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

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 ====== The Current Picture of Physics ======
-[00:41:47] 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? [[Edward Witten|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.
+''[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.
-[00:42:13] And they correspond to the three great equations. So the first one is, is that somehow physics takes place in an arena and that arena is a [[manifold]] $$X$$ together with some kind of [[semi-Riemannian]] [[metric structure]], something that allows us to take length and angle. So that we can perform measurements at every point in this space-time or higher-dimensional structure, leaving us a little bit of head room. The equation most associated with this is the [[Einstein field equation]].
+<blockquote style="background: #f3f3ff; border-color: #ddd;">If one wants to summarize our knowledge of physics in the briefest possible terms, there are three really fundamental observations:  (i)  Space-time is a pseudo-Riemannian manifold \(M\), endowed with a metric tensor and governed by geometrical laws.  (ii)  Over \(M\) is a vector bundle \(X\) with a nonabelian gauge group \(G\).  (iii)  Fermions are sections of \((\hat{S}{+} \otimes V{R}) \oplus (\hat{S}{-} \otimes V{\tilde{R}})\).  \(R\) and \(\tilde{R}\) not isomorphic; their failure to be isomorphic explains why the light fermions are light and presumably has its origins in a representation difference \(\Delta\) in some underlying theory.  All of this must be supplemented with the understanding that the geometrical laws obeyed by the metric tensor, the gauge fields, and the fermions are to interpreted in quantum mechanical terms.<br>
+<br>
+Edward Witten, ''[https://cds.cern.ch/record/181783/files/cer-000093203.pdf Physics and Geometry]''</blockquote>
-[00:43:12] And, of course, I'm running into the margin. Okay.
+''[https://youtu.be/Z7rd04KzLcg?t=2536 00:42:16]''<br>
+So the first one is that somehow, physics takes place in an arena, and that arena is a manifold \(X\), together with some kind of semi-Riemannian metric structure: something that allows us to take length and angle, so that we can perform measurements at every point in this spacetime or higher-dimensional structure, leaving us a little bit of head room. The equation most associated with this is the Einstein field equation. And of course, I've run into the margin. So it says that a piece of the Riemann curvature tensor, the Ricci tensor minus an even smaller piece, the scalar curvature multiplied by the metric, is equal—plus the cosmological constant—is equal to some amount of matter and energy, the stress-energy tensor. So it's intrinsically a curvature equation.
-[00:43:18] So, it says that a piece of the [[Riemann curvature tensor]], or the Ricci tensor minus an even smaller piece, the scalar curvature multiplied by the metric is equal, plus the cosmological constant, is equal to some amount of matter and energy, the stress energy tensor. So it's intrinsically a curvature equation.
-[00:43:47] The second fundamental insight... I'm going to begin to start drawing pictures here as well.
+$$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = 8 \pi T_{\mu\nu}$$
-[00:43:55] So, if this is the space-time manifold, "the arena"; the second one concerns symmetry groups which cannot necessarily be deduced from any structure inside of "the arena". They are additional data that come to us out of the blue without explanation and these symmetries from a non-Abelian group, which is currently SU(3) "color" x SU(2) "weak" x U(1) "weak hypercharge", which breaks down to SU(3)xU(1), where the broken U(1) is the electromagnetic symmetry.
-[00:44:54] This equation is also a curvature equation. The corresponding equation, the curvature of an auxiliary structure known as a gauge potential when differentiated in a particular way is equal, again, to the amount of stuff in the system that is not directly involved in the left-hand side of the equation. So, it has many similarities to the above equation. Both involve curvature. One involves a projection or a series of projections. The other involves a differential operator.
+''[https://youtu.be/Z7rd04KzLcg?t=2627 00:43:47]''<br>
+The second fundamental insight—I'm going to begin to start drawing pictures here as well. So, if this is the spacetime manifold, the arena; the second one concerns symmetry groups which cannot, necessarily, be deduced from any structure inside of the arena. They are additional data that come to us out of the blue without explanation. And these symmetries from a non-Abelian group, which is currently:
-[00:45:44] The third point surrounds the matter in the system.
-[00:45:58] And here we have a Dirac equation. Again, coupled to a connection.
+$$ \text{SU}(3) \text{ (color)} \times \text{SU}(2) \text{ (weak isospin)} \times \text{U}(1) \text{ (weak hypercharge)}$$
-[00:46:12] One of the great insights is that the reason for the lightness of matter in the natural mass scale of physics has to do with the fact that this $$\psi$$ really should have two components and the differential operators should map to one component on the other side of the equation, but the mass operators should map to another.
-[00:46:33] And so if one of the components is missing, if the equation is intrinsically lopsided, chiral, asymmetric; then the mass term and the differential term have difficulty interacting, which is sort of overcompensating for the mass scale of the universe, so you get to a point where you actually have to define a massless equation. But then, just like overshooting a putt, it's easier to knock it back by putting in a [[Higgs field]] in order to generate an "as-if" fundamental mass through the [[Yukawa couplings]].
+Which breaks down to \(\text{SU}(3) \times \text{U}(1)\), where the broken \(\text{U}(1)\) is the electromagnetic symmetry. This equation is also a curvature equation—the corresponding equation—and it says that this time, the curvature of an auxiliary structure known as a gauge potential, when differentiated in a particular way is equal, again, to the amount of stuff in the system that is not directly involved in the left-hand side of the equation. So it has many similarities to the above equation, both involve curvature. One involves a projection, or a series of projections. The other involves a differential operator.
-[00:47:15] Let me, for consistency, say "matter is asymmetric", okay. "and, therefore [also], light".
-[00:47:35] And then interestingly, he went on to say one more thing. He said, of course, these three central observations must be supplemented with the idea that this all [be] treated in quantum mechanical fashion or quantum field theoretic [fashion]. So it's a bit of an after-market modification rather than, in his opinion at the time, one of the core insights.
+$$d^*_A F_A = J(\psi)$$
-[00:48:07] I actually think that that's in some sense about right. No. One of my differences with the [modern-day physics] community -- in some sense -- is I question whether the quantum isn't in good enough shape. We don't know whether we have a serious quantum mechanical problem or not. We know that we have a quantum mechanical problem, a quantum field theoretic problem, [but only] relative to the current formulations of these theories.
-[00:48:31] But we know that in some other cases, the quantum becomes incredibly natural, sometimes sort of almost magically natural, and we don't know whether the true theories that we will need to be generalizing, in some sense, have beautiful quantum mechanical treatments. Whereas the effective theories that we're dealing with now may not survive the quantization.
+''[https://youtu.be/Z7rd04KzLcg?t=2744 00:45:44]''<br>
+The third point surrounds the matter in the system. And here we have a Dirac equation, again coupled to a connection. But one of the great insights is that the reason for the lightness of matter in the natural mass scale of physics has to do with the fact that this \(\psi\) really should have two components, and the differential operator should map to one component on the other side of the equation, but the mass operator should map to another. And so if one of the components is missing, if the equation is intrinsically lopsided—chiral, asymmetric—then the mass term and the differential term have difficulty interacting, which is sort of overcompensating for the mass scale of the universe, so you get to a point where you actually have to define a massless equation. But then, just like overshooting a putt, it's easier to knock it back by putting in a Higgs field in order to generate an as-if fundamental mass through the Yukawa couplings. So matter is asymmetric, and therefore light.
-====== Connecting the Three Observations of Witten ======
-[00:48:54] So what I want to do is I want to imagine a different sort of incompatibilities. So let's take our great three theories and just visually treat them as the vertices of a triangle.
-[00:49:14] So I'm going to put general relativity and Einstein's formulation. And, I'm going to put, the probably won't write this again. Yang-Mills, Maxwell, [Philip Warren] Anderson, Higgs theory, over here, and I'm going to write the Dirac theory.
+$$ \unicode{x2215}\kern-0.7em D_A \psi = m \psi $$
-[00:49:39] What I want to explore is the incompatibilities, not at the quantum level. But the geometric input. All three of these are geometric theories. And the question is, what are the compatibilities or incompatibilities at the level of geometry -- before the theory is treated quantum mechanically? Well, in the case of Einstein's general relativity, we can rewrite the Einstein theory by saying that there's a projection map due to Einstein of a curvature tensor, where I'm going to write that curvature tensor as I would in the Yang-Mills theory.
-[00:50:23] That should be an "LC" for Levi-Civita. So, the Einstein projection ($$P_{E}$$) of the curvature tensor of the Levi-Civita connection ($$F_{\nabla^{LC}}$$) of the metric on this side, and on this side, I'm going to write down this differential operator: the adjoint of the exterior derivative coupled to a connection.
+''[https://youtu.be/Z7rd04KzLcg?t=2855 00:47:35]''<br>
+And then, interestingly, he went on to say one more thing. He said of course, these three central observations must be supplemented with the idea that this all takes place treated in quantum mechanical fashion, or quantum field theoretic. So it's a bit of an after-market modification rather than, in his opinion at the time, one of the core insights. I actually think that that's, in some sense, about right. Now, one of my differences with the community, in some sense, is that I question whether the quantum is in good enough shape, that we don't know whether we have a serious quantum mechanical problem or not. We know that we have a quantum mechanical problem, a quantum field theoretic problem, relative to the current formulations of these theories, but we know that in some other cases, the quantum becomes incredibly natural, sometimes sort of almost magically natural. And we don't know whether the true theories that we will need to be generalizing, in some sense, have beautiful quantum mechanical treatments, whereas the effective theories that we're dealing with now may not survive the quantization.
-[00:50:47] And you begin to see that we're missing an opportunity, potentially. What if the $$F_A$$s were the same in both contexts? Then you're applying two separate operators: 1) zeroth-order and destructive, in the sense that it doesn't see the entire curvature tensor; the other) inclusive, but of first-order. And so the question is, is there any opportunity to do anything that combines these two?
+[[File:GU Presentation Theory Triangle.png|thumb|right]]
-[00:51:15] But the problem is, is that the hallmark of the Yang-Mills theory is the freedom to choose the data, the internal quantum numbers that give all the particles their personalities beyond the mass and the spin.
+''[https://youtu.be/Z7rd04KzLcg?t=2934 00:48:54]''<br>
+So what I want to do is I want to imagine a different sort of incompatibility. So let's take our great three theories and just visually treat them as the vertices of a triangle. So I'm going to put general relativity (GR) in Einstein's formulation, and I'm going to put the—I probably won't write this again—Yang-Mills-Maxwell-Anderson-Higgs (YMMAH) theory over here, and I'm going to write the Dirac theory. What I want to explore is the incompatibilities, not at the quantum level, but the geometric input. All three of these are geometric theories. And the question is, what are the compatibilities or incompatibilities at the level of geometry—before the theory is treated quantum mechanically?
-[00:51:34] In addition to all of that freedom is some means of taking away some of the redundancy that comes with that freedom, which is the action of the gauge group. Now we can allow the gauge group of symmetries to act on both sides of the equation, but the key problem is that.
+''[https://youtu.be/Z7rd04KzLcg?t=3001 00:50:01]''<br>
+Well, in the case of Einstein's general relativity, we can rewrite the Einstein theory by saying that there's a projection map due to Einstein of a curvature tensor, where I'm going to write that curvature tensor as I would in the Yang-Mills theory. That should be an LC for Levi-Civita. So, the Einstein projection of the curvature tensor of the Levi-Civita connection of the metric on this side, and on this side, I'm going to write down this differential operator, the adjoint of the exterior derivative coupled to a connection. And you begin to see that we're missing an opportunity, potentially. What if the \(F_A\)s were the same in both contexts? Then you're applying two separate operators, one zeroth-order and destructive, in the sense that it doesn't see the entire curvature tensor; the other inclusive, but of first-order. And so the question is, is there any opportunity to do anything that combines these two?
-[00:52:03] If I act on connections on the right and then take the Einstein projection, this is not equal to first taking the projection and then conjugating with the gauge action. So the problem is, is that the projection is based on the fact that you have a relationship between the intrinsic geometry. If this is an ad[joint]-valued two-form, the two-form portion of this and the adjoint portion of this are both associated to the structure group of the tangent bundle.
-[00:52:52] But the gauge rotation is only acting on one of the two factors. Yet the projection is making use of both of them. So there is a fundamental incompatibility in the claim that Einstein's theory is a gauge theory relies more on analogy than an exact mapping between the two theories. What about the incompatibilities between the Einstein theory of general relativity and the Dirac theory of matter?
+$$ P_E(F_{\nabla^{LC} h}) \neq h^{-1}P_E(F_{\nabla^{LC}}) h $$
-[00:53:20] I was very struck that if we're going to try to quantize gravity and we associate gravity with the spin-2 field $$G_{\mu \nu}$$, we actually have a pretty serious problem, which is, if you think about spinors, electrons, quarks as being waves in a medium. And you think about photons as being waves in a different medium. [The] photon’s medium does not depend on the existence of a metric. One-forms are defined whether or not a metric is present, it's spinors or not.
-[00:54:02] So if we're going to take the spin-2 $$G_{\mu \nu}$$ field to be quantum mechanical, if it blinks out and does whatever the quantum does between observations. In the case of the photon, it is saying that the waves may blink out, but the ocean need not blink out. In the case of the Dirac theory, it is the ocean, the medium, in which the waves live that becomes uncertain itself. So even if you're comfortable with the quantum, to me, this becomes a bridge too far. So the question is: "How do we liberate the definition?"
+''[https://youtu.be/Z7rd04KzLcg?t=3075 00:51:15]''<br>
+But the problem is that the hallmark of the Yang-Mills theory is the freedom to choose the data, the internal quantum numbers that give all the particles their personalities beyond the mass and the spin. In addition to all of that freedom is some means of taking away some of the redundancy that comes with that freedom, which is the action of the gauge group. Now we can allow the gauge group of symmetries to act on both sides of the equation, but the key problem is that if I act on connections on the right and then take the Einstein projection, this is not equal to first taking the projection and then conjugating with the gauge action. So the problem is that the projection is based on the fact that you have a relationship between the intrinsic geometry—if this is an ad-valued two-form—the two-form portion of this and the adjoint portion of this are both associated to the structure group of the tangent bundle. But the gauge rotation is only acting on one of the two factors, yet the projection is making use of both of them. So there is a fundamental incompatibility, and the claim that Einstein's theory is a gauge theory relies more on analogy than on an exact mapping between the two theories.
-[00:54:47] How do we get the metric out from its responsibilities? It's been assigned far too many responsibilities. It is responsible for a volume form; for differential operators; it's responsible for measurement; it's responsible for being a dynamical field, part of the field content of the system. Lastly, we have the compatibilities and incompatibility between Yang-Mills and in the Dirac theory, these may be the most mild of the various incompatibilities, but it is an incompatibility of naturality where the Dirac field, Einstein's field, and the connection fields are all geometrically well-motivated, we push a lot of the artificiality that we do not understand into the potential for the scalar field that gives everything its mass.
+''[https://youtu.be/Z7rd04KzLcg?t=3191 00:53:11]''<br>
+What about the incompatibilities between the Einstein theory of general relativity and the Dirac theory of matter? I was very struck that, if we're going to try to quantize gravity and we associate gravity with the spin-2 field \(g_{\mu \nu}\), we actually have a pretty serious problem, which is if you think about spinors, electrons, quarks, as being waves in a medium, and you think about photons as being waves in a different medium, [the] photon's medium does not depend on the existence of a metric. One-forms are defined whether or not a metric is present: it's spinors or not. So if we're going to take the spin-2 \(g_{\mu \nu}\) field to be quantum mechanical, if it blinks out and does whatever the quantum does between observations, in the case of the photon, it is saying that the waves may blink out, but the ocean need not blink out. In the case of the Dirac theory, it is the ocean, the medium, in which the waves live that becomes uncertain itself. So even if you're comfortable with the quantum, to me, this becomes a bridge too far. So the question is how do we liberate the definition? How do we get the metric out from its responsibilities? It's been assigned far too many responsibilities. It is responsible for a volume form, for differential operators, it's responsible for measurement, it's responsible for being a dynamical field, part of the field content of the system.
-[00:55:33] Yes. So we tend to treat it as something of a mysterious fudge factor. So the question is, if we have a Higgs field: "why is it here and why is it geometric?"
+''[https://youtu.be/Z7rd04KzLcg?t=3306 00:55:06]''<br>
+Lastly, we have the compatibilities and incompatibilities between Yang-Mills and in the Dirac theory. These may be the most mild of the various incompatibilities, but it is an incompatibility of naturality. Where the Dirac field, Einstein's field, and the connection fields are all geometrically well-motivated, we push a lot of the artificiality that we do not understand into the potential for the scalar field that gives everything its mass, so we tend to treat it as something of a mysterious fudge factor. So the question is if we have a Higgs field, why is it here and why is it geometric? It has long been the most artificial sector of our models.
-[00:55:52] It has long been the most artificial sector of our models.
+''[https://youtu.be/Z7rd04KzLcg?t=3359 00:55:59]''<br>
+The proposal that I want to put to you today is that one of the reasons that we may be having trouble with unification is that the duty, our duty, may be to generalize all three vertices before we can make progress. That's daunting because in each case, it would appear that we can make an argument that this, that, and the other vertex are the simplest possible theories that could live at that vertex. We know, for example, the Dirac operator is the most fundamental of all the elliptic operators in Euclidean signature, generating all of the Atiyah-Singer theory. We know that Einstein's theory is, in some sense, a unique spin-2 massless field capable of communicating gravity, which can be arrived at from field-theoretic rather than geometric considerations. In the Yang-Mills case, it can be argued that the Yang-Mills theory is the simplest theory that can possibly result. Where we're taking the simplest Lagrangian in the Einstein case, looking only at the scalar curvature, in the Yang-Mills case, we have no substructure, and so we're doing the most simple-minded thing we can do by taking the norm-square of the curvature and saying whatever the field strength is, let's measure that size. So if each one of these is simplest possible, doesn't Occam's razor tell us that if we wish to remain in geometric field theory, that we've already reached bottom, and that what we're being asked to do is to abandon this as merely an effective theory? That's possible. And I would say that that, in some sense, represents a lot of conventional wisdom. But there are other possibilities.
-[00:55:59] The proposal that I want to put to you today is that one of the reasons that we may be having trouble with unification this is that the duty, our duty, may be to generalize all three vertices before we can make progress.
+[[File:GU Oxford Lecture Square Roots Slide.png|thumb|right]]
-[00:56:23] That's daunting because in each case, it would appear that we can make an argument. That this, that, and the other Vertex are the simplest possible theories that could live at that vertex. We know, for example, the Dirac operator is the most fundamental of all the elliptic operators and Euclidean signature generating all of the Atiyah-Singer theory.
+''[https://youtu.be/Z7rd04KzLcg?t=3461 00:57:41]''<br>
+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?"
-[00:56:47] We know that Einstein's theory is, in some sense, a unique spin to massless field capable of communicating gravity, which can be arrived at from field-theoretic rather than geometric consideration. In the Yang-Mills case, it can be argued that the Yang-Mills theory is the simplest theory that can possibly result where we're taking the simplest Lagrangian in the Einstein case, looking only at the scalar curvature. In the Yang-Mills case, we have no substructure, and so we're doing the most simple-minded thing we can do by taking the norm-squared of the curvature and saying whatever the field strength is, let's measure that size. So if each one of these is simplest possible, doesn't Occam’s razor tell us that if we wish to remain in geometric field theory, that we've already reached bottom? And that what we're being asked to do is to abandon this as merely an effective theory? That's possible. And I would say that that in some sense represents a lot of conventional wisdom. But there are other possibilities.
-[00:57:41] There are other possibilities that while each of these may be simplest in its category, they are not simplest in their interaction.
+=====Motivations for Geometric Unity=====
-[00:57:54] For example, we know that Dirac famously took the square root of the [[Klein-Gordon equation]] to achieve the Dirac equation. You 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?
-[00:58:28] So, let's imagine the following: we replaced the standard model with a true second-order theory. We imagine the 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.
-[00:58:50] 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?"
-====== Introduction to Geometric Unity (GU) ======
 [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.