Physical and Geometrical Motivation

Abstract

It sometimes transpires that mathematics and physics, pursuing quite different agendas, find that their intellectual wanderings have converged upon the same fundamental idea and that, once it is recognized that this has occurred, each breathes new life into the other. The classic example is the symbiosis between General Relativity and Differential Geometry. As the Singularity Theorems of Hawking and Penrose (see [N2]) amply attest, the results of such an interaction can be spectacular. The story we have to tell is of another such confluence of ideas, more recent and perhaps even more profound. Our purpose in this preliminary chapter is to trace the physical and geometrical origins of the notion of a “gauge field” (known to mathematicians as the “curvature” of a “connection on a principal bundle” ). We will not be much concerned yet with rigorously defining the terms we use, nor will we bother to prove most of our assertions. Indeed, much of the remainder of the book is devoted to these very tasks. We hope only to offer something in the way of motivation.