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