Bosonization of Nonrelativistic Fermions and W-infinity Algebra

TL;DR

This paper explores the bosonization of non-relativistic fermions in one dimension using bilocal operators linked to the W-infinity algebra, introducing a new three-dimensional action with gauge invariance.

Contribution

It presents a novel bosonization approach based on W-infinity algebra and formulates a new three-dimensional gauge-invariant action for the system.

Findings

Establishes a connection between bilocal operators and W-infinity algebra generators.

Derives a classical action with H gauge invariance for the bosonized system.

Relates the new action to previous Euclidean formulations.

Abstract

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.