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Difference between revisions of "A Portal Special Presentation- Geometric Unity: A First Look"
A Portal Special Presentation- Geometric Unity: A First Look (view source)
Revision as of 20:33, 15 March 2021
, 20:33, 15 March 2021→Lecture at the University of Oxford
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==== The Current Picture of Physics ==== | ==== The Current Picture of Physics ==== | ||
''[https://youtu.be/Z7rd04KzLcg?t=2507 00:41:47]''<br> | ''[https://youtu.be/Z7rd04KzLcg?t=2507 00:41:47]''<br> | ||
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? 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. And they correspond to the three great equations. | 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? 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. And they correspond to the three great equations. | ||
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==== Motivations for Geometric Unity ==== | ==== Motivations for Geometric Unity ==== | ||
[[File:GU Presentation Intrinsic-Auxiliary Diagram.png|thumb|right]] | [[File:GU Presentation Intrinsic-Auxiliary Diagram.png|thumb|right]] | ||
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==== The Observerse and Four Versions of GU ==== | ==== The Observerse and Four Versions of GU ==== | ||
[[File:GU Presentation Flavors Diagram.png|thumb|right]] | [[File:GU Presentation Flavors Diagram.png|thumb|right]] | ||
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====== Constructing Endogenous GU: Choosing All Metrics ====== | ====== Constructing Endogenous GU: Choosing All Metrics ====== | ||
''[https://youtu.be/Z7rd04KzLcg?t=4224 01:10:24]''<br> | ''[https://youtu.be/Z7rd04KzLcg?t=4224 01:10:24]''<br> | ||
We take \(X^4\). We need metrics, we have none. We're not allowed to choose one, so we do the standard trick: we choose them all. | We take \(X^4\). We need metrics, we have none. We're not allowed to choose one, so we do the standard trick: we choose them all. | ||
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====== Chimeric Bundle ====== | ====== Chimeric Bundle ====== | ||
''[https://youtu.be/Z7rd04KzLcg?t=4337 01:12:17]''<br> | ''[https://youtu.be/Z7rd04KzLcg?t=4337 01:12:17]''<br> | ||
We now define the '''chimeric bundle''', right? And the chimeric bundle is this direct sum of the vertical tangent bundle along the fibers with the pullback, which we're going to call the horizontal bundle, from the base space. So the chimeric bundle is going to be the vertical tangent space of 10 dimensions to \(U\), direct sum the cotangent space, which we're going to call horizontal, to \(U\). And the great thing about the chimeric bundle is that it has an a priori metric. It's got a metric on the 4, a metric on the 10, and we can always decide that the two of them are naturally perpendicular to each other. | We now define the '''chimeric bundle''', right? And the chimeric bundle is this direct sum of the vertical tangent bundle along the fibers with the pullback, which we're going to call the horizontal bundle, from the base space. So the chimeric bundle is going to be the vertical tangent space of 10 dimensions to \(U\), direct sum the cotangent space, which we're going to call horizontal, to \(U\). And the great thing about the chimeric bundle is that it has an a priori metric. It's got a metric on the 4, a metric on the 10, and we can always decide that the two of them are naturally perpendicular to each other. | ||
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====== Observerse Conclusion ====== | ====== Observerse Conclusion ====== | ||
''[https://youtu.be/Z7rd04KzLcg?t=4467 01:14:27]''<br> | ''[https://youtu.be/Z7rd04KzLcg?t=4467 01:14:27]''<br> | ||
But, and I want to emphasize this: one thing most of us—we think a lot about final theories and about unification, but until you actually start daring to try to do it, you don't realize what the process of it feels like. And, try to imagine conducting your life where you have no children, let's say, and no philanthropic urges, and what you want to do is you want to use all of your money for yourself and die penniless, right? Like a perfect finish. Assuming that that's what you wanted to do, it would be pretty nerve-wracking at the end, right? How many days left do I have? How many dollars left do I have? This is the process of unification in physics. You start giving away all of your most valuable possessions, and you don't know whether you've given them away too early, whether you husbanded them too long, and so in this process, what we've just done is we've started to paint ourselves into a corner. We got something we wanted, but we've given away freedom. We're now dealing with a 14-dimensional world. | But, and I want to emphasize this: one thing most of us—we think a lot about final theories and about unification, but until you actually start daring to try to do it, you don't realize what the process of it feels like. And, try to imagine conducting your life where you have no children, let's say, and no philanthropic urges, and what you want to do is you want to use all of your money for yourself and die penniless, right? Like a perfect finish. Assuming that that's what you wanted to do, it would be pretty nerve-wracking at the end, right? How many days left do I have? How many dollars left do I have? This is the process of unification in physics. You start giving away all of your most valuable possessions, and you don't know whether you've given them away too early, whether you husbanded them too long, and so in this process, what we've just done is we've started to paint ourselves into a corner. We got something we wanted, but we've given away freedom. We're now dealing with a 14-dimensional world. | ||
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Well, let me just sum this up by saying: between fundamental and emergent, standard model and GR... Let's do GR. Fundamental is the metric, emergent is the connection. Here in GU, it is the connection that's fundamental and the metric that's emergent. | Well, let me just sum this up by saying: between fundamental and emergent, standard model and GR... Let's do GR. Fundamental is the metric, emergent is the connection. Here in GU, it is the connection that's fundamental and the metric that's emergent. | ||
==== | ==== Unified Field Content ==== | ||
[01:16:23] | ''[https://youtu.be/Z7rd04KzLcg?t=4583 01:16:23]''<br> | ||
In the next unit of GU—so this is sort of the first unit of GU. Are there any quick questions having to do with confusion, or may I proceed to the next unit? | |||
''[https://youtu.be/Z7rd04KzLcg?t=4596 01:16:36]''<br> | |||
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 \(\theta\). | |||
===== The Trade ===== | |||
''[https://youtu.be/Z7rd04KzLcg?t=4652 01:17:32]''<br> | |||
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. | |||
# You need a catalyst. | |||
Our view is going to be that somebody doesn't understand what trade is possible and we're going to make a trade that looks like one of the worst trades of all time, and hopefully, if we have enough conviction, we're going to have a catalyst to show that we actually got the better part of the deal. What is that trade? What is it that we think has been blocking progress? | |||
''[https://youtu.be/Z7rd04KzLcg?t=4696 01:18:16]''<br> | |||
In GR and Riemannian geometry, as we've said, we have the projection operators and we also have the Levi-Civita connection. In the auxiliary theory, we have freedom to choose our field content, and we have the ability to get rid of much excess through the symmetries of the gauge group. We're going to take particle theory, and 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. And we're going to lose the ability to use the gauge group, because we're going to trade it all. | |||
''[https://youtu.be/Z7rd04KzLcg?t=4767 01:19:27]''<br> | |||
You have the family cow and you have some magic beans. | |||
''[https://youtu.be/Z7rd04KzLcg?t=4780 01:19:40]''<br> | |||
So it's now time to trade the family cow for the magic beans and bring them home, and see whether or not we got the better of the deal. | |||
''[https://youtu.be/Z7rd04KzLcg?t=4801 01:20:01]''<br> | |||
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, you pick out one point, which then becomes the origin. That means that any connection \(A\) has a torsion tensor \(T_A\), 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 are only defined up to a choice of gauge. | |||
<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ A \hookrightarrow T_A = A - \nabla^{LC} $$</div> | |||
''[https://youtu.be/Z7rd04KzLcg?t=4860 01:21:00]''<br> | |||
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 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. | |||
''[https://youtu.be/Z7rd04KzLcg?t=4881 01:21:21]''<br> | |||
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? | |||
''[https://youtu.be/Z7rd04KzLcg?t=4908 01:21:48]''<br> | |||
[The] first thing we need to do is we still have the right to choose intrinsic field content. We have an intrinsic field theory, so if you consider the structure bundle of the spinors—we've 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. Right? So \(2^{14}\) over \(2^7\) is \(128\). | |||
<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ C ⇝ \unicode{x2215}\kern-0.55em S(C) $$</div> | |||
''[https://youtu.be/Z7rd04KzLcg?t=4944 01:22:24]''<br> | |||
So we have a map into a structure group of \(U(128)\), at least in Euclidean signature—we can get to mixed signatures later. From that, we can form the associated bundle: | |||
<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \Gamma^\infty(P_{U(8)} \times _{\text{Ad}}U(\unicode{x2215}\kern-0.55em S)) $$</div> | |||
''[https://youtu.be/Z7rd04KzLcg?t=4974 01:22:54]''<br> | |||
And sections of this bundle are either, depending upon how you want to think about it, the gauge group \(\mathcal{H}\), or \(\Xi\), a space of \(\sigma\) fields—nonlinear. | |||
<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \Gamma^\infty(P_{U(8)} \times _{\text{Ad}}U(\unicode{x2215}\kern-0.55em S)) = \begin{cases} \mathcal{H} \text{ gauge group}\\<br>\Xi \text{ sigma fields (non-linear)} \end{cases} $$</div> | |||
[01: | ''[https://youtu.be/Z7rd04KzLcg?t=5005 01:23:25]''<br> | ||
There's no reason that we can't choose this as field content. Again, we're being led by the nose like a bull. If we want to make use of the symmetries of the theory, we have to promote some symmetry to being part of the theory, and we have to let it be subjected to dynamical laws. We're going to lose control over it. 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... | |||
===== Inhomogeneous Gauge Group and Tilted Gauge Group ===== | |||
[ | [[File:GU Oxford Lecture Shiab Broken Slide.png|thumb|right]] | ||
[01: | ''[https://youtu.be/Z7rd04KzLcg?t=5046 01:24:06]''<br> | ||
So when I was thinking about this, I used to be amazed by ships in bottles, and 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 mast is the Ricci curvature. If you just try to shove it into the bottle, you're undoubtedly going to snap the mast. So, you imagine that you've transformed your gauge fields, you've kept track of where the Ricci curvature was, you try to push it from one space, like ad-valued two-forms, into another space, like ad-valued one-forms, where connections live. | |||
[ | [[File:GU Oxford Lecture Shiab Unbroken Slide.png|thumb|right]] | ||
[01: | ''[https://youtu.be/Z7rd04KzLcg?t=5094 01:24:54]''<br> | ||
That's not a good idea. Instead, what we do is the following: imagine that you're carrying around group theoretic information, and what you do is you do a transformation based on the group theory. So you lower the mast, you push it through the neck, having some string attached to the mast, and then you undo the transformation on the other side. | |||
[01: | ''[https://youtu.be/Z7rd04KzLcg?t=5118 01:25:18]''<br> | ||
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 field 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 commutative situation. But then, once we've had this idea, we start to get a little bolder. | |||
[01: | ''[https://youtu.be/Z7rd04KzLcg?t=5143 01:25:43]''<br> | ||
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 ad-valued one-forms as a vector space. The gauge group represents on ad-valued one-forms. | |||
<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ A = \Omega^1(ad) $$</div> | |||
[01: | ''[https://youtu.be/Z7rd04KzLcg?t=5168 01:26:08]''<br> | ||
So, if we also have the gauge group, but we think of that instead as a space of \(\sigma\) fields, what if we take the semi-direct product at a group theoretic level between the two and call this our group of interest? | |||
<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \mathcal{H} = \Xi ⋉ \Omega^1(ad) = A $$</div> | |||
[01: | ''[https://youtu.be/Z7rd04KzLcg?t=5185 01:26:25]''<br> | ||
Well by analogy, we've always had a problem with the Poincaré group being too intrinsically tied to rigid, flat Minkowski space. What if we wanted to do quantum field theory in some situation which was more amenable to a curved space situation?''' '''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 Minkowski space, and ad-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 Poincaré group, or rather its double cover to allow spin. | |||
[01: | ''[https://youtu.be/Z7rd04KzLcg?t=5232 01:27:12]''<br> | ||
So we're going to call this the '''inhomogeneous gauge group''', or '''iggy'''. And this is going to be a really interesting space, because it has a couple of properties. One is it has a very interesting subgroup. Now, of course, \(H\) includes into \(G\) by just including onto the first factor, but in fact, there's a more interesting homomorphism brought to you by the Levi-Civita connection. So, this magic bean trade is going to start to enter more and more into our consciousness. | |||
<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \mathcal{H} \hookrightarrow \mathcal{G} $$</div> | |||
= | <div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \tau_{\mathcal{H^i}} \mathcal{H} \hookrightarrow \mathcal{G} $$</div> | ||
[01: | ''[https://youtu.be/Z7rd04KzLcg?t=5274 01:27:54]''<br> | ||
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 when I wish I remembered more of this stuff—into the second factor, it turns out that this is actually a group homomorphism. | |||
= | <div style="text-align: center; margin-left: auto; margin-right: auto;">$$ h \rightarrow (h, h^{-1} d_{A_0} h) $$</div> | ||
[01: | ''[https://youtu.be/Z7rd04KzLcg?t=5305 01:28:25]''<br> | ||
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 GmodHτ, and if we have any interesting representation of \(H\), we can form homogeneous vector bundles and work with induced representations. And that's what the fermions are going to be. So the fermions in our theory are going to be \(H\) modules, and the idea is that we're going to work with vector bundles of the form inhomogeneous gauge group producted over the tilted gauge group. | |||
<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \mathcal{G} / \mathcal{H_{\tau}} $$</div> | |||
= | <div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \epsilon = \mathcal{G} \times_{H_{\tau}} \Upsilon $$</div> | ||
<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \Upsilon_H = \text{Fermi Cover} $$</div> | |||
[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 | ''[https://youtu.be/Z7rd04KzLcg?t=5387 01:29:47]''<br> | ||
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 spinorial 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. So just the way when we look at supersymmetry, we can take products of the spin-1/2 fields and map them into the linear sector, we can do the same thing here. | |||
[01:30: | ''[https://youtu.be/Z7rd04KzLcg?t=5419 01:30:19]''<br> | ||
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 Lorentz or Poincaré 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. So it's perfectly built for representation theory, and if you think back to Wigner's classification, and the concept that a particle should correspond to an irreducible representation of the inhomogeneous Lorentz group, we may be able to play the same games here, up to the issue of infinite-dimensionality. | |||
[01: | ''[https://youtu.be/Z7rd04KzLcg?t=5475 01:31:15]''<br> | ||
So right now, our field content is looking pretty good. It's looking unified, in the sense that it has an algebraic structure that is not usually enjoyed by field content, and the field content from different sectors can interact and know about each other, provided we can drag something out of this with meaning. | |||
==== Toolkit for the Unified Field Content ==== | |||
[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 $$\mathcal{G}$$ let's say to the $$\mathbb{R}$$. Invariant, not under the full group, but under the tilted gauge subgroup, $$\mathcal{H_{\tau}}$$. | [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 $$\mathcal{G}$$ let's say to the $$\mathbb{R}$$. Invariant, not under the full group, but under the tilted gauge subgroup, $$\mathcal{H_{\tau}}$$. | ||
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[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. | [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. | ||
[01:33:43] So we've, we've restricted ourselves to one gauge group, this big unitary group on the spinors using whatever sort of inner product naturally exists on the spinors. And not spinors valued in an auxiliary structure, but intrinsic spinors. | [01:33:43] So we've, we've restricted ourselves to one gauge group, this big unitary group on the spinors using whatever sort of inner product naturally exists on the spinors. And not spinors valued in an auxiliary structure, but intrinsic spinors. | ||