Description
Quantum field theory is a mathematical framework that brings naturally together quantum mechanics and special relativity. In physics it provides the formalism that we use to describe the fundamental laws of nature, and a plethora of phenomena in condensed matter systems and statistical mechanics.
The typical formulation of a statistical system, or a quantum theory of high energy physics (e.g. the theory of quarks and gluons), involves at very high energies (equivalently, very small length scales) a microscopic description. This could be a quantum field theory based on a Lagrangian of elementary bosonic and/or fermionic fields. At low energies (equivalently, large length scales) the effective description of the system, which involves only the light degrees of freedom, can be considerably different. The high-energy and low-energy quantum field theories enjoy an extended set of symmetries. Besides the usual symmetries of translation and rotation, these theories are also scale-invariant. Scale invariance typically enhances further to a larger symmetry called conformal symmetry. With this symmetry a theory looks like a fractal: it exhibits the same type of behaviour at all length scales. Quantum field theories with conformal symmetry are called conformal field theories.
Since conformal field theories appear naturally in the long- and short-distance descriptions of many quantum systems (and in critical phenomena), they can be viewed as the most basic examples of quantum field theories and thus play a prominent role in the study of quantum mechanical systems capturing universal behaviours.
In this project we will introduce the basic formulation of quantum field theory, extending familiar notions from quantum mechanics. We will explore simple, concrete examples, of classical and quantum field theories and will analyse how symmetries affect their properties. Applying these lessons we will examine how conformal symmetry constrains a quantum field theory and how the special properties of conformal field theory are applied in physics and in modern, more mathematical, explorations of quantum field theory.
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Corequisite