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

So, with that as a beginning, I'm just going to say one disclaimer, which is that this is not a usual talk. And whatever contract a speaker usually has with the audience, right now we're going to break that contract. This is a talk about ideas, and some of these ideas are bold, some of them may offend some people because there's a sense that you don't have a right to be considering those ideas, but I go back to the admonition of Jim Watson that said if you're going to try to make progress, big progress, you are by definition unqualified to be doing whatever it is that you're doing. So, in that spirit, let us begin.

''[https://youtu.be/Z7rd04KzLcg?t=2507 00:41:47]''<br>

What is physics to physicists today? How do they see it different from the way in which we might imagine the lay person sees physics? Ed Witten was asked this question in a talk he gave on physics and geometry many years ago, and he pointed us to three fundamental insights, which were his big three insights in physics. And they correspond to the three great equations.

There are other possibilities that while each of these may be simplest in its category, they are not simplest in their interaction. For example, we know that Dirac famously took the square root of the Klein-Gordon equation to achieve the Dirac equation—he actually took two square roots, one of the differential operator, and another of the algebra on which it acts. But could we not do the same thing by re-interpreting what we saw in Donaldson theory and Chern-Simons theory, and finding that there are first-order equations that imply second-order equations that are nonlinear in the curvature? So let's imagine the following: we replace the standard model with a true second-order theory. We imagine that general relativity is replaced by a true first-order theory. And then we find that the true second-order theory admits of a square root and can be linked with the true first-order theory. This would be a program for some kind of unification of Dirac's type, but in the force sector. The question is, "Does this really make any sense? Are there any possibilities to do any such thing?"

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And last, to proceed without loss of generality, we have the tautological model. In that case, \(X^4 = U\), and the immersion is the identity, and without loss of generality, we simply play our games on one space. Okay?

''[https://youtu.be/Z7rd04KzLcg?t=4116 01:08:36]''<br>

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. Let's get started.

''[https://youtu.be/Z7rd04KzLcg?t=4224 01:10:24]''<br>

We take \(X^4\). We need metrics, we have none. We're not allowed to choose one, so we do the standard trick: we choose them all.

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 \(X\).

''[https://youtu.be/Z7rd04KzLcg?t=4337 01:12:17]''<br>

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 cotangent space, which we're going to call horizontal, to \(U\). And the great thing about the chimeric bundle is that it has an a priori metric. It's got a metric on the 4, a metric on the 10, and we can always decide that the two of them are naturally perpendicular to each other.

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.

''[https://youtu.be/Z7rd04KzLcg?t=4467 01:14:27]''<br>

But, and I want to emphasize this: one thing most of us—we think a lot about final theories 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, try to imagine conducting your life where you have no children, let's say, and no philanthropic urges, 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 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 husbanded them too long, and so in this process, what we've just done is we've started to paint ourselves into a corner. We got something we wanted, but we've given away freedom. We're now dealing with a 14-dimensional world.