Inequivalent quantizations of the N=3 Calogero model with scale and mirror-S3 symmetry

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 ' 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.

Zero-curvature condition in Calogero model

“…. < x N instead of whole space and demand that curve x k (t) belongs to that simplex, but careful analysis [14] shows that correct choice of boundary conditions for wave function when approaching to singular point leads to consistent quantization scheme. The good illustration is the N = 2 case, after excluding the center of mass, it reduces to the study of the 1-dimensional Schrödinger operator:…”

Supersymmetric many-particle quantum systems with inverse-square interactions

“…(i) Self-adjoint extensions. The self-adjoint extension of the rational Calogero model has been studied in [38,39]. The scale invariance of the rational Calogero model with purely inversesquare interaction gets broken at the quantum level due to the imposition of modified boundary conditions and the Hamiltonian admits bound states.…”

“…The study of Calogero-Moser-Sutherland systems have produced many interesting results which are relevant in the context of a diverse branche of physics, including exclusion statistics [18,19], quantum Hall effect [20], Tomonaga-Luttinger liquid [21], quantum chaos [22], electric transport in mesoscopic systems [23], novel correlations [24], spin-chains [25][26][27][28][29][30][31][32][33], etc. These developments are also important in the context of mathematical physics, for example, algebraic and integrable structure [34][35][36], mapping of rational model to Calogero model with Coulomb-like potential [37], self-adjoint extensions [38,39], equivalence to a system of free oscillators [40,41], collective field formulation of many-particle systems [42], etc.…”

Discrete scale-invariant boson-fermion duality in one dimension

“…In contrast, there is no such constraint for distinguishable particles, meaning that there are much more variety of two-body contact interactions for nonidentical particles than those for identical particles. In fact, in one spatial dimension, the most general two-body contact interaction consistent with the probability conservation (unitarity) is known to be described by a 2 × 2 unitary matrix ∈ (2) [4,5], which contains dim (2) = 4 independent real parameters. However, scale-invariance breaking under this (2) family of two-body interactions has never been studied in the literature.…”