Covariant derivative vs. Lie derivative

lavinia

Gold Member
Alright then, Electrodynamics is surely the easiest example.
Take R^4 as a manifold and assign an additional degree of freedom to each point*. In the EM case, that is an element of U(1), think of some ring everywhere in spacetime. There's now a connection on this bundle, and the connection coefficients turn out to be 1-forms that describe the electromagnetic potential. The associated curvature is F=dA, the electromagnetic field strength.
*: I guess that's the fiber, but I haven't found it stated like that explicitly

Tomsk

Interesting things happen when you have a charged field that interacts with the EM field. I'll try to describe scalar electrodynamics, describing electrons requires spinors which is just extra complication at this stage.

The main object of study is the action functional, which is the integral of the lagrangian over all of spacetime. The lagrangian has terms that go like $\eta^{\mu\nu}\partial_\mu \phi(x)^* \partial_\nu \phi(x)$ and also $m^2 \phi(x)^* \phi(x)$. (* is the complex conjugate) The way we make this field phi interact with the EM field is by demanding that the action is invariant under 'local U(1) symmetry', which I guess in the language of fibre bundles means we have a U(1) fibre bundle sort of... sitting there... doing... something.

The term $\phi(x)^* \phi(x)$ is unchanged when we transform it like this: $\phi(x) \rightarrow e^{iq\theta(x)}\phi(x)$. The field transforms under a *rep* of U(1) hence the q, which is the charge of the field. But the term $\eta^{\mu\nu}\partial_\mu \phi(x)^* \partial_\nu \phi(x)$ picks up extra terms when the partial derivatives act on the theta(x), so it's not invariant. The way this is fixed is by replacing the partial derivative with the covariant derivative on the U(1) bundle, which includes the connection, and this extra bit represents an interaction between the EM field and the scalar field. This is called 'minimal coupling'.

So we end up with an 'interaction lagrangian' that looks something like $q^2 \eta^{\mu\nu} A_\mu A_\nu \phi(x)^* \phi(x)$, which in terms of Feynman diagrams represents an antiparticle (the phi*) interacting with a particle (the phi) creating two photons (the As). I don't know it thoroughly enough to explain it in any more detail, and some of it might be wrong, I can't remember. I hope it helps a bit.

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