Editing Theory of Geometric Unity (section)

=== Proposed Solution ===


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[[File:Geometric unity puzzle pieces.png|500px|right]]

[[File:GU proposal2.png|right|500px|right]]

We may have to generalize all three vertices before we can make progress. That's daunting because in each case, it would appear that we can make an argument that the three vertices are already the simplest possible theories that could live at these vertices.

* We know, for example, the Dirac operator is the most fundamental of all the elliptic operators and Euclidean signature generating all of the Atiyah-Singer theory.
* We know that Einstein's theory describes, in some sense, a unique spin two massless field capable of communicating gravity, which can be arrived at from field-theoretic rather than geometric consideration. 
* In the Yang-Mills case, it can also be argued that the Yang-Mills theory is the simplest theory that we can write down. In the Yang-Mills case, we have no substructure, and so we're doing the most simple-minded thing we can do by taking the norm-squared of the curvature and saying whatever the field strength is, let's measure that size. 

So if each one of these is simplest possible, doesn't Occam’s razor tell us that if we wish to remain in geometric field theory, that we've already reached bottom? 

I would say that there are other possibilities that while each of these may be simplest in its category, they are not simplest in their interaction.

For example, we know that Dirac famously took the square root of the Klein-Gordon equation to achieve the Dirac equation. He actually took two square roots, one of the differential operator, and another of the algebra on which it acts. But could we not do the same thing by re-interpreting what we saw in Donaldson theory and Chern-Simons theory and finding that there are first-order equations that imply second-order equations that are nonlinear in the curvature?

So, let's imagine the following: we replaced the standard model with a true second-order theory. We imagine the general relativity is replaced by a true first-order theory. And then we find that the true second-order theory admits of a square root and can be linked with the true first order theory.

''This would be a program for some kind of unification of Dirac's type, but in the force sector. The question is, "does this really make any sense? Are there any possibilities to do any such thing?"''



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<blockquote >
[[File:GU vision.png|right|400px|right]]

Let's talk about what the Geometric Unity (GU) proposal is. First of all, we observe that we have a division into intrinsic theories and auxiliary theory and between physics and mathematics. An intrinsic physical theory would be general relativity. An auxiliary physical theory would be the Yang-Mills theory, with the freedom to choose internal quantum numbers. 

At the mathematical level, an intrinsic theory would be, the older semi-Riemannian geometry. The study of manifolds with length and angle. Auxiliary geometry is what we're going to call fiber bundle theory or modern gauge theory. 

''Geometric Unity is the search for some way to break down the walls between these four boxes.''

What we'd like to come up with is some theory that is intrinsic, but allows us to play some of the games that exist in other boxes. How can we fit? How can we try to have our cake and eat it too? And use all the full suite of techniques that are available to us?

Our perspective is that the quantum that may be the comparatively easy part and that the unification of the geometry, which has not occurred, may be what we're being asked to do. 
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