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

Avinash Dhar

TL;DR

The paper shows that free 2D non-relativistic fermions admit an exact bosonization in terms of a noncommutative Wigner phase-space density with $u*u=u$ and dynamics $ rac{∂u}{∂t}= rac{1}{i\hbar}(h*u-u*h)$. In the classical limit, the collective boundary action describes droplets whose dynamics are governed by area-preserving diffeomorphisms, with the boundary degrees of freedom obeying Hamilton’s equations for the single-particle Hamiltonian; this is exactly solvable for the harmonic oscillator potential. Quantization of the boundary theory reproduces the fermion spectrum only in the limit $N\to\infty$; for finite $N$, the collective theory fails to reproduce the correct spectrum or the detailed phase-space density, unless an ad hoc cutoff at $N$ is imposed. The results emphasize the gap between exact noncommutative bosonization and the emergent collective boundary theory, suggesting future directions toward corrected actions or finite-$N$ oscillator formulations and highlighting potential links to quantum gravity in the AdS/CFT context.

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.

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

TL;DR

The paper shows that free 2D non-relativistic fermions admit an exact bosonization in terms of a noncommutative Wigner phase-space density with and dynamics . In the classical limit, the collective boundary action describes droplets whose dynamics are governed by area-preserving diffeomorphisms, with the boundary degrees of freedom obeying Hamilton’s equations for the single-particle Hamiltonian; this is exactly solvable for the harmonic oscillator potential. Quantization of the boundary theory reproduces the fermion spectrum only in the limit ; for finite , the collective theory fails to reproduce the correct spectrum or the detailed phase-space density, unless an ad hoc cutoff at is imposed. The results emphasize the gap between exact noncommutative bosonization and the emergent collective boundary theory, suggesting future directions toward corrected actions or finite- oscillator formulations and highlighting potential links to quantum gravity in the AdS/CFT context.

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.

Paper Structure

This paper contains 16 sections, 58 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of exact bosonization
  3. Bosonization in terms of Wigner density
  4. The role of W$_\infty$ symmetry
  5. The action of the bosonized theory
  6. Consistency of bosonization
  7. Classical limit and collective action
  8. Classical motion is area-preserving diffeomorphisms
  9. The boundary action
  10. Boundary reparametrizations
  11. Gauge-fixing boundary reparametrizations
  12. Conserved charges and area-preserving diffeomorphisms
  13. Quantization of the collective theory
  14. Boundary profiles with tangent nowhere radially directed
  15. Boundary profiles with radially directed tangents
  16. ...and 1 more sections

Figures (3)

  • Figure 1: (a) A simple configuration with a single connected filled region $\Sigma$ in phase space. The droplet is centered at the origin in the phase plane. (b) A general configuration of several disconnected fluid droplets on the phase plane.
  • Figure 2: (a) A fluid boundary profile with radially directed tangents at two points. (b) A fluid boundary profile with a radially directed tangent at one point which is also a point of inflexion. Both droplets are centered at the origin O of the phase plane.
  • Figure 3: (a) A fluid boundary profile with just the two bounding radially directed tangents. (b) A fluid boundary profile with radially directed tangents other than the two bounding tangents. The origin O of the phase plane is outside both droplets.