Gauge theory

[Qfwfq]

Originally Posted by Ben

I have been working on an argument that the vector potential is a 1-form connection for a theory

It is!

This, of course, means that garvitation is also a gauge theory. The affine connection is the gauge and defines the covariant derivative (which is tensorial, whereas the connection and the partial derivative taken singly are not).

[Ben]

Originally Posted by Qfwfq

It is!

So my texts tell me. Please argue the point; that is, just because the vector potential is a 1-form, and just because, essentially by definition, the connection is also a 1-form, the two must coincide. This is not at all obvious to me.

This, of course, means that garvitation is also a gauge theory.

How does the "of course" follow?

[Qfwfq]

Actually they do not "coincide" at all, they simply share a similar pattern in describing how something changes for a given displacement. In the EM case the something is a simple phase, in the gravitation case, it's the tensors of rank greater than zero.

[Ben]

Originally Posted by Ben

I have been working on an argument that the vector potential A is a 1-form connection for a U(1) theory

Originally Posted by you

It is!

Originally Posted by Qfwfq

Actually they do not "coincide" at all,

I must be dim, but I cannot square these two assertions.

I have a lengthy (and extremely boring) geometric argument that defines the connection, but it still doesn't give me the vector potential as the connection for a theory or its generalization to a theory.

So what is it to be? Is the vector potential literally the connection on the bundle, or is it not? If not, then it seems like the generalization to would be unsafe on the same grounds.

I have to say, I am not happy with this - I lost a lot of dogs when walking them - literally - trying to figure this out (happily they know their way home). Did I waste my time?

[Erasmus00]

The vector potential A is LITERALLY the connection on the U(1) bundle. You can show this by creating a Lorentz invariant action over the manifold, and working out some equations of motion. They turn out to be electricity and magnetism. Crazy. As to "why"? :: Why does any model turn out to be the model it is?

[Qfwfq]

Originally Posted by Ben

I must be dim, but I cannot square these two assertions.

Well, maybe I'm the dim one I thought you meant coinciding with .

Try contracting a diplacement with Jay-Qu's definitions of the gauge-covariant derivative. You should see how the extra term, added to the differential, gives the algebraic sum which doesn't depend on a local change of gauge.