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 05:32, 11 April 2020
, 05:32, 11 April 2020→Part III: Unified Field Content Plus a Toolkit
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<p>[01:36:44] So in this case, if I have a $$\Phi$$, which is one of these invariants in the form piece, I can either take a contraction or I can take a wedge product. In the Lie algebra piece, I can either take a Lie product or because I'm looking at the unitary group, there's a second possibility, which is I can multiply everything by $$i$$ and go from [[Skew-Hermitian]] to [[Hermitian]] and take a [[Jordan product]] using anti-commutators rather than commutators. | <p>[01:36:44] So in this case, if I have a $$\Phi$$, which is one of these invariants in the form piece, I can either take a contraction or I can take a wedge product. In the Lie algebra piece, I can either take a Lie product or because I'm looking at the unitary group, there's a second possibility, which is I can multiply everything by $$i$$ and go from [[Skew-Hermitian]] to [[Hermitian]] and take a [[Jordan product]] using anti-commutators rather than commutators. | ||
<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, knows what it wants to be is does it | <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, | <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>[01:37:59] To much higher degree forms valued in the adjoint bundle. So for | <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 | <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 they 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 D minus three plus I, | <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, | ||