Introduction to Quantum Field Theory

An online video sketch “Why Learn QFT?” is available at MacVideo (here), so you can taste this course’s ideas and try on the online lecture format. Pre-recordings of the other lectures can also be found at the same MacVideo channel. Links to recordings of the synchronous lectures will either be posted here or at this same MacVideo channel.

The zoom link to use for the synchronous lectures has been emailed to you using the address listed for you in mosaic.

This course aims to provide an introduction to quantum field theory for undergraduates. Quantum field theory is normally not taught until graduate school, which is a shame because it is the language used at the frontier in almost every discipline of modern physics. Yet there is no fundamental reason why it could not be taught at the undergraduate level, apart from the obvious observation that an undergraduate physics curriculum is already pretty full.

Students are assumed to have had a first exposure to quantum mechanics and electromagnetism. (Some knowledge of statistical physics is also useful, but not required.)

In essence, quantum field theory is the formulation of quantum mechanics that allows one to describe processes that can change the number of particles involved. Because it turns out to very efficiently bake in a few core properties shared by all known physical laws, quantum field theory is also widely used even for systems whose total number of particles never changes.

As the course will show, once relativity and quantum mechanics are combined it is a basic fact that all interactions necessarily change the number of particles. (This is related to the reason why antiparticles must exist.) For this reason quantum field theory seems to be the language in terms of which Nature’s fundamental laws are necessarily written.

Although I work from my own notes (that are in progress, but available here) sometimes it is useful to have alternative points of view. Two qualitative approaches to the subject that might be useful (but is not required) are “Quantum Field Theory for the Gifted Amateur” by Blundell and Lancaster, or “Quantum Field Theory in a Nutshell” by Zee. See below for the course syllabus and more detailed course information.

SYLLABUS

  1. Multiparticle Quantum Mechanics 

    1.1 Bosons
    1.2 Fermions
    1.3 Quantum Statistical Mechanics

  2. Creation and Annihilation Operators   

    2.1 Heisenberg’s Treatment of the Harmonic Oscillator
    2.2 Creation and Annihilation Operators
    2.3 Fermions and Anticommutation Relations
    2.4 Completeness of Operators

  3. Interactions   

    3.1 Perturbation Theory
    3.2 Emission and Absorption
    3.3 Stimulated Emission
    3.4 Pauli Blocking and 2-body Scattering
    3.5 Equilibrium and Detailed Balance
    3.6 Coherent States
    3.7 Bosons and Macroscopic Forces

  4. Fields   

    4.1 Locality and Cluster Decomposition
    4.2 The Schrodinger Field and Second Quantization
    4.3 Bose-Einstein Condensation
    4.4 Spontaneous Symmetry Breaking

  5. Relativistic Quantum Mechanics   

    5.1. The Electromagnetic Field
    5.2 Relativity of Simultaneity and Antiparticles
    5.3 Crossing Symmetry, CPT Invariance
    5.4 Spin-Statistics Connection

  6. Quantum Electrodynamics   

    6.1 The Dirac Field
    6.2 Gauge Invariance
    6.3 Relativistic Feynman Rules

COURSE INFORMATION

The procedural information relevant to this course is given below.

Lectures

Classes are online in the Winter 2021 term, so the pandemic version of this class has an online and synchronous part. Many of the lectures will be pre-recorded on a lightboard so they can be viewed at leisure. These are available at MacVideo (here), including a “Why Learn QFT?” preliminary, so you can try them on before enrolling.

The synchronous lectures meet online (via zoom) Mondays, Wednesdays and Thursdays from 10:30 to 11:20, starting in January. The zoom link will be emailed to you, using your email address as listed in mosaic.

The idea will be for the class to pre-read sections of the lecture notes and/or pre-view some of the online lectures, which we can discuss tutorial style during the synchronous classes. These synchronous sessions may involve very short quizzes, to provide incentive for students to watch any online material before the interaction sessions.

Attendance to the synchronous lectures is certainly not compulsory, but if you come I do ask you to pay attention and not disrupt the class with personal conversation or social media. (For zoom my preference is for you to turn on your camera so that we all have a more interactive experience.) I will do what I can to ensure that you do not have to gnaw your own arm off to stay awake.

Textbook

There is no official course textbook; I provide lecture notes which lay out the course much as I will present it in the lectures. The goal of these notes is to provide more detail and to provide a written record to allow you to focus on what is being said in the lectures. (I do recommend making your own notes as well as you watch the lectures, though, since writing your understanding helps absorb the material more robustly. A preliminary version of my notes (that I continually update as the term progresses) can be found here.

For those seeking alternative viewpoints, David Tong also has a good set of lecture notes (though at a graduate level) at his Cambridge University webpage.

Office Hours

Because I spend half my time at Perimeter Institute I may be hard to find in my office (and this is even more true when we are locked down). So it is worth setting up any appointments in advance, and the best way to do so is to contact me by email (cburgess – at – physics.mcmaster.ca). I will also linger online immediately after the scheduled lecture times (after the interaction sessions if we are online). Otherwise, feel free to arrange another time with me on an individual basis. (I will make a point of being there for scheduled appointments, so if you do set up an appointment, please show up or give me adequate warning that you cannot make it!)

TA

There will be a course TA (Mathew Schneider) who will also hold office hours, and meetings with him are probably best arranged by email at the address schnem6 (at) mcmaster.ca.

Assignments

The course work involves completing a (roughly) weekly assignment. Like any worthwhile subject, Quantum Field Theory is a contact sport and so is only really learned by doing. It is very very strongly recommended to work the assignments even if you only audit the course.

I will be issuing assignments during class on a class day to be mutually agreed at the first lecture, and they are due in class the following week. Since we are online they should be submitted as pdf files, which can be scans of hand-written pages if you wish. If you wish to use some other format please negotiate this directly with Mathew (our TA). It is his choice whether he does so, and also how many marks are available for work that is handed in late (or assesses a penalty for late work).

Assignment 1 (due Mon. Jan. 25) can be found here. A solution set is here.

Assignment 2 (due Mon. Feb. 01) can be found here. A solution set is here.

Assignment 3 (due Mon. Feb. 08) can be found here. A solution set is here.

Assignment 4 (due Mon. Mar. 01) can be found here. A solution set is here.

Assignment 5 (due Mon. Mar 15) can be found here. A solution set is here.

Assignment 6 (due Mon. Mar 22) can be found here. A solution set is here.

Assignment 7 (due Mon. Mar 29) can be found here. A solution set is here.

Assignment 8 (due Mon. Apr 12) can be found here. A solution set is here.

There will not be assignments in the weeks of Feb. 22 or of Mar 1 due to the midterm.

You are welcome to work together on figuring out the assignments, though everyone must write up and submit their own solutions. It should go without saying, while you are allowed (for assignments) to consult internet material about conceptual information, you are never permitted to plagiarize material from the internet (or to use internet cheating tools like chegg) when preparing any work for marks, including assignments.

Mini Quizzes

The best way to stay abreast of the class is to read ahead in the lecture notes. To encourage this I will suggest sections to read and then have a short in-class mini-quiz about the content. This is meant to help motivate you to do the readings (I know you are all busy), since you get more out of the lectures (and can use them to ask questions) if you have given the material a little thought. It also allows me to focus in class either on special topics or on working through problems (and if you would like this to be done for specific problems, let me know).

Term Project

The term project is to summarize in your own words one of the classic papers of quantum field theory. Describe both what the paper’s intended point was, and why it was important (these are not always the same thing). You can work in groups if you wish, but if so when you submit your paper choice you should also submit a list of the others in your group.

Procedural details and a lengthy list of papers from which to choose are given here. Here are the deadlines (though do not wait for the deadlines before starting!):

Step one: choose a paper from the list (or you can choose one not on the list if you first get my approval for the paper you have in mind) and tell the TA which it is (and who your fellow group members are, if any) by Thursday February 25, 2021.

Step two: hand in your finished essay in class by Thursday April 8, 2021.

When preparing this essay you are allowed to consult internet sources for conceptual and historical information, provided that you cite this properly in the essay. You are never permitted to plagiarize material from the internet when preparing any marked work, including essays.

Midterm Exam

Since the course is online, the midterm test will be held in the week of March 1st, 2021. Since the university is online the midterm will be a take-home exam which you will be given several days to complete. (The exam itself should really only take a few hours to complete, so your being given a number of days to do it is meant to allow you to coordinate it with the obligations of your other courses and the rest of your life.)

The midterm provides the best possible practice for the final exam, so it would be silly not to write it. Those who do not write the midterm for whatever reason can avail themselves of marking Option B below. 

Perhaps needless to say, you are never permitted to plagiarize material from the internet when performing a midterm exam.

Final Exam

The Final Exam will be held during the April examination session. Since the university is online the final exam would be a take home exam. The exam should only take an afternoon (a few hours) to complete, but you will be given at least 4 days to complete it. The university has scheduled the exam for this course to be on Monday April 26, and many students have theses due by Thursday April 22. On the other hand marks must be submitted by May 1, and so the exam’s timing is chosen to minimize conflict with these constraints, and will be made available midnight on the evening of Thursday April 22 and will be due at midnight Tueday April 27. I will email you a link from which a pdf of the exam can be viewed.

You are never permitted to plagiarize material from the internet when preparing any marked work, and this is especially true for final exams.

** You ARE entitled to consult the course lecture notes, should you so wish, and to use calculators to compute numerical results. A formula sheet is also provided with the midterm, and any formulae you use from that sheet do not need to be rederived — just quote the formula sheet. **

Completed exams should be scanned or saved as PDF files and emailed to cburgess (at) physics.mcmaster.ca with Mathew cced (schnem6 (at) mcmaster.ca). I will acknowledge receipt and so if you do not hear from me, please recheck by email to verify that it got through OK. Good luck!

Marking Scheme

The course marks are completely based on the weekly assignments, the mini-quizzes, the midterm test, the term project and the final exam. The term mark will be computed from these according to whichever of the following formulae maximizes your final mark:

  • A) Ass./Quz: 20% Midterm: 20% Term Project: 20% Final Exam: 40%
  • B) Ass./Quz: 20% Midterm: 0% Term Project: 20% Final Exam: 60%
  • C) Ass./Quz: 20% Midterm: 20% Term Project: 10% Final Exam: 50%
  • D) Ass./Quz: 20% Midterm: 0% Term Project: 10% Final Exam: 70%

You do not have to choose in advance which one you want; I will compute all four and give you the maximum.

Part of the reason for providing you this menu of alternatives is to allow you to accommodate the imponderables of your own life, such as unexpected illnesses and the like. The purpose of doing so is to make this flexibility open to everybody in the class, and not just to those who wish to make special arrangements with me, or with the Associate Dean. In particular, this my preferred way to deal with MSAF applications in this class.

Additional Work and Supplemental Exam:

Additional work will NOT be available for students who might wish to improve their marks. The standard McMaster rules apply regarding the availability of supplemental exams.

Reading you your rights:

The Centre for Student Development offers free academic skill support.

Although hopefully it does not need saying, be warned that the University does not tolerate cheating, plagiarism and the like:

THE UNIVERSITY VALUES ACADEMIC INTEGRITY. THEREFORE ALL STUDENTS MUST UNDERSTAND THE MEANING AND CONSEQUENCES OF CHEATING, PLAGIARISM AND OTHER ACADEMIC OFFENCES UNDER THE CODE OF STUDENT CONDUCT AND DISCIPLINARY PROCEDURES

(see http://www.mcmaster.ca/academicintegrity for more information).