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

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 =====Motivations for Geometric Unity=====
+[[File:GU Presentation Intrinsic-Auxiliary Diagram.png|thumb|right]]
-[00:59:12] So what I'd like to do is I'd like to talk a little bit about what the Geometric Unity (GU) proposal is.
+''[https://youtu.be/Z7rd04KzLcg?t=3552 00:59:12]''<br>
+So what I'd like to do is I'd like to talk a little bit about what the Geometric Unity (GU) proposal is. So we have a division into intrinsic theories and auxiliary theories, between physics and mathematics, more specifically geometry. An 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—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 Is Singer brought insights from Stony Brook to the UK. And so we're going to call this fiber bundle theory, or modern gauge theory.
-[00:59:34] So we have a division into intrinsic theories and auxiliary theory. 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.
+''[https://youtu.be/Z7rd04KzLcg?t=3648 01:00:48]''<br>
+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 to use the Levi-Civita connection. 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 came as something of a shock to the mathematicians who had missed that structure, and have since exploited it to great effect. 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 try to have our cake, eat it, and use the full suite of techniques that are available to us? So, our perspective 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?
-[01:00:20] At the mathematical level, an intrinsic theory would be, let's be a little fastidious: 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 Is Singer brought insights from Stony Brook to the U.K.
+''[https://youtu.be/Z7rd04KzLcg?t=3732 01:02:12]''<br>
+When I was a bit younger, I remember reading this question of Einstein, in which he said I'm not really interested in some of the details of physics. What really concerns me is whether the creator had any choice in how the world was constructed. And some people may have read that as a philosophical statement, but I took that as an actual call for a research program. So I'd like to describe that research program and try to unpack what I think he was saying.
-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 to use the Levi-Civita connection.
+''[https://youtu.be/Z7rd04KzLcg?t=3766 01:02:46]''<br>
+We talk a lot about unification, but we hardly ever actually imagine if we had a unified theory, what would it look like? 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 something has as 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. If we have calculus and linear algebra, I'm going to imagine that we have some manifold, at least one of dimension four—but it's not a spacetime. It doesn't have a metric. It's not broken down into two different kinds of coordinates, which then have some bleed into each other, but still maintains a distinction. It's just some sort of flabby proto-spacetime, and in the end it has got to fill up with stuff and give us some kind of an equation. So let me write an equation.
-[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. [This] came as something of a shock to the mathematicians who had missed that structure, and have since exploited it to great effect.
-[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. And use all the full suite of techniques that are available to us?
+<div style="text-align: center; margin-left: auto; margin-right: auto;">$$\delta^\omega(\mathscr{D}^\omega_2 \mathscr{D}^\omega_1) = 0$$</div>
-[01:01:54] So our perspective is that 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?
-====== A research program for a Unified Theory ======
+''[https://youtu.be/Z7rd04KzLcg?t=3853 01:04:13]''<br>
+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 of zeroth-order, and some sort of further differential operator saying that whatever those two operators are in composition is in some sense harmonic. 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.
-[01:02:12] When I was a bit younger, I remember reading this question of Einstein, which he said, I'm not really interested in some of the details of physics. What really concerns me is whether the creator had any choice in how the world was constructed. And some people may have read that as a philosophical statement, but I took that as an actual call for a research program.
+[[File:GU Presentation Sandbox Diagram.png|thumb|right]]
-[01:02:35] So I'd like to describe that research program and try to unpack what I think he was saying.
+''[https://youtu.be/Z7rd04KzLcg?t=3911 01:05:11]''<br>
+So, if here is physical reality, 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 strait-jacketed task of seeing how close we can come to dragging out a model that looks like the natural world that follows this trajectory. While it may appear that that is not a particularly smart thing to do, I would like to think that we could agree that it is quite possible that if that were to be the case, we might say that this is what Einstein meant by a creator, which was his anthropomorphic concept for necessity and elegance and design, having no choice in the making of the world.
-[01:02:47] We talked a lot about unification, but we hardly ever actually imagine if we had a unified theory, what would it look like?
-[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 that has as 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.
-[01:03:17] If we have calculus and linear algebra, I mean, imagine that we have some manifold, at least one of dimension four, but it's not a spacetime. It doesn't have a metric. It's not broken down into two different kinds of coordinates, which then have some bleed into each other, but still maintains a distinction.
-[01:03:36] It's just some sort of flabby proto-spacetime, and in the end it has got to fill up with stuff and give us some kind of an equation. So let me write an equation.
-[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.
-[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.
-[01:05:11] So, if here this physical reality.
-[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.
-[01:05:56] While it may appear that that is not a particularly smart thing to do, I would like to think that we could agree that it is quite possible. That if that were to be the case, we might say that this is what Einstein meant by a creator, which was his anthropomorphic concept for necessity and elegance and design having no choice in the making of the world.
 ====== Four flavors of GU with a focus on the endogenous version ======