Non-Perturbative Methods in 2 Dimensional Quantum Field Theory

The following sections are included:

• Preface

• Contents

The development of Relativistic Quantum Field Theory started in 1932 as a natural extension of Quantum Mechanics to the relativistic domain [1]. The work of Feynman in the late forties provided a powerful tool for the calculation of processes in Quantum Electrodynamics. 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, and later on completely understood [2]…

The following sections are included:

• Introduction

• Bosonic Free Fields

• Fermionic Free Fields

• Bosonization of Massless Fermions

• The RS-Model

• Conclusions

• Bibliography

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

• Bibliography

The following sections are included:

• Introduction

• Functional Determinant, one-loop diagram
  • Determinants and the Generalized Zeta-Function
  • One Point Compactification
  • The associated Dirac operator

• Calculating Seeley Coefficients
  • The perturbative approach
  • The Schwinger–DeWitt method
  • The Fujikawa method

• Computing Functional Determinants
  • ζ-function regularization
  • Proper-time regularization
  • The Fujikawa point of view

• A Theorem on a one parameter family of factorizable operators

• The QCD₂ functional determinant

• Zero-modes
  • Axial anomaly equation in the presence of zero-modes
  • Atiyah–Singer Index Theorem

• Ambiguities in Functional Determinants
  • Ambiguities in the regularization
  • Dependence on the scale parameter

• Mass expansion in proper-time regularization

• The Finite Temperature Heat Kernel
  • Scalar field in a static background potential
  • Scalar field in a static background gauge potential

• Conclusion

• Bibliography

The following sections are included:

• Introduction

• The O(N) Invariant Gross–Neveu Model
  • Classical conservation laws
  • Effective potential and β-function in a expansion
  • The 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

• Bibliography

The following sections are included:

• Historical development

• Sigma models and current algebra

• Two-dimensional σ models: preliminaries

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

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

• Analogies with 4D Gauge Theories

• Concluding Remarks

• Bibliography

The following sections are included:

• Introduction

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

• Grassmannian Models and 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

• Algebra of non-local charges
  • Bosonic O(N)-symmetric sigma models

• Non-local charges in the WZNW model

• Perturbative Renormalization
  • Background Field Method
  • Parallelizable manifolds; applications to string theory

• Anomalous Non-Linear σ Models in four dimensions

• Conclusion

• Bibliography

The following sections are included:

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

• S-matrices and Conservation Laws
  • SU(N) invariant S-matrices
  • Sine-Gordon and massive Thirring models
  • Exact S-matrix for O(N) symmetry
  • The Z_N invariant S-matrix

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

• Boundary S-matrices

• Further Developments

• Conclusion

• Bibliography

The following sections are included:

• Introduction

• Existence of a Critical Point

• Properties at the Critical Point
  • The Polyakov–Wiegmann formula
  • The Affine 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 WZNW action
  • On the solution off the critical point
  • Supersymmetric WZW model

• Conclusion

• Bibliography

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 flavour
  • 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

• Bibliography

The following sections are included:

• Introduction

• The 1/N expansion: 't Hooft model

• Currents, Green functions and determinants
  • Tree graph expansion of the current
  • Recovering the QCD₂ effective action
  • Fermion Green Function

• Local decoupled formulation and BRST constraints
  • Local decoupled partition function and BRST symmetries
  • Systematic derivation of the constraints

• Non-local decoupled formulation and BRST constraints
  • Non-local decoupled partition function and BRST symmetries

• The physical Hilbert space

• The QCD₂ vacuum

• Massive two-dimensional QCD

• Screening in two-dimensional QCD

• Further algebraic aspects

• Conclusions

• Bibliography

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 Correlators

• Screening versus Confinement

• Quasi-Periodic Boundary Conditions and the θ-Vacuum

• Axial anomaly and the Dirac sea

• Functional Representation of Tunneling Amplitudes

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

• Eigenvalue Spectrum of the Dirac Operator

• Zero Modes and Boundary-Value Problem
  • Free Dirac operator and non-local boundary conditions
  • The little Dirac operator

• The U(1) Problem Revisited

• Conclusion

• Bibliography

The following sections are included:

• Introduction

• Heat kernel and Seeley expansion

• The Atiyah–Singer Index theorem

• Fermions in an Instanton potential

• Chiral condensate and symmetry breaking

• Polyakov loop-operator and screening

• Conclusion

• Bibliography

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
  • Gauge-invariant embedding
  • External Field Ward Identities
  • 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]
  • BRST Quantization of GI Formulation
  • Chiral QCD₂ in Terms of Chiral Bosons

• Constraint Structure from the Fermionic Hamiltonian

• Chiral QCD₂ in the local decoupled formulation
  • Gauge non-invariant formulation
  • Gauge-invariant formulation

• Conclusion

• Bibliography

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 fermion field
  • Energy-momentum tensor
  • Vector-field two-point function
  • Fermionic two-point function

• Quantization in the GI Formulation
  • Hamiltonian and constraints
  • Implementation of gauge conditions
  • Isomorphism between GI and GNI formulations: phase space view
  • WZ term and BFT Hamiltonian embedding
  • Alternative approach to quantization
  • Operator solution in Lorentz-type gauges

• 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

• Bibliography

The following sections are included:

• Introduction

• Conformal transformations and conformal group
  • Dilatations
  • The conformal group in D dimensions

• The conformal group in two dimensions
  • Möbius transformations

• The BPZ construction
  • Primary and quasi-primary fields
  • Radial quantization
  • Descendants of primary fields
  • Virasoro algebra

• Realization of Conformal Algebra for c < 1

• Superconformal Symmetry

• Conclusion

• Bibliography

The following sections are included:

• Introduction

• Conformal algebra and Ward identities

• Realizations of non-Abelian conformal algebra
  • The Wess–Zumino–Witten field
  • The non-Abelian Thirring field at the Critical Point

• Coset description of CQFT
  • Coset realization of the FQS minimal unitary series
  • Fermionic coset realization of SU(N)₁
  • Fermionic coset realization of FQS series
  • Reduction formula for negative level WZW fields

• Critical statistical models
  • Fermionic coset description of the critical Ising model

• Conclusions

• Bibliography

The following sections are included:

• Introduction

• The Nambu–Goto string

• The effective action of 2D quantum gravity
  • Uniqueness of the Polyakov action
  • Quantum Gravity

• The Liouville theory
  • The classical Liouville theory
  • The quantum Liouville theory

• Gravity in the light-cone gauge
  • Canonical quantization and SL(2,R) symmetry
  • Operator product expansions and Ward identities
  • Interaction of matter fields with gravity
  • Two-Dimensional Supergravity

• Conclusion

• Bibliography

Two-dimensional quantum field theory provides a very powerful laboratory for gaining 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 provides 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 Fermionic Kinetic Term

• Appendix F Classical Integrability in the Massive Thirring Model

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

• Appendix H S-Matrices

• Appendix I Complete S-matrix of the Gross—Neveu Model

• Appendix J Poisson Brackets and Commutators

• Appendix K Chiral Bosons

• Appendix L Axial anomaly from dispersion relations

• Appendix M Loop expansion in QCD₂

• Index