A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions

We then choose to add some stuff that we can't see at all, that's dark. And this matter would be governed by forces that were dark too. There might be dark electromagnetism, and dark strong, and dark weak. It might be that things break in that sector completely differently, and it doesn't break down to an \(\text{SU}(3) \times \text{SU}(2) \times \text{U}(1)\) because these are different \(\text{SU}(3)\)s, \(\text{SU}(2)\)s, and \(\text{U}(1)\)s, and it may be that there would be like a high energy \(\text{SU}(5)\), or some Pati-Salam model. Imagine then that chirality was not fundamental, but it was emergent—that you had some complex, and as long as there were cross terms these two halves would talk to each other. But if the cross terms went away, the two terms would become decoupled.

So in other words, just to recap, starting with nothing other than a four-manifold, we built a bundle \(U\). The bundle \(U\) had no metric, but it almost had a metric. It had a metric up to a connection. There was another bundle on top of that bundle called the chimeric bundle. The chimeric bundle had an intrinsic metric. We built our spinors on that. We restricted ourselves to those spinors. We moved most of our attention to the emergent metric on \(U^{14}\), which gave us a map between the chimeric bundle and the tangent bundle of \(U^{14}\). We built a toolkit, allowing us to choose symmetric field content, to define equations of motion on the cotangent space of that field content, to form a homogeneous vector bundle with the fermions, to come up with unifications of the Einstein field equations, Yang-Mills equations, and Dirac equations. We then broke those things apart under decomposition, pulling things back from \(U^{14}\), and we found a three-generation model where nothing has been put in by hand, and we have a 10-dimensional normal component, which looks like the \(\text{Spin}(10)\) theory.

Now there's a huge problem in the spinorial sector, which I don't know why more people don't worry about, which is that spinors aren't defined for representations of the double cover of \(\text{GL}(4, \mathbb{R})\), the general linear group's effective spin analog. And as such, if we imagine that we will one day quantize gravity, we will lose our definition, not of the electrons but, let's say, of the medium in which the electrons operate. That is, there will be no spinorial bundle until we have an observation of a metric. So, one thing we can do is to take a manifold \(X^d\) as the starting point and see if we can create an entire universe from no other data—not even with a metric. So, since we don't choose a metric, what we instead do is to work over the space of all possible point-wise metrics. So not quite in the Feynman sense, but in the sense that we will work over a bundle that is of a quite larger, quite a bit larger dimension.

So that gets rid of the biggest problem, because the internal symmetry group is what causes the failure, I think, of supersymmetric particles to be seen in nature, which is we have two different origin stories, which is a little bit like Lilith and Genesis. We can't easily say we have a unified theory if spacetime and the \(\text{SU}(3) \times \text{SU}(2) \times \text{U}(1)\) group that lives on spacetime have different origins and cannot be related. In this situation, we tie our hands and we have no choice over the group content.

And I should say that the Pati-Salam theory, which is usually advertised as, I think as \(\text{SU}(4) \times \text{SU}(2) \times \text{SU}(2)\), is really much more naturally \(\text{Spin}(6) \times \text{Spin}(4)\) when the trace portion of the space of metrics is put in with the proper sign if you're trying to generate the sector that begins as \(X(1,3)\). Remember \(X^d\), where \(d = 4\), is the generic situation. But you have all these different sectors. I believe that these sectors probably exist if this model's correct, but we are trapped in the \((1,3)\) sector, so you have to figure out what the implications are for pushing that indefinite signature up into an indefinite signature on the \(Y\) manifold. And, there are signatures that make it look like the Pati-Salam rather than directly in the \(\text{Spin}(10)\), \(\text{SU}(5)\) line of thinking.