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, namely path integrals, perturbation theory via Feynman diagrams and the application of these concepts to second quantisation (including Grassmannian variables). 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.

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:

  • construct and use of the classical action for particles and fields;
  • derive and evaluate the quantum mechanical path integral for particles with simple Lagrngians or Hamiltonians;
  • derive and evaluate functional integrals for fields, in particular for fields accessible as a continuum limit of systems with point particles;
  • understand and be able to apply the ideas of second quantisation, with an emphasis on systems with discrete sites;
  • use perturbation theory (Wick's theorem, Feynman diagrams) to compute functional integrals for free and interacting theories;
  • define Grassmann variables and describe their properties;
  • describe the concept of supersymmetry quantum mechanics and explain how it plays out in simple examples;
  • use Grassmannian variables and 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.

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 quantum mechanics in terms of
trajectories, we will also discuss its application to fields)

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

Lecture notes will be provided. The unit does not follow a particular book but useful references are:
  • 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)
Unit director: Dr Sebastian Muller
Lecturer: Dr Sebastian Muller

Pre-requisites

MATH11005 (Linear Algebra and Geometry), MATH 11007 (Calculus 1), either MATH 21900 (Mechanics 2) or 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

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 five of these weeks.

Thie will be complemented by lecturer 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% from a 1 hour 30 minute exam in May/June

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.

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