Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exact operator bosonization of finite number of fermions in one space dimension

Avinash Dhar, Gautam Mandal, Nemani V Suryanarayana

TL;DR

This work presents an exact operator bosonization for a finite number N of fermions in one spatial dimension, establishing two complementary bosonic pictures that obey the Heisenberg algebra. The finitely ranked single-particle Hilbert space in the first bosonization yields a fuzzy, noncommutative, and lattice-like phase space, with ultraviolet cutoffs arising naturally from N. The second bosonization uses a dual bosonic system with a fixed total number of bosons, providing a parallel, dual description that aligns with giant and dual giant gravitons in holographic settings. The formalism is applied to the half-BPS sector of AdS/CFT via LLM geometries and to the c=1 matrix model, highlighting finite-N graininess and potential nonperturbative gravity realizations. Together, these results suggest that finite bulk mode counts induce noncommutative geometry and offer a concrete operator framework connecting fermionic systems to gravitationally relevant bosonic degrees of freedom. The approach opens pathways to higher-dimensional extensions and deeper connections between microscopic fermionic descriptions and bulk gravity phenomena.

Abstract

We derive an exact operator bosonization of a finite number of fermions in one space dimension. The fermions can be interacting or noninteracting and can have an arbitrary hamiltonian, as long as there is a countable basis of states in the Hilbert space. In the bosonized theory the finiteness of the number of fermions appears as an ultraviolet cut-off. We discuss implications of this for the bosonized theory. We also discuss applications of our bosonization to one-dimensional fermion systems dual to (sectors of) string theory such as LLM geometries and c=1 matrix model.

Exact operator bosonization of finite number of fermions in one space dimension

TL;DR

This work presents an exact operator bosonization for a finite number N of fermions in one spatial dimension, establishing two complementary bosonic pictures that obey the Heisenberg algebra. The finitely ranked single-particle Hilbert space in the first bosonization yields a fuzzy, noncommutative, and lattice-like phase space, with ultraviolet cutoffs arising naturally from N. The second bosonization uses a dual bosonic system with a fixed total number of bosons, providing a parallel, dual description that aligns with giant and dual giant gravitons in holographic settings. The formalism is applied to the half-BPS sector of AdS/CFT via LLM geometries and to the c=1 matrix model, highlighting finite-N graininess and potential nonperturbative gravity realizations. Together, these results suggest that finite bulk mode counts induce noncommutative geometry and offer a concrete operator framework connecting fermionic systems to gravitationally relevant bosonic degrees of freedom. The approach opens pathways to higher-dimensional extensions and deeper connections between microscopic fermionic descriptions and bulk gravity phenomena.

Abstract

We derive an exact operator bosonization of a finite number of fermions in one space dimension. The fermions can be interacting or noninteracting and can have an arbitrary hamiltonian, as long as there is a countable basis of states in the Hilbert space. In the bosonized theory the finiteness of the number of fermions appears as an ultraviolet cut-off. We discuss implications of this for the bosonized theory. We also discuss applications of our bosonization to one-dimensional fermion systems dual to (sectors of) string theory such as LLM geometries and c=1 matrix model.

Paper Structure

This paper contains 35 sections, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The First Bosonization
  3. The bosonization formulae
  4. Fermionic representation of bosonic oscillators
  5. Fermion bilinears $\psi^\dagger_m\psi_n$ in terms of bosonic oscillators
  6. The bosonized hamiltonian
  7. Interacting fermion models
  8. General properties of the bosonized theory
  9. Single particle: Quantum phase space is fuzzy and compact
  10. Second quantization: the bosonic phase space density
  11. Lattice interpretation
  12. More on Second quantization
  13. Action for $W_B$
  14. Application of our bosonization to LLM
  15. Giant graviton phase space density
  16. ...and 20 more sections

Figures (1)

  • Figure 1: The bosonic phase space density $\langle W_B(x,p) \rangle$, in the state (\ref{['bose-state']}), as a function of the $(x,p)$ plane. In the LLM example, this corresponds to the density of giant gravitons in the $(x_1,x_2)$ plane.