Quantum Field Theory II


Textbook, notes, and assessment

QFT textbooks I recommend: Matthew Schwartz, Michael Peskin and Daniel Schroeder, Thomas Banks, Anthony Zee, Lewis Ryder, Pierre Ramond, and the encyclopaedic Steven Weinberg.

Lecture notes are provided online as an accessibility measure. [expect a few typos!]

Assessment will be 60% on homeworks, 40% on the final exam. There will be four homework assignments, due on Jan.29, Feb.19, Mar.12, and Apr.02. Please carefully read my deadline policy, noting in particular my Grace Days policy. Lateness beyond the due date (plus Grace Days) will be penalized 5% per day, and the latest you may hand in an assigment is one week after its due date. Assignments must not exceed 25 pages in length. The open book final exam will involve both individual and group work, and will be held over two adjacent days.


Symmetries and conservation laws
Noether's theorem and continuous (Lie) groups. The Poincaré group and the origin of wave equations, spin, and helicity. Abelian gauge symmetry and QED. Non-Abelian Yang-Mills theory and QCD. Chirality and the electroweak sector of the Standard Model.
Spontaneous symmetry breaking
Goldstone's theorem. SSB with global symmetry: linear sigma model. SSB with local gauge symmetry: Abelian Higgs. SSB for non-Abelian gauge symmetry, and how the Higgs boson gives mass to vectors and fermions.
The Feynman Path Integral
FPI for non-relativistic point particles. Functional quantization for scalar fields. Correlation functions and the Feynman propagator. Functional determinants. General potentials and sources.
Generating functionals (GFs) for spin zero
GF for all Feynman graphs. Schwinger-Dyson equations. Feynman rules. GFs for connected and one-particle-irreducible graphs. The S-matrix and the LSZ reduction formula. Spectral representation for interacting QFTs.
Functional quantization for spin half
FPI quantization for fermion fields. Grassmann variables and Grassmann integration. Functional determinants. Dirac fields and the generating functional with fermionic sources. Grassmann differentiation. Weyl and Majorana fields.
Functional quantization for spin one
Defining the measure of the FPI in the presence of gauge symmetry. The Fadeev-Popov ghost procedure for Abelian and non-Abelian cases. Lorentz gauge Feynman rules. BRST invariance and unitarity.
Renormalization and quartic scalar field theory
Length scales and UV cutoffs. 1PI graphs in $\phi^4$ scalar field theory. Divergences in $\phi^4$ theory in $D=4$. Divergences in general, power counting, and Weinberg's theorem. Dimensional regularization. Propagator and vertex corrections for $\phi^4$.
Callan-Symanzik equation and the Wilsonian renormalization (semi)group
Counterterms. The Callan-Symanzik equation, beta functions, and anomalous dimensions. Fixed points. Wilsonian RG: UV cutoffs, integrating out degrees of freedom, RG flow, relevant and irrelevant operators.
One loop renormalization of QED
Photon and electron self-energies and QED vertex correction. Ward-Takahashi identities. Photon masslessness and charge renormalization. Optical theorem and Cutkosky rules. Counterterms and the QED beta function.
An introduction to chiral anomalies
Anomalies in path integral quantization. The triangle anomaly via functional methods and Feynman graphs. Anomaly cancellation for chiral gauge field theories.

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