## Topic outline

### General

## Welcome to the Relativistic Waves and Quantum Fields (SPA7018U/P) WEBPAGE

SYNOPSIS

This course provides an introduction to Relativistic Quantum Field Theory, which unifies two of last century's greatest discoveries in Physics: Special Relativity and Quantum Mechanics.

## AIMS

Relativistic wave equations for particles of various spins will be derived and studied, and the physical interpretations of their solutions will be analysed. After an introduction to classical field theory, and the role of symmetries in field theory (including the beautiful Noether's theorem) students will learn the fundamental concepts of quantum field theory, including the quantisation of the free Klein-Gordon and Dirac fields and the derivation of the Feynman propagator. Interactions are introduced and a systematic procedure to calculate scattering amplitudes using Feynman diagrams is derived. We will also compute some explicit tree-level scattering amplitudes in a number of simple examples.

## OUTCOMES

Students successfully completing this course will be able to analyse the relativistic wave equations for particles of various spins, and to discuss the physical interpretations of their basic solutions. They will become familiar with various concepts in classical field theory (Noether theorem, stress-energy tensor, symmetries and conserved currents) and quantum field theory (including canonical quantisation of the Klein-Gordon and Dirac fields, creation and annihilation operators, spin-statistics connection, commutators and time ordered products, the Feynman propagator).

## RECOMMENDED BOOKS

Quantum Field Theory is a fascinating and vast subject, and there is a large variety of texts on it. Below you can find a selection of the books that you may find useful for this and later courses in the subject.

- M. Maggiore,
*A modern introduction to Quantum Field Theory*, Oxford University Press. A very nice and modern introduction to Quantum Field Theory. - F. Mandl and G. Shaw,
*Quantum Field Theory*, J Wiley. This is a pedagogical, clear introduction to Quantum Field Theory. We will mostly follow Mandl's and Maggiore's books in this course. - M.E. Peskin and D.V. Schroeder,
*An Introduction to Quantum Field Theory*, Addison-Wesley. A more advanced book on the subject, which covers much more material than the course. If you are interested in theoretical particle physics and wish to study in more detail some topics or have a look at more advanced topic, this is an excellent reference.

- M. Schwartz,
*Quantum Field Theory and the Standard Model,*Cambridge University Press. Another excellent, more recent book. It contains material beyond the scope of this course, but very useful for related courses also on applications to Particle Physics.

- P. Ramond,
*Field Theory: a modern primer,*Addison-Wesley. It contains a very nice introduction to the Lorentz group, and a complete presentation of Noether's theorem. Highly recommended. Very useful also for more advanced topics (beyond the scope of this course) including symmetry factors in Feynman diagrams.

- S. Pokorski,
*Gauge Field Theories*, Cambridge University Press. Another more advanced book on the subject, very clear and complete. A good book to have if you are keen on this subject. It also contains useful applications to the Standar Model.

- S. Weinberg,
*The Quantum Theory of Fields,*in 3 Volumes., Cambridge University Press. An advanced book on Quantum Field Theory, written by a Nobel laureate. Probably overwhelming as a first QFT book. The first chapters of the first volume are relevant for the course.

- C. Itzykson and J.-B. Zuber,
*Quantum Field Theory*. A classic text, but less pedagogical than Peskin-Schroeder; in particular Sections 1, 2 & 3 have considerable overlap with this course.

- J. Bjorken and S. Drell,
*Relativistic quantum mechanics*and*Relativistic quantum fields*, McGraw-Hill. An old but reliable source, in particular the first volume is relevant to the first part of this course. - S. Coleman, lecture notes from his Quantum Field Theory course in Harvard. These can be downloaded here. A deep, fascinating presentation of the subject from a superb teacher. These have now been published with the title
*Quantum Field Theory Lectures of Sidney Coleman*.

- M. Maggiore,

### Syllabus

**A very detailed programme of the course can be found here**. It contains a precise list of all the topics covered in every single lecture of the course.Below you can find a more succinct outline of the course, with references to textbook chapters (MS stands for Mandl-Shaw).

1. Introduction to the Lorentz group.

2. Elements of classical field theory: relativistic notation [MS Section 2.1], variational principle, equations of motion and Noether's theorem [MS Section 2.2, part in Section 2.4]. Natural units [MS Sections 6.1].

3. Classical scalar fields: real and complex Klein-Gordon action and the associated equation [MS Sections 3.1 and 3.2 (just the classical part)].

4. Second quantisation (free theories): canonical quantisation and Fock space interpretation of the free real and complex Klein-Gordon field; normal ordering [MS Sections 3.1 and 3.2]. Causality, commutators and time-ordered products, the Feynman propagator [MS Sections 3.3 and 3.4]. Advanced, retarded and Feynman Green's functions.

5. Classical spinor fields: Dirac equation [MS Section 4.2 (just the classical part)], Gamma matrices [MS Section A.8]. Spin of the Dirac field. Relativistic invariance of the Dirac equation [MS Section A.7]. Action principle for Dirac's field [MS Section 4.2 (just the classical part)], plane wave solutions [MS Section A.4]. Minimal coupling to an electromagnetic field [MS Section 4.5], non-relativistic limit of the Dirac equation, gyromagnetic ratio.

6. Fock space interpretation of the Dirac equation [MS Sections 4.1 and 4.3], Hamiltonian, momentum and U(1) charge in terms of the oscillators. The spin of the Dirac field in the second-quantised language. Dirac propagator [MS Sections 4.4].

7. Interacting theories. Schroedinger, Heisenberg and Interaction pictures [MS Appendix 1.5]. The evolution operator in the interaction picture and the S-matrix. The Dyson expansion of the S-matrix [MS Sections 6.2]. Feynman diagrams. Tree-level diagrams. Examples in phi^3 and phi^4 theories. Superficial degree of divergence of a Feynman diagram.

### Homework Sheets and Solutions

There will be ten exercise sheets, which will be posted on the web each Thursday. Assignments are due a week later (I will collect them during the lecture). Late assignments will be not be marked unless you have medical or other valid reasons. Homework solutions will be available here shortly after the deadline. Note that only five of them will be marked but all ten are very important!

### Lecture Notes

Here you can find my lecture notes for the 2014-2015 course, as well as lecture notes from previous lecturers. Each file correspond to a different lecture. Please refer to the

**programme**for a complete list of the topics covered in the course.**Notes for the 2018 Course - Prof. Gabriele Travaglini****Foundations of mathematical physics - Dr PAUL COOK (KCL). Contains a really nice revision of special relativity and classical mechanics****Notes for the 2011 Course - Dr RODOLFO Russo****Notes for the 2010 Course - Prof. ANDREAS Brandhuber**

### Past Exam Papers

Some of the past exam papers can be downloaded below: