10 - Applications to Gauge Field Theory

Summary

The quantization of classical electromagnetic theory (see the appendix to Chapter 2) leads to the description of the photon, the quantum of electromagnetic radiation. One of the aims of quantum field theory is to explain all elementary particles as quanta of appropriate classical field theories. In the search for such field theories, non-Abelian gauge theories have become leading candidates since their introduction in 1954 by Yang and Mills. Only in the 1970s did it become generally recognized that the mathematical theory of connections on vector bundles is the proper context for gauge theory.

In this chapter we shall start out by giving a geometric description of the Yang–Mills Lagrangian, which must be minimized in order to construct the desired classical field. The minimization is actually carried out here using a special class of connections which are called “self-dual.” We reformulate electromagnetism as an Abelian gauge field theory, and give a detailed account of a non-Abelian SU(2) gauge theory over S⁴, including the formula for “instantons,” the connections which minimize the Yang–Mills Lagrangian. Much of this chapter is based on Lawson [1985].

The Role of Connections in Field Theory

Imagine a structured particle, that is, a particle located at some point p in a four-dimensional manifold M (“space-time”), and with an internal structure, or set of states (“spin”, etc.) labeled by elements of a complex Lie group G (e.g., SU (2)).