School of Mathematics and Statistics (SoMaS)

MAS415 Topics in Mathematical Physics

This module introduces important concepts in modern mathematical physics, based on special relativity, analytical (Lagrangian and Hamiltonian) dynamics and quantum theory, whose prior knowledge is assumed. We will combine these foundations to formulate the language of quantum field theory, which forms the basis of our current understanding of fundamental particles and their interactions, as described by the Standard Model of particle physics. We will review the connection between symmetries and conserved quantities, which plays a crucial role both in the classical and quantum formulation of field theories. We will categorise different types of fields, such as the scalar field and the electromagnetic field, and learn how to quantise both free and interacting field theories. We will compute scattering amplitudes between particles, using the technology of Feynman diagrams, which can be used to interpret particle physics experiments.

Prerequisites: MAS324 (Quantum Theory) Or physics equivalent, e.g. PHY349 PHY472/432.
Corequisites: MAS413 (Analytical Dynamics and Classical Field Theory) MAS413/6431 or MAS410, or equivalent.
No other modules have this module as a prerequisite.

Outline syllabus

• Review of classical field theory, symmetries and Noether's theorem;
• Canonical quantisation of a free scalar field, introduction of Fock space;
• Green's functions and Feynman propagator;
• Interacting fields, Wick's theorem, and Feynman diagrams;
• Particles as representations of the Lorentz group;
• Quantisation of electromagnetic field.

Aims

• to develop an understanding of how quantum fields are more fundamental than particles;
• to demonstrate the significance of fundamental symmetries and their associated conserved quantities;
• to develop a mathematical Hilbert space formalism for quantum fields;
• to develop the language of Feynman diagrams that allows calculating observable quantities;
• to understand the significance of representation theory of the Lorentz group in particle physics.

Learning outcomes

• derive equations of motion and conserved quantities from a field theory action;
• construct the Hilbert space for a quantum field theory and study physically relevant operators on it;
• write down Feynman rules for an interacting field theory, and compute transition amplitudes;
• extend these methods from scalars to the electromagnetic field.

Teaching methods

Lectures (20) and Independent Study (130 hours).

Assessment

One formal 2 hour written examination.

Reading list