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

<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, I wish I remembered more of this stuff. Into the second factor. It turns out that this is actually a group homomomorphism 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.

<p>[01:31:38] Now, what would it mean to be able to use a gauge group in an intrinsic theory like this? We would be talking. About something like an action, let's say a first order action and it would take the group G no, let's say to the real numbers. Invariant,  not under the full group, but under the tilted gauge subgroup. And now the question is, do we have any such actions that are particularly nice? And could we recognize them the way Einstein did by trying to write down, not the action and Hilbert was the first one to write that down. But I, you know, I always feel defensive, uh, because I think Einstein and Grossman did so much more to begin the theory in that the Lagrangian that got written down was really just an inevitability.

<p>[01:33:43] So we've, we've restricted ourselves to one gauge group, this big unitary group on the spinors using whatever sort of inner product naturally exists on the spinors and not spinors value in an auxiliary structure, but intrinsic spinors.

<p>[01:34:28] Which is just looking like the exterior algebra on the chimeric bundle. tThat means that it's graded by degrees. chimeric bundle has dimension 14 so there's a zero part, a one part, a two part, all the way up to 14 plus. We have forms in the manifold, and so the question is, if I want to look at $$Omega^i$$ okay valued in the adjunct bundle.

<p>[01:35:05] Which is pure trace. Okay.

<p>[01:35:12] Right. Because it's the same representations appearing where in the usually auxiliary directions as well as the geometric directions. So we get an entire suite. Of invariance together with trivially, uh, associated variants that come from using the Hodge star operator on the forms. I'm just going to call them for completeness.

<p>[01:36:04] I can take field content. Epsilon and Pi, where Epsilon, where these are elements of the inhomogeneous gauge group. In other words, Epsilon is a gauge transformation and Pi is an, is a gauge potential

<p>[01:36:27] and I can start to define operators.

<p>[01:36:44] I'm used. So in this case, if I have a phi, which is one of these invariants in the form piece, I can either take a contraction or I can take a wedge product. In the Lie algebra piece, I can either take a lead product. Or because I'm looking at, um, the unitary group, there's a second possibility, which is I can multiply everything by i and go from scew hermitian to hermitian and take a Jordan product using anti commutators rather than commutators.

<p>[01:37:15] So I actually have a fair amount of freedom and I'm going to use a magic bracket notation, which in whatever situation I'm looking for, knows what it wants to be is does it want to do contraction? Does want to do wedge product, Lie product, Jordan product. But the point is, I now have a suite of ways of moving forms around.