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

''<a href="https://youtu.be/Z7rd04KzLcg?t=5498" data-type="URL" data-id="https://youtu.be/Z7rd04KzLcg?t=5498">01:31:38</a>''<br>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.

''<a href="https://youtu.be/Z7rd04KzLcg?t=5533" data-type="URL" data-id="https://youtu.be/Z7rd04KzLcg?t=5533">01:32:13</a>''<br>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, because I think Einstein and Grossmann did so much more to begin the theory, and that the Lagrangian that got written down was really just an inevitability. 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.

''<a href="https://youtu.be/Z7rd04KzLcg?t=5602" data-type="URL" data-id="https://youtu.be/Z7rd04KzLcg?t=5602">01:33:22</a>''<br>Now in this section of GU, unified field content is only one part of it. But what we really want is unified field content plus a toolkit. So 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 valued in an auxiliary structure, but intrinsic spinors.