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 23:06, 11 April 2020
, 11 April 2020→Part II: Unified Field Content
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<p>[01:18:31] In the auxiliary theory, we have freedom to choose our field content. And we have the ability to get rid of much excess through the symmetries of the gauge group. | <p>[01:18:31] In the auxiliary theory, we have freedom to choose our field content. And we have the ability to get rid of much excess through the symmetries of the gauge group. | ||
<p>[01:18:50] We're going to take particle theory, we're going to make a bad trade, or what appears to be a bad trade, which is that we are going to give away the freedom to choose our field content, which is already extremely, as I think I said in the abstract, | <p>[01:18:50] We're going to take particle theory, we're going to make a bad trade, or what appears to be a bad trade, which is that we are going to give away the freedom to choose our field content, which is already extremely, as I think I said in the abstract, baroque with all of the different particle properties. | ||
<p>[01:19:09] And we're going to lose the ability to use the gauge group because we're going to trade it all. | <p>[01:19:09] And we're going to lose the ability to use the gauge group because we're going to trade it all. | ||
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<p>[01:19:40] So it's now time to trade the family cow for the magic beans and bring them home and see whether or not we got the better of the deal. | <p>[01:19:40] So it's now time to trade the family cow for the magic beans and bring them home and see whether or not we got the better of the deal. | ||
<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 | <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. But, in fact, if we let the gauge group act, there's going to be an affine shift. | <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. | ||
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<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. Nonlinear. | ||
<p>[01:23:25] There's no reason that we can't choose this as field content. Again, we're being led by the nose like a bull. If we want to make use of the symmetries of the theory, we have to promote some symmetry to being part of the theory, and we have to let it be subjected to dynamical laws. We're going to lose control over it. | <p>[01:23:25] There's no reason that we can't choose this as field content. Again, we're being led by the nose like a bull. If we want to make use of the symmetries of the theory, we have to promote some symmetry to being part of the theory, and we have to let it be subjected to dynamical laws. We're going to lose control over it. | ||
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<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 mass does is the Ricci curvature. | <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 mass does 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 | <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. You try to push it from one space, like add-value two-forms into another space, like add-value one-forms where connections live. | ||
<p>[01:24:54] That's not a good idea. Instead, what we do is the following | <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 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. | <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. | ||
<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 add | <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 add-value one-forms as a vector space. The gauge group represents an add-valued one-forms. So, if we also have the gauge group, what we think of that instead as a space of sigma fields. | ||
<p>[01:26:16] What if we take the semi direct product at a group theoretic level between the two and call this our group of interests. Well by analogy, we've always had a problem with the | <p>[01:26:16] What if we take the semi-direct product at a group theoretic level between the two and call this our group of interests. Well by analogy, we've always had a problem with the Poincaré group being too intrinsically tied to rigid flat Minkowski space. What if we wanted to do quantum field theory in some situation, which was more amenable to a curved space situation? | ||
<p>[01:26:44] It's possible that we should be basing it around something more akin to the gauge group, and in this case, we're mimicking the construction where Xi here would be analogous to the Lorentz group, fixing a point in Mankowski space, and add valued one forms would be analogous to the four momentums. We take in the semi direct product to create the inhomogeneous Lorentz group, otherwise known as the Poincare group, or rather, it's a double cover to allow spin. | <p>[01:26:44] It's possible that we should be basing it around something more akin to the gauge group, and in this case, we're mimicking the construction where Xi here would be analogous to the Lorentz group, fixing a point in Mankowski space, and add valued one forms would be analogous to the four momentums. We take in the semi direct product to create the inhomogeneous Lorentz group, otherwise known as the Poincare group, or rather, it's a double cover to allow spin. | ||