Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:hep-th/0502150 · Physical Review Letters

Bosonization in Higher Dimensions via Noncommutative Field Theory

Alexios P. Polychronakos

Published May 8, 2006• 11 citations
arXivHTMLPDF

Abstract

We propose the bosonization of a many-body fermion theory in D spatial dimensions through a noncommutative field theory on a (2D-1)-dimensional space. This theory leads to a chiral current algebra over the noncommutative space and reproduces the correct perturbative Hilbert space and excitation energies for the fermions. The validity of the method is demonstrated by bosonizing a two-dimensional gas of fermions in a harmonic trap.

DOI
10.1103/physrevlett.96.186401

Related papers

Current Algebra and Bosonization in Three Dimensions

J. C. Le Guillou, Carlos Núñez, F Schaposnik +1 more

We consider the fermion-boson mapping in three dimensional space-time, in the Abelian case, from the current algebra point of view. We show that in a path-integral framework one can derive a general bosonization recipe leading, in the bosonic language, to the correct equal-time current commutators of the original free fermionic theory.

•Jan 1, 1996•0 citations

Fermionic current algebra in three dimensions

Carlos Núñez

We consider the fermion-boson mapping in three dimensional space-time, in the Abelian case, from the current algebra point of view. It is shown that in a path-integral framework one can derive a general bosonization recipe leading, in the bosonic language, to the correct equal-time current commutators of the original free fermionic theory.

AIP conference proceedings•Jan 1, 1998•0 citations

Geometric Algebras and Fermion Quantum Field Theory

Stanley Gudder

Corresponding to a finite dimensional Hilbert space $H$ with $\dim H=n$, we define a geometric algebra $\mathcal{G}(H)$ with $\dim\left[\mathcal{G}(H)\right]=2^n$. The algebra $\mathcal{G}(H)$ is a Hilbert space that contains $H$ as a subspace. We interpret the unit vectors of $H$ as states of individual fermions of the same type and $\mathcal{G}(H)$ as a fermion quantum field whose unit vectors represent states of collections of interacting fermions. We discuss creation operators on $\mathcal{G}(H)$ and provide their matrix representations. Evolution operators provided by self-adjoint Hamiltonians on $H$ and $\mathcal{G}(H)$ are considered. Boson-Fermion quantum fields are constructed. Extensions of operators from $H$ to $\mathcal{G}(H)$ are studied. Finally, we present a generalization of our work to infinite dimensional separable Hilbert spaces. Quanta 2025; 14: 48–65.

arXiv:2507.20394•Quanta•Jul 26, 2025•0 citations

Chiral sound waves from a gauge theory of 1D generalized statistics

Silvio Rabello

A topological gauge field theory in one spatial dimension is studied, with the gauge fields as generators of two commuting U(1) Kauc-Moody algebras. Coupling of these gauge fields to nonrelativistic bosonic matter fields, produces a statistical transmutation of the later, as in the Chern-Simons theory in two dimensions. The sound waves of the model are investigated and proven to be chiral bosonic excitations, with the same spectrum as the density fluctuations of the Luttinger model.

arXiv:cond-mat/9604040•Apr 6, 1996•0 citations

Bosonization and fermion liquids in dimensions greater than one

A. Houghton, John Bradley Marston

We develop and describe new approaches to the problem of interacting fermions in spatial dimensions greater than 1. These approaches are based on generalizations of powerful tools previously applied to problems in one spatial dimension. We begin with a review of one-dimensional interacting fermions. We then introduce a simplified model in two spatial dimensions to study the role that spin and perfect nesting play in destabilizing fermion liquids. The complicated functional renormalization-group equations of the full problem are made tractable in our model by replacing the continuum of points that make up the closed Fermi line with four Fermi points. Despite this drastic approximation, the model exhibits physically reasonable behavior both at half-filling (where instabilities occur) and away from half-filling (where a Luttinger liquid arises). Next we implement the bosonization of higher-dimensional Fermi surfaces introduced by Luther and advocated most recently by Haldane. Bosonization incorporates the phase-space and small-angle scattering processes neglected in our model (but does not, as yet, address questions of stability). The charge sector is equivalent to an exactly solvable Gaussian quantum field theory; the spin sector, however, must be solved semiclassically. Using the Luther-Haldane approach we recover the collective mode equation of Fermi-liquid theory and in three dimensions reproduce the ${\mathit{T}}^{3}$ln(T) contribution to the specific heat due to small-angle scattering processes. We conclude with a discussion of our results and some speculation about future possibilities.

arXiv:cond-mat/9210007•Physical review. B, Condensed matter•Sep 15, 1993•186 citations

Canonical approach to 2D WZNW model, non-Abelian bosonization, and anomalies

BRANISLAV SAZDOVIĆ

The gauged WZNW model has been derived as an effective action, whose Poisson brackets algebra of the constraints is isomorphic to the commutator algebra of operators in quantized fermionic theory. As a consequence, the Hamiltonian as well as the usual Lagrangian non-Abelian bosonization rules have been obtained for the chiral currents and chiral densities. The expression for the anomaly has been obtained as a function of the Schwinger term, using canonical methods.

arXiv:hep-th/9912216•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jul 20, 2000•6 citations
We propose the bosonization of a many-body fermion theory in D spatial dimensions through a noncommutative field theory on a (2D-1)-dimensional space. This theory leads to a chiral current algebra over the noncommutative space and reproduces the correct perturbative Hilbert space and excitation energies for the fermions. The validity of the method is demonstrated by bosonizing a two-dimensional gas of fermions in a harmonic trap.
Corresponding to a finite dimensional Hilbert space HHH with dim⁡H=n\dim H=ndimH=n, we define a geometric algebra G(H)\mathcal{G}(H)G(H) with dim⁡[G(H)]=2n\dim\left[\mathcal{G}(H)\right]=2^ndim[G(H)]=2n. The algebra G(H)\mathcal{G}(H)G(H) is a Hilbert space that contains HHH as a subspace. We interpret the unit vectors of HHH as states of individual fermions of the same type and G(H)\mathcal{G}(H)G(H) as a fermion quantum field whose unit vectors represent states of collections of interacting fermions. We discuss creation operators on G(H)\mathcal{G}(H)G(H) and provide their matrix representations. Evolution operators provided by self-adjoint Hamiltonians on HHH and G(H)\mathcal{G}(H)G(H) are considered. Boson-Fermion quantum fields are constructed. Extensions of operators from HHH to G(H)\mathcal{G}(H)G(H) are studied. Finally, we present a generalization of our work to infinite dimensional separable Hilbert spaces. Quanta 2025; 14: 48–65.
We develop and describe new approaches to the problem of interacting fermions in spatial dimensions greater than 1. These approaches are based on generalizations of powerful tools previously applied to problems in one spatial dimension. We begin with a review of one-dimensional interacting fermions. We then introduce a simplified model in two spatial dimensions to study the role that spin and perfect nesting play in destabilizing fermion liquids. The complicated functional renormalization-group equations of the full problem are made tractable in our model by replacing the continuum of points that make up the closed Fermi line with four Fermi points. Despite this drastic approximation, the model exhibits physically reasonable behavior both at half-filling (where instabilities occur) and away from half-filling (where a Luttinger liquid arises). Next we implement the bosonization of higher-dimensional Fermi surfaces introduced by Luther and advocated most recently by Haldane. Bosonization incorporates the phase-space and small-angle scattering processes neglected in our model (but does not, as yet, address questions of stability). The charge sector is equivalent to an exactly solvable Gaussian quantum field theory; the spin sector, however, must be solved semiclassically. Using the Luther-Haldane approach we recover the collective mode equation of Fermi-liquid theory and in three dimensions reproduce the T3{\mathit{T}}^{3}T3ln(T) contribution to the specific heat due to small-angle scattering processes. We conclude with a discussion of our results and some speculation about future possibilities.