Lorentz Group and Oriented MICZ-Kepler Orbits

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The MICZ-Kepler orbits are the non-colliding orbits of the MICZ Kepler problems (the magnetized versions of the Kepler problem). The oriented MICZ-Kepler orbits can be parametrized by the canonical angular momentum L and the Lenz vector A, with the parameter space consisting of the pairs of 3D vectors (A, L) with L· L > (L· A)². The recent 4D perspective of the Kepler problem yields a new parametrization, with the parameter space consisting of the pairs of Minkowski vectors (a,l) with l· l =-1, a· l =0, a₀>0. This new parametrization of orbits implies that the MICZ-Kepler orbits of different magnetic charges are related to each other by symmetries: SO+(1,3)× R+ acts transitively on both the set of oriented elliptic MICZ-Kepler orbits and the set of oriented parabolic MICZ-Kepler orbits. This action extends to O+(1,3)× R+, the structure group for the rank-two Euclidean Jordan algebra whose underlying Lorentz space is the Minkowski space.