Description

Gauge theory is the theoretical framework which describes most of the interactions between elementary particles in the context of relativistic quantum field theory. 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 interaction occurs by the exchange of new 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 interactions and the spectrum of gauge bosons. On the mathematical side, you will learn about Lie algebras, their classification via root systems and 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.

If time allows, you can pursue several further directions, such as investigating fibre bundles or Lie groups on the mathematical side, and special classical solutions of the equations of motion (e.g. monopoles and instantons) or the quantization of gauge theories on the physical side.

Prerequisites

Mathematical Physics II

Resources

Reading material:

• W. Fulton, J. Harris, Representation Theory - A First Course, Springer, 2004.
• R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, 1984.
• 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.
• H. Osborn, Symmetries and Groups, DAMTP Cambridge course.