Unit information: Advanced Quantum Theory in 2013/14

Unit name	Advanced Quantum Theory
Unit code	MATHM0013
Credit points	10
Level of study	M/7
Teaching block(s)	Teaching Block 2C (weeks 13 - 18)
Unit director	Dr. Muller
Open unit status	Not open
Pre-requisites	MATH11005 (Linear Algebra and Geometry), MATH11006 (Analysis 1), MATH 11007 (Calculus 1), MATH 31910 (Mechanics 23), MATH 35500 (Quantum Mechanics), or comparable units. Students will be expected to have attained a degree of mathematical maturity and facility at least at the level of a beginning Level M/7 student.
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

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, namely path integrals, perturbation theory via Feynman diagrams and supersymmetry. 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 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 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 theoretical or mathematical physics as well as range of other disciplines.

Intended Learning Outcomes

A student successfully completing this unit will be able to:

• construct and use of the classical action for particles and fields;
• write down the quantum mechanical path integral for particles in 1d and be able to compute it for simple cases;
• construct and use of the functional integral for quantum field theory and statistical mechanics;
• compute functional integrals for free theories using the key tool: Wicks theorem;
• write the perturbation theory for interacting theories using the Feynman diagram expansion;
• define Grassmann variables and describe their properties;
• perform integration over Grassmann variables;
• describe the concept of supersymmetry quantum mechanics and explain how it plays out in simple examples;
• use ideas from supersymmetry to compute functional integrals;
• appreciate how the subject relates to some other areas of mathematics and physics, including, for example, statistical physics, particle physics, solid state physics and quantum information theory; be able to apply results from the course to problems in these areas.

Teaching Information

The unit will be delivered through lectures. 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.

Assessment Information

Formative homework exercises will be assigned throughout the unit.

The final assessment mark will be based on a 1½-hour written examination (100%).

Reading and References

• 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)
• Condensed matter field theory, A Altland and B Simons. 2nd ed (Cambridge University Press, 2010)
• Quantum signatures of chaos, F Haake. 2nd rev (Springer, 2001)
• Path integral methods in quantum field theory, R Rivers (Cambridge University Press, 1987)