Course syllabus
Course-PM
FUF020 / FIM430 Quantum field theory lp4 VT20 (7.5 hp)
Course is offered by the department of Physics
NOTE 1: The course will be given as lectures via Zoom which is based on email invitations from the host, i.e., the teacher. All lectures in QFT starts at 13.15 and you will receive the invitation at 13.00 so you have 15 minutes to get the Zoom connection working.
If you have problems with Zoom contact me asap but not after the lecture has started.
NOTE 2: It will be essential that all students have access to the text book by Peskin and Schroeder (see below) and that you consult these Canvas pages more or less daily. Some information may also be distributed via email. Read the lecture notes carefully (see the Doc files below) and follow the reading instructions for the book. This last point is very important since we will jump back and forth in it a lot. You can contact me on email at any time if you have questions! Please check Canvas before each lecture!
NOTE 3: All students that plan to follow the course but do not appear in the list of participants (see People to the left) must send me an email asap so that I can put you on my Zoom mailing list.
Contact details
- examiner: Bengt E.W. Nilsson, tfebn@chalmers.se, office Origo N6.109B, phone 0704-101283
- lecturer: Bengt E.W. Nilsson
- teachers: -
- supervisors: -
- student representatives: Erik Andersson, Boel Brandström, Henric Ernbrink, Tim Johansson Nero och Simon Pettersson Fors.
Course purpose
This course will introduce the student to relativistic quantum field theory, QFT, and the most fundamental processes involving elementary particles. The methods studied are perturbation theory and Feynman integral techniques both at tree level and 1-loop level.
These methods are also applicable in condensed matter theory but that will only very briefly be commented upon.
The analysis at loop-level will require regularisation and renormalisation methods which are crucial for a proper understanding of how physics is extracted from ordinary field theories. The fields discussed are spin 0 (Klein-Gordon), spin 1/2 (Dirac) and spin 1 (Maxwell and Yang-Mills) paving the way for a basic understanding of the Standard Model of particle physics. Also Einstein's theory of gravity (general relativity) will be discussed and its problematic features in relation to QFT emphasised.
We will also discuss symmetries and spontaneous symmetry breaking, including the Higgs effect, since these aspects are an essential part of the Standard Model mentioned above.
Schedule
Course literature
The text book we will use is: "An introduction to quantum field theory", Michael Peskin and Daniel V. Schroeder (AW 1995). Available at Cremona.
Other excellent books (at a slightly higher level):
"Quantum Field Theory and the Standard Model", Matthew D. Schwartz (2014)
"The Quantum Theory of Fields",Volumes 1 and 2, Steven Weinberg, (CUP 2005)
"Quantum Field Theory", Mark Srednicki (CUP 2007)
"Field Theory - A Modern Primer", Pierre Ramond (AW 1999)
On-line lecture notes: Look up the authors
a) Tong b) Beisert c) Alvarez-Gaume'
Course design
General reading instructions:
It is very important that you follow the reading instructions attached to each lecture below. The textbook is very pedagogical and most derivations quite detailed but since the goal of this course is to cover the renormalisation of QED we will sometimes be forced to jump back and forth in the book.
This Canvas document will be updated at least once a week, which could involve more reading instructions and new links to things to read.
Lecture 1 (24/3): Intro to QFT and the course. Overview of classical field theories plus some introductory group theory. Interacting Lagrangian field theories, covariant derivatives and Noether's theorem.
Reading instructions: Read the lecture notes and "Doc1-Intro to QFT" carefully together with sections 2.2 and 4.1 and have a look at sect 11.1 plus the chapter 11 intro-page (p. 347). This is all discussed in "Doc2-Field theory". The lecture notes on Noether's theorem differs slightly from the book (in sect 2.2) but is probably easier. Note that we will come back to the very important discussion about " renormalisability" (mentioned on pages 81-82) later! Read also Doc3: "Group theory 1" containing some basic group theory.
Doc3-Home problem 1- The Higgs mechanism.pdf Deadline: 18.00 Friday, April 3
Doc4-Weinberg's view on path integrals.pdf (sorry for the rotation!)
Lecture 2 (26/3): cont. from lecture 1 plus PS Chapter 1. Intro to scattering with a brief account of the cross section of the electron - positron scattering process into muon - antimuon (PS Chapter 1) and the role of loop corrections studied in detail later in the course.
Reading instructions: . Read the lecture notes and chapter 1 in PS carefully and try to get the logic as well as possible! Don't worry if you don't understand the details in PS Chap 1. They will be explained later in the course. Read also " Notations and Conventions" in PS, pages xix--xxi, and review from previous courses in QM the definitions of "differential cross section", the "Born approximation" and its first correction.
Lecture 3 (27/3): The quantized free Klein-Gordon field, causality and the Feynman propagator.
Reading instructions: Read PS Chapter 2 plus lecture notes. Most things in this chapter are of fundamental importance for the conceptual understanding of QFT discussed here in the most simple context possible, the scalar field (Klein-Gordon theory). Read this chapter together with the lecture notes (which differs from the book at some points) carefully and try to check as many calculations as possible, in particular those in the lecture notes (Doc6-file) and on pages 20-24 and 28-31!
Important conceptual equations and paragraphs are: eq 2.38, the paragraph with eqs 2.41 and 2.42, last paragraph on page 28, the main paragraph on page 29 ("Causality...") and the last two paragraphs on page 31.
Doc6-The Klein-Gordon quantum field.pdf
Home problem 2: Do problem 2.2 in PS. Deadline: 18.00 Friday, April 17
Lecture 4 (31/3): cont. Chapter 2: Causality and the Feynman propagator.
Reading instruction: Make sure you understand the differences between the retarded and Feynman propagators! Compare also eqs (2.46) and (2.48) to eq (2.38).
Doc7-Causality and the Feynman propagator.pdf
Lecture 5 (2/4): PS Chapter 3. The Dirac field. Sections 3.1 - 3.4 and lecture notes plus "Group theory 2 " (useful but not required). The free quantized Dirac field (lecture notes).
Reading instructions for Chapter 3: See lecture notes for what are the most important points in this (rather difficult) chapter! Read your notes from the lectures and make sure you understand the gamma matrices and how Lorentz transformations work (Doc-file: "Group theory 2" might be useful here). Read also the talk by David Gross at Solvay 2011.
Doc8-Dirac field theory.pdf (two lectures)
Doc9-David Gross at Solvay 2011.pdf
Doc10-Solvay 1911 (picture).pdf
Lecture 6 (3/4): cont. PS Chapter 3
3/4: Hand-in deadline for home problem 1 at 18.00-> Monday 6/4 at 12.00.
Home problem 3: Solve problem 3.4 in PS on Majorana fermions. Deadline: Friday April 24 at 18.00.
Easter holiday
Lecture 7 (16/4): Sections 3.5 - 3.6 and summary of Chap. 3.
Reading instructions for Chapter 3: The discussion about "How not to quantize the Dirac field:..." (bottom page 52 - top page 58) may be better to read after the correct way to quantize it has been understood in detail (pages 58 - 63).
Doc11-Dirac quantum field theory.pdf
Below are three files on group theory (on request): not required for the oral!
Doc12-Group theory 1.pdf (2 pages )
Doc13-Group theory 2-Basic Lie algebras and representations.pdf (7 pages, most of the important parts appear already in the lecture notes)
Doc14-Group theory 3-Yang-Mills gauge theory.pdf (3 pages)
Lecture 8 (17/4): PS sections 4.1 and 4.2: Interactions in field theory and the perturbation expansion.
Doc15-Perturbation theory 1 .pdf
Reading instructions for Chapter 4: In section 4.1 try again to get the point about renormalisable Lagrangians. In section 4.2 the main equation to understand is (4.31) and in section 4.3 the connection between time ordering and normal ordering, i.e., eq (4.38) known as "Wick's theorem". Note the comment after eq (4.36). Section 4.4 is very important for the over-all picture of how Feynman diagrams and perturbation theory work. Important concepts here are: symmetry factors, Feynman rules, exponentiation of disconnected diagrams and connected diagrams. The final result is eq (4.57).
Instructions for the second half of Chap. 4, see below.
17/4: Hand-in deadline for home problem 2 at 18.00.
Lecture 9 (21/4): PS sections 4.3 and 4.4.
Doc16-Perturbation theory 2.pdf
Wick's theorem and the Feynman diagram expansion. The main result is eq. (4.57).
Lecture 10 (23/4): PS sections 4.5 and 4.6. Cross-sections and S-matrix elements. The main results are eqs (4.79) and (4.104).
Doc17-Perturbation theory 3.pdf
Reading instructions: Section 4.5 will be presented in the lecture in condensed form and you can skip most of this section in the book. However, make sure you understand eq. (4.79) and the rest of sect. 4.5. The whole section 4.6 is crucial with the main results given in eq. (4.90) (or (4.104)) and the final form of the Feynman rules for a real scalar field on pages 114 - 115. Read both the lecture notes and the book!
Home problem 4: Do problem 4.3 in PS. Deadline 5/5 at 18.00.
Lecture 11 (24/4): PS sections 4.7 and 4.8.
Doc18-Perturbation theory 4.pdf
The important results here are the Feynman rules for fermions and the vector potential in EM. Read the lecture notes first and go to the book if you need more information. This will certainly be needed for the understanding of the discussion on pages 125 and 126.
24/4: Hand-in deadline for home problem 3.
Lecture 12 (28/4): PS Chapter 5, sections 5.1 and comments on bound states (from section 5.3 which is not part of the course except for the comments). See "Reading instructions for Chapter 5 below!
Doc19-Cross section calculation-unpolarised.pdf
Lecture 13 (5/5): cont. PS Chapter 5, section 5.2 plus beginning of sect. 5.4.
Doc20-Cross section calculation-polarised.pdf
Home problem 5: Do problem 5.2 in PS on (unpolarised) Bhabha scattering. Deadline 14/5 at 18.00.
Hand-in deadline for home problem 4 at 18.00.
Lecture 14 (7/5): cont. PS Chapter 5, section 5.4 and 5.5.
Doc21-Crossing symmetry-Ward identity.pdf
Reading instructions for Chapter 5:
a) There is a lot of Dirac matrix algebra in Chapter 5: try to use the tricks introduced in the lectures. These tricks will speed up the calculations! See also the Appendix in PS!
b) NOTE: Section 5.3 is not included in the course, just the comments in the lecture.
c) The key points in sect. 5.5. are the Klein-Nishina, the Thomson cross section formulas (page 163) and the discussion leading up to the Ward identity on page 160 (together with the short paragraph at the top of p. 161). Also important is the pair annihilation on pages 168-169.
Lecture 15 (8/5): Regularisation and renormalisation.
Doc22-Renormalisation Lect. 15-16.pdf
Parts of Chap. 6, sections 6.2 and 6.3, and Chapter 10, section 10.2 plus Chapter 7, p. 249 - 251.
Reading instructions: You should try to read the following pages in this order. Chap 6: pages 175-176, skip section 6.1 (not included) and read 2nd half of page 184 in section 6.2. The rest of this section plus section 6.3 will be done later. Sections 6.4 and 6.5 are not included.
Then jump to Chapter 10, read the first half page p. 315 before section 10.1 and then the whole of section 10.2. Note that here "Feynman parameters", pages 189 - 190 in section 6.3, and "Dimensional regularisation", pages 249-251 in section 7.5, are used). We'll come back to sections 10.1 and 10.3 later. (Sections 10.4 and 10.5 are not included in the course.)
Home problem 6: Do problem 10.2 in PS. Deadline Monday May 25 at noon sharp!
Lecture 16 (12/5): cont. Chap 10: section 10. 2.
14/5: Hand-in deadline home problem 5 at 18.00.
Lecture 17 (15/5): cont. Chapter 10, sections 10.1 and 10.3.
Doc22-Renormalisation Lects. 15-16-17.pdf
Lecture 18 (19/5): Chapter 6, sections 6.2 and 6.3.
25/5 at noon sharp: Hand-in deadline for home problem 6.
Lecture 19 (26/5): cont. Chapter 6, intro, sections 6.2 and 6.3 (the rest of the chapter is not included).
Doc22-Renormalisation Lects. 15-16-17-18-19.pdf
Lecture 20 (28/5): Chapter 7, sections 7.1 and 7.5 in detail, and some comments (see lectures) on the optical theorem (sects 7.2, 7.3 and 7.4 are not included).
Doc22-Renormalisation Lects. 15-16-17-18-19 - 20.pdf
Reading instructions Chapter 7: Try to get the general idea about the structure of the exact propagator in sect 7.1, pages 211 - 216 (the mathematical details are not that important). The rest of sect 7.1 is important (see also the lecture notes) including the mathematical details. The optical theorem in sect 7.3 is only included in relation to the result in eq (7.92) (see fig. 7.6, page 235) as explained in the lecture. Sect 7.5 contains the main results of the whole course, in particular the physics discussed at the end, pages 255 and 256!
Changes made since the last occasion
No major changes. Minor changes done to adopt to the new master program.
Learning objectives and syllabus
After the successful completion of the course the student will be able to use QFT to derive the basic fundamental dynamical properties of the following processes: Electron - electron scattering (Moeller scattering), electron - positron scattering (Bhabha scattering), Compton scattering, muon and hadronic production in electron - positron annihilation. Besides the above specific processes the student will have reached a level of knowledge where he or she will be able to derive Feynman rules for more general processes and study their dynamics to leading order in perturbation theory. Some more advanced aspects like Yang-Mills theoires and loop corrections will also be discussed. The student will be well equipped for taking more advanced courses involving renormalisation and radiative corrections.
Examination form
The examination consists of two parts, both mandatory, with different weights in the final grade:
1. A number of home problems (roughly one per week): 40% of the final grade.
2. An oral exam (40 to 50 minutes, in the examination week): 60% of the final grade.
Course summary:
| Date | Details | Due |
|---|---|---|