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 23:12, 11 April 2020
, 11 April 2020→Part II: Unified Field Content
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<p>[01:18:31] In the auxiliary theory, we have freedom to choose our field content. And we have the ability to get rid of much excess through the symmetries of the gauge group. | <p>[01:18:31] In the auxiliary theory, we have freedom to choose our field content. And we have the ability to get rid of much excess through the symmetries of the gauge group. | ||
<p>[01:18:50] We're going to take particle theory | <p>[01:18:50] We're going to take particle theory; we're going to make a bad trade, or what appears to be a bad trade, which is that we are going to give away the freedom to choose our field content, which is already extremely, as I think I said in the abstract, baroque with all of the different particle properties. | ||
<p>[01:19:09] And we're going to lose the ability to use the gauge group because we're going to trade it all. | <p>[01:19:09] And we're going to lose the ability to use the gauge group because we're going to trade it all. | ||
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<p>[01:26:16] What if we take the semi-direct product at a group theoretic level between the two and call this our group of interests. Well by analogy, we've always had a problem with the Poincaré group being too intrinsically tied to rigid flat Minkowski space. What if we wanted to do quantum field theory in some situation, which was more amenable to a curved space situation? | <p>[01:26:16] What if we take the semi-direct product at a group theoretic level between the two and call this our group of interests. Well by analogy, we've always had a problem with the Poincaré group being too intrinsically tied to rigid flat Minkowski space. What if we wanted to do quantum field theory in some situation, which was more amenable to a curved space situation? | ||
<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 | <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 | <p>[01:27:12] So we're going to call this the inhomogeneous gauge group or IGG. | ||
<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 | <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>[01:27:49] 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 | <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>[01:29:14] 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. | ||