Explicit solution to the N-body Calogero problem

Abstract We solve the N -body Calogero problem, i.e., N particles in one dimension subject to a two-body interaction of the form 1 2 Σ i,j [(x i −x j ) 2 + g (x i −x j ) 2 ] , by constructing annihilation and creation operators of the form ⊣ 1 = ( 1 √2 )(x i ± i p i ) , where 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 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.