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 01:30, 17 April 2020
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<p>[00:49:39] What I want to explore is the incompatibilities, not at the quantum level. But the geometric input. All three of these are geometric theories. And the question is, what are the compatibilities or incompatibilities at the level of geometry -- before the theory is treated quantum mechanically? Well, in the case of Einstein's general relativity, we can rewrite the Einstein theory by saying that there's a projection map due to Einstein of a curvature tensor, where I'm going to write that curvature tensor as I would in the Yang-Mills theory. | <p>[00:49:39] What I want to explore is the incompatibilities, not at the quantum level. But the geometric input. All three of these are geometric theories. And the question is, what are the compatibilities or incompatibilities at the level of geometry -- before the theory is treated quantum mechanically? Well, in the case of Einstein's general relativity, we can rewrite the Einstein theory by saying that there's a projection map due to Einstein of a curvature tensor, where I'm going to write that curvature tensor as I would in the Yang-Mills theory. | ||
<p>[00:50:23] That should be an "LC" for Levi-Civita. So, the Einstein projection of the curvature tensor of the Levi-Civita connection of the metric on this side, and on this side, I'm going to write down this differential operator: the adjoint of the exterior derivative coupled to a connection. | <p>[00:50:23] That should be an "LC" for Levi-Civita. So, the Einstein projection ($$P_{E}$$$) of the curvature tensor of the Levi-Civita connection ($$F_{\Delta_{LC}}$$) of the metric on this side, and on this side, I'm going to write down this differential operator: the adjoint of the exterior derivative coupled to a connection. | ||
<p>[00:50:47] And you begin to see that we're missing an opportunity, potentially. What if the $$F_A$$s were the same in both contexts? Then you're applying two separate operators: 1) zeroth-order and destructive, in the sense that it doesn't see the entire curvature tensor; the other) inclusive, but of first-order. And so the question is, is there any opportunity to do anything that combines these two? | <p>[00:50:47] And you begin to see that we're missing an opportunity, potentially. What if the $$F_A$$s were the same in both contexts? Then you're applying two separate operators: 1) zeroth-order and destructive, in the sense that it doesn't see the entire curvature tensor; the other) inclusive, but of first-order. And so the question is, is there any opportunity to do anything that combines these two? | ||