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<p>[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? [[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.
<p>[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? [[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.


<p>[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 spacetime or higher dimensional structure, leaving us a little bit of head room. The equation most associated with this is the [[Einstein field equation]].
<p>[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]].


<p>[00:43:12] And of course I'm running into the margin. Okay.
<p>[00:43:12] And, of course, I'm running into the margin. Okay.


<p>[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.  
<p>[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.  


<p>[00:43:47] The second fundamental insight, I'm going to begin to start drawing pictures here as well.
<p>[00:43:47] The second fundamental insight... I'm going to begin to start drawing pictures here as well.


<p>[00:43:55] 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 for a non-Abelian group, which is currently SU(3) "color" cross  SU(2) "weak" cross U(1) "weak hypercharge", which breaks down to SU(3) cross U(1), where the broken U(1) is the electromagnetic symmetry.
<p>[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 for a non-Abelian group, which is currently SU(3) "color" cross  SU(2) "weak" cross U(1) "weak hypercharge", which breaks down to SU(3) cross U(1), where the broken U(1) is the electromagnetic symmetry.


<p>[00:44:54] This equation is also a curvature equation, the corresponding equation, and it says. But 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.
<p>[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.


<p>[00:45:44] The third point surrounds the matter in the system.
<p>[00:45:44] The third point surrounds the matter in the system.
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<p>[00:45:58] And here we have a Dirac equation. Again, coupled to a connection.
<p>[00:45:58] And here we have a Dirac equation. Again, coupled to a connection.


<p>[00:46:12] One of the great insights is, 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.
<p>[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.


<p>[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]].
<p>[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]].


<p>[00:47:15] Let me for consistency so matter is asymmetric, okay. And therefore light.
<p>[00:47:15] Let me, for consistency, say "matter is asymmetric", okay. "and therefore light".


<p>[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 takes place, treated in quantum mechanical fashion or quantum field theoretic. So it's a bit of a, an aftermarket modification rather than in his opinion at the time one of the core insights.
<p>[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 aftermarket modification, rather than his opinion at the time, [or] one of the core insights.


<p>[00:48:07] I actually think that that's in some sense about right. No. One of my differences with the community in some sense is that I questioned whether the quantum isn't 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.
<p>[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.


<p>[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.
<p>[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.


=== Connecting the Three Observations of Witten ===
<p>[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.
<p>[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.


<p>[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, Anderson, Higgs theory, uh, over here, and I'm going to write the Dirac theory.
<p>[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/Anderson/Higgs theory, uh, over here, and I'm going to write the Dirac theory.


<p>[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.
<p>[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.


<p>[00:50:23] 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.
<p>[00:50:23] 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.


<p>[00:50:47] And you begin to see that we're missing an opportunity, potentially. What if the FAs 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?
<p>[00:50:47] And you begin to see that we're missing an opportunity, potentially. What if the FAs 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?


<p>[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.
<p>[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.
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