7Gauge theories

III Symmetries, Fields and Particles

7 Gauge theories

We are now going to do something rather amazing. We are going to show that

Maxwell’s equation is “forced” upon us by requiring that our (charged) fields

possess something known as a U(1) gauge symmetry. We are going to write

down essentially the only possible theory under the assumption of this gauge

symmetry, and unwrapping it gives us Maxwell’s equation.

When we do it for Maxwell’s equation, it is not entirely clear that what we

wrote down was “the only possible thing”, but it will be when we try to do it

for general gauge theories.

Before we begin, it is important to note that everything we do here has a very

nice geometric interpretation in terms of connections on principal

G

-bundles,

and there are very good geometric reasons to pick the definitions (and names) we

use here. Unfortunately, we do not have the time to introduce all the necessary

geometry in order to do so.