TY - JOUR
T1 - The classical dynamic symmetry for the U(1)-Kepler problems
AU - Bouarroudj, Sofiane
AU - Meng, Guowu
N1 - Funding Information:
The authors were supported by the Hong Kong Research Grants Council under RGC Project No. 16304014 ; SB was also supported by the grant NYUAD-065.
Publisher Copyright:
© 2017 Elsevier B.V.
PY - 2018/1
Y1 - 2018/1
N2 - 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: T∗X→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 T∗X. Let ωX be the canonical symplectic form on T∗X and μ a real number. A standard fact says that ωμ≔ωX+2μωK turns T∗X 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 (T∗X,{,}μ), i.e., a Lie algebra homomorphism from su(n,n) to C∞(T∗X,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.
AB - 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: T∗X→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 T∗X. Let ωX be the canonical symplectic form on T∗X and μ a real number. A standard fact says that ωμ≔ωX+2μωK turns T∗X 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 (T∗X,{,}μ), i.e., a Lie algebra homomorphism from su(n,n) to C∞(T∗X,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.
KW - Dynamic symmetry
KW - Jordan algebra
KW - Kepler problem
KW - Laplace–Runge–Lenz vector
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U2 - 10.1016/j.geomphys.2017.10.012
DO - 10.1016/j.geomphys.2017.10.012
M3 - Article
AN - SCOPUS:85033570723
VL - 124
SP - 1
EP - 15
JO - Journal of Geometry and Physics
JF - Journal of Geometry and Physics
SN - 0393-0440
ER -