A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions

<p>[01:06:19] So with that, let us begin to think about what we mean today by Geometric Unity. GU comes in four flavors, but I'm only getting one shot to do this, so I'm going to do the most exciting of them to me. There's a completely exogenous flavor.

<p>[01:07:15] The idea of an observerse is a bit like a stadium. You have a playing field and you have stands. They aren't distinct entities, they're coupled. And so fundamentally we're going to replace one space with two. Exogenous model simply means that U is unrestricted, although larger than $$X^4$$ so any manifold of four dimension higher that is capable of admitting $$X^4$$ as an immersion.

<p>[01:07:41] The next model we have is the bundle theoretic, in which case U sits over X as a fiber bundle.

<p>[01:07:58] The most exciting, which is the one we'll deal with today, is the endogenous model where $$X^4$$ actually grows the space U where the activity takes place. So we talked about extra dimensions, but these are in some sense not extra dimensions. They are implicit dimensions within $$X^4$$. And last, to proceed without loss of generality we have the tautological model. In that case, $$X^4$$ equals U. And the immersion is the identity. And without loss of generality, we simply play our games on one space. Okay? Now we need rules. The rules are, sorry. Okay. Feedback. Um. No choice of fundamental metric. So we imagine that Einstein was presented with a fork in the road, and it's always disturbing not to follow Einstein's path, but we're in fact going to turn Einstein's game on its head and see if we can get anywhere with that. Right. It's also a possibility that because Einstein's theory is so perfect that if there's anything wrong with it, it's very hard to unthink it because everything is built on top of it. So let's make no choice of fundamental metric.

<p>[01:09:56] And lastly, we want to liberate matter from its dependence on the metric for its very existence. So we now need to build [[fermions]] onto our four dimensional manifold in some way without ever choosing a metric, if we're even going to have any hope of playing a game involving matter, starting from this perspective of no information other than the most bare information.

<p>[01:10:58] At a point wise tensorial level.

<p>[01:11:03] Is there a metric on $$U^14$$? Well, we both want one and don't want one. If we had a metric from the space of all metrics, we could define fermions, but we would also lock out any ability to do dynamics. We want some choice over what this metric is, but we don't want full choice because we want enough to be able to define the matter fields to begin with.

<p>[01:11:22] It turns out. That if this is $$X^4$$ and this is this particular endogenous choice of $$U^14$$ we have a 10 dimensional metric along the fibers. So we have a $$G^{10}_{\mu nu}$$. Further for every point in the fibers, we get a metric downstairs on the base space. So if we pull back the cotangent bundle, we get a metric $$G^4_{\mu \nu}$$ on $$\Pi^*$$ of the cotangent bundle of X. We now define the chimeric bundle, right? And the chimeric bundle is this direct sum of the vertical tangent bundle along the fibers with the pullback, which we're going to call the horizontal bundle from the base space. So the chimeric bundle is going to be the vertical tangent space of 10 dimensions to U direct sum, the four dimensional cotangent space which we're going to call horizontal to U. And the great thing about the chimeric bundle is, is that it has an a priori metric. It's got a metric on the four a metric on the 10 and we can always decide that the two of them are naturally perpendicular to each other.

<p>[01:13:00] Furthermore, it is almost canonically isomorphic to the tangent bundle or the cotangent bundle, because we either have 4 out of 14 or 10 out of 14 dimensions on the nose. So the question is, what are we missing. And the answer is that we're missing exactly the data of a connection. So this bundle, chimeric, C. We have C is equal to the tangent bundle of U up to a choice of a connection theta.

<p>[01:13:32] And this is exactly what we wanted. We have a situation where we have some field on the manifold X in the form of a connection which is amenable, more friendly to quantization, which is now determining a metric turning around the Levi-Civita game. And the only problem is, is that we've had to buy ourselves into a different space than the one we thought we wanted to work on.

<p>[01:13:55] But now as theta changes, the fermions are defined on the chimeric bundle, and it's the isomorphism from the chimeric bundle to the tangent bundle of the space U, which is variant, which means that the fermions no longer depend on the metric. They no longer depend on the theta connection. They are there if things go quantum mechanical, and we've achieved our objective of putting the matter fields and the spin one fields on something of the same footing.

<p>[01:14:27] But, and I want to emphasize this: One thing, most of us, we think a lot about final theories and and about unification, but until you actually start daring to try to do it, you don't realize what the process of it feels like. And I try to imagine conducting your life where you have no children, let's say, and no philanthropic urges.

<p>[01:14:54] And what you want to do is you want to use all of your money for yourself. And die penniless, right? Like a perfect finish. Assuming that that's what you wanted to do, it would be pretty nerve wracking at the end, right? How many days left do I have? How many dollars left do I have? This is the process of unification.

<p>[01:15:12] In physics, you start giving away all of your most valuable possessions, and you don't know whether you've given them away too early, whether you husband them too long. And so in this process, what we've just done is we've started to paint ourselves into a corner. And we got something we wanted, but we've given away freedom.

<p>[01:15:39] Well, let me just sum this up by saying: between fundamental and emergent, standard model and GR. Let's do GR. Fundamental is the metric. Emergent is the connection. Here in GU, it is the connection that's fundamental and the metric that's emergent.