A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions
| Line 407: | Line 407: | ||
===== Part II: Unified Field Content ===== | ===== Part II: Unified Field Content ===== | ||
=== Magic Beans trade === | ===== Magic Beans trade ===== | ||
<p>[01:16:36] Okay. The next unit of GU is the unified field content. What does it mean for our fields to become unified? There are, in fact, only at this moment, two fields that know about $$X$$. $$\theta$$, which is the connection that we've just talked about, and a section, $$\sigma$$ that takes us back so that we can communicate back and forth between $$U$$ and $$X$$. We now need field content that only knows about $$U$$, which now has a metric depending on $$\theta$$. | <p>[01:16:36] Okay. The next unit of GU is the unified field content. What does it mean for our fields to become unified? There are, in fact, only at this moment, two fields that know about $$X$$. $$\theta$$, which is the connection that we've just talked about, and a section, $$\sigma$$ that takes us back so that we can communicate back and forth between $$U$$ and $$X$$. We now need field content that only knows about $$U$$, which now has a metric depending on $$\theta$$. | ||
| Line 428: | Line 428: | ||
<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. | ||
===== The Levi-Civita Connection ===== | |||
<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>[01:20:01] 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. | ||