Abdalla Elcio, Abdalla M. Cristina B., Rothe Klaus D. Non-perturbative Methods in 2 Dimensional Quantum Field Theory. Second Edition

2 ed. — World Scientific, Singapore, New Jersey, London, Hong Kong, 2001. — 834 p.While other — more phenomenologically motivated — branches of high energy physics, such as Quantum Chromodynamics, the electroweak model of Glashow, Salam and Weinberg, as well as purely group-theoretical studies have by now been extensively portrayed in a large number of research reports and books, the same has not been the case for two-dimensional quantum field theory. This book is intended to fill this gap. Its aim is to give a fairly detailed survey of the developments in two-dimensional quantum field theory since the pioneering work of Thirring, without
loosing sight of their relevance to the four-dimensional world. Though many of the properties and techniques to be used are peculiar to two-dimensional space-time, the structural richness of the models portrays the complexity to be expected in realistic quantum field theories. Our emphasis therefore lies on the non-perturbative aspects of the two-dimensional models and our tools shall correspondingly be operator and functional methods, although we shall occasionally make also use of Feynman diagram techniques.Free FieldsBosonic Free Fields
Fermionic Free Fields
Bosonization of Massless Fermions
The RS-Model
Conclusions
The Thirring modelThe 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 ParameterDeterminants and Heat KernelsFunctional 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
|-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 potentialSelf-Interacting fermionic modelsThe O(N) Invariant Gross-Neveu Model
Classical conservation laws
Effective potential and b-function in a 1/N expansion
The 1/N Expansion: Feynman rules
Leading order S-matrix elements
Quantization of the non-local charge
Chiral Gross-Neveu Model
Cancellation of infrared singularities
The 1/N expansion
Operator formulation
Quantization of non-local charge
Conclusions and Physical Interpretation
Non-linear s Models: Classical Aspects
Historical development
Sigma models and current algebra
Two-dimensional s models: preliminaries
Purely Bosonic Non-linear a Models
Formal developments
Dual symmetry and higher conservation laws
An explicit example: the Grassmannians
Non-linear s Models with Fermions
Definition and properties
Dual symmetry and higher conservation laws
Construction of an explicit example
Analogies with 4D Gauge Theories
Concluding Remarks
Non-linear s Models - Quantum AspectsGrassmannian Bosonic Models
1/N expansion
Renormalization
Infrared divergencies
Physical interpretation of the results
Grassmannian Models and Fermions
1/N 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 a Models in four dimensionsExact S-matrices of 2D ModelsConsequences 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 ZN 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 DevelopmentsThe Wess-Zumino-Witten TheoryExistence 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 modelQED2: Operator ApproachThe 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 theoryQuantum ChromodynamicsThe 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 symmetries
The physical Hilbert space
The QCD2 vacuum
Massive two-dimensional QCD
Screening in two-dimensional QCD
Further algebraic aspects
Conclusions
QED2: Functional ApproachEquivalent 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 Theta-Vacuum
Axial anomaly and the Dirac sea
Functional Representation of Tunneling
Amplitudes
Interpretation of the Result
Zero modes
Calculation of det i Jfi 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(l) Problem RevisitedThe Finite Temperature Schwinger ModelHeat kernel and Seeley expansion
The Atiyah-Singer Index theorem
Fermions in an Instanton potential
Chiral condensate and symmetry breaking
Polyakov loop-operator and screeningNon-Abelian Chiral Gauge TheoriesAnomalies 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 a1[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 formulationChiral Quantum ElectrodynamicsThe 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 BosonsConformally Invariant Field TheoryConformal transformations and conformal group
Dilatations
The conformal group in D dimensions
The conformal group in two dimensions
Mobius 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 SymmetryConformal Field Theory with Internal SymmetryConformal 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
2D gravity and string related topicsThe 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 SupergravityFinal Remarks