Bosonization of Nonrelativistic Fermions and W -infinity Algebra

We discuss the bosonization of non-relativistic fermions in one space dimension in terms of bilocal operators which are naturally related to the generators of W infinity algebra. The resulting system is analogous to the problem of a spin in a magnetic field for the group W -infinity. The new dynamical variables turn out to be W -infinity group elements valued in the coset W -infinity/H where H is a Cartan subalgebra. A classical action with an H gauge invariance is presented. This action is three-dimensional. It turns out to be similiar to the action that describes the colour degrees of freedom of a Yang-Mills particle in a fixed external field. We also discuss the relation of this action with the one we recently arrived at in the Euclidean continuation of the theory using different coordinates.