Attached my solution.

I'd just like to add something to this, because it confused me for a little while:



i.e. ,

Since is restricted to U(1), this means that derivatives of have to have a particular form. For example, along a path parameterised by , we have:



In general, then, derivatives of a U(1) field must have the form:




The bundle (guage) connection applied to :



clearly fits this pattern, when and is a real-valued one-form, so it does indeed define a valid derivative on U(1).

Note that although the spacetime fibre bundle is unique, as is the electromagnetic potential (or alternatively the Maxwell tensor ), the bundle connection is different for each particle, as it depends on the particle's charge, e.

(Alternatively we could instead regard each particle or particle-type as defining a completely different set of U(1) fibres over the same base Minkowski space , with a specific bundle connection then being unique to each such fibre set)...

This is in contrast to the metric (Riemannian) connection of §14.7, which can be regarded as an intrinsic property of spacetime, and doesn't vary depending on what you are applying it to!