Difference between revisions of "A Portal Special Presentation- Geometric Unity: A First Look"
A Portal Special Presentation- Geometric Unity: A First Look (view source)
Revision as of 18:17, 25 April 2020
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<p>[01:35:41] Now, this is a tremendous amount of freedom that we've just gained. Normally we keep losing freedom, but this is the first time we actually begin to see that we have a lot of freedom and we're going to actually retain some of this freedom to the end of the talk. But the idea being that I can now start to define operators which correspond to the "Ship in the Bottle" problem. | <p>[01:35:41] Now, this is a tremendous amount of freedom that we've just gained. Normally we keep losing freedom, but this is the first time we actually begin to see that we have a lot of freedom and we're going to actually retain some of this freedom to the end of the talk. But the idea being that I can now start to define operators which correspond to the "Ship in the Bottle" problem. | ||
<p>[01:36:04] I can take field content: $$\epsilon$$ and $$\pi$$ | <p>[01:36:04] I can take field content: $$\epsilon$$ and $$\pi$$ where these are elements of the inhomogeneous gauge group. In other words, where $$\epsilon$$, is a gauge transformation and $$\pi$$ is a gauge potential. | ||
<p>[01:36:27] And I can start to define operators. | <p>[01:36:27] And I can start to define operators. | ||
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<p>[01:37:15] So I actually have a fair amount of freedom and I'm going to use a "magic bracket" notation, which in whatever situation I'm looking for, [the operator] knows what it wants to be, is does it. Want to do contraction? Does want to do wedge product, Lie product, Jordan product? But the point is, I now have a suite of ways of moving forms around. | <p>[01:37:15] So I actually have a fair amount of freedom and I'm going to use a "magic bracket" notation, which in whatever situation I'm looking for, [the operator] knows what it wants to be, is does it. Want to do contraction? Does want to do wedge product, Lie product, Jordan product? But the point is, I now have a suite of ways of moving forms around. | ||
<p>[01:37:34] So for example, I can define a Shiab ("Ship in a bottle") operator | <p>[01:37:34] So, for example, I can define a Shiab ("Ship in a bottle") operator that takes [$$\Omega_{i}$$] $$i$$-forms valued in the adjoint bundle to much higher-degree forms valued in the adjoint bundle. | ||
<p>[01: | <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>[01:38:38] This is exactly what Einstein was doing. He took the curvature, which was large, and he bent it back and he sheared off the [[Weyl curvature]] and he took that part and he pushed it back along the space of metrics to give us something | <p>[01:38:38] This is exactly what Einstein was doing. He took the curvature, which was large, and he bent it back and he sheared off the [[Weyl curvature]] and he took that part and he pushed it back along the space of metrics to give us something which we nowadays call [[Ricci Flow]] and ability for the curvature to direct us to the next structure. | ||
<p>[01:38:59] Well, we're doing the same thing here. We're taking the curvature and we can now push it back onto the space of connections. | <p>[01:38:59] Well, we're doing the same thing here. We're taking the curvature and we can now push it back onto the space of connections. | ||
<p>[01:39:28] Can you just clarify | <p>[01:39:28] Can you just clarify the index on the Shiab is? So you take i-forms to d-minus-three-plus-1? [No,] d-minus-three-plus-i. | ||
<p>[01:39:43] | <p>[01:39:43] So, in this case, the idea is, is that we've actually got something for our magic beans. We have an ability now to get equations of motion, which go along the group. In some sense, it's as if it was a gradient vector field, except we're using forms rather than vectors. But now what are the transformation properties? | ||
<p>[01:40:08] Well, because the curvature. | <p>[01:40:08] Well, because the curvature... so, you have Shiab. Because the curvature of a connection hit by a gauge transformation is equal on-the-nose to the adjoint action on the Lie algebra of the curvature. We know that if we have two possible actions of conjugation under a bracket and the bracket respects the action of the gauge group, we know that this is going to be well-preserved. In other words, we're going to get a form that is gauged invariant relative to the tilted gauge group. | ||
<p>[01: | <p>[01:41:08] And so, as a result, we now have the possibility for equations of motion, which are well-defined even though they involve projection operators, because we've built the symmetries into the theory and we're working on a group manifold to begin with. | ||
<p>[01: | <p>[01:41:26] What about the torsion, $$T_{\epsilon, \pi}$$? Can we rescue the torsion? Here again, we have good news. The torsion is problematic. But if I look at a different field which I'm going to call the augmented torsion and I define it to be the regular torsion, which would be $$\Pi$$ minus this expression. Okay? This turns out to be beautifully invariant again. | ||
<p>[01:41: | <p>[01:41:58] So, neither this term nor this term is invariant. But they fail to be gauge invariant in exactly the same way. So, an important principle of life, which I took too long to realize is that if you have one disease, you're in real trouble. But if you have two diseases, you always have the possibility of having one disease kill the other disease. | ||
<p>[01: | <p>[01:42:16] This is true in [Renormalization theory]. It's true in [Black-Scholes] theory. It's true all over the place. So, what we have here is we've gotten more diseases into the theory, but an even number of diseases allow us to have no disease at all. So, now we have two great tensors. We've got one tensor coming from the curvature and Shiab operator. | ||
<p>[01:42:34] We have another tensor coming from the torsion and its augmentation. | |||
<p>[01:42:34] We have another tensor coming from the torsion and | |||
<p>[01:42:54] We're doing okay. | <p>[01:42:54] We're doing okay. | ||