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

<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>[01:26:44] 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 valued one forms would be analogous to the four momentums. We take in the semi direct product to create the inhomogeneous Lorentz group, otherwise known as the Poincare 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>[01:27:23] 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 homomomorphism 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 what 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, 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.