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

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[https://theportal.wiki/images/b/b9/Geometric-Unity-A-First-Look_-_YouTube.vtt Raw transcript file]

<p>[00:16:05] Now, when she listened to [[episode 19]] about [[Bret Weinstein]], she found that it was a very compelling episode, but strangely, even though the point of the episode was to surface Bret's long forgotten theory, because Bret had not been acknowledged as having predicted that laboratory mice would in particular have radically elongated [[telomere]]s where it was thought that all mice had long, radically elongated telomeres which has incredible potential implications for drug testing and all of the work that is done on laboratory, rodents as model organisms.

<p>[00:32:56] I'd like to think Raoul Bott, who's no longer with us. Who I should have invited to my wedding. I was very angry at him, but I didn't realize that he was saving me in a very difficult situation against the department that probably just wanted to see me gone. I'd like to thank Peter Thiel. One of my closest friends is like a brother to me for allowing me these seven years since this lecture to lick my wounds, to get strong, to have a 401(k), to buy a house. And, to begin coming back to regular society after a very difficult and strained career.  

<p>[00:36:24] And we began, he began to explain the mathematics teaching working on in private for the last 20 years. As he took me through the equations he had been formulating, I began to see emerging before my eyes potential answers to many of the major problems in physics. It was an extremely exciting, daring proposal, and also mathematically so natural that it started to work it's magic on me. Over the last two years, I have had the privilege of being taken through the twists and turns of Eric's ideas. After our postdocs in Israel when I went the academic route getting my professorship here in Oxford, Eric went a more independent route working in economics, government, and finance.

<p>[00:39:47] Marcus asked me to begin presenting these ideas here. And hopefully this is the first opportunity, but if the ideas are not good then lighting on the aisles will lead you to safety and your exits may be behind you. But, in the event of a good flight , hopefully this will begin a conversation rather than be the end of one.

<p>[00:40:10] I feel in some sense that I'm presenting the works of another man, a younger man. Someone who came of age right in the middle of the great string theory boom. With the anomaly cancellation in 1984. And I look at this work and I see a young person struggling with the idea, why can't I see that string theory is going to answer all of these questions over the next ten years, as we were told at the time, and making a very dangerous decision, which was, I think I'm not going to follow that particular path, and I'm going to follow another.

<p>[00:43:18] So, it says that a piece of the [[Riemann curvature tensor]] or the Ricci tensor minus an even smaller piece, the scalar curvature multiplied by the metric is equal plus the cosmological constant is equal to some amount of matter and energy, the stress energy tensor. So it's intrinsically a curvature equation.  

<p>[00:43:55] So, if this is the space-time manifold, "the arena"; the second one concerns symmetry groups which cannot necessarily be deduced from any structure inside of "the arena". They are additional data that come to us out of the blue without explanation and these symmetries for a non-Abelian group, which is currently SU(3) "color" x SU(2) "weak" x U(1) "weak hypercharge", which breaks down to SU(3)xU(1), where the broken U(1) is the electromagnetic symmetry.

<p>[00:44:54] This equation is also a curvature equation, the corresponding equation, 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.

<p>[00:46:33] And so if one of the components is missing, if the equation is intrinsically lopsided, chiral, asymmetric, then the mass term and the differential term have difficulty interacting, which is sort of overcompensating for the mass scale of the universe so you get to a point where you actually have to define a massless equation, but then just like overshooting a putt, it's easier to knock it back by putting in a [[Higgs field]] in order to generate an "as-if" fundamental mass through the [[Yukawa couplings]].

<p>[00:47:15] Let me, for consistency, say "matter is asymmetric", okay. "and therefore light".

<p>[00:47:35] And then interestingly, he went on to say one more thing. He said, of course, these three central observations must be supplemented with the idea that this all [be] treated in quantum mechanical fashion or quantum field theoretic [fashion]. So it's a bit of an after-market modification, rather than his opinion at the time, [or] one of the core insights.

<p>[00:48:07] I actually think that that's in some sense about right. No. One of my differences with the [modern-day physics] community in some sense is I question whether the quantum isn't in good enough shape. We don't know whether we have a serious quantum mechanical problem or not. We know that we have a quantum mechanical problem, a quantum field theoretic problem, [but only] relative to the current formulations of these theories.

<p>[00:52:03] If I 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. So the problem is, is that the projection is based on the fact that you have a relationship between the intrinsic geometry. If this is an ad-valued two-form, the two-form portion of this and the adjoint portion of this are both associated to the structure group of the tangent bundle.

<p>[01:24:30] If you just try to shove it into the bottle, you're undoubtedly going to snap the mast. So, you imagine that you've transformed your gauge fields, you've kept track of where the Ricci curvature was, uou try to push it from one space, like ad-valued two-forms into another space like ad-valued one-forms, where connections live.

<p>[01:25:43] Let's think about unified content. We know that we want a space of connections, $$A$$ for our field theory, but we know because we have a Levi-Civita connection, that this is going to be equal on-the-nose to ad-valued one-forms $$(\Omega^{1}(Ad))$$ as a vector space. The gauge group represents an ad-valued one-forms. So, if we also have the gauge group ($$\mathcal{H}$$), but we think of that instead as a space of sigma fields.

<p>[01:26:44] It's possible that we should be basing it around something more akin to the gauge group. And in this case, we're mimicking the construction where $$\Xi$$ here would be analogous to the Lorentz group fixing a point in Mankowski space. And the ad-valued one-forms would be analogous to the four momentums we take in the semi-direct product to create the inhomogeneous Lorentz group, otherwise known as the Poincaré group, or rather its double cover to allow spin.

<p>[01:38:00] So, for in this case, for example [where i = 2], it would take a two-form to a d-minus-three-plus-two or a d-minus-one-form. So, curvature is an ad-valued two-form. And, if I had such a Shiab operator, it would take ad-valued two-forms to ad-valued d-minus-one-forms, which is exactly the right space to be an $$\alpha$$ coming from the derivative of an action.

<p>[01:50:15] So in other words, I have two derivative operators here. I have two ad-value one-forms. The difference between them has been to be a zero-th order, and it's going to be precisely the augmented torsion. And that's the same game I'm going to repeat here.

<p>[02:28:42] So our summary diagram looks something like this. Take a look at the Taus of $$A_0$$. We will find a homeomorphism of the gauge group into its inhomogeneous extension that isn't simply inclusion under the first factor. And I'm realizing that I have the wrong pi production. That should just be as simple $$\pi$$, projecting down, we have a map from the inhomogeneous gauge group, via the bi-connection to A cross A connections cross connections, and that that behaves well according to the difference operator $$\delta$$ that takes the difference of two connections and gives an honest ad-valued one-form.

<p>[02:31:17] I hope I remember the terminology right. It's been a long time. The map is injective because it's injective under the first factor, but it actually gives us a nontrivial embedding of the gauge group in its inhomogeneous extension, which makes the whole theory work. We then get to Shiab operators. Now, a Shiab operator a map from the group crossed the ad-valued i forms.

<p>[02:31:43] In this case, the particular Shiab operators we’re interested in is mapping i form is to d minus three, plus i forms. So, for example, you would map a two form to a d minus three plus i. So if d, for example, were 14, ..., and i was equal to two. Then 14 minus three is equal to 11 plus two is equal to 13. So that would be an ad-valued 14 minus one form, which is exactly the right place for something to form a current. That is the differential of a Lagrangian on the space.