A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions

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[01:26:44] It's possible that we should be basing it around something more akin to the gauge group, and in this case, we're mimicking the construction where $$\Xi$$ here would be analogous to the Lorentz group, fixing a point in Mankowski space, and add-value one-forms would be analogous to the [each of the] four momentums. We take in the semi-direct product to create the inhomogeneous Lorentz group, otherwise known as the Poincaré group, or rather, it's a double cover to allow spin.

[01:27:12] So we're going to call this the inhomogeneous gauge group or ~~IGG~~.

[01:27:12] So we're going to call this the inhomogeneous gauge group or IGGy.

[01:27:23] And this is going to be a really interesting space because it has a couple of properties. One is it has a very interesting subgroup. Now, of course, H includes into G by just including onto the first factor, but in fact, there's a more interesting homomorphism brought to you by the Levi-Civita connection.

[01:27:23] And this is going to be a really interesting space because it has a couple of properties. One is it has a very interesting subgroup. Now, of course, $$\mathcal{H}$$ includes into $$\mathcal{G}$$ by just including onto the first factor, but in fact, there's a more interesting homomorphism brought to you by the Levi-Civita connection.

[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 G mod [modulo] the tilted gauge group, and if we have any interesting representation of 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 H modules.

[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 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 H modules.

[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.

@@ Line 462: / Line 462: @@
 <p>[01:26:44] It's possible that we should be basing it around something more akin to the gauge group, and in this case, we're mimicking the construction where $$\Xi$$ here would be analogous to the Lorentz group, fixing a point in Mankowski space, and add-value one-forms would be analogous to the [each of the] four momentums. We take in the semi-direct product to create the inhomogeneous Lorentz group, otherwise known as the Poincaré group, or rather, it's a double cover to allow spin.
-<p>[01:27:12] So we're going to call this the inhomogeneous gauge group or IGG.
+<p>[01:27:12] So we're going to call this the inhomogeneous gauge group or IGGy.
-<p>[01:27:23] And this is going to be a really interesting space because it has a couple of properties. One is it has a very interesting subgroup. Now, of course, H includes into G by just including onto the first factor, but in fact, there's a more interesting homomorphism brought to you by the Levi-Civita connection.
+<p>[01:27:23] And this is going to be a really interesting space because it has a couple of properties. One is it has a very interesting subgroup. Now, of course, $$\mathcal{H}$$ includes into $$\mathcal{G}$$ by just including onto the first factor, but in fact, there's a more interesting homomorphism brought to you by the Levi-Civita connection.
-<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 G mod [modulo] the tilted gauge group, and if we have any interesting representation of 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 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 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 H modules.
 <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.