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 04:28, 11 April 2020
, 11 April 2020→Part II: Unified Field Content
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<p>[01:29:14] And the idea is that we're going to work with vector bundles, curly E of the form inhomogeneous gauge group producted over the tilted gauge group. | <p>[01:29:14] And the idea is that we're going to work with vector bundles, curly E of the form inhomogeneous gauge group producted over the tilted gauge group. | ||
<p>[01:29:47] Now, just as in the finite dimensional case, we have a linear and a nonlinear component, right? Because at the topological level, this is just a Cartesian product. So if we wished to take products of fermions | <p>[01:29:47] Now, just as in the finite dimensional case, we have a linear and a nonlinear component, right? Because at the topological level, this is just a Cartesian product. So if we wished to take products of fermions; of spinorial fields. We have a place to accept them. We can't figure out necessarily how to map them into the nonlinear sector, but we don't want to. | ||
<p>[01:30:09] So just the way, when we look at supersymmetry, we can take products of the spin one half fields and map them into the linear sector. We can do the same thing here. So what we're talking about is something like a supersymmetric extension of the inhomogeneous gauge group analogous to supersymmetric extensions of the double cover of the inhomogeneous | <p>[01:30:09] So just the way, when we look at supersymmetry, we can take products of the spin-one-half fields and map them into the linear sector. We can do the same thing here. So what we're talking about is something like a supersymmetric extension of the inhomogeneous gauge group analogous to supersymmetric extensions of the double cover of the inhomogeneous Lorentz or Poincaré group. Further, because this construction is at the level of groups, we've left a slot on the left hand side on which to act. So for example, if we want to take regular representations on the group, we can act by the group G on the left-hand side, because we're allowing the tilted gauge group to act on the right-hand side. | ||
<p>[01:30:56] So it's perfectly built for representation theory. And if you think back to Wigner’s | <p>[01:30:56] So it's perfectly built for representation theory. And if you think back to Wigner’s classification and the concept that a particle should correspond to an irreducible representation of the inhomogeneous gauge group, uh, inhomogeneous Lorentz group, we may be able to play the same games here up to the issue of infinite dimentionality. | ||
<p>[01:31:15] So right now, our field content is looking pretty good. It's looking unified in the sense. That it has an algebraic structure that is not usually enjoyed by field content and the field content from different sectors can interact and know about each other. Mmm. Provided we can drag something of this out of this with meaning. | <p>[01:31:15] So right now, our field content is looking pretty good. It's looking unified in the sense. That it has an algebraic structure that is not usually enjoyed by field content and the field content from different sectors can interact and know about each other. Mmm. Provided we can drag something of this out of this with meaning. | ||
<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, | <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 Grossmann 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-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>[01:32:38] 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. | ||
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<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 algebra 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. [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, | <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, there'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. | ||