Mathematical challenge for unification of gravity and electromagnetism in classical theory?

Question

I am trying to better understand the mathematical foundations of a possible reconciliation between quantum field theory and gravity in general relativity.

However, before the application of the quantum theory, I need to understand the mathematics problem at the classical level.

We know that Yang-Mills field and Yang-Mills field strength are essentially the same in the classical electromagnetism and general relativity.

They are identical in the sense that electromagnetic field $A_{\mu}$ is a real valued one form sitting on the principle bundle $\omega$ that has been pulled back on the base manifold to have a local theory through a section map $\sigma$. Christoffel symbol $\Gamma^{i}_{j\mu}$ is also a one form that again has been pulled back on the base manifold but it is Lie algebra valued based on the fiber Lie group $G$. The $i$ and $j$ indices are coming from the representation of the Lie algebra and $\mu$ index is coming from the dimension of the manifold.

Also, field strength $F_{\mu\nu}$ is indeed a two form sitting on the principle bundle pulled back and curvature tensor is indeed a Lie algebra valued two form again $R^{i}_{j\mu\nu}$ pulled back on the base manifold.

My questions

1. Is the above summary a correct understanding?

2. If we are in the classic theory and want to unify or to reconcile gravity with electromagnetism should we create a new structure beyond and above the principle bundle so that in the case of electromagnetism it gives real valued forms and in the case of gravity it gives Lie algebra valued forms?

Answer 1

Are you familiar with Kaluza-Klein theory? The earliest unification attempt? It seems that you are assuming a structure that you know you need. A unification may need to embed that structure in something else.

K-K theory assumes a diff geom structure in 4+1 dimensions (rather than 3+1 that we live in). The extra off-diagonal metric tensor components act as the gauge potential and the Christoffel symbol as the field tensor. Assuming away the terms you don't want, or applying compactification of the extra dimension, the transform properties work out as expected.

K-K theory has been largely abandoned (I think, not sure if there has been a revival in the presence of the standard model, string theory etc). But the idea I am trying to express is simply that the structures we need sometimes come from unexpected places. My interpretation of your post is that it is a description of what exists and how it can be described as a fiber bundle (very important and useful language). But in fact, unification may be embedded in a completely different structure and require a different language to describe.

1. Other than Yang-Mills being non-abelian your description may be okay.
2. Yes, you may need or want to abandon the above structure all together but keep in mind that something similar to it needs to emerge as an approximation, just as Galilean invariance is an approximate symmetry for slow relative motion.