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

=== Lecture at the University of Oxford ===

''[https://youtu.be/Z7rd04KzLcg?t=2123 00:35:23]''<br>

And then, interestingly, he went on to say one more thing. He said of course, these three central observations must be supplemented with the idea that this all takes place treated in quantum mechanical fashion, or quantum field theoretic. So it's a bit of an after-market modification rather than, in his opinion at the time, one of the core insights. I actually think that that's, in some sense, about right. Now, one of my differences with the community, in some sense, is that I question whether the quantum is in good enough shape, that we don't know whether we have a serious quantum mechanical problem or not. We know that we have a quantum mechanical problem, a quantum field theoretic problem, relative to the current formulations of these theories, but we know that in some other cases, the quantum becomes incredibly natural, sometimes sort of almost magically natural. And we don't know whether the true theories that we will need to be generalizing, in some sense, have beautiful quantum mechanical treatments, whereas the effective theories that we're dealing with now may not survive the quantization.

''[https://youtu.be/Z7rd04KzLcg?t=2934 00:48:54]''<br>

The proposal that I want to put to you today is that one of the reasons that we may be having trouble with unification is that the duty, our duty, may be to generalize all three vertices before we can make progress. 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 in Euclidean signature, generating all of the Atiyah-Singer theory. We know that Einstein's theory is, in some sense, a unique spin-2 massless field capable of communicating gravity, which can be arrived at from field-theoretic rather than geometric considerations. 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.

''[https://youtu.be/Z7rd04KzLcg?t=3461 00:57:41]''<br>

==== Motivations for Geometric Unity ====

''[https://youtu.be/Z7rd04KzLcg?t=3552 00:59:12]''<br>

So I have in mind differential operators, parameterized by some fields \(\omega\), which when composed are not of second-order if these are first-order operators, but of zeroth-order, and some sort of further differential operator saying that whatever those two operators are in composition is in some sense harmonic. Is such a program even possible? So if the universe is in fact capable of being the fire that lights itself, is it capable of managing its own flame as well and closing up? What I'd like to do is to set ourselves an almost impossible task, which is to begin with this little data in a sandbox, to use a computer science concept.

''[https://youtu.be/Z7rd04KzLcg?t=3911 01:05:11]''<br>

==== The Observerse and Four Versions of GU ====

''[https://youtu.be/Z7rd04KzLcg?t=3979 01:06:19]''<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.

''[https://youtu.be/Z7rd04KzLcg?t=4539 01:15:39]''<br>

===== Inhomogeneous Gauge Group and Tilted Gauge Group =====

''[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.

''[https://youtu.be/Z7rd04KzLcg?t=5094 01:24:54]''<br>

Well, that's pretty good, if true. Can you go farther? Well, look at how close to this field content is to the picture from deformation theory that we learned about in low dimensions. The low-dimensional world works by saying that symmetries map to field content map to equations, usually in the curvature.

''[https://youtu.be/Z7rd04KzLcg?t=6774 01:52:54]''<br>

We've got problems. We're not in four dimensions, we're in 14. We don't have great field content because we've just got these unadorned spinors, and we're doing gauge transformations effectively on the intrinsic geometric quantities, not on some safe auxiliary data that's tensor producted with what our spinors are. How is it that we're going to find anything realistic? And then we have to remember everything we've been doing recently has been done on \(U\). We've forgotten about \(X\). How does all of this look to \(X\)?

''[https://youtu.be/Z7rd04KzLcg?t=7504 02:05:04]''<br>

So \(X\) is sitting down here, and all the action is happening up here on \(U^{14}\). There's a projection operator—I've used \(\pi\) twice, it's not, here, the field content, it's just projection. And I've got a \(\sigma\), which is a section. What does \(\zeta\) pulled back or \(\nu\) pulled back look like on \(X^4\)?

''[https://youtu.be/Z7rd04KzLcg?t=7565 02:06:05]''<br>

Okay. Let's try to think about how we would come up with this field content starting from first principles. Let's imagine that there's nothing to begin with. Then you have one copy of matter: whatever it is that we see in our world, the first generation. In order for that to become interesting, it has to have an equation, so it has to get mapped somewhere.

''[https://youtu.be/Z7rd04KzLcg?t=7597 02:06:37]''<br>

Then we see the muon and all the rest of the matter that comes with it, and we have a second generation.

''[https://youtu.be/Z7rd04KzLcg?t=7604 02:06:44]''<br>

===== Third Generation is an Imposter =====

''[https://youtu.be/Z7rd04KzLcg?t=7614 02:06:54]''<br>

One thing we could do is we could move these equations around a little bit and move the equation for the first generation back, and then we can start adding particles. Let's imagine that we could guess what particles we'd add.

''[https://youtu.be/Z7rd04KzLcg?t=7630 02:07:10]''<br>

We'd add a pseudo-generation of 16 particles, spin-3/2, never before seen. Not necessarily superpartners, but Rarita-Schwinger matter with familiar internal quantum numbers, but potentially, so that they're flipped, so that matter looks like anti-matter to this generation. Then we add, just for the heck of it, 144 spin-1/2 fermions, which contain a bunch of particles with familiar quantum numbers, but also some very exotic looking particles that nobody's ever seen before.

''[https://youtu.be/Z7rd04KzLcg?t=7666 02:07:46]''<br>

Now we start doing something different. We make an accusation. One of our generations isn't a regular generation: it's an impostor. At low energy, in a cooled state, potentially, it looks just the same as these other generations, but were we somehow able to turn up the energy, imagine that it would unify differently with this new matter that we've posited rather than simply unifying onto itself. So two of the generations would unify onto themselves, but this third generation would fuse with the new particles that we've already added. We consolidate geometrically. We can add some zeroth-order terms, and we imagine that there is an elliptic complex that would govern the state of affairs.

''[https://youtu.be/Z7rd04KzLcg?t=7716 02:08:36]''<br>

=== Powerpoint ===

''[https://youtu.be/Z7rd04KzLcg?t=8004 02:13:24]''<br>

====Preliminary====

''[https://youtu.be/Z7rd04KzLcg?t=8055 02:14:15]''<br>

First of all, I think the most important thing to begin with is to ask what new hard problems arise when you're trying to think about a fundamental theory that aren't found in any earlier theory. Now, every time you have an effective theory, which is a partial theory, there is always the idea that you can have recourse to a lower-level strata. So you don't have to explain, in some sense, everything coming from very little, or nothing. I think that the really difficult issue that people don't talk enough about is the problem of the fire that lights itself. And I think this was beautifully demonstrated by M.C. Escher in his famous lithograph ''Drawing Hands'', where he takes the idea of the canvas, or the paper, as a given, but somehow he imagines that the canvas could will into existence the ink needed to draw the hands that move the pen that draw the hands. That concept is actually the super tricky part, in my opinion, about going from effective theories to any attempt at a fundamental theory. So, with that said, what I want to think about is what antecedents does this concept have in physics? And I find that there really aren't any candidate theories of everything, or unified field theories, that I can find that plausibly give us an idea of how a canvas would will an entire universe into being. And so, that really, to me, is the conceptual problem that I think bedevils this, and makes the step quite a bit more difficult than some of the previous technical steps.

''[https://youtu.be/Z7rd04KzLcg?t=8159 02:15:59]''<br>

==== Sector I: The Observerse ====

''[https://youtu.be/Z7rd04KzLcg?t=8248 02:17:28]''<br>

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, "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.

''[https://youtu.be/Z7rd04KzLcg?t=8467 02:21:07]''<br>

==== Sector II: Unified Field Content ====

''[https://youtu.be/Z7rd04KzLcg?t=8610 02:23:30]''<br>

So that gets rid of the biggest problem, because the internal symmetry group is what causes the failure, I think, of supersymmetric particles to be seen in nature, which is we have two different origin stories, which is a little bit like Lilith and Genesis. We can't easily say we have a unified theory if spacetime and the \(\text{SU}(3) \times \text{SU}(2) \times \text{U}(1)\) group that lives on spacetime have different origins and cannot be related. In this situation, we tie our hands and we have no choice over the group content.

''[https://youtu.be/Z7rd04KzLcg?t=8746 02:25:46]''<br>

So just to fix bundle notation, we let \(H\) be the structure group of a bundle \(P_H\) over a base space \(B\). We use \(\pi\) for the projection map. We've reserved the variation in the \(\pi\) orthography for the field content, and we try to use right principal actions. I'm terrible with left and right, but we do our best. We use \(H\) here, not \(G\), because we want to reserve \(G\) for the inhomogeneous extension of \(H\) once we move to function spaces.

''[https://youtu.be/Z7rd04KzLcg?t=8783 02:26:23]''<br>

So with function spaces, we can take the bundle of groups using the adjoint action of \(H\) on itself and form the associated bundle, and then move to \(C^\infty\) sections to get the so-called gauge group of automorphisms. We have a space of connections, typically denoted by \(A\), and we're going to promote \(\Omega^1(B,\text{ ad}(P_H))\) to a notation of script \(\mathcal{N}\) as the affine group, which acts directly on the space of connections.

''[https://youtu.be/Z7rd04KzLcg?t=8827 02:27:07]''<br>

Now, the '''inhomogeneous gauge group''' is formed as the semi-direct product of the gauge group of automorphisms together with the affine group of translations of the space of connections. And you can see here is a version of an explicit group multiplication rule, I hope I got that one right.

''[https://youtu.be/Z7rd04KzLcg?t=8847 02:27:27]''<br>

And then we have an action of \(G\), that is the inhomogeneous gauge group, on the space of connections, because we have two different ways to act on connections. We can either act by gauge transformations or we can act by affine translations. So putting them together gives us something for the inhomogeneous gauge group to do.

''[https://youtu.be/Z7rd04KzLcg?t=8866 02:27:46]''<br>

We then get a bi-connection. In other words, because we have two different ways of pushing a connection around, if we have a choice of a base connection, we can push the base connection around in two different ways according to the portion of it that is coming from the gauge transformations of curly \(\mathcal{H}\) or the affine translations coming from curly \(\mathcal{N}\). We can call this map the '''bi-connection''', which gives us two separate connections for any point in the inhomogeneous gauge group. And we can notice that it can be viewed as a section of a bundle over the base space to come when we find an interesting embedding of the gauge group that is not the standard one in the inhomogeneous gauge group.

''[https://youtu.be/Z7rd04KzLcg?t=8922 02:28:42]''<br>

So our summary diagram looks something like this. Take a look at the \(\tau_{A_0}\). We will find a homomorphism of the gauge group into its inhomogeneous extension that isn't simply inclusion under the first factor. We—and I'm realizing that I have the wrong pi projection, that should just be a simple \(\pi\) projecting down. We have a map from the inhomogeneous gauge group via the bi-connection to \(A \times A\), connections cross connections, and that behaves well according to the difference operator \(\delta\) that takes the difference of two connections and gives an honest ad-valued one-form.

''[https://youtu.be/Z7rd04KzLcg?t=8963 02:29:23]''<br>

==== Sector III: Toolkit for the Unified Field Content ====

''[https://youtu.be/Z7rd04KzLcg?t=9009 02:30:09]''<br>

Now, in sector III, there are payoffs to the magic beans trade. The big issue here is that we've forgone the privilege of being able to choose and dial in our own field content, and we've decided to remain restricted to anything we can generate only from \(X^d\), in this case \(X^4\). So we generated \(Y^{14}\) from \(X^4\), and then we generated chimeric tangent bundles on top of that. We built spinors off of the chimeric tangent bundle, and we have not made any other choices. So we're dealing with, I think it's \(U^{128}\), \(U^{2^7}\). That is our structure group, and it's fixed by the choice of \(X^4\), not anything else.

''[https://youtu.be/Z7rd04KzLcg?t=9058 02:30:58]''<br>

===== Shiab Operators =====

''[https://youtu.be/Z7rd04KzLcg?t=9094 02:31:34]''<br>

We then get to shiab operators. Now, a '''shiab operator''' is a map from the group crossed the ad-valued i-forms. In this case, the particular shiab operator we're interested in is mapping i-forms to d-minus-three-plus-i-forms. So for example, you would map a two-form to d-minus-three-plus-i. So if d, for example, were 14, and i were equal to two, then 14 minus three is equal to 11 plus two is equal to 13. So that would be an ad-valued 14-minus-one-form, which is exactly the right place for something to form a current, that is, the differential of a Lagrangian on the space.

''[https://youtu.be/Z7rd04KzLcg?t=9158 02:32:38]''<br>

So the augmented torsion is relatively well behaved, relative to this particular slanted, or tilted, embedding of the gauge group in its inhomogeneous extension. This is very nice, because now we actually have a use for the torsion. We have an understanding of why it may never have figured, particularly into geometry, is that you need two connections rather than one to see the advantages of torsion at all.

''[https://youtu.be/Z7rd04KzLcg?t=9223 02:33:43]''<br>

===== GU Equations: Swervature and Displasion =====

''[https://youtu.be/Z7rd04KzLcg?t=9310 02:35:10]''<br>

So, given that you've been on a long journey, here is something of what Geometric Unity equations might look like. So in the first place, you have the swerved curvature, the shiab applied to the curvature tensor. That, in general, does not work out to be exact, so you can't have it as the differential of a Lagrangian. And in fact, you know, we talked about swirls, swerves, twirls, eddies—there has to be a quadratic '''eddy tensor''', that I occasionally forget when I pull this thing out of mothballs, and the two of those together make up what I call the '''total swervature'''. And, on the other side of that equation, you have the displaced torsion, which I've called the '''displasion'''. And to get rid of the pesky, sort of, minus sign and Hodge star operator... This would be the replacement for the Einstein equation, not on \(X\) where we would perceive it, but on \(Y\) before being pulled back onto the manifold \(X\).

''[https://youtu.be/Z7rd04KzLcg?t=9372 02:36:12]''<br>

==== Sector IV: Fermionic Field Content ====

''[https://youtu.be/Z7rd04KzLcg?t=9393 02:36:33]''<br>

Next is the sketch of the fermionic field content. I'm not sure whether that should have been sector IV or sector III, but it's going to be very brief. I showed some pictures during the lecture, and I'm not going to go back through them, but I wanted to just give you an idea of where this mysterious third generation I think comes from.

''[https://youtu.be/Z7rd04KzLcg?t=9415 02:36:55]''<br>

Now recalling that, when I started my career, we did not know that neutrinos were massive. And I figured that they probably had to be massive because I desperately wanted a 16-dimensional space of internal quantum numbers, not 15, because my ideas only work if the space of internal quantum numbers is of dimension \(2^n\). And, one of my favorite equations at the time was \(15 = 2^4\). Not literally true, but almost true. And thankfully in the late 1990s, the case for 16 particles in a generation was strengthened when neutrinos were found to have mass. But that remaining term in the southeast corner, the spinors on \(X\) tensor spinors on \(Y\) looks like the term above it in line 2.15. And that, in fact, is the third generation of matter, in my opinion. That is, it is not a true generation. It is broken off, and would unify very differently if we were able to heat the universe to the proper temperature.

''[https://youtu.be/Z7rd04KzLcg?t=9604 02:40:04]''<br>

=== Closing Thoughts ===

''[https://youtu.be/Z7rd04KzLcg?t=9798 02:43:18]''<br>

And most especially, I just want to say that I've asked a tremendous amount from my family to stick with me on this quixotic quest. I want to thank Pia Malaney, Naila Weinstein, and Zev Weinstein. I love you all very much, and thank you for making this possible.

''[https://youtu.be/Z7rd04KzLcg?t=9888 02:44:48]''<br>