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

<p>[01:30:09] So just the way, when we look at supersymmetry, we can take products of the spin-1/2 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 Lorentz or Poincaré 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 classification]] and the concept that a particle should correspond to an irreducible representation of the inhomogeneous Lorentz group, we may be able to play the same games here up to the issue of infinite dimentionality.

<p>[01:31:15] So right now, our field content is looking pretty good. It's looking unified in the sense. That it has an algebraic structure that is not usually enjoyed by field content and the field content from different sectors can interact and know about each other. Mmm. Provided we can drag something of this out of this with meaning.

<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 $$\mathcal{G}$$, let's say to the $$\mathbb{R}$$. Invariant, not under the full group, but under the tilted gauge subgroup, $$\mathcal{H_{\tau}}$$. 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 Grossmann did so much more to begin the theory in that the Lagrangian that got written down was really just an inevitability.

<p>[01:32:38] So just humor me for this talk, and let me call it the Einstein-Grossmann Lagrangian. Hilbert's certainly done fantastic things and has a lot of credit elsewhere, and he did do it first. But here what we had was that Einstein thought in terms of the differential of the action, not the action itself. So what we're looking for is equations of motion or some field, $$\alpha$$ where $$\alpha$$ belongs to the one-forms on the group.