Explicit Solution to the N-Body Calogero Problem

TL;DR

This paper provides an explicit solution to the N-body Calogero problem by constructing specialized operators, linking the model to fractional statistics and anyons, and offering insights into its quantum structure.

Contribution

The paper introduces a novel explicit operator-based solution to the Calogero model, connecting it to fractional statistics and covariant derivatives.

Findings

Explicit construction of creation and annihilation operators for the Calogero model

Identification of a covariant derivative interpretation of momentum operators

Discussion of the relation to fractional statistics and anyons

Abstract

We solve the N-body Calogero problem, \ie N particles in 1 dimension subject to a two-body interaction of the form $\half \sum_{i,j}[ (x_i - x_j)^2 + g/ {(x_i - x_j)^2}]$, by constructing annihilation and creation operators of the form $ a_i^\mp =\frac 1 {\sqrt 2} (x _i \pm i\hat{p}_i )$, where $\hat{p}_i$ is a modified momentum operator obeying %!!!!!!! Heisenberg-type commutation relations with $x_i$, involving explicitly permutation operators. On the other hand, $ D_j =i\,\hat{p}_j$ can be interpreted as a covariant derivative corresponding to a flat connection. The relation to fractional statistics in 1+1 dimensions and anyons in a strong magnetic field is briefly discussed.