Phys 151: Introduction to Quantum Field Theory

Fall 2023

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Basic Info

Lecturer: Petr Hořava (email: horava@berkeley.edu)
Office: 401 Physics South.
Office hours: 401 Physics South, Tuesdays 2-3:30pm.

GSI: Emil Albrychiewicz (email: ealbrych@berkeley.edu)
GSI office hours: Mondays, 2-4pm, 420K Physics South
Discussion sessions:
Mondays 4-5pm, Wheeler 204,
Fridays 2-3pm, Barker 110.

Our basic philosophy in this course will be based on the following: Over the past several decades, our scientific community has recognized that quantum field theory (QFT) -- more generally, many-body theory -- represents the leading paradigm and universal language in modern theoretical physics, and an absolutely essential ingredient in our current understanding of the Universe on an astonishingly diverse range of scales. The basic ideas and techniques of QFT are at the core of our understanding of high-energy particle physics, inflationary cosmology and cosmological structure formation, as well as phenomena in condensed matter, statistical mechanics, quantum information, nonequilibrium mesoscopic dynamics, AMO physics, and even finance. QFT also naturally leads to its logical extension -- string theory -- which in turn provides a unified framework for reconciling the quantum paradigm with the other leading paradigm of the 20th century physics: that of general relativity, in wich gravity is understood as the geometry of spacetime.

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.

This course will provide the introduction to the principles of QFT, mostly in -- but not limited to -- the special case of the relativistic regime. The focus will be two-fold: Primarily, on developing a "big-picture" understanding of the basic ideas and concepts of QFT, and secondly, on developing the techniques of QFT (albeit at a slightly less ambitious and detailed level than in our graduate-level QFT courses 232A and 232B).

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

• Path integral re-formulation of Quantum Mechanics.
• From quantum particles to quantum fields.
• 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, etc., without requiring any prior knowledge of those examples.

The main required textbook is going to be:

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

The second book is a 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, ofter in the form of an arXiv paper.

Week 1: After covering the logistics of the course, the scope and goals for this semester, stressing the interdisciplinary and universal nature of QFT, we jumped right in, asking for a tentative answer to the question of "what even is Quantum Field Theory?" There is no universally agreed-on answer, different subfields of physics and math may stress different aspects, but we will (at least for now) view QFT as a (Wilsonian) "Calculus for understanding the behavior of systems with many interacting and fluctuating degrees of freedom, aimed to identify what is relevant and what is irrelevant." In any case, the historical origins were different: QFT originated from asking how to reconcile quantum mechanics and special relativity. My discussion of the Klein paradox followed the beginning chapter of another little introductory book about QFT:
L. Alvarez-Gaumé and M.A. Vázquez-Mozo, Invitation to Quantum Field Theory (Springer, 2012).
A beautiful and very inspiring book on quantum mechanical paradoxes is:
Y. Aharonov and D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed (Wiley, 2005).

Week 2: Following up on our relativistic Klein paradox, we first discussed additional reasons why even non-relativistic physicists need quantum field theory: To understand the ground state and lowest excited states (i.e., presumably the properties most accessible and interesting to a low-energy observer), it is often necessary to reconsider how to think about the microscopic versus effective degrees of freedom (think of the example of collective normal mode excitations of that proverbial mattress as discussed in Zee and in lecture). Having dispensed with our philosophical motivations, we jumped directly into the technical parts, of introducing a path-integral reformulation of Quantum Mechanics, working our way towards QFT. We stressed the importance of Gaussian integrals; those who wish to examine in detail the rigorou way how to treat the various analytic continuations of the Gaussian integral relevant to us (such as the "Fresnel integral"), can consult Howie Haber's short note from UCSC on this subject (very readable to a physicist, while also very rigorously formulated)

Week 3: First, we made sure to go over various properties of the all-important Gaussian integrals with one and few variables, including the definition of correlation functions that will be extremely useful later in field theory. Then we practiced our understanding of path integrals in QM by a sequence of simple examples, each of which illustrates an important point. The first nontrivial example showed how much more fundamental the Hamiltonian form of the path integral in phase space is, compared to the naive path integral in configuration space. A free particle in D dimensions then revealed that a generic history contributing to the path integral is nowhere-differentiable, and that the free quantum particle is effectively spanning a two-dimensional fluctuating fractal object in spacetime! We used this to predict the critical dimension D=4 of the scattering of two weakly-coupled particles to two particles.
These examples are not in [Zee]. My favorite references for them are two books by J. Zinn-Justin, first a thin one devoted completely to QM path integrals,
J. Zinn-Justin, Path Integrals in Quantum Mechanics (Oxford U.P., 2010)
and then an epic and encyclopedic volume
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford U.P., 2002).
In particular, my presentation of the fractal dimension of the typical history of the quantum particle was based on the "Fundamental Comments" paragraph on page 24 (at the end of Section 2.2) of the latter Zinn-Justin reference; or see also the Discussion paragraph on page 36 (at the end of Section 2.3.1) of the first Zinn-Justin book cited above.
Finally, we studied how to take the semiclassical limit in the path integral, and saw that in the so-called "steepest descent" approximation (simply approximating the full integral by Gaussians around extrema) we recover the classical Euler-Lagrange equations, plus small quantum corrections.

Week 4: On Wednesday, we continued with a few examples of path integrals in QM that are not in [Zee]. Here are the references:
The finite-temperature partition function of the harmonic oscillator is discussed from the path-integral perspective in Chapter 2.7 of the Quantum Mechanics book by Zinn-Justin cited above. The case of the free particle on a circle, which so beautifully illustrates the importance of topology and topological quantum numbers in the path integral, is discussed (pretty much in the same form as we did in class) in Chapter 5.6 of that same Quantum Mechanics book by Zinn-Justin. You can also compare with Chapters 2.3, 2.4 and with Chapter 3.3 of the big QFT book by Zinn-Justin cited above.

Week 5: This week, we first worked on developing more intuition behind the notion of a propagator. We discussed the important point that all four distinct propagators (Feynman, advanced, retarded, anti-Feynman) that solve the same defining Green's function equation for the Klein-Gordon field play an important role in quantum field theory: When one does not make the simplifying assumption of the existence of a static, stable, eternal vacuum state (an assumption that would be inappropriate in most nonequilibrium systems, or in cosmology), one is led naturally to the so-called "Schwinger-Keldysh formalism" (which I think should be more properly called "QFT without simplifying assumptions about the static vacuum"), in which all four propagators will appear in perturbation theory. Only with the assumption of the static vacuum can one get away solely with the Feynman propagator, and call it "causal" for those circumstances. A great recent textbook that covers the Schwinger-Keldysh formalism both from the perspective of systems away from equilibrium and from relativistic QFT, is
F. Gelis, Quantum Field Theory. From Basics to Modern Topics (Cambridge U.P., 2019),
especially in Chapters 2.4 and then 17 and 18 of that book. (Disclaimer: Reading this material would require prior knowledge of more advanced parts of QFT we haven't covered yet.)
In our second lecture this week, we found out how a quantum field implies a propagator, which implies the existence of "resonances" that we can interpret as particles of a certain mass, which in turn implies the existence of forces between external sources, mediated by the virtual particles associated with the original field.

Week 6: We continued illustrating some very important general phenomena using the simplest examples, relativistic free scalar fields. We focused on the classical dimensional analysis, and determined the dimensions (measured in the units of mass or energy) of various objects in the classical theory, including the dimension of the scalar field itself. We found that this dimension of the scalar field depends significantly on the number D of spacetime dimensions, and we discovered the notion of a critical dimension. Then, focusing on the behavior of the mass of the free field, we introduced the idea of the "renormalization group" (or RG for short), and the associated renormalization group equation, as a way of quantifying how important or unimportant any term in the action becomes as we change the energy scale at which we experimentally probe the system (the "RG scale" for short). This has lead to the very important notion of "relevant", "marginal" and "irrelevant" couplings, and explained the notion of the RG fixed points, and the RG flow. Using two scalar fields, we illustrated the notion of crossover, when more than two RG fixed points are important for describing the system as we vary the RG scale. We also performed a similar dimensional analysis for the electromagnetic field, and discovered the result of Section I.5 of [Zee], that electromagnetic interaction between equal-sign electric charges is repulsive. Week 7: In our first lecture, we performed the detailed dimensional analysis of electromagnetism as a function of the spacetime dimension, and found -- classically -- that the electric charge is classically marginal in four spacetime dimensions. We also studied a nonrelativistic scalar field example, to show how the dimensional analysis generalizes in the absence of Lorentz invariance. Finally, we analyzed the case of the gravitational field, as described by the Einstein-Hilbert Lagrangian for the metric field on spacetime representing gravity. We reproduces [Zee]'s result that the gravitational force must always be attractive, by studying in detail the structure of the Feynman propagator for the spin-2 particle/field. In our second lecture that week, we introduced anharmonicity in the field-theory action, and the concept of Feynman diagrams as combinatorial tools how to keep track of the terms in the perturbative expansion of physical observables in the strength of the interaction/anharmonicity. We studied the simplest baby example first, and the full relativistic scalar field theory second, all following Section I.7 of [Zee].

HW 1: (posted on Thu, Aug 31; due on Thu, Sept 7): Problem I.2.1 from [Zee] (on page 16). This asks you to derive Eqn. (5) from Section I.2. Since we will derive the closely related Eqn. (4) from Section I.2 in the lecture on Friday, Sept 1, you may want to postpone working on this HW assignment until after seeing that derivation in the lecture.

HW 2: (posted on Thu, Sept 7; due on Thu, Sept 14): This week's assignment contains two problems, each involving some practice with Gaussian integrals: First, Problem I.2.2 from [Zee] (on p. 16), which asks you to prove Eqn. (24) from Section I.2, thereby effectively proving what we will later call Wick's theorem. In the second problem this week, I am asking you to derive Eqn. (28) on p. 16 in [Zee], which will prepare us to take the semiclassical limit of path integrals.

HW 3: (posted on Thu, Sept 14; due on Thu, Sept 21): Problems I.3.1 and I.3.3 from [Zee] (on pages 24 and 25). This homework set is related to the material that will be discussed in detail in our next lecture, on Friday Sept 15; you may wish to wait until after that lecture, before you start solving the two problems.

HW 4: (posted on Thu, Sept 21; due on Thu, Sept 28): Problems I.4.1 and I.3.2 from [Zee] (on pages 31 and 24, respectively).

HW 5: (posted on Thu, Sept 28; due on Thu, Oct 5): UPDATE on Sept 29: When this HW assignment was first posted, Problem I.5.1 (on p. 39) was assigned as one of the problems this week. However, since we did not cover properly the relevant gravity case in our Friday lecture on Sept 29, I will now postpone this Problem I.5.1 by one week; it will be assigned as a part of HW 6. This leaves the remaining two problems for HW 5: Problem III.2.1 (on p. 172) from [Zee]. Finally, read first the unnumbered subchapter entitled "Renormalization group flow" on pages 359-360 of Chapter VI.8 of [Zee]; then solve Problem VI.8.1 (on p. 368).

HW 6: (posted on Thu, Oct 5; due on Thu, Oct 12): First, we inherited Problem 1.5.1 from [Zee] (p. 39) from last week, so it is now a part of this week's assignment. In addition, two more problems from [Zee]: Problem I.7.1 and Problem I.7.2 (p. 60). These two problems are dealing with Feynman diagrams, which will be discussed thoroughly in our lecture on Friday, Oct 6.

Thu, Oct 12: No Homework Assignment will be posted this week. Instead, on Monday, Oct 16, Emil will distribute the Midterm Exam (as a pdf file) to all students, sometime after the Monday discussion session. You can use the rest of this week and the upcoming weekend to review the material that we have covered so far, as a preparation for the midterm; Emil will also use the Friday and Monday discussion sessions to review specifically some material relevant to the midterm. The midterm will be due on Saturday, Oct 21.

HW 7: (posted on Sat, Oct 21; due on Thu, Oct 26): Problems III.1.1, III.1.3 (p.168 of [Zee]), and Problem III.3.2 (p. 181).

HW 8: (posted on Thu, Oct 26; due on Thu, Nov 2): Problems III.3.3 on p. 181 (but only the part that asks about Eq. (14), NOT the part where [Zee] is asking for an extension to quantum electrodynamics), and Problems II.1.1 (on p. 105) and II.1.12 (on p. 106), all from [Zee].

HW 9: (posted on Thu, Nov 2; due on Thu, Nov 9): First, read Section II.5 of [Zee], and convince yourselves by an explicit calculation as outlined in the text that Eqns (12) and (13) (on p. 127 of [Zee]) are correct. Then, Problem II.5.2 (on p. 131), Problems I.8.2, I.8.3 and I.8.4 (on p. 69), and finally Problem II.4.1 (on p. 122).

HW 10: (posted on Thu, Nov 9; due on Thu, Nov 16): Since this week we only had one lecture because of a holiday, the HW assignment is shorter: Problems I.10.1 (p. 80) and I.11.2 (p. 87), from [Zee].

HW 11: (posted on Thu, Nov 16; due IN TWO WEEKS, after the Thanksgiving break, on Thu, Nov 30): Problems IV.3.1, IV.3.2, IV.3.3 and IV.3.4 from Zee (pp. 243-4).

HW 12: (posted on Thu, Nov 30; due on Thu, Dec 7): Problems VI.6.2 (on p. 349), VI.8.3 and VI.8.5 (p. 368).

horava@berkeley.edu