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
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.