#### Syllabus

Chapters 1,2

- Introduction and notation
- Relativistic notation
- Classical mechanics and action principle
- Classical fields and functional differentiation
- Solutions of the Klein-Gordon and Maxwell equations
- The stat mech analogue

Chapter 3

- Symmetries and Conservation Laws
- Currents for translation, Lorentz or rotational, and internal symmetries
- Lie groups, Lie algebras and representations
- Gauge invariant actions

Chapter 4

- PoincarĂ© group and representations
- Spinors: Lorentz properties and actions for Weyl and Dirac spinors
- Particle states and little groups
- Solutions of the free Dirac equation

Chapter 5

- Canonical quantization
- 1d lattice models and the continuum and thermodynamic limits
- Relativistic scalar quantum field theories
- Quantization of the Dirac theory (3+1)
- Propagators and causality; contour prescriptions

Chapter 6

- Functional integral quantization
- Quantum mechanics; path integral representations for correlation functions; perturbation Theory
- Vacuum to vacuum transitions and time ordering
- Scalar field theories in Lorentzian and Euclidean spacetime
- The mass gap
- Schwinger-Dyson equations
- The generating functional and perturbation theory
- Feynman rules
- The S-matrix and LSZ reduction
- Tree-level scattering in the Yukawa model
- The Spectral Representation

Chapter 7

- Gauge theory quantization
- Propagators and their gauge dependence
- Functional quantization: the Faddeev-Popov method
- Gauge fixing and ghosts
- BRST quantization

Chapter 8

- One-loop Amplitudes in QED
- Dimensional regularization and the structure of UV divergences
- Vacuum polarization
- Cutkosky rules, unitarity and the non-relativistic Coulomb potential
- Fermion self-energy; pole mass and residues
- The vertex correction; structure functions and the anomalous magnetic moment

Chapter 9

- Renormalized Perturbation Theory
- Power counting, local counterterms
- Effective Actions
- Background Field Methods

Chapter 10

- Renormalized Couplings
- The Wilson RG
- Fixed points, RG flows, critical exponents, epsilon expansion
- The Exact RG
- The Beta Function in Gauge Theories
- Chiral Anomalies

Chapter 11

- Symmetry Breaking
- Spontaneous Breaking of Global Symmetries, Goldstone's Theorem
- Spontaneous Breaking of Local Symmetries, Higgs phenomena
- Quantization and Renormalization