In a lecture held live at UCSD in April 2025 at the Astrophysics and Cosmology Seminar, Eric Weinstein presented an update to his theory of Geometric Unity.
Abstract
The Einstein Field Equations (EFE) traditionally treat the dark energy term Λ gᵤᵥ as a non-geometric addition to the curvature-based Einstein tensor Gᵤᵥ. In contrast, the framework of Geometric Unity (GU) reformulates the EFE as part of a covector field on an inhomogeneous gauge group over the Dirac spinor bundle of a 14-dimensional space of Lorentzian metrics. This setting introduces a new tensor ϑ_ω with exceptional equivariance properties, promoting dark energy to a fully geometric, gauge-theoretic quantity. This geometric approach naturally extends to a supersymmetric version of the Einstein–Dirac system and yields three Pati–Salam generations when compactification is imposed. The resulting structure aligns the Einstein–Dirac sector with both General Relativity and the Seiberg–Witten monopole equations, offering a unified and geometrically grounded view of spacetime, matter, and gauge fields.
Transcript
00:00:00
Eric Weinstein: Quantum gravity is a mental disease which theoretical physics needs to rid itself of, and people have to be willing to say that in public.
00:00:08
Brian Keating: Eric Weinstein is one of the most revered thinkers of our generation.
00:00:13
Eric Weinstein: The great nap is now over. Right now, where we are is four light-years from the nearest star. There is no way to get to the speed of light or even close. The problem is that the culmination of all human theory about the base reality stagnated abruptly and mysteriously in nineteen seventy-three.
00:00:29
Brian Keating: Is the cosmological constant really necessary?
00:00:32
Eric Weinstein: So we have a cosmological constant term without an explanation and an explanation torsion without a term.
00:00:39
Brian Keating: And a radical new prediction that could explain the recent results from the DESI project, a startling claim that Einstein’s cosmological constant is not at all what Einstein thought it was.
00:00:50
Eric Weinstein: This is the term which everyone loves. This is the term which we keep going back and forth. Is it a blunder? Is it genius? Is it a blunder? Is it genius? Einstein compared this to a building with fine marble, think Leaning Tower of Pisa, his greatest blunder, and cheap wood on the other side. The field is not producing new results. This is also terrible, but it turns wonderful because if we can find the problem, we can make progress and reach the stars.
00:01:18
Brian Keating: In today’s talk, you’re about to witness an update to his groundbreaking theory first presented at Oxford in twenty thirteen. This lecture was held live at the UC San Diego Physics Department in the Mayer Room in April twenty twenty-five at UCSD’s prestigious Astroparticle Cosmology Seminar. Here, one of the most brilliant mathematical physicists of our time presents his revolutionary theory of everything to an audience of awed skeptics and supporters. Can geometric unity actually solve the deepest mysteries of our universe? Or will it join a host of others who have tried in the past unsuccessfully to create a new unified theory?
00:01:58
Eric Weinstein: Thank you guys so much. Uh, what I wanted to talk about is the cosmological sector has a very different character than either the general r-relativistic, um, attempt at an equation for fie- for the gravitational field, uh, or the standard model because in essence the standard model, uh, got codified in Ehresmannian bundle theoretic geometry. So only the Higgs sector sort of has this kind of hobbyist flavor. Everything else is pretty much kind of, uh, locked in. So what I have is three basic equations. The central one is from geometric unity. This is the bosonic part. This is the fermionic part. My, uh, difficulty with this field concerns the bottom equation, which in nineteen eighty-seven or thereabouts was called insufficiently nonlinear. It later became sufficiently [chuckles] nonlinear in nineteen ninety-four when Ed Witten and Natty Seiberg did it. And on top I have the Einstein field equations. So what I wanna talk about is the fact that we can’t continue with dark energy as a constant lambda times the metric just for the purpose of maintaining divergence free across the various terms of the equation. In case any of you have to leave early, my claim is, is that this is going to end up as the formula for dark energy, what currently is lambda times g mu nu. Epsilon sub omega is gonna be a gauge transformation. This is gonna be an exterior derivative minimally coupled to a connection that will come from something called alpha. And this is actually a pi, which we don’t use all that much, which is an add valued one form or a gauge potential. So basically, this whole thing is gonna live in add valued one forms, and it’s gonna replace the cosmological constant times the metric, and you have to ask yourself, on what kind of a gadget does that live? So the claim is that what we’re going to be doing is taking a semidirect product. So if you f-are familiar with the Poincaré group, think about the group of gauge transformations as what the Lorentz group always wanted to be and, um, the space of add valued one forms or gauge potentials being the natural, uh, linear space upon which an affine space of connections is modeled, so that’ll be playing the role of the m- four momenta. So the idea is that you form the semidirect product as a group to begin with. That object seems to be wildly understudied, which I find very strange. If you have a single connection, you can push it around either by gauge transformations, or you can push it around by adding gauge potentials to it. So you have two different ways to take a single connection from every element in this semidirect product and to create two connections, uh, from which you can examine curvature, and you can also look at their differences. Y fourteen is going to be the space of pointwise Lorentzian metrics on an X four that has not yet become space-time. So imagine that you’re given space-time, which is a four dimensions– four-dimensional structure, but you’re going to use a, what mathematicians would call a forgetful functor, forget the metric, at least initially, pass to the f-frame bundle, the vierbeins, take a double cover of those, so you’re in GL four R double cover, mod out by spin one comma three, and that will give you a fourteen-dimensional, uh, object naturally. Um, and, and the idea is that that will end up replacing X four as the place where we do our quantum work, and then X four will be the place where we do our classical work, and so you’ll keep this separated in two different spaces. So unlike other branches of physics which keep progressing, what we know in fundamental physics is measured by the Lagrangian or the direct Euler-Lagrange equations has not been moving very much at all, so that the CERN and related mugs don’t need to be changed in the merch shop. And the question is, what is going wrong that somehow we used to be great at this stuff, and then we discovered Ken Wilson, and then we became really bad at it because we believed that we couldn’t figure out, uh, how to do anything, uh, w- that doesn’t have a unique UV completion? So today’s talk is the dark energy fragment of a larger theory. To your point, and you’re just anticipating this, imagine that you have general relativity in the standard model, first two rows. You wanna know in what– over what ambient space are they phrased? What bundle will concern us most? What will be the structure group of that bundle? What are the– what is the field content of the theory? At least in this case, uh, A would be a space of connections, and S would be the space of spinors. I’m leaving out the Higgs for the moment. And you have an action, in this case, Yang-Mills plus Dirac plus Higgs. In GU, there’s a first-order theory and then a second-order theory that’s built from the first-order theory. So the first-order theory encapsulates the Einsteinian and Dirac components, and then the second-order theory effectively is its square. So you have a square and a square root, think double copy. Uh, you’ve got Yang-Mills, Link-Neroitz, Laplacians, et cetera, et cetera. So this is the schematic for comparing. The X four is contained here in this Y fourteen that is endogenous. It doesn’t– It’s not extra dimensions. It’s not Kaluza-Klein. The space that is four-dimensional births its own fourteen-dimensional ambient space. So if you think about the Leaning Tower of Pisa, you, you’ve already made a mistake. And what is that? If you have a problem in a system and you only see it in one place, you try to fix that problem. You might try to fix the cosmological constant. What’s not as well known is that all the towers in Pisa seem to lean. In fact, the Leaning Tower isn’t the leanest of all of the towers in Pisa because they’ve got a soil problem from the Arno River. And so effectively, what I’m talking about is not complaining about these beautiful structures, but moving them wholesale to a different world which doesn’t have all of these problems of, uh, let’s say, the diffeomorphism group is notoriously badly behaved as an infinite dimensional function space group. You can’t quantize spin two fields very easily. There are all sorts of problems with going through, uh, the degeneracy point that t-takes a Eucli-Euclidean signature metric into a Lorentzian one. And so effectively, what you need to do is you need to take all of the stuff that physics has done really well and resituate it on totally different soil according to GU. Many of you will think that the purpose of theoretical physics, if you, if you came up after Ed Witten, and I, I was there at the first lecture Ed Witten ever gave on D equals ten supersymmetry at the University of Pennsylvania in nineteen eighty-three. I’m the only person I think in my fifties still who was at that lecture. Um, everything changed around that time. We started hearing a perseverant cry that quantum gravity is the Holy Grail of theoretical physics. And I just wanna say categorically that the fact that no one stands up against this wholesale and says, “This is a complete bait and switch in the history of physics. This has never been the Holy Grail of physics. This is a pet project of Bright’s DeWitt, and it is not something that is intrinsic to this field.” This is the youngest theorist, uh, with a Nobel Prize, uh, in fundamental physics. You’ll notice that it changes character around nineteen eighty-four. That person was always below fifty years old. Currently, that person is Frank Wilczek in his early to mid-seventies, born in nineteen fifty-one. Um, what’s going on? This is a search on quantum gravity, all books published in English. If this was the Holy Grail of theoretical physics, somebody explain to me why there’s no trace of this phrase before we stagnated in nineteen seventy-three. I would submit to you that quantum gravity is a mental disease which th-theoretical physics needs to rid itself of, and people have to be willing to say that in public and not simply continue to spend decade after decade spinning our wheels, getting nothing done. Let’s talk about Einstein and the Einstein field equations. Supposedly, this is one of the most beautiful and powerful equations that landed Einstein the Man of the Century, uh, for Time Magazine back in the twentieth century. This is the term which everyone loves. This is the term which we keep going back and forth. Is it a blunder? Is it genius? Is it a blunder? Is it genius? Uh, and then there’s this sort of ad hoc term that just get– you get from taking random field content and a Lagrangian varying the metric. Now, Einstein compared this, and I, I don’t speak German, so forgive me, to a building with fine marble, think Leaning Tower of Pisa, his greatest blunder, [chuckles] and cheap wood on the other side. So basically, one out of three terms is perfect, and I would say artificially so, and the other two terms are sort of unsalvageable. And the question is, why would we have this situation? Well, this satisfies an automatic differential equation. There’s an in– an intertwining operation where if you take the divergence operator attacking g mu nu not as a tensor but as an operator on add value two forms where the adjoint bundle is that of the Lorentz group, you get an R I J K L, or in this case, uh, with rhos and mus and nus. And if you pass this differential operators through this, you get the exterior derivative minimally coupled to the Levi-Civita connection of its own curvature has to equal zero by the Bianchi identity, so you get a contracted Bianchi identity. That means that if this gadget over here is zero and you throw the dark energy term to the other side, you need an automatic reason why it will also be divergence free. Now, what you have is you have the metric, which is always annihilated by its own Levi-Civita connection. And so if you take a product, by the product rule you have to have that the derivative of lambda as a field has to die, and that’s how we ended up with a cosmological constant. And once it’s constant, it has no explanation. It can’t rise and fall to meet the needs of the Riemann curvature tensor in its Einsteinian form. And so as a result We’re left with a term that does satisfy an automatic differential equation just like this one, but it’s completely preposterous, and we can’t figure out how to do better. That’s why it’s the greatest blunder, because it’s sitting inside of this beautiful equation, but it in fact has a lousy reason for being divergence-free. So why is it that Einstein only embraced his own tensor as being made of marble? Well, it’s dynamic and natural. We love that about it. It’s interpretable. It measures something that we, we care about, which is curvature, even if it takes a little bit of, uh, effort to feel Ricci curvature. It’s second order or less, it’s divergence-free from what we were just talking about. And it turned out that when Einstein corrected it with the minus, uh, scalar curvature over two times the metric, that it became divergence-free, which, uh, Hilbert pounced on and said, “Well, that’s because it comes from the simplest possible Lagrangian.” And as we’ve just said, such a curvature term appears to be unique, and there appears to be no other ways to get dark energy, so long as it’s sitting on the lousy foundation of the space of all metrics. That’s an infinite dimensional, badly behaved function space. So conclusion is we are likely not working in the right place. Have any of you heard of geometric unity? Do you have any understanding of what it is? In essence, geometric unity is a claim that the two theories that are thought to be incompatible, the standard model and general relativity, are, after Jim Simons and C.N. Yang in Stony Brook, both based on differential geometry, but they’re based on two different flavors. One is Arismanian geometry, in the case of the standard model, only known since around nineteen seventy-five, and the other one is Riemannian or pseudo-Riemannian geometry based since inception of general relativity, uh, around nineteen thirteen through fifteen with Grossman. The key features that I wanna call is that you have complete content freedom. You can dial in SU3 cross SU2 cross U1. You just have to worry about things like anomalies. Uh, you don’t need to know where that comes from in the case of Arismanian geometry. But in the case of Riemannian geometry, you have a distinguished connection, which we don’t actually use all that much in the theory of general relativity unless you’re in the Palatini, uh, sort of a school. But what Einstein did that we all should think about is he used a contraction. He used the fact that he had a Riemann curvature tensor that was an ad-valued two-form, and he said, “You know, ad is just another copy of the two-form, so I have a two-form valued two-form. I can contract one index on either side of a tensor product to get a symmetric two tensor.” And that move is not allowed in the Arismanian world because it says you’re treating the two different– two, two, uh, forms differently. You’re gauge rotating one, you’re holding the other fixed because it’s tied to the manifold. So in essence, you have two different geometries, and rather than it being a fight about the quantum and the children of Bohr versus the children of Einstein, it’s really about two different versions of differential geometry. It’s an unacknowledged battle between Charles Arisman, the Alsatian, and Bernhard Riemann, the German. Those two people had two different flavors that they based this theory on, and the key issue is, is that general relativity is not compatible with the classical version of the standard model before quantization, because you can’t gauge general relativity. And there’s a fake meme that goes through the physics community that says, “Oh, the different– the diffeomorphism group is just like the gauge group, and it all looks the same.” That is not gauge theory. That is an attempt to make everything look the same when it really isn’t. Okay, so we have a, a const– a cosmological constant term without an explanation and an explanation torsion without a term. So we have these three basic tensors that pervade differential geometry, the metric tensor, which we use all the time, the Riemann curvature tensor, which we use all the time, and the torsion tensor that you briefly learn about during your first week in differential geometry and then is studied by somebody in Uruguay or Botswana. The question is: Why is torsion the weak sister in this, uh, triumvirate? So the obvious guess is that if you wanted to rebase the cosmological constant, you would find some way to integrate the torsion, which is sitting there neglected like a wallflower where everyone else is at the dance. So here’s some simple observations that I wanna make clear. Certain that you have a theory in which there’s a distinguished connection, A sub olive, on a principal bundle over the space Y. If you ask, what is the formula for a random connection that can be thought of relative to your base connection? The formula for a gauge transformation is that if I take A, the connection A, which is, uh, up to you to choose, minus the distinguished connection, that gives me an ad-valued one-form. I can conjugate and take an adjoint representation based on a gauge transformation, and then I get this other term over here that is not so nice because it spoils gauge invariance. That’s why we can’t have a bare term where we can’t just give mass easily to, let’s say, photons and gluons and things like that, because that would spoil gauge invariance. But what if we had a second connection? What if you had a theory not of one connection, but of two connections? You’d repeat exactly the same statement. But the funny part is that the thing that doesn’t look good from the point of view of gauge equivariance is exactly the same. It has no dependence on the connection that you’re looking at. It’s simply a feature of gauge transformations of connections. So there’s a rule. Anytime you have a disease, you should either try to get rid of the disease and go for zero or to find an even number of diseases so you can have a Mexican standoff th– where every disease kills every other If you take a difference of these objects, the resulting difference in the space on which the affine space is modeled will be perfectly gauge equivariant. So the key problem is that we have a theory in the standard model where we have a single connection. But what if you have a distinguished connection and two different ways of pushing it around? You can either push it around by taking a gauge transformation, or you can add a random gauge potential to it. So in other words, if I have an element of the inhomogeneous gauge group, I have two sub-elements that can both push that one connection into different places, and then I can take a difference. And by the magic of the inhomogeneous gauge group, both of those connections are going to transform properly as well as their difference is going to be perfectly gauge equivariant. So imagine that you’re in this inhomogeneous gauge group. Then you have a map, tau, which takes the ordinary gauge group, let’s make it tau plus, into the inhomogeneous extension where G goes to tau plus of G equal to G. Now, I could just put in a zero here, uh, and that would be the, the sort of obvious homomorphism, but I can do a little bit better if I have a distinguished connection, which is D aleph G, and then I pre-multiply by G inverse. So once you have this copy of the gauge group sitting inside diagonally of the inhomogeneous gauge group, I can multiply on the right. So in other words, I have W– uh, curly mathcal W for those of you in LaTeX head, um, going to omega one of Ad P of G for a principle bundle. Equivariant says that this map, theta, is G equivariant under this subgroup. In other words, I can multiply this, uh, this inhomogeneous gauge group by its subgroup, and I can represent the gauge group on the space of Ad-valued one-forms, and those are compatible under this map theta that we’re going to take. So that’s what G equivariant means. G equivariant means I’ve got a map between two spaces, both of them have G actions, and it doesn’t matter whether I first rotate and then map or first map and then rotate. It’s a commutativity concept. Thanks for asking, Brian. Appreciate it. Are you always going to get torsion in this? Well, no. You could have– Your, you always have a place for torsion, but the torsion can be zero. My claim is the reason none of us ever really use torsion is that it’s slightly the wrong concept. Torsion is something called contorsion, there’s a slight difference, is usually the difference of any connection minus the Levi-Civita. Okay? That’s wrong. It should be any connection minus the gauge transformed Levi-Civita. If you make that little adjustment, torsion’s your best friend. And so there’s this weird way in which, uh, I, I guess I– It’s a weird claim to make. We’ve been using slightly the wrong notion of torsion our entire lives. So which is better, a theory with one or two diseases? Um, here we have this inhomogeneous gauge group. We actually have two separate connections. So for any element omega sitting inside of curly mathcal W. If I have a distinguished connection, I can either add the part of this that’s an Ad-valued one-form, or I can gauge transform the Levi-Civita connection as per your question. This is the transformed, um, displaced base place where you’re going to take the torsion with the displaced version of the Levi-Civita, not the Levi-Civita naked. And that that thing is going to turn out to be exactly what we want. By the way, this is also the rule for letting the entire inhomogeneous gauge group act on the space of connections. Remember, for some reason we didn’t do this. You have two different ways of acting on connections. You put them together in an inhomogeneous gauge group, and then you have to say, “Well, does that thing continue to act on the space of connections?” And it absolutely does. So the distortion with superior equivariance is intended to replace the well-known but often useless torsion, uh, and you see the sort of worse version of it here that should be gauge transformed. And for those of you who are true enthusiasts, you might think about the Stuckelberg trick and how to maintain gauge invariance under difficult circumstances. So can we recover dark energy on A mod G after all? Now, some of you will know that there was an attempt in the seventies by McDowell and Mansouri, I did not know about this, um, where they attempted to reformulate, uh, general relativity as a gauge theory of, uh, of gauge potentials directly, but it doesn’t work. So what you need to do is you need to recognize that there’s a double coset where you’re pre– you’re multiplying on either physical side of the inhomogeneous gauge group by either the tau plus homomorphism on the right side or the inverse so that everything remains a right action on the left-hand physical side of that gauge group. If you take the double quotient, you’re in something that’s equivalent to A mod G. Then you get this first really cool payoff Which, forgive me, you’re not supposed to read this, except if you wanted to, you’d start off here and you’d say theta, which is given by pi minus epsilon inverse d epsilon. Pre and post multiply by two separate elements, GA and GB, under the tau plus homomorphism. If you go through the long derivation, you end up with a very simple statement that it’s just the adjoint based on the second of these two transformations, and the first one actually has no effect. In other words, you’ve got a tremendous object with great equivariant properties, and equivariant is what leads to divergence-free. It, it went in the other direction where Einstein first said R mu nu was the right curvature tensor, but then he had to be told, “Oh, no, no, you need it to be divergence-free.” And then he said, “Okay, so I– if it has to be perpendicular to orbits under the diffeomorphism group, I can correct it.” And then Hilbert said, “Well, the reason that that worked is, is that you’re now exact for the integral of the scalar curvature.” That became the action. Here, what you’re finding is I’ve got a great tensor on this different object that I’ve never thought about and I’ve never heard about with beautiful invariance properties. Which tells you that in a schematic, you’re going to have something like a divergence operator. There’s gonna be a curvature term, which is gonna replace G mu nu. And there’s going to be lambda times little g mu nu, the metric, which is gonna get replaced by this gadget. And so the question is, if I annihil- if I try to operate on both of these gadgets, imagine there was an equal sign in the middle, and you put a negative in front of this term. Then you’d have, with no stress-energy tensor, so sort of a vacuum, only dark energy and the manifold itself, you’d have an attempt to use the divergence operator on these two terms, and you’d get zero zero. In other words, you’ve successfully found a candidate to replace the Einstein field equations where there’s a curvature term and there’s a dark energy term, but they’re not… the second thing is not constant. It’s free to respond, to gain a VEV. If you have curvature stuck in your system, this thing can come roaring out of the vacuum. And as a result, you don’t have this problem about, oh, the greatest problem in physics, a hundred and twenty o- orders of magnitude. Yeah, of course you’re gonna have that problem because it’s not lambda times the metric. It’s a field. All right, so now what? So having successfully changed our field content for a new dark energy candidate from metrics to parameterized torsion, can we rescue Einstein’s curvature tensor? And what I want you to think about is the following. Assume that you have the Lorentz curvature tensor where you have a two-form valued in the two-forms. Now, for some reason, many of you don’t know how this breaks up, which I think is, is criminal. We need to teach this to our students. It breaks up into six pieces, uh, when the Lorentz group gets large enough so that you don’t get accidental splittings and things. Two of those pieces, the scalar curvature and the traceless Ricci, are depicted over here. This top thing is the Weyl curvature, which it gets killed off by Einstein’s capital G mu nu. And then you’ve got three terms that you don’t see because of identities. They’ll show up if you start allowing torsion, but they won’t show up if you use the Levi-Civita connection. Now, the thing is, if you allow for torsion with just the Lorentz group, you see these three gadgets here, which is the decomposition of irreducible components, and they would really fit here, here, and here. So there’s no way of mapping curvature into gauge potentials for the Lorentz group. So that’s what I mean to show you, which is that the representation theoretically, you’re not even in the right ballpark. So here’s an idea. We can first try to augment general relativity to Einstein-Cartan, de Sitter, AdS, or any theory with a copy of one-forms in the ad bundle. Now, most of you who’ve gotten frustrated and bored by standard geometry will probably have spent a little bit of time in, let’s say, Cartan theory. And so what you do is you add potentials that are valued in the translations. So that’s one-forms valued in the one-forms, but the two-forms valued in the one-forms, that is, the relevant curvature, doesn’t map to the right space. So there’s no way of getting a map from curvature forms yet into the right place, uh, in terms of gauge potentials. But here’s the– I just find this really mind-blowing. Nobody remarks on it. Einstein effectively taught us that we can treat a four-manifold like a three-manifold. What’s the best thing about a three-manifold from an Ed Witten style position? It’s that the Hodge star operator maps something that you know and care about, curvature tensors, to something else that you know and care about, gauge potentials. And the idea is that’s because two is dual to one on a three-manifold. But what Einstein did, if you allow him me– the liberty of expanding to the Poincaré group rather than just the Lorentz group, is he gave you a map which maps the curvature to the gauge potentials on a four-manifold. He just doesn’t use the Hodge star operator. He uses his own contraction through the tensor product. Therefore, what I would submit to you is that Einstein, by about sixty-five years, is really anticipating Chern-Simons. He’s telling us that if you restrict your field content to things that have to do with tangent bundles, you don’t need to be on a three f- manifold to relate Curvature two forms to gauge potential one forms. By the way, I am not a physicist, so I have no idea whether this is all standard to you guys or not. I am a humble podcast host. If you wanna stop me, I will be happy to slow down. Now, the point is, is that the Poincaré group, the de Sitter group, which would be like s- de Sit- de Sitter and anti-de Sitter would be spin one four and spin two three. So you’ve got three different groups that share the same Lie algebra as a vector space with different brackets on them. You’ve got the Poincaré, de Sitter, and anti-s- de Sitter groups. Should you use any of them? Absolutely not. What you should use is you should use a spinor group because the spinor group has a Lie algebra that effectively up to, you know, among friends, just looks like the exterior algebra. So you get all the degree forms, including the two forms which give you the Lorentz and in- including the one forms which give you the magic of the Einstein version of the star operator. So if we’re not on the space of metrics and we’re not on the tangent bundle because we’re on the spinor bundle, are we on X four at all? And the answer to me is absolutely not. I don’t believe we are sitting here in a four-dimensional world. I don’t think we live in space time. I don’t think any of that’s true, and I think it’s clearly not true. I think we are stuck as a slice of a fourteen-dimensional object. And what this is, is imagine… So if I’m gonna call the metric upstairs on this Y fourteen manifold little g, I’m gonna use gimel to indicate I’m downstairs on X four. A metric is a section of its own bundle of metrics. If something is going on upstairs in the bundle of metrics, you can pull back data. You don’t have to compactify because you’re not in a situation with a random space. You’ve got a bundle. You can take a section, and if you pull back ordinary spinors, uh, zero forms valued in the positive spinors, direct sum one forms valued in the negative spinors on that top space, you’re gonna get three generations of standard model fermions. In other words, I haven’t specified weak hypercharge, weak isospin. I’ve just said, “Go to the bundle of metrics, pull back spinors,” and you’ll find that you’re already in the standard model. One of the cool things about having a podcast and not being a scientist is that you get to talk to interesting people like Frank Wilczek. Frank wrote this in a book. A particularly intriguing feature of SO ten, which by the way should be spin ten, I have no idea why you guys call it SO ten, is its spinor representation used to house the quarks and leptons. Now, he says, “Perhaps this suggests that both the internal and the space-time degrees of freedom are spinors. Perhaps this suggests composite structure,” but I really wanna call your attention to this sentence. “Alternatively, one could wonder whether the occurrence of spinors in both internal space and in space-time is more than a coincidence. These are just intriguing facts and not presently [chuckles] incorporated in any compelling theoretical framework as far as I know.” Um, I found this vaguely offensive since I tried to talk to Frank about this in the nineteen-eighties, uh, but he clearly doesn’t remember it. What this is is a description of the fact that you’re just pulling back Weyl spinors from the space of Lorentz metrics. So according to GU, it is telling us that we don’t live in four D, we live in fourteen. So Einstein made a four-manifold look like a thr- a D equals three Hodge star. GU makes a fourteen manifold do the same and creates a de Rham-Dirac-Einstein complex. So in three dimensions, you can take the ordinary de Rham sequence tensored with spinors. You can rewrite that. Instead of omega two and omega three, you can write omega D minus one and omega D for D equals three. And then if you can find some way of filling in this middle map, you can bring that to a fourteen manifold, a two thousand and forty-seven manifold, and that’s going to be what’s gonna generate three generations, the CKM and the PNMS matrix. So this is an exterior derivative coupled to connection information, uh, that’s housed in the inhomogeneous gauge group. So for example, part of the inhomogeneous gauge group looks like gauge potentials, so imagine that you take your special connection, you add a potential. So there, there’s connection information in the inhomogeneous gauge group, and you’re mining that for a minimally coupled exterior derivative. Now, the problem is, how the hell do you get from omega one to omega D minus one with a differential? That’s, that’s really gonna be your issue. It’s not up top, it’s down bottom where it gets complicated. So this ultimately leads to a rolled up Dirac de Rham-Rarita-Schwinger shape familiar from seesaw theory. In other words, if you roll up a de Rham complex on a three manifold, think about this as one forms, think about this as zero forms. They’re valued in another vector bundle, the spinors. This thing here is the rolling up of what would normally be an elliptic sequence if there is no obstruction to D squ- if, if D squared equals zero. You roll this up, and you can create a Dirac de Rham-Rarita-Schwinger gadget, which will r- yield you three families, really two plus one. The third family is an imposter for representation theoretic reasons, but at low energy it’ll look the same as the other two. And this symbol is the only thing that you need, which takes a two form- valued in the spinors and maps it back into one forms valued in the spinors. So effectively, what I’m claiming, it’s just the ordinary derivative which would take you from one forms to two forms, and then you knock it back from two forms to one forms with this ship-in-a-bottle operator, and then that’s what gives you your rolled-up complex. And that’s also what gives you that sort of famous, uh, structure from the… If you want different, wildly different masses of your neutrinos, let’s say, um, you want a zero in a, um, self-adjoint operator that looks like that in order to get wildly different eigenvalues. Now, a spinor in an ambient space pulls back to a spinor on an embedded or immersed subspace. Tensor a spinor on the normal bundle. If you think about grand unification, what are the numbers involved? SO ten. Ten is ten times one real dimension. SU five. Five times two complex dimensions. Five times two equals ten. Then you– The third most popular one is Petit Salon. That doesn’t fit. It’s SU four cross SU two cross SU two. But that’s not what it really is. It’s spin six, which is SU four cross spin four, six plus four, ten. Why is the number ten suffused throughout all of grand unification, and why doesn’t grand unification work? There is no [laughs] grand unification. It’s just a normal bundle in your ambient space. You’re picking it up because you’re pulling back spinors from the space of pointwise metrics, and you’re confusing the normal bundle as if it fell out of the sky, uh, in mitten der linnen, which it, which it didn’t do. So if we’ve talked about the lambda in lambda CDM, we should also talk about the dark matter. If I take zero forms tensor spinors and one form tensor spinors, and I make that this entire column, these three representations are exactly what we s- now see in the standard model. The reason that I called this one an imposter is you’ll see that it is parenthetically linked to two other representations. My interpretation is that if you were to turn up the heat sufficiently high, these two things would continue to behave the same way with the same internal quantum numbers. And this one will surprise the hell out of you because it’ll reunify with all of these other particles from which it’s become disconnected. So many of you don’t know, and I don’t know why this is, that the spinors have an exponential property, that the spinors on a direct sum of vector spaces are the tensor products of the spinors on the individual summands. There’s a slightly more complicated rule that looks vaguely like a product rule for the Rarita-Schwinger three-halves representation, and that’s where this thing comes from. In other words, there’s this extra term where it’s like Rarita V tensor spinors on W, spinors on V tensor Rarita-Schwinger on W– uh, tensor Rarita-Schwinger on W plus spinors on V tensor spinors on W. So that’s where you get your third generation of matter from. Everything below the line is dark. So it can say quite clearly what this matter is structurally in terms of groups. And these two things here are luminous, but you haven’t seen them yet. Now, as, as our dear friend Sabina has pointed out, there’s sort of three reasons why you don’t see something. It’s too massive, and you haven’t gotten enough energy to see it yet. It’s too weakly coupled, and you don’t have instruments that are sensitive enough yet. Or the thing has to be in some special configuration like Bohm-Arenov where you only get to see the effect if you con-contrived your laboratory to be just so. So in GU, there’s one family of sixteen flipped chiral spin three-halves particles. That is, there is a sort of spin three-halves family which, aside from being spin three-halves, is just the conjugate of the internal symmetry representation. But there’s a lot more left to discover, and if you wanted the exact, um, representations in terms of SU three, SU two, uh, and the electric charge distilling the weak hypercharge into electric charge after symmetry breaking, you can say exactly what these things are. Some of these things will be electrically neutral, but lots of them won’t be. Then it becomes a challenge, why is it that we haven’t seen the things that are predicted in the model? But one of the things online that I just find funny is people who don’t read things say, “Well, this makes no new predictions.” Um, in general, almost everything said about GU is untrue. You know, the-these would be the analogs of quarks. These would be the analogs of antiquarks. These would be leptonic.
00:41:35
Audience: So what limits are there on spin– fundamental spin three-halves coupling?
00:41:40
Eric Weinstein: Well, Velos-Wanziger is the big one. Velos-Wanziger says that if you have spin three-halves matter that is coupled, uh, to some sort of nontrivial acting group, you b- have to be very careful you acquire tachyons or failures of unitarity. Causality goes out the window. Um, but again, you know, one of the things you have to remember about physics is that physicists tend to remember the conclusions of their no-go theorems. They don’t tend to remember exactly what the assumptions are. So if your model differs by having no e- internal symmetry groups, I have no idea whether it has a- any kind of a Velos-Wanziger problem. But I would start with Velos-Wanziger.
00:42:19
Audience: Are there constraints on spin three-halves from grow, you know, the growth factor or spin sta- G factors, spin statistics, things? Or nothing? I still don’t know where the mass is, uh, from the physical structure. Do you have some hints that somehow some-
00:42:33
Eric Weinstein: Sure. But there’s no Higgs. The Higgs is an illusion. If you look at the Yang-Mills sector of the standard model versus the Higgs It’s almost exactly the same. They both have a Klein-Gordon kinetic term. They both have a quartic term. You have that A wedge A in the ex-perturbative expansion of a curvature tensor, so when you take its norm square, you get a quartic. If you take the norm square, uh, you also get a term that looks like the unperturbed curvature, uh, inner producted with A wedge A, which is a quadratic. So if your curvature is negative, now you start to get a Mexican hat potential. Minimal coupling and Yukawa coupling are the same thing. The only thing that’s really different is the spin. So on the Y fourteen, you have a vertical tangent space, which is a ten-dimensional space. You have a four-dimensional space, which is the pullback under the projection map of the cotangent bundle downstairs, which lives inside of the cotangent bundle upstairs. Both of those separately have metrics automagically because it’s the space of metrics. You trace reverse the Frobenius metric along the fibers, which gets you from a seven-three signature to a six-four. And then you combine these two, and suddenly you have spinors because you have a, a, a bundle that is semi-canonically equivalent to the tangent bundle upstairs with a God-given metric without ever choosing a metric. So part of the whole point of GU is that your quantum gravity escapade will never work as long as you have fermions because you don’t have a metric bundle… if you don’t have a metric between observations of the metric in a quantum theory. In the case of integral spin fields, you have the bundles, but you don’t know where the wave is. In the case of fractional spin bundles, you can’t even define spin one-half without a metric. Standard Model answers the question: What is the maximal compact subgroup of SU three comma two? And that’s SU three cross SU two cross U one. In other words, the punchline comes first. What is SU three cross [chuckles] SU two cross U one? And it’s an answer to the question: What is maximal compact of SU three comma two? Same question, what is, uh, the Petit Salam group? It’s not SU four cross SU two cross SU two. It’s spin six cross spin four, and it’s the maximal compact subgroup of spin six comma spin four. So you can see this chain. Everything is contained in spin ten C, which mathematicians care about, my guess is physicists less, unless they’re string theorists. And what you see is that s- this spin ten is not right. We wasted the seventies work because we wanted to avoid indefinite signature on the Killing form, and I don’t know what to do because we’re in a maximally compact subgroup. We’re shielded experimentally from understanding how nature handles the, uh, indeterminacy of the Killing form. But this is the right chain. Spin six four, spin three comma two, SU three cross SU two cross U one b- w- Brian, i-in terms of the axis of evil in certain, uh, Lorentz-breaking directions in space, if you take the one dimension that’s distinguished in the space of all metrics, and this has a complex structure, you can ask where that gets sent to, and that will actually break, in a certain sense, uh, your Lorentz invariance. Okay. We will never find space-time SUSY. We fed Salam’s strategy, which always needs to eat an affine space, the wrong affine space. Don’t feed it Minkowski space. Feed it the space of connections. Then the Lorentz group is the gauge group. The space of four-momentum becomes the space of gauge potentials. And what you find is that the fermionic extension gives you exactly three families of chiral fermions if you have a decreased VEV in the total space taking a Dirac equation into two Weyl equations because the mass is actually a variable to your point. So astounding, simple, little-known fact, general relativity knows Petit Salam. That is, I don’t need to talk about weak hypercharge, weak isospin. I can just say the following facts. I have a four-manifold passed to its bundle of metrics. Take the Frobenius metric, reverse the trace, reduce the maximal compact subgroups along the fibers, pull back Weyl spinors, and you have one grand unified generation where the lepton, um, the electron, and the electron neutrino become the fourth flavor of quark. I don’t have to specify quark content. I don’t specify weak isospin. I don’t specify weak hypercharge. It comes out that simply, and yet we don’t talk about it. I’d like to hear why this is such a dumb idea. Let me just make a claim. Uh, four days ago, this gentleman, Curt Jaimungal, dropped a three-hour, uh, discussion of this. It’s now at about a hundred thousand, uh, views. In four words, Einstein knows Petit Salam. In under thirty words, he just said what I said. Now, we can pretend that I’m not saying what I’m saying. We can pretend that I don’t know what I’m talking about, that this is all nonsense, it’s coming from outside the community, peer review, blah, blah, blah, blah. Fine. Let me tell you what’s about to happen. The LLMs are about to be good enough to do the work that physics isn’t doing for itself. We’ve been stuck and stalled listening to the same voices for forty or fifty years, and it’s time to say it’s possible that Leonard Susskind, Ed Witten, and company just don’t know what they’re talking about. It’s possible that Bryce DeWitt and Ed Witten and Lewis Witten led us astray, that we’re not supposed to be quantizing gravity, that we’re supposed to be looking for a unified field, and that all efforts in order to do this are going to come to naught, and that’s why the Lagrangian doesn’t move. So in conclusion, it is emergent from GU that unified algebraic field theory is far more important than quantum gravity, assuming that this approach is valid. Uh, if it’s fool’s gold, at least give me that it’s pretty interesting fool’s gold. The unified field sought by Einstein is the observational graded inhomogeneous gauge group of the unitary kymetric– chimeric spin bundle. In other words, you have X four, it grows a space of metrics. You do this construction of the vertical direct sum, the horizontal that I was just talking about. It has an automatic metric. You haven’t chosen a metric. You form spinors on that because you can form it because it has a metric. You take the unitary group of those spinors. Then what you do is you take the inhomogeneous gauge group on that group, and you extend it to, through supersymmetry. Now, that’s a mouthful, but it’s also the entire universe without making any choices. So I would, I would represent to you that this is the best candidate we have for Einstein’s unified field. And what you can see is that it’s super simple in terms of the linearized field content. It’s zero forms and one forms valued either in ad or in the spinors, and that’s it. It’s very symmetrical. So in some sense, it has to be intricate and baroque because when you unpack it, it has to explain the universe if that’s at all a valid approach. On the other hand, it’s basically the result of choosing four degrees of freedom, one dimension of time on those four degrees of freedom, and a spin structure. In other words, everything seems to unpack from that. Thank you for your time. [audience applauding]
00:50:10
Brian Keating: [upbeat music] But wait, there’s more, a lot more. Eric and I recorded an exclusive conversation right after this lecture, and let’s just say we went even deeper down a rabbit hole. Some say we’re still there, but you can watch the full interview from Eric’s previous podcast at UC San Diego right here, or check out his conversation with Dan Green, also recorded in his last visit to UCSD right here. Smash that like button and subscribe even harder than Eric smashes conventional physics
