,n)$ to $\left(C^{\infty }(T^{\ast }X,\mathbb{R}),\{,{\}}_{\mu }\right)$. Consequently one obtains the Laplace-Runge-Lenz vector for the classical $\mathrm{U}(1)$-Kepler problem with level $n$ and magnetic charge $\mu$. Since the McIntosh-Cisneros-Zwanziger-Kepler problems (MICZ-Kepler Problems) are the $\mathrm{U}(1)$-Kepler problems with level $2$, the work presented here is a direct generalization of the work by A. Barut and G. Bornzin [ J. Math. Phys. $12$ (1971), 841-843] on the classical dynamic symmetry for the MICZ-Kepler problems.

19 pages

References in corpus (6)

• Euclidean Jordan Algebras, Hidden Actions, and $J$-Kepler Problems
• Generalized Kepler Problems I: Without Magnetic Charges
• The Poisson Realization of so(2, 2k+2) on Magnetic Leaves
• The Universal Kepler Problem
• Lorentz Group and Oriented MICZ-Kepler Orbits
• The classical dynamic symmetry for the $\mathrm{Sp}(1)$-Kepler problems

Cited by in corpus (1)

• Perturbed $(2n-1)$-Dimensional Kepler Problem and the Nilpotent Adjoint Orbits of $U(n,n)$