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

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ SU(3) \text{ (color)} \times SU(2) \text{ (weak isospin)} \times U(1) \text{ (weak hypercharge)}$$</div>  

Which breaks down to \(SU(3) \times U(1)\), where the broken \(U(1)\) is the electromagnetic symmetry. This equation is also a curvature equation—the corresponding equation—and it says that this time, the curvature of an auxiliary structure known as a gauge potential, when differentiated in a particular way is equal, again, to the amount of stuff in the system that is not directly involved in the left-hand side of the equation. So it has many similarities to the above equation, both involve curvature. One involves a projection, or a series of projections. The other involves a differential operator.

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 \(SU(3) \times SU(2) \times U(1)\) because these are different \(SU(3)\)s, \(SU(2)\)s, and \(U(1)\)s, and it may be that there would be like a high energy \(SU(5)\), 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 \(Spin(10)\) theory.

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 \(SU(3) \times SU(2) \times U(1)\) 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 \(SU(4) \times SU(2) \times SU(2)\), is really much more naturally \(Spin(6) \times Spin(4)\) 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 \(Spin(10)\), \(SU(5)\) line of thinking.

I do want to leave you with one thought. I really think that we've gotten completely bent out of shape about trying to formalize and routinize science, and it doesn't work. You cannot mandate science as social engineering, you can't decide that science is always in the zeitgeist and done by committee. In fact, it is essential to understand that science will not conform to what you want. One of the things that I'm very proud of, and I think is quite true, is the saying that great science has the scientific method as its radio edit. I don't think that great science is actually done the way we say it's done, and I think that <a href="https://blogs.scientificamerican.com/guest-blog/the-evolution-of-the-physicists-picture-of-nature/">Dirac's 1963 Scientific American article</a> should be read by absolutely everyone.