Editing Eric Weinstein: A Conversation (YouTube Content)

So I think that you have a situation by which new ideas are always not properly instantiated, and the community that is constantly trying to make sure that... I think that the idea is that people are foolish enough to play this game with the most aggressive members of the community, because the implication is if you won't come up with a testable prediction that invalidates your theory, you're anti scientific and we have no time for this. And so people, well like, you know, with the <math>\text{SU}(5)</math> theory, they immediately said okay, well it predicts proton decay. Well, grand unification is a larger idea, and some versions and instantiations do predict proton decay, and some do not. So what are you going to say about that? I think that the problem is that we're not in an adult phase where we've faced up to the fact that we have almost 50 years of stagnation, and what you're seeing with this proliferation of new claimants to have fundamental theories is, in part, that string theory has finally weakened itself, and the aging of the particular cohort—which is Baby Boomers, who are the string theory proponents—they've gotten weak enough that effectively other people feel emboldened. And I think Stephen Wolfram said this recently, that in a previous era, he would have expected to have been attacked. But we've been waiting around for so long that perhaps the political economy of unification and wild ideas has changed somewhat.

'''Eric Weinstein:''' So effectively, what I'm asking is, can a manifold <math>X^4</math> produce the baroque structure of the Standard Model? Now—and gravity. And if you think back to the famous mug popular in the CERN gift shop, there really isn't that much going on in the Standard Model if you group terms in particular ways. But there's a lot of weirdness. Why the Lorentz group, why <math>\text{SU}(3) \times \text{SU}(2) \times \text{U}(1)</math> for the internal symmetries generating the forces, why three families? I thought that something that many younger viewers may not be aware of is that things really changed around 1983, '84. If you think about the original anomaly cancellation of Green and Schwarz in 1984, I believe, you could ask what was physics like right before that moment? And I think it's absolutely shocking, because we don't realize the extent to which the string theorists really redefined what the major problems in physics were. I think most people in the post-string era somehow believe that the major issue is quantum gravity. And I don't really, I just find it astounding, because that's really what the string theorists were selling.  

Okay, so what are his big questions? Why this particular structure for the families? In particular, why flavor chiral with left- and right-handed particles being treated differently by the weak force, rather than say vectorlike ones left and right transformable into being treated the same? Next, why three families? That generalizes Robbie's famous question "Who ordered that?" as if the universe was a Jewish deli, commenting on the muon. How many sets of Higgs bosons are there? We talk about the Higgs boson, but maybe there are multiple sets and there are multiple different scales at which symmetry is broken and mass is imparted through soft mass mechanisms. Lastly, why <math>\text{SU}(3) \times \text{SU}(2) \times \text{U}(1)</math>? Remember, <math>\text{SU}(3)</math> is the color force for the strong force, but <math>\text{SU}(2)</math> here is weak isospin, which has not yet become the W and Z's. And this <math>\text{U}(1)</math> is weak hypercharge, which has not yet become electromagnetism through symmetry breaking. And in some sense, I just feel sort of sad that we don't think of these as questions because we know not to ask them.  

'''Eric Weinstein:''' You have <math>X^n</math> for a manifold of n-dimensions. Make it orientable with a particular orientation, make it have a unique spin structure, whatever you need to do to set it up as a decent manifold. Replace that manifold, momentarily, by the bundle of all metric tensors pointwise on the same space. And that way, spacetime would be a particular section of that bundle. Let me see if I can find a...

Okay. So right here, I've got a 4-dimensional manifold. Imagine that I'm interested in looking at the bundle of all pointwise metrics, which is going to be—if the base space is 4-dimensional, make <math>4 = n</math>—it will be of dimension <math>n^{\frac{n^2 + 3n}{2}}</math>. So <math>4^2</math> is 16, plus <math>3n</math>, <math>3 \times 4 = 12</math>. So <math>16 + 12 = 28</math>, divided by 2 is 14. If you have a <math>(1,3)</math> metric downstairs, I believe that you are naturally courting a <math>(7, 7)</math> or <math>(9, 5)</math> metric upstairs. And that is the first step in GU, which is that you replace a single space with one particular metric by a pair of spaces, a total space and a base space of a fiber bundle—this is in the strong form of GU—and physics mostly happens upstairs on the bundle of all metrics, not downstairs on the particular space that got you started.  

Here, <math>U^4</math> is an open set in <math>X^4</math>. Okay, so effectively, what are we saying? We're saying that physics is going to dance on not only the space of four coordinates, typically <math>x</math>, <math>y</math>, <math>z</math>, and <math>t</math>, or thinking in a coordinate-independent fashion, simply four parameters, it's also going to dance on the space of rulers and protractors at every given point. And so that structure is the beginning of GU, and then you can recover Einstein, spacetime, by simply saying that if I have a section of that bundle, that's a spacetime metric.

'''Eric Weinstein:''' Well, I gave three forms of GU. One form is the trivial form, in which you have the second space <math>Y</math> the same as the first space <math>X</math>. That means that you can easily recover everything Einstein did as a form of Geometric Unity by trivially making the observerse irrelevant. You're just repeating the same space twice, and you've got one map between them called the identity, and now you're back in your old world. So without loss of generality, you cover that. Another one is a completely general world, which I think—What did we call it here... Well, I called the middle one the Einsteinian one, where you actually make the second space <math>Y</math> the space of metrics. And that's the one that I think is the most interesting, but I don't want to box myself in, because I don't want to play these games of Simon's "You said this," or "You said that." You know, I can play the lawyerly game as well as anyone if that's what we are really trying to do. I thought we were trying to do physics.  

The thing that I'm trying to get at here is that I believe you and I are somehow having a pullback of a 14-dimensional conversation right now. My guess is that there is a space, with a <math>(7, 7)</math> metric, probably more likely than a <math>(9, 5)</math> metric, on 14 dimensions, where not only are the waves that are relevant going over the original coordinates <math>x_1</math> through <math>x_4</math>, they're also going through four ruler coordinates on the tangent bundle of the original <math>x</math> coordinates. So there are 4 rulers to measure the 4 directions, and then there are also going to be 6 protractors. Because if you name the directions John, Paul, George, and Ringo, you'd have John with Paul, John with George, John with Ringo, Paul with George, Paul with Ringo, George with Ringo. Right? And so, those 6 protractors are actually degrees of freedom for the fields, and the fields live on that space.  

Then the question is why do we perceive 4 dimensions and complicated fields? And the answer is pullbacks. When you have a metric, you have a map from the base space into the total space, so Einstein—we don't think of it this way—is embedding a lifeless space which is without form, <math>X^4</math>, into a 14-dimensional space before Geometric Unity ever even got on the scene, and giving him the ability to pull back information, which he may say is only happening on that tiny little slice, that little filament that is the 4-dimensional manifold swimming in a 14-dimensional world with a 10-dimensional normal bundle. But why not imagine that actually the fields are actually spread out over all 14 dimensions, and then all you're seeing is pullback information downstairs. Now the metric is doing something new that it wasn't doing before. It's pulling back data that is natural to <math>Y^{14}</math> as if it was natural on X, but I call this invasive fields versus native fields, just because some species are invasive, and some species are endemic, or native. The interesting thing about the bundle of all spinors, sorry, the bundle of all metrics, is that it almost has a metric on it. I don't know if I've ever heard anyone mention this.

'''Eric Weinstein:''' So in other words, assume that you haven't chosen a metric on <math>X^4</math>. What you have then is you have a 10-dimensional subspace along the fibers, which we can call the vertical space. And that 10-dimensional space at every point upstairs, every point is, in fact, a metric downstairs, being by construction. So that means it imparts a metric on 10-dimensional vectors along the fibers. Now those are symmetric 2-tensors, effectively, because it's a space of metrics. You have this really interesting space here, call that <math>V</math>. Well that <math>V</math> has a Frobenius metric based on the particular metric at which you are looking at the tangent space, which has got a 10-dimensional subspace picked out. If you map that 10-dimensional subspace into the 14-dimensional tangent space of the manifold <math>Y^{14}</math>, you can take a quotient and call that <math>H</math>. And that <math>H</math> will also have a metric because it's isomorphic to the dual of the pullback of the cotangent bundle downstairs. And the cotangent bundle has a metric because at that point that you picked in <math>Y^{14}</math> is itself a metric downstairs.  

So now you've got a metric on <math>V</math>, you've got a metric on <math>H^*</math>, and you just don't know how <math>H^*</math> becomes the complement to <math>V</math> and <math>T</math>. That's the only piece of data you're missing for a metric. So you've got a 4-metric, you've got a 10-metric, the 10-metric is sitting inside of the tangent. The 4-metric is naturally sitting inside of the cotangent bundle. They're weirdly complementary, you've got a metric on the nose but for one piece of data, which we call a connection. So up to a connection, the manifold <math>Y^{14}</math> has a metric on it without ever having chosen a metric because it's made out of metric data.

'''Eric Weinstein:''' Well, that's true for any—no, it's true for the spin representation. It's not true generically, for any representation. But it allows you to build the spinors on what should be the total space, because now you've got a 4-dimensional... So, I think it's here at 3.12. If the spinors of a sum are the tensor products of the spinors on the summands, and I create a new bundle, which is the 10-dimensional vertical bundle inside the tangent bundle direct sum the 4-dimensional bundle inside the cotangent bundle, then the spinors on that thing—which is isomorphic and in fact semi-canonically isomorphic to both the tangent bundle and the cotangent bundle, being chimeric, it's isomorphic, but it's not fully canonically. It's only semi-canonically. So spinors on that will be identifiable with the spinors on <math>Y</math> as soon as you have a connection that completes this and makes it fully canonically isomorphic.  

'''Eric Weinstein:''' So we've got a metric. The metric has a connection, the connection produces curvature that's Riemannian. We find that by identities, it's got three components. It tries to go towards metrics and the Weyl curvature is snapped off. Afterwards, the scalar curvature is lowered somewhat, or adjusted, by scalar curvature over 2 times <math>g_{\mu \nu}</math>. And so symbolically, what we've done is we've said Einstein threw away the Weyl curvature, readjusted the Ricci scalar curvature, and fed metric information through to the Levi-Civita connection, through to the Riemann curvature tensor, and then played these projection games to feed it back to the space of metrics. And that particular combination is perpendicular to the action of the diffeomorphism group on the space of all metrics, leading to a divergence free condition via our friend the Bianchi identity.  

The problem here is the gauge transformations act on the Lie algebra component and don't touch the form component. But Einsteinian projection, or contraction, or summing over <math>g_{\mu \nu}</math> indices, is democratic: it deals simultaneously with the form piece and the Lie algebra piece. So if you treat only the Lie algebra piece under a gauge transformation and you don't touch the form piece, then contraction followed by gauge transformation will never be the same thing as gauge transformation followed by contraction. And so that's the puzzle, which is if Geometric Unity is really about the idea of trying to say maybe it's not so much quantizing gravity, maybe it's a fight between the different geometry of Riemann and Ehresmann, because gauge transformations are Ehresmannian geometry but contractions are Riemannian geometry.  

So here's a GU approach, how do you get geometric harmony between General Relativity and gauge theory when you have the ship in a bottle problem? This is almost a tight analogy. You've got the curvature tensor, you apply a gauge transformation to two of the masts and you pass them through into ad-valued <math>(d-1)</math>-forms, and then you do an inverse gauge transformation, which is exactly how you do the ship in the bottle trick—by the way, Brian gave me a wonderful ship in a bottle, thank you very much—raising the mast inside. And then you can potentially, if need be, adjust one of the two masts again in order to get agreement.  

'''Eric Weinstein:''' Alright. Imagine that that torus that you see in the lower left corner of the screen is a 2-dimensional model, toy model, of spacetime. So going around through the center is like Groundhog Day, you come back to the same place and it's a repeating time cycle, and space is simply a circle. Now in such a world, we would normally think of quantum field theory or gravity as taking place on that object. You'd have fields, you'd have effectively functions called sections on that object, and what you're seeing here is something that's very hard to picture because it's 5-dimensional, but one trick here is because the torus has a property called parallelizability... The object on the right is a depiction of a metric. Each point that isn't on one of those two sheets is a potential metric at any given point on the torus. So in other words, if a metric is a symmetric non-degenerate 2-tensor, if you think of it as a matrix, it would be of the form <math>\begin{bmatrix} x & z \\ z & y \\ \end{bmatrix}</math>. Non-degenerate means that <math>xy - z^2 \ne 0</math>. So that's what's cutting out that variety, if you will, the zeros of the of the determinant would be points, given that there are 3 degrees of freedom in the metric.

So every point in that region is in play, and if you mapped—imagine that the stuff in that weird hourglassy region on the far right was like very warm and on the far left was very cold. Then if you map the torus in to the far left region, it would show up as being cold. If you mapped it into the far right region, you'd see it as being very hot. So every way of mapping the torus in pulls back different information from that hourglassy region. That is in large measure, in part, one of the things that may be going on with the illusion of many worlds, is that what you're seeing is that the metric may be capable of pulling back data that is dancing on the space of all metrics as well as the space of all points on the original manifold <math>X</math>. So in this case, you've got 2 degrees of freedom on the torus, you've got 3 degrees of freedom around the hourglass, and <math>2 + 3 = 5</math>.  

So the big take home from the restrictive version of GU that we're exploring here is that if you allow fields to dance on the space of metric apparatus—measurement apparatus—then the paradoxes of measurement start to make a lot more sense. You could also, potentially, try to keep the metric classical, because we have two spaces. We have a space downstairs <math>X</math>, which is just the torus, and we have a space upstairs, which is the torus, in this case, cross the hourglass region, as long as it doesn't touch the two sheets. So you've got a 5-dimensional manifold hovering over a 2-dimensional manifold, and fields on the 5-dimensional manifold will be perceived on the 2-dimensional manifold when you pull them back via a particular Einsteinian spacetime as fields on the tangent bundle of what you will call spacetime, together with fields on the normal bundle inside of the 5 dimensions.  

The normal bundle of a 2-dimensional manifold in a 5-dimensional space is 3-dimensional, so you're gonna see fields that look like spinors on 2 dimensions tensor spinors on 3 dimensions. If you were in 4 dimensions, make that torus in your mind represent a 4-dimensional spacetime, then that Diablo region would be a 10-dimensional region of metrics, because 4x4 matrices that are symmetric have <math>\frac{4^2 + 4}{2}</math> [Inaudible] for different degrees of freedom. In other words, you get a 10-dimensional normal bundle.  

Now you'll notice that if you have ordinary spinors on 14-dimensional space and you pull them back via a metric, which is a mapping of 4 into 14, it looks like spinors on the 4-dimensional space tensor spinors on the 10-dimensional normal bundle. If the normal bundle inherits the Frobenius metric from <math>X(1,3)</math>, and you glue in the trace piece in the right way—well, if you glue it in the wrong way, you'd get a <math>(7,3)</math> metric on the normal bundle. But if you glue it in the right way, you'd get a <math>(6,4)</math> metric on the normal bundle.  

<math>\text{Spin}(6,4)</math> is a sort of nasty non-compact group, so you might want to break to its maximal compact subgroup like Witten and Bar-Natan discuss. And the interesting thing about <math>\text{Spin}(6,4)</math> is that it has different names. By low-dimensional isomorphisms, <math>\text{Spin}(6)</math> is the same thing as <math>\text{SU}(4)</math>. <math>\text{Spin}(4)</math> is the same thing as <math>\text{SU}(2) \times \text{SU}(2)</math>. And <math>\text{SU}(4) \times \text{SU}(2) \times \text{SU}(2)</math> is the Pati–Salam theory. So you can argue that ordinary spinors on the induced metric in 14 dimensions, glued in the right way, pull back as Pati–Salam. And I don't know if anyone's ever discussed the connection between Einstein and Pati and Salam.  

'''Eric Weinstein:''' And whether you're talking about <math>\text{Spin}(10)</math> models, <math>\text{SU}(5)</math> models, or <math>\text{SU}(4) \times \text{SU}(2) \times \text{SU}(2)</math>, which is <math>\text{Spin}(6) \times \text{Spin}(4)</math>, isn't that exactly what we see in the Standard Model? So Frank Wilczek—let me just see if I can find this beautiful quote from him, because he definitely brought this up. And what I recently did when I had him on my podcast, which we haven't released—so, if we go over to my screen share...

'''Eric Weinstein:''' Let me read it. "A particularly intriguing feature of <math>\text{SO}(10)</math>," which is really <math>\text{Spin}(10)</math>spin 10, or it could be <math>\text{Spin}(6,4)</math>, "is its spinor representation, used to house the quarks and leptons, in which the states have a simple representation in terms of basis states labeled by a set of "+" and "-" signs. Perhaps this suggests composite structure." Now here's the sentence that just floored me. "Alternatively, one could wonder whether the occurrence of spinors both in internal space and in space-time is more than a coincidence." And then he pulls back immediately, "These are just intriguing facts; they are not presently incorporated in any compelling theoretical framework as far as I know." Geometric Unity is that compelling framework.