A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions
A Portal Special Presentation- Geometric Unity: A First Look (edit)
Revision as of 03:40, 11 April 2020
, 11 April 2020→Connecting the Three Observations of Witten
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<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. [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. | <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. [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. | ||
<p>[00:54:02] So if we're going to take the spin | <p>[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?" | ||
<p>[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. | <p>[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. | ||
<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 | <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>[00:55:52] It has long been the most artificial sector of our models. | <p>[00:55:52] It has long been the most artificial sector of our models. | ||
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<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>[00:57:41] 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 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? | <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 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? | ||
<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-order theory admits of a square root and can be linked with the true first order theory. | <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-order theory admits of a square root and can be linked with the true first order theory. | ||