Physics 232B -- Quantum Field Theory II

Spring 2024

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Basic Info

Discussion sessions:
Time: Thu, 3:40-4:30pm
Place: 402 Physics South

Lecturer: Petr Hořava (email: horava@berkeley.edu)
Office: 401 Physics South.
Office hours: TBD, likely on Tue, 2:00-3:00pm, or a meeting can always be arranged individually by an email request.

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, 2022 and 2023 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. Even though AdS/CFT and holography really originated from string theory, we can now teach its basics in the "bottom-up" way, 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). We will focus primarily on the effective field theory of cosmological inflation, and quantum gravity, as examples of EFT. Technical Naturalness and hierarchy puzzles in Nature (examples: Higgs mass hierarchy problem, cosmological constant problem, the 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 primary 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 2021, 2022 or 2023 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.

Grading, Reading Assignments, Discussion Sessions

Besides the participation in class, the main portion of the grade consists of Reading Assignments: After selecting their Reading Assignment from the official list, we will switch the format of the Discussion-Session starting from Thursday, April 11, to consists of Reading Assignments presentations by the students; each session will contain three laptop presentations with a projector, each presentation being assigned 20 minutes plus 5 minutes for discussion/questions. The list of available Reading Assignments has been posted here. This list has been chosen so that it significantly enhances the scope of material we can cover in lectures, and it mostly contains original arXiv papers. Now that the list of Reading Assignments has been posted, you will have a peaceful few days until Tuesday, April 2, 12 noon to review the list and see which papers might be your favorites. After this moratorium, each student will sign up for their choice of one paper by email, on the first-come first-served basis. Please wait with sending your email request with your selection until April 2, 12 noon; all earlier requests will be ignored. I will then confirm by return email whether you have been assigned your chosen paper. 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, and send it to me by email so that I can share it with all the students. No paper will be assigned to more than one student. The final grade will be a combination of the participation in class and discussions, and the presentation of the Reading Assignment in the Discussion.

References and Recommended Reading

Week 1: We covered the logistics for the course, I presented my vision for the interdisciplinary topics I wish to cover this semester, and then we had an interactive discussion in which the students themselves expressed their intersts and stressed which particular topics and areas they would like to learn in this class. Week 2: Before we could move to the more advanced topics, such as systematics of renormalization and the Renormalization Group, or quantization of Yang-Mills gauge theories, we began with Part 0: Review of the Path-Integral Method of Quantization. The recommended reading is first of all Chapter I of [Zee], and beginning sections of Chapter 9 of [Peskin-Schroeder]. In addition, some extra material that I also covered in lectures and our discussion session this week can be found in one of my favorite books on this topic, the elegant and accurate book J. Zinn-Justin, Path Integrals in Quantum Mechanics (Oxford U.P., 2005), and/or in the much more encyclopedic Quantum Field Theory and Critical Phenomena (Oxford U.P., 2002), from the same author. Although these books may be a bit tangential to the scope of this course, they are 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; another nice and careful discussion of the precise way how to analytically continue the Gaussian integral is in Section 8 of V. Guillemin and S. Sternberg, Symplectic Techniques in Physics (Cambridge U.P., 1990).

horava@berkeley.edu