**Homepage for 171.701-2 (Fall 2022, Spring 2023): Quantum Field theory**

**Instructor: ** Marc Kamionkowski

Bloomberg 439

**Class times/location: ** MW 10:30am--11:45, Bloomberg 464

**Office Hours: **Thursday, 10am-11am or email for an appointment

**COURSE OVERVIEW AND COMMENTS:**

In the first semester, we will aim to get through quantization of the scalar field, the Dirac field, and the photon field. We will develop quantum electrodynamics (QED) fully and also get started with renormalization. The second semester will focus on the rich physics associated with gauge theories with connections made to particle theory and some condensed-matter theory.

Quantum field theory is *hard*. It is probably the most technically and conceptually sophisticated subject that is regularly offered in a typical physics curriculum. It is also quite broad, and so its hard to feel that you've ever mastered it. Some of the most eminent theoretical physicists will tell you that they're still trying to understand things in QFT, even after decades of study.

The upshot is that you should not get frustrated if things don't fall into place as quickly as they did in other classes you've taken. But as with other things, with focus and effort, and after some period of time, you start to get the hang of it and develop some intuition for it. I realize that different students have different aims in taking this class. If you are taking it for exposure/breadth, then try to get the gist of things without necessarily getting bogged down in the details. If you are planning to make a living as a particle, mathematical, or CM theorist, you should probably make more effort to work through as many problems as you can.

**Prerequisites:** I am assuming that you have a grasp of quantum mechanics, electromagnetism, and special relativity. Some basic familiarity with general relativity might come in handy in the second semester, but you can probably get by without it.

**Homework/Grades: **I will assign three problem sets throughout the semester. The class grade will be based on completion of these problem sets. Feel free to work with other students and consult with other books, more senior students, patient postdocs, and the internet for help, but then try, once you’ve figured things out, to write the solutions on your own. Grading will be flexible. If you plan to make a living in particle theory, quantum field theory, or perhaps CM theory, then you might want to try to keep up with the homeworks. If not, do as many as your schedule/sanity allow. If your time for this class is indeed limited, find some fraction of each homework to complete, so that you keep up with what we're doing.

**RECOMMENDED TEXTBOOKS:**

**First Semester:**

"Quantum Field Theory in a Nutshell, 2nd “Edition by A. Zee (“Z” in the syllabus below)

"Quantum Field Theory," by Mark Srednicki (“S” in the syllabus below)

David Tong: Lectures on Statistical Field Theory, https://www.damtp.cam.ac.uk/user/tong/sft.html (“SFT” in the syllabus below)

**Second Semester:**

We will follow David Tong's lecture notes on Gauge Theories (http://www.damtp.cam.ac.uk/user/tong/teaching.html)

Zee’s book will be useful

"Quantum Field Theory: An Integrated Approach," by Eduardo Fradkin may also be useful

**Comment about books: **There are probably hundreds of books on quantum field theory and the related subjects of statistical field theory and renormalization group. There is no "best" one---the scope of the subject is just too large and rapidly evolving to allow it to all be put in one place. I'd say a good fraction of QFT books are by particle theorists and are presented with the primary aim of developing the mathematical tools required to describe the Standard Model and its phenomenology. There are then others that are written more with the aim of developing applications to statistical mechanics or condensed-matter physics. And plenty of others are less categorizable. Here are my take on the books listed above as well as some others that may be useful.

Zee: Is a very colloquial/informal presentation with some great insights and simple explanations for things that are often confusing. It does so, however, by sacrificing rigor. The earlier parts of the book are focussed more on tools of relativistic field theory relevant for particle physics, and the later sections are quite nice at describing applications to CM physics.

Srednicki: This book is a great book for learning the QFT required to describe the Standard Model and the background for extensions like GUTs, supersymmetry, and neutrino masses

Fradkin's book is focused more on the development of QFT as a unified subject with applications to particle theory and to statistical mechanics and condensed-matter. It provides more in the way of subjects relevant to current research in QFT.

Tong's lectures are famous for their colloquial style and clarity at presenting advanced subjects in QFT and related areas. The lectures on Gauge Theories describe many ingredients of QFT that appear in the modern study of CM systems, particle theory, string theory, and entanglement.

Weinberg has a set of three volumes entitled "Quantum Field Theory" that are impressively complete and rigorous. They are focussed primarily on the development of relativistic field theory relevant for particle physics. They are, however, quite dense. Weinberg moreover develops everything from scratch in his own way, often in his own notation. It is a great reference but not necessarily a great textbook.

The book by Peskin and Schroeder is also driven primarily by particle physics, but it is a great book and was, for some time, *the* book for many QFT classes.

Matthew Schwartz has a QFT book that is also now widely adopted for relativstic QFT classes. It is a bit more modern, has a bit more in the way of advanced subjects than Peskin, and is written in the language/style that many particle theorists now use.

I first learned the subject from a little book by Mandl and Shaw that presents all the tools required for basic SM phenomenology in the simplest way possible. It does not provide, however, a great platform for understanding QFT more generally.

**SYLLABUS AND READING ASSIGNMENTS**** **

SPRING 2023

Chapter numbers refer to Tong’s lecture notes on gauge theories

Lecture Date Subject/Reading

1 1/23 Ch 1: monopoles and theta terms

2 1/25

3 1/30 Ch 2: Yang-Mills theory

4 2/1

5 2/6

6 2/8

7 2/13

8 2/15

9 2/20 Guest lecture by Nitu on relevant mathematics

10 2/22 Guest lecture by Nitu on relevant mathematics

11 2/27 Ch. 3: Anomalies

12 3/1

13 3/6

14 3/8 Ch 4: Lattice Gauge Theory

15 3/13 Guest lecture by Ibou Bah on lattice fermions and anomaly inflow

16 3/15 Ch 5: Chiral Symmetry Breaking

17 3/27

18 3/29

19 4/3 Ch 6: Large N

20 4/5

21 4/10

22 4/12 Ch 7: QFT on the line

23 4/17

24 4/19

25 4/24 Ch 8: QFT on the plane

26 4/26

**SPRING HOMEWORK ASSIGNMENTS:**

Homework 1 (due 27 February 2023)

Homework 2 (due 31 March 2023)

FALL SEMESTER

Pre-reading (before the first class): S: 1-2; Z: I.1. Also, Chapters 1-2 in Fradkin’s book are nice.

Lecture Date Subject/Reading

1 8/29 Canonical quantization

Z:I.8-9; S:3

2 8/31 Path-integral formulation of quantum mechanics

S: 6-7; Z:I.2

3 9/7 The free scalar field

S:8 Z:I.3

4 9/12 Connected vs disconnected diagrams

S:9; Z:I.4,

5 9/14 Perturbation theory and Feynam diagrams

Z:I.7; S:9

6 9/19 More pert theory and some divergences

S:9,10,11,12; Z:I.7

7 9/21 Quick intro to the vector field

Z:I.5,I.6

8 9/26 Symmetries

S:22,23,24; Z:I.10

9 9/28 Dirac spinor

Z:II.1; S:34,35

10 10/3 Quantizing the Dirac spinor

S:36-39; Z:II.2

11 10/5 Representations of the Lorentz group

Z:II.3; S:33-35

12 10/10 Grassman variables and Feynman rules for Dirac

S:43-45,53; Z:II.5

13 10/12 Gauge invariance and Ward-Takahashi identities

Z:II.7; S:67,68

14 10/17 regularization of divergences

S:14,15; Z:III.1

15 10/19 renormalizability

Z:III.2; S:15,16

16 10/24 counterterms

Z:III.3; S:17,18

17 10/26 Fadeev-Popov technique

Z:III.4

18 10/31 magnetic moment of electron

Z:III.6,S:64

19 11/2 charge renormalization in QED

Z:III.7,S:66

20 11/7 Ising model, phase transitions, Landau-Ginzburg theory

Tong: lecture on Statistical Field Theory, Part 1:

https://www.damtp.cam.ac.uk/user/tong/sft.html

21 11/9 Path integral in statistical mechanics

Tong SFT lectures, Part 2

22 11/14 renormalization group

Tong SFT lectures Part 3

23 11/16 renormalization group

Tong SFT lectures Part 3

24 11/28 continuous symmetries

Tong SFT lectures Part 4

25 11/30 continuous symmetries

Tong SFT lectures Part 4

26 12/5 non-abelian symmetries

S:24

27 12/7 non-abelian symmetries

S:70

**FALL HOMEWORK ASSIGNMENTS:**

Homework 1 (due 3 October 2022)

Homework 2 (due 7 November 2022)

Homework 3 (due 9 December 2022)

Last updated 12 April 2023