Phys 151: Introduction to Quantum Field Theory

Fall 2025

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Basic Info

Lecturer: Petr Hořava (email: horava@berkeley.edu)
Office: 401 Physics South.
Office hours: 401 Physics South, Tuesdays, 2:00-3:00 pm.

Reader: Viola Z. Zhao (email: zhaozixin@berkeley.edu).

Discussion sessions:
Mondays 4:10-5:00 pm, 402 Physics South. Discussion sessions will start on Monday, Sept 8. (Note that the location has been changed to Physics South, instead of the originally scheduled option somewhere way across campus.)

In the process of learning the basic concepts of QFT, we will discuss a vast array of examples, in which this paradigm plays a role -- not just in relativistic particle physics, but in nonrelativistic condensed matter, nonequilibrium statistical physics, ideas and fundamental questions of quantum gravity and cosmology, in revolutionizing parts of pure mathematics, and even beyond. My aim is to stress this universality of the fundamental concepts of QFT, whenever we are dealing with systems of many fluctuating, interacting degrees of freedom.

At the core of the modern understanding of QFT is the so-called Wilsonian framework: A way of understanding how interacting systems with many degrees of freedom reorganize themselves as we change the scale at which we observe the system. This makes concepts and techniques of QFT remarkably universal, and applicable to just about every area of physics. As a result, a solid understanding of the basic structure, ideas and techniques of QFT is indispensable not only to high-energy particle theorists and experimentalists, or condensed matter theorists, but also to string theorists, astrophysicists and cosmologists, as well as an increasing number of mathematicians.

The course in Fall 2025 will be somewhat similar to Physics 151: Introduction to Quantum Field Theory, which I taught in Fall 2023 and in Fall 2022. More specifically, the main themes that we will focus on in this semester will be the following:

• Path-integral reformulation of Quantum Mechanics.
• From quantum particles to quantum fields, QFT as a description of systems with many interacting, fluctuating degrees of freedom.
• Quantization of free fields: Canonical and path-integral formulations. Bosonic and fermionic fields.
• Interactions: Perturbation theory, the logistics of Feynman diagrams.
• Importance of topological invariants in QFT.
• Basics of the renormalization group ideas.
• Renormalization process in perturbation theory.
• Basics of quantization in theories with gauge invariance.

In the process, we will illustrate the abstract concepts using concrete examples, ranging from the relativistic fields of the Standard Model, to relativistic and non-relativistic field theories in low spacetime dimensions, to Fermi liquids, topological insulators, nonequilibrium statistical systems (such as surface growth phenomena), and even holographic dualities with the celebrated AdS/CFT correspondence, involving classical and quantum gravity, and more -- without requiring any prior knowledge of those examples. We will stress the universality and practicality of the basic QFT methods (including Wilsoninan renormalization group concepts) across this amazingly rich plethora of disparate areas across modern physics.

The main required textbook is going to be:

A. Zee, Quantum Field Theory in a Nutshell. 2nd edition (Princeton U.P., 2010).

I also recommend the fairly new and intriguing introductory text,

J. Donoghue and L. Sorbo, A Prelude to Quantum Field Theory (Princeton U.P., 2022),

which is aimed at precisely the right level at which our course will begin: To provide a bridge from the ideas and concepts of Quantum Mechanics (assumed to be known to the student) to the ideas and concepts of Quantum Field Theory, comparing and contrasting what is similar and what is different in these two ways of thinking about quantum systems. This book is very readable and short (less than 150 pages!), and therefore can serve as a possible life-boat if any student starts feeling in danger of drowning in the vast sea of Quantum Field Theory as covered in the incredibly large number of textbooks on this subject.

I also strongly recommend

M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory (Perseus, 1995),

a textbook that is often used as the go-to text in many graduate-level courses on relativistic QFT, and which can serve as a "second-opinion book" for the students in our Physics 151 when they encouter a new topic in Zee and want to see more.

Occasionally, I will introduce examples or concepts not covered in any of these three books, in which case I will provide a specific reference to a good source in the literature, often in the form of an arXiv paper. I will post the relevant references on a weekly basis on this page, see below.

Week 1 (Aug. 27 and 29): This week, we discussed a broad-brush overview and preview of what to expect from this course, and why Quantum Field Theory (QFT) is so universal and impactful across so many areas of physics, math and beyond. We pointed out the beautiful example of the simplest surface-growth problem in nonequilibrium classical statistical mechanics, as a problem that is efficiently phrased and solved in the language of QFT. This indicates how QFT is not at all only a field applicable to relativistic particle physics (which may be the area where the term first historically originated from.) We will encounter the surface growth problem (and its KPZ equation) through the semester in a more detailed, technical form, as one of our many diverse examples of how the same methods of QFT can be found useful, and how they help us determine what is important and what is not in understanding many-body interacting, fluctuating systems. I introduced the term "Wilsonian renormalization" and "renormalization group" (RG), at this stage still without any technical definitions -- these concepts will be dominant themes o the entire semester, and we will get a more technical understanding of what they mean as we move along. For now, if you wish to read more about surface growth, I strongly recommend the beautiful book
A.-L. Barabási and H.E. Stanley, Fractal Concepts in Surface Growth (Cambridge U.P., 1995).
This book never really uses the words "quantum field theory", but it will introduce you to the ideas and techniques that you will be encountering throughout the semester under the umbrella name of QFT. It is a great bed-side table read, especially its first few chapters give a great intuitive sense for what we will see later. (The surface growth problem and the KPZ equation are also discussed in Chapters VI.6 and VI.8 in [Zee], and we will encounter them when we get to those chapters.)

We also discussed the Klein paradox; my presentation of this topic was following Chapter 1 of the nice little introductory book on QFT,
L. Alvarez-Gaumé and M.A. Vázquez-Mozo, Invitation to Quantum Field Theory (Springer, 2012).

Week 2 (Sept. 3 and 5): We started our step-by-step way towards quantum field theory. First, we will develop a detailed understanding of a new method for quantum mechanical systems, which is not one of the pre-requisites for this course: The path-integral method. We first introduced the ideas and logic of path integrals while using nonrelativistic quantum mechanics of a single particle as an example. In the process of deriving the path-integral representation of the transition amplitude for the particle, we briefly introduced the technique of Gaussian integrals. Since Gaussian integrals (in various disguises) will play a central role throughout our development of QFT during this entire semester, we will return to this concept in more detail in Week 3. An important subtlety will be the analytic continuation of the Gaussian integral between "real" and "imaginary" time; this concept of "Wick rotation" will also be discussed in Week 3, in a somewhat heuristif form. As a preview, however, you can read a much more mathematically rigorous direct treatment of the analytic continuation(s) of the Gaussian integral in 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).)

The path-integral method for quantum mechanics (and other fluctuating systems) was primarily developed by Richard Feynman, in the late 1950's and early 1960's. To this day, one of the most beautiful introductions into this idea is the original book by Feynman and Hibbs, published already 60 years ago:
R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).
This book is an incredibly entertaining first-hand account of the fundamental ideas and features of path integrals, written with great lucidity and clarity, and staying surprisingly fresh and modern even to this day.

Week 3 (Sept. 10 and 12): In Week 2, we talked about the importance of Lagrangians (and configuration spaces), and Hamiltonians (and phase spaces), already for the case of the classical mechanics. This topic should be taught in Physics 105, which I consider a pre-requisite for Quantum Mechanics. We will talk about Lagrangians and Hamiltonians all the time throughout the semester. If you are not familiar with these topics in classical mechanics, or if you would like a refresher, I recommend the following:
L.N. Hand and J.D. Finch, Analytical Mechanics(Cambridge U.P., 1998);
this book covers the basics of Lagrangian and Hamiltonian formulations, plus extra chapters that you will not need this semester. It is the classic book that matches the expectations of our Physics 105.
I also strongly recommend the sleek historical classic,
L.D. Landau and E.M. Lifshitz, Mechanics (Courses on Theoretical Physics, Volume 1; Elsevier, 3rd edition, 1976).

Lots of great details (both mathematically correct and physically accurate, focusing on important things) are in the books by Jean Zinn-Justin,
J. Zinn-Justin, Path Integrals in Quantum Mechanics (Oxford Graduate Texts, 2005),
and the more advanced and more comprehensive book by the same author,
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford, Fifth edition, 2021).
In particular, I discussed in some detail how the typical history of a quantum nonrelativistic particle in the quantum regime of the path integral is effectively a trajectory of "fractal" dimension two, not the naively expected classical value of 1. This particular point is discussed in careful detail in Section 2.2 (and then some more again in Section 3.1) of Zinn-Justin's QFT book cited above.

This week we also started moving from the path inegral description of quantum mechanics to quantum field theory. This material is well covered in early chapters of [Zee].

Week 4 (Sept. 17 and 19): We continued with the basics of path inegrals in relativistic scalar field theory. Besides, we discussed the important method of evaluating the path integrals by the steepest descent method, which gives the semiclassical approximation. I presented a cute example of a system which is still described exactly by the Gaussian evaluation of the path integral, but requires a discretely infinite number of classical extrema to be included: The case of a free nonrelativistic particle on a circle. The infinite classical solutions that appear in the semiclassical method can be viewed as the simplest example of a much more general type of solutions in QFT, known as "instantons", and they illustrate the ubiquitous importance of topological invariants in QFT. This case of the particle on a circle is very nicely described in Zinn-Justin's QFT book, Chapter 3.3, where it is referred to not as a particle on a circle, but with a fancy name "the O(2) symmetric rigid rotator".

I also discussed qualitatively the simple but important fact, that when we try to go beyond the Gaussians and develop perturbative approximations in a "small" coupling constant describing the strength of the unharmonic terms in the path integral expeonential term, this perturbative expansion is only an asymptotic expansion, almost never converging. This indicates the importance of various nonperturbative effects in general QFTs, a topic we will not address in detail until we develop a sound understanding of the methods of the perturbation theory first. This topic of perturbative expansions being only asymptotic expansions can also be found in late chapters of Zinn-Justin.

On Friday, we talked in more detail about the propagators, largely for the case of the relativistic scalar field in four spacetime dimensions, and largely in the style of [Zee]. In addition, I also talked about the extended case of the so-called "Schwinger-Keldysh formalism", which is required if you are interested in more general transitions than just the relativistic vacuum-to-vacuuum transition amplitude, and cannot rely ahead of time on the existence of a stable, static facuum. In this Schwinger-Keldysh formalism (sometimes also known as the "in-in" formalism), the propagarors that are needed are more general than the famous "causal" Feynman propagator: In fact, in that more general case, all the different possible ways of closing the integration contour in the momentum space while finding the propagator play a role. If you wish to read more about this more general form of QFT, the Schwinger-Keldysh formalism is described for example in the intedisciplinary book
F. Gelis, Quantum Field Theory: From Basics to Modern Topics (Cambridge U.P., 2019).

In the process of finding the propagators, we analytically continued into the complexified space in the time component of the momentum, and then used the classic Cauchy formula from the theory of integration of contours in compex space. If you need some background on this Cauchy integral formula (often also referred to as Cauchy's theorem), you can first consult the nice, sufficiently detailed presentation (with the outlined proof) in
the Wikipedia page of Cauchy's integral formula,
and if you wish to read more (together with more basics of the elements of complex analysis and analytic functions), you can try the nicely sleek classic book by Henri Cartan,
H. Cartan, Elementrary Theory of Analytic Functions of One or Several Complex Variables (Dover, 1973 edition).

Week 5 (Sept. 24 and 26): This week, there are fewer extra references beyond [Zee], simply because we followed fairly straightforwardly the relevant Sections of [Zee] that explain the idea, structure, properties and consequences of (free-field) propagators, and then we started systematically introducing non-Gaussian (anharmonic) terms in the action of our path integrals, leading to a systematic development of a perturbative expansion in the powers of the strength of the coupling, and its combinatorial language of "Feynman diagrams" which provide a systematic math shortcut for getting the correct coefficients in the systematic Taylor expansion of the generating functional (these coefficients we started referring to as the "s-point Green's functions"). This graphical shortcut gives the most practical (so far) technique for not having to do all the combinatorial derivations from scratch, and provides a nice intuitive picture depicting the interesting observables we try to calculate. The relevant Chapters of [Zee] that we covered were I.3, I.4, I.5 and we started Chapter I.7, the Feynman diagrams. Week 6 (Oct. 1 and 3): On Wednesday, we continued with Feynman diagrams, by considering specific examples of this combinatorial and graphical technique, mostly focusing on more details of the examples from Ch. 1.7 of [Zee]. In addition, on Monday in our Discussion Session, I recommended two additional nice Chapters from [Zee]: Chapters I.6 and VIII.1. Chapter I.6 we won't have the time to discuss in detail in lectures, but it deals with the cool idea of "braneworlds" -- what if we live on a 3+1 dimensional membrane in a higher-dimensional spacetime, and gravity lives everywhere? In contrast, Chapter VIII.1 gives a detailed exposition of the case of gravity, and in particular what goes wrong when you make the "graviton" slightly massive and then try to take the limit of the mass going to zero: There is a famous discontinuity, which doesn't on the face of it give you the correct massless result, and needs to be treated in a rather subtle way by discovering the famous Vainshtein radius and the need to perform a resummation of an infinite sequence of Feynman diagrams to get the correct physics. This is indeed a great warning example, pointing out that it is dangerous to take naive unsubstantiated limits and shortcuts in our calculations.

Homeworks will be often assigned from Zee's book mentioned above, which I will simply refer to as [Zee].

HW 1 (due on Friday, September 12, in class): Problems I.2.1 and I.2.2 from [Zee] (on page 16, of the 2nd edition of the book). Comments: In the first of those two problems, you are asked to re-derive the path integral for a free particle in one dimension as discussed in class, for the slightly more complicated case of a particle responding to a non-zero potential V(q). To prepare for solving this problem properly, you may want to review in detail the steps that led to the solution of Eqn (4) on pages 10 and 11, and apply the same logic for the case with a non-zero potential. The second problem deals with a Gaussian integral with more than one integration variables, a topic which we will discuss in lectures on Wednesday next week; if you are not comfortable addressing this problem now, you can wait with solving it until you hear more about the Gaussians in next Wednesday's lecture.

HW 2 (posted on Sept. 12, due on MONDAY, Sept. 22): First, read carefully the entire chapter I.3 of [Zee], and then solve Problems I.3.1, I.3.2 and I.3.3 (on pp. 24-25 of [Zee]).

HW 3 (posted on Sept. 19, due on MONDAY, Sept. 29, 10 am): Problem I.4.1 (on p. 31), and Problem I.5.1 (from p. 39), both from [Zee]. With an attempt to solve the second problem, you may want to wait until after our next lecture (on Wed., Sept. 24) where we will discuss in detail the concept of summing over independent polarizations of on-shell states for vector and symmetric 2-tensor fields, as they correspond to electrodynamics and gravity.

HW 4 (posted on Sept. 27, due on MONDAY, Oct. 6): Problems I.7.2 and I.7.3 (on p. 60) from [Zee]. You may want to wait with this hw set until after our next lecture, Wednesday Oct. 1, when we will discuss in more detail the basic principles of the diagrammatics of Feynman diagrams.

HW 5 (posted on Oct. 5, due on MONDAY, Oct. 13): Problems I.7.4 (on p. 60), I.8.1 and I.8.2 (on p. 69 of [Zee]).

horava@berkeley.edu