$W_\infty$ coherent states and path-integral derivation of bosonization of non-relativistic fermions in one dimension

TL;DR

This paper derives the bosonized action for nonrelativistic fermions in one dimension using $W_$ coherent states within a path-integral framework, completing previous proofs and clarifying the classical limit.

Contribution

It provides a path-integral derivation of bosonization for 1D fermions using $W_$ coherent states, complementing earlier coadjoint orbit methods.

Findings

Derived the bosonized action via path-integral with $W_$ states

Clarified the classical limit and truncation to collective field theory

Connected the derivation to previous coadjoint orbit results

Abstract

We complete the proof of bosonization of noninteracting nonrelativistic fermions in one space dimension by deriving the bosonized action using $W_\infty$ coherent states in the fermion path-integral. This action was earlier derived by us using the method of coadjoint orbits. We also discuss the classical limit of the bosonized theory and indicate the precise nature of the truncation of the full theory that leads to the collective field theory.