Quantum Field Theory

Preface
General References
1. Classical Theory
1.1 Principle of Least Action
1.1.1 Classical Motion
1.1.2 Electromagnetic Field as an Infinite Dynamical System
1.1.3 Electromagnetic Interaction of a Point Particle
1.2 Symmetries and Conservation Laws
1.2.1 Fundamental Invariants
1.2.2 Energy Momentum Tensor
1.2.3 Internal Symmetries
1.3 Propagation and Radiation
1.3.1 Green Functions
1.3.2 Radiation
2. The Dirac Equation
2.1 Toward a Relativistic Wave Equation
2.1.1 Quantum Mechanics and Relativity
2.1.2 The Dirac Equation
2.1.3 Relativistic Covariance
2.2 Physical Content
2.2.1 Plane Wave Solutions and Projectors
2.2.2 Wave Packets
2.2.3 Electromagnetic Coupling
2.2.4 Foldy-Wouthuysen Transformation
2.3 Hydrogen-like Atoms
2.3.1 Nonrelativistic versus Relativistic Spectrum
2.3.2 Dirac Theory
2.4 Hole Theory and Charge Conjugation
2.4.1 Reinterpretation of Negative Energy Solutions
2.4.2 Charge Conjugation
2.4.3 Zero-Mass Particles
2.5 Dirac Propagator
2.5.1 Free Propagator
2.5.2 Propagation in an Arbitrary External Electromagnetic Field
2.5.3 Application to the Coulomb Scattering
2.5.4 Fock-Schwinger Proper Time Method
3. Quantization--Free Fields
3.1 Canonical Quantization
3.1.1 General Formulation
3.1.2 Scalar Field
3.1.3 Charged Scalar Field
3.1.4 Time-Ordered Product
3.1.5 Thermodynamic Equilibrium
3.2 Quantized Radiation Field
3.2.1 Indefinite Metric
3.2.2 Propagator
3.2.3 Massive Vector Field
3.2.4 Vacuum Fluctuations
3.3 Dirac Field and Exclusion Principle
3.3.1 Anticommutators
3.3.2 Fock Space for Fermions
3.3.3 Relation between Spin and Statistics--Propagator
3.4 Discrete Symmetries
3.4.1 Parity
3.4.2 Charge Conjugation
3.4.3 Time Reversal
3.4.4 Summary
4. Interaction with an External Field
4.1 Quantized Electromagnetic Field Interacting with a Classical Source
4.1.1 Emission Probabilities
4.1.2 Emitted Energy and the Infrared Catastrophe
4.1.3 Induced Absorption and Emission
4.1.4 S Matrix and Evolution Operator
4.2 Wick's Theorem
4.2.1 Bose Fields
4.2.2 Fermi Fields
4.2.3 General Case
4.3 Quantized Dirac Field Interacting with a Classical Potential
4.3.1 General Formalism
4.3.2 Emission Rate to Lowest Order
4.3.3 Pair Creation in a Constant Uniform Electric Field
4.3.4 The Euler-Heisenberg Effective Lagrangian
5. Elementary Processes
5.1 S Matrix and Asymptotic Theory
5.1.1 Cross Sections
5.1.2 Asymptotic Theory
5.1.3 Reduction Formulas
5.1.4 Generating Functional
5.1.5 Connected Parts
5.1.6 Fermions
5.1.7 Photons
5.2 Applications
5.2.1 Compton Effect
5.2.2 Pair Annihilation
5.2.3 Positronium Lifetime
5.2.4 Bremsstrahlung
5.3 Unitarity and Causality
5.3.1 Unitarity and Partial Wave Decomposition
5.3.2 Causality and Analyticity
5.3.3 The Jost-Lehmann-Dyson Representation
5.3.4 Forward Dispersion Relations
5.3.5 Momentum Transfer Analyticity
6. Perturbation Theory
6.1 Interaction Representation and Feynman Rules
6.1.1 Self-Interacting Scalar Field
6.1.2 Feynman Rules for Spinor Electrodynamics
6.1.3 Electron-Electron and Electron-Positron Scattering
6.1.4 Scalar Electrodynamics
6.2 Diagrammatics
6.2.1 Loopwise Expansion
6.2.2 Truncated and Proper Diagrams
6.2.3 Parametric Representation
6.2.4 Euclidean Green Functions
6.3 Analyticity Properties
6.3.1 Landau Equations
6.3.2 Real Singularities
6.3.3 Real Singularities of Simple Diagrams
6.3.4 Physical-Region Singularities. Cutkosky Rules
7. Radiative Corrections
7.1 One-Loop Renormalization
7.1.1 Vacuum Polarization
7.1.2 Electron Propagator
7.1.3 Vertex Function
7.1.4 Summary
7.2 Radiative Corrections to the Interaction with an External Field
7.2.1 Effective Interaction and Anomalous Magnetic Moment
7.2.2 Radiative Corrections to Coulomb Scattering
7.2.3 Soft Bremsstrahlung
7.2.4 Finite Inclusive Cross Section
7.3 New Effects
7.3.1 Photon-Photon Scattering
7.3.2 Lamb Shift
7.3.3 Van der Waals Forces at Large Distances
8. Renormalization
8.1 Regularization and Power Counting
8.1.1 Introduction
8.1.2 Regularization
8.1.3 Power Counting
8.1.4 Convergence Theorem
8.2 Renormalization
8.2.1 Normalization Conditions and Structure of the Counterterms
8.2.2 Bogoliubov's Recursion Formula
8.2.3 Zimmermann's Explicit Solution
8.2.4 Renormalization in Parametric Space
8.2.5 Finite Renormalizations
8.2.6 Composite Operators
8.3 Zero-Mass Limit, Asymptotic Behavior, and Weinberg's Theorem
8.3.1 Massless Theories
8.3.2 Ultraviolet Behavior and Weinberg's Theorem
8.4 The Case of Quantum Electrodynamics
8.4.1 Formal Derivation of the Ward-Takahashi Identities
8.4.2 Pauli-Villars Regularization to All Orders
8.4.3 Renormalization
8.4.4 Two-Loop Vacuum Polarization
9. Functional Methods
9.1 Path Integrals
9.1.1 The Role of the Classical Action in Quantum Mechanics
9.1.2 Trajectories in the Bargmann-Fock Space
9.1.3 Fermion Systems
9.2 Relativistic Formulation
9.2.1 S Matrix and Green Functions in Terms of Path Integrals
9.2.2 Effective Action and Steepest-Descent Method
9.3 Constrained Systems
9.3.1 General Discussion
9.3.2 The Electromagnetic Field as an Example
9.4 Large Orders in Perturbation Theory
9.4.1 Introduction
9.4.2 Anharmonic Oscillator
10. Integral Equations and Bound-State Problems
10.1 The Dyson-Schwinger Equations
10.1.1 Field Equations
10.1.2 Renormalization
10.2 Relativistic Bound States
10.2.1 Homogeneous Bethe-Salpeter Equation
10.2.2 The Wick Rotation
10.2.3 Scalar Massless Exchange in the Ladder Approximation
&n
12.3 The Effective Action at the One-Loop Order
12.3.1 General Form
12.3.2 Two-Point Function
12.3.3 Other Functions
12.3.4 One-Loop Renormalization
12.4 Renormalization
12.4.1 Slavnov-Taylor Identities
12.4.2 Identities for Proper Functions
12.4.3 Recursive Construction of the Counterterms
12.4.4 Gauge Dependence of Green Functions
12.4.5 Anomalies
12.5 Massive Gauge Fields
12.5.1 Historical Background
12.5.2 Massive Gauge Theory
12.5.3 Spontaneous Symmetry Breaking
12.5.4 Renormalization of Spontaneously Broken Gauge
12.5.5 Gauge Independence and Unitarity of the S Matrix
12.6 The Weinberg-Salam Model
12.6.1 The Model for Leptons
12.6.2 Electron-Neutrino Cross Sections
12.6.3 Higher-Order Corrections
12.6.4 Incorporation of Hadrons
13. Asymptotic Behavior
13.1 Effective Charge in Electrodynamics
13.1.1 The Gell-Mann and Low Function
13.1.2 The Callan-Symanzik Equation
13.2 Broken Scale Invariance
13.2.1 Scale and Conformal Invariance
13.2.2 Modified Ward Identities
13.2.3 Callan-Symanzik Coefficients to Lowest Order
13.3 Scale Invariance Recovered
13.3.1 Coupling Constant Flow
13.3.2 Asymptotic Freedom
13.3.3 Mass Corrections
13.4 Deep Inelastic Lepton-Hadron Scattering and Electron-Positron Annihilation into Hadrons
13.4.1 Electroproduction
13.4.2 Light-Cone Dynamics
13.4.3 Electron-Positron Annihilation
13.5 Operator Product Expansions
13.5.1 Short-Distance Expansion
13.5.2 Dominant and Subdominant Operators, Operator Mixing, and Conservation Laws
13.5.3 Light-Cone Expansion
Appendix
A-1 Metric
A-2 Dirac Matrices and Spinors
A-3 Normalization of States, S Matrix, Unitarity, and Cross Sections
A-4 Feynman Rules
Index