# Advanced Quantum Theory

## Unit aims

The aims of this unit are to introduce and develop some key ideas and techniques of modern quantum theory. These ideas – functional integration, perturbation theory via Feynman diagrams and concepts leading up to supersymmetry – are central concepts with extremely wide applicability within modern physics. The aim is to introduce the ideas and also to enable the student to be able to do example calculations with these sophisticated tools. This unit provides essential techniques for any graduate who intends to start research in mathematical or theoretical physics as well as range of other disciplines.

## Unit description

Quantum theory is the fundamental framework within which a vast section of modern physics is cast: this includes atomic, molecular and particle physics as well as condensed matter and statistical physics, and modern quantum chemistry. In recent years it has also had unexpected and deep impact on pure mathematics. Fundamental to applying quantum theory in these areas are the more sophisticated techniques and ideas introduced in this course. These ideas not only allow quantum theory to be applied to these areas but also introduce a raft of concepts which have become a standard language for these fields.

The course starts by introducing path integrals. These provide a way to describe quantum mechanical time evolution in terms of classical trajectories. Crucially, the integration runs over all trajectories with a given initial and final point including those that do not satisfy the classical laws of motion. Path integrals for simple systems such as the harmonic oscillator will be computed exactly. We will then introduce perturbation theory and Feynman diagrams, which provide a powerful method to approximately evaluate path integrals of more complicated systems. We will also generalise the path integral formalism to many-particle systems. To do this we will first introduce second quantisation, a formalism to study many-particle systems that is technically similar to the treatment of the harmonic oscillator in terms of raising and lowering operators. Then this approach will be connected to the path integral formalism. Here the treatment of fermionic many-particle systems requires particular attention as the corresponding path integral has to be formulated in terms of anticommuting (Grassmannian) variables. In this context we will also discuss the important concept of supersymmetry.

NOTE: This unit is also part of the Oxford-led Taught Course Centre (TCC), and is taken by first- and second-year PhD students in Bristol and its TCC partner departments. The unit has been designed primarily with a postgraduate audience in mind. Undergraduate students should not normally take more than one TCC unit per semester.

## Relation to other units

Quantum Mechanics and Mechanics 2/23 or equivalent units are prerequisites.

The methods introduced in this course are used in current research in several areas of mathematical and theoretical physics. Units giving an introduction into some of these areas are Statistical Mechanics, Quantum Information, Quantum Chaos, and Random Matrix Theory in Mathematics, and Relativistic Field Theory as well as several courses dealing with Condensed Matter in Physics. The Physics unit Advanced Quantum Physics includes complementary material about the Feynman path integral outside a field theoretical context.

## Learning objectives

A student successfully completing this unit will be able to

- derive and evaluate quantum mechanical path integrals and field integrals for simple Lagrangians and Hamiltonians
- use perturbation theory (Wick's theorem, Feynman diagrams) to evaluate path integrals
- understand and apply the ideas of second quantisation in simple examples
- appreciate the role of Grassmannian variables and supersymmetry
- appreciate how the subject relates to other areas of Mathematics and Physics, and apply results from the course in these areas

## Transferable skills

- Clear, logical thinking.
- Problem solving techniques.
- Assimilation and use of complex and novel ideas.
- Appreciating connections between and unifying principles behind different areas of research.

## Syllabus

Brief review of basics (Hamilton’s principle in mechanics, resolution of the identity and time evolution in quantum mechanics)

Feynman path integral (a formulation of time evolution in quantum mechanics in terms of trajectories, we will also discuss its application to fields and problems from statistical mechanics)

Perturbation theory (an approximation method to treat complex problems in the context of path integrals)

Second quantisation (a formalism to treat quantum many particle systems) and its connection to path integrals

Grassmann variables and supersymmetry (this topic deals with anti-commuting variables needed to write down the field integral for fermionic particles, and their connection to conventional commuting variables)

Examples for applications to different areas of research will be given throughout the course, however we are not aiming for completeness in any application area.

There may be minor changes to this syllabus.

## Reading and References

- Condensed matter field theory, A Altland and B Simons. 2nd ed (Cambridge University Press, 2010)
- Path integral methods in quantum field theory, R Rivers (Cambridge University Press, 1987)
- Quantum field theory for the gifted amateur, T. Lancaster and S. J. Blundell (Oxford University Press, 2014)
- Quantum Field Theory in a Nutshell, A Zee (Princeton University Press, 2003)
- Quantum Mechanics and Path Integrals: Emended Edition, RP Feynman, AR Hibbs and DF Styer (Dover, 2010)
- Quantum signatures of chaos, F Haake. 2nd rev (Springer, 2001)

**Unit code:** MATHM0013

**Level of study:** M/7

**Credit points:** 10

**Teaching block (weeks):** 2 (13-20)

**Lecturer:** Dr Sebastian Müller

## Pre-requisites

MATH11005 Linear Algebra and Geometry, MATH11007 Calculus 1, either MATH21900/MATH31910 Mechanics 2/3 or PHYS30008 Analytical Mechanics, MATH35500 Quantum Mechanics, or comparable units.

## Co-requisites

None

## Methods of teaching

The schedule is slightly different from other courses. There will be a weekly two-hour slot. The lectures will be transmitted over the internet as part of the Taught Course Centre (TCC). The TCC is a consortium of five mathematics departments, including Bath, Bristol, Imperial College, Oxford and Warwick. With 15 lectures the course will thus run over eight weeks.

In addition there will be one-hour problem classes, not transmitted over the internet, in about four of these weeks.

Thie will be complemented by lecture notes, problem sheets, and a revision class.

## Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

- 100% Exam

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.