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

<p>[01:34:28] Which is just looking like the exterior algebra on the chimeric bundle. That means that it's graded by degrees. [The] 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 $$Omega^i$$ valued in the adjunct bundle, there's going to be some element $$\Phi_{i}$$, which is pure trace."

<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 invariants that come from using the Hodge star operator on the forms. I'm just going to call them for completeness.

<p>[01:35:41] Now, this is a tremendous amount of freedom that we've just gained. Normally we keep losing freedom, but this is the first time we actually begin to see that we have a lot of freedom and we're going to actually retain some of this freedom through the to the end of the talk. But the idea being that I can now start to define operators which correspond to the ship in the bottle problem.

<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>[01:36:44] 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 Lie 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 [[Skew-Hermitian]] to [[Hermitian]] and take a [[Jordan product]] using anti commutators rather than commutators.