Calogero models and nonlocal conformal transformations

Abstract

We propose a universal method of relating the Calogero model to a set of decoupled particles on the real line, which can be uniformly applied to both the conformal and nonconformal versions as well as to supersymmetric extensions. For conformal models the simplification is achieved at the price of a nonlocal realization of the full conformal symmetry in the Hilbert space of the resulting free theory. As an application, we construct two different N=2 superconformal extensions.Comment: 1+9 pages, v2: tex macros fixe

Remark on quantum mechanics with conformal Galilean symmetry

“…The purpose of this note is to generalize the results previously obtained for one-dimensional systems [19] to higher dimensions, i.e. to the case of the conformal Galilei algebra.…”

“…An interesting peculiarity of the Calogero model with the oscillator potential is that it can be transformed into a set of decoupled oscillators by applying an appropriate similarity transformation [18,19], the fact anticipated by Calogero in [20]. In particular, this explains why the spectra of the Calogero model and the decoupled oscillators are so alike.…”

“…It also provided an efficient means for constructing various N = 2 and N = 4 superconformal many-body models in one dimension [19,22,23,24]. An elegant geometric interpretation of the similarity transformation as the inversion of the Klein model of the Lobachevsky plane was proposed in [25].…”

Hierarchy of mechanics models

“…the integrability condition of which is the equation 14) which is just the dimension-7 Laplace equation (3.7) for the considered particular case. It is of interest to present a few first polynomial solutions of this equation, together with the relevant conformal factorsG (7) = 4(xL xx + 2L x ) = −4(yL yy + 3 2 L y ):…”

“…[4]), and as such provide a simplified setting for studying salient features of these theories. Other models of this kind represent supersymmetric extensions of certain intrinsically one-dimensional systems, like conformal mechanics [5]- [10], [2], or Calogero-Moser integrable models [11]- [14]. From the mathematical point of view, extended d=1 supersymmetry, as compared with its higher-dimensional counterpart, exhibits rather unusual features, such as the so-called automorphic duality between multiplets with the same number of fermions but with different distributions of the bosons to the physical and the auxiliary sector [15,16,17].…”