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

[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.

[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]].

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

[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.

[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]].

[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.

[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.

[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.

[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?

[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.

[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.

[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?"

[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.

[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?"

[00:55:52] It has long been the most artificial sector of our models.

[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.

[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.

 

 

 

 

====== Introduction to Geometric Unity (GU) ======