Physics 232B -- Quantum Field Theory II

Spring 2025

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Basic Info

Discussion sessions:
We have now been granted our request for a more practical time and location of our Discussion Sessions: Starting from Thursday, Feb 6, the Discussion Sessions during the rest of the semester will always be on Thursdays, 3:40-4:30pm, at 402 Physics South (instead of the time and location that was originally assigned to us at the beginning of the semester, which was Thu, 12:40-1:30pm in 174 Social Sciences Building, and which was clashing with other important lectures taught in BCTP).

Lecturer: Petr Hořava (email: horava@berkeley.edu)
Office: 401 Physics South.
Office hours: We agreed interactively that this semester, the office hours that work for all students will be on Tue, 1:00-2:00 pm, in 401 Physics South. (Outside the office hours, a meeting can always be arranged individually by email.)

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.

This semester, we will also devote an extra focus to an excellent (and quite recent) book on the extremely important, interdisciplinary and universal concept of effective field theory, written by Cliff Burgess:

C.P. Burgess, Introduction to Effective Field Theory (Cambridge U.P., 2021).

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 not only the important textbook references to Effective Field Theories, but specically also on Effective Field Theories of Cosmological Inflation (and quantum gravity).

Prerequisites

Physics 232A: Quantum Field Theory I (or equivalent). In particular, any of the Berkeley one-semester courses 232A from previous Fall semesters 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 somewhat useful.

Grading, Reading Assignments, Discussion Sessions

TBD.

References and Recommended Reading

Week 1: We looked first at the big picture, what is the vast landscape of applications of Quantum Field Theory (QFT) and what specifically are the (necessarily selected) topics we wish to cover in this vast semester. In the second lecture, the students had the opportunity to discuss their own interests and wishes, leading us to the interactive fine-tuning of the intended lecture material for this semester. Specifically, we will cover the following:
0. The Path Integral Method (both in Quantum Mechanics and in QFT). This important technique is necessary, as a preamble before we get to our main topics, since it was not covered in 232A: QFT I for most of the students taking our class.
1. Quantization of Gauge Theories; the BRST Quantization Method. We will see the origin of Faddeev-Popov ghosts in general gauge theories, and focus on the detailed quantization of non-Abelian Yang-Mills gauge theories, primarily being led to the central method common to most gauge theories: The BRST quantization method, and its natural mathematical structure of BRST cohomology. Having developed this method, we will apply it to other gauge theories, including the topological Yang-Mills and its consequences for pure mathematics.
2. Selected Nonperturbative Techniques. Here we will discuss first some semi-classical methods, leading us to concepts of solitons and instantons. We will discuss the importance of topology, with some condensed-matter examples, including the concept of topological insulators. We will discover the first hints of the usefulness of another nonperturbative technique: The large-N approximation, and the 1/N expansion, primarily preparing for its role in the next chapter,
3. Holographic Duality and the AdS/CFT Correspondence. This important topic will be presented solely using QFT, in the bottom-up approach; no string theory required!
4. Effective Field Theories. This final chapter is perhaps the most central one of the whole semester. We will discuss the logic, structure and implications of EFT. We will focus on several intriguing and important examples, including Quantum Gravity, and the Effective Field Theory of Cosmological Inflation.

Week 2: This week, we jumped directly into Chapter 0: The Path-Integral Method, first focusing on its role in Quantum Mechanics. The text is parallel to Chapter 9 of [Peskin-Schroeder] (denoted [PS] here and from now on), and Chapters I.2 and following in [Zee].
Some of the various tidbits and extra insights into various subtler aspects of path integrals (several of which I presented in class) can be found in the two beautiful books by Jean Zinn-Justin,
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 5th edition (Oxford University Press, 2021).
J. Zinn-Justin, Path Integrals in Quantum Mechanics (Oxford University Press, 2011).
The precise explanation how to perform carefully the analytic continuation from purely real Gaussian integrals to their purely imaginary counterparts (often referred to as "Fresnel integrals", and directly relevant to the process of "Wick rotation" in relativistic QFT) can be found in Howie Haber's notes at UC Santa Cruz, here.

Week 3: We have continued the systematic development of the path integral method, mostly first in quantum mechanics, before applying it to QFT. The basic logic can be found in [Zee] and in [PS] as cited above. In the Discussion Sessions of Weeks 2 and 3, we discussed the case of a free particle on a circle. This is an intriguing example, illustrating many important features, whose more complex features reappear again in many more advanced examples of path integrals in QFT and beyond. In Week 2, we formulated the theory in the path integral form, indicating the exact solvability of the Gaussian path integral, and the need for a countably infinite extrema to be evaluated in the exact solution. The path integral for the particle on a circle is discussed in detail (under the fancy name of an "O(2) invariant rigid rotator") in Chapter 3.3 (on page 50) of Zinn-Justin's book cited above, J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th edition (Oxford University Press), and even in some more detail in Chapter 5.6 (on page 126) of the shorter Zinn-Justin book that focuses on QM: J. Zinn-Justin, Path Integrals in Quantum Mechanics (Oxford University Press, 2011). In Week 3, we used the path integral in an intriguing way to evaluate the spectrum of quantum states of the particle on the circle, stressing how the correct calculation requires the Poisson resummation formula. This formula is discusssed in detail in Zinn-Justin, and can be further found for example here. This formula can be intuitively understood as a certain discrete version of the Gaussian integrals, and plays an important role in all kinds of physics contexts, including in string theory (see for example the book by K. Becker, M. Becker and J.H. Schwarz, String Theory and M-Theory. A Modern Introduction, Cambridge, 2007). Week 4: We extended the path integral method to full relativistic quantum field theory. Then, we considered the first questions about the path integral method that we face if our theory is a gauge theory. The prime example for us will be non-Abelian Yang-Mills gauge theories. A good reference for their perturbative quantum properties (including the treatment of "renormalized perturbation theory") that we will at least partially follow in lectures is M.D. Schwartz, Quantum Field Theory and the Standard Model (Cambridge, 2014). Of course, one can also benefit from reading the appropriate chapters from [PS] and/or [Zee]. Additional texts may be recommended later, as needed.

horava@berkeley.edu