Inequivalent Quantizations of the N = 3 Calogero model with Scale and Mirror-S_3 Symmetry

Details Inequivalent Quantizations of the N = 3 Calogero model with Scale and Mirror-S_3 Symmetry Inequivalent Quantizations of the N = 3 Calogero model with Scale and Mirror-S_3 Symmetry

2005-10-13

arxiv.org/abs/hep-th/0510106v2

J.Math.Phys. 47 (2006) 012104

Physics

High Energy Physics

High Energy Physics - Theory

24 pages, LaTeX

Scientific paper

10.1063/1.2162821

We study the inequivalent quantizations of the N = 3 Calogero model by separation of variables, in which the model decomposes into the angular and the radial parts. Our inequivalent quantizations respect the ` mirror-S_3\rq\ invariance (which realizes the symmetry under the cyclic permutations of the particles) and the scale invariance in the limit of vanishing harmonic potential. We find a two-parameter family of novel quantizations in the angular part and classify the eigenstates in terms of the irreducible representations of the S_3 group. The scale invariance restricts the quantization in the radial part uniquely, except for the eigenstates coupled to the lowest two angular levels for which two types of boundary conditions are allowed independently from all upper levels. It is also found that the eigenvalues corresponding to the singlet representations of the S_3 are universal (parameter-independent) in the family, whereas those corresponding to the doublets of the S_3 are dependent on one of the parameters. These properties are shown to be a consequence of the spectral preserving SU(2) (or its subrgoup U(1)) transformations allowed in the family of inequivalent quantizations.

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