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A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions
A Portal Special Presentation- Geometric Unity: A First Look (edit)
Revision as of 20:45, 15 March 2021
, 15 March 2021→Toolkit for the Unified Field Content
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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. | 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. | ||
'' | ''[https://youtu.be/Z7rd04KzLcg?t=5640 01:34:00]''<br> | ||
The toolkit that we have is that the adjoint bundle looks like the Clifford algebra at the level of vector spaces, which is just looking like the exterior algebra on the chimeric bundle. | |||
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'' | ''[https://youtu.be/Z7rd04KzLcg?t=5674 01:34:34]''<br> | ||
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 adjoint bundle, there's going to be some element \(\Phi_i\), which is pure trace. | |||
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'' | ''[https://youtu.be/Z7rd04KzLcg?t=5712 01:35:12]''<br> | ||
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 associated invariants that come from using the Hodge star operator on the forms. I'm just going to call them, for completeness, \(\tilde{\Phi}_i\). I'm not going to deal with them. | |||
===== Ship in a Bottle (Shiab) Operator ===== | ===== Ship in a Bottle (Shiab) Operator ===== | ||
'' | ''[https://youtu.be/Z7rd04KzLcg?t=5741 01:35:41]''<br> | ||
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 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. I can take field content \(\epsilon\) and \(\pi\), where these are elements of the inhomogeneous gauge group. In other words, \(\epsilon\) is a gauge transformation, and \(\pi\) is a gauge potential. | |||
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'' | ''[https://youtu.be/Z7rd04KzLcg?t=5787 01:36:27]''<br> | ||
And I can start to define operators: | |||
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'' | ''[https://youtu.be/Z7rd04KzLcg?t=5804 01:36:44]''<br> | ||
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 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. 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, [the operator] knows what it wants to be. 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. | |||
'' | ''[https://youtu.be/Z7rd04KzLcg?t=5854 01:37:34]''<br> | ||
So for example, I can define a '''shiab operator '''(ship in a bottle operator) that takes i-forms valued in the adjoint bundle to much higher-degree forms valued in the adjoint bundle. | |||