# RELATIVISTIC QUANTUM MECHANICS - 2023/4

Module code: MATM054

## Module Overview

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

### Module provider

Mathematics

WOLF Martin (Maths & Phys)

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

Independent Learning Hours: 106

Tutorial Hours: 22

Captured Content: 22

Semester 2

## Prerequisites / Co-requisites

MAT3039 or PHY3044

## Module content

1. Foundations of Special Relativity:

1. Lorentz transformations and consequences such as relativistic velocity addition, time dilation, length construction, Doppler effect;

2. Four-vectors, Minkowski space, and Lorentz group;

3. Relativistic mechanics including four-momenta, energy-momentum conservation, equivalence of mass and energy;

2. Free Relativistic Scalar Fields:

1. States and observables, quantum symmetries and Wigner’s theorem;

2. Single-particle theory, causality, and the need for quantum fields;

3. Multi-particle theory, Fock space, creation and annihilation operators, construction of the causal scalar field, Klein–Gordon equation;

4. Classical field theory, canonical quantisation, classical symmetries and Noether’s theorem;

3. Introduction to Scattering Theory:

1. S-matrix, pictures of quantum mechanics, time-dependent perturbation theory, Wick contractions and Wick’s theorem, Wick diagrams and Feynman diagrams, counter terms

2. Basic examples including scalar Yukawa theory;

3. General properties of the S-matrix including crossing symmetry, transition probabilities and rates, decay rates, cross sections;

4. Green’s functions, renormalisation and regularisation, Lehmann–Symanzik–Zimmermann reduction formula, Kaellen–Lehmann spectral decomposition, 1PI functions;

4. Optional Additional Topics – if time permits, some of the following topics may be covered:

1. Representations of the Poincare group and construction of general causal fields including fermionic and vector fields;

2. Path integral methods, Dyson–Schwinger equation;

3. Loop expansion, divergences and regularisation, and power counting

4. Renormalisation group, Callan–Symanzik equation.

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

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test Test (1 hour) 20
Examination Examination (2 hours) 80

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.

• Ability to apply subject knowledge to solve problems similar to the ones covered in the lectures.

Thus, the summative assessment for this module consists of:

• One two-hour end-of-semester examination 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 number of marked formative coursework assignments over the entire duration of the module. Students are also encouraged to arrange meetings with the module convenor for questions and verbal feedback on the weekly comprehension of the material.

## 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 as how to apply the mathematical techniques to concrete physical examples KT 003 Be able to analyse quantum field theories and compute relevant quantities explicitly KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## 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.

The learning and teaching methods include:

• There are four hours per week of captured content, tutorials, and lectures.

• A number of marked formative coursework assignments over 11 weeks with feedback given to the students.

• Additional tutorial sessions and/or office hours may be arranged as and when needed.

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.