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

TL;DR

This paper explores the different possible quantizations of the N=3 Calogero model that respect mirror-S_3 symmetry and scale invariance, revealing a two-parameter family of novel angular quantizations and classifying eigenstates by S_3 representations.

Contribution

It introduces a two-parameter family of inequivalent quantizations for the angular part of the N=3 Calogero model respecting specific symmetries and classifies eigenstates accordingly.

Findings

Identifies a two-parameter family of angular quantizations.

Classifies eigenstates using S_3 group representations.

Eigenvalues for singlet states are universal, doublets depend on parameters.

Abstract

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.