The MIC-Kepler problem and its symmetry group for zero energy both in classical and quantum mechanics
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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 groupR3⋉SO(3) acting on the zero-energy manifold diffeomorphic toR3×S2. In quantum theory, the quantized MIC-Kepler problem with zeroenergy, assigned by an integerm, turns out to carry a unitary irreducible representation ofR3⋉SU(2), the double cover ofR3⋉SO(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 ofR3⋉SU(2), assigned bym, are equivalent to the Mackey’s induced representations ofR3⋉SU(2).
KeywordsPACS 02.20 Group theory PACS 02.40 Geometry differential geometry and topology PACS 03.20 Classical mechanics of discrete systems general mathematical aspects PACS 03.65 Quantum theory quantum mechanics
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- G. W. Mackey:Induced Representation (Benjamin-Editore Boringhieri, New York-Torino, 1968).Google Scholar