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 22:23, 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, | <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. | ||
<p>[01:30:09] So just the way, when we look at supersymmetry, we can take products of the spin- | <p>[01:30:09] So just the way, when we look at supersymmetry, we can take products of the spin-1/2 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 classification]] and the concept that a particle should correspond to an irreducible representation of the | <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 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. | ||