A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions
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<p>[02:17:56] If we're going to replace that. It's very tricky because it's almost impossible to think about what would be underneath Einstein’s theory. Now, there's a huge problem in the spinorial sector, which I don't why more people don't worry about. Which is that spinors aren't defined for representations of the double cover of GL four R the general linear group’s effective spin analog. And as such, if we imagine that we will one day quantize gravity, we will lose our definition, not of the electrons, but let's say of the medium in which the electrons operate. That is, there will be no spinorial bundle until we have an observation of a metric. So one thing we can do is to take a manifold $$X^d$$ as the starting point and see if we can create an entire universe from no other data. Not even with a metric. So since we don't choose a metric, what we instead do is to work over the space of all possible point-wise metrics. So not quite in the Feynman sense, but in the sense that, um, we will work over a bundle that is of a quite larger, quite a bit larger dimension. | <p>[02:17:56] If we're going to replace that. It's very tricky because it's almost impossible to think about what would be underneath Einstein’s theory. Now, there's a huge problem in the spinorial sector, which I don't why more people don't worry about. Which is that spinors aren't defined for representations of the double cover of GL four R the general linear group’s effective spin analog. And as such, if we imagine that we will one day quantize gravity, we will lose our definition, not of the electrons, but let's say of the medium in which the electrons operate. That is, there will be no spinorial bundle until we have an observation of a metric. So one thing we can do is to take a manifold $$X^d$$ as the starting point and see if we can create an entire universe from no other data. Not even with a metric. So since we don't choose a metric, what we instead do is to work over the space of all possible point-wise metrics. So not quite in the Feynman sense, but in the sense that, um, we will work over a bundle that is of a quite larger, quite a bit larger dimension. | ||
<p>[02:19:10] So for example, if X | <p>[02:19:10] So for example, if $$X$$ was a four dimensional, therefore d equals four, then $$Y$$, in this case, would be $$d^2$$, which would be 16 plus 3d, which would be 12 making 28 divided by two, which would be 14. So in other words, a four-dimensional proto-spacetime, not a spacetime, but a proto-spacetime with no metric would give rise to a 14-dimensional "observerse" portion called $$Y$$. Now, I believe that in the lecture in Oxford, I called that $$U$$, so I'm sorry for the confusion, but of course, this shifts around every time I take it out of the garage. And that's one of the problems with working on a theory in solitude for many years. So, we have two separate spaces and we have fields on the two spaces. | ||
<p>[02:20:02] Now what I'm going to do as I'm going to refer to fields on the $$X^d$$ space by Hebrew letters. So instead of $$G_{\mu \nu}$$ for a metric, I just wrote gimmel mem nun. And the idea being that I want to separate Latin and Greek fields on the Y space from the | <p>[02:20:02] Now what I'm going to do as I'm going to refer to fields on the $$X^d$$ space by Hebrew letters. So instead of $$G_{\mu \nu}$$ for a metric, I just wrote gimmel mem nun. And the idea being that I want to separate Latin and Greek fields on the $$Y$$ space from the rather rarer fields that actually live directly on $$X$$. | ||
<p>[02:20:27] So this is a little bit confusing. One way of thinking about it is to think of the observerse as the stands plus the pitch in a stadium. I think I may have said that | <p>[02:20:27] So, this is a little bit confusing. One way of thinking about it is to think of the "observerse" as the stands plus the pitch in a stadium. I think I may have said that in the lecture, but this is what replaces the questions of where and when in the newspaper story that is a fundamental theory. Where and when correspond to space and time. | ||
<p> | <p> | ||
[[File:Slide3.jpg|thumb]] | [[File:Slide3.jpg|thumb]] | ||
<p>[02:20:49] Who and what correspond to bosons and fermions. And how and why correspond to equations and the Lagrangian that generates them. So if you think about those six quantities, you'll realize that that's really what the content of a fundamental theory is | <p>[02:20:49] Who and what correspond to bosons and fermions. And how and why correspond to equations and the Lagrangian that generates them. So, if you think about those six quantities, you'll realize that that's really what the content of a fundamental theory is assuming that it can be quantized properly. Most fields and, in this case, we're going to call the collection of two-tuples $$\Omega$$. So the inside of $$\Omega$$ that will be in the first tuple will have $$\epsilon$$ and $$\pi$$ written sort of an nontraditional variation of how we write this symbol for $$\pi$$. In the second tuple, we'll have the letters, $$\nu$$ and $$\zeta$$. And I would like them not to move because they honor particular people who are important (NB: children are named N. and Z.). | ||
<p>[02:21:36] So most fields, in this case, Omega | <p>[02:21:36] So most fields, in this case, $$\Omega$$, are dancing on $$Y$$, which was called $$U$$ in the lecture, unfortunately, but they are observed via pullback as if they lived on $$X$$. In other words, if you're sitting in the stands, you might feel that you're actually literally on the pitch, even though that's not true. So what we've done is we've taken the U of Wheeler, we've put it on its back and created a double-U ("W") structure. | ||
<p>[02:21:59] And the W structure is meant to say that there's a bundle on top of a bundle. Again, geometric unity is more flexible than this, but I wanted to make the most concrete approach to this possible for at least this introduction | <p>[02:21:59] And the W structure is meant to say that there's a bundle on top of a bundle. Again, geometric unity is more flexible than this, but I wanted to make the most concrete approach to this possible for at least this introduction. We don't need to state that Y would, in fact, be a bundle A. It could be an immersion of $$X$$ into any old manifold. But I, I'd like to go with the most ambitious version of GU first. So the two projection maps are $$\pi_2$$ and $$\pi_1$$. And what we're going to say up top is that we're going to have a symbol $$Z$$ and an action of a group $$\rho$$ on $$Z$$, standing in for any bundle associated to the principle bundle, which is generated as the unitary bundle of the spin of the spinors on the chimeric tangent bundle to $$Y$$. | ||
<p>[02:22: | <p>[02:22:54] It's a bit of a mouthful, but the key issue was that on the manifold $$Y$$, there happens to be a bundle, which is isomorphic non-canonicaly to the tangent bundle of Y, which has a definite and canonical metric. And, in fact, there, that carries the spinors. So, this is the way in which we get spinors without ever having to choose a metric. | ||
<p>[02: | <p>[02:23:17] But we pick up some technical debt to use the computer science concept by actually having to now work on two different spaces, $$X$$ and $$Y$$. And we're not merely working on $$X$$ anymore. | ||
<p>[02:23: | <p>[02:23:30] This leads to the [[Mark of Zorro]]. That is, we know that whenever we have a metric by the fundamental theorem of Riemannian geometry, we always get a connection. It happens, however, that what is missing to turn the canonical chimeric bundle on $$Y$$ into the tangent bundle of $$Y$$ is, in fact, a connection on the space $$X$$. | ||
<p>[02:23:52] So there is one way in which we've reversed the fundamental theorem of Riemannian in geometry where a connection on $$X$$ leads to a metric on $$Y$$. So if we do the full transmission mechanism out, a Gimmel on X leads to alpha sub Gimmel for the Levi-Civita connection on X. Alpha sub Gimmel live leads to a G sub alpha, which is, or sorry, the G sub Alef. | |||
<p>[02:23:52] So there is one way in which we've reversed the fundamental theorem of Riemannian in geometry where a connection on X leads to a metric on Y. So if we do the full transmission mechanism out, a Gimmel on X leads to alpha sub Gimmel for the Levi-Civita connection on X. Alpha sub Gimmel live leads to a G sub alpha, which is, or sorry, the G sub Alef. | |||
<p>[02:24:17] Um, I'm not used to using Hebrew, uh, in, in math. Um, so G sub Alef, uh, then is a metric on Y and that creates a Levi-Civita connection of the metric on the space Y as well, which then induces one on the spinorial levels. In sector two the inhomogenous gauge group on Y replaces the Poincare. Group and the internal symmetries that are found on X. | <p>[02:24:17] Um, I'm not used to using Hebrew, uh, in, in math. Um, so G sub Alef, uh, then is a metric on Y and that creates a Levi-Civita connection of the metric on the space Y as well, which then induces one on the spinorial levels. In sector two the inhomogenous gauge group on Y replaces the Poincare. Group and the internal symmetries that are found on X. | ||