# PHY2404S

## Quantum Field Theory II

### Announcements

• Apr.24: congratulations, everyone, for a really fine performance on the QCD beta function final exam. I was also very pleased with your homeworks. It is hard to assign statistical weight to such a small experiment, but I believe that your cooperation skills improved over the semester. At any rate, the quality of this year's homework solutions surpassed previous years'. A rising tide raises all boats, indeed!
• Apr.12b: Throughout the semester, I have mentioned in class what would be in the final exam. I have just emailed you the first page of the exam for Monday and Tuesday next week which summarizes the instructions in one place. You will receive your Feynman diagram assignments at 9am on Monday morning by email.
• Apr.12a: I hope HW4 is going OK for everyone. As mentioned previously, it is important preparation for the final exam, because it helps you rehearse computing beta functions with a simpler theory than QCD. The due date was Apr.03; please do your best to hand in HW4 before the final exam. If you need to consult me on HW4, my best availability this week is Fri.14.Apr (a holiday).
• Mar.22: I am super sorry that my grading of homeworks has been so delayed. I have had some mental health issues arise (PTSD recurrence) and I had to take some of my discretionary physics time for healing from that. After some hard work I have conquered the bulk of it, so I should be catching up with backlogged grading pretty quickly from Friday onwards. I will be clearing backlogs for my more junior classes first; I really appreciate your patience.
• Mar.20: I have booked a room for the second day of the final exam, to foster easier working together as a group. MP408 will be at your disposal from 9am-3pm.
• Mar.15: HW4 is posted. It is due Apr.03, the last Monday of term. I will send pod assignments by email shortly.
• Feb.27: HW3 is posted. It is due Mar.13. I will send pod assignments by email shortly.
• Feb.14: Happy Valentine's Day: I updated the lecture notes. There are quite a few improvements, although still sadly not all typo bugs have been squashed.
On Thursday, I will start discussing how to do Fadeev-Popov ghosts (correctly path integral quantize a gauge field). We will take three whole lectures to develop this stuff slowly, because it is nuanced and important. In the first, on Thu.16.Feb, I will discuss ghosts for U(1) electromagnetism. Then, after Reading Week, on Mon.27.Feb, I will discuss ghosts for Yang-Mills gauge fields. On Thu.02.Mar I will describe how to work out the Feynman rules in an arbitrary gauge. I will also describe the even more modern BRST method for tracking which states are physical in path integral quantization when you have gauge symmetries lurking about. Then after that, we will get stuck into one-loop renormalization. This will occupy us for the rest of the course and prepare us for the final exam.
• Feb.08: here's HW2 due Feb.20. Pod assignments will be transmitted shortly by e-mail.
• Feb.07: here's the note on Legendre transformations as promised in class yesterday.
• Feb.01: thanks for all your HW1s! (I'm waiting for the last one to start grading.)
• Jan.31: only one pod turned in HW1 on the due date. Other pods: please email me to let me know when to expect your assignments.
• Jan.24: I re-fixed a HW1 typo. My apologies for the sign confusion!
• Jan.17: HW1 is posted. Pod assignments are being transmitted by email: check your inbox!
• Jan.11: The plan is for my PhD student J. Cresswell to teach my QFT class tomorrow Jan.12 while I am away at a conference at Harvard. He will introduce gauge symmetry, first the Abelian kind and then the non-Abelian kind. I will resume lecturing next Monday.
• Please send me a quick test email, to ensure that I have the addresses of everyone interested in the class (not only those taking it for credit). If you like the idea, please share with me in your email (a) your motivation for taking the class, (b) what you hope to learn, and (c) what approximate grade you are aiming for. That way I can do a better job of teaching to your specific needs/interests.
• Jan.09: Today I will introduce a little bit of group theory, and the Poincare group of symmetries of flat Minkowski spacetime. Then I will explain how wave equations, spin angular momentum, and helicity all arise from symmetry considerations.
• Jan.08: The distribution of students taking QFT2 this year is different from historical averages. Accordingly, I will alter my teaching style, to allow me to best serve the students in the whole class. In particular, I plan to make the material somewhat less formal, and the homework assignments less work than last year. Lecture notes will be updated accordingly - usually the day/night before lecture or the morning of.
• Jan.05: thank you to everyone for filling out the form in class today, especially the part about your intended field of specialization.
• Dec.07: Our first class will be in MP1115 from 11:10am-12:00noon on Thursday 5th January 2017 (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

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