Generalized Kepler problems. I. Without magnetic charges

Abstract

For each simple euclidean Jordan algebra V of rank ρ and degree δ, we introduce a family of classical dynamic problems. These dynamical problems all share the characteristic features of the Kepler problem for planetary motions, such as the existence of Laplace-Runge-Lenz vector and hidden symmetry. After suitable quantizations, a family of quantum dynamic problems, parametrized by the nontrivial Wallach parameter ν, is obtained. Here, $\nu \in {\mathcal {W}}(V):=\lbrace k \frac{\delta}{2}\mid k=1, \ldots , (\rho -1)\rbrace \cup ((\rho -1)\frac{\delta }{ 2}, \infty )$ν∈W(V):={kδ2∣k=1,...,(ρ−1)}∪((ρ−1)δ2,∞) and was introduced by N. Wallach to parametrize the set of nontrivial scalar-type unitary lowest weight representations of the conformal group of V. For the quantum dynamic problem labelled by ν, the bound state spectra is $-\frac{1/2}{ (I+\nu \frac{\rho }{ 2})^2}$−1/2(I+νρ2)2, I = 0, 1, … and its Hilbert space of bound states gives a new realization for the afore-mentioned representation labelled by ν. A few results in the literature about these representations become more explicit and more refined. The Lagrangian for a classical Kepler-type dynamic problem introduced here is still of the simple form: $\frac{1}{ 2} \Vert \dot{x}\Vert ^2+\frac{1}{ r}$12‖ẋ‖2+1r. Here, $\dot{x}$ẋ is the velocity of a unit-mass particle moving on the space consisting of V’s semi-positive elements of a fixed rank, and r is the inner product of x with the identity element of V.