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

<p>[01:16:23] And the next unit of GU. So this is sort of the first unit of G U. Are there any quick questions having to do with confusion or may I proceed to the next unit?

<p>[01:16:36] Okay. The next unit of GU is the unified field content. What does it mean for our fields to become unified? There are, in fact, only at this moment, two fields that know about X. Theta, which is the connection that we've just talked about, and a section sigma that takes us back so that we can communicate back and forth between U and X. We now need field content that only knows about U, which now has a metric depending on theta.

<p>[01:17:32] A particular member of the audience is a hedge fund manager who taught me that there is something of a universal trade and a universal trade has four components. You have to have a view, you have to have a trade expression, you have to be able to calculate your cost of carry and you need a catalyst.

<p>[01:18:03] What is that trade? What is it that we think has been blocking progress?

<p>[01:18:50] We're going to take particle theory, we're going to make a bad trade, or what appears to be a bad trade, which is that we are going to give away the freedom to choose our field content, which is already extremely, as I think I said in the abstract, Baroque with all of the different particle properties.

<p>[01:20:01] Okay. What is it that we get for the Levi-Civita connection? Well, not much. One thing we get is that normally the space of connections is an affine space. Not a vector space, but an affine space, almost a vector space, a vector space up to a choice of origin. But with the Levi-Civita connection, rather than having an infinite plane with an ability to take differences, but no real ability to have a group structure.

<p>[01:20:35] You pick out one point, which then becomes the origin. That means that any connection A has a torsion tensor A which is equal to the connection minus the Levi-Civita connection. So we get a tensor that we don't usually have. Gauge potentials are not usually well-defined. They're only defined up to a choice of gauge.

<p>[01:21:00] So that's one of the things we get for our Levi-Civita connection, but because the gauge group is going to go missing, this has terrible properties from with respect to the gauge group. It almost looks like a representation, but in fact, if we let the gauge group act, there's going to be an affine shift.

<p>[01:21:21] Furthermore, as we've said before. The ability to use projection operators together with the gauge group is frustrated by virtue of the fact that these two things do not commute with each other. So now the question is, how are we going to prove that we're actually making a good trade?

<p>[01:21:48] First thing we need to do is, we still have the right to choose intrinsic field content. Have an intrinsic field theory. So if you consider the structure bundle of the spinors, right, we built the chimeric bundle, so we can define Dirac spinors on the chimeric bundle if we're in Euclidean signature. A 14 dimensional manifold has Dirac spinors of dimension two to the dimension of the space divided by two.

<p>[01:22:20] Right? So two to the 14 over two to the seventh is 128 so we have a map into a structured group of $$U^128$$

<p>[01:22:36] At least in Euclidean signature. We can get to mixed signatures later. From that, we can form the associated bundle.

<p>[01:22:54] And sections of this bundle are either, depending upon how you want to think about it, the gauge group H or Xi, a space of sigma fields.

<p>[01:23:44] But we're not dead yet, right? We're fighting for our life to make sure that this trade has some hope. So potentially by including symmetries as field content, we will have some opportunity to make use of the projections. So for those of you who...

<p>[01:24:06] So when I was thinking about this, I used to be amazed by ships in bottles. I must confess that I never figured out what the trick was for ships in bottles, but once I saw it, I remembered thinking, that's really clever. So if you've never seen it, you have a ship, which is like a curvature tensor. And imagine that the mass does is the Ricci curvature.

<p>[01:25:18] This is exactly what we're going to hope is going to save us in this bad trade that we've made because we're going to add fields, content that has the ability to lower the mast and bring the mast back up. We're going to hope to have a theory which is going to create a communitive situation. But then once we've had this idea, we start to get a little bolder.

<p>[01:25:43] Let's think about unified content. We know that we want a space of connections, A for our field theory, but we know because we have a Levi-Civita connection, that this is going to be equal on the nose to add valued one forms as a vector space. The gauge group represents an add valued one forms. So if we also have the gauge group, what we think of that instead as a space of Sigma fields.

<p>[01:26:16] What if we take the semi direct product at a group theoretic level between the two and call this our group of interests. Well by analogy, we've always had a problem with the Poincare group being too intrinsically tied to rigid flat Mankowski space. What if we wanted to do quantum field theory in some situation, which was more amenable to a curved space situation?

<p>[01:26:44] It's possible that we should be basing it around something more akin to the gauge group, and in this case, we're mimicking the construction where Xi here would be analogous to the Lorentz group, fixing a point in Mankowski space, and add valued one forms would be analogous to the four momentums. We take in the semi direct product to create the inhomogeneous Lorentz group, otherwise known as the Poincare group, or rather, it's a double cover to allow spin.

<p>[01:27:49] So this magic being trade is going to start to enter more and more into our consciousness. If I take an element H and I map that in the obvious way into the first factor, but I map it onto the Maurer-Cartan form, I think that's what I, I wish I remembered more of this stuff. Into the second factor. It turns out that this is actually a group homomomorphism and so we have a nontrivial embedding, which is in some sense diagonal between the two factors. That subgroup we are going to refer to as the tilted gauge group, and now our field content, at least in the Bosonic sector, is going to be a group manifold, an infinite dimensional function space [[Lie Group]], but a group nonetheless. And we can now look at G mod, the tilted gauge group, and if we have any interesting representation of H, we can form homogeneous vector bundles and work with induced representation. And that's what the fermions are going to be. So the fermions in our theory are going to be H modules.

<p>[01:29:14] And the idea is that we're going to work with vector bundles, curly E of the form inhomogeneous gauge group producted over the tilted gauge group.

<p>[01:29:47] Now, just as in the finite dimensional case, we have a linear and a nonlinear component, right? Because at the topological level, this is just a Cartesian product. So if we wished to take products of fermions, of spinnorial fields. We have a place to accept them. We can't figure out necessarily how to map them into the nonlinear sector, but we don't want to.

<p>[01:30:09] So just the way, when we look at supersymmetry, we can take products of the spin one half fields and map them into the linear sector. We can do the same thing here. So what we're talking about is something like a supersymmetric extension of the inhomogeneous gauge group analogous to supersymmetric extensions of the double cover of the inhomogeneous Lorenz or poincare group. Further, because this construction is at the level of groups, we've left a slot on the left hand side on which to act. So for example, if we want to take regular representations on the group, we can act by the group G on the left hand side, because we're allowing the tilted gauge group to act on the right hand side.

<p>[01:31:38] Now, what would it mean to be able to use a gauge group in an intrinsic theory like this? We would be talking. About something like an action, let's say a first order action and it would take the group G no, let's say to the real numbers. Invariant, not under the full group, but under the tilted gauge subgroup. And now the question is, do we have any such actions that are particularly nice? And could we recognize them the way Einstein did by trying to write down, not the action and Hilbert was the first one to write that down. But I, you know, I always feel defensive, uh, because I think Einstein and Grossman did so much more to begin the theory in that the Lagrangian that got written down was really just an inevitability.

<p>[01:32:38] So just humor me for this talk, and let me call it the Einstein Grossman Lagrangian. Uh. Hubbard's certainly done fantastic things and has a lot of credit elsewhere. And he did do it first. But here what we had was that Einstein thought in terms of the differential of the action, not the action itself. So what we're looking for is equations of motion or some field alpha where alpha belongs to the one forms on the group.

<p>[01:33:22] Now in this section of GU unified field content is only one part of it, but what we really want, is unified field content plus a toolkit.

<p>[01:34:00] The toolkit that we have is that the adjoint bundle looks like the Clifford algebra. At the level of vector space.

<p>[01:34:28] Which is just looking like the exterior algebra on the chimeric bundle. tThat means that it's graded by degrees. chimeric bundle has dimension 14 so there's a zero part, a one part, a two part, all the way up to 14 plus. We have forms in the manifold, and so the question is, if I want to look at $$Omega^i$$ okay valued in the adjunct bundle.