The MIC-Kepler problem and its symmetry group for positive energies both in classical and quantum mechanics

Abstract

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, 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- (respectively, zero-) energy cases, to admit the same symmetry groupSO(4) (respectively, the Euclidean motion groupE(3)) as the Kepler problem does in classical theory, and to carry all the irreducible representations ofSU(2)×SU(2) (respectively,R3blSU(2)), the double cover ofSO(4) (respectively,E(3)), in quantum theory. This paper is a continuation of the previous ones and intended for the study of the symmetry group in the positive-energy case. In classical theory, the symmetry group of the MIC-Kepler problem of positive energy proves to be the proper Lorentz groupSO+ (1,3) acting on the positive-energy manifold diffeomorphic toR3×S2. In quantum theory, the quantised MIC-Kepler problem with positive energy, assigned by a positive numbers and an integerm, turns out to carry a unitary irreducible representation ofSL(2,C), the double cover ofSO+(1,3), in a Hilbert space, which is isomorphic with the space ofL2-cross-sections in the complex line bundleLm associated with the principalS1 bundleS3→S2. These representations ofSL(2,C) are equivalent to the principal series representations ofSL(2,C).