# 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. 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 (18/5/17)**

** **

The PS4 answers (18/5/17) for Q1 and the answers to Q2 in the QFT Exam 2015 answers (18/5/17) have been updated to include the x and y dependence in the commutators being calculated.

The answers to PS6 have been updated again with the correct loop momenta for the vacuum diagrams H and I in table in equation (67) in Q5.iii (the symmetry factor for diagram C was corrected earlier in the week). Note that the diagram with a single line is the exceptional case and does not obey the L=I-(V-C) formula so focus on non-trivial subdiagrams or use common sense when counting loops. This is now noted in the answers for PS6. For the same reason, the single line diagram is the only example which cannot be constructed correctly from the momentum space rules as stated in the course.

The office hours are Tuesdays and Fridays at 10am in my office until the exam.

A final definitive version of the Syllabus for the Autumn 2016 QFT course has been posted (note that the version dated 13th December 2016 is final and so is not provisional as suggested by the title of an earlier version).

The script for the QFFF MSc student Test for 2017 is online along with general comments on typical problems seen in answers. This will be useful for revision for the summer exam.

Response to QFT SOLE Autumn 2016 (6/2/17) from SOLE is online.

## QFT

### Latest Information for Autumn 2017 Course

- A Summary of key facts (12/5/16) 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 in week 2 on PS1 (13/7/16). This does not draw on material from the lecture course but it illustrates the level of algebraic manipulations with annihilation and creation operators needed for the course. It acts as a warm up and can be done before the course starts. PS1 Answers 26/9/16 are online.
- The library has a nice version of my QFT bibliography with links to the actual texts. Information from my Bibliography (19/6/15) is largely duplicated there except for my comments on each text
- Response to QFT SOLE Autumn 2016 (6/2/17).

### Outline

Below is a rough outline of the course. A detailed Syllabus for the Autumn 2016 QFT course is online. Note that Fermions will *not* be included in the Autumn 2016 course but otherwise no major changes are expected from 2015. Check my Bibliography (19/6/15) 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 there will be no lectures on fermion fields in the Autumn 2016 QFT course.

### Office Hours

Until the exam, I will be holding the two office hours every week on Tuesdays and Fridays at 10.00 in my office H609.

### Rapid Feedback Classes

The QFT Rapid Feedback Classes for Autumn 2016 are on *alternate* Fridays in **Lecture Theatre 3 **at 13.00 *except* for the first one in week 2 when it is at 13.00 on Thursday 13th October. They are run by Jonathon Baird from the High Energy Physics group (who ran these last year) or Scott Melville 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 by 12.00 on the Wednesday before the presentation (*except* for the first rapid feedback where hand in is Monday 10th October). The office will provide baskets for this. Jonathan 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 classes are optional, this work does **not** contribute to the course assessment. However I may ask Jonathan and Scott to provide a grade on scripts as part of the written feedback. 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.

**Timetable Autumn 2016**

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 16th October, PS1, Q2 and 4. Note hand in for feedback is by
**noon**on**Monday 10th October**to UG office. - Week 4: Friday 28th October, PS2, Q6 and 7.
- Week 6: Friday 11th November, PS4, Q1 and 3. This will be run by Scott Melville from Theoretical Physics.
- Week 8: Friday 25th November, PS5, Q2 and 3. This will also be run by Scott Melville from Theoretical Physics.
- Week 10: Friday 10th December, PS6, Q1 and 4.

### Problem Sheets and Handouts

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

Some of my lecture notes will also be made available but are not intended for student use. These are scanned in from my personal hand written notes and are full of opaque annotations. I do not always stick to my script so the content of lectures will vary too.

The final Syllabus for the Autumn 2016 QFT course (13/12/16) is also available and defines the content of the course.

### Handouts

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

- Summary of key facts (12/5/16)
- Bibliography (19/6/15) or Library's QFT Course Bibliography - the former has my comments on each text, the latter links texts to library resources.
- Notation (26/9/16)
- Delta Functions and Theta Functions (15/10/15)
- The Feynman Propagator and Cauchy’s Theorem (16/10/15)
- Matrix Element to Green Function (18/10/16) includes definition of SYTh (Scalar Yukawa Theory)
- Operator Ordering (29/11/16)
- Feynman Rules for SYTh in Coordinate Space (23/12/16) (labels now included more carefully)
- Notes on Feynman Diagram Types (17/11/16)
- Cross Sections (Major update, draft version 9/12/16)

Problem Sheets and Solutions

Problem Sheets:

- PS1 (13/7/16) 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 in the course. See my handout for more details. - PS2 (10/10/16) Classical Field Theory
- PS3 Fermions. This is no longer issued in this course.
- PS4 (25/10/16) Free Quantum Field Theory,
- PS5 (9/11/16) Generic Interacting Quantum Field Theory,
- PS6 (2/12/16) Scalar Yukawa Theory (updated by moving confusing definition of Green function in four-momentum space)
- PS7 (29/11/16) Interacting Quantum Field Theory in Momentum Space - phi4 theory.

Solutions: (Available as course develops) PS1 Answers 26/9/16 Annihilation and Creation Operators, PS2 Answers 10/10/16 classical Field Theory, PS4 answers (18/5/17) Free Quantum Field Theory, PS5 Answers (9/11/16) Generic Interacting Quantum Field Theory, PS6 Answers (16/05/17) Scalar Yukawa Theory, PS7 Answers (29/11/16) Quantum Field Theory in Momentum Space - phi4 theory.

### Supplementary Material

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

- Note the videos on Panopto (links elsewhere on this web site).
- 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.

- QFT exam 2016, Comments on 2016 QFT Exam.
- Summer qftexam2015.pdf, 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 can *not* take this test formally, 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. I expect the 2017 test will be of the exactly same format and style as 2016.

- QFT Test 2017 and General Comments on QFT Test 2017
- 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 also 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. This site may give limited access but I adjust the settings for each video by hand (typically a few days after the lecture) 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.

The definitive and most up to date list of videos is on the Imperial panopto web site - search for "Quantum Field Theory".

#### Autumn 2016 Course

- 6/10/12 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.