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Department of Mathematical Sciences

# MATH4061 Advanced Quantum Theory IV

The course will introduce Quantum Field Theory (QFT) by bringing together concepts from classical Lagrangian and Hamiltonian mechanics, quantum mechanics and special relativity. It also provides an elementary introduction to string theory, both as a simple two-dimensional QFT and as a way to go beyond QFT concepts.

The course will begin with a reminder of classical field theory concepts. We then go on to a discussion of free and interacting quantum field theories in the operator formalism, explain Feynman diagrams and the computation of scattering amplitudes. Throughout, simple examples will be used to emphasise the conceptual ingredients rather than the computational technicalities.

In the second term, we begin with an explanation of the path integral formulation of quantum theory, both in relativistic quantum mechanics and in quantum field theory. String theory follows next, and we will discuss its spectrum, symmetries and dualities. We will see how self-consistency conditions lead to a preferred number of spacetime dimensions and the existence of gauge fields and gravitons (providing a connection to general relativity). The last part of the course is about scale dependence, renormalisation and the renormalisation group.

The course is complementary to the PHYS4181 Particle Theory module in the sense that it focuses more on conceptual and mathematical foundations. It should prove interesting and useful especially to students who want to continue to pursue an interest in what is currently understood of the fundamental nature of matter.

## Outline of Course

Aim: To introduce quantum field theory using the operator formalism as well as path integrals, and to apply it to string theory, developing it sufficiently to show that its spectrum includes all elementary particles thus unifying the fundamental forces.

### Term 1

• Action principles and classical theory: Review of Lagrangian formulation of classical field theory. Symmetries, equations of motion, Lagrangian and Hamiltonian methods, Noether's theorem.
• Quantisation of free scalar fields: Multi-particle quantum mechanics, canonical quantisation of free scalar fields, Fock space, anti-particles, propagators, causality.
• Interacting quantum fields: Evolution operators, perturbative expansion, Wick's theorem, Feynman diagrams in position and momentum space, LSZ reduction, scattering matrix, cross sections.

### Term 2

• Path integrals: Relativistic particle in the world-line formulation, generating functionals, diagrammatic expansion, zero-dimensional quantum field theory.
• String theory: World-line action for free relativistic particle, formulation with intrinsic metric. Nambu-Goto action and Polyakov action and equations of motion, symmetries, boundary conditions, simple classical solutions, quantisation, Virasoro algebra, physical states, connection to general relativity, T-duality.
• Renormalisation and scale dependence: Regularisation methods, renormalisation, power counting, renormalisation group flow.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.