Physics 232B -- Quantum Field Theory II

Spring 2023

shortcut to Homework Assignments
shortcut to List of Reading Assignments
shortcut to References and Reading Recommendations

Basic Info

Discussion sessions:
Time: Thu, 3:40-4:30pm
Place: 402 Physics South (actually, we have the room reserved from 3:30pm to 5pm).

Lecturer: Petr Hořava (email: horava@berkeley.edu)
Office: 401 Physics South.
Office hours: Tue, 2:00-3:00pm; in addition, during most weeks, a secondary option on Wed, 1:00-2:00pm.

GSI: Unlikely.

In this advanced course, we will develop a more systematic understanding of Quantum Field Theory, building on the basics that you have learned in Physics 232A (or equivalent), as taught for example in Fall 2021 or 2022 by Yasunori or in Fall 2020 by me. 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" (especially, in its form centering on Wilsonian approach to renormalization) across subfields.

The focus will again be two-fold: To continue developing a strong technical understanding and mastery of the theoretical techniques involved, while simultaneously getting the "big picture" of understanding the role of QFT in describing the behavior of many-body systems and cooperative phenomena. The "big picture" that I will focus on will stress two main concepts in QFT:
(1) Renormalization (especially as covered by the concept of the renormalization group),
and
(2) Symmetries (both global and gauge symmetries).
Much of modern QFT deals with the mutual interplay of these two concepts, and we will illustrate this by focusing on several specific themes more deeply. Here are the five major themes, with some more details on the specific topics that we are planning to discuss:

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

2. Symmetries in QFT. Global symmetries, gauge symmetries, and their interplay with renormalization. The main focus will be on non-Abelian Yang-Mills gauge theories, and their quantization. Faddeev-Popov ghosts, BRST quantization (including a brief look at the anti-bracket and the BRST-BFV approach). Asymptotic freedom. Spontaneous symmetry breaking, Higgs mechanism. Renormalization of Yang-Mills. Quantum anomalies. Topological quantum field theories and their mathematical applications.

3. Holographic dualities, intro to AdS/CFT correspondence. While AdS/CFT and holography originated from string theory, we can now teach its basics without requiring any string-theory pre-requisites. Elements of gauge-gravity duality, holographic renormalization, condensed-matter aspects of AdS/CFT correspondence, connection to Quantum Information Theory etc.

4. Nonperturbative techniques. Focus on importance of topological invariants. Examples: Solitons, instantons, large-N expansion. Nonperturbative dualities between QFTs. Stable Fermi surfaces in Fermi-like liquids. Basics of topological insulators and edge states in condensed matter.

5. Effective field theories (EFTs). 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 high-temperature superconductors).

Overall, I expect that the exact focus of the various parts of this course will be determined interactively, based on my polls of the research interests of the students who will register. Based on the level of interest, additional topics can be added (or subtracted) from this list, depending on the early feedback that I will get at the beginning of the semester from the course participants.

Required and Recommended Textbooks

There are two required textbooks:

M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory (Perseus, 1995),
and
A. Zee, Quantum Field Theory in a Nutshell, 2nd Edition (Princeton U. Press, 2010).

I shall refer to them as [PS] and [Zee] from now on.

In addition, there are many possible recommended texts, of which I will now mention only three. First,
S. Weinberg, The Quantum Theory of Fields, Volumes 1 and 2 (Cambridge U. Press),
is a brilliant resource, focused on the fundamental principles and properties of relativistic QFT as developed primarily in particle physics. This semester, it will be Volume 2 that will be most relevant to us.

M.D. Schwartz, Quantum Field Theory and the Standard Model (Cambridge U.P., 2014).
This book can be viewed as an updated version of [PS], with many new modern aspects of QFT and particle theory covered in extra detail (including for example Effective QFT).

F. Gelis, Quantum Field Theory: From Basics to Modern Topics (Cambridge, 2019).
This book is great at stressing the interplay between equilibrium and non-equilibrium QFT, Schwinger-Keldysh formalism and its uses, and the connections between similar QFT concepts in condensed matter, particle physics, etc.

Later on, in the second half of the semester ,I will add to this list two important textbook references to Effective Field Theories in general, and Effective Field Theories of Cosmological Inflation in particular.

Prerequisites

Physics 232A: Quantum Field Theory I (or equivalent). In particular, Yasunori's 232A from Fall 2021or 2022 is the precise level of the required pre-requisite this semester.

And, I should say, no prior knowledge of string theory required (even for the part of the course where we will discuss AdS/CFT correspondence)! On the other hand, some rudimentary knowledge of classical general relativity will at times be useful. In particular, because we will spend time discussing holography and AdS/CFT correspondence (from the field-theoretical, "bottom-up" approach), this course would be particularly suitable for those students who were hoping to sign up for String Theory I (234A) in Fall 2022, when it unfortunately was not offered. This course might give at least a partial way how to catch up on this fascinating material!

Discussion Sessions, Homeworks, Reading Assignments

The structure of the Discussion Sessions will be very similar to my 232A: QFT II that I taught in Spring 2022, and before that in Spring 2021: First we will have a few weekly Homework Assignments, which will be due in one week, and whose solutions will be discussed in our Discussion Sessions. After that, starting after Spring Break, we will switch to Discussion-Session presentations of Reading Assignments by the enrolled students. The list of available Reading Assignments will be posted on this webpage sometime before Spring break. Once it is posted, students will sign up for their choice by email, on the first-come first-served basis. Each paper will have an assigned presentation day and time during the Discussion sessions, and each student will prepare a two-page typed-up summary of the paper before their presentation. No paper will be assigned to more than one student. The final grade will be a combination of the participation in class, performance on homeworks, and the presentation of the Reading Assignment in the Discussion.

References and Recommended Reading

Week 1: We will discuss the logistics and the possible topics to cover in this semester, so no recommended reading material for this week.

Week 2: On the topic of path integrals in quantum mechanics (and QFT), one of my favorite books is the elegant and accurate book
J. Zinn-Justin, Path Integrals in Quantum Mechanics (Oxford U.P., 2005).)
Although this book may be a bit tangential to the scope of this course, it is an extremely valuable source of insights on some of the more subtle aspects of the path integral method.
When we used Gaussian integrals, we treated the important case in which the Gaussian has a purely imaginary coefficient of the quadratic term in the exponential by naive brute-force analytic continuation (as in [Zee]). In the math literature, this continued version of the Gaussian integral is often referred to as the "Fresnel integral".) For a much more mathematically rigorous direct treatment of the Fresnel version of the Gaussian integral, see Howie Haber's notes at UCSC.

In the following weeks, all the way until Spring break, our main references for the lectures are the corresponding chapters from [Peskin-Schroeder] and [Zee] (plus sometimes [Schwartz] or [Weinberg]). Which Chapters align with the material of each given week is self-explanatory from the assignments of the Homework sets HW1-HW7. In addition, the following references may be useful for the material covered before Spring break: BRST symmetry and BRST quantization: Chapter 4 in Volume 1 of J. Polchinski, String Theory (CUP, 1998). BRST-BV formalism is explained clearly in Chapters 15-17 in Volume 2 of S. Weinberg, Quantum Theory of Fields (CUP, 1996). For applications of BRST cohomology to quantum anomalies, including the origin of the Wess-Zumino consistency conditions for the anomalies, see also Chapter 22 of [Weinberg]. Our treatment of the one-loop renormalization and RG flow in Yang-Mills gauge theories with matter followed Chapters 25 and 26 of Matt Schwartz's Quantum Field Theory and the Standard Model (cited above). A lot more information about the mathematics and physics of BRST can be found in the very deep, thorough and detailed book: M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (PUP, 1994). After Spring break, we no longer followed any standard textbooks, and here I will list the most useful references to the specific topics that we have covered: Intro to large-N expansion, holography and AdS/CFT correspondence: For the most part, I followed Maldacena's lectures J.M. Maldacena, TASI 2003 Lectures on AdS/CFT, arXiv:hep-th/0309246. A lot more information about the technicalities of AdS/CFT correspondence can be found in the thorough and carefully written book: M. Ammon and J. Erdmenger, Gauge/Gravity Duality (CUP, 2015). The big-picture and big ideas about applying holography and AdS/CFT in condensed matter can be found in a very readable and exciting book: J. Zaanen et al, Holographic Duality in Condensed Matter Physics (CUP, 2015). For methods of Effective Field Theory, in quantum gravity and otherwise, there is a great new book C.P. Burgess, Introduction to Effective Field Theory (CUP, 2020). In my lectures on EFT approach to quantum gravity specifically, I followed closely John Donoghue Petropolis lectures, J. Donoghue, Effective Field Theory Treatment of Quantum Gravity, arXiv:1209.3511[gr-qc]. The important topic of the EFT approach to cosmological inflation is covered in detail in D. Baumann and L. McAllister, Inflation and String Theory (CUP, 2015).

Homework Assignments

HW1 (posted on Tue, Jan 24; due on Tue, Jan 31 in class): Problems I.2.1 and I.2.2 from [Zee] (on p. 16 of the 2nd edition).

HW2 (posted on Tue, Jan 31; due on Tue, Feb 7 in class): Problems 9.2(c) and 9.2(d) from Peskin-Schroeder (or [PS] for short; on p. 313), and Problem I.3.3 from [Zee] (on p. 24).

HW3 (posted on Tue, Feb 7; due on Tue, Feb 14 in class): Problems III.2.1 (on p. 172 of [Zee]), III.3.1 and III.3.2 (on p. 181), and Problem 10.2(a) on pp. 344-5 of [PS].

HW4 (posted on Tue, Feb 14; due on Tue, Feb 21 in class): Problem III.1.3 (on p. 168 of [Zee]), Problem 11.3(a,b,c) on p. 390 of [PS], and then attempt Problems IV.3.3 and IV.3.4 (on p. 244 of [Zee]).

There is no new homework for the week of Feb 21.

HW5 (posted on Tue, Feb 28, due on Tue, March 7 in class): All problems from this set deal with the non-equilibrium surface growth system as introduced today in class. First, read Section VI.6 of [Zee], and solve Problem VI.6.2 (on p. 349). Then, solve Problems VI.8.3, VI.8.4 and VI.8.5 (on p. 368). Finally, as an optional bonus problem, I also include Problem VI.8.6 from [Zee].

HW6 (posted on Tue, March 7, due on Tue, March 14 in class): I recommend that you first review Chapter 15 of [PS], especially if some of the material about Yang-Mills as covered in the lecture today was new to you. Then, solve Problems IV.5.1, IV.5.2, IV.5.3, IV.5.4 and IV.5.6 from [Zee] (pp. 261-2).

HW7 (posted on Tue, March 14, due on Tue, March 21 in class): First, prove by an explicit calculation that the BRST transformations for Yang-Mills as defined by Eqn. (16.45) in [PS] (on p. 518) are such that the BRST charge squares to zero. Then, demonstrate that the gauge-fixed action in Eqn. (16.44) is invariant under the BRST charge. Finally, solve Problem 16.1 (from p. 544) in [PS].

HW7 was the final homework set of this semester. Now we are moving on to the Reading Assignments, and their presentations in the Discussion Sessions (after Spring break). The list of available Reading Assignments has been posted here. The sign-up time, after which you can request your choice of a Reading Assignment (by email, on a first-come, first-served basis) is Friday, March 24, 1pm, PDT. (All email requests received before that cut-off time will be ignored.)

horava@berkeley.edu