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 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 nu, is obtained. Here, nu is an element of W(V) := {k delta/2 vertical bar k = 1, ... , (rho - 1)} boolean OR ((rho - 1)delta/2, infinity) 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/(I+nu rho/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: 12 parallel to x parallel to(2) + 1/r. Here, (x) over dot 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. (C) 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4775343