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

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?

• Are you trying to formulate GR as a Yang-Mills theory? – mmeent May 24 '18 at 9:54
• Not familiar with Yang-Mills theory. But only Yang-Mills field. For details see this video @minute 34:00 youtube.com/… – user56963 May 24 '18 at 10:00