Explicit Solution to the N-Body Calogero Problem

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.Comment: 6 p., latex, USITP-92-