Introduction to Quantum Field Theory

This document is the syllabus for the Quantum Field Theory II class as taught in Spring 2019 (unique number 55740) by Dr. Vadim Kaplunovsky. Note that future offering of this class may vary.

The QFT II class is the second half of a two-semester course Introduction to Quantum Field Theory. 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.

Normally, the QFT I class is offered every year in the Fall semester, while the QFT II class is only offered every other year in the Spring semester, so some students have to wait a year between the two halves of the QFT class. In particular, in Fall 2017 and again in Fall 2018 professor Duane Dicus taught the QFT I class, and was also expected to teach the QFT II class in Spring 2019. Alas, he is currently too ill to teach, so professor Vadim Kaplunovsky took over the QFT II class this Spring.

Prerequisite Knowledge

The formal pre-requisites for the QFT (II) class is graduate standing and the QFT (I) class (PHY 396 K). 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.

I presume all students in my class have already made it through the QFT I class, which also implies graduate-level knowledge of Quantum Mechanics as well good-undergraduate level (or better) knowledge of Classical Mechanics and Special Relativity. Math-wise, I expect all the students to be familiar with complex analysis, especially the contour integrals and how to take them.

In addition, Statistical Mechanics (at the good undergraduate level or better) is very useful for understanding the functional quantization of field theories, while graduate-level StatMech would help with the Wilsonian renormalization. On the Math side, learning a bit of continuous group theory would be quite helpful, but this is not a pre-requisit as I shall explain the basics during the class.

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.

I presume the QFT (I) class prof. Dicus taught in Fall 2017 and in Fall 2018 has covered the bosonic and the fermionic fields, the symmetries, the perturbation theory and the Feynman graphs, and the elementary processes in QED. So my QFT (II) class this Spring is going to cover the remaining subjects:

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. 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 contains 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 sometimes teach, but I won't cover it in this class.)

Besides the textbooks, I wrote a bunch of supplementary notes (an I might write afew 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.


Regular Lectures

The regular lecture are on Mondays and Wednesdays, from 5 to 7 PM, in room RLM 9.222 (the brown bag room). Note 4 academic hours of regular lectures per week.

Starting January 30, Wednesday lectures will start at 5:15 PM since the departmental colloquium ends at 5:00 and sometimes a few minutes later than that. Likewise, if there is a special colloquium on Monday from 4 to 5 PM — and we expect quite a few this semester — then the QFT lecture that Monday will start at 5:15 PM. But when there is no colloqium on Monday, then the QFT lecture will start at 5:00 PM sharp..

Extra 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 lattice gauge theory 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.

The extra lectures will be on Mondays, from 1 to 2 PM, in the regular classroom RLM 9.222. Here is the tentaive 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.


Professor Vadim Kaplunovsky.
Tyler Guglielmo.

Last Modified: April 18, 2019.
Vadim Kaplunovsky