Non-Perturbative Methods in Two-Dimensional Quantum Field Theory

The following sections are included:

• FOREWORD

• CONTENTS

The development of Relativistic Quantum Field Theory started in 1932 as a natural extension of Quantum Mechanics to the relativistic domain¹. The work of Feynman in the late forties provided a powerful tool for the calculation of processes in Quantum Electrodynamics². Second quantization led, however to new conceptual and technical difficulties. Quantum fields had to be regarded as operator-valued distributions, their local products being ill defined as a result of ultraviolet divergencies, which plagued the higher order computations in perturbation theory. This problem was partially mastered via the techniques of renormalization^3,4, and later on completely understood^5,6…

The following sections are included:

• Introduction

• Bosonic Free Fields

• Fermionic Free Fields

• Bosonization of Fermions

• The RS-Model

• Conclusions

The following sections are included:

• Introduction

• The Massless Thirring Model

• The Massive Thirring Model
  • Equivalence with sine-Gordon equation
  • Classical conservation laws
  • Quantum conservation laws

• Bosonization Revisited
  • Fermions in terms of bosons

• The Soliton as a Disorder Parameter

• Conclusion

The following sections are included:

• Definition and General Remarks

• Determinants and the Generalized Zeta-Function
  • Elliptic operators
  • The zeta-function

• One Point Compactification
  • Stereographic projection of Dirac operator
  • The associated Dirac operator
  • Vortex on the stereographic sphere

• Calculation of Seeley Coefficients

• Methods for Computing Determinants
  • ζ-function regularization
  • Proper-time regularization
  • Fujikawa method
  • Determinant from current-anomaly
  • Pauli-representation of determinant

• Ambiguities in the Calculation of the Determinant
  • Ambiguities in the regularization
  • Dependence on R

• Atiyah-Singer Index Theorem

• Conclusion

The following sections are included:

• Introduction

• The O(N) Invariant Gross-Neveu Model
  • Classical conservation laws
  • Effective potential and β-function in a expansion
  • expansion: Feynman rules
  • Leading order S-matrix elements
  • Quantization of the non-local charge

• Chiral Gross-Neveu Model
  • Cancellation of infrared singularities
  • The expansion
  • Operator formulation
  • Quantization of non-local charge

• Conclusions and Physical Interpretation

The following sections are included:

• Introduction
  • Historical development
  • Sigma models and current algebra
  • Two-dimensional sigma models: preliminaries

• Purely Bosonic Non-linear Sigma Models
  • Formal development
  • Dual symmetry and higher conservation laws
  • An explicit example: the Grassmannians

• Non-linear Sigma Models with Fermions
  • Definition and properties
  • Dual symmetry and higher conservation laws
  • Construction of an explicit example

• Analogies with Four Dimensional Gauge Theories

• Concluding Remarks

The following sections are included:

• Introduction

• Grassmannian Bosonic Models
  • expansion
  • Renormalization
  • Infrared divergencies
  • Physical interpretation of the results

• Grassmannian Models Interacting with Fermions
  • expansion and Feynman rules
  • Physical interpretation of the results

• Quantization of Higher Conservation Laws
  • Purely bosonic sigma models and anomalies
  • Fermionic interaction and anomaly cancellation

• Perturbative Renormalization and the Background Field Method
  • Background field method applied to the sigma model
  • Parallelizable manifolds, and applications to string theory

• Anomalous Non-Linear Sigma Models in Four Dimensions

• Conclusion

The following sections are included:

• Introduction
  • Consequences of higher conservation laws
  • Factorizable S-matrix
  • Fusion rules
  • Bound state scattering

• Classification of Exact S-matrices from the Local Conservation Laws
  • SU(N) invariant S-matrix
  • S-matrix of the sine-Gordon and massive Thirring model
  • Exact S-matrix for O(N) symmetry
  • The Z_N invariant S-matrix

• Quantum Non Local Charges, and Exact S-Matrices
  • S-matrices of purely fermionic models
  • S-matrices of non-linear sigma models

• Further Developments and Conclusion

The following sections are included:

• Introduction

• Existence of a Critical Point

• Properties at the Critical Point
  • The Kac Moody algebra
  • The WZW fields in terms of fermions
  • The Sugawara form of the energy momentum tensor
  • The non-abelian bosonization in the operator language

• Properties off the Critical Point
  • Integrability of the Wess-Zumino-Witten action
  • On the solution off the critical point
  • Supersymmetric WZW model

• Conclusion

The following sections are included:

• Introduction

• The Massless Schwinger Model
  • Quantum solution
  • The Maxwell current
  • Chiral densities
  • Vacuum structure
  • Gauge transformations
  • Correlation functions and violation of clustering
  • Absence of charged states (screening)
  • The quark-antiquark potential
  • Adding flavor
  • Fractional winding number and the U(1) problem

• The Massive Schwinger Model
  • Equivalent bosonic formulation
  • The quantum Dirac equation
  • Vacuum structure and all that
  • Screening versus confinement
  • Adding flavour
  • Lorentz transformation properties
  • The MSM as the limit of a massive vector theory

• Conclusion

The following sections are included:

• Introduction

• The External Field Current
  • Tree graph expansion
  • Loop expansion

• QCD₂ Determinant
  • Heat kernel method
  • Polyakov-Wiegman Method
  • A further invariance of the effective action

• Bosonization of the QCD₂ Fermionic Action
  • Recovering the U(1) case
  • QCD₂ bosonized currents

• Fermion Greens Function
  • Specializing to the U(1) case

• Conclusion

The following sections are included:

• Introduction

• Equivalent Bosonic Action

• Gauge Invariant Correlation Functions
  • The external field current, and chiral densities

• Vacuum Structure
  • Chirality of the vacuum

• Why Study Gauge Invariant Correlation Functions?

• Screening versus Confinement

• Quasi Periodic Boundary Conditions and the θ Vacuum

• Functional Representation of Tunneling Amplitudes

• Interpretation of the Result
  • Zero modes
  • Calculation of det from the anomaly equation

• Eigenvalue Spectrum of the Associated Dirac Operator

• Zero Modes of the Dirac Operator as a Boundary-Valued Problem

• The U(1) Problem Revisited

• Conclusion

The following sections are included:

• Introduction

• Anomalies and Cocycles
  • Consistent anomaly
  • More about cocycles
  • Gauss anomaly
  • Relation between consistent and covariant anomaly

• Isomorphic Representations of Chiral QCD₂

• External Field Ward Identities

• Ward-Takahashi Identities

• The Effective Bosonic Action in the GNI Formulation

• Construction of the one-Cocycle from the Anomaly

• Bosonic Action in the GNI and GI Formulation

• Symmetries of the Model

• Relation of Source Currents in GNI and GI Formulations

• Poisson Algebra of the Currents

• Hamiltonian Quantization

• Fermionization of α₁[A,g⁻¹]

• BRS Quantization of GI Formulation

• Chiral QCD₂ in Terms of Chiral Bosons

• Constraint Structure from Fermionic Hamiltonian

• Conclusion

The following sections are included:

• Introduction

• The JR Model

• Quantization in the GNI Formulation
  • Hamiltonian and constraints
  • Commutation relations
  • Current–potential and bosonic representation of the fermion field
  • Energy-momentum tensor
  • A_μ two-point function
  • Fermionic two-point function

• Quantization in the GI Formulation
  • Hamiltonian and constraints
  • Implementation of gauge conditions
  • Isomorphism between GI and GNI formulation: phase space view
  • Alternative approach to quantization
  • Path-Integral Formulation

• Perturbative Analysis in the Fermionic Formulation
  • Perturbative analysis in the GNI formulation
  • Perturbative analysis in the GI formulation

• Anomalous Poisson Brackets Revisited
  • Operator view of anomalous Poisson brackets
  • Bjorken-Johnson-Low view of anomalous Poisson brackets
  • Reconstruction of commutators of the GNI formulation

• Chiral QED₂ in terms of Chiral Bosons

• Conclusion

The following sections are included:

• Introduction

• Conformal Transformations in Field Theory
  • The conformal algebra
  • The BPZ construction

• Operator Realization of the Conformal Algebra

• Non Abelian Conformal Theory
  • WZNW theory revisited
  • Kac Moody algebra
  • Ward identities and equations of motion
  • Correlation functions

• The Non Abelian Thirring Model at the Critical Point, and the WZW Theory
  • Definition of the conformally invariant theory
  • The solution at the conformally invariant fixed point

• More on Kac Moody Algebras

• Further Applications to String Theories

• Superconformal Symmetry

• Conformally Invariant Two Dimensional Gravity
  • The general theory of gravity in two dimensions
  • Induced gravity; contribution from the matter fields to the determinant
  • Canonical quantization and SL(2,R) symmetry
  • Operator product expansions and quantum solution
  • Interaction of matter fields with gravity in two dimensions

• Two Dimensional Supergravity
  • General theory of supergravity in two dimensions
  • OPE and quantum solution
  • Interaction of matter fields and supergravity

• Non Perturbative 2D Gravity, Surfaces, and Critical Behavior

• Conclusions

Two dimensional quantum field theory provides a very powerful laboratory for gaining a non perturbative understanding of quantum field theory. The kinematical simplifications resulting from two dimensional space time have allowed for the complete solution of a variety of models involving interacting fields. The non trivial nature of these solutions provide a deeper insight into the structure of quantum field theory, and has found useful applications in several areas of research, such as string theories and systems in statistical mechanics at criticality…

The following sections are included:

• Appendix A Notation (Minkowski Space)

• Appendix B Notation (Euclidean Space)

• Appendix C Further Conventions

• Appendix D Functional Bosonization of the Massive Thirring Model

• Appendix E Bosonization of the Kinetic Term

• Appendix F Classical Integrability in the Massive Thirring Model

• Appendix G Quantum Non-Local Charge: Action on Asymptotic States

• Appendix H Non Linear Sigma Models

• Appendix I Conservation Laws in NLS Models

• Appendix J S-Matrices

• Appendix K Complete S-matrix of the Gross-Neveu Model

• Appendix L Poisson Brackets and Commutators

• Appendix M Chiral Bosons

• Appendix N Teichmüller Parameters

• Appendix O Dirac Quantization

• REFERENCES

• INDEX