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 17:34, 25 April 2020
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<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 $$(\Omega^{1}(Ad))$$ as a vector space. The gauge group represents an ad-valued one-forms. So, if we also have the gauge group ($$\mathcal{H}$$), but 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 $$(\Omega^{1}(Ad))$$ as a vector space. The gauge group represents an ad-valued one-forms. So, if we also have the gauge group ($$\mathcal{H}$$), but we think of that instead as a space of sigma fields. | ||
<p>[01:26:16] What if we take the semi-direct product ($$\ltimes$$) at a group theoretic level between the two and call this our group of | <p>[01:26:16] What if we take the semi-direct product ($$\ltimes$$) at a group theoretic level between the two and call this our group of interest. 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 | <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 the ad-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 Poincaré group, or rather its double cover to allow spin. | ||
<p>[01:27:12] So we're going to call this the inhomogeneous gauge group or | <p>[01:27:12] So we're going to call this the inhomogeneous gauge group, or iggy. | ||
===== Properties of the Inhomogeneous Gauge Group (IGG) ===== | |||
<p>[01:27:23] And this is going to be a really interesting space because it has a couple of properties. One is it has a very interesting subgroup. Now, of course, $$\mathcal{H}$$ includes into $$\mathcal{G}$$ by just including onto the first factor, but in fact, there's a more interesting homomorphism brought to you by the Levi-Civita connection. | <p>[01:27:23] And this is going to be a really interesting space because it has a couple of properties. One is it has a very interesting subgroup. Now, of course, $$\mathcal{H}$$ includes into $$\mathcal{G}$$ by just including onto the first factor, but in fact, there's a more interesting homomorphism brought to you by the Levi-Civita connection. | ||