Bosonization

The following sections are included:

• PREFACE

• CONTENTS

• INTRODUCTION

In 1975 the particle physics community was startled by a short paper written by Sidney Coleman [rep. 8] in which he demonstrated that the quantum Sine-Gordon model, a theory of interacting bosonic “mesons”, was equivalent to the Thirring model of interacting massive fermions. Coleman further conjectured that the Sine-Gordon soliton was the fundamental fermion of the Thirring model. This assertion, that a “lump” in a Bose field might be a fermion, seems to have surprised even Coleman himself. A similar claim had been made by Skyrme in 1958 [rep. 2] but, like Skyrme's now familiar soliton model of the nucleon [1], the idea seemed too strange to be true. The new conjecture was more convincingly argued, but in the acknowledgments section of [rep. 8] Coleman thanks a colleague for reassuring him of his sanity. I myself remember being a second-year graduate student listening to Coleman begin a seminar in Cambridge (England) with the words “I may be crazy”. He went on, of course, to give a characteristically sane and lucid exposition of his work…

The following sections are included:

• Introduction

• Free Fermi Fields

• Free Bosons

• The Bosonization Rules

• A Quantum Pythagoras Theorem

• Appendix 1A. Complex Coordinates

• Appendix IB. Conformal Symmetry

• References

The following sections are included:

• Introduction

• Dirac Equation for Metals

• Thirring Model

• Hamiltonian Solution of the Luttinger-Thirring Model

• Schwinger Model

• Appendix 2A. Thirring Model via Chiral Rotations

• Appendix 2B. Solving the Schwinger Model by Chiral Rotation

• References

The following sections are included:

• Introduction

• Charge-Density Waves

• Zero Temperature Effective Action for BCS Superconductors

• Appendix 3A. Pair Creation in an External Field

• Appendix 3B. Madelung Fluids

• References

The following sections are included:

• Introduction

• Slater Determinants and Schur Functions

• Symmetric Polynomials

• S(z) as a Hilbert Space

• The LU(1) Group Action

• Appendix 4A. The Very Useful Identity

• Appendix 4B. Jacobi's Triple Product Formula

• References

The fact implied by Bloch several years ago that in some approximate sense the behavior of an assembly of Fermi particles can be described by a quantized field of sound waves in the Fermi gas, where the sound field obeys Bose statistics, is proved in the one-dimensional case. This fact provides us with a new possibility of treating an assembly of Fermi particles in terms of the equivalent assembly of Bose particles, namely, the assembly of sound quanta. The field equation for the sound wave is found to be linear irrespective of the absence or presence of mutual interaction between particles, so that this method is a very useful means of dealing with many-Fermion problems. It is also applicable to the case where the interparticle force is not weak. In the case of force of too short a range this method fails.

A simple non-linear field theory is considered as the model for a recently proposed classical field theory of mesons and their particle sources. Quantization may be made according to canonical procedures; the problem is to show the existence of quantum states corresponding with the particle-like solutions of the classical field equations. A plausible way to do this is suggested.

Luttinger's exactly soluble model of a one-dimensional many-fermion system is discussed. We show that he did not solve his model properly because of the paradoxical fact that the density operator commutators [ρ(p), ρ(−p′)], which always vanish for any finite number of particles, no longer vanish in the field-theoretic limit of a filled Dirac sea. In fact the operators ρ(p) define a boson field which is ipso facto associated with the Fermi–Dirac field. We then use this observation to solve the model, and obtain the exact (and now nontrivial) spectrum, free energy, and dielectric constant. This we also extend to more realistic interactions in an Appendix. We calculate the Fermi surface parameter , and find: (i.e., there exists a sharp Fermi surface) only in the case of a sufficiently weak interaction.

Singularities near the threshold of the soft x-ray spectra of metals have been predicted by Mahan and have recently been calculated by Nozièrcs et al. using the model of a localized core hole. We show that the singular behavior can be understood in terms of density waves of the conduction electrons which are excited when, in the absorption process, the core hole is created, providing an attractive potential for the conduction electrons. For the description of the conduction electrons in terms of density waves, Tomonaga's model is adopted.

We show that a large class of backward-scattering matrix elements involving Δk ∼ ±2k_F vanish for fermions interacting with two-body attractive forces in one dimension. (These same matrix elements are finite for noninteracting particles and infinite for particles interacting with two-body repulsive forces.) Our results demonstrate the possibility of persistent currents in one dimension at T = 0, and are a strong indication of a metal-to-insulator transition al T = 0 for repulsive forces. They are obtained by use of a convenient representation of the wave operator in terms of density-fluctuation operators.

We compute the single-particle spectral density, susceptibility near the Kohn anomaly, and pair propagator for a one-dimensional interacting-electron gas. With an attractive interaction, the pair propagator is divergent in the zero-temperature limit and the Kohn singularity is removed. For repulsive interactions, the Kohn singularity is stronger than the free-particle case and the pair propagator is finite. The low-temperature behavior of the interacting system is not consistent with the usual Ginzburg-Landau functional because the frequency, temperature, and momentum dependences are characterized by power-law behavior with the exponent dependent on the interaction strength. Similarly, the energy dependence of the single-particle spectral density obeys a power law whose exponent depends on the interaction and exhibits no quasiparticle character. Our calculations are exact for the Luttinger or Tomonaga model of the one-dimensional interacting system.

An exact solution to the one-dimensional electron gas with a particular attractive-interaction strength for scattering across the Fermi “surface” is given. It is shown that conductivity enhancement occurs for physically interesting values of the coupling constants. Scaling arguments are advanced to demonstrate that this solution applies generally for attractive backward scattering. In addition, the spinless problem is solved exactly for arbitrary couplings.

The sine-Gordon equation is the theory of a massless scalar field in one space and one time dimension with interaction density proportional to cosβφ, where β is a real parameter. I show that if β² exceeds 8π, the energy density of the theory is unbounded below; If β² equals 4π, the theory is equivalent to the zero-charge sector of the theory of a free massive Fermi field; for other values of β, the theory is equivalent to the zero-charge sector of the massive Thirring model. The sine-Gordon soliton is identified with the fundamental fermion of the Thirring model.

Operators for the creation and annihilation of quantum sine-Gordon solitons are constructed. The operators satisfy the anticommutation relations and field equations of the massive Thirring model. The results of Coleman are thus reestablished without the use of perturbation theory. It is hoped that the method is more generally applicable to a quantum-mechanical treatment of extended solutions of field theories.

In two dimensions, we find a construction for an SU(N) quark field in terms of N real Bose fields. Hence, equivalence is shown between certain massive SU(N) Thirring models and systems of quantum sine-Gordon–type equations. From the point of view of the bosons, the “soliton-quark” SU(N) is topological. To minimize guesswork in the development of such correspondences, we employ a systematic blend of Mandelstam's operator approach with the interaction picture.

Bosonization is applied to the SU(N) Thirring models, and interesting relations between various two-dimensional field theories arise. In particular, we show that the SU(2) model is equivalent to a version of the Sine-Gordon equation plus a free massless field.

The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called ‘Luttinger liquid theory’, by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.

We present an exact theory of an O(4)-σ-model based on its relation to a certain fermionic model. The S-matrix and the vacuum energy in a constant external field are computed.

A non-abelian generalization of the usual formulas for bosonization of fermions in 1 + 1 dimensions is presented. Any fermi theory in 1 + 1 dimensions is equivalent to a local bose theory which manifestly possesses all the symmetries of the fermi theory.

Using the functional technique we prove the bosonization rules of Witten for the currents in a non-abelian two-dimensional theory with a particular regularization of the Fermi theory.

Using recent results on fermionic determinants in two-dimensional non-abelian background fields we give a very simple path integral demonstration of the equivalence between the free Fermi theory in this background and a corresponding chiral Bose theory with Wess–Zumino action. The result is compared to previously proposed bosonization rules and certain limitations to the general validity of these are found.

Abelian and non-Abelian bosonization of two-dimensional models is discussed within the path-integral framework. Concerning the Abelian case, the equivalence between the massive Thirring and the sine-Gordon models is rederived in a very simple way by making a chiral change in the fermionic path-integral variables. The massive Schwinger model is also studied using the same technique. The extension of this bosonization approach to the solution of non-Abelian models is performed in a very natural way, showing the appearance of the Wess-Zumino functional through the Jacobian associated with the non-Abelian chiral change of variables. Relevant features of massless two-dimensional QCD are discussed in this context.

A field theory is considered on a Riemann surface of genus p, where the field is a section of a holomorphic line bundle ξ with Chern number p − 1. When ξ has no holomorphic section, one can calculate the propagator explicitly. As an application the expectation value of the energy–momentum tensor is computed.

We study the bosonization of chiral fermion theories on arbitrary compact Riemann surfaces. We express the fermionic and bosonic correlation functions in terms of theta functions and prove their equality. This is used to obtain explicit expressions for a class of chiral determinants relevant to string theory. The anomaly structure of these determinants and their behaviour on degenerate Riemann surfaces is analysed. We apply these results to multi-loop calculations of the bosonic string.

We point our that the basic addition theorem of θ-functions, Fay's identity, implies an equivalence between bosons and chiral fermions on Riemann surfaces with arbitrary genus. We present a rule for a bosonized calculation of correlation functions. We also discuss ghost systems of n and (1 − n) tensors and derive formulas for their chiral determinants.

We use Quillen's theorem and algebraic geometry to investigate the modular transformation properties of some quantities of interest in string theory. In particular, we show that the spin structure dependence of the chiral Dirac determinant on a Riemann surface is given by Riemann's theta function. We use this result to investigate the modular invariance of multiloop heterotic string amplitudes.

We prove the equivalence between certain fermionic and bosonic theories in two spacetime dimensions. The theories have fields of arbitrary spin on compact surfaces with any number of handles. Global considerations require that we add new topological terms to the bosonic action. The proof that our prescription is correct relies on methods of complex algebraic geometry.

The equivalence is proved between fermionic and scalar field theories on Riemann surfaces of arbitrary topology. The effects of global topology include a modification of the bosonic action.

We use symmetries of the path-integral on Riemann surfaces with boundaries to develop an operator formulation for higher loop Riemann surfaces.

The theory of representations of loop groups provides a framework where one can consider Riemann surfaces with arbitrary numbers of handles and nodes on the same footing. Using infinite grassmanians we present a general formulation of some conformal field theories on arbitrary surfaces in terms of an operator formalism. As a by-product, one can obtain some general results for the chiral bosonization of fermions using the vertex operator representation of infinite dimensional groups. We believe that this set-up provides the natural arena where the recent proposal of Friedan and Shenker of formulating string theory in the universal moduli space can be discussed.

We present a general method for constructing path intergrals for the nuclear many-body problem. This method uses continuous and overcomplete sets of vectors in the Hilbert space. The state labels play the role of classical coordinates which are quantized as bosons. The equations of motion for the classical coordinates are obtained by calculating the functional integral in the saddle point approximation. In the particular case where the over-complete set considered is the set of all Slater determinants, the classical equations of motion are the time-dependent Hartree-Fock equations. The functional integral provides a way of requantizing these classical equations. This quantization involves boson degrees of freedom and is in some cases very similar to the method of boson expansion. It is shown that the functional integral formalism provides a unifying framework to describe various approaches to the nuclear many-body problem.

The following sections are included:

• Introduction

• The Kadomtsev-Petviashivili equation
  • Introduction
  • KdV equation
  • KP equation
  • The wave function and the adjoint wave function
  • Bilinear identity
  • The τ function for the KP hierarchy
  • Bilinear equations for the τ function
  • Vertex operators and soliton solutions
  • BKP hierarchy
  • Quasi-periodic solution

• The operator Theory for the KP and the BKP hierarchy
  • The operator algebra
  • The Fock representation
  • The τ functions
  • The character polynomials
  • Realization of the Fock space
  • Realization of the Fock representation
  • The bilinear equations
  • The Lie algebra 𝔙(∞)
  • The BKP hierarchy

• Reductions of the KP and the BKP hierarchies, Euclidean Lie algebras and the number of Hirota's bilinear equations

• References

The functional integral for the quantization of the coadjoint orbits of the unitary and orthogonal groups is given by means of an explicit construction of the corresponding «Darbouxi» variables.

After a discussion of coherent states for the loop group LU(N), we use them to write down a bosonic path integral which describes the level-one representations of LU(N). The construction uses the description of the representations in terms of free right-going Weyl fermions and so provides one with an explicit geometrical interpretation for a Fermi–Bose equivalence. The bosonic system is easily seen to have a single Kac–Moody algebra as its Poisson brackets and coincides with the non-abelian chiral boson model that has been introduced by Sonnenschein et al. [Nucl. Phys. B301 (1988) 346; B309 (1988) 752].