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

<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>[00:46:12] 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>[00:46:33] 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>[00:47:35] 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>[00:48:31] 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>[00:48:54] 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>[00:49:39] 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>[00:51:15] 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>[00:51:34] 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.

<p>[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 add value 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.

<p>[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. 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.

<p>[00:54:02] So if we're going to take the spin two $$G_{mu nu}$$ field to be quantum mechanical, if it blinks out and does whatever the quantum does between observations. In the case of the photon, it's 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?

<p>[00:54:47] How do we get the metric out from its responsibilities? It's been assigned far too many responsibilities. It's 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.

<p>[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?

<p>[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.

<p>[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.

<p>[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 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.

<p>[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.

<p>[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, uh, on which it acts. But could not, could we not do the same thing by re-interpreting what we saw in Donaldson theory and Chern- Simon's theory and finding that there are first order equations that imply second order equations that are nonlinear in the curvature?

<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>[01:02:12] 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.

<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>[01:03:36] It's just some sort of flabby proto space time, 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.

<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.