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

<p>[00:42:13] And they correspond to the three great equations. So the first one is, is that somehow physics takes place in an arena and that arena is a [[manifold]] X together with some kind of [[semi-Riemannian]] [[metric structure]], something that allows us to take length and angle. So that we can perform measurements at every point in this spacetime or higher dimensional structure, leaving us a little bit of head room. The equation most associated with this is the [[Einstein field equation]].

<p>[00:43:12] And of course I'm running into the margin. Okay.

<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>[00:43:47] The second fundamental insight, I'm going to begin to start drawing pictures here as well.

<p>[00:43:55] So if this is the spacetime 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" cross  SU(2) "weak" cross U(1) "weak hypercharge", which breaks down to SU(3) cross U(1), where the broken U(1) is the electromagnetic symmetry.

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

<p>[00:46:12] One of the great insights is, is that the reason for the lightness of matter in the natural mass scale of physics has to do with the fact that this psi really should have two components and the differential operators should map to one component on the other side of the equation, but the mass operators should map to another.

<p>[00:47:15] Let me for consistency so 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 takes place, treated in quantum mechanical fashion or quantum field theoretic. So it's a bit of a, an aftermarket modification rather than in his opinion at the time 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 community in some sense is that I questioned whether the quantum isn't in good enough shape that 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 relative to the current formulations of these theories.

<p>[00:49:14] So I'm going to put general relativity and Einstein's formulation. And I'm going to put the probably won't write this again. Yang, Mills, Maxwell, Anderson, Higgs theory, uh, over here, and I'm going to write the Dirac theory.

<p>[00:49:39] What I want to explore is the incompatibilities, not at the quantum level. But the geometric input, all three of these are geometric theories. And the question is, what are the compatibilities or incompatibilities at the level of geometry before the theory is treated quantum mechanically? Well, in the case of Einstein's general relativity, we can rewrite the Einstein theory by saying that there's a projection map due to Einstein of a curvature tensor where I'm going to write that curvature tensor as I would in the Yang Mills theory.

<p>[00:50:23] That should be an LC for Levi-Civita. So the Einstein projection of the curvature tensor of the Levi-Civita connection of the metric on this side, and on this side, I'm going to write down this differential operator, the adjoint of the exterior derivative coupled to a connection.

<p>[00:50:47] And you begin to see that we're missing an opportunity, potentially. What if the FAs were the same in both contexts? Then you're applying two separate operators, one zeroth order and destructive in the sense that it doesn't see the entire curvature tensor, the other inclusive, but of first order. And so the question is, is there any opportunity to do anything that combines these two?