The classical dynamic symmetry for the U(1)-Kepler problems

Sofiane Bouarroudj, Guowu Meng

Research output: Contribution to journalArticlepeer-review

Abstract

For the Jordan algebra of hermitian matrices of order n≥2, we let X be its submanifold consisting of rank-one semi-positive definite elements. The composition of the cotangent bundle map πX: TX→X with the canonical map X→CPn−1 (i.e., the map that sends a given hermitian matrix to its column space), pulls back the Kähler form of the Fubini–Study metric on CPn−1 to a real closed differential two-form ωK on TX. Let ωX be the canonical symplectic form on TX and μ a real number. A standard fact says that ωμ≔ωX+2μωK turns TX into a symplectic manifold, hence a Poisson manifold with Poisson bracket {,}μ. In this article we exhibit a Poisson realization of the simple real Lie algebra su(n,n) on the Poisson manifold (TX,{,}μ), i.e., a Lie algebra homomorphism from su(n,n) to C(TX,R),{,}μ. Consequently one obtains the Laplace–Runge–Lenz vector for the classical U(1)-Kepler problem of level n and magnetic charge μ. Since the McIntosh–Cisneros–Zwanziger–Kepler problems (MICZ-Kepler Problems) are the U(1)-Kepler problems of level 2, the work presented here is a direct generalization of the work by A. Barut and G. Bornzin (1971) on the classical dynamic symmetry for the MICZ-Kepler problems.

Original languageEnglish (US)
Pages (from-to)1-15
Number of pages15
JournalJournal of Geometry and Physics
Volume124
DOIs
StatePublished - Jan 2018

Keywords

  • Dynamic symmetry
  • Jordan algebra
  • Kepler problem
  • Laplace–Runge–Lenz vector

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Geometry and Topology

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