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 17:26, 25 April 2020
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<p>[01:23:44] But we're not dead yet, right? We're fighting for our life to make sure that this trade has some hope. So potentially by including symmetries as field content, we will have some opportunity to make use of the projections. So for those of you who... | <p>[01:23:44] But we're not dead yet, right? We're fighting for our life to make sure that this trade has some hope. So potentially by including symmetries as field content, we will have some opportunity to make use of the projections. So for those of you who... | ||
<p>[01:24:06] So when I was thinking about this, I used to be amazed by ships in bottles. I must confess that I never figured out what the trick was for ships in bottles, but once I saw it, I remembered thinking, that's really clever. So if you've never seen it, you have a ship, which is like a curvature tensor. And imagine that the | ===== Ship in a Bottle (Shiab) Operator ===== | ||
<p>[01:24:06] So when I was thinking about this, I used to be amazed by ships in bottles. I must confess that I never figured out what the trick was for ships in bottles, but once I saw it, I remembered thinking, that's really clever. So if you've never seen it, you have a ship, which is like a curvature tensor. And imagine that the mast is the Ricci curvature. | |||
<p>[01:24:30] If you just try to shove it into the bottle, you're undoubtedly going to snap the mast. So you imagine that you've transformed your gauge fields, you've kept track of where the Ricci curvature was | <p>[01:24:30] If you just try to shove it into the bottle, you're undoubtedly going to snap the mast. So, you imagine that you've transformed your gauge fields, you've kept track of where the Ricci curvature was, uou try to push it from one space, like ad-valued two-forms into another space like ad-valued one-forms, where connections live. | ||
<p>[01:24:54] That's not a good idea. Instead, what we do is the following: imagine that you're carrying around group theoretic information and what you do is you do a transformation based on the group theory. So you lower the mast. You push it through the neck, having some string attached to the mast, and then you undo the transformation on the other side. | <p>[01:24:54] That's not a good idea. Instead, what we do is the following: imagine that you're carrying around group theoretic information and what you do is you do a transformation based on the group theory. So you lower the mast. You push it through the neck, having some string attached to the mast, and then you undo the transformation on the other side. | ||
<p>[01:25:18] This is exactly what we're going to hope is going to save us in this bad trade that we've made because we're going to add fields, content that has the ability to lower the mast | <p>[01:25:18] This is exactly what we're going to hope is going to save us in this bad trade that we've made because we're going to add fields, content that has the ability to lower the mast and bring the mast back up. We're going to hope to have a theory which is going to create a communitive situation. But then once we've had this idea, we start to get a little bolder. | ||
===== Unified Content ===== | |||
<p>[01:25:43] Let's think about unified content. We know that we want a space of connections, $$A$$ for our field theory, but we know because we have a Levi-Civita connection, that this is going to be equal on-the-nose to ad-valued one-forms as a vector space. The gauge group represents an ad-valued one-forms. So, if we also have the gauge group, what we think of that instead as a space of sigma fields. | <p>[01:25:43] Let's think about unified content. We know that we want a space of connections, $$A$$ for our field theory, but we know because we have a Levi-Civita connection, that this is going to be equal on-the-nose to ad-valued one-forms as a vector space. The gauge group represents an ad-valued one-forms. So, if we also have the gauge group, what we think of that instead as a space of sigma fields. | ||