Quantum effective dynamics of the Calogero-Sutherland model

We consider the quantum dynamics described by the effective field theory of the Calogero-Sutherland model in terms of fields that carry representations of the W-infinity algebra. Our formulation of the effective theory is based in the standard procedure in conformal field theory, in which the Hilbert space is completely known a priori in terms of irreducible representations of the symmetry algebra, and not by the diagonalization of the hamiltonian. For the purposes of comparison with alternative effective theories based on quantum hydrodynamics, we consider one chirality and study the hamiltonian equation of motion of the density field, which is shown to obey the quantum Benjamin-Ono equation, in agreement with previous hydrodynamic results. In this study, all calculations are performed at the quantum operator level, without making any assumption on the semiclassical limit of the fields and their equations of motion. We consider this result to be a reliable indication of the equivalence between the quantum field theoric and quantum hydrodynamical formulations of the effective theories of the Calogero-Sutherland model.