Editing Theory of Geometric Unity (section)

=== Problem Nr. 1: Einstein's Theory of General Relativity is not a proper Gauge Theory ===

* From Einstein's general relativity, we take the Einstein projection of the curvature tensor of the Levi-Civita connection <math>\nabla</math> of the metric <math>P_E(F_{\nabla})</math>
* From Yang-Mills-Maxwell-Anderson-Higgs theory of gauge fields, we take the adjoint exterior derivative coupled to a connection <math>d^\star_A F_A</math>

'''Idea:''' What if the <math>F</math>'s are the same in both contexts?

Further, supposing these <math>F</math>'s are the same, then why apply two different operators? 

'''Thus the question becomes:''' Is there any opportunity to combine these two operators?

A problem is that the hallmark of the Yang-Mills theory is the freedom to choose the data, the internal quantum numbers that give all the particles their personalities beyond the mass and the spin. We can allow the gauge group of symmetries to act on both sides of the equation, but the key problem is that: <math>P_E(F_{\nabla h}) \neq  h^{-1} P_E(F_{\nabla}) h </math>. If we act on connections on the right and then take the Einstein projection, this is not equal to first taking the projection and then conjugating with the gauge action. The gauge rotation is only acting on one of the two factors. Yet the projection is making use of both of them. So there is a fundamental incompatibility in the claim that Einstein's theory is a gauge theory relies more on analogy than an exact mapping between the two theories.