QUANTUM FIELD THEORY I: PHYSICS 662 Outline I. Special Relativity, Quantum Mechanics and Their Compatibility Review of Special Relativity Postulates of Quantum Theory Appendix: The Need for Particle Number Changing Interactions and Anti-Particles II. Poincar´eInvariance Poincar´eGroup Unitary Operator Representations of the Restricted Lorentz Group Space-Time Partity Poincar´eTransformation Properties of the Field Operators Representation of the Translation Group Representations of the Homogeneous Lorentz Group Relationship Between (A; B) Representations and Tensor Representations of the Lorentz Group Lorentz Covariant Free Field Equations Free Klein-Gordon Equation Free Weyl and Dirac Equations Free Proca and Maxwell Equations III. Classical Field Theory and Canonical Quantization Hamilton's Principle and the Euler-Lagrange Equations Noether's Theorem and Symmetries Space-Time Translations: Canonical Energy-Momentum Tensor Free Field Actions, Canonical Energy-Momentum Tensors and Hamiltonia Klein-Gordon Theory Dirac Theory Maxwell Theory Canonical Quantization 1 IV. Free Klein-Gordon Field Canonical Quantization Normal Mode Expansion and the Hamiltonian Eigenvalue Problem Wightman Functions Commutator Function Free Klein-Gordon Field Propagator Internal Symmetries Charged Spin-0 System V. Interacting Scalar Field Theory Spin Zero Two-Point Wightman Function Lehmann-Kallen Representation Lehmann-Kallen Representation for the Scalar Propagator Interacting Scalar Field Theory Two Point Wightman Functions as x0 ! ±∞ Yang-Feldman Equations Weak Asymptotic Condition S-Matrix Elements: General Formalism LSZ Reduction Formula: Spin-Zero States Schwinger (Quantum) Action Principle Vacuum Functional Gell-Mann Low Theorem Free Field Theory Green Function Generating Functional Potential Function Between Static External Sources in Free Klein-Gordon Theory VI. λφ4 Perturbation Theory: Feynman Rules 2 Vacuum Functional Through O(λ0) Wick's Theorem for the Vacuum Functional 2 Full Two-Point Green Function to O(λ0) 2 Full Four-Point Green Function to O(λ0) Feynman Rules for Green Functions: Coordinate Space Feynman Rules for Green Functions: Momentum Space T-Matrix Elements 4 Feynman Rules for the T -Matrix Elements for n ! m Scattering in λ0φ0 2 2 ! 2 Scattering Cross Section to O(λ0) VII. Path Integral Quantization: Scalars Functional Integral Gaussian Functional Integral Change of Variables Functional Determinants Change of Variables: Continued Feynman Path Integral Scalars Gell-Mann Low Theorem Revisited Free Field Vacuum Functional Feynman Rules Complex (Charged) Scalars VIII. Loop Expansion and the Effective Action Loop Expansion Tree Approximation One-Loop Correction Connected Green Function Generating Function Classical Field Quantum Effective Action: 1PI Generating Functional Effective Potential IX. Renormalization of λφ4 Field Theory Renormalized Two-Point Function to One Loop Renormalized Four-Point 1PI Function to One Loop Dimensional Regularization Feynman Parametrization Wick Rotation On the Renormalization of λφ4 Theory in Higher Orders Renormalized Perturbation Theory and Counterterm Lagrangian: λφ4 X. Renormalization Group 3 Renormalization Group Equation: Massless λφ4 Solution of Renormalization Group Equation: Running Couplings Inclusion of Dimensionful Couplings: Relevant, Irrelevant and Marginal Op- erators Renormalization of Local Composite Alien Operators Renormalization Group Running in Arbitrary Dimensions Wilson Renormalization Group XI. Global Symmetries and Spontaneously Broken Global Symme- tries Global Space-time Symmetries Translation Invariance Ward Identity Lorentz Invariance Ward Identity Continuous Global Internal Symmetry Ward Identities Wigner-Weyl Realization of Symmetry O(2) Linear σ-Model Nambu-Goldstone Realization U(1) Goldstone Model Lee-Symanzik Theorem 1-Loop Effective Potential 1-Loop Effective Potential: O(N) Sigma Model Renormalization Group Improved Effective Potential Non-Linear O(N) Sigma Model 4.