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On symmetry groups of the MIC-Kepler problem and their unitary irreducible representations. (English) Zbl 0816.22006

Shiohama, K. (ed.), Progress in differential geometry. Tokyo: Kinokuniya Company Ltd.. Adv. Stud. Pure Math. 22, 69-87 (1993).
The Hamiltonian system \((M,\omega_ \mu,H)\), where \(M = T^* R^ 3 \setminus \{0\}\), \[ \omega_ \mu = \sum_{i = 1}^ 3dp_ i \wedge dq_ i - {\mu \over 2| q|^ 3} \sum_{i,j,k = 1}^ 3 \varepsilon_{ijk} q_ i dq_ j \wedge dq_ k, \quad | q|^ 2 = q_ 1^ 2 + q_ 2^ 2 + q_ 3^ 2\text{ and }H = {| p|^ 2\over 2} - {\alpha\over | q|} + {\mu^ 2\over 2| q|^ 2} \] is known as MIC-Kepler problem and can be considered as one-parameter deformation family of the standard Kepler problem with the remarkable property of retaining its high geometric and dynamical symmetries. The deformation parameter is interpreted as the magnetic charge of the particle in rest. The (global) symmetry group of the problem is known to be either \(\text{SO}(4)\), \(\text{E}(3)\) or \(\text{SO}(3,1)\) depending on energy being negative, zero or positive. An interesting moment is that the Hilbert spaces associated with the quantized problem carry almost all unitary irreducible representations of the respective covering groups, the only exception being the \(\text{SL}(2,\mathbb{C})\) group for which solely principal series representations arise. These results are proved using reduction procedures. For completeness we should mention that \(\text{SL}(2,\mathbb{C})\) and \(\text{E}(3)\) representations associated with the MIC-Kepler problem have been used in the reviewer’s work [J. Phys. A, Math. Gen. 21, L1–L4 (1988)] for derivation of the differential cross section of charged particles scattering in a dyon’s field.
For the entire collection see [Zbl 0779.00011].
Reviewer: I.Mladenov (Sofia)

MSC:

22E70 Applications of Lie groups to the sciences; explicit representations
53D50 Geometric quantization
58H15 Deformations of general structures on manifolds
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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