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 04:21, 11 April 2020
, 11 April 2020→Part II: Unified Field Content
| Line 479: | Line 479: | ||
<p>[01:31:38] Now, what would it mean to be able to use a gauge group in an intrinsic theory like this? We would be talking. About something like an action, let's say a first order action and it would take the group G, let's say to the real numbers. Invariant, not under the full group, but under the tilted gauge subgroup. And now the question is, do we have any such actions that are particularly nice? And could we recognize them the way Einstein did by trying to write down, not the action and Hilbert was the first one to write that down. But I, you know, I always feel defensive, uh, because I think Einstein and Grossman did so much more to begin the theory in that the Lagrangian that got written down was really just an inevitability. | <p>[01:31:38] Now, what would it mean to be able to use a gauge group in an intrinsic theory like this? We would be talking. About something like an action, let's say a first order action and it would take the group G, let's say to the real numbers. Invariant, not under the full group, but under the tilted gauge subgroup. And now the question is, do we have any such actions that are particularly nice? And could we recognize them the way Einstein did by trying to write down, not the action and Hilbert was the first one to write that down. But I, you know, I always feel defensive, uh, because I think Einstein and Grossman did so much more to begin the theory in that the Lagrangian that got written down was really just an inevitability. | ||
<p>[01:32:38] So just humor me for this talk, and let me call it the Einstein | <p>[01:32:38] So just humor me for this talk, and let me call it the Einstein-Grossmann Lagrangian. Hilbert's certainly done fantastic things and has a lot of credit elsewhere, and he did do it first. But here what we had was that Einstein thought in terms of the differential of the action, not the action itself. So what we're looking for is equations of motion or some field, $$\alpha$$ where $$\alpha$$ belongs to the one-forms on the group. | ||
<p>[01:33:22] Now in this section of GU unified field content is only one part of it, but what we really want | <p>[01:33:22] Now in this section of GU unified field content is only one part of it, but what we really want is unified field content plus a toolkit. | ||
<p>[01:33:43] So we've, we've restricted ourselves to one gauge group, this big unitary group on the spinors using whatever sort of inner product naturally exists on the spinors and not spinors valued in an auxiliary structure, but intrinsic spinors. | <p>[01:33:43] So we've, we've restricted ourselves to one gauge group, this big unitary group on the spinors using whatever sort of inner product naturally exists on the spinors, and not spinors valued in an auxiliary structure, but intrinsic spinors. | ||
<p>[01:34:00] The toolkit that we have is that the adjoint bundle looks like the Clifford | <p>[01:34:00] The toolkit that we have is that the adjoint bundle looks like the Clifford algebr at the level of vector space. | ||
<p>[01:34:28] Which is just looking like the exterior algebra on the chimeric bundle. That means that it's graded by degrees. chimeric bundle has dimension 14 so there's a zero part, a one part, a two part, all the way up to 14. Plus we have forms in the manifold, and so the question is | <p>[01:34:28] Which is just looking like the exterior algebra on the chimeric bundle. That means that it's graded by degrees. [The] chimeric bundle has dimension 14, so there's a zero-part, a one-part, a two-part, all the way up to 14. Plus, we have forms in the manifold, and so the question is "If I want to look at $$Omega^i$$ valued in the adjunct bundle,tThere's going to be some element $$Phi_i$$, which is pure trace." | ||
<p>[01:35:12] Right? Because it's the same representations appearing where in the usually auxiliary directions as well as the geometric directions. So we get an entire suite of invariance together with trivially, uh, associated invariants that come from using the Hodge star operator on the forms. I'm just going to call them for completeness. | <p>[01:35:12] Right? Because it's the same representations appearing where in the usually auxiliary directions as well as the geometric directions. So we get an entire suite of invariance together with trivially, uh, associated invariants that come from using the Hodge star operator on the forms. I'm just going to call them for completeness. | ||