Difference between revisions of "A Portal Special Presentation- Geometric Unity: A First Look"
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<p>[01:46:17] So that's pretty cool. If that works, we've now taken the Einstein equation and we've put it not on the space of metrics, but we've put a generalization and an analog on the space of gauge potentials, much more amenable to quantization with much more algebraic structure and symmetry in the form of the inhomogeneous gauge group and its homogeneous vector bundle, some of which may be super symmetric. | <p>[01:46:17] So that's pretty cool. If that works, we've now taken the Einstein equation and we've put it not on the space of metrics, but we've put a generalization and an analog on the space of gauge potentials, much more amenable to quantization with much more algebraic structure and symmetry in the form of the inhomogeneous gauge group and its homogeneous vector bundle, some of which may be super symmetric. | ||
<p>[01:46:44] Now the question is | <p>[01:46:44] Now the question is we've integrated so tight with the matter field, we have to ask ourselves the question, can we see unification here? | ||
<p>[01:47:01] Let's define matter content in the form of Omega zero tensor spinors, which is a fancy way of saying spinors together. Okay. | <p>[01:47:01] Let's define matter content in the form of Omega zero tensor spinors, which is a fancy way of saying spinors together. Okay. | ||
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<p>[01:47:43] And , whereas a little kid, I had the [https://en.wikipedia.org/wiki/Soma_cube Soma cube]. I don't know if you've ever played with one of these things. They're fantastic. And, uh, I later found out that this guy who invented the Soma cube, which you had to put together as pieces, there was one piece that looked like this, this object. And he was like this amazing guy in the Resistance during World War II. | <p>[01:47:43] And , whereas a little kid, I had the [https://en.wikipedia.org/wiki/Soma_cube Soma cube]. I don't know if you've ever played with one of these things. They're fantastic. And, uh, I later found out that this guy who invented the Soma cube, which you had to put together as pieces, there was one piece that looked like this, this object. And he was like this amazing guy in the Resistance during World War II. | ||
<p>[01:48:03] So I would like to name this, the Somatic Complex | <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 spinorial matter with the intrinsic replacement for the curvature equations. | <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. | ||
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<p>[01:52:21] Well, that's pretty good, if true. | <p>[01:52:21] Well, that's pretty good, if true. | ||
<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 | <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] 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>[01:53:12] 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 | <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 $$\Omega^{0}(ad)$$, $$\Omega^{0}(ad)$$ direct sum $$\Omega^{0}(ad)$$ | ||
<p>[01:54:00] Omega d | <p>[01:54:00] $$\Omega^{d-1}(ad)$$ and it's almost the same operators. | ||
<p>[01:54:09] And this is now not just a great guess, it's actually the information for the deformation theory of the linearized replacement of the Einstein field equations. So this is computing the [ | <p>[01:54:09] And this is now not just a great guess, it's actually the information for the deformation theory of the linearized replacement of the Einstein field equations. So this is computing the [[[Zariski tangent space]] just as if you were doing self-dual theory or Chern-Simons theory. You've got two somatic complexes, right? | ||
<p>[01:54:33] One of them is Bose. One of them is Fermi. The obstruction to both of them is a common generalization of the Einstein field equations. What's more is if you, if you start to think about this, this is some version of Hodge theory with funky operators, so you can ask yourself | <p>[01:54:33] One of them is [[Bose]]. One of them is [[Fermi]]. The obstruction to both of them is a common generalization of the Einstein field equations. What's more is if you, if you start to think about this, this is some version of [[Hodge theory]] with funky operators, so you can ask yourself: "Well, what are the harmonic forms in a fractional spin context?" | ||
<p>[01:55:03] Well, there are different, depending upon whether you take the degree-zero piece together with the degree-two piece, or you take the degree-one piece, let's just take the degree-one piece. | <p>[01:55:03] Well, there are different, depending upon whether you take the degree-zero piece together with the degree-two piece, or you take the degree-one piece, let's just take the degree-one piece. | ||
<p>[01:55:16] You get some kind of equation. So I'm going to decide that I have a Zeta field, which is an Omega | <p>[01:55:16] You get some kind of equation. So I'm going to decide that I have a $$\Zeta$$ field, which is an $$\Omega^{1}$$ tensor spinors and a field $$\nu$$, which always strikes me as a Yiddish field. $$\nu$$ is $$\Omega^{0}$$ tensor S. | ||
<p>[01:55:47] What equation would they solve if we were doing Hodge theory relative to this complex? The equation would look something like this. | |||
<p>[01:55:47] What equation would they solve if we were doing Hodge theory relative to this complex | |||
<p>[01:56:10] There'd be one equation that was very simple, and then there'd be one equation that would be like really hard to guess. | <p>[01:56:10] There'd be one equation that was very simple, and then there'd be one equation that would be like really hard to guess. | ||
<p>[01:56:56] Yeah, look all these boards and I still feel like I'm managing to run out of room. | <p>[01:56:56] Yeah, look all these boards and I still feel like I'm managing to run out of room. | ||
<p>[01:57:48] Now, if you have something like that, that would be a hell of a Dirac equation. | <p>[01:57:48] Now, if you have something like that, that would be a hell of a Dirac equation. Right. You've got differential operators over here. You've got differential operators. I guess I didn't write them in. But you would have two differential operators over here, and you'd have this differential operator coming from this [[Maurer-Cartan form]]. So I apologize, I'm being a little loose here, but the idea is you have two of these terms are zeroth-order. Three of these terms would be first order, and on this side, one term would be first-order. | ||
<p>[01: | <p>[01:58:27] And that's not there. That's fine. That was a mistake. Oh, no, sorry. That was a mistake, calling it a mistake. These are two separate equations, right? So you have two separate fields, $$\Nu$$ and $$\Zeta$$, and you have a coupled set of differential equations that are playing the role of the Dirac theory. Coming from the Hodge theory of a complex, whose obstruction to being cohomology theory, would be the replacement to the Einstein field equations, which would be rendered gauge invariant on a group relative to a tilted subgroup. | ||
<p>[01: | <p>[01:58:50] So now we've dealt with two of the three sectors. Is there any generalization of the Yang-Mills equation? Well, if we were to take the Einstein field equation generalization and take the norm square of it. Oh, there's some point I should make here. | ||
<p>[01:59: | <p>[01:59:29] Just one second. Yeah. I've been treating this as if everything is first-order, but what really happens here is that you've got symmetries. You've got symmetric field content, you've got ordinary connections. | ||
<p>[02:00:01] And we're neglecting to draw the fact that there have to be equations here too. These equations are first order. So why do we get to call this a first-order theory? If there are equations here, which are of second-order, well, it's not a pure first-order theory, but when I say a first-order theory in this context, what I really mean. | <p>[02:00:01] And we're neglecting to draw the fact that there have to be equations here too. These equations are first order. So why do we get to call this a first-order theory? If there are equations here, which are of second-order, well, it's not a pure first-order theory, but when I say a first-order theory in this context, what I really mean. | ||
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<p>[02:00:49] So I can now look. Let's call that entire replacement, | <p>[02:00:49] So I can now look. Let's call that entire replacement, | ||
<p>[02:00:59] which we previously called Alpha. I mean that Alpha equal to Upsilon because I've actually been using Upsilon. The portion of that is just the first-order equations and take the norm square of that. That gives me a new Lagrangian, and if I solve that new Lagrangian, it leads to equations of motion. | <p>[02:00:59] which we previously called $$\Alpha$$. I mean that $$\Alpha$$ equal to $$\Upsilon$$ because I've actually been using $$\Upsilon$$. The portion of that is just the first-order equations and take the norm square of that. That gives me a new Lagrangian, and if I solve that new Lagrangian, it leads to equations of motion. | ||
<p>[02:01:17] That look like exactly what we said before. | <p>[02:01:17] That look like exactly what we said before. | ||
<p>[02:01:28] And it ends up defining an operator that looks something like this, $$d_A^*$$, the adjoint of the operator. | <p>[02:01:28] And it ends up defining an operator that looks something like this, $$d_A^*$$, the adjoint of the Shiab operator. | ||
<p>[02:02:01] So in other words, this piece gives you some portion that looks like right from the swervature tensor there’s going to be some component that's playing the role of Einstein's field equations directly and the Ricci tensor, but generalized. And then you're going to have some differential operator here so that the replacement for the Yang-Mills term instead of $$d_A^*$$ of FAA, you've got these two an F | <p>[02:02:01] So in other words, this piece gives you some portion that looks like right from the swervature tensor there’s going to be some component that's playing the role of Einstein's field equations directly and the Ricci tensor, but generalized. And then you're going to have some differential operator here so that the replacement for the Yang-Mills term instead of $$d_A^*$$ of FAA, you've got these two an F |