Introduction to Quantum Field Theory

The Introduction to Quantum Field Theory is a two-semester course. Content-wise, this is a continious 29-week long course, but for administrative purposes it is split in two:

Physics-wise, the split is rather arbitrary, so students seriously interested in the Quantum Field Theory should take both halves of the course.

Unfortunately, the UT Physics Department is unable to offer the QFT II class every year, so the students who took QFT I (396 K) last Fall (2019) will have to wait till Spring of 2021 to take the QFT II (396 L) class.

This web page is the syllabus for the whole course as taught in the academic years 2019/20/21 (that is, 396K taught in Fall 2019 and again in Fall 2020, and 396L taught in Spring 2021) by Dr. Vadim Kaplunovsky, although the logistical detail are specific to the Fall 2020 and Spring 2021 semesters.

Covid-19 Notice

If you come to any in-person sessions (in any class, not just QFT), please make sure to follow the covid-19 safety measures. In particular, wear a face mask at all times you are inside a building or walking a crowded street. Measure your temperature in the morning: If you have a fever stay home. And to keep the UT administration happy, get a covid-19 symptom tracking app, and whenever you come to UT campus, post a no-symptom certificate to Canvas.

QFT I class in Fall 2020: Formally, there were two parallel sections: 55854 (Internet only) and 55850 (hybrid or blended). However, due to high infection risk throughout the semester, both sections ended up being internet only.

QFT II class in Spring 2021: This time, there is only one section 57340, formally Internet-only. In practice, this means Internet-only unless/until it becomes safe to meet in a classroom without masks. If this happens before the end of the Spring semester, I shall try to make the class hybrid/blended and schedule some in-person classes (but with a zoom mirror for the out-of-town students). Otherwise, the class will stay Internet-only till the end of the semester.

Prerequisite Knowledge


The formal pre-requisites for the QFT (I) class is graduate standing and the PHY 389 K class (graduate Quantum Mechanics (I)). However, what I care about is your knowledge rather than your status or grades. If you have the pre-requisite knowledge — however you have learned it — I'll sign the paperwork to let you into my class even if you are an undergraduate student.

Understanding Quantum Field Theory requires serious knowledge of quantum mechanics at graduate or advanced undergraduate level. Besides the QM basics — like knowing how to solve the hydrogen atom — you must be familiar with the multi-oscillator systems, the rotational symmetry and the angular momenta as its generators, the identical particles, the perturbation theory, and the basics of scattering theory. For the UT undergraduate students, you should complete the undergraduate QM sequence of 373 + 362K + 362L classes before taking the QFT class. For students who have learned their QM elsewhere, you need either 120 hours of undergraduate QM classes (not counting the inroductory Modern Physics class), or basic undergraduate QM followed by a gradute-level QM class. In any case, read J. J. Sakirai's book Modern Quantum Mechanics in the summer; if you understand everything in it, you are ready for my QFT class, but if the book looks all Japanese to you, you should beef up your Quanum Mechanics before taking Quantum Field Theory.

Besides QM, you would need good undergraduate-level knowledge of Classical Mechanics (especially the Lagrangian and the Hamiltonian formalisms), Classical Electrodynamics (the vector potential A, the gauge transforms, the EM stress-energy tensor, etc.), and basic special relativity (the Lorentz transforms, the 4–vectors, and the tensors). Make sure you are familiar with both 3D and 4D index notations, so expressions like FμνFμν do not confuse you or slow you down. You do not need general relativity for the QFT classes. In terms of the UT undergraduate classes, the 336 + 352K + 352L classes should give you adequate background. If you have not taken the 352L class, read the Griffith's textbook during the summer and focus on chapters/sections 7, 8, 9.1-2, 10.1, and 12.

The undegraduate-level Statistical Mechanics would be very useful for the second semester of QFT (396L), but you would not need it for the first semester (396K).

On the Math side, you would need basic complex analysis, especially the contour integrals and how to calculate them from the residues. In terms of UT undergraduate classes, the M 361 class (not to be confused with M 361K) should be good enough. If you have not taken this class (or its equivalent elsewhere), get a textbook and read it during the summer; focus on the applications rather then the proofs.

Finally, if you have time learn a bit of continuous group theory. This is not a pre-requisite, as I shall explain in class the basics of continuous groups, their generators, and their representations from scratch. But it would help if you already know something when I do.


For the QFT (II) class, the pre-rrequisite is QFT (I).

I presume most of QFT (II) students this Spring have taken my QFT (I) class in Fall 2020 or Fall 2019, so I know exactly what you have learned. (Or rather, what you should have learned.) The students who took their QFT (I) class from another professor — here at UT or at another University — should contact me during the first week of Spring classes to determine the level of their preparation.

Course Content (QFT I and QFT II combined)

Bosonic Fields:
Classical field theory; relativistic fields; identical bosons and quantum fields; Klein–Gordon propagator and relativistic causality; quantum electromagnetic fields and photons.
Fermionic fields:
Lorentz symmetry and spinor fields; Dirac equation and its solutions; second quantization of fermions and particle-hole formalism; quantum Dirac field; Weyl and Majorana spinor fields.
Symmetries in QFT:
Continuous symmetries and conserved currents; spontaneous symmetry breaking and Goldstone bosons; local (gauge) symmetry and QED; Higgs mechanism and superconductivity; non-abelian gauge symmetries and the Yang-Mills theory; discrete symmetries.
Interacting Fields and Feynman Rules:
Perturbation theory; correlation functions and Feynman diagrams; S-matrix and cross-sections; Feynman rules for the QED, QCD, and other important firld theories.
Quantum Electrodynamics:
Some elementary processes; radiative corrections; infrared and ultraviolet divergencies; renormalization of fields and of the electric charge; Ward identities.
Functional Methods:
Path integrals in quantum mechanics; "path" integrals for classical fields and functional quantization; functional quantization of QED; QFT and statistical mechanics; quantum symmetries and conservation laws.
Renormalization Theory:
Systematics of renormalization; `integration out' and the Wilsonian renormalization; `running' of the coupling constants and the renormalization group.
Non-Abelian Gauge Theories:
Non-abelian gauge symmetries and the Yang-Mills theory; interactions of gauge bosons and Feynman rules; Fadde'ev-Popov ghosts and BRST; renormalization of the YM theories and the asymptotic freedom; chiral gauge symmetries; the Standard Model; confinement and other non-perturbative effects.

In the first semester (the 396 K class) I shall focus on:

The second semester (the 396 L class) will be dedicated to the remaining subjects, namely:

Textbooks and Supplementary Notes

The primary textbook for both semesters of QFT is An Introduction to Quantum Field Theory by Michael Peskin and Daniel Schroeder. The UT library has a digital copy. To a large extent, the course is based on this book and should follow it fairly closely, but don't expect a 100% match.

Since both the course and the main textbook are introductory in nature, many questions would be left an-answered. The best reference book for finding the answers is The Quantum Theory of Fields by Steven Weinberg. The first two volumes of this three-volume series are based on a two-year course Dr. Weinberg used to teach here at UT — but of course they also contain much additional material. To a first approximation, Dr. Weinberg's book teaches you everything you ever wanted to know about QFT and more — which is unfortunately way too much for a one-year intoductory course. (Weinberg's volume 3 is about supersymmetry, a fascinating subject I used to teach, but I won't cover it in this class.)

Besides the textbooks, I wrote a bunch of supplementary notes (and I might write a few more). All these supplemantary notes are linked to this page.

Homeworks, Exams, and Grades

The homeworks are absolutely essential for understanding the course material. Often, due to the time pressure, I will explain the general theory in class and leave the examples for the homework assignment. It is extremely important for you to work them out by yourselves; otherwise, you might think you understand the class material but you would not! Be warned: The homeworks will be very hard.

I shall post homework assignments each week on this page. The solutions will be linked to the same page after the due date of each assignment.

The homeworks are assigned on the honor system: I shall not collect or grade the homeworks, but you should endeavor to finish them on time and check each other's solutions.

The solutions to previous years' homeworks — often quite similar to this year's — are available on the web, even on my own web server. On the honor system, I will keep them available at all times. But you should do your best to do the homework yourself, and only then read the solutions I post.

There will be separate final grades for each semester. Each grade is based on two take-home tests, one in the middle of the semester, the other at the end; the mid-term test contributes half of the grade and the end-term test the other half. There will be no in-class final exams.


I shall try using Canvas to schedule Zoom sessions for all the lectures; this way, the lectures would be recorded and you would find the records on Canvas. Apart from that (and the mandatory posting of this syllabus to Canvas), I shall not use Canvas for anything else. In particular, all homeworks and exams will be posted to the homework page rather than to Canvas, and the exam grades shall be emailed directly to the students.


Regular Lectures

Fall 2020 semester:
Spring 2021 semester:
UT Zoom notice:
To access the lectures, make sure to use the UT-authorized version of Zoom from, and login to your UT account before the class. (Your login can stay active for a month, so you do not need to do it often, but do not forget). If you do not have a UT account (or forget to login), Zoom would put you in a waiting room until I notice (which might take time) and let you in.

Extra and Make-Up Lectures

Besides the regular lectures, I shall give a few extra lectures about subjects that are somewhat ouside the main focus of the course but are interesting for their own sake, such as vortices or grand unified theories. The students are strongly encoraged to attend the supplementary lectures, but there is no penalty for missing them. The issues covered by supplementary lectures will not be necessary to understand the regular lectures and will not appear on exams.

In the Fall 2020 semester, the extra lectures were on Fridays, from 5 to 6 PM (right after the regular Friday lecture), at Zoom link Note: different Zoom link from the regular lectures!

Fall 2020 schedule:

In the Spring 2021 semester, the extra lectures will be on Wednesdays, from 1 to 2 PM, at Zoom link Again, different Zoom link from the regular lectures!

Update 2/17: Since all classes of the week of 2/15 through 2/19 were canceled due to bad weather, I am going to make them up using the same Wednesday 1–2 time slot — and the same Zoom link — as the extra lectures. The make-up lectures would be every week until we make up for all the canceled classes.

Unlike the extra lectures, the make-up lectures are requred, and the material covered there may well appear in a homework or an exam. If a student has other obligations at the time of a make-up lecture, he or she should watch the recording of the Zoom session; a link should appear on Canvas about an hour after the lecture's end.

Update 2/20: UT adminsitration canceled all regular classes on 2/22-23 but allows for optional extra classes. Therefore, I shall use the Tuesday 2/23 3:30 to 5 time slot for an extra lecture, while the Wednesday 2/24 1-2 time slot — and few more Wednesday after that — will be used for make-up regular lectures.

The Zoom link for a lecture follows its time slot, not the purpose. The extra lecture on Tuesday 2/23 will be at while the make-up lectures on Wednesdays will be at

Tentative Spring 2021 schedule:

Lecture Log

For students' convenience, I shall keep a log of lectures and their subjects on this page. Since the pace of the course may change according to the students' understanding, I will not make a complete schedule at the beginning of the class. Instead, I will simply log every lecture after I give it. This way, if you miss a lecture, you will know what you should read in the textbook and other students' notes.

Cloud Recordings of Zoom Lectures

To help QFT 2 students (in Spring 2021) who took QFT 1 from another professor, here are links to cloud recordings of my QFT 1 lectures in Fall 2020.


Professor Vadim Kaplunovsky.
Please use email for all questions which allow short answers (a few lines), especially simple homework questions, administrivia, or typos in my notes. The questions which take more elaborate answers should be asked in person or via zoom; you may ask them via email, but the anwer would have to wait till next office hour.
PMA 9.314 A, phone (512) 471–4918.
But unless I get to teach in-class (instead of online-only), I probably will not come to my office this year.
Other contact info:
I am not publishing my cellphone or home phone, but if you need to talk to me, email me your phone number and I'll call you back. Also, I am not publishing my Skype address online, but I shall give it to you in class.
Virtual office hours:

Last Modified: April 1, 2021.
Vadim Kaplunovsky