Description

Gauge theory is the theoretical framework which describes most of the interactions between elementary particles. The electromagnetic force which keep electrons bound to nuclei in atoms, the weak force which is responsible for radioactive decay, and the strong force which keeps quarks together in protons and neutrons, and protons and neutrons together in nuclei, can all be understood using gauge theory. The interactions occur by the exchange of elementary particles, like the photons that mediate the electromagnetic force, which are called gauge bosons.

This project explores the formulation of classical gauge theories and the mathematics of Lie algebras, which governs the structure of gauge bosons and their interactions. The theory of Lie algebras encompasses the so-called classical Lie algebras, which can be realized in terms of matrices and their commutators, but also exceptional Lie algebras, which are defined in an alternative and more abstract way. On the mathematical side, you will learn about simple Lie algebras, their classification via root systems and graphs called Dynkin diagrams, and how to construct their representations using weights. On the physical side, starting form the Lagrangian formulation of classical relativistic field theory and the concepts of global and local symmetries, you will learn how to formulate abelian gauge theories like the theory of electromagnetism, how to extend this to non-abelian gauge theories based on semi-simple Lie algebras (like the theory of weak and strong interactions) following Yang and Mills, and how to incorporate the interactions of gauge bosons with matter like electrons or quarks.

A few further directions that you may pursue in the second half of the project are the theory of simple Lie groups or the geometry of gauge theories on the mathematical side, and special classical gauge field configurations like magnetic monopoles and instantons on the physical side.

Prerequisites

2H Mathematical Physics

Resources

We will draw from some of these resources:

• R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, 1984.
• W. Fulton, J. Harris, Representation Theory - A First Course, Springer, 2004.
• M. Nakahara, Geometry, Topology and Physics, Institute of Physics, 2003.
• P. Ramond, Field Theory: A Modern Primer, Addison-Wesley, 1989.
• H. Liu, Relativistic Quantum Field Theory II, MIT course, lectures 1-4.
• N. Dorey (written up by J. Kirkin), Symmetries, Fields and Particles, Cambridge Part III course.