Difference between revisions of "A Portal Special Presentation- Geometric Unity: A First Look"
A Portal Special Presentation- Geometric Unity: A First Look (view source)
Revision as of 03:11, 14 April 2020
, 03:11, 14 April 2020→Supplementary Explainer Presentation
| Line 749: | Line 749: | ||
<p>[02:27:26] I hope I got that one right. And then we have an action of G that is the inhomogeneous gauge group on the space of connections, because we have two different ways to act on connections. We can either act by gauge trans transformations, or we can act by affine translations. So putting them together gives us something for the inhomogeneous gauge group too. | <p>[02:27:26] I hope I got that one right. And then we have an action of G that is the inhomogeneous gauge group on the space of connections, because we have two different ways to act on connections. We can either act by gauge trans transformations, or we can act by affine translations. So putting them together gives us something for the inhomogeneous gauge group too. | ||
<p>[02:27:46] We then get a bi connection. In other words, because we have two different ways of pushing a connection around. If we have a choice of a base connection, we can push the base connection around in two different ways according to the portion of it that is | <p>[02:27:46] We then get a bi connection. In other words, because we have two different ways of pushing a connection around. If we have a choice of a base connection, we can push the base connection around in two different ways according to the portion of it that is coming from the gauge transformations curly H or the affine translations coming from curly N. | ||
<p>[02:28:10] Yeah. We can call this map the bi connection, which gives us two separate connections for any point in the | <p>[02:28:10] Yeah. We can call this map the bi-connection, which gives us two separate connections for any point in the inhomogeneous gauge group. And we can notice that it can be viewed as a section of a bundle over the base space to come. We find an interesting embedding of the gauge group that is not the standard one in the inhomogeneous gauge group. | ||
<p>[02:28:42] So our summary diagram looks something like this. Take a look at the Taus of $$A_0$$. We will find a homeomorphism of the gauge group into its inhomogeneous extension that isn't simply inclusion under the first factor. And I'm realizing that I have the wrong pi production. That should just be as simple $$\pi$$, projecting down, we have a map from the inhomogeneous gauge group, via the bi-connection to A cross A connections cross connections, and that that behaves well according to the difference operator $$\delta$$ that takes the difference of two connections and gives an honest add-value one-form. | <p>[02:28:42] So our summary diagram looks something like this. Take a look at the Taus of $$A_0$$. We will find a homeomorphism of the gauge group into its inhomogeneous extension that isn't simply inclusion under the first factor. And I'm realizing that I have the wrong pi production. That should just be as simple $$\pi$$, projecting down, we have a map from the inhomogeneous gauge group, via the bi-connection to A cross A connections cross connections, and that that behaves well according to the difference operator $$\delta$$ that takes the difference of two connections and gives an honest add-value one-form. | ||
| Line 765: | Line 765: | ||
<p>[02:31:17] I hope I remember the terminology right. It's been a long time. The map is injective because it's injective under the first factor, but it actually gives us a nontrivial embedding of the gauge group in its inhomogeneous extension, which makes the whole theory work. We then get to Shiab operators now, a Shiab operator a map from the group crossed the add-valued i forms. | <p>[02:31:17] I hope I remember the terminology right. It's been a long time. The map is injective because it's injective under the first factor, but it actually gives us a nontrivial embedding of the gauge group in its inhomogeneous extension, which makes the whole theory work. We then get to Shiab operators now, a Shiab operator a map from the group crossed the add-valued i forms. | ||
<p>[02:31:43] In this case, the particular Shiab operators we’re interested in is mapping i form is to d minus three, plus i forms. So, for example | <p>[02:31:43] In this case, the particular Shiab operators we’re interested in is mapping i form is to d minus three, plus i forms. So, for example, you would map a two form to a d minus three plus i. So if d, for example, were 14, ..., and i was equal to two. Then 14 minus three is equal to 11 plus two is equal to 13. So that would be an add-valued 14 minus one form, which is exactly the right place for something to form a current. That is the differential of a Lagrangian on the space. | ||
<p>[02:32:38] Now the augmented torsion, the torsion is a very strange object. It's introduced sort of right at the beginning of learning a differential geometry, but it really doesn't get used very much. One of the reasons it doesn't get used, gauge theory is that it's not gauge invariant. It has a gauge and variant piece to it, but then a piece that spoils the gauge invariant. | <p>[02:32:38] Now the augmented torsion, the torsion is a very strange object. It's introduced sort of right at the beginning of learning a differential geometry, but it really doesn't get used very much. One of the reasons it doesn't get used, gauge theory is that it's not gauge invariant. It has a gauge and variant piece to it, but then a piece that spoils the gauge invariant. | ||
| Line 801: | Line 801: | ||
<p>[02:38:53] Now recalling that. Um, when I started my career, we did not know that neutrinos were massive. And I figured that they probably had to be massive because I desperately wanted a 16 dimensional space of internal quantum numbers, not 15, because my, my ideas only work if the space of internal quantum numbers as a dimension to to the n. | <p>[02:38:53] Now recalling that. Um, when I started my career, we did not know that neutrinos were massive. And I figured that they probably had to be massive because I desperately wanted a 16 dimensional space of internal quantum numbers, not 15, because my, my ideas only work if the space of internal quantum numbers as a dimension to to the n. | ||
<p>[02:39:14] And | <p>[02:39:14] And, one of my favorite equations at the time was 15 equals two to the four. Not literally true, but almost true. And thankfully in the late 1990s, the case for 16 particles in a generation was strengthened when neutrinos were found to have mass. But that remaining term in the southeast corner, the spinors on $$X$$ tensor spinors on $$Y$$ looks like the term above it in line 2.15. | ||
<p>[02:39:46] And that, in fact, is the third generation of matter, in my opinion. That is, it is not a true generation. It is broken off and would unify very differently if we were able to heat the universe to the proper temperature. So, starting to sum up, this is not the full theory. I'm just presenting this in part to dip my toe back into the water. | <p>[02:39:46] And that, in fact, is the third generation of matter, in my opinion. That is, it is not a true generation. It is broken off and would unify very differently if we were able to heat the universe to the proper temperature. So, starting to sum up, this is not the full theory. I'm just presenting this in part to dip my toe back into the water. | ||