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:

- PHY 396 K -- Quantum Field Theory I, usually taught in the Fall, and
- PHY 396 L -- Quantum Field Theory II, usually taught in the Spring.

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 take QFT I (396 K) last Fall (2015) will have to wait for the Spring of 2017 for the QFT II (396 L) course.

This document is the syllabus for the whole course as taught in the academic years 2015/16/17
(that is, **396K** taught in Fall 2015 and again in Fall 2016, and **396L** taught in Spring 2017)
by
**Dr. Vadim Kaplunovsky**.
Note that future offering of the Quantum Field Theory course may vary.

Understanding Quantum Field Theory requires serious knowledge of quantum mechanics
at graduate or *advanced* undergraduate level.
Knowing how to solve the hydrogen atom is not enough — a student must be
familiar with multi-oscillator systems, spin, identical particles, perturbation theory,
and scattering.

The formal prerequisite for the 396 K class is 389 K (Graduate Quantum Mechanics I), but the
real prerequisite is the knowledge rather than the grade.
**If** you have already taken a graduate-level QM course elsewhere,
or took two undergraduate semesters (80+ hours) of QM
(not counting an introductory *Modern Physics* class or
applied QM classes such as *Atoms and Molecules*)
— or if you have learned enough QM by yourself —
I shall waive the prerequisite.
But **if** your QM knowledge stops with the single-electron wave-function and the hydrogen atom
but J. J. Sakurai's Modern Quantum Mechanics is all Japanese to you,
**then** taking my QFT class right away would be rather unwise
and you should really take the 389K course first.

Besides QM, you would need good undergraduate-level knowledge of Classical Mechanics —
the Lagrangian, the Hamiltonian, the canonical variables, etc. — and Classical Electrodynamics —
the vector potential **A**, the gauge transforms, the EM stress-energy tensor, etc.
You would also need the basic special relativity, especially the Lorentz transforms, the 4–vectors, and the tensors.
Make sure you are familiar with both 3D and 4D index notations, so expressions like
**F**_{μν}**F**^{μν} do not confuse you or slow you down.

Finally, 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**).

- 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 fermions; Feynman rules for QED.
- 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 cover the bosonic and the fermionic fields, the symmetries (including the Higgs mechanism and the non-abelian gauge symmetries at the semi-classical level), the perturbation theory and the Feynman graphs, and the elementary processes in QED. The remaining subjects will be covered in the second semester (the 396 L class).

The primary textbook for this course (both semesters) 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 would not be able to cover at all in this course.)

I have told the campus bookstore that I use Peskin's book as a textbook for both 396 K and 396 L (Fall 2015, Fall 2016, and Spring 2017), Weinberg's vol.1 as a supplementary texbook for the 396 K (Fall 2015 and Fall 2016) and vol.2 as a supplementary textbook for the 396 L (Spring 2017). I hope the store have stocked the books accordingly, but you should buy them while the supply lasts.

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 page http://www.ph.utexas.edu/~vadim/Classes/2015f/homeworks.html (for the Fall 2015 class). 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.

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.

- Fall 2015 mid-term test will be given to students on October 27 and due on November 3.
- Fall 2015 end-term test will be given to students on December 3 (last class) and due on December 10.

- Tuesdays and Thursdays, 3:30 to 5:30, in room RLM 5.114.
- Note: Each lecture is 2 hours rather than an hour and a half.
- I may miss a lecture or two due to travel to other universities, dates TBA. All missed lectures will be made up.
- No lecture on
**October 8**(Thursday): I am going to Stanford University.

Besides the regular lectures, I shall give a few supplementary lectures about subjects that are somewhat ouside the main focus of the course but are interesting for their own sake, such as magnetic monopoles or superconductivity. 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 supplementary lectures will be on Fridays, from 5 to 6 PM, in room RLM 5.112.
- Expect a lecture roughly every other week, tentatively on
**9/11, 9/25, 10/2, 10/16, 10/23, 11/6, 11/20.** - Lecture on
**9/11**:*Seeing classical motion in QM and classical fields in QFT.* - Lecture on
**9/25**:*Field theory of superfluidity.* - Lecture on
**10/2**:*More Superfluidity.* - Lecture on
**10/16**:*Spin-statistics Theorem.* - Lecture on
**10/23**:*Spinor fields in different spacetime dimensions.* - Lecture on
**11/06**:*Bound states and resonances.* - Lecture on
**11/20**:*Superconductivity.*

For students' convenience, I shall keep a log of lectures and their subjects on a separate web page http://www.ph.utexas.edu/~vadim/Classes/2015f/lecturelog.html (for the Fall 2015 classes). 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.

- Office Location: RLM 9.314A.
- Reserved office hour: Friday 3 to 4:30.
- At other times, the students are welcome whenever I'm in my office and not too busy. The best times to look for me are late afternoons and early evening hours, and also Thursdays between the brown bag seminar and the class.
- E-mail: vadim@physics.utexas.edu.

Please use email for*simple*homework questions or administrivia. Complicated physics questions should be asked in person. - Office phone: (512) 471-4918.

Last Modified: November 14, 2015. Vadim Kaplunovsky

vadim@physics.utexas.edu