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

<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>[00:55:59] 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>[00:57:54] 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>[00:58:28] 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 orders theory admits of a square root and can be linked with the true first order theory.

<p>[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?