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 22:24, 15 April 2020
, 15 April 2020→Part II: Unified Field Content
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<p>[01:27:49] So, this magic being trade is going to start to enter more and more into our consciousness. If I take an element h and I map that in the obvious way into the first factor, but I map it onto the [[Maurer-Cartan]] form, I think that's when I wish I remembered more of this stuff, into the second factor. It turns out that this is actually a group homomorphism and so we have a nontrivial embedding, which is, in some sense, diagonal between the two factors. That subgroup we are going to refer to as the Tilted Gauge Group, and now our field content, at least in the Bosonic sector, is going to be a group manifold, an infinite dimensional function space [[Lie Group]], but a group nonetheless. And we can now look at $$\mathcal{G}$$ mod [modulo] $$\mathcal{H}$$ (the tilted gauge group), and if we have any interesting representation of $$\mathcal{H}$$, we can form homogeneous vector bundles and work with induced representation. And that's what the fermions are going to be. So the fermions in our theory are going to be $$\mathcal{H}$$ modules. | <p>[01:27:49] So, this magic being trade is going to start to enter more and more into our consciousness. If I take an element h and I map that in the obvious way into the first factor, but I map it onto the [[Maurer-Cartan]] form, I think that's when I wish I remembered more of this stuff, into the second factor. It turns out that this is actually a group homomorphism and so we have a nontrivial embedding, which is, in some sense, diagonal between the two factors. That subgroup we are going to refer to as the Tilted Gauge Group, and now our field content, at least in the Bosonic sector, is going to be a group manifold, an infinite dimensional function space [[Lie Group]], but a group nonetheless. And we can now look at $$\mathcal{G}$$ mod [modulo] $$\mathcal{H}$$ (the tilted gauge group), and if we have any interesting representation of $$\mathcal{H}$$, we can form homogeneous vector bundles and work with induced representation. And that's what the fermions are going to be. So the fermions in our theory are going to be $$\mathcal{H}$$ modules. | ||
<p>[01:29:14] And the idea is that we're going to work with vector bundles, $$\mathcal{E}$$ 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, $$\mathcal{E}$$, 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; 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>[01:29:47] 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. | ||