What is the analogy between the gauge covariant derivative and the covariant derivative in General Relativity?

Question

A particle in the Dirac field can be described with the following equation $$i\gamma^\mu\partial_\mu\psi-m\psi=0$$ This is if the particle is non-interacting. However, if we impose a local $U(1)$ symmetry such that $\psi'=\psi e^{i\Lambda(x)}$ we should expect that the equations of motion should be invariant $$i\gamma^\mu\partial_\mu\psi'-m\psi'=0$$ However this equation is not invariant under a $U(1)$ local symmetry instead the equations of motion should be $$i\gamma^\mu(\partial_\mu +iqA_\mu)\psi-m\psi=0$$ To impose a local U(1) gauge invariance. My question is if the gauge covariant derivative stays invariant under local $U(1)$ transformation, what does this really represent. In the context of General relativity the covariant derivative $$\nabla_\mu A^\alpha = \partial_\mu A^\alpha + \Gamma_{\gamma\alpha}^{\mu} A^\gamma$$ describes the change of the basis vectors when moving through space-time. Does the covariant derivative in the Dirac equation represent the change of phase moving through spacetime? With this covariant derivative can we come up with a geodesic equation, the curvature of the field, and also can we derive a metric from the gauge covariant derivative. Also in general relativity, the manifold that is curved is describing the manifold of space-time, what manifold does the gauge covariant derivative explain? Thanks

Answer 1

As you can see, the covariant derivative has two components: $$\nabla_{\mu} =\partial_{\mu} +\Gamma_{\gamma \alpha}^{\mu}, $$ in which the first describes the change of the vector field and the second is a linear transformation of the vector space, related with the possibility of making different choices for the basis of the vector space in each point of the spacetime (manifold).

In the context of Yang-Mills, the covariant derivative does exactly the same: $$\nabla_{\mu} = \partial_{\mu}+i q A_{\mu},$$ in which the first term is the change of your filed and the second term is a linear transformation in the gauge group instead of the vector space.

This similarity arises from the fact that, "gluing" a vector space or a gauge group to each point on your manifold is very similar (from the point of view of fiber bundles).

From the point of view of fiber bundles, these two situations are equal (there might be some technical differences with which I am not too familiar, I have just started studying it myself). In particular, you can see that if you try to define a derivative for these kinds of objects, you will always have two terms in each one relates to the change of the field and the other is associated with the freedom you have to parameterize the object that you "glued" to each point of the manifold (for a vector space you have the freedom to choose the basis, as you also have for the gauge groups, as you can see from the existence of gauge transformations, which, in this case, reduces to changing the phase).

In fact, and do not quote me on this, the difference between GR and Yang-Mills (from the perspective of fiber bundles) might only be the details of the theories (e.g. in one you have vector spaces and in the other you have groups) and the definition of the action (i.e the mathematical structure of both theories is the same).

(Feel free to comment if you notice any mistake or imprecision).