Difference between revisions of "A Portal Special Presentation- Geometric Unity: A First Look"
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A Portal Special Presentation- Geometric Unity: A First Look (view source)
Revision as of 00:24, 12 April 2020
, 00:24, 12 April 2020→Part III: Starting in on Physics
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<p>[01:43:23] We've built ourselves an projection operators. We've picked up some symmetric, nonlinear Sigma field. | <p>[01:43:23] We've built ourselves an projection operators. We've picked up some symmetric, nonlinear Sigma field. | ||
<p>[01:43:33] What can we write down in terms of equations of motion. Let's start with Einstein's concept. Okay. If we do, of the curvature tensor, but the gauge potential hit with an operator. Okay. Defined by the | <p>[01:43:33] What can we write down in terms of equations of motion. Let's start with Einstein's concept. Okay. If we do, of the curvature tensor, but the gauge potential hit with an operator. Okay. Defined by the epsilon sigma field plus the star operator. Acting on the augmented torsion of the pair. This contains all of the information when $$\pi$$ is zero in Einstein's tensor. | ||
<p>[01:44:16] In other words, there is a spinorial analog of the Weyl curvature, a spinorial analog of the traceless Ricci curvature, and a spinorial analog of the scalar curvature. This operator should shear off the analog of the Weyl curvature. Just the way the Einstein's projection shears off the Weyl curvature, when you're looking at the tangent bundle, and this term, which is now gauge invariant, may be considered as containing a piece that looks like $$\lambda$$ times $$G_{\mu \nu}$$, or a cosmological constant. | <p>[01:44:16] In other words, there is a spinorial analog of the Weyl curvature, a spinorial analog of the traceless Ricci curvature, and a spinorial analog of the scalar curvature. This operator should shear off the analog of the Weyl curvature. Just the way the Einstein's projection shears off the Weyl curvature, when you're looking at the tangent bundle, and this term, which is now gauge invariant, may be considered as containing a piece that looks like $$\lambda$$ times $$G_{\mu \nu}$$, or a cosmological constant. | ||
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<p>[01:48:03] So I would like to name this, the Somatic Complex. after, I guess his name is Piet Hein, I think. So this complex, I'm going to choose to start filling in some operators, the exterior derivative coupled to a connection, but on the case of spinors, we're going to put a slash through it. Let's make this the identity. | <p>[01:48:03] So I would like to name this, the Somatic Complex. after, I guess his name is Piet Hein, I think. So this complex, I'm going to choose to start filling in some operators, the exterior derivative coupled to a connection, but on the case of spinors, we're going to put a slash through it. Let's make this the identity. | ||
<p>[01:48:26] We'd now like to come up with a second operator here. Here in this second operator here should have the property that the complex should be exact and the obstruction to it being a true complex to nillpotency should be exactly the generalization of the Einstein equations. Thus unifying the | <p>[01:48:26] We'd now like to come up with a second operator here. Here in this second operator here should have the property that the complex should be exact and the obstruction to it being a true complex, to nillpotency, should be exactly the generalization of the Einstein equations. Thus, unifying the spinorial matter with the intrinsic replacement for the curvature equations. | ||
<p>[01:48:59] Well, we know that | <p>[01:48:59] Well, we know that $d_A$ composed with itself is going to be the curvature, and we know that we want that to be hit by a Shiab operator. And if is a derivation, you can start to see that that's going to be curvature. So you want something like $$F_A$$ followed by Shiab over here, to cancel. Then you think, okay, well how am I going to get at getting this augmented torsion? | ||
<p>[01:49:32] And then you realize that the information in the inhomogeneous gauge group, you actually have information, not for one connection, but for two connections. | <p>[01:49:32] And then you realize that the information in the inhomogeneous gauge group, you actually have information, not for one connection, but for two connections. | ||
<p>[01:49:44] I can do | <p>[01:49:44] So in one case I can do plus star to pick up the $A_\pi$$. | ||
<p>[01:49: | <p>[01:49:59] But I'm also going to have a derivative operator, if I just do a star operation. So I need another derivative operator to kill it off here. So I'm going to take minus the derivative with respect to the connection, $$H^{-1}$$ $$d_{A_0}$$ $$H$$, which defines a connection one form as well as having the same. | ||
<p>[01: | <p>[01:50:21] Derivative coming from the Levi-Civita connection on $$U$$. So in other words, I have two derivative operators here. I have two add value one forms. The difference between them has been to be a zero with order, and it's going to be precisely the augmented torsion. And that's the same game I'm going to repeat here. | ||
<p>[01:50: | <p>[01:50:52] So I'm going to do the same thing here. I'm going to define a bunch of terms where in the numerator, I'm going to pick up the pie as well as the derivative in the denominator, because I have no derivative here. I'm going to pick up this $$H^{-1}$$ $$d_{A_0}$$ $$H$$. | ||
<p>[01:51:17] I'm going to do that again. On the other side. There are going to be plus and minus signs, but it's a magic bracket that knows whether or not should be a plus sign or a minus sign, and I apologize for that, but I'm not able to keep that straight. And then there's going to be one extra term. | |||
<p>[01:51:17] I'm going to do that again. On the other side | |||
<p>[01:51:48] Where all these Ts have the epsilon and pis. Okay. So some crazy series of differential operators on the northern route. So if you take the high road or you take the low road, when you take the composition of the two, the differential operators fall out and you're left with an obstruction term that looks like the Einstein field equation. | <p>[01:51:48] Where all these Ts have the epsilon and pis. Okay. So some crazy series of differential operators on the northern route. So if you take the high road or you take the low road, when you take the composition of the two, the differential operators fall out and you're left with an obstruction term that looks like the Einstein field equation. | ||
<p>[01:52:21] Well, that's pretty good | <p>[01:52:21] Well, that's pretty good, if true. | ||
<p>[01:52: | <p>[01:52:26] Can you go farther? Well, look it up. Close to this field. Content is to the picture from [[Deformation theory]] that we learned about in low dimensions. The low dimensional world works by saying that symmetries map to field content map to equations, usually in the curvature. And when you linearize that if you're in low enough dimensions, you have $$\Omega^{0}$$, $$\Omega^{1}$$. Sometimes $$\Omega^{0}$$ again and then something that comes from $$\Omega^{3}$$ | ||
<p>[01:53:12] | <p>[01:53:12] and if you can get that sequence to terminate by looking at something like a half-signature theorem or a bent-back De Rahm complex. In the case of dimension three, you have Atiyah-Singer theory, and remember, we need some way to get out of infinite dimensional trouble, right? You have to have someone to call when things go wrong overseas, and you have to be able to get your way home. And in some sense, we call on Atiyah-Singer and say, we're in some infinite dimensional space. | ||
<p>[01:53:30] Can't you cut out some finite dimensional problem that we can solve even though we start getting ourselves into serious trouble? And so we're going to do the same thing down here. We're going to have Omega-zero add Omega-one add direct um Omega-zero add | <p>[01:53:30] Can't you cut out some finite dimensional problem that we can solve even though we start getting ourselves into serious trouble? And so we're going to do the same thing down here. We're going to have Omega-zero add Omega-one add direct um Omega-zero add | ||