A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions
A Portal Special Presentation- Geometric Unity: A First Look (edit)
Revision as of 05:34, 11 April 2020
, 11 April 2020→Part III: Unified Field Content Plus a Toolkit
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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 $$\ | <p>[01:36:04] I can take field content: $$\epsilon$$ and $$\pi$$, where these are elements of the inhomogeneous gauge group. In other words, where $$\epsilon$$, is a gauge transformation and $$\pi$$ is a gauge potential. | ||
<p>[01:36:27] And I can start to define operators. | <p>[01:36:27] And I can start to define operators. | ||
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<p>[01:37:15] So I actually have a fair amount of freedom and I'm going to use a "magic bracket" notation, which in whatever situation I'm looking for, [the operator] knows what it wants to be, is does it. Want to do contraction? Does want to do wedge product, Lie product, Jordan product? But the point is, I now have a suite of ways of moving forms around. | <p>[01:37:15] So I actually have a fair amount of freedom and I'm going to use a "magic bracket" notation, which in whatever situation I'm looking for, [the operator] knows what it wants to be, is does it. Want to do contraction? Does want to do wedge product, Lie product, Jordan product? But the point is, I now have a suite of ways of moving forms around. | ||
<p>[01:37:34] So for example, I can define a Shiab ("Ship in a bottle") operator. Okay. That takes, [$$\Omega_{i}$$] forms valued in the adjoint bundle. | <p>[01:37:34] So for example, I can define a Shiab ("Ship in a bottle") operator. Okay. That takes, [$$\Omega_{i}$$] $$i$$-forms valued in the adjoint bundle. | ||
<p>[01:37:59] To much higher-degree forms valued in the adjoint bundle. So for in this case, for example, it would take a two-form to a d-minus-three-plus-two or a d-minus one-form. So curvature is an add-value two-form. And if I had such a Shiab operator, it would take add-value two-forms to add-value d-minus one-forms, which is exactly the right space to be an $$\alpha$$ coming from the derivative of an action. | <p>[01:37:59] To much higher-degree forms valued in the adjoint bundle. So for in this case, for example, it would take a two-form to a d-minus-three-plus-two or a d-minus one-form. So curvature is an add-value two-form. And if I had such a Shiab operator, it would take add-value two-forms to add-value d-minus one-forms, which is exactly the right space to be an $$\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 | <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 what the index of the is? So you take all your forms to | <p>[01:39:28] Can you just clarify what the index of the is? So you take all your forms to d-minus-three-plus-i, | ||
<p>[01:39:43] right? So in this case, the idea is, is that we've actually got something for our magic. 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>[01:39:43] right? So in this case, the idea is, is that we've actually got something for our magic. 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? | ||