# Quantum Field Theory

## PT4.4

Lecturer Tim Evans (see Tim Evans' QFT research).

26 hours of lectures, Rapid Feedback classes (see below) every two weeks plus two office hours each week, all during the Autumn term. Seven problem sheets will be handed out but are not assessed, along with numerous additional handouts. A final revision lecture and exam office hours will be given in the summer term.

The physics undergraduate lecture schedule on the undergraduate website is the definitive timetable for this course, not the MSc timetable available elsewhere.

**Recent Updates (12/1/18)**

(12/01/18) Three small but annoying annoying typos in answers to PS6 corrected, new version PS6 Answers (12/01/18)

(9/1/18) The New Year Test for QFFF students in QFT New Year Test 2018 (8/1/18) is now online. May be of use for revision by other students as questions are typical exam questions. Note this paper includes a final page of formulae which I expect to be used in the main QFT exams.

(2/1/18) Updated handout The Feynman Propagator and Cauchy’s Theorem (2/1/18) with additional note on time-ordering of fields at equal times. A couple of very minor changes to other handouts.

(21/12/17) Answers to PS7 online. Also online, Feynman Rules in Momentum Space (21/12/17) a draft version of new handout, including a discussion of loop momenta and the special case of the single propgator diagram.

(11/12/17) **THANKS** to everyone who sat through the course, pointed out my mistakes or just asked questions in office hours. It really helps me improve the course and makes it so much more enjoyable for me to teach. QFT is my favourite theory but it is far from easy to follow. I never understood it from my first course, I'm just hoping my course helps you along that road.

(11/12/17) The 2017 exam script and comments on the exam are online. The QFT Syllabus Autumn 2017 (8/12/17) was handed out at the last lecture along with PS7 but all material is now are online. Last two office hours are this week. There will be more in the summer term, along with a revision lecture, before the exam.

(17/10/17) Crowded Lecture Theatre.** **The QFT course is unusual as it takes significant numbers of students from a wide range of UG and MSc courses, probably a wider range than any other Physics course. It seems numbers have fluctuated up this year which is making for a crowded lecture theatre. I'm pretty sure that the numbers of students taking the final exam will be between 110 and 120. That is because we had 105 sitting the summer 2017 exam of which around 60% were not on BSc or MSci Imperial undergraduate courses, I can't see MSc numbers rising by more than 25% in one year. So you will all fit into the lecture theatre but it may be a squeeze which is where we are now. There is no alternative lecture theatre available for the QFT lectures on a regular basis and it is too late to change the timing of QFT lectures because of so many different student timetables. The department and I are actively looking at this issue both for this year and for future cohorts. In the meantime, I can suggest a few ways to deal with this problem:-

- The lectures are recorded. They often become available within 30 minutes but I have to give approval by hand to publish them which may take longer. Please look on the Imperial Panopto website and search for "Quantum Field Theory". More information in the "Videos" tab on this page but that is updated by me so maybe less up to date.
- I will put up copies of my handwritten notes on this website, under the "Handwritten Notes" tab. Not designed to be read by students but I am trying to put a recent version online in advance of each lecture and to leave an older version (probably close to what I will say anyway) in the same directory so they are accessible even if not with an explicit link on my web page.

## QFT

### General Information for Autumn 2017 Course

- A Summary of key facts (25/9/17) provides general overview and guidance on the prerequisites and knowledge needed for this course. Checkout Problem Sheet 1 (PS1) below as that is a good test to see if you are ready for the course.
- Syllabus for the Autumn 2016 QFT course is available and is unlikely to change in any significant way for Autumn 2017.
- There will be an early rapid feedback on
**Thursday 12th October**on Q2 and 4 of PS1 (11/10/17). This problem sheet does not draw on material from the lecture course but it illustrates the level of algebraic manipulations of annihilation and creation operators needed for the course. The problem sheet acts as a warm up for the course and can be done before the course starts. For written feedback hand in by**noon, Monday 9th October**, as detailed under the rapid feedback section of this website. - The library has a nice version of my QFT bibliography with links to the actual texts. Information from my Bibliography (25/9/17) is largely duplicated there except for my comments on each text.

### Outline

Below is a rough outline of the course. A detailed Syllabus for the Autumn 2016 QFT course is online and no major changes are expected in 2017 from the 2016 course. Check my Bibliography (25/9/17) with more detailed comments on a range of books.

**Introduction. **(1 lecture) Basic aims and ideas of field theory. Types of field and relationship to symmetries (relativistic, non-relativistic symmetry and other particle symmetries).

**Classical Field Theory and Scalar Fields.** (5 lectures) Lagrangian and Hamiltonian descriptions of field theory. Linearity and interactions. Scalar Field Plane wave solutions of the Klein Gordon equation. Internal symmetries and Noether's theorem.

**Free QFT.** (5 lectures) Quantisation and commutation relations. Creation and annihilation operators. Time and QFT. Complex scalar fields and their conserved charge.

**Interacting QFT.** (14 lectures) Interaction Picture and S matrix. Wick's theorem and normal ordering. Simple Feynman diagrams in coordinate and momentum spaces. Cross sections, matrix elements and vacuum expectation values. Vacuum diagrams. Basic example of mass renormalisation.

Note that Fermions were included by previous lecturers so you will find some older exam papers contain questions no longer covered by this course though the QFFF MSc will cover that material in other courses.

### Office Hours

Tuesdays and Fridays at 10.00 in my office Huxley 609.

**NOTE** that the office hour on **Tuesday 24th October** will start half an hour later at **10.30am** and end at 11.20am.

Students are welcome to drop into my office at these times when I'll be happy to talk about anything related to the QFT course.

### Rapid Feedback Classes

The QFT Rapid Feedback Classes for Autumn 2016 are on *alternate* weeks on **Fridays** in **Lecture Theatre 3 **at **17.00** (but check your timetable for the lastest information). There will also be a special initial rapid feedback session in week 2 on **Thursday 12th October at 1pm** (again check your timetable). The rapid feedback sessions are run by Lucas Wallis, a PhD student from Theoretical Physics. For each rapid feedback class the lecturer will nominate a few questions from one of the problem sheets. If students wish written feedback on their answers to these nominated questions only, they hand their work to the Physics Undergraduate Student Administrative Office on Level 3 in Blackett by **12.00 on Monday 9th October** for the first session and then **12.00 on Wednesday** before every other presentation. The UG office on level 3 Blackett will provide boxes for this. The PhD student will provide written feedback on answers (for the nominated questions only) and will return these scripts, either at his presentation on the following Friday or via the student office again.

Handing in the work and/or attending the rapid feedback classes are optional. This work does **not** contribute to the course assessment. However, I may ask the Rapid Feedback presenter to provide a grade on scripts as part of the written feedback to students. I will collect these at the end of the course as they can be useful in many informal circumstances when students need some measure of their progress before the exam. For instance, QFFF masters students often need references long before the exams. I will keep these informal marks private, only sharing them with other staff when appropriate. Students are strongly advised to make use of this opportunity for feedback and extra tuition.

Note that **only** the nominated questions are dealt with by the PhD student in charge of rapid feedback. I provide full answers for all questions and am happy to discuss the problem sheets in my office hours.

**Timetable Autumn 2017**

Please refer to the official timetable which is the definitive guide. To gain written feedback, hand in your work to the Physics Undergraduate Student Administrative Office on Level 3 by **noon** on the Wednesday before the presentation unless stated below.

- Week 2: Thursday 12th October, PS1, Q2 and 4. Note hand in for feedback is by
**noon**on**Monday 9th October**to UG office. - Week 4: Friday 27th October, PS2, Q3 and 4.
- Week 6: Friday 10th November, PS4, Q1 and 3.
- Week 8: Friday 24th November, PS5, Q2 and 3.
- Week 10: Friday 9th December, PS6, Q3 and 4.

### Problem Sheets and Handouts

David Tong's QFT lecture notes match this course well.

Tim Evans' handwritten QFT notes from this year are added before each lecture (usually) see below.

The QFT Syllabus Autumn 2017 (8/12/17) is available for the Autumn 2017 course.

### Handouts

Individual handouts (also see below for supplementary material not handed out):-

- Summary of key facts (25/9/17)
- Bibliography (25/9/17) or Library's QFT Course Bibliography - the former has my comments on each text, the latter links texts to library resources.
- Notation (7/10/16)
- Fourier Transforms, Delta Functions and Theta Functions (3/10/17)
- The Feynman Propagator and Cauchy’s Theorem (2/1/18)
- Different Pictures in QFT (3/11/17)
- Scalar Yukawa Theory - SYTh (10/11/17)
- Recipe For Perturbative QFT (17/11/17)
- Matrix Element to Green Function (13/11/17)
- Notes on Wick's Theorem (Draft version, 29/11/17) see also Haisch's notes on Wick's theorem
- Feynman Rules for SYTh in Coordinate Space (24/11/17)
- 'Diagramology' - Types of Feynman Diagram (2/12/18)
- The Free Vacuum and the Physical Vacuum (30/11/17)
- Feynman Rules in Momentum Space (21/12/17)
- Cross Sections (30/11/17)

Problem Sheets and Solutions

Problem Sheets:

- PS1 (11/10/17) Annihilation and Creation Operators.

Students must be able to handle this problem sheet**before**taking this course. I will assume students are already experienced in handling such operators and I will not be covering this aspect in the course. See my course summary handout for more details. - PS2 (10/10/17) Classical Field Theory (mostly).
- PS4 (31/10/17) Free Scalar Quantum Field Theory
- PS5 (9/11/16) Generic Interacting Quantum Field Theory
- PS6 (20/11/17) Feynman Diagrams in Coordinate Space
- PS7 (30/11/17) Interacting Quantum Field Theory: λφ4

Solutions: PS1 Answers 11/10/17, PS2 Answers 10/10/17, PS4 answers (17/11/17), PS5 Answers (9/11/16), PS6 Answers (12/01/18) , PS7 Answers (29/11/16)

### Supplementary Material

This may be of interest or use to supplement other material.

- Note the videos on Panopto (links elsewhere on this website).
- These notes on Classical Field Theory are taken from a longer set of notes I made for my old Unification course. Those doing the Unification course might find these whole set of notes on Symmetry and Unification useful. They contain a brief overview of the background particle physics and group theory needed for Unification so may be of use more generally. The notes are unfinished in places so use with care.
- Haisch's notes on Wick's theorem, taken from Interacting Fields notes.

### Past Exams

### QFT examination

This is a final exam which is the same for undergraduates and postgraduates. The assessment for this course is based only on this exam.

The Syllabus for the Autumn 2016 QFT course lists the examinable material. All material covered in the lectures is examinable except for the lecture on Fermions. Material in the problem sheets is examinable except for those questions marked as optional. The questions marked with a star in the problem sheets are core material, those unmarked are important, but I will not assume any knowledge of questions marked optional. I could draw on material contained in the optional questions but this would only to be for the last few marks of a paper and I will not assume any prior knowledge.

The current course will be reflected in exams from May 2015 onwards. Earlier exams will contain some material no longer in the syllabus. However, the structure of the 2015 exam will be the same as recent years. That is students will have to answer three out of four questions, each question marked out of 30. Note that the department's policy is to put out solutions for undergraduate exams only every three years. However much of the material is derived from the problem sheets so you should find most answers in their solutions.

- QFT Exam 2017 and QFT Exam 2017 post exam comments.
- QFT exam 2016Comments on 2016 QFT Exam.
- qftexam2015.pdf, the solutions for 2015 QFT exam (12/5/16) and comments on 2015 QFT exam

Previous less relevant exams are *2006, **2007, **2008, **2009, **2010, **2011 *and* **2012. *Solutions for undergraduate exams for the 2009 and 2012 are provided here, reproducing what is on the department's model answers web page.

### New Year test

The **QFFF MSc students only** have the option to take a test on the QFT course early in the New Year (typically one of the first few days of Spring term). Students on other courses *cannot* take this test, but the material may be of use for revision purposes. This test is taken in exam conditions but the marks are only used to provide feedback to QFFF students. The test does *not* contribute to the final mark for the course. The New Year test is also used to support QFFF students by providing staff with useful information when writing references as many QFFF students have only been here for only a few months (though this is not a service provided by most MSc courses). The test will be made available here as soon as the formal test has finished as the test material may be of use to anyone revising for the final exam. The test is *not* a mock exam and it is not necessarily in the style of the final exam but the test is likely to share several features with the final exam. In particular, it covers exactly the same material as detailed for the exam. The January 2018 test was three exam style questions with no choicem while the final exam requires three questions out of four.

- QFT New Year Test 2018 (8/1/18)
- QFT Test 2017 and General Comments on QFT Test 2017 (see relevant problem sheets for answers)
- 2016 QFT New Year Test and 2016 QFT New Year Test answers.

Note these are*not*model answers. The answers contain the minimum needed but they also contain extra material such as a longer discussion to illustrate how I am thinking through a problem. - 2015 QFT Test (see relevant problem sheets for answers).

Like the final exam, only tests from 2015 onwards will reflect the current syllabus exactly, previous tests may contain material no longer on the syllabus but are provided below in case they are of use.

2006 Test, 2007 Test, 2008 Test, 2009 Test, 2010 test, 2011 test, 2012 test,

### Videos of Lectures

The primary source is the Imperial College Panopto site which is the definitive and most up to date source - search for "Quantum Field Theory".

I adjust the settings for each video by hand (hopefully soon after the lecture but sometimes there may be a delay of a day or two) to maximise availability within College but I am limited by College policy in what I can do. The access lists are maintained by hand so may not be always up to date so if you have a problem contact me and I will pass your name on. In some cases sound may be poor - do remind me to wear the microphone and to make sure they are not muted. In some rare cases, the lecture may not have been recorded for some technical reason. Feedback on these videos (were they useful, how did you use them, ideas for improvements) welcome.

This list of links below is updated by hand often a few days after lectures so try the main Panopto site for latest videos.

#### Autumn 2017 Course

- 5/10/17 QFT Lecture 1 - 1. Introduction
- 6/10/17 QFT Lecture 2 - 2.1 Classical Dynamics, 2.2 One Dimensional Ring Model
- 9/10/17 QFT Lecture 3 - 2.2 One Dimensional Ring Model
- 10/10/17 QFT Lecture 4 - 2.3 Symmetry and Lagrangians (see also my notes on Classical Field Theory)
- 13/10/17 QFT Lecture 5 - 2.4 The Klein-Gordon Equation (see also my notes on Classical Field Theory)
- 16/10/17 QFT Lecture 6 - 2.5 Noether's theorem (note there is no section 3 in the current course)
- 17/10/17 QFT Lecture 7 - 4.1 QHO, 4.2 Free Real Scalar Field QFT
- 20/10/17 QFT Lecture 8 - 4.2 Free Real Scalar Field QFT
- 23/10/17 QFT Lecture 9 - 4.3 Time and QFT
- 27/10/17 QFT Lecture 10 - 4.4 Propagators and Two-Point Green Functions: Retarded.
- 30/10/17 QFT Lecture 11 - 4.4. Propagators and Two-Point Green Functions: Retarded, Wightman, Feynman.
- 31/10/17 QFT Lecture 12 - 4.5. Free Quantised Complex Fields
- 3/11/17 QFT Lecture 13 - 5.1 Interaction Picture
- 6/11/17 QFT Lecture 14 - 5.1 Interaction Picture, start of 5.2
- 10/11/17 QFT Lecture 15 - 5.2 Matrix Elements, state normalisation, definition of Scalar Yukawa Theory
- 13/11/17 QFT Lecture 16 - 5.2 Matrix Elements to Green functions
- 17/11/17 QFT Lecture 17 - 5.3 Normal Ordering and Contractions
- 20/11/17 QFT Lecture 18 - 5.3 Wick's Theorem: definition, use, and start of proof.
- 24/11/17 QFT Lecture 19 - 5.3 Wick's Theorem, end of proof; 5.4 Feynman rules.
- 27/11/16 QFT Lecture 20 - 5.4 Feynman Diagrams in Coordinate Space.
- 28/11/16 QFT Lecture 21- 5.4 Feynman Diagrams in Coordinate Space (cont.)
- 1/12/16 QFT Lecture 22 - 5.4 Feynman Diagrams and the Symmetry Factor, 5.5 Vacuums.
- 4/12/16 QFT Lecture 23 - 5.5 Feynman Diagrams, the Physical Vacuum and Vaccum Diagrams
- 5/12/16 QFT Lecture 24 - 5.6 Feynman Diagrams in Momentum space and Loop Momenta
- 8/12/16 QFT Lecture 25 - 5.7 Cross Section example
- 11/12/16 QFT Lecture 26 - 5.8 Renormalisation

#### Autumn 2016 Course

- 6/10/16 QFT Lecture 1
- 7/10/16 QFT Lecture 2
- 10/10/16 QFT Lecture 3
- 11/10/16 QFT Lecture 4
- 14/10/16 QFT Lecture 5
- 17/10/16 QFT Lecture 6
- 18/10/16 QFT Lecture 7
- 21/10/16 QFT Lecture 8
- 24/10/16 QFT Lecture 9
- 28/10/16 QFT Lecture 10 - 4.4. Two-Point Functions, Propagators
- 31/10/16 QFT Lecture 11 - 4.4. Retarded and Feynman Propagators
- 1/11/16 QFT Lecture 12 - 4.5. Complex Scalar Free Field Theory
- 4/11/16 QFT Lecture 13 - 5. Interacting QFT, 5.1 Interaction Picture
- 7/11/16 QFT Lecture 14 - 5.1 Interaction Picture, 5.2 Scalar Yukawa Theory Definition
- 11/11/16 QFT Lecture 15 - 5.2 Matrix Elements and State Normalisation
- 14/11/16 QFT Lecture 16 - 5.2 Matrix Elements from Green functions
- 18/11/16 QFT Lecture 17 - 5.3 Wick's Theorem, Normal Ordering and Contractions
- 21/11/16 QFT Lecture 18 - 5.3 Wick's Theorem proof
- 25/11/16 QFT Lecture 19 - 5.3 Wick's Theorem, 5.4 Feynman Diagrams
- 28/11/16 QFT Lecture 20 - 5.4 Feynman Diagrams in Coordinate Space
- 29/11/16 QFT Lecture 21- 5.4 Feynman Diagrams in Coordinate Space
- 2/12/16 QFT Lecture 22 - 5.4 Feynman Diagrams and the Symmetry Factor
- 5/12/16 QFT Lecture 23 - 5.4 Feynman Diagrams, the Physical Vacuum and Vaccum Diagrams
- 6/12/16 QFT Lecture 24 - 5.4 Feynman Diagrams in Momentum space and Loop Momenta
- 9/12/16 QFT Lecture 25 - 5.4 Cross Section example
- 12/12/16 QFT Lecture 26 - 5.5 Renormalisation

#### Autumn 2015 Course

Another list of the Autumn QFT 2015 Course Videos is available. Just ignore the fermion lecture for 2016 onwards.

- Monday 12th October. QFT Lecture 1, Introduction
- Tuesday 13th October (morning). QFT Lecture 2, Classical Field Theory
- Tuesday 13th October (afternoon). QFT Lecture 3, Classical Field Theory
- Friday 16th October. QFT Lecture 4, Classical Field Theory
- Tuesday 20th October. QFT Lecture 5, Classical Field Theory
- Wednesday 21st October. QFT Lecture 6, Classical Field Theory
- Friday 23rd October. QFT Lecture 7, Fermions
- Friday 23rd October. QFT Lecture 8, Free QFT
- Monday 26th October. QFT Lecture 8, Free QFT
- Tuesday 27th October. QFT Lecture 9, Free QFT
- Friday 30th October. QFT Lecture 10, Free QFT
- Monday 2nd November. QFT Lecture 11, Free QFT
- Friday 6th November. QFT Lecture 12, Free QFT
- Monday 9th November. QFT Lecture 13, Free Complex Scalar QFT
- Friday 13th November (morning). QFT Lecture 14, Interaction Picture
- Friday 13th November (afternoon). QFT Lecture 15, Interaction Picture
- Monday 16th November. QFT Lecture 16, Matrix Elements
- Friday 20th November. QFT Lecture 17, Wick's Theorem (no recording, technical failure)
- Monday 23rd November. QFT Lecture 18, Wicks Theorem
- Friday 27th November. QFT Lecture 19, Wick's Theorem
- Monday 30th November. QFT Lecture 20, 5.4, Feynman Diagrams
- Friday 4th December. QFT Lecture 21, 5.4, Feynman Diagrams
- Monday 7th December. QFT Lecture 22, 5.4, Feynman Diagrams - Symmetry Factors
- Thursday 10th December. QFT Lecture 23, 5.4, Feynman Diagrams - Vacuum Diagrams
- Friday 11th December. QFT Lecture 24, 5.4, Feynman Diagrams - Momentum space rules
- Monday 14th December. QFT Lecture 25, 5.4, Feynman Diagrams - Cross Sections
- Tuesday 15th December. QFT Lecture 26, 5.5 Renormalisation. Not recorded.

Some of the Autumn QFT 2014 Course Videos are available too.

### Handwritten Notes

### Handwritten Notes 2017

These are not intended for student use and may not reflect what was actually said. I provide them **as is** in case they are any use. The notes have been corrected over the last few years so most errors have been eliminated from the older notes. At the same time, I do rewrite sections every year so new errors creep in. Do flag up any queries with me, I'm happy to correct mistakes or explain issues but I would rely on David Tong's typed notes for factors of ** i** or signs.

A list of Tim Evans' handwritten QFT notes or try links below (lecture notes also online)

- Section 1. Introduction
- Section 2.1. Classical Dynamics
- Section 2.2. One-Dimensional Chain
- Section 2.3. Symmetry and Classical Fields
- Section 2.3. Klein-Gordon Equation and Section 2.4 Complex Fields
- Section 2.5. Noether's Theorem
- Section 4.1-4.3: 4.1 QHO, 4.2 Free Scalar Field QFT, 4.3 Time and QFT
- Section 4.4 Propagators
- Section 4.5 Quantised Free Complex Fields
- Section 5.1 Interaction Picture
- Section 5.2 Matrix Elements
- Section 5.3 Wick's Theorem
- Section 5.4 Feynman Diagrams
- Section 5.4 Feynman Diagrams in Coordinate Space
- Section 5.5 The Vacuum in QFT
- Section 5.6 Feynman Diagrams in Momentum Space
- Section 5.7 Cross Sections
- Section 5.7 Renormalisation