Bosonization of non-relativstic fermions in 2-dimensions and collective field theory

TL;DR

This paper revisits the bosonization of non-relativistic fermions in one dimension, deriving a collective field theory action that captures boundary dynamics and connects to classical and quantum descriptions, with exact results for harmonic potentials.

Contribution

It introduces a boundary-invariant collective field action for non-relativistic fermions, linking classical boundary dynamics to Hamiltonian equations and providing exact quantization for harmonic oscillator potentials.

Findings

Classical boundary dynamics follow Hamilton's equations.

Quantization is exact for harmonic oscillator potential.

Quantum collective theory differs from noncommutative bosonization for finite fermion numbers.

Abstract

We revisit bosonization of non-relativistic fermions in one space dimension. Our motivation is the recent work on bubbling half-BPS geometries by Lin, Lunin and Maldacena (hep-th/0409174). After reviewing earlier work on exact bosonization in terms of a noncommutative theory, we derive an action for the collective field which lives on the droplet boundaries in the classical limit. Our action is manifestly invariant under time-dependent reparametrizations of the boundary. We show that, in an appropriate gauge, the classical collective field equations imply that each point on the boundary satisfies Hamilton's equations for a classical particle in the appropriate potential. For the harmonic oscillator potential, a straightforward quantization of this action can be carried out exactly for any boundary profile. For a finite number of fermions, the quantum collective field theory does not reproduce the results of the exact noncommutative bosonization, while the latter are in complete agreement with the results computed directly in the fermi theory.