# PHY2404S

## Quantum Field Theory II

### Announcements

• Apr.04: I will be holding office hours tomorrow from 11am-noon (in MP1115), to assist students with any last-minute questions about remaining graded work.
• Mar.13b: HW4 is posted. It is due by 11:00am on Mon.02.Apr.2018.
• Mar.13a: oops. I accidentally mis-named the lecture notes .pdf when uploading the new version last night. This situation is now fixed.
• Mar.12: I have made some nontrivial updates to the lecture notes sections on one-loop renormalization of quartic scalar field theory, to remove a number of unfortunate typos and reduce confusions arising from those typos. I will reiterate a couple key aspects of this before I start discussing one-loop renormalization of QED starting on Thursday. (It will take us about four lectures to achieve it, and the final two lectures will be on nonperturbative physics.)
• Mar.02: I have booked MP1115 from 9am-3pm on April 20th to facilitate the group work part of the final exam. Please let me know by the start of class tomorrow if you cannot make it for the proposed Apr.19-20 exam timeframe.
• Mar.01: Please mark your calendars with the following two new pieces of data. (1) The lecture of March 8th will instead be held on March 9th from 14-15h, because I will be speaking on a panel at York U. in the morning at a clashing time. (2) We decided that the exam would be held on April 19th and 20th. If you skipped class today and cannot make those exam dates, tell me immediately.
• Feb.26: HW3 is posted.
• Feb.01: HW2 is posted. For now, concentrate on Q1 and Q2. You will soon also be able to do Q3.
• Jan.15b: IMPORTANT! If you are taking the course for credit, please fill out this form showing me your handwritten alphabets, scan it, and email it to me. This will enable me to grade your work without making mistakes. Thank you.
• Jan.15: Please note: I am happy to give out hints for homework assignments. A good time to pop by to ask questions about QFT2 problems is right after lectures. Or feel free to make a private appointment.
• Jan.14: I have updated the lecture notes to make them clearer and more useful. Here are the remaining two problems for HW1: HW1 Q2 and HW1 Q3. They are designed to familiarize you with pertinent Yang-Mills-ology. The due date for all three problems is Jan.29 at 11am and the page limit is 25 pages.
• Jan.08b: I like to give out homework problems as soon as I have covered the material in lectures, to give you as much time as possible to complete them. Here is HW1 Q1. It is designed to familiarize you with the algebra of the Poincare group of symmetries of flat Minkowski spacetime.
• Jan.08a: A few words about lecture notes. I rarely cover more than six pages of notes in one lecture, and recommend reading ahead as a good way to stay on top of the material. I usually do minor surgery on my lecture notes (a) the day/night before each lecture or the morning of, and (b) immediately afterwards when I fix typos or add needed clarifications.
• Jan.04: It was great to meet you all in class today. I have updated the lecture notes to make the discussion of today's material clearer.
• Aug.03: Our first class will be in MP1115 from 11:10am-12:00noon on Thursday 4th January 2018 (the first day of Winter/Spring semester). I will begin with organizational matters, and then start discussing the physics of Noether's Theorem. See you there!

### 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.

### Syllabus

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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