# RELATIVISTIC QUANTUM MECHANICS - 2020/1

Module code: MATM054

Module Overview

This module introduces the basic concepts and techniques of the Quantum Theory of Fields.

Module provider

Mathematics

TORRIELLI Alessandro (Maths)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 7

JACs code: F342

Module cap (Maximum number of students): N/A

Module Availability

Semester 2

Prerequisites / Co-requisites

MAT3039 or PHY3044

Module content

Topics covered will include some or all of:

1 Preliminaries (1.5 weeks)

• Summary of electromagnetism, without and with 4-vectors

• Summary of relativistic invariance

• Reminder of quantum harmonic oscillator. Second quantisation

• The need for fields

2 Classical Fields (2.5 weeks)

• Summary of Lagrangian and Hamiltonian, application to fields

• Summary of Noether theorem and symmetries

• Examples (real and complex scalars, vectors)

• The Klein-Gordon equation

• Spinorial representations of the Lorentz group

3 Canonical quantisation of free fields (2 weeks)

• The real scalar field

• Fock space

• Locality, causality

4 Interacting fields (2 weeks)

• Perturbation theory of phi^4

• Wick's theorem

• Feynman graphs and rules

• Scattering: S-matrix, LSZ formula, Mandelstam variables, Fermi Golden rule

• Infinities and renormalisation

5 Quantum electrodynamics (3 weeks)

• Bosonic vs Fermionic statistics

• Gamma matrices and Clifford algebras

• Schroedinger-Pauli Hamiltonian

• Dirac equation

• CPT and all that

• Famous tree-level QED processes

• Gauge invariance: from abelian to non-abelian

Students intending to take the module are encouraged to contact the module coordinator at their earliest convenience for an informal discussion and introduction to the structure of the course, and for possible preliminary reading material if they wish to.

Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test One-hour in-semester test 20
Examination EXAMINATION 80

Alternative Assessment

N/A

Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of and ability to interpret and manipulate mathematical statements.

• Subject knowledge through the recall of key postulates, theorems and their proofs.

• Analytical ability through the solution of unseen problems in the exam and the analysis of slightly more advanced topics in the project.

Thus, the summative assessment for this module consists of:

• One two-hour examination (three out of four best answers contribute to the exam mark) at the end of the Semester; worth 80% of the module mark.

• One one-hour in-semester test; worth 20% of the module mark.

Formative assessment and feedback

Students receive written feedback via a mid-Semester un-assessed coursework assignment. Students are encouraged to arrange meetings with the module coordinator for questions and verbal feedback on the weekly comprehension of the material throughout the entire duration of the course.

Module aims

• Introduce students to the mathematical description of relativistic quantum mechanics.
• Enable students to understand the foundations and basic tools of quantum field theory and their applications to the physical world.
• Illustrate standard applications of the theory of quantum fields.

Learning outcomes

Attributes Developed
001 Have a firm understanding of the concepts, theorems and techniques of the quantum theory of fields. KC
002 Have a clear understanding of how to apply the mathematical techniques to concrete physical examples (simple scattering processes between elementary particles, symmetry analysis of specific quantum field theories, spinorial calculus and the Dirac equation). KT
003 Be able to explicitly derive the Feynman rules for simple toy-model systems. KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

Independent Study Hours: 117

Lecture Hours: 33

Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• A detailed introduction to the relevant theory and its tenets, and to the appropriate mathematical tools for their implementation

• Experience (through demonstration) of the methods used to interpret, understand and solve concrete problems, especially for simple toy-model examples

The learning and teaching methods include:

• 3 x 1 hour lectures per week x 11 weeks, with black/whiteboard written notes to supplement the module notes and question/answer opportunities for students.

• Support during the preparation of the mini group-project report and presentation.

A complete set of self-contained notes will be provided in advance to any topics to be treated.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.