Quantum Field Theory II


Textbook, notes, and assessment

I recommend QFT textbooks by Michael Peskin & Daniel Schroeder, Lewis Ryder, Pierre Ramond, Tom Banks, Steven Weinberg, Mark Srednicki, Ta-Pei Cheng and Ling-Fong Li, and Tony Zee.

I provide my own frequently updated lecture notes for free online.    [old notes: expect a few typos!]

Assessment will be based on five equally weighted components: four homework assignments and a final exam. Homeworks will be due on Jan.30, Feb.20, Mar.13, and Apr.03. The final exam will be to compute the one-loop beta function of QCD together, and will involve both individual and group work, held over over two adjacent days.

Syllabus (approximate)

Symmetries and Conservation Laws
Noether's Theorem, Continuous (Lie) Groups, The Lorentz Algebra, Casimir Operators and Lorentz Irreps, Abelian gauge symmetry and QED, Nonabelian gauge symmetry, Yang-Mills Lagrangian, Yang-Mills equations of motion, Yang-Mills vs General Relativity, Chirality and the Standard Model, Gauging isospin and hypercharge
Spontaneous Symmetry Breaking
Goldstone's Theorem and Spontaneous Symmetry Breaking (SSB), SSB with global symmetry: Linear Sigma Model, SSB with local gauge symmetry: Abelian Higgs, Summary for U(1) Goldstone and Higgs, SSB, Non-Abelian Gauge Symmetry, and the Standard Model, The Higgs boson, SSB: Higgsings and Vector Bosons, SSB: Higgsing and Fermion Masses
The Feynman Path Integral
FPI for non-relativistic point particles, Functional quantization for scalar fields, How to pick off correlation functions, Free massive scalar field, Functional determinants, Free two-point correlation function, Free four-point correlation function, General potentials
Generating functionals and the Schwinger-Dyson equations
The generating functional for all Feynman graphs, The generating functional for free scalar field theory, Analogy between Statistical Mechanics and QFT, The generating functional and the Schwinger-Dyson equations, Feynman rules for scalar field theory, Generating functional for connected graphs W[J], Semiclassically speaking, Quantum [effective] action and Legendre trees, The S-matrix and the LSZ reduction formula, Kaellen-Lehmann spectral representation for interacting QFTs
Functional Quantization for Spin Half
FPI Quantization for Fermions: Grassmann Variables, Grassmann Integration, Gaussians, Functional determinants, Dirac Fields, Generating Functional for Dirac Fermions, Fermionic Sources, Grassmann Derivatives, Weyl and Majorana fields
Functional Quantization for Spin One
U(1) Quantum Electrodynamics, Gauge Invariance, Fadeev-Popov Procedure: Abelian case, Gauge-Fixed Photon Action at Tree Level, Functional quantization for Yang-Mills fields: quick version, Fadeev-Popov ghosts for Yang-Mills fields, Lorentz gauge Feynman rules for Yang-Mills, BRST invariance and unitarity
Renormalization and Quartic Scalar Field Theory
Length Scales, Cutoffs, Focus: 1PI Diagrams in $\phi^4$, Divergences in $\phi^4$ in D=4, Divergences in General, Weinberg's Theorem, Dimensional Regularization and the Propagator Correction, Vertex Correction, Counterterms
Callan-Symanzik equation and Wilsonian Renormalization Group
Counterterms, Callan-Symanzik Equation, Fixed Points, Wilsonian RG and UV cutoffs
One loop renormalization of QED
Power counting, Photon self-energy a.k.a. Vacuum Polarization, Electron self-energy, QED Vertex Correction, Ward-Takahashi Identities in QED, Photon masslessness and Charge Renormalization, The Optical Theorem and Cutkosky Rules, Counterterms and the QED beta function
An introduction to chiral anomalies
Anomalies in Path Integral Quantization, Triangle anomaly: the Feynman Diagram approach, Anomaly cancellation for chiral gauge field theories