Generalized Kepler Problems I: Without Magnetic Charges

For each simple euclidean Jordan algebra $V$ of rank $\rho$ and degree $\delta$, 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 existence of Laplace-Runge-Lenz vector and hidden symmetry. After suitable quantizations, a family of quantum dynamic problems, parametrized by the nontrivial Wallach parameter $\nu$, is obtained. Here, $\nu\in{\mathcal W}(V):=\{k {\delta\over 2}\mid k=1, ..., (\rho-1)\}\cup((\rho-1){\delta\over 2}, \infty)$ 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 $\nu$, the bound state spectra is $-{1/2\over (I+\nu{\rho\over 2})^2}$, I=0, 1, ... and its Hilbert space of bound states gives a new realization for the afore-mentioned representation labelled by $\nu$. 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: ${1\over 2} ||\dot x||^2+{1\over r}$. Here, $\dot x$ 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 $V$.