Difference between revisions of "A Portal Special Presentation- Geometric Unity: A First Look"
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A Portal Special Presentation- Geometric Unity: A First Look (view source)
Revision as of 15:51, 10 April 2020
, 15:51, 10 April 2020→Part II: Unified Field Content
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<p>[01:20:01] Okay. What is it that we get for the Levi-Civita connection? Well, not much. One thing we get is that normally the space of connections is an affine space. Not a vector space, but an affine space, almost a vector space, a vector space up to a choice of origin. But with the Levi-Civita connection, rather than having an infinite plane with an ability to take differences, but no real ability to have a group structure. | <p>[01:20:01] Okay. What is it that we get for the Levi-Civita connection? Well, not much. One thing we get is that normally the space of connections is an affine space. Not a vector space, but an affine space, almost a vector space, a vector space up to a choice of origin. But with the Levi-Civita connection, rather than having an infinite plane with an ability to take differences, but no real ability to have a group structure. | ||
<p>[01:20:35] You pick out one point, which then becomes the origin. That means that any connection A has a torsion tensor A which is equal to the connection minus the Levi-Civita connection. So we get a tensor that we don't usually have. Gauge potentials are not usually well-defined | <p>[01:20:35] You pick out one point, which then becomes the origin. That means that any connection A has a torsion tensor, $$A$$, which is equal to the connection minus the Levi-Civita connection. So we get a tensor that we don't usually have. Gauge potentials are not usually well-defined they're are only defined up to a choice of gauge. | ||
<p>[01:21:00] So that's one of the things we get for our Levi-Civita connection, but because the gauge group is going to go missing, this has terrible properties from with respect to the gauge group. It almost looks like a representation, | <p>[01:21:00] So that's one of the things we get for our Levi-Civita connection, but because the gauge group is going to go missing, this has terrible properties from with respect to the gauge group. It almost looks like a representation. But, in fact, if we let the gauge group act, there's going to be an affine shift. | ||
<p>[01:21:21] Furthermore, as we've said before | <p>[01:21:21] Furthermore, as we've said before the ability to use projection operators together with the gauge group is frustrated by virtue of the fact that these two things do not commute with each other. So now the question is, how are we going to prove that we're actually making a good trade? | ||
<p>[01:21: | <p>[01:21:48] [The] first thing we need to do is we still have the right to choose intrinsic field content. [We] have an intrinsic field theory. So, if you consider the structure bundle of the spinors, we built the chimeric bundle, so we can define Dirac spinors on the chimeric bundle if we're in Euclidean signature. A 14-dimensional manifold has Dirac spinors of dimension-two to the dimension of the space divided by two. | ||
<p>[01:22:20] Right? So 2^14 over 2^7 is 128, so we have a map into a structured group of $$U^128$$ | |||
<p>[01:22:20] Right? So | |||
<p>[01:22:36] At least in Euclidean signature. We can get to mixed signatures later. From that, we can form the associated bundle. | <p>[01:22:36] At least in Euclidean signature. We can get to mixed signatures later. From that, we can form the associated bundle. | ||
<p>[01:22:54] And sections of this bundle are either, depending upon how you want to think about it, the gauge group H or Xi, a space of sigma fields. | <p>[01:22:54] And sections of this bundle are either, depending upon how you want to think about it, the gauge group $\H$ or $$\Xi$$, a space of sigma fields. | ||
<p>[01:23:18] Nonlinear. | <p>[01:23:18] Nonlinear. | ||