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

<p>[01:29:47] Now, just as in the finite dimensional case, we have a linear and a nonlinear component, right? Because at the topological level, this is just a Cartesian product. So if we wished to take products of fermions, of spinnorial fields. We have a place to accept them. We can't figure out necessarily how to map them into the nonlinear sector, but we don't want to.

<p>[01:30:09] So just the way, when we look at supersymmetry, we can take products of the spin one half fields and map them into the linear sector. We can do the same thing here. So what we're talking about is something like a supersymmetric extension of the inhomogeneous gauge group analogous to supersymmetric extensions of the double cover of the inhomogeneous Lorenz or poincare group. Further, because this construction is at the level of groups, we've left a slot on the left hand side on which to act. So for example, if we want to take regular representations on the group, we can act by the group G on the left hand side, because we're allowing the tilted gauge group to act on the right hand side.

<p>[01:30:56] So it's perfectly built for representation theory. And if you think back to Wigner’s uh, classification and the concept that a particle should correspond to an irreducible representation of the inhomogeneous gauge group, uh, inhomogeneous Lorens group, we may be able to play the same games here up to the issue of infinite dimentionality.

<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, 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>[01:34:00] The toolkit that we have is that the adjoint bundle looks like the Clifford algebr at the level of vector space.

<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,tThere's going to be some element $$Phi_i$$, which is pure trace."