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

===== Introduction to GU =====

<p>[00:59:12] So what I'd like to do is I'd like to talk a little bit about what the Geometric Unity proposal is.

<p>[00:59:44] Between physics and mathematics. More specifically, geometry. And intrinsic physical theory would be general relativity. An auxiliary physical theory would be the Yang-Mills theory with the freedom to choose internal quantum numbers. At the mathematical level an intrinsic theory would be, let's be a little fastidious.

<p>[01:00:20] The older semi-Riemannian geometry. The study of manifolds with length and angle. But auxiliary geometry is really what's taken off of late since the revolution partially begun at Oxford when Iz singer brought insights from Stony Brook to the U.K. And so we're going to call this fiber bundle theory or modern gauge theory. Geometric Unity is the search for some way to break down the walls between these four boxes. What's natural to one theory is unnatural to another. Semi-Riemannian geometry is dominated by these projection operators as well as the ability, uh, to use the Levi-Civita connection.

<p>[01:01:08] Now, some aspects of this are less explored. Torsion tensors are definable in Semi-Riemannian geometry, but they are not used to the extent that you might imagine. In the case of fiber bundle theory, the discovery of physicists that the gauge group was fantastically important. Um. Came as something of a shock to the mathematicians who had missed that structure, uh, and have since exploited it to great effect.

<p>[01:01:34] So what we'd like to do is we'd like to come up with some theory that is intrinsic, but allows us to play some of the games that exist in other boxes. How can we fit? How can we try to have our cake eat it, uh, and use all the full suite of techniques that are available to us?

<p>[01:01:54] So our perspective is, is that it is the quantum that may be the comparatively easy part, and that the unification of the geometry, which has not occurred, may be what we're being asked to do. So let's try to figure out what would a final theory even look like?

<p>[01:02:57] Let us imagine that we cannot figure out the puzzle of why is there something rather than nothing. But if we do have a something that that's something has his little structure as possible, but it still invites us to work mathematically. So to me, the two great theories that we have mathematically and in physics are calculus and linear algebra.

<p>[01:04:13] So I have in mind differential operators parameterized by some fields, omega, which when composed are not of second order, if these are first order operators, but as zeroth order in some sort of further differential operator saying that whatever those two operators are in composition is in some sense harmonic.

<p>[01:04:41] Is such a program even possible? So if the universe is in fact capable of being the fire that lights itself, is it capable of managing its own flame as well, and closing up? What I'd like to do is to set ourselves an almost impossible task, which is to begin with this little data in a sandbox, to use a computer science concept.

<p>[01:05:11] So if here this physical reality.

<p>[01:05:20] Standard physics is over here. We're going to start with the sandbox and all we're going to put in it is $$X^4$$. And we're going to set ourselves a straight jacketed task of seeing how close we can come to dragging out a model that looks like the natural world that follows this trajectory.

<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.