The MIC-Kepler problem and its symmetry group for zero energy both in classical and quantum mechanics

Summary

The symmetry of the Kepler problem has been well known in classical as well as quantum mechanics on the level of Lie algebra, while little is known of global symmetry. In previous papers[1, 2], the MIC-Kepler problem was introduced, which is the Kepler problem along with a centrifugal potential and Dirac’s monopole field. This system was shown, in the negative energy case, to admit the same symmetry groupSO(4) as the Kepler problem does in classical theory, and to carry all the irreducible representations ofSU(2)×SU(2), the double cover ofSO(4), in qunatum theory. This paper is a continuation of the previous ones and intended for the study of the symmetry group in the zero-energy case. In classical theory, the symmetry group of the MIC-Kepler problem of zero energy proves to be a semi-direct product groupR ³⋉SO(3) acting on the zero-energy manifold diffeomorphic toR ³×S ². In quantum theory, the quantized MIC-Kepler problem with zeroenergy, assigned by an integerm, turns out to carry a unitary irreducible representation ofR ³⋉SU(2), the double cover ofR ³⋉SO(3), in a Hilbert space, which is isomorphic with the space ofL ²-cross-sections in the complex line bundleL _m associated with the principalS ¹ bundleS ³→S ². These representations ofR ³⋉SU(2), assigned bym, are equivalent to the Mackey’s induced representations ofR ³⋉SU(2).