University of St Andrews

Key information

SCOTCAT credits : 15

ECTS credits : 7

Level : SCQF level 11

Semester: 1

Availability restrictions: Normally only taken in the final year of an MPhys or MSci programme involving the School

This module presents an introductory account of the ideas of quantum field theory and of simple applications thereof, including quantization of classical field theories, second quantization of bosons and fermions, solving simple models using second quantization, path integral approach to quantum mechanics and its relation to classical action principles, field integrals for bosons and fermions, the relationship between path integral methods and second quantization, solving many-body quantum problems with mean-field theory, and applications of field theoretic methods to models of magnetism.

Relationship to other modules

Pre-requisite(s): Before taking this module you must pass PH3012 and pass PH3061 and pass PH3062 and pass 1 module from {PH4038, MT4507} and pass 1 module from {PH4028, MT3503}

Learning and teaching methods and delivery

Weekly contact: 3 lectures or tutorials

Scheduled learning hours: 30

Guided independent study hours: 120

Assessment pattern

As used by St Andrews: 2-hour Written Examination = 85%, Coursework = 15%


Re-assessment: Oral Re-assessment, capped at grade 7

Personnel

Additional information from school

Aims & Objectives

To present an introductory account of the ideas of quantum field theory and of simple applications thereof, including:

• The coordinate path integral approach to quantum mechanics and its relation to classical action principles
• Quantization of classical field theories.
• Second quantization as a tool for many-particle quantum problems
• Solving simple models using second quantization
• Field integrals for bosons and fermions including Matsubara summation - Relationship between path integral methods and second quantization.
• Bosonic representations of spin problems

Learning Outcomes

By the end of the module, students will have a comprehensive knowledge of the topics covered in the lectures and of their application in solving simple problems. They will be able to:

• Understand the conceptual basis of path integrals.
• Calculate simple path integrals.
• Understand the canonical quantization of classical field theories.
• Diagonalize simple quantum field theories using unitary transformations, Bogoliubov transformations and Fourier transforms.
• Identify the mode energies in such theories and understand the connection of symmetry breaking to mode energies.
• Understand path integrals and their relation to second quantization.
• Calculate path integrals to find partition functions for general fermionic and boson theories.
• Apply second quantization techniques to spin models.

Synopsis

Coupled quantum harmonic oscillators, canonical quantization of classical field theory, bosonic quantum field theory. Second quantization for bosons.

The path integral treatment of quantum mechanics. Relation to classical principle of least action. Relationship to statistical mechanics.

Second quantization and many body problems. Diagonalizing simple quantum field theories by unitary rotations, Bogoliubov transformation and Fourier transformations. Identifying normal modes. Examples: tight binding problems, Bose condensation of weakly interacting bosons, BCS theory of superconductivity.

Bosonic coherent states and bosonic field integrals. Grassmann variables, fermionic coherent states and fermionic field integrals. Matsubara frequencies, Matsubara summation and solving simple problems using functional integration. Application to the same examples as for second quantization.

Models of interacting spins. The Heisenberg model. Representing spins in terms of Holstein-Primakov bosons. Spin wave theory for ferromagnets and antiferromagnets. Goldstone’s theorem and the origin of the gapless spectrum for continuous symmetry breaking.

Additional information on continuous assessment etc.

Please note that the definitive comments on continuous assessment will be communicated within the module. This section is intended to give an indication of the likely breakdown and timing of the continuous assessment.

As with many other modules, it is working with the ideas covered in lectures that brings a major part of the learning gains. Four assessed problem sets are provided during the semester, and some of the questions are expected to require substantial synthesis of ideas from different parts of this module and prior work. These assessed problem sets will contribute to the 15% of the module grade that comes from this work. In addition, each problem set contains further non-assessed problems for practice in problem solving during revision.

Recommended Books

Please view University online record: http://resourcelists.st-andrews.ac.uk/modules/ph5004.html

General Information

Please also read the general information in the School's honours handbook that is available via https://www.st-andrews.ac.uk/physics-astronomy/students/ug/