Physics 232B -- Quantum Field Theory II

Discussion sessions:
Time and Zoom link TBD (run by our GSI, Kevin).

Lecturer: Petr Hořava (email: horava@berkeley.edu)
Office: Pre-pandemic in 401 Le Conte Hall, now at home.
Office hours: Wed and Fri, 1:00-2:00pm (via Zoom, under a separate link). I am hopeful that the office hours will be a great platform for all of us to get to know each other better, and to discuss any questions related to the course, to physics, or to our lives under the pandemic in general. I strongly encourage the students to participate in the office hours, regardless of whether or not they have a specific question about the lecture material.

GSI: Kevin Langhoff (email: klanghoff@berkeley.edu).

In this advanced course, we will develop a more systematic understanding of Quantum Field Theory, building on the basics that we have learned in 232A (or equivalent). The subject of Quantum Field Theory is vast, with applications in virtually all areas of physics (and beyond) -- wherever many-body systems with fluctuations are involved. Ideas, methods and techniqes of QFT are now the prevalent language of theoretical physics, no longer confined only to high-energy particle physics: QFT is the go-to language and tool in particle phenomenology, condensed matter physics, equilibrium and non-equilibrium statistical mechanics, mesoscopic and AMO physics, quantum gravity, string theory and cosmology, with ramifications in mathematics and other fields. My aim is to stress this interdisciplinary nature of this fundamental theoretical "calculus of QFT" across subfields.

While teaching QFT, one could spend semesters upon semesters discussing more advanced topics, without ever repeating the same topics. In order to keep our thinking focused, my intention in this course is to concentrate on five main themes. The focus will again be two-fold: To develop a technical understanding of the techniques involved, while simultaneously getting the "big picture" (and trying to answer the "why?" question) for each theme. Compared to our previous 232A, we will provide a substantially more in-depth survey of each chosen topic. Here are the five major themes, with some more details on the specific topics that we will discuss:

Part 1. Systematics of renormalization. Renormalized perturbation theory. Evaluating loops and counterterms. Nonperturbative propagators, spectral representation. Unitarity of the S-matrix, Cutkosky rules, relation to Schwinger-Keldysh nonequilibrium formalism. Renormalization group, Callan-Symanzik equation. Relation of QFT to critical phenomena. QFT at finite temperature.

Part 2. Symmetries in QFT. Global symmetries, gauge symmetries, and their interplay with renormalization. QED and non-Abelian Yang-Mills gauge theories. Ward and Slavnov-Taylor identities. Faddeev-Popov ghosts, BRST quantization (including a brief look at the anti-bracket and the BRST-BV approach). Asymptotic freedom. Spontaneous symmetry breaking, Higgs mechanism. Renormalization of Yang-Mills. Quantum anomalies.

Part 3. Nonperturbative techniques. Focus on importance of topological invariants. Examples: Solitons, instantons, large-N expansion. Stable Fermi surfaces in Fermi-like liquids. Topological insulators and edge states in condensed matter.

Part 4. Effective field theories (EFTs). More on the Wilsonian approach to EFT and renormalization group. Quantum gravity and cosmological inflation as examples of EFT. Technical Naturalness and hierarchy puzzles in Nature (examples: Higgs mass hierarchy problem, cosmological constant problem, eta problem of cosmological inflation, resistivity in hgh-temperature superconductors). Time permitting: Hydrodynamics and EFT.

Part 5. Bonus topics. Here we can interactively determine the exact topics, based on the feedback from students during the semester. Some of the possible topics include: Holography, elements of gauge-gravity duality, condensed-matter aspects of AdS/CFT correspondence, interplay with hydrodynamics, topological quantum field theories and their use in mathematics or condensed matter, or even applications of QFT in financial markets. Any additional suggestions from the students are welcome.

In order to achieve a higher level of transparency and predictability, here is the approximate timeline on the weekly basis. I reserve the right to adjust the timeline as the course evolves forward, and we adjust the time we spend on individual topics based on the needs of the students.

Week 1 to 3: Part 1 -- Systematics of renormalization.

Week 4 to 7: Part 2 -- Symmetries in QFT.

Week 8 to 9: Part 3 -- Nonperturbative techniques.

Week 10: Spring break!

Week 11 to 13: Part 4 -- EFTs.

Week 14 to 15: Part 5 -- Bonus topics.

Week 16 (RRR week): Likely to be devoted to extra lectures with more of Part 5 -- Bonus topics.

Prerequisites

Physics 232A: Quantum Field Theory I (or equivalent).

Grading, Homeworks, Reading Assignments

We will have a few weekly Homework Assignments, after that we will move towards Discussion-Session presentations of Reading Assignments by the enrolled students. The list of Reading Assignments has now been posted here. At the beginning of the semester, before we were given a GSI, my plan was to spend less time on Homework Sets and have the presentations of Reading Assignments start earlier in the semester, but I revised this plan once we were given a GSI. Indeed, having a strong GSI benefits the students, and we were able to introduce a more balance between the Homework Sets and Discussions with someone who can provide a second opinion on the course material, versus just focusing our attention on the Reading Assignments. Please make your selection of your chosen Reading Assignment, I will accept your sign-ups by email starting on Friday, March 19, 2pm PDT.

Homework Assignments

HW1 (no due date): This first homework assignment of the semester was distributed by email on Jan 28.

HW2 (due on Thu, Feb 11): Problem 7.3(a,b) from [PS] (on pages 257-8); and Problem 7.1 from [PS] (The part of Problem 7.1 where you are asked to check the validity of the optical theorem (for this particular diagram) you may wish to postpone until after our Tue Feb 9 lecture, when we will be discussing the unitarity of the S-matrix and its relation to the optical theorem.)

HW3 (due on Thu, Feb 18): Problem 11.3(a,b,c,d) from [PS] (on pages 390-391). (Parts (a,b,c) are a good review of some topics we learned in QFT I last Fall; in contrast, the background material for Part (d) -- the notion of the effective potential, and how to calculate it -- will be covered in our next lecture, Tue, Feb 16.) I also recommend Problem III.8.2 from [Zee], page 218-219 -- its technical side can get quite tedious at times, but it is a good excercise for seeing unitarity and the Cutkosky rules at work.

HW4 (due on Thu, Feb 25): Problem 11.3(e,f) from [PS] (on page 391). Problem IV.3.4 from [Zee] (on p. 244). In addition, read the small paragraph entitled "Breaking by quantum fluctuations" at the bottom of p. 242 of [Zee], and do the small checks and calculations that he recommends there.

HW5 (due on Thu, March 4): Problems IV.1.1 and IV.1.2 from [Zee] (on p. 230). Then, read Chapter V.1 of [Zee] ("Superfluids"), and solve Problem V.1.1 (on p. 286).

There will be no more official HW sets for the rest of the semester. Instead, the students will concentrate on their Reading Assignments and prepare their presentations in the indicated Discussion Session.

horava@berkeley.edu