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Curt Jaimungal (Me): The Geometric Unity Iceberg... Oh Boy.
April 23, 2025
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Welcome to the iceberg of geometric unity, a comprehensive and technical edition. This iceberg format is one that will guide you through the intricacies of this theory of everything, beginning with foundational concepts and then advancing into the more sophisticated hinterlands. In this special episode, we rigorously explore Eric Weinstein's geometric unity, moving beyond metaphorical explanations to engage directly with the mathematical underpinnings of the theory.
If you skip the rigor and opt for explanations aimed at a five-year-old, well, I'm not sure how many five-year-olds you've spoken to, but sure, it's cute.
You can't explain what a Dirac operator is to them outside of making a TikTok video that gives the impression of knowing without actually understanding. My name's Kurt Jaimungal, and on Theories of Everything, I use my background in mathematical physics from the University of Toronto to explore the unification of gravity with the Standard Model, and I've also become interested in fundamental laws in general as they relate to explanations of some of the largest philosophical questions we have, such as what is consciousness and how does it arise?
In other words, it's a peregrination into the all-encompassing nature of the universe. Today we'll cover the abstruse math of bundle theory, of index theory, of course the standard model with general relativity. Just so you know, this episode took a combined 250 hours across three different editors and several rewrites on my part. It's on par with the most labor that's gone into any single theories of everything video, comparable to the iceberg of string theory, and that's saying something.
If you're confused at any point by the exposition, don't worry, GU may seem like a formidable subject. That's what I thought before I started reading what Eric's write-ups were. And then I realized that it only uses standard notions in differential geometry.
the primary challenge of which lies in the novel construction and the terminology introduced by Eric, yet these are accessible to those with a graduate level understanding of mathematical physics. Even if you're not at that level, don't worry because I'll explain and I'll re-explain several points. First, I'll provide a quick overview of geometric unity, followed by an overview of modern physics, then I'll give a more detailed explanation of GU to thoroughly explain the derivations,
Finally, I'll relate it back to modern physics. There are timestamps in the description to help navigate around. Don't worry if you get lost, this video is meant to be watched and rewatched, where each time you'll glean something new. So let's begin with the first layer of the iceberg. Layer 1. Firstly, let's ask, what is a theory of everything?
Now, most of the lay public thinks that it has something to do with quantum gravity. However, that's just a single approach to reconciling general relativity with the quantum world.
Furthermore, quantum gravity isn't a toe, it's not a theory of everything. A theory of everything in the physics sense is a framework that encompasses both the standard model of particle physics as well as general relativity. In other words, it's not just about something being quantum. You can then ask the question, okay, well, what's the minimal input that such a model has in order to recover the particles that we see, the gauge groups, the Lorentz group, the Yang-Mills action, and other ingredients of modern physics?
There's always the temptation to make your theory more tortuous in terms of what's added to it as elements to the stew. But the goal of a toe has always been an elegant one. This means that you start with a tiny set of assumptions and you recover a plethora. Now, this iceberg isn't going to be hand wavy or vague. It will give you analogies, yes, to help you if you don't understand the math.
But if you do know these topics on screen, then that's enough to understand all of the conclusions, the derivations, and the claims of geometric unity. By the end of this iceberg, you'll not only be familiar with geometric unity, but also with the current state of fundamental physics as a whole.
I'll explain GU in four words, in 30 words, and then in 20,000 words. In four words, Einstein knows petit salam. Not terribly informative to people without a physics background. However, you can see my notes here on this substack on eschewing simplistic explanations, as you only get to choose two of these three, simplicity, accuracy, and succinctness.
Now slightly more accurate is the 30 word explanation. General relativity grand unifies the standard models first generation by pulling back vile spinners from the space of metrics
after trace reversing the Frobenius metric on the fibers. Again, that's a handful, that's actually only 28 words, and that will make sense to someone who has a differential geometric background, but maybe not to you yet. Now the 20,000-word explanation is the rest of this iceberg. One problem is that there are three legs to this mathematical physics stool, geometry, algebra, and analysis, or in other words, calculus. The issue with quantum field theory is that it developed in an unbalanced manner.
It's predominantly analysis, and we discovered super late that we'd been neglecting certain topological, geometric and algebraic aspects.
Starting in the 1970s with Jim Simons, for the last 50 years, people like Witten, Seagull, Quillen, Singer, Attia, Hitchens, Donaldson, Dan Freed, C.N. Yang, and Alvarez-Gomay have been making quantum field theory more geometric. When you're taught quantum field theory in graduate school, it's generally from the point of view of effectively a generalization of multivariate calculus.
But this tool of analysis is too crude of a tool to bear the responsibility of advancing fundamental physics. Now I'll give some simple examples. How do you avoid issues with pseudotensors by ensuring that physical quantities are tensorial and coordinate independent?
2. How do you ensure that the time evolution of a quantum state preserves the non-negativity of the probability density during propagation? 3. How can you tell if you have an anomaly in quantum field theory? 4. How do non-local spectral contributions arise in ostensibly local theories?
And number five, of course, how do you formulate quantum theory on curved spaces? None of these considerations that I've just mentioned are natural in an analytic framework and analysis, but geometric principles ensure that all of these conditions are met. I won't say the answers now, but I will address them later in the iceberg.
Needless to say, it's exactly the same issue where phenomenon that are difficult to prove regarding convergence and analyticity in the real case become completely obvious or even trivial when you extend to the complex case. For now, let's look at the current state of fundamental physics.
Hi, everyone. Hope you're enjoying today's episode. If you're hungry for deeper dives into physics, AI, consciousness, philosophy, along with my personal reflections, you'll find it all on my sub stack. Subscribers get first access to new episodes, new posts as well, behind the scenes insights, and the chance to be a part of a thriving community of like minded pilgrimers. By joining, you'll directly be supporting my work and helping keep these conversations at the cutting edge. So click the link on screen here.
Hit subscribe and let's keep pushing the boundaries of knowledge together. Thank you and enjoy the show. Just so you know, if you're listening, it's C-U-R-T-J-A-I-M-U-N-G-A-L dot org, KurtJaimengel dot org.
What's on screen are terms which are the ingredients of modern physics in the same way that you have flour, sugar, eggs, butter, and milk to form a cake. The following are ingredients of the universe as we know it. The universe combines these all, but we don't know how. And further, we don't know of a source that could give rise to all of these. Is there something like the four simple base pairs of DNA that give rise to the protein bonanza that we call life?
Let me go through these one by one. Firstly, we have Einstein. Now specifically, this term represents the Ritchie curvature tensor defined as the contraction of the Riemann tensor.
Most people don't realize that this is quite odd, as the Riemann tensor has its anti-symmetry in the last two factors coming from it being a two-form, whereas the first two factors have anti-symmetry coming from the Lie algebra, which is actually after you pull down the i with the metric. More on this later, but intuitively, the Ricci scalar measures how volume changes in a curved spacetime, capturing the gravitational effects of matter.
Next is Yang-Mills-Maxwell, and this term is the Yang-Mills action for gauge fields, where F A is the field strength, also known as the curvature, associated with the gauge connection A, given by this formula here. And this inner product is something called the killing form on the Lie algebra of the gauge group. Intuitively, this term represents the energy stored in the gauge fields. The next term describes the action for fermionic fields,
psi and psi bar, where psi bar is the Dirac adjoint of psi, and this D with a slash is the Dirac operator coupled to the gauge connection. The gamma matrices here satisfy the Clifford algebra, and this guy describes how particles like electrons and quarks move and interact with gauge fields within spacetime. Confusingly, this psi here isn't the same psi as a wave function. This is just one of the many ambiguities in physics. You're just going to have to get used to it.
The next is Higgs and this governs the dynamics of the Higgs field and through the potential here it gives mass to gauge bosons like the W and Z ones via spontaneous symmetry breaking when this Higgs field acquires a vacuum expectation value. And this next factor here is when the Higgs field acquires a VEV of vacuum expectation value, this time giving rise to mass for fermions.
which is in proportion to the strength of the Yukawa coupling. At this point, don't worry if you're confused. Again, this video is meant to be watched and rewatched. And furthermore, this is standard physics, so nothing here is specific to geometric unity. All of this will make sense and I'll explain and re-explain. Just keep watching.
Now this next term, spin 1,3, I could have also said SL2C, technically, which is the double cover of the proper orthochronous Lorentz group SO plus 1 comma 3, which is actually necessary for representing spinner fields, and anyhow, this is what allows us to correctly describe particles with half-integer spins.
Then there's this, which you hear of plenty, SU3 x SU2 x U1. This product group is the gauge symmetry group of the standard model, where roughly speaking SU3 corresponds to the strong force, also known as quantum chromodynamics,
and SU2 corresponds to the weak force, technically isospin, and U1 corresponds to electromagnetism, though technically hypercharge. Next, we have the family quantum numbers. These will be outlined later in this talk, but intuitively speaking, this space encodes all the intrinsic properties that distinguish different types of particles within a particular generation. Now next,
Speaking of these generations, this denotes that there are three generations of matter. Now, some people call them three families. I've also heard it called three flavors of matter, confusingly again. This field here, psi, can be decomposed as follows. That's what this symbol here means, which represents these three generations or families or flavors or whatever you call it. This accounts for the existence of particles like the electron, the muon and the tau.
which are all similar in behavior, but they differ in mass. And lastly, please don't make me pronounce these, the CKM matrix explains why quarks can change types or flavors during weak decays leading to phenomenon like the decay of a strange quark into an up quark. And the PMNS does something similar, but for neutrino mixing or oscillations as they're sometimes called.
There are two key equations which can be derived from these terms, but I'll include them on screen anyhow for completeness. So one is the Einstein field equation and the other is the Yang-Mills equation. Since the Einstein tensor on the left-hand side can be thought of as an operator that acts on the Riemann tensor, you can rewrite it into this form. Now I'm going to write both of these equations suggestively as Weinstein suggests in a suggestive manner.
If you squint, you'll see an analogy here between Yang-Mills and Einstein. Both theories involve operations acting on curvature tensors to map them back into field variables and sources. However, one huge difference is how these operators interact with symmetries. In Yang-Mills theory, gauge covariance is preserved under gauge transformations due to the structure of the covariant derivative and field strength. In contrast,
The linear contraction operator, which I've denoted P, which is used to form the Einstein tensor, denoted G, in general relativity, doesn't commute with gauge transformations. I'll leave my proof of this on screen for those who don't believe me. You can just pause. But also, there are show notes in the description with a full breakdown of everything in this video. This, by the way, is partly what Eric means when he talks about the twin origin problems. But more on that later.
In Yang-Mills theory, this involves Lie algebra-valued one-forms, which are also known as the gauge potentials or connections, while in general relativity, it involves symmetric rank two tensors, or the metric tensor as you may know it. What other analogies exist? And how can we make these analogies not only precise, but derived? Furthermore, how can this be done with the smallest set of simple assumptions possible?
That's what Geometric Unity aims to achieve. Layer 2. Geometric Unity begins with a four-dimensional manifold, X4, and then it constructs a bundle of metrics called Y14.
This 14-dimensional space comes about naturally. How? Well, at each point in the original manifold, you can assign a symmetric bilinear form, which is a metric of 10 independent components,
Plus, of course, the four dimensions of the base space to account for each point in the manifold. Rather than choosing a metric which is what's ordinarily done, GU instead works with the space of all possible metrics simultaneously point-wise. This is extremely important because traditionally, spinners require a metric for their definition, and this creates a chicken-and-the-egg problem with quantum gravity.
How is it that matter can exist for mionic matter when a metric isn't well defined? We'll speak more about this later. The key innovation is what Erich calls the chimeric bundle over the space, constructed as follows, where V is the vertical bundle, so changes in the metric at a point, and H star is the dual of the horizontal bundle, so movement in the base space. The vertical space inherits a natural metric
via something called the Frobenius inner product, which is a fancy word for this formula on screen here. Again, all of this will be explained with examples. Now, what about that horizontal space? Well, it gets its structure from a connection choice. This allows us to define spinors without first choosing a metric on X4. Now, the structure group of GU is precipitated from spin 7, 7 or spin 5, 9.
And this signature, by the way, is because we have this decomposition here between the vertical and horizontal components. This leads to a complex spinner representation. This dimension 64, by the way, comes about because the spinner representation of spin 7 comma 7 has dimension 2 to the 7, which is 128.
and that splits into two 64-dimensional pieces. Note, these vial spinners are not of definite signature. Importantly, they are two vial spinners of split signature, 32, 32, and then, of course, another 32, 32. Eric then introduces something called the inhomogeneous gauge group, and that combines gauge transformations, which are H, with so-called translations in the space of connections. That's just an analogy, and we call that N.
This structure parallels how the Poincare group combines Lorentz transformations with spacetime ones. But this time we're in the context of gauge theory. Later on, Eric then introduces something called the augmented torsion tensor. Again, this is plenty of jargon and new terminology. It does need to be introduced because when you name something, you can then use that concept on its own rather than having to construct it from scratch every single time.
Now this augmented torsion tensor, which again is on screen but will be explained more later, is what combines aspects of gravity and maintains that gauge covariance that we talked about before. This is what resolves Einstein's quote-unquote twin origins problem, or more specifically, Eric's coining of the twin origins problem. Now where does the standard model's gauge group SU3 cross SU2 cross SU1 come about? Well, what Eric does is instead of using a compact group, he uses a larger non-compact one
which has as a maximal compact subgroup the standard model. In other words, the gauge group of the standard model comes about not by an arbitrary choice, certainly not three arbitrary choices, but rather as a necessary consequence of the geometry itself. The three generations of matter come about from decomposing spinner-valued forms on this larger space Y14, with something special happening for the two generations, and then the third is more like a remnant. So the first two come about from what's on screen here,
a function, a spinner-valued field, so spinners, and then a spinner-valued one-form. However, the third generation is actually a Rarita-Schwinger field in the decomposition of zeta. And all of this comes about from minimal assumptions on the original manifold x4.
Again, Eric's theory, geometric unity, makes unification not by adding new structures, but via recognizing that the ingredients of modern physics, remember that Einstein equation, Yang-Mills theory, fermions, Higgs, et cetera, they all come about from geometry and moreover from X for itself. Just a moment. Don't go anywhere. Hey, I see you inching away.
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Let's begin with what's familiar, that is a four-dimensional manifold.
This is that tiny input. It's the priming of the pump necessary to construct the rest of geometric unity. There's no metric needed. There's no connection. There's no additional structure yet other than being spin or orientable or connected, et cetera. But there are generalizations of GU without these facets. I personally find it easier to assume these. However, instead of working with X4 directly, Eric constructs what's called the observer's.
Now this observer is actually a triple, even though before I denoted it simply as Y14, it's technically the base space X4, the total space Y14, and then the map pi that projects down between them. So recall, at every point in the manifold, there's a fiber, and this consists of all possible metrics that one could have had at that point. Now let's be precise, what the heck is meant when I say all possible metric tensors at this point?
Okay, at any point x in x4, a metric tensor is a symmetric non-degenerate bilinear form g sub x, say, which takes in two vectors at that same point and then outputs linearly something from the real numbers. The space of such metrics forms a ten-dimensional manifold because a symmetric four by four matrix has ten independent components and then you get the extra four because of the base space. So what's different about this construction?
Rather than fixing a metric which is what we ordinarily do in general relativity we consider all possible metrics simultaneously and then this allows one to study how would physics look if you didn't make any specific choice of a metric eric maneuver which i haven't seen done before.
is instead of quantizing gravity directly on the base space X4, which as I'm sure you know, has problems with renormalization. Instead, Eric works on this larger space Y14 and then he pulls back information to X4 via what he calls observation maps. So I should spell out what a pullback is and what an observation map is. Eric, confusingly to me initially, calls the local sections of this bundle observations. So that's all it is. It's nothing more.
Then you look at a local patch of the manifold and you think, okay, what am I smoothly going to assign as a metric from the larger space Y14? Okay, let's clarify what's meant by pullback. Anytime you have a map, let's say something that goes from A to B, and then you have some structure on B in that target space, whether it's a differential form, a tensor field or a function, a pullback is the corresponding structure
Step 2. The frame bundle and its double cover. Okay, let's construct one of the other essentials. The frame bundle over our base space.
At each point in our base space for this frame bundle, the fibers consist of all possible frames or bases for the tangent space. The group structure here is GL4R. Nothing here is unique to GU. This is standard in differential geometry. But what exactly is a frame? Well, again, at each point, a frame is an ordered basis for the tangent space, and I'll give you some examples.
Here is the standard basis that you know probably by x, y, z, and t in some coordinate system, and then another frame could look like so. In differential geometry, by the way, your bases are vectors, which are differential operators. These frames are related to one another via an element of GL4R. In other words, if you had a base and another base, there exists a transformation between them, which is a member of GL4 in order to get you from one to the other. But here we hit our first snag.
GL4 does not have a finite dimensional spinner representation. Why?
If you take an element of it, say A, and you square it and you get negative i, any finite dimensional representation A would need to satisfy that A also squares to negative i, which is impossible over the reals. This is an extremely important point, so it deserves some elaboration. Let's consider this matrix, which satisfies that condition of A squared equals minus i. The eigenvalues of this matrix would also need to be square roots of minus 1, which don't exist.
So to fix this, we need to do something called passing to the double cover. The map is on screen here. And then we do something called the lifted frame bundle, which has the structure group of this double cover of GL4R, also known as the meta linear group.
Step 3. We construct observation maps. Now what exactly do we mean by observation map? I understand this is plenty of new terminology as we have an observer so now we have observation maps and you may have heard of the Shiab operator or the Chimeric bundle.
Eric is evidently an enormous fan of these neologisms, and this is just something we'll have to get used to. There is a method to the madness though, because when we give a name to something, it means that we can then pick it out again and reference it without having to construct it anew each time. So back to the question, what are observation maps? These are local sections, so some map from a neighborhood of a point X going into the larger space Y, but why do we call these observations?
More on this later, but for now, you can think of it as how one measures geometry. Each of these IOTAs, each of these maps, picks out a specific metric at each point in its domain. For any such of these maps, we get a pullback map that brings geometric data from y back down to x4. Specifically speaking, if you have any tensor, say omega, on y, then you can pull back the corresponding tensor on x4.
Think of this like a probe into the space of all possible metrics. The spin 1 comma 3 bundle comes about as follows. At each point little y in the larger space big Y, we have a metric g little y on x4. This metric determines a principal s o 1 comma 3 bundle of orthonormal frames. The insight from Eric is that this bundle naturally admits a canonical double cover by a spin 1 comma 3 bundle by this construction on screen here.
The other insight from Eric is that this bundle has a 14 dimensional matrix representation broken up into the 10 vertical and then four horizontal actually asterisk on that dual horizontal and that is the bundle that we're calling C the chimeric bundle. Now this C has a metric on it and part of it is actually canonically isomorphic to the original 14 dimensional tangent bundle.
The other part requires some extra structure. By the way, when someone says a bundle admits something, they use this word admit, what's meant is that there then exists something. In this case, when we say that a bundle admits a double cover, we mean there exists a bundle map that's locally two to one. More precisely, for any point in here, there are exactly two points in here that map from here to here. And this mapping here preserves the bundle structure.
This is analogous to how the complex function, say z which maps to z squared, gives a 2 to 1 map from the circle to itself. Why do we need this double cover? Because SO1,3 doesn't admit spinner representations. More precisely, there's no finite dimensional representations of SO1,3 that, when restricted to rotations, gives the spin-half representations of SO3.
However, spin 1,3 does admit such representations, and this is how Eric will eventually describe fermions. Step 4. We construct the tangent bundle and its dual. At each point in the larger observer's capital Y, we have a tangent space which has the same amount of dimensions, namely 14. And again, 10 of them come from the symmetric matrices at each point, which represent all the possible metrics
And we have the other four dimensions coming from the base space below. The dual space consists of linear functionals from the tangent space to whichever is your underlying field, namely the reals in this context. To be specific, at a particular point in the observer's Y, it can be decomposed
as the base space X and then the metric G. Specifically speaking, you can see this isomorphism on screen here, where this guy represents the symmetric zero comma two tensors. And recall that in four dimensions, symmetric matrices have 10 independent components.
Now you may ask, why do we need both the tangent space and its dual tangent space? The answer lies in how Eric defines field contents later. Some fields will naturally live in one bundle and others in its dual. In fact, we'll see that without a metric on why these bundles are not naturally isomorphic. And this is what leads Eric to later construct the Chimeric bundle.
I'll be using this phrase field content so i should define it in physics when people talk about field content we just mean the collection of fields that appear in the theory so these could be a scalar field like the Higgs for instance could also be a vector field like the electromagnetic potential it could also be tensor fields like the metric all the previous examples were tensor fields as well or spinner fields like electrons.
The word content just means collection or set. It's like the complete list of ingredients in our theory.
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Step 5. The vertical and horizontal bundles over Y. At each point in the observer's Y, we have a splitting of the tangent space, decomposed as follows. The V is the vertical space, which is tangent to the fiber, and the H is the horizontal space, which is non-canonical, meaning one needs to make a choice.
Precisely speaking, the vertical bundle is defined as follows. To visualize this, you can think of the vertical subspace at y as the space of all possible velocities for the changing metric at x while staying in the same fiber over the x.
In other words, if h of t is a path in this metric space where h of zero is g, your initial g, then taking the derivative with respect to time and setting it to zero is an element of the vertical tangent space. This is all standard in differential geometry.
Step 6. The Zorro construction is named for its zigzag pattern and this is what provides a canonical manner of defining the horizontal distribution. Now when I say canonical here, what I mean or what is generally meant by canonical is that there's a natural choice or natural option that doesn't depend on arbitrary decisions. It's similar to how when you have a vector space, there is no canonical basis.
However, you do have a canonical dual space. Here, the Zorah construction gives us a natural method to split T y, T y being the tangent space of y or the observer's or the metric bundle, in other words, without making loathsome arbitrary choices.
Here's how it works. You'll see this construction here, which looks like the backward Z of Zorro, which is why Eric denoted it the Zorro construction. And this funny symbol here is a Gimel, which is a Hebrew letter. And then this symbol here, which looks like an N is an Aleph.
Note, I'm unfamiliar with using Hebrew letters in math unless it's Aleph for cardinality. Now these two symbols on screen here look almost identical to me, which are Gimel and Beth respectively. You may see me confuse these symbols throughout, but don't worry because anytime they're referenced, they mean the same thing.
namely, a section of the metric bundle. Note that the augmented torsion tensor is now called the displacement torsion tensor by Eric. I've also heard him call it distortion, but I'm going to continue to call it the augmented torsion tensor for the remainder of this iceberg. To me, I wouldn't use these Hebrew letters, and in fact, I'd change this gimbal to the iota from before. The only difference is that the gimbal is a global section, whereas the iota from before is a local one.
Now the Aleph represents the Levi-Chevita connection, G sub-Aleph is the induced metric on Y, and A sub-Aleph is the resulting connection on Y. However, I should point out that Eric uses this Gimmel symbol and the lowercase g to prevent confusion that was coming about from calling two different metrics on two different spaces both by the traditional G. Thus I understand that the Gimmel and Aleph aren't there arbitrarily, as the way Eric sees it,
All of the drama takes place on this larger observers. The reason Eric has this induction from the Zorro construction is that he wants the freedom to later not have a metric on the base space when not observing the system, for instance. And this is Eric's move to make the quantum metric make sense later. Let's break this down step by step. We start with a metric, which is a choice of a global section on X four akin to iota from before.
This determines a unique Levy-Chevita connection, which Eric denotes as Aleph on X4, and this comes about by the fundamental theorem of Riemannian geometry, nothing not standard here. Next, this Aleph then introduces a metric on the larger space Y, through the Frobenius inner product on symmetric matrices. And finally, the G of Aleph determines its own Levy-Chevita connection, A sub Aleph on Y.
This process gives one a canonical manner of splitting Ty, so the tangent space.
into the
From X for.
to y, the larger space, without making choices. The horizontal subspace is precisely the space of such lifted vectors. When I say lift, by the way, what I mean is we take something that's defined on a lower space and we find a corresponding object to it in a higher space such that it projects downward to give us what we started with. More precisely, if this here is a fiber bundle and we have a vector that's a tangent vector at say b, the base space,
Then a lift of the vector from the base space is another vector in this larger space such that when you project down, you get the same vector. Now the horizontal distribution or the choice of horizontal subspace gives us an approach to choose such lifts. Step seven, the chimeric bundle. You'll notice we're introducing plenty of terminology. There's chimeric bundle. There's the observers. This isn't just jargon for jargon sake.
Instead, we're actually enhancing the clarity because we're avoiding repetitive exposition, having to define these over and over. These terms will become familiar as we proceed and recall this entire iceberg is meant to be watched and rewatched where you learn something new every single time. Alright, let's define this beast. First of all, notice that if we take Y14 as its own bundle, the Observer as its own entire bundle,
and we take the tangent space at a particular point in it, it can always be decomposed as follows, a vertical component and a horizontal one. You know from differential geometry, a choice of a horizontal subspace is a choice, that is the same as a connection
which then becomes something like curvature. However, in GU, you're always trying to minimize the amount of choices you make. You can even think of summing up GU as, if you cannot have one, then you must have them all. Anyhow, let's take a look at that Frobenius inner product that we referenced before, and let me just give you an example of two matrices. So, this isn't actually from the bundle, it's just what I could write on screen.
Let's imagine you have 1, 2, 2, 3, 0, 1, 1, minus 1. Well, you can just do the math and compute its Frobenius inner product, and it works out to minus 1. Now, this chimeric bundle differs in the second component, its H-dual. It's not the horizontal bundle, but the dual of it. This asymmetry will turn out to be required for the theory's ability to unify gravity with gauge theory.
Hi, Kurt here. If you're enjoying this conversation, please take a second to like and to share this video with someone who may appreciate it. It actually makes a difference in getting these ideas out there. Subscribe, of course. Thank you. Step 8. The Frobenius inner product. At each point in the large space Y, we need to define an inner product on the chimeric bundle.
So how do we do this without already having a metric on Y? The way Eric goes about doing this is by noticing that V inherits a natural metric via the Frobenius inner product. Again, for symmetric matrices A and B, the Frobenius inner product is defined as follows on screen. Also notice that you can decompose the trace and the traceless parts of a symmetric two tensor. The trace part has dimension one,
and the traceless part has all the other dimensions. To see why this decomposition is reasonable and valid, consider that for any symmetric matrix A, you can always write it as follows, where you have a traceless part and a traceless part. The traceless part has a signature, in this case, 3,6.
whereas the trace itself contributes either a 1 comma 0 or a 0 comma 1, depending on a choice. Here is where you make a specific choice. So we can either choose 4 comma 6 or 3 comma 7. For geometric unity, Eric chooses 4 comma 6 for reasons that will become clear later, though there are generalizations of geometric unity with other choices. Step 9, choosing a signature.
Why is it that we have to be so careful about this signature? The answer is representation theory. The signature determines which spinner representations are possible.
With our choice of 4,6 for the vertical space, and 1,3 for the horizontal space from spacetime, we get a total signature of 7,7. Note, GU could have had a signature of 5,9, and I believe Eric isn't sure which of these is the sector of our universe, in his theory. But for the remainder of this iceberg, we're going to select spin 7,7. But why these particular signatures, Kurt, you may ask?
Now the magic lies in representation theory, the representation theory of spin seven comma seven.
When we have a metric of a signature, an arbitrary one, P, 6, the real spinner representation has dimensions 2 raised to the floor of P plus Q over 2. Now for 7, 7, it gives us 2 to the power of 7. This signature is essential because if you take the dimension of a spinner bundle, it equals this formula on screen here where you get 2 raised to some floor function. In this case, it becomes 2 raised to 7, which equals 128.
And this 128 splits not into C64 plus C64, but into the equally split signature that we talked about before, C32, 32, and then another C32, 32. Remember, these are vile spinners of split signature. Again, the same is also true for a spin 5, 9 bundle matching what's required for the standard model. Again, more later, this is somewhat of a flyby overview.
Step 10, defining spinors without a metric. Here's where everything so far comes together. The spinner bundle on the chimeric bundle decomposes as follows. Now what's so special about this decomposition? Well, it's the exponential property of spinors at work.
For any direct sum of vectors, v, say, direct sum with w, as long as they have metrics, we have the following. Think of it like if you have a particle that can move in two independent directions, its quantum states multiply rather than adding.
This is the quote-unquote exponential property of spinors that Eric mentions. When you pull this back via an observation map, IOTA, you get the following decomposition into tensor products. This decomposition eventuates in both spacetime spinors and internal quantum numbers exactly what's required for the standard model fermions.
Now this alchemy happens because the vertical part V contributes internal symmetries, whereas the horizontal part, or more specifically the dual to the horizontal part, gives us spacetime properties. When one pulls this back to X4, the spinners decompose perfectly to give both the spacetime transformations of the particles and their internal quantum numbers like color and isospin. The implication? Eric has now constructed spinners without choosing a metric on spacetime.
Instead, they're fomented from the geometry of the observers. This resolves a long-standing chicken-and-egg problem in quantum gravity, which is how can matter exist between measurements if a metric is required for that matter to be defined? The answer, according to Eric, is that matter lives in the observers, namely that larger y-space where spinners exist prior to any choice of a metric on spacetime. Step 11, the structure group.
The structure group of our theory originates from the spinner representations of spin 7,7. Why these dimensions? Recall, 7,7 comes about from combining 4,6 with 1,3. And you can see that one is vertical and then the latter is horizontal. Let's pause and ask, why do these signatures even matter? Again, the 4,6 signature comes from the Frobenius inner product on symmetric matrices, while the 1,3 comes from spacetime.
It's actually here that we make a choice. Because we could have had anything that's summed to 4, and we're just choosing 1, 3. This is one of the only places in this entire theory that I can see a choice being made. Now you may be wondering, why can't we just use any group here? And the answer lies in representation theory. We would like a group whose representations can accommodate both gauge fields and fermions.
The spinner representations of spin 7,7 do exactly that. Why do we need a structure group at all? When we started, we had a base space, an X4 manifold, and then we immediately got the frame bundle with the structure group GL4. But now we're working on Y14, and we require a group that preserves the structure that we've built. The signature 7,7 is not arbitrary. It comes from the natural metric structure on the chimeric bundle.
Now you may be wondering, why can't we just use any old group here? We need a group whose representations can accommodate both gauge fields and fermions. The spinner representations of spin 7,7 do precisely that. So why 7,7 and not say 14? The key is that we would like to preserve the signature that comes about naturally from the vertical and horizontal decomposition. But here's the rub.
The Frobenius inner product, which by the way, Eric sometimes calls the Frobenius metric, but I'm going to continue to call it the inner product, is given by this formula on screen here in components. When you have 4x4 symmetric matrices, this naturally gives a 4,6 signature because the space of symmetric matrices compose into what's called a trace.
and a traceless part.
it naturally gives a 3,7. And this isn't often remarked on. Seeing this 4,6 signature here is extremely subtle because it requires remembering that you can do something called flipping the trace, which is what Eric says, and technically that's a trace reversal. And that's, by the way, what Einstein did when he realized that his equation couldn't just be the Ricci tensor equal to the stress energy tensor. Instead, you require the minus r over 2 correction.
So let's take a look here. Recall the trace is a single number, which is why you can represent it by something of dimension one, which is just the real numbers here. And then the rest becomes the traceless component. And then you wonder, well, what's a representation space of seven comma seven, and it's U 64 comma 64. And that 64 again comes from half of two to the power of seven. Thus, we preserve the Z two grading on the spinners. Step 12, the principal bundle construction.
Let's pause again. We've constructed this elaborate chimeric bundle with its spinner representations, but how does one actually implement gauge theory here? You may think, well, let's just use this spin 7 comma 7 directly. However, there is a subtler approach that Eric takes.
His idea is that spin 7 comma 7 acts on that 128 dimensional complex vector space via its spinner representation. This space splits into two 64 dimensional pieces as we've said before and we wonder why is it we're using this unitary 64 comma 64 rather than just U of 128.
It's because the spinner representation preserves a metric of signature 64,64. This brings us to our principal bundle on screen here, where we finally have a gauge group or a structure group, namely U64,64.
And then we have what's called an associated bundle. This is from taking the frame bundle of the chimeric bundle and then doing what's called a lift to its double cover. We then use that row representation to convert the spin 7 comma 7 transformations into U 64 comma 64 transformations. Step 13, the inhomogeneous gauge group. OK, let's carefully build up our gauge structure. First, what do we mean by gauge group?
In physics, gauge groups represent redundancies in the descriptions of nature. In other words, they're different mathematical descriptions of the same underlying physical reality, which is unobservable. So for instance, you can measure your height in inches, you can measure it in centimeters, you can measure it in meters. It doesn't change your height. Those are just different representations of your height. Now let's think about electromagnetism. You can add a gradient to the vector potential without changing the physics. That's a gauge transformation.
But there's something deeper going on here that we're going to explore. So let's clarify an important distinction in gauge theory. There are two related but distinct concepts that are often confused and so I'd like to spell it out.
There's a choice of connection and then there's gauge transformations. The choice of connection, let's call it a one-form A on a principal G bundle, which has a total space of P going down to M, is a Lie algebra-valued one-form satisfying certain properties. The space of all of these connections, calligraphic A, is an affine space. Here's what we mean by affine space, by the way.
A vector space has an origin. Some people like to say a vector space has a preferred origin. Actually, anything that has an origin is not an affine space, so you don't even need to put the word preferred there. Okay, now what about gauge transformations? Gauge transformations are bundle automorphisms that preserve the fiber structure. Again, they're not just bundle automorphisms, which are often said, they have to preserve the fiber structure. They form a group, calligraphic H,
acting on connections via what we see here. Now let's unpack this. The first term here you can think of as a rotation of a sort of the connection and the second term is a correction to the connection that you need in order to preserve the transformational properties. In physics language this ensures that the quote-unquote covariant derivative transforms properly.
For a concrete example, let's just take the non-abelian gauge theory case of QCD. If this A mu here is a gluon field and this g of x here is some varying spacetime dependent element of SU3 and that SU3 comes from the SU3 cross SU2 cross SU1.
Then you have this situation over here being satisfied. And importantly, this describes the same physics, which is why people call it a redundancy. So the choice of connection is not redundant, but the gauge transformation is. And that's something that's quite confusing when you first learn about it. Now here, this calligraphic H, by the way, is the smooth sections of the associated bundle P sub H with the adjoint action of H, where H, again, not calligraphic,
is U of 64 comma 64. In fact, I'm going to stop just calling it H because that's confusing. I'm going to say U 64 comma 64 from now on. In simpler terms, we have a principal U 64 comma 64 bundle and these are our gauge transformations of it. Again, I can be even more precise. Let's say we have a Lie group G and we have its adjoint representation where G goes into the automorphisms on the Lie algebra, which acts on the Lie algebra by conjugation.
Then the adjoint bundle with the lowercase a is the vector bundle associated to the principal bundle P subscript U 6464 via this representation. The inhomogenous gauge group is then this calligraphic G which has the calligraphic H semi-direct producted with this calligraphic scripted N. Now this is like translations in the space of connections but not in a trivial manner. There's a multiplication rule here.
And this is what shows how gauge transformations act on the translation part. It seems like Eric is creating the structure to parallel Poincare's group's combination of Lorentz transformations with spacetime translations, but this time in the context of gauge theory rather than spacetime symmetry.
It should be noted here that this is quite a large move. We're neither working on a space of metrics like Einstein, nor on a space of connections like Yang Mills or Yang Mills Maxwell as Eric says. Instead, what I'll do is I'll overlay a dictionary here that you can take a screenshot of.
But recall, there are further notes in the substack at c-u-r-t-j-a-i-m-u-n-g-a-l dot org, curtjymongle dot org, myname dot org. So you can sign up there if you'd like the PDF. This is, as far as I can tell, Eric's interpretation of Einstein's unified field. Unified means algebraic in the eyes of G-U. Step 14. Defining a right action on connections.
Now we need to understand how our gauge group acts on connections. For any connection A in this calligraphic A and any element lowercase g in this calligraphic g, we define the following. And you may ask, Kurt, are you going to read that? No, it's tedious. Just look, take a screenshot. Why this complicated formula? Well, let's be like Kurtis Blow and break it down. This term here is the familiar gauge transformation.
This next term represents translations in the space of connections, and this term here is a correction term to ensure the consistency. This right action was constructed to satisfy this specific property, making calligraphic A into a right calligraphic G space. So what happens when we try to combine gauge theory with gravity?
Well, there's several problems, but an immediate one is that gauge transformations don't play well with Einstein's way of contracting indices. However, what if we could find a quantity that transforms correctly under both?
That's what Eric has found with the augmented torsion tensor, defined on screen as follows. And now this variational pi, at first I thought this was an omega, it's technically a variational pi, is a member of the adjoint valued one-forms on the larger space y. And that's going to be the gauge potential. Whereas this variation on epsilon here, belonging to calligraphic H, is a gauge transformation.
Now the key property is that under gauge transformations, we have this formula on screen here. So let's break this down piece by piece. What is firstly this variational pi? Well, it's an adjoint valued one form, as I said before, which means it takes in vectors from T Y and it returns elements of the Lie algebra H of the gauge group H, which again is U 64 comma 64. And now this other term here looks complicated.
However, it's just the gauge transformation of the base connection A0. By the way, when I use the word variational pi, it's not in reference to anything about variational calculus.
It's instead because Eric's notation is to use this symbol right here. And in LaTeX, it's written with a slash and then var pi. It's a variation of pi in the same way that there's a variational phi. So var phi or var phi as some people call it. Just letting you know because you'll be hearing me use this term variational pi and upon rewatching this iceberg for maybe the fourth, fifth time, too many times to count, I realized that this can be confusing. Now let me make a concrete example.
Let's say we have a U1 gauge theory and our variational epsilon is e to the i theta, then what we have is this on screen here. This is exactly the gauge covariant combination that appears in electromagnetism. Now this is quite interesting because it combines aspects of both gravity, so torsion, and gauge theory, so covariant derivatives, while maintaining gauge covariance.
The way that I see it is it's like finding a method to make Einstein's gravitational theory speak the language of Yang-Mills theory. How do we generalize Einstein's contraction of the Riemann tensor in a gauge covariant manner? The answer lies in what Eric calls the Shiab operator that's spelled S-H-I-A-B.
Now for a gauge covariant two-form C, which some people call cassis, but I'm just going to say C, and it's not the letter C, it's this symbol on screen. You have this formula here, where the Shiab operator acts on this two-form and gives you a Ritchie-like term, which we'll explain more later, and then a scalar curvature-like term. Again, more will be explained later.
Note, the circle with a dot in the center is my notation for the shiab operator. Eric actually writes two concentric circles with a dot, but I wasn't able to get this to consistently render in my workflow, so just note that whenever you see this shiab online outside of this video, it will likely have two circles and a dot. Let's further break it down, piece by piece. First, what is this operator doing? It's taking a gauge covariant 2 form, and then it's returning another differential form
that transforms this time properly under gauge transformations. So why do we need such an operator? Think about Einstein's theory. When you contract the Riemann tensor to get the Ricci tensor, you're using the metric to raise and to lower indices. However, this operation doesn't respect gauge symmetries. It treats all copies of two forms in the same way. The Shiab operator fixes this by incorporating the gauge transformation epsilon explicitly.
Here's how it works. These forms phi, phi 1, phi 2, for instance, are invariant under the action of spin 7, 7. When one conjugates them by an epsilon, we get objects that transform covariantly under gauge transformations. And now you also may ask, hey, Kurt, why these particular combinations of wedge products and Hodge stars? And again, the answer lies in representation theory, just as the Einstein tensor
For a concrete example, consider the case of U1 gauge theory. Here, the C would be the electromagnetic field strength.
The general non-abelian case is more subtle, but the principle is analogous. Eric is building an operator that combines the metric and gauge structures consistently. The action principle in GU takes a form reminiscent of both Einstein-Hilbert and Chern-Simons.
Now you may look at this and say, this looks nothing like Einstein, it looks nothing like Chern-Simons. What's remarkable about this action? First, notice that it's first order in its derivatives, like Chern-Simons theory, but unlike the second order of Einstein-Hilbert action.
Also notice that the field variables are omega, which actually comprise this epsilon and then this variational pi here, where epsilon is a gauge transformation and pi is a gauge potential. The first term combines the augmented torsion tensor with the curvature through the Shiab operator. This generalizes both the Einstein-Hilbert term and the Yang-Mills term. The second term is like a mass term for the torsion with coupling constant kappa.
Remember, in vanilla gauge theory, one can't put the gauge potential directly in the action because it's not gauge covariant. So in general relativity, you can't put the connection directly in the action because it's not diffeomorphism invariant. But here, the augmented torsion gives us a covariant object we can use directly.
Note you also have to recall that every element omega that comes from the inhomogeneous gauge group actually produces two connections. One is A and another is B. The difference between these two is called T. Also you should note that at this point the theory is purely bosonic. The fermions haven't come about yet. This reminds me of how string theory was initially bosonic prior to being fermionic or having both. Step 18. Field equations.
From our action principle, we derive the field equations through a variational principle. The result is deceptively simple. It's on screen here, so what is going on? The Shiab operator acts on the curvature here, much like Einstein's contraction acts on the Riemann tensor. The primary difference is that this operation preserves gauge covariance. The augmented torsion tensor term T omega enters with a coupling constant kappa.
This E term here is essentially something that helps the theory's consistency as it makes the equation properly reflect the variational principle from which it's derived. I think of it like an error term. Eric demands this because it's necessary for recovering both Einstein's equations and Yang-Mills theory in the appropriate limits. Step 19. Fermions and supersymmetry. Where do fermions enter?
Recall that fermions act as quote-unquote square roots of gauge potentials. For spinner-valued forms, which we see here as a zero-valued spinner form and a one-valued spinner form, we get a Dirac-like operator. Now this is fascinating because traditional supersymmetry relates bosons and fermions through spacetime translations
Here we're seeing a different sort of supersymmetry based on the affine space of connections. The operator here combines aspects of the Dirac operator with our gauge structure. The upper left block involves the Shiab operator acting on the derivative of a spinner-valued one-form. This is like the square root of the Yang-Mills operator. The off-diagonal blocks couple scalar spinners to vector spinners similar to how supersymmetry transforms fermions
into bosons and vice versa. When one decomposes the spinner representations under spin 7 comma 7, one finds the following formula. Now this is how the three generations of fermions come about. Notice that upper index of three there. The first two generations come from a spinner-valued zero form and a spinner-valued one form directly, whereas the third generation comes about from Rarita-Schwinger fields in the decomposition of zeta.
Step 20, the deformation complex. How do we study small perturbations around solutions? We require a complex. Now this complex is what's necessary for understanding the physical content or the field content of the theory. The first map here encodes infinitesimal gauge transformations. It tells us how the fields change under small symmetry transformations. The second map gives us the linearized field equations. It tells us how the field propagates.
The cohomology of this complex, which is the kernel module of the images, describes the physical degrees of freedom. At the first level, we have the first homology, and it gives us the gauge-inequivalent perturbations. This is analogous to how in electromagnetism, two gauge potentials differing by a gradient describe the same physics. Explicitly, these operations take the form on screen here.
And we also have the d squared equals zero property that makes this a complex, ensuring the gauge invariance of the linearized theory. Note that the second term is actually slightly more complicated, but I will cover that later and or in the PDF notes on my sub stack and or with the upcoming podcast with Eric Weinstein himself. Step 21, the seesaw mechanism. Now, here's where it gets fascinating if it wasn't already.
Our Dirac-Rarita-Schwinger complex leads us to an operator of the form, which is on screen here, that we've talked about before. Now why is this interesting at all? It's because this structure mirrors the neutrino-seasaw mechanism. So the seesaw mechanism explains neutrino masses through the mixing between light and heavy states. Here, we're mixing between different spinaural sectors.
The zero block in the lower right corner is actually essential because this is what allows for the hierarchy between different types of fermions and this potentially explains why we see three generations of matter with such different masses.
I debated whether this section should go earlier in the script or later just because it's representation theory and there's nothing specific to GU here. However, I placed it here due to its length. The reduction of spin 7 comma 7 to the standard model gauge group follows a path through intermediate subgroups. So let's go over how does this reduction work. Let's peel back the layers.
Firstly, you have a spin seven comma seven acting on a 128 dimensional space of spinners that we've talked about ad nauseum and it splits into positive and negative chirality parts. Now here's where we get something different. You can reduce this to a maximal compact subgroup spin seven cross spin seven. Why this particular reduction of all the reductions we could make? Well,
Experimentalists haven't ever observed non-compact internal symmetries in particle physics. Indeed, there are compelling theoretical reasons why physicists don't consider non-compact groups. So for instance, the famous or infamous Coleman-Mendula theorem, which essentially states that the symmetries of the S matrix, which describes particle interactions, must be a direct product of the Poincare group and an internal symmetry group, though there are some assumptions here.
This internal group must be compact for unitary representations, which means you need this for consistent quantum theory. But why must it be compact? Well, it boils down to these two reasons, unitarity that we mentioned already, and then positive energy. Non-compact groups often lead to these theories with negative energy states, which are physically problematic.
We'll discuss the reasons why Eric's model circumvents these objections later. For now, let's break this down into the standard model's gauge group step by step. Again, the complete structure group reduction path begins with U64 comma 64. In low gravity and in Eric's model, this decouples into two vial halves, bringing us to spin 7 comma 7. This contains a spin 1 comma 3 cross spin 6 comma 4,
where the first term, the spin 1,3 represents spacetime, or specifically the spacetime symmetries. Then you have to notice that the spin 6,4 part has spin 6 cross spin 4 as its unique maximal compact subgroup. And then this gives us SU4 cross SU2 cross SU2
via an isomorphism, which is precisely the Pati-Salam model. This answers a constitutional question. What is the maximal compact subgroup of the fiber structure group of our observable universe? The final reduction down to SU3 cross SU2 cross U1, our standard model, comes about when the metrics carry an additional special unitary structure. Keep in mind that much of the above is somewhat standard in GUT circles, so Grand Unified Theory circles.
Eric, though, follows a different path by using the non-compact group SU3,2, which is a real form of SL5,C.
The standard model gauge group comes about now as the maximal compact subgroup of SU3,2. The specific real form corresponds to the A4-Dinckin diagram and this distinction is what allows Eric to resolve the proton decay controversy and other issues that plague 1970s grand unified theory schemes and approaches or what have you because they used real forms that in Eric's eyes were incorrect. So how do we get this SU3,2 while spin 7,7
has a spacetime split given on screen here. One of those, the 6, 4, has SU3, 2 as a complex structure inside. We can then take the maximal compact subgroup of that, which is taking the special part of U3 cross U2. And we can further make an isomorphism of that to the standard model.
Keep in mind that each step involves symmetry breaking, which in physics corresponds to the vacuum state not respecting the full symmetry that used to be there in the Lagrangian. This breaking can be spontaneous, dynamical or explicit, put it into the Lagrangian by hand. Now, in this section, there are a plethora of subgroups being taken. So one question naturally comes up, which is, can we just take any subgroup willy nilly in physics? And the short answer is no.
Each time you take a subgroup, you're saying that the symmetry is broken and thus you're introducing new physics. So for instance, the breaking from SU4 to SU3 cross SU1 in step four corresponds to the separation of leptons and quarks. This is both a boon and a curse. Since there's new physics, it means you're deviating from what's standard. However, it also means that there are predictions which can be falsified. Step 23.
Again, we're going to make this painfully clear. The Dirac-Rarita-Schwinger complex on Y14 gives us this generalization of the Diram complex, which is what we need to deal with the spinner-valued forms. On screen here, you take a one-form with the Shiab operator, technically the Hodge star of the Shiab operator and then some other differential,
which takes you from a one-form to one dimension less than the manifold, so 13 in this case. Note, you may be wondering why you haven't seen this complex before, and that's because as far as I can tell, it's a novel Dirac-Rarita-Schwinger-like complex introduced by Eric. This is brand new in the physics literature. This complex yields three distinct sets of fermions. How? Well, the scalar spinner here, new, gives the first generation.
The zeta vector spinner here splits into two parts, a gamma traceless part, which gives the second generation, and a gamma trace part. And that's what gives the third generation. So the first generation again, this space is the scalar valued spinners on Y14. And when you pull it back to X4, these correspond to the familiar first generation fermions like the electron, the electron neutrino and up and the down quark.
The second and third come from zeta, which is a spinner value to one form. The decomposition of this is where it gets interesting. The first term here on the left side of the direct sum gives what Eric considers to be the second generation. And the last term, which is on the right hand side here of the direct sum, is where the third generation comes from. These decompose to give us distinct generations because of how these spaces transform
Step 24. Higgs from Yang-Mills. So where's the Higgs field at? Well, it's lurking in the gauge potential variational pi here. Here's how. Firstly, we decompose this form as follows. The second term here on the right-hand side contains the Higgs field.
because symmetric two tensors decompose, remember, into a trace and a traceless part that we talked about ad nauseum again. Let's unpack this further. The gauge potential is a one-form on the Y14 space, which is valued in the adjoint bundle of the principal bundle P sub G. When we decompose the one-forms on this space, we're essentially splitting it into parts that live on X
and parts that live in the vertical direction in the fiber. The key term is this one. This is the space of symmetric two tensors on X4, and it contains scalar fields from the perspective of X4. Among these scalar fields is our Higgs field, according to Eric. But what's the justification for even calling it the Higgs field? Well, it behaves like a Higgs field because of how it transforms under gauge transformations and a few morphisms of X4.
The trace component R in the decomposition transforms as a scalar under diffeomorphisms just like the Higgs field should. Moreover, under gauge transformations of the structure group G, this component transforms in the adjoint representation exactly how one would expect the Higgs to transform in gauge theories. Step 25 Trace and Traceless Contributions
The decomposition of the symmetric tensors into trace and traceless parts sounds like some mathematical pedantry, but it's not. Why? Consider the symmetric tensor aij, which you can write as follows, where you decompose it into trace and traceless parts, done this many times. The traceless part gives spin two contributions, and the trace part gives spin zero contributions. This mirrors exactly what happens with gravitons and the Higgs.
Now we've discussed the trace and traceless decompositions before in the context of the Frobenius inner product, but let's go further. So we have this gravitational sector here where the traceless part is on screen and it corresponds to the spin two field in general relativity. This represents gravitational waves or gravitons in quantum theory. So why is it spin two? Because it's a symmetric traceless rank two tensor, which transforms under the spin two representation of the Lorentz group.
Now the scalar sector here is what engenders the Higgs field, in geometric unity at least. This decomposition has these as a consequence. It suggests that the graviton which is spin 2 and the Higgs field which is spin 0 are intimately related. Interestingly, it's called geometric unity for a reason because these two are different aspects of the same geometric object on Y14.
Step 26 Natural Quartic Potential Why does the Higgs field need that particular Mexican hat potential? Is it possible to get it to emerge in something that resembles something natural? Well, the Yang-Mills action contains terms like the following, where you take the normed squared and you get quartic terms. This matches the structure of the Higgs potential.
Is it possible that the emergence of the Higgs potential comes from the Yang-Mills structure? Let's continue to explore this. In standard Yang-Mills theory, we have this term here, and that represents the self-interaction of gauge fields. However, in GU, remember that A contains components that we identify as the Higgs field. Let's write this out explicitly. Here, phi represents Higgs-like components.
Now then, we expand the a wedge a squared term to get terms like the following. Notice this last term here. This looks precisely like a quartic term in phi. Is this the origin of the famous Mexican hat potential? Eric says, of course bro. You may wonder, where's the negative mass that gives the potential its characteristic shape?
Now this comes from the coupling between phi and the other components of A, specifically from terms like ADX wedge phi comma ADX wedge phi. This geometric unity specific derivation is to me phenomenal because it shows that the Higgs potential, far from being an ad hoc addition to the standard model, emanates inevitably from the geometry of gauge fields.
Step 27, Yukawa as minimal coupling. The traditional Yukawa coupling also looks ad hoc, at least to me and to most other physicists and mathematicians. But in GU, it comes about inevitably. This time, it stems from viewing the Yukawa coupling as minimal coupling. Minimal coupling, by the way, to a mathematician just means a gauge covariant derivative. This A mu term, when it contains the Higgs component, gives the Yukawa interaction.
This is the reinterpretation of the Yukawa coupling, and it's a prime example of how geometric unity unifies seemingly disparate aspects of particle physics. To be specific, in the Standard Model, the Yukawa coupling is introduced by hand to give fermions mass. In contrast to geometric unity, this Yukawa coupling comes about from the geometry. Here's how. Recall that in GU, the Higgs field, phi, is part of the gauge potential A.
This Dirac operator coupled to A is D slash A here on screen. Expanding this you get this plus phi mu where this new A with a little tilde on top are the usual gauge fields and this Higgs is the component and we get this full formula on screen here. This last term is precisely the Yukawa coupling. Step 28. The correspondence between the Higgs and the Yang-Mills sectors.
First, let's write out the Yang-Mills Dirac action in the language of geometric unity. Here this alpha is our gauge potential and f sub a is the field strength as usual and we also have some left and right-handed fermions. The covariant derivative acts as follows. Now compare this to the Higgs-Yukawa action. Here this capital phi is traditionally viewed as a scalar field which is valued in a representation of the gauge group.
but in geometric unity it comes about as a component of this variational pi under the decomposition of one forms when pulled back to x4. This correspondence is surprising because we have kinetic terms like d sub a alpha squared and d sub a phi squared and they match and we also have this quartic term which correspond to one another. We also have this quadratic coupling which parallels this. We also have the fermion couplings and it takes on analogous forms
But how can a gauge field component act like a scalar field? Just remember, there either is right now or is going to be an accompanying PDF to this Geometric Unity iceberg. So if you'd like more notes, such as expansions on these topics and proofs that I wasn't able to get to in this iceberg, then subscribe to my sub stack as that's where I'll publish it. It's curtjamungle.org or C-U-R-T-J-A-I-M-U-N-G-A-L dot org.
Step 29, the missing quadratic term. The Einstein field equations have traditionally been written on screen here with g mu nu plus the cosmological constant and we set that all as something proportional to the stress energy tensor. However, in GU, one needs to include a quadratic term in the field equations. So you get a modified equation which takes the form on screen here.
This term here with the T omega comma T omega is the self-interaction of the augmented torsion, similar to how the Yang-Mills field strength contains A comma A. And it means that the theory maintains gauge covariance while it preserves Einstein's intuition about geometric contraction. Think about it. In Yang-Mills theory, the field strength is F sub A, which equals DA plus A comma A, and it needs a quadratic term to be covariant, namely that last part.
Similarly, our augmented torsion needs its quadratic interactions to maintain gauge covariance while allowing for some Einstein-like contraction. Step 30, the emergence of the cosmological constant and CKM matrix. Lastly, both the cosmological constant and the CKM matrix come about from components of the gauge potential, variational pi here. How does that work?
The gauge potential decomposes as follows. You see that it splits into these components based on how they transform under the structure group when pulled back via an observation, which recall is iota. So the first part is what gives the standard model gauge fields. The second is what gives the Higgs field as we've discussed earlier.
The third contributes a constant term into the Einstein equations, and the fourth determines mixing between generations. When one pulls this back to x4 via an observation, the component variational pi sub lambda gives the cosmological constant term in Einstein's equations, and then this variational pi sub ckm is what gives the mixing angles between the quark generations. Seemingly disparate physical phenomenon like dark energy, quark mixing,
All derived from the geometry of the observers. Ordinarily, we think of these as independent parameters that we need to add in by hand. However, in Geometric Unity, they're intrinsic parts of the geometric structure. Just so you know, if you have any questions, which you likely do, that's all right, I'm going to have a large solo podcast with Eric just on Geometric Unity. It will be unlike any other podcast because we'll delve into the particularities of it.
Layer 3 Welcome to Layer 3 of the Geometric Unity iceberg. Let's recap what's been done so far. In Layer 1, we gave a brief overview of Geometric Unity, as well as the universe as we know it. In Layer 2, just now,
We went over geometric unity in four minutes and then I gave the longer one hour or so version of it as well. It's been quite a journey and now in layer three, here's where it all starts to come together. Again, if you haven't understood much of this so far, then that's entirely fine because this iceberg is designed to be watched and rewatched where you glean something new every single time, not only from geometric unity, but perhaps from physics and math as well.
I've worked on this like i would work on a documentary with hundreds of hours put in also note that i will be recapitulating often so that your hand is held metaphorically so. Unless you're into that now as i mentioned in layer one we talked about the universe as it's currently known and we went over symbols and equations that are at the heart of our models of the physical universe.
They were the Richie Scaler, which is at the heart of the Einstein-Hilbert action, as well as for other actions, there's the Yang-Mills Maxwell one, there's the Dirac one, there's the Higgs one, there's also the Yukawa coupling, there's also the Lorentz group, or the double cover of it, namely spin one comma three,
There's the internal gauge group of the standard model, so SU3 across SU2 across SU1. There's also the family of quantum numbers. There are also the three generations of matter or families of matter or flavors of matter. I've heard some people call it. There's also the CKM matrix and the PMNS matrix for quark and neutrino mixing respectively.
Now there are also these which are alternative writings of what I've just mentioned. So there's the Einstein field equations, there's the Dirac equation, the Klein-Gordon equation, the Yang-Mills equation, and the Higgs field equation. While I've reviewed the different steps of geometric unity as I see it, the question is, well, how do we integrate these concepts that I've just mentioned within geometric unity? Exactly. Now, GU's explanation for these equations and concepts encompasses essentially everything in modern physics.
Einstein-Hilbert action in GU
In standard GR, one writes the Einstein-Hilbert action as follows, where the Ricci tensor is computed by contracting the Riemann curvature tensor with the inverse metric, and then you further do a trace. In geometric unity, however, there's no single metric that's privileged, as you know, and instead, Eric begins with the curvature of a distinguished connection, A0, which is defined on the frame bundle over the manifold lifted to the double cover,
so that spinners can exist. Now this A0 is obtained via the Zoro construction that we talked about earlier and more precisely one starts with the frame bundle of X4 lifts it to its double cover
Notice I use the term lifted here, and I do so in the sense of a standard bundle theoretic lift.
That is to say, given a projection from F tilde to F, we can look at a connection which is A0 on the regular F, and that can be lifted, quote-unquote, to F tilde by composing with the covering map. Precisely this means what's on screen here, which is a 2 to 1 covering, and A0 is a connection on the base F,
which is then lifted to an A tilde on F tilde, and it satisfies this equation here. In other words, for any vector in the tangent bundle of F tilde, you can define a connection by just pushing that vector forward. This allows for the proper transformation properties that Eric wants for spinors.
As I'm reviewing this iceberg several weeks later, recall this is many months, many, many months in the making, I'm realizing that I sometimes use A0 and I sometimes use A aleph. You may be wondering what is the relationship between these. These are the same. My mistake is that in ordinary math notation, one would use A0, but in GU, because Eric is referencing something specific with the aleph, then the A sub aleph is used and it refers to the choice made in the Zoro construction.
Now, geometric unity reinterprets this contraction process as coming from an algebraic operator, which compresses the full curvature two form on Y, the metric bundle, the observer's, into a symmetric two tensor. You'll notice that the domain of P E consists of a tensor product with two fundamentally different factors. So this first component is a differential form, which is intrinsically tied to the manifold Y itself. The second factor here has a Lie algebraic character.
In Einstein, it's SO1,3, the Lie algebra. So what's interesting is that this P-E operator contracts mathematically distinct structures using a metric on y. So the operator P sub e, it contracts these two using the metric on y
much in the same spirit as how Einstein in his original approach obtained the Ricci tensor by contracting the indices of the Riemann tensor like we talked about just a minute ago. Now, where did these two forms come from? Well, one comes directly from the curvature of the distinguished connection, namely what's on screen here, while the other is an internal two-form from the contraction mechanism itself induced by the metric on y. Thus, both copies of the two-form are essentially the same curvature data, however,
Eric regards them as having different transformation behaviors under the gauge group. This is reminiscent, again, to how Einstein in his approach also had two appearances of two forms in the Riemann tensor treated identically by the metric contraction. In fact, one way that I think of this algebraic operator is that it encapsulates the same idea as forming inner products via the Frobenius metric. If you take two symmetric matrices and multiply them together using the natural inner product on screen here, you obtain a contraction that produces a scalar.
Here, however, this algebraic operator acts extended bilinearly to the tensor product to yield an element of the symmetric 0,2 tensors, thus compressing the curvature information into a form that mimics the Einstein tensor.
What you're doing is you're allowing an algebraic operator, which is defined to act on a curvature tensor, and then it's supposed to return something from the parameter space of field content. In Yang-Mills, this involves a Lie algebra-valued one-form, while in general relativity, it involves symmetric two tensors. After this algebraic operator on y, we then pull back the result along the section, which was that observation map, so that
the effective Einstein tensor on X4 is given by this formula on screen here. So you begin again with FA0, which is a two-form. The contraction operator is defined locally such that we have this formula on screen here so that we can then pull back with an observation and we get the formula on screen here. And that's where the R comes from in geometric unity, re-derived from a gauge covariate contraction on Y.
the Yang-Mills-Maxwell term. Next, we will derive the Yang-Mills-Maxwell term within geometric unity.
In conventional settings on a fixed spacetime x4, the Yang-Mills action is written as follows, where the fA here is the curvature as usual of the connection A on a principal G bundle, and the inner product denotes the invariant bilinear form, also known as the killing form, on the Lie algebra G. Editors note, in a local trivialization, that is, on a patch of the base space where the bundle is trivial, each connection is represented by a Lie algebra-valued one-form A,
Its curvature is the following, but when you're comparing two different connections, say A and B, then you often write it as follows. The connection B here can be viewed as a reference or a base connection. Now when you set B to equal zero, that is the trivial connection, you then recover the usual single connection formula. This will become important later in the Higgs sector portion of this geometric unity iceberg.
Eric considers the principal bundle here with the structure group of U64,64 coming from that spinner representation of spin 7,7.
The gauge field is defined on y as a curvature 2 form, with the transformation property as follows. Now recall that the key step is that this inner product here, which we use to contract F A, is actually induced from the metric on the larger space y, which is in turn built from the vertical Frobenius metric on the symmetric 2 tensors from the base manifold.
and the horizontal lift via the Zoro construction. Explicitly for local coordinates on y, we can define the following slightly hairy formula at least to look at on screen. This ensures that the contraction is performed in a manner that's compatible with the dynamical nature of y. Now in our current context, A represents the full gauge field on y and A0 denotes that distinguished background connection which is chosen by the Zoro construction. Therefore, you should actually understand that this connection
is supposed to be expressed as A equals A zero plus alpha, where alpha is like a correction or a fluctuation. In contrast, this F A zero that appears in some derivations is simply the curvature of the background connection alone. Our full curvature, F A, is thus containing extra pieces.
and then the contraction via the Shiab operator is engineered to handle these extra terms. Moreover, the domain of this operator is the space of Lie algebra value two forms on y, and its target space is typically the space of scalar fields or symmetric two tensors, for instance, because it compresses the two forms into a lower degree object using the
But under the gauge transformations H, for instance, we have the following, which preserves gauge covariance. Technically, it's the below, but the above is much more succinct. Now, you may be confused. Look, is P E an example of the Shiab operator or vice versa or nothing? Now, the answer is that the P E in Einstein's theory is an algebraic contraction that maps two of something. This is a two form and then some Lie algebra data. Their domains and target are different.
Also, you'll see that this lambda with a bullet is what represents, in Eric's theory, the U64 comma 64 Lie algebra structure, while the Shiab operator maps these two forms, which locally involve both a two-form part and a Lie algebra data part, to a one-form or another suitable counterpart. The idea here is that the Shiab operator is a generalization of Einstein's contraction operator.
except Eric constructs it to be compatible with both the base manifold and the fiber gauge structure. At this point, I should point out that the function space for the bosonic theory is the inhomogeneous gauge group. It has two components. One looks like the gauge potential and functions like a connection. The other piece does the symmetry work. It allows the chiab to be defined without destroying equivariance in contraction. Why don't we just do a step-by-step derivation? So step A, let's say,
you get that distinguished connection from the Zoro construction. Step B, you define the gauge potential correction. Step C, you compute the curvature. Step D, you then apply the Shiab operator. Now you'll notice that this form is analogous to the standard Yang-Mills action, except contracted by the Shiab operator here and replacing the usual inner product contraction on FA. Now one has to show that in a suitable limit after pulling back to X4,
that this actually reduces to the familiar expression. For an explicit demonstration, you would take a local section from x4 to y, then you would note that the induced metric on x4 satisfies that the metric is a pulled back metric and that the Shiab operator on the curvature when pulled back is the same as the inner product of the curvature with itself. That completes our derivation of the Yang-Mills-Maxwell term in GU.
Eric derives the Dirac action in the framework of geometric unity, solving the previously unsolved problem of defining spinors without a prior metric. Now, in conventional formulations, the Dirac action is given as follows, with the Dirac operator also written on screen here. And the gamma matrices, of course, satisfy that Clifford algebra so equals twice the metric. Even though many people think Clifford algebras don't rely on a metric, there is an implicit pre-assigned metric, and that's the dilemma.
The very existence of spinors demand a metric. However, one would like to work in a framework where the metric is allowed to be dynamic, allowed to vary. Now, geometric unity sidesteps this by constructing the chimeric bundle, which recall is the vertical component, direct summed with the dual of the horizontal.
Eric then uses the exponential property of spinors, which you can recall if you have different vector spaces V and W direct summed together, that their spinner bundle is isomorphic to the tensor of the individual components. Now in our case, because our components are V and the dual of H, we have the following. The prowess of this construction is that it's like liberating. It's like liberating the definition of fermions from an a priori choice of a metric.
Instead, the metrics are actually determined by the point that they are in Y. This means that the Dirac operator can be defined on Y, the larger space, rather than X4, the base space. Concretely speaking, you construct a Dirac operator acting on the sections of the spinner bundle by creating something familiar here, where the nabla is of course the covariant derivative associated with the distinguished connection A0, which is from the Zoro construction.
The gammas are the gamma matrices or more specifically the corresponding gamma matrices to the 7,7 signature determined by the vertical Frobenius metric on V and the induced metric on the horizontal space or the dual of the horizontal space H star. The I of course runs over the entire indices of 14 dimensions and the alpha represents additional potential coming from the inhomogeneous gauge group structure. We're going to talk more about that.
Now one uses these observation maps to pull back the spinner fields from the larger space y to x4, and this is done using the standard pullback of vector bundles on screen here. You'll notice that this pullback yields a decomposition, and what happens is that the v part becomes the internal quantum numbers, whereas the h part becomes the spacetime Lorentz group spin 1 comma 3.
Now, to spell it out in detail, if you look at the chimeric bundle, and let's just pick a single point Y, then we have the following, and we then construct the spinner bundle as follows, again at a single point, and we take the pullback, which here I'll just show on screen every element. By this construction, this last factor here is equivalent to the conventional spinner bundle on X4, since H star evaluated at the pullback corresponds via the Zorro construction
to the cotangent space on the base manifold, which we denote by that S and then the slash and spacetime. Meanwhile, the other factor here with the V, that's where the internal symmetry space comes from, so that's why I placed a S with a slash through it and said internal as an indice.
whose dimension encodes the family quantum numbers, thus that decomposition that you see above. Finally, in GU, the action, or the Dirac action to be specific, is defined as follows, where that field there is a section of the chimeric spinner bundle, and mu is the natural measure on y. What I mean by natural is that the measure actually comes from the induced metric on y itself.
specifically the square root of the absolute value of the determinant of G with the 14 dimensions dy. This measure has the required properties, that is, it's locally given by the Lebesgue measure in suitable coordinates and it transforms appropriately under coordinate changes so that the integrals such as the Dirac action are well defined and gauge invariant. The Higgs sector, the derivation of the Higgs kinetic and potential terms.
In geometric unity, the origin of the Higgs sector is the gauge potential written on screen here with this variational pi, and the origin of the Higgs sector is when we decompose this according to the splitting of the observers. Now here, let me clarify what this gauge potential means. In GU, this gauge potential variational pi here is a one-form defined on the larger observers
whose domain is the tangent space of Y, and whose target is the Lie algebra of the gauge group, specifically lowercase u, 64, 64. You can think of this as a rule that goes from here to here, that tells you how to differentiate sections of the associated bundle. There's nothing miraculous about this specific one-form. It's the canonical gauge potential coming from the inhomogeneous gauge group structure on Y.
In other words, any connection on the principal bundle Y provides an adjoint-valued one-form, but this variational pi is our chosen representative that encodes fluctuations over the distinguished background connection. Now, the gauge potential will decompose when pulled back to X4 via a local section. See it decomposing as follows with alpha and phi.
where the alpha represents the Lie algebra valued one form component corresponding to the Yang-Mills fields and the phi is a scalar field or at least appears as one coming from additional degrees of freedom in the vertical direction.
When one pulls back a one-form, variational pi from y, via the section, we then decompose, or Eric then decomposes the tangent vector into horizontal and vertical components. The horizontal part gives a one-form on the base manifold, since it involves directions along the base, and we denote that by alpha. In contrast, the vertical part is associated with the variations along the fiber.
So that is the variations of the metric at a fixed point, which when pulled back effectively lose their quote-unquote direction along x4, becoming functions. That is to say, they become zero forms valued in the symmetric two tensors, which we denote by phi.
Editors note, in a more fine-grained analysis, and this is as Eric points out, you have to decompose both the one-form part, the horizontal versus the vertical, and the adjoint part, which is the pure horizontal, pure vertical and mixed. And concretely, the pulled back one-form is written as a direct sum of six different pieces. Each of the summands here are different sectors in 4D.
So some give a spin-1 field, like the usual gauge bosons, and others give a spin-0 field, like the Higgs scalar, and it just depends on how it transforms. This is a refinement about the splitting of what I've just showed of alpha plus phi, but my initial wave writing it was much more compact. As you can see, it's quite tortuous otherwise to show every pairing of horizontal versus vertical in both the one form and the adjoint representation.
Note that the vertical component corresponds to the infinitesimal changes in the metric at a fixed point. Given that they're symmetric matrices, they contain both a trace and a traceless part. In our construction, or more specifically in Eric's construction, we use a contraction, so the trace, to isolate the scalar degree of freedom, which is then identified with the Higgs field. To derive the Higgs Lagrangian,
Eric starts with the Yang-Mills action on Y as follows, where the curvature is here. When we decompose A as A0 plus variational pi, with A0 or A0 being that distinguished connection induced by the Levi-Chevita connection on the base manifold, the field strength splits into parts involving our gauge connection.
In particular, the terms that are quadratic in the gauge connection lead upon pullback to terms like D phi squared and phi wedge phi squared. Now remember, A zero is that classical background connection that encodes the standard geometric data, while the variational pi, the full gauge potential, represents the fluctuations. Now by writing A equals A naught plus variational pi, you separate this unchanging classical piece
To explain this clearly, we begin with a gauge connection that gets decomposed as follows. Then you insert that into the curvature and that leads to this formula on screen here.
When you then pull back this variational pi via the local observations, its vertical component yields phi. The derivative then contains a term that projects onto the derivative of phi, giving a kinetic term here. Meanwhile, the gauge potential wedged with itself when restricted to the vertical directions and then contracted via the Frobenius inner product provides a quartic self-interaction term
of the form on screen here. In GU's language, after the appropriate contractions on the metric on X4, these pieces match the standard Higgs kinetic term
and the Mexican hat potential, typically written as follows. What I find remarkable, personally, is that the gauge potential, when viewed through the lens of the observer and decomposed via the pullback, bifurcates into these two sectors. One is that alpha, which reproduces conventional Yang-Mills, as we'll see, and the other is the phi, which embodies the scalar fluctuations of the metric, which then become the Higgs field.
In this way, GU unifies the disparate sectors of gauge theory and spontaneous symmetry breaking within a geometric framework, all following from the metric. Now, every step as follows. Let me just repeat that as you start with the full gauge potential. You then split it into horizontal and vertical components using iota to pull it back so that the vertical part, which carries the metric fluctuation degrees of freedom, provide a natural candidate for the Higgs field.
The kinetic term is then given by the norm squared here, while the quartic self-interaction is from the contraction, and both are computed via the Frobenius inner product on the space of symmetric 0,2 tensors. Yukawa coupling, the derivation via minimal coupling. The Yukawa coupling term that gives mass to fermions come out as part of the minimal coupling of the Dirac operator on the spinner bundle to the gauge potential.
Start with the Dirac action on the Observer. So we have this action here, and of course this D slash is as follows, which is the Dirac operator coupled to the full gauge connection A, and psi is a section of the Chimeric Spinner Bundle, S of C, or S slash of C, actually. Now, the minimal coupling is another way of saying that the derivative in the Dirac operator is replaced by a covariant derivative.
Inserting the decomposition of A we saw just a minute ago, we obtain the following. In standard physics, the term here, phi psi, isn't actually present in the definition of a derivative because the Higgs field is taken to be a separate independent scalar.
However, Eric's identification is that the Higgs field originates from a particular component of that variational pi. Now in practice, one arranges the theory so that the coupling of phi to the fermions takes the form on screen here, where y is the Yukawa coupling constant. Let's clarify the connection between this guy and this guy.
Now in GU, the gauge connection again decomposes as follows, so that the minimal coupling replaces the ordinary derivative as follows, and next one decomposes the chimeric spinner into its left and right components. This is so that the Dirac operator then splits into off-diagonal blocks as follows.
Here, the splitting into the left and the right-handed parts is fomented from the Z2 grading of the chimeric spinner bundle S of C as constructed via that exponential property we talked about before ad nauseum. In more detail, once a spinner bundle is defined on a space with a metric of indefinite signature, that is 7,7, the associated Clifford algebra admits a decomposition into even and odd parts. This decomposition
allows one to identify chiral or vial sub-bundles, usually denoted by S slash L and S slash R. Some people put a plus and a minus instead. Thus, psi is a section of the chimeric spin bundle which decomposes into its chiral components. Here the operator D minus A and D plus A incorporate the corresponding parts of the connection.
The Higgs field contribution appears precisely in the following blocks, where the first part, the A0 parts, denote the chiral pieces of the Dirac operator associated with the background connection A0, and phi, after some projection or contraction, behaves like a scalar.
Inserting these into the Dirac action yield the following. Now these two terms here combine after appropriate normalization, absorbing constants into the Yukawa coupling to yield the Yukawa action as follows. Thus, geometric unity derives the Yukawa action from the original Dirac action by expanding the covariant derivative to include the gauge potential's extra component phi, which is associated with the Higgs field.
Now this derivation is entirely quote-unquote minimal once one posits that the gauge potential contains both the conventional gauge fields and an extra component phi, the Lorentz group and the standard model gauge group.
The Chimeric bundle is defined as the vertical component plus the dual of the horizontal component. With the overall signature of 7,7, now the relevant real spin group is spin 7,7, however, Eric works with complex Dirac spinners, thus the natural symmetry group is U64,64. The real spinner representation of spin 7,7
has of real dimension 128. However, once you complexify and then you split into chiral components, it decomposes into two 64 dimensional complex of vial representations. In this complex setting, the indefinite unitary group U 64 comma 64 is what preserves the Hermitian form. And this is what acts as the full symmetry group of the spinner bundle, even though the spin seven comma seven governs the underlying real Clifford structure at signature seven comma seven.
All of this is as far as I understand it, and I could be incorrect, so I'm just telling you what I've understood. Now, to recover spacetime versus internal degrees of freedom, one uses the observation map, written on screen here, and the differential of this, or the push forward of this, maps from the tangent space on the base space to the upper space, so that the image is isomorphic to a four-dimensional subspace, the dual of H.
while the complement V of dimension 10 corresponds to the vertical variations. Now this splitting is what is allowing this reduction of the structure group. So we start with spin 7 comma 7, that then gets reduced to spin 1 comma 3 cross spin 6 comma 4. Now this spin 1 comma 3 acts as the spacetime horizontal part and it's isomorphic to the double cover SL2C of the Lorentz group.
the remaining factor 6,4 acts on the vertical 10-dimensional space. Now since 10 equals 2 mod 4, this is just a result, that means that the vertical space admits a complex structure, and you can reduce 6,4 in at least two ways.
So one approach is to reduce it to a non-compact real form, SU3,2. Alternatively, you can reduce spin 6,4 to its maximal compact subgroup. And then you'll find spin 6 cross spin 4. Now there are some isomorphic coincidences where spin 6 is SU4 and spin 4 is two copies of SU2, and then that is precisely the Patiss-Lomb group.
If you know anything about grand unified theories, this is quite remarkable and should be surprising. At least it was to me. Now, to obtain the standard model gauge group, we then do a further reduction. And from here, this is standard in GUT folklore. So, provided you choose a correct embedding of the residual U1 factor as the difference between the two SU2 factors or equivalently by requiring certain trace conditions on the symmetric tensors,
Then the physical gauge group comes about. So you take U3 cross U2 and demand that the determinant equals 1 and then you get something that's isomorphic to the standard model gauge group SU3 cross SU2 cross U1 up to discrete identifications. Note that the left-hand side is the maximal compact subgroup of SU3 comma 2.
Now, this identification is enforced by the requirement that the internal quantum numbers, which ultimately turn out to be 16-dimensional for a single family, and we'll see more about that soon, they match the observed hypercharge and color assignments of elementary particles. So, let me just sum up. First, we start with the full symmetry group, spin 7, 7, on the chimeric bundle.
Then, by application of the observation map, we split the geometry into a four-dimensional spacetime with the spin 1, 3 symmetry, and then a 10-dimensional vertical part which has a different structure group, spin 6, 4, which then can get reduced to SU4 across SU2, SU2, and that further gets reduced to the standard model. Now, GU is broke with
many alternate paths like a labyrinthine roguelike video game as far as I can see. So one path that I mentioned is that pati salam first which is spin six comma four reducing to pati salam providing an immediate quark lepton unification picture. Now the other path is the one that goes through what I mentioned before where you take u three cross u two and you take the special part or you enforce speciality which means the determinant equals one
And then you recover up to discrete factors, the standard model gauge group. Neither of these, as far as I can tell, is more fundamental. They're just complementary ways to see how the indefinite group spin 7,7 breaks into the spin 1,3 spacetime group and the standard model gauge group. The family of quantum numbers. In conventional SL10 grand unification,
The fundamental spinner representation is 16-dimensional. Here, however, in GU, the 16 is reinterpreted as coming from the structure of the metric bundle. So let's recall that the chimeric bundle has that 7,7 signature.
In Clifford algebra theory, for a space of signature PQ, I'm being general now, the real spinner dimension is given by 2 to the P plus Q over 2, and you have to take the floor function of that. Now for 7 comma 7, that works out to 2 to the power of 7, which is 128. This 128 dimensional complex representation on Dirac spinners naturally splits by chirality into two 64 dimensional Weyl spinners. By the way, you may wonder,
Why is this with a W pronounced vial? It's just German. As I mentioned a couple minutes ago, the properties of the Clifford algebra are such that there's a Z2 grading into positive and negative chirality parts, each of that 64 dimension. In technical terms, consider that for Clifford algebra CLPQ, the real spinner representation, if it exists, has this dimension here.
and it decomposes as follows. Notice that this time I'm calling it S plus and then S minus. This operator is unique up to scaling so that the splitting is canonical. Now this action is defined by the standard representation induced by the Clifford algebra multiplication on this space.
In our context, the full chimeric bundle's spinner representation later will be pulled back and partially reduced through projection onto the internal degrees of freedom to yield a 16-dimensional space corresponding to exactly the family of quantum numbers. Let me explain this reduction in more detail. Here, think of the spin bundle on the vertical part as encoding the internal degrees of freedom associated with the variations of the metric, a kind of quote-unquote gauge content.
and then the spin bundle associated with the dual horizontal as encoding space-time properties. When one performs an observation and one pulls back via this observation back onto your base manifold, then you get the decomposition on screen as follows. The careful matching of these components with the quantum numbers like weak hypercharge and isospin forces the internal part to be 16-dimensional.
Now, this matching process exploits the fact that for any given element, say k of u1, one can assign it a weight. And the sum of these weights in a chiral multiplet typically adds up to produce the correct quantization conditions. Now, for instance, representing all of these components as little quote unquote bit strings, so not in the string theory sense, but in the computer science sense, as in conventional SU-5 language,
You'll find that the total number of independent states is exactly 64. You can actually prove this to yourself by looking at the exterior algebra of C5, which has dimensions 2 to the power 5, which is 32, and if particles split into particles and then antiparticles, then you get 16, one for each chirality.
However, GU doesn't just postulate SU5, instead it recovers the same number from its more fundamental chimeric construction. Explaining the Three Generations of Matter Some of you may know that in Hodge theory you obtain cohomological information from harmonic forms, but this requires a metric.
However, there's also the Dirac theory. Now, in Durham theory, there's the metric independence. So, where is the Durham version of the Dirac operator? What I mean to say is that Nigel Hitchens showed that the dimension of the space of harmonic forms changes with respect to variations in the metric. It's only the difference between these dimensions that's topologically determined by the A-roof genus. Now, unlike harmonic forms,
The individual kernels don't behave consistently. In Hodge theory, regardless of the metric chosen, the dimension of the kernel of the Laplacian remains constant and it's topologically determined. So this then raises the question, can you make Dirac theory resemble Diram theory? The way that I see it is that in answering this question, Eric derives the three generations of matter.
To explain the three generations, we studied the Dirac-Rarita-Schwinger complex on the full metric bundle Y14, also known as the Observer, so I know I keep saying this over and over. By the way, the way I'm calling it a complex may give you the false impression that this complex exists in the literature, but as far as I can tell, this is a novel construction by Eric himself. Instead of merely considering spinner-valued zero forms, which are just regular spinner fields,
Eric assembles the combined complex as follows and then he defines a Dirac operator or a Dirac-like operator because it's not exactly Dirac where here the D with the subscript A with the further subscript omega is a gauge covariant exterior derivative involving the full connection of A and then the D with the star is the adjoint constructed with the Hodge star and then this little dot circle guy is our Shiab operator that everyone knows and loves
It ensures that the quote unquote projection is gauge invariant. I call it a contraction. I've heard Eric repeatedly call it a projection. Also, this operator itself is another novel and central contribution of geometric unity. You can tell that by the presence of the Shiab operator, but I'm just hammering the point home anyhow. What does it mean for this operator to quote unquote mix different spinner valued forms? Okay, let's take a look at the block matrix form.
The off-diagonal operators couple zero forms and one forms. Consequently, any solution, like let's just say there's a psi from the kernel of this operator, it will take the form of these two guys here with one guy coming from a spinner-valued zero form and the other guy coming from a spinner-valued one form. Now, a detailed index theoretic analysis, which uses the Atiya-Singer index theorem adapted to this context,
shows that the solution space, so the kernel of this, splits into three gauge inequivalent sectors. Note that I do need to be careful as the T.S. Singer index theorem only applies to Euclidean signature, thus more work needs to be done here, which I've glossed over. In simple terms, the first generation comes directly from the zero-form spinors, and I'll just rewrite that here with psi 1. Now the second part is where it's interesting, because the second part is actually a one-form,
and this decomposes under the action of the Clifford algebra into two components. So you'll see here there's one that's traceless or gamma traceless, then there's another gamma trace part. The gamma traceless piece is gotten to by contracting with the gamma matrices and requiring that this contraction vanish, yielding the second generation, according to Eric. And the complement is the gamma trace part, which produces the third generation.
Also known as the imposter generation by Eric. Those are Eric's words. What you may not know is why he calls it an imposter. So he calls it an imposter in GU because unlike the first two generations in Eric's framework, the third one has different unification properties as one goes up toward U64 comma 64.
Now, I should emphasize that each of these components is isomorphic to the 16-dimensional representation predicted by grand unification schemes. That is, isomorphic at the level of reduction to spin 6 cross spin 4 representations. But how do we know these pieces are exactly 16-dimensional?
Recall that the internal part of the Chimeric Spinner Bundle here comes from the vertical bundle of metrics, and by representation theory of the Clifford algebra for signature 7,7, you get that the full spinner representation's dimension is 128, and then you just divide by 2 to get the 264 dimensional Vial representations.
Now you have to then pull this back, and when you do, the horizontal, so spacetime, component factors out as the standard 4-dimensional spinner representation of spin 1, 3, leaving out a factor of 4, which you then just divide 64 by to get 16.
Thus, you get the full matter field on spacetime as follows. Now, not only does this yield the correct field content for a single generation in conventional SL10, but you get the other three. You get this splitting of the three generations. Again, this is as far as my reading of Eric's theory goes. Now, why does this Rarita-Dirac-Schwinger complex give three dimensions instead of, let's say, two or four?
Well, let's just take a precise look at the structure of the differential complex and its interaction with the Clifford algebra. The zero form part here contributes one block. The one form part, through the action of the gamma matrices, splits into a part that vanishes upon contraction, so the gamma traceless part,
and a complementary gamma trace part. An index theory argument, which I'll leave as a PDF, shows that the net index forces the total kernel to split into three parts, and then you use spectral theory computation to confirm that this has 16 dimensions.
You can show step by step that this Dirac-like operator, when acting on the 0 form plus the 1 form part, is Fredholm, and that its index, which is the difference between the dimensions of its kernel and co-kernel, is topologically determined by the Arouf genus of the observers. Now I should repeat that this Dirac-like operator is novel to GU, so perhaps we should call it the GU fermionic operator.
Now let me just recapitulate this all. The operator structure on the complex here implies that its kernel splits as follows, with each of these guys here individually being 16-dimensional. By the way, to clarify the distinction between operators and complexes in the context of Dirac, Rowida, Schwinger, etc., and these quote-unquote Dirac-Rowida-Schwinger types, formalisms, and so on, let me just go through this one by one.
First, let's take a look at the Dirac operator on n-dimensional spin manifold. Now, this is standard. Nothing here I'm saying is new. We have a spinner bundle S with sections as follows, and the Dirac operator is an endomorphism on this space. And the gammas here are the regular gamma matrices, and that's the spin connection. In even dimensions, S splits into its chirality parts, its chiral parts, and you have this as follows, where it's not exactly an endomorphism. You map instead the left to the right and the right to the left.
Now D alone is just an operator, a single first order map actually, and one often embeds it into a Dirac complex.
So you'll see this on screen here. Complexes just mean that you have many vector spaces and you tie them together by differential maps that eventually terminate to zero. And then there's some other conditions such that you want subsequent compositions to equal zero. And this is what allows people to study the cohomological properties and the index. Now, if none of those words meant anything, it doesn't actually matter. Next is the Rarita-Schwinger operator, R, which deals with spin three half fields.
And these are usually realized as vector spinners. Concretely, you can write it as follows, plus some other possible constraints. These are rarely encountered in standard physics because there isn't any known Rorita-Schwinger fundamental particle. But this too, anyhow, can be placed into a complex, as follows here, where you see this sequence here. And again, you have these consecutive maps that go to zero.
actually if you pause here this should confuse you because the one form spinner part is not spin three halves the reason is that the one form spinner part it's that spinner trace sector that's the ordinary spinners and only the spinner traceless part has the spin three halves and it's that part that gets mapped to the two forms of the spinner bundle this two form of the spinner bundle contains a part that looks like pure spinners so
the zero forms valued in spinners then it has a piece that looks like a rarita swinger spin three half part however it has another piece that we haven't seen yet and that part looks like pure two forms valued in spinners and these will vanish under any contraction
in Eric's framework. So this becomes analogous to scalar, traceless Ritchie and vial sectors of the curvature. See now in geometric unity, you actually merge the spin half and the spin three half fields into a single structure. And it's that that gives this Dirac Rarita Schwinger type operator here, where the Zeta is what contains the Rarita Schwinger part. It's just that it also has a trace piece that has ordinary spinners as well. And the new is the Dirac part.
This block matrix operator generalizes both Dirac and Rarita Schwinger simultaneously. Why? Because it mixes the zero-form or the one-form spinner fields. It's neither purely Dirac nor is it purely Rarita. Hence, I've heard Eric call it a Dirac-Rarita-Schwinger-type system. The CKM and PMS Mixing Matrices
In the standard model, the CKM matrix comes about when you consider flavor mixing of quarks, also known as generation mixing, which happens under the weak interaction. Now in conventional math terms, the CKM matrix is a unitary three by three matrix, and it comes about because the weak interaction eigenstates, so that is the states that participate in the charged weak currents, don't align with the mass eigenstates, that is the states with definite mass.
So I actually talked about this here in this podcast with Lawrence Krauss if you're interested. Link is on screen and in the description. Let's say U, C and T are the weak eigenstates and I'm just putting a subscript of L here because we're dealing with left chirality. Now these are weak eigenstates of quote-unquote up-type and now we have the DSB for the down-type quarks. Then the physical mass eigenstate down quarks are in a linear combination of these via the CKM matrix.
Mathematically, we express it as follows. Where the primed guys here are the mass eigenstates and the DSB unprimed are the weak eigenstates. Now the question is whether this has an interpretation or an explanation or a derivation in geometric unity. Let's proceed by recalling again that GU constructs this unified field associated with the gauge interactions from an inhomogeneous gauge group. And it's on screen here with the variational pi.
This then gets pulled back via the observation and after we perform this pullback, the spinner representation gets decomposed into those three generations that we talked about a few minutes ago. Concretely, we can write it as follows.
Now what about this indicates mixing well because as you move in the internal space determined by the fourteen dimensional construction the observation construction the representation of the unified spinner field isn't fixed in a way that preserves the labels of the individual generations.
So in other words, that internal bundle that we talked about before, which is the spin bundle of the vertical part, has extra substructure. If you change trivializations or perform gauge transformations that act non-trivially on these three families, then the mass term acquires off-diagonal contributions, generating a non-diagonal mass matrix MAB.
In standard physics, you usually allow MAB to be non-diagonal by fiat, meaning that you just impose it. However, from the GU standpoint, this non-diagonally is viewed as a consequence of leftover gauge freedom, the residual transformations that don't block diagonalize the three different families.
So let's say if you write the following, then you have a Yukawa coupling here, which is the capital Phi, and the diagonalization of M A B is gotten to by finding two unitary matrices that separate the uptype and the downtype quark sectors.
The physical misalignment between these two guys here, these two block diagonalizations, that is what gives a unitary matrix the CKM matrix. So geometrically, you can see the gauge freedom in the internal 16 dimensional representation spaces permitting these off diagonal transformations among, say, Psi 1, Psi 2, Psi 3.
The way that I understand it is that suppose you write a gauge transformation as epsilon, like a three by three array of blocks. Let's say epsilon a b. Each block acts between the spaces corresponding to the different generations. In the absence of any of these off diagonal blocks, the three families are unmixed. They do not mix. However, if some of these guys with a not equal to b, so the off diagonal parts are non zero,
in which the mass matrix appears block diagonal can be altered producing mixing. So we have under a gauge transformation epsilon the following here and similarly for say psi 2 and psi 3.
Because weak interaction eigenstates differ from the mass eigenstates by these transformations, the resulting mixing is codified in the CKM matrix. You can view the CKM matrix as the relative rotation required to diagonalize the up and the down type quark mass matrices so that we have this equation here as the overall residual rotation.
So why would this be unitary? Well, again, the way that I understand this is that because the gauge transformations in geometric unity, as in typical quantum gauge theories, are unitary at the relevant stage, the resulting mixing matrix must itself be unitary, preserving probability. That's the advantage or the main want of unitarity.
In effect, you no longer impose by hand that psi phi psi can be off diagonal in generation space. It's forced by the leftover gauge transformations that don't preserve the decomposition here.
The GU derivation of neutrino mixing is essentially a copy and paste of the quark sector derivation just applied to leptons, particularly the neutrino mass matrix. The only difference is that quarks and leptons reside in different parts of the 16-dimensional multiplets. Now, because of this, the unitary matrix that diagonalizes the quark mass matrix is called the CKM matrix, while the one that diagonalizes the neutrino mass matrix is called the PMNS matrix.
The Dirac equation in geometric unity. Let's begin by recalling the conventional Dirac equation, written as follows, where psi is a spinner field defined on a four-dimensional spacetime. Here, the gamma matrices satisfy the Clifford relation, and that's what ensures compatibility with the spacetime metric. See, there's always an implicit metric used to define spinors.
Next, any other connection A on this bundle can be expressed in the affine space of connections as A0 plus an alpha. Again, this alpha is an adjoint valued one-form, representing fluctuations. In conventional Dirac theory, you usually couple a spinner, psi, to a connection A with the derivative, so the Dirac operator here. However, in GU, the Dirac operator must be defined on the chimeric-spinner bundle.
And it takes the form of this calligraphic D, where the omega, the subscript, has two components, an epsilon and the variational pi. And that represents the overall gauge data, where we have a gauge transformation, which is epsilon, and a gauge potential, which is that variational pi. And precisely, we define or Eric defines this calligraphic D operator as follows.
You can clearly see that this acts on a two-component object. Which one? Well, let's just write it as follows, zeta and nu. The top one, zeta, is a one-form and the bottom one is a zero-form, so just regular spinners. Here, the operator is the shiab, which is designed or engineered to perform that generalization of a contraction that Einstein used when forming the Einstein tensor. But at least this time, it's gauge covariant.
Now notice that D, the exterior covariant derivative, defined relative to the connection A omega, itself is built from that distinguished A zero and a fluctuation, so the variational pi.
Now see this decomposition into zeta and nu. It seems like it's different than the original decomposition into this tensor, the spinner bundle of H-dual. It actually is consistent because the decomposition into zero and one forms come about because of the exterior algebra structure on y and it later plays a role in distinguishing the different generations. Recall just a couple minutes ago, maybe 10 or 20 minutes ago at most, I talked about the calligraphic d's
kernel decomposition into subspaces of the first and the second and third generation. Now when someone says that fermions are quote-unquote square roots of gauge potential, someone like Eric for instance, this is because the Dirac operator is a linear first-order differential operator whose square yields the Laplacian up to curvature terms and it's exactly like taking the square root of the Klein-Gordon operator which leads to the Dirac operator. This isn't language that I use personally.
In GU, the Dirac operator, so the calligraphic D, plays a dual role. It not only governs the dynamics of the matter fields for a given connection, but it also encodes via the square, the meshing of gravitational and gauge degrees of freedom,
Let's now derive the gauge covariant form of the Dirac operator in GU step by step. So firstly, we start with the chimeric spinner, which is the decomposition of zeta and nu, the one form and the zero form. Perspectively, you can think of them as a vector-valued spinner versus just a regular spinner. Step two, you introduce that distinguished connection A0 on the principal bundle, and you can get that however you like with an observation or what have you.
Step 3, you define the covariant differential coupled to a connection A omega, acting on differential forms valued in the spinner bundle, and the operator satisfies what you see here, which is just you move it up a form. This obeys the Leibniz rule, and when I say you move it up a form, I mean if you are a P-form, you become a P plus 1 form.
Step 4 is you construct the adjoint with the Hodgstar operator, which is actually defined because you do have a metric now. The Hodgstar requires a metric, or at least it requires a volume form, which usually comes from a metric, which we do have here on Y. Step 5, you introduce the Shiab operator, which is a contraction map.
And it's constructed to mimic the contraction of the Riemann curvature tensor into the Einstein tensor while maintaining gauge covariance. And when I say the SIAP, I mean Eric's SIAP because this is something Eric is introducing.
and it's not a standard term in the physics or math literature. Neither is the following. Step 6, you assemble the Dirac-Rarita-Schwinger operator as follows. Also note that this could be called the GU-Fermionic operator. Step 7, you verify that the composition of the lower and upper blocks satisfy that the square is approximately the following, analogous to the standard property that we have with the regular Dirac operator, Modulo Curvature Corrections.
Step 8, you notice that under gauge transformations of the entire operator that it transforms covariantly. In other words, if you take a capital Psi transformed by an element of the inhomogenous gauge group, calligraphic G, then the calligraphic D, capital Psi, transforms in the same way, preserving the physical content. Step 9, you conclude that the Dirac equation in GU may be written as follows, which covers both the propagation of the fermionic fields
Many of the ideas I'm about to lay out are inspired by geometric unity. However, I wasn't able to find specific source notes on it, so this is my best attempt to piece things together. It may not align with how Eric sees it. Let's now turn our attention to the Klein-Gordon equation, which is the conventional way that we talk about spin zero fields.
In standard quantum field theory, the Klein-Gordon equation is written as follows, where the square is that double Laplacian, also known as the de Lambertian operator, and phi is the scalar field. This equation is inherently second order in its derivatives, and it comes about from the Euler-Lagrange equation, which is on screen here, and of course we're up to a potential term.
So how does this come about in geometric unity? Well first, remember that we decompose the symmetric 0,2 tensors as follows with a trace component and a traceless one, and the one-dimensional R subspace, so the trace subspace, is going to be the Higgs-like scalar field. Now in GU, the gauge potential A with a subscript omega includes both the usual Yang-Mills vector components and the scalar components coming from the vertical decomposition.
When we form the kinetic term, which is the square of taking the derivative of phi, it comes about from the Yang-Mills-Lagrangian term on screen here after decomposing the curvature into the parts that are quote-unquote purely horizontal and those that are mixed or vertical. More concretely, begin with the gauge curvature here. This is standard. Now note that A decomposes into the following.
Now, the first part represents the conventional gauge fields, like I mentioned, Yang-Mills, and the second part is going to be something like the Higgs component. So, then you expand this expression and you obtain cross terms. Cross terms that are linear with respect to the Higgs, so the variational pi subscript H, and quadratic in A. You'll get terms like the following. That's actually quark tick in the variational pi.
Now this quartic term then plays the role of the Mexican hat potential in the Higgs sector, typically written as follows. Keep in mind that in Eric's framework, the structure and normalization of this potential are not arbitrary. They instead come from these contraction rules. When one projects the curvature, so F A,
onto the vertical subspace of the chimeric bundle. To be precise, we can define an algebraic operator as follows, which compresses the curvature tensor in a gauge covariant fashion. When applied to the curvature, the operator, P e in this case, yields a symmetric two tensor whose trace reproduces the scalar curvature relevant to gravitational dynamics. And then there's some deviation, and that deviation from the idealized form
is what encodes the self-interaction of the Higgs. You can vary the action with respectify and it leads to the following equation, which one recognizes as the Klein-Gordon equation except with a nonlinear potential. To say this differently, the same contraction mechanism that transforms the full gauge curvature
into a form suitable for gravity also when applied to the vertical fluctuations yields the kinetic and the potential via the quartic self-interaction and these terms are characteristic of the Klein-Gordon equation for a scalar field. Again, much of this is based on my own reading of Eric's notes and discussions about this but this is my current understanding so I'll share it with you.
This is about Einstein's field equations. So in standard form, they read as follows, g plus the lambda with the lowercase g equals the kappa times t. It's quite familiar to most of you. Now in geometric unity, you still want to write some Einstein-like tensor, capital G, but defined on the 14-dimensional observer's or the metric bundle. Instead of simply writing the following, Eric introduces a projection operator, which I'm calling P
Erick adjusts the vertical Frobenius metric via trace reversal.
is treated so as to preserve the covariance. Now the result is a modified Einstein tensor g, satisfying g plus lambda lowercase g equals kappa t, where the t comes about from the matter fields that are on y except when pulled back via an observation. Now something I was confused about is what is the relationship between this p and the Shiab operator.
Well, the Shiab operator is a more general mechanism for contracting curvature in a gauge covariant manner. In GU, one usually denotes the Shiab operator that mixes the torsion type terms with the curvature to project out a Ritchie-like combination. Thus, this P operator
can be seen as an Einstein-specific restriction of the broader Shiab framework, focusing on producing the usual Ricci tensor minus half the Ricci scalar g structure, except while remaining compatible with the inhomogeneous gauge group. Consequently, the final Einstein equation on y, the larger space, is not, like I mentioned, a naive contraction. Instead, it's from the Shiab operator or a specific instance of it to get this Einstein-type decomposition.
This modified scheme is what recovers a gauge covariant version of G, the Einstein tensor, plus a cosmological term equated to a derived stress energy tensor. The Yang-Mills equation in geometric unity.
Finally, let's examine the Yang-Mills equation in the context of geometric unity, beginning with the classical expression of dF equals J, where F is the curvature of the gauge connection A and we're on a principal G bundle over spacetime in the standard formulation. Now again in standard Yang-Mills theory, we're in four dimensions and the connection A is a Lie algebra G, so a lowercase g algebra valued one form,
and its curvature is given as we've seen here ad nauseum, where the wedge product incorporates both the exterior derivative and the Lie algebraic structure. The gauge covariant derivative acts on sections of the adjoint bundle as follows. Now under a gauge transformation, the connection and curvature transform like so, which guarantees that the Df transforms homogeneously and makes it already gauge invariant.
In Geometric Unity, however, Eric sees the Yang-Mills as recast, except on the observers, of course, where the gauge fields aren't extensions of some external field, but a component of the Chimeric bundle associated with the space of metrics.
More precisely, the connection on the principal bundle over y is written as follows, where we have, like I mentioned, the distinguished levy-chevita connection coming about from the Zoro construction, and the variational pi, which is just a one-form that is Lie algebra-valued, representing gauge potential fluctuations.
The Yang-Mills curvature in Eric's framework is then given as follows, which is a Lie algebra value 2 form, so there's nothing new here, except that in GU, Eric's trying to solve the problem that the usual contraction of the curvature tensor that appears in Einstein's equations don't straightforwardly commute with gauge transformations as we discussed earlier, and something similar occurs in Yang-Mills theory if one tries to insert the gauge potential directly into the action. To remedy this,
Eric treats the gauge potential as living in an affine space, which is modeled on the one-forms, and defines its action indirectly via the curvature, which is manifestly gauge covariant. Now, the Yang-Mills action in GU is defined as follows with the inner product defined by the killing form on the Lie algebra and the volume being the volume form on Y, which is induced by the Chimeric metric. This affine space that I mentioned
The structure guarantees that every term, especially the curvature, transforms correctly under the inhomogeneous gauge group, and I'll just write it here again for reminder. In other words, while the final expression mirrors the familiar Yang-Mills action, its derivation is fully rooted in GU's framework, so Eric ensures gauge covariance and a natural incorporation of the horizontal and vertical degrees of freedom. When you try and vary that action with respect to a sub-omega,
The Euler-Lagrange equation gives a familiar Yang-Mills equation form, where the D with the star is the adjoint of the covariant derivative, and J will represent the current from derived matter fields. Now there's a subtlety here in that GU with the curvature has to further be processed by the Shiab operator.
For clarity, the Shiap operator is defined as an algebraic contraction map designed to replace what I've called the naive metric contraction present in Einstein's quote-unquote projection, although I see it as a contraction. Concretely, let's say we have a two-form Kazai. The Shiap operator combines other forms,
phi 1 and phi 2 which are built from the geometric data of y via the construction as follows where the star is the hodge star operator associated with the induced metric on y and the conjugation by epsilon is what ensures gauge covariance.
In effect, this operator is what compresses the two-form curvature into the lower degree structure analogous to a symmetric two tensor in a way that commutes with the residual gauge transformations. This construction is baroque and carefully done. It's necessary because Einstein's usual contraction doesn't commute with gauge transformations, leading to inconsistencies when unifying gravity with gauge theory. It's another reason why Eric calls it the ship in the bottle
When I was telling him, hey, isn't this just conjugation, which is just something viewed from another perspective? And it's from here that you can see why the ship in the bottle reference is more appropriate because of, like I mentioned, it's quite baroque and carefully constructed. My current understanding is that the effective Yang-Mills equation in GU takes the form as follows. I'm writing it like this.
But also know that the following equations are already gauge invariant by construction, and that's why they don't require the same specialized operator. There will be an accompanying PDF to this, so if you would like more notes, such as expansions on these topics and proofs that I wasn't able to get to in this iceberg, then subscribe to the substag, as that's where I will publish it. And that's it. That's all of physics explained in geometric unity.
You can breathe now. The rest of this iceberg will just be reprises of the derivations and the concepts that you've already encountered. So congratulations. Before we move on to the next and final layer, layer four, which will be summaries and open questions, I want to cover some notes on the simplicity of the equations that we've just outlined that underlie theoretical physics. The Dirac equation, for instance, it's the quote unquote simplest of its kind.
Why? Well, in part, because it generates k-theory.
So, what is K-theory? K-theory is about the differences between vector bundles. So, sure we can add vector bundles, but then the question is, what does it mean for one vector bundle to be subtracted by another? In order to make vector bundles into a complete abelian group, you require a way of talking about their quote-unquote formal differences. This is done by Grothendieck, called the Grothendieck completion of the monoid of isomorphism classes of vector bundles under direct sum.
It's quite a hairy term, but the point is that you have something called the index, which generates the k-theory classification of vector bundles through something called the Atiya-Singer index theorem, which we've talked about, and the Dirac operator is the simplest operator in k-theory. Well, now how about the Yang-Mills equation? Well, it's the simplest geometric equation involving the curvature of a connection.
What about the Einstein field equation? Well, it's simple in the sense that it's derived from the simplest Lagrangian possible. It's built not even from the curvature tensor, but just the scalar curvature. And also the Klein-Gordon is seen as one of the simplest equations, if not the simplest of its class.
Paradoxically though, the Klein-Gordon equation, it appears to be the most geometric one with its metric hat potential, is actually the least intrinsically geometric of the four equations that I've just mentioned. Differential geometers often overlook the Higgs sector. Note, what I mean is that the Mexican hat is easily seen to be visually geometric, but not easily seen to be differential geometric.
However, to Eric, the Higgs sector comes about as a vertical component of a connection form. So in GU, when decomposing the gauge potential, the Higgs field appears naturally, and the Yukawa coupling is just minimal coupling. Now conventional field theories introduce various vector bundles that are seemingly disconnected structures from the derived space-time geometry. In general relativity, symmetric zero-two tensors decompose, like we mentioned, into the trace and the traceless component, with the trace component
Carrying an eric's formulation to spin zero so eric's innovation is that he lifts the frame bundle to the double cover with the gl double cover also known as the metal linear group and this structure may be new to most of you even though you're familiar with physics and it's because mathematicians tend to avoid the double cover as there's no
Finite dimensional representation that can do the job of representing spinners. A fundamental group's obstruction exists because the gl4,r is real. Now, GU overcomes this by quickly passing to the spin subgroup and moving to the observer's rather than working on the base x4 directly.
This then is topological rather than metric dependent. Now the connection between these spaces allow the vertical components of the connection to appear as the Higgs field when pulled back to space-time.
Eric constructs these chimeric versions of spinner representations in indefinite signature space with the spin 7,7 that we mentioned before. And then you reduce it through its maximal compact reduction in one of the paths that I mentioned. And that's how he unifies previously disconnected geometric structures.
Just so you know, if you have any questions, which you likely do, that's all right. I'm going to have a large solo podcast with Eric just on geometric unity. It will be unlike any other podcast because we'll delve into the particularities of it, especially now that they've been explored in a fair amount of detail. Therefore, feel free to subscribe to be notified of this upcoming podcast with Eric Weinstein. Layer four.
Finally, we're in the deepest layer, which is actually the most accessible now that you've had the previous two or three hours of background. Man, you don't see this, but this iceberg took hundreds of hours of work across 10 months on and off. I hope you enjoy it. The feeling that I get from Geometric Unity, I figured out how to make it into an analogy, is that firstly, if you take a look at the standard model, it resembles a jigsaw puzzle.
It's broke, the edges aren't clean, there are little protrusions. It's unclear where these pieces came from or what the source of the jagged edges, the spikiness, the messiness is. What is clear is that it fits together and works somehow, except there's this other piece beside it, which is a pristine disc, also known as gravity.
And it's unclear how to combine these, they look like they're different elements. One is like that colorful, baroque object that I talked about, the jigsaw, and then the other is the gravity, the nice porcelain disc.
What geometric unity looks or feels like to me is if you start with this disk, generalize it, you get, say, a sphere, a perfect pearl. Then, if you allow this pearl to drop on the floor, it will shatter into different pieces, but what's remarkable is that some of these pieces exactly outline the aforementioned messy jigsaw puzzle.
Then you wonder, what are the odds? Look, I was trying to combine these two different pieces, the jigsaw puzzle and this disc, and I couldn't make it work. But actually, they're not just meant to naively be combined. Instead, they all fall out of the same structure. And you don't need to do any work to get it to do so. You just let it drop on the floor and examine the pieces.
Well, that's the feeling of geometric unity, at least to me. Now enough about feelings. Let's get to this layer. Right now, I'd like to give you an explanation at four different levels. So one is the explain like I'm five level, even though I have my issues with that. And you can read this sub stack here, where I go into detail about how misleading and foolish this enterprise of hey, if you can't explain it to a five year old, you don't understand it is.
after the ELI 5 you'll get the ELI U so that is explain it like I'm an undergrad and then after it's explained at the undergrad level I'll explain it at the graduate level and then I'll explain it at the PhD level then we're going to get to open questions and then there's one more treat
Hi, everyone. Hope you're enjoying today's episode. If you're hungry for deeper dives into physics, AI, consciousness, philosophy, along with my personal reflections, you'll find it all on my sub stack. Subscribers get first access to new episodes, new posts as well, behind the scenes insights and the chance to be a part of a thriving community of like minded pilgrimers. By joining, you'll directly be supporting my work and helping keep these conversations at the cutting edge. So click the link on screen here.
Explaining geometric unity to a five-year-old
In physics today, we have these two primary theories that don't get along well. One is about gravity, so general relativity, and one is also about particles, so it's the standard model. The universe seems governed by particles, but it also seems to be governed by gravity, since you're made up of both particles and you stick to the ground, and you orbit the sun, etc.
However both of these theories even though they describe the universe they don't combine well together and the prefix of universe is unique which means one so is there a unification that combines these that's what geometric unity attempts the key insight from eric weinstein.
is to take a look at this 4D space time and instead of putting a metric on it, so it's actually not even a space time, it's just a 4D space, you think of what are all the possible Einstein theories that can be placed on this? Mathematicians sometimes call this a moduli space, but technically in this case, it's a 14 dimensional manifold.
What Eric calls the observers now in this higher dimensional space forces and even the three generations of matter that we observe in the standard model they aren't added in by hand instead there in gendered from the geometry of this fourteen dimensional space itself. Explain geometric unity like i'm an undergrad in math or physics.
Start with a four manifold. You then think about how do I make it geometric since currently it's topological. Now geometric in this case means metrical, so adding a metric. However, you want to be general. You want to think about all metrics.
One way of thinking about this is by attaching all possible metrics at a single point and that is technically called the metric bundle. You then think about if this metric bundle itself carries a metric that would be meta and it turns out it doesn't but there's a way that you can metricize part of it.
namely the vertical parts of the tangent space. You do so with a certain type of metric called a Frobenius metric. The name doesn't matter. It's actually a somewhat natural metric and there are two choices here. There's a regular Frobenius metric and then there's something called a
Trace reversed one or just the reversed one doesn't matter they're both choices the reversed one is preferred for various reasons from there you can then think about what is the signature because you have a metric now and now that you have the signature you can think about the spin group of the signature so spin seven comma seven.
And now that you have a spin group, you can think about what does this spin group act on, and it turns out one of the spaces it can act on is a real dimensional space except of 128 dimensions, so R to the 128. Now since chiral fermions are complex chiral, and since real vector spaces can always be complexified, Eric complexifies here.
and that also splits into two copies of C32, 32. Note, I'm not saying anything special like a Newlander, Nuremberg integrability condition or that there's the vanishing of a certain nine house tensor. We're not dealing with complex manifolds. I'm just making the observation that anytime you have copies of R, you can complexify it. That ability is there in the same way that when you have a manifold, you get with it a tangent space.
You don't have to provide anything special to the manifold, it just comes with a tangent space. We also know that a manifold has a cotangent space, just like I said it has a tangent space. However, there is not an isomorphism between the tangent and the cotangent without a metric.
Or without additional structure like a symplectic form a metric in this case for the base manifold would be a choice of a section of the metric bundle you can think about why that is and you'll be able to convince yourself that is an equivalent notion so let's assume that you make that selection of a section.
Now although we won't fix what type of section, we'll just say that you choose some section. At that point, you can make an isomorphism between the tangent and the cotangent. And from there, given that your tangent space of the full metric bundle will split into vertical and horizontal parts, you can think of the dual of the horizontal now. I should say at this point that there's plenty of twists and turns, like there's so much to remember, but the point is that you start from something simple and you look at what structure is contained within.
In many ways you can think of the claim of geometric unity as the claim of what we think of as simple actually has extreme complexity inside it and furthermore a subset of that complexity matches the standard model almost verbatim.
okay so now we keep moving around in our little space of complexity and we use a result from differential geometry about spinner bundles which is that if you have a direct sum b has bundles and you take the spinner bundle of that it's the same as the spinner bundle of a tensor the spinner bundle of b you can use this along with the isomorphism provided before to form spinners and now you wonder well
Okay these spinners are built on the metric bundle and we live in a base space or at least we supposedly do so what do these spinners look like from the perspective of the base space now the question about perspective from the base space is the same as a pullback operation.
In this instance so that's what we do when we're allowed this pullback operation because we've already chosen a section of the bundle once this is pulled back we then get what matches the standard model spinners as for the gauge group this one you can see because there's a reduction from spin seven comma seven.
down to spin 1 comma 3 cross spin 6 comma 4, and from there, after moving to the maximal compact subgroup, the second factor goes down to spin 6 cross spin 4, which is Pati Salam's grand unified theory.
In this way, the quantum numbers and the standard model gauge group actually have different origins and the quantum numbers are fixed while the gauge group could have been different or been broken down differently, theoretically, at least in geometric unity. Now as for the Higgs and other parts of the standard model, those can be seen as different connections on the full metric bundle, but also pulled back.
In some ways you can think of this as wondering about how we have these two different theories one that describes particles and one that describes gravity gravity is like a simple dove in that it's beautiful and innocent and the standard model is like this torturous snake.
in that the standard model is effective but it's convoluted. People have been trying to put these together for decades and what Eric has found is that if you look at gravity itself and generalize it, the particles come out. You can visualize the four manifold as a chiapet which grows fibers naturally and that living on the blades of grass are the different particles we're looking for. Be as innocent as doves and as wise as serpents.
Explaining Geometric Unity to a Graduate Student
Firstly, we have to think about what's the goal. So to explain to a graduate student, I'm going to use Eric's frame of mind. What's the goal of Eric? Eric is thinking, how do I resolve the chicken and egg problem of quantum gravity? That is, how can matter, which is described by spinors, exist between metric measurements if you require a metric in order to define the spinors? The answer, according to geometric unity, is that matter lives in the observer's in this 14 dimensional metric bundle.
and spinners can exist there even though you're not making a canonical choice of space-time metric. So you start with a four manifold x four, GU then constructs a metric bundle which is on screen and at each point you have the decomposition in the tangent space as follows with the vertical space representing the metric variations.
Eric then uses a certain construction, where if you choose a section of this metric bundle, it then gives you a connection, the Levi-Chevita connection. Now that metric on y induces a connection itself. And this is what a choice of a horizontal subspace means, a connection. Now the Chimeric bundle combines the vertical space, which has the signature 4,6, with the dual of the horizontal space, so the signature 1,3, and that gives the total signature of 7,7.
I should note once more that spinners are derived on this chimeric bundle, which has a metric without requiring a connection. Now because this chimeric bundle is isomorphic to both tangent and cotangent bundles of Y, you get spinner bundles to exist prior to choosing a metric on the base space X, or even a connection on Y. But once you do select a connection, you become canonically isomorphic.
This is that Zorro construction and it's a constitutional mechanism that powers geometric unity. The spinner bundle on the space then decomposes as follows. There is something else called the augmented torsion tensor and that is supposed to resolve. Again, we have to think what is Eric thinking is the problem? What is the goal?
Well, Eric's thinking I'm looking for a map from the inhomogeneous gauge group to the space of connections, which is equivariant under this right action and equivariant on the left hand side as well. This is what becomes dark energy. Eric sees another problem which he calls the gauge incompatibility problem. What that is is that in Einstein's formulation, the Riemann curvature tensor is viewed as a two-form taking values in a two-form.
So essentially you have two copies of a two-form, one from the connection and then one from the metric structure. When you contract this tensor to produce the Einstein tensor, you're treating both copies symmetrically. However, that makes it not transform properly under gauge transformations. This mismatch in the origins of the curvature components is what Weinstein, Eric Weinstein, refers to as the twin origins problem.
So this augmented torsion tensor is one way of solving the problem with gravity that is not gauge covariant, but gauge theory is gauge covariant. So how do you make gravity gauge covariant? Eric then introduces a Shiab operator, which generalizes that contraction I referenced about Einstein in a gauge covariant manner.
The formula is on screen here. And then there's a Dirac-Rarita-Schwinger complex on this larger space. This generalizes the Diram complex, twisted by spinors in dimension three. Now this complex gives three distinct sets of fermions.
The scalar spinner, which gives the first generation, and then there's a vector spinner, which splits into two parts, a gamma trace part, which gives the second generation, and a gamma traceless part, which gives the third. Each of these is 16-dimensional, and that matches the standard model's Fermion structure. In particular, it matches spin 6, 4, which is an alternative real form of spin 10 complexified, and that's closely related to the well-known SO10 real form from grand unification theory.
Hi, Kurt here. If you're enjoying this conversation, please take a second to like and to share this video with someone who may appreciate it. It actually makes a difference in getting these ideas out there. Subscribe, of course. Thank you.
How about rather than quantizing gravity directly on X4, which gives renormalization problems, how about we work with fields on Y14, that metric bundle. Then from there, we're going to pull it back to X4 via sections of the metric bundle.
Geometric Unity's foundation is in constructing principal bundles over this Y14, so principal bundles on principal bundles, this time with groups spin1,3 and spin7,7 and u64,64 as the structure groups, and we have a chain of inclusions on screen here. There's also something called the inhomogeneous gauge group, which is on screen here.
where the first factor is a genuine gauge transformation and the second is actually just gauge potentials. Now this inhomogeneous gauge group calligraphic G combines the gauge transformations with the translation quote unquote in the space of connections with a certain product structure given on screen as follows. Now let's say for any connection and any element inside this inhomogeneous gauge group we also have a right action that right action
is defined as follows. The augmented torsion tensor, that is what gives a gauge covariant object that transforms correctly under both gauge and diffeomorphism symmetries, resolving that gauge incompatibility problem of the Riemann tensor. The Shiab operator generalizes Einstein's contraction in a gauge covariant manner, and I'll put the formula on screen here again. The action principle, because we all want to know what is the action, what's the Lagrangian,
That takes the form of a first order equation reminiscent or it rhymes with Einstein-Hilbert and even Chern-Simons theories and that's on screen here. The field equations are derived from this action. The term involving the torsion T is what ensures the Euler-Lagrange current remains exact.
This exactness property is what allows Eric the geometric formulation of the field equations. There's also a quadratic term, and that maintains the gauge covariance. The Dirac-Rarita-Schwinger operator takes the form on screen.
and it acts on regular spinners plus spinner-valued spinners. This operator on this space here comes from Eric thinking differently about supersymmetry. There is something pioneered in the late 70s by Salam and Strotti about constructing superfields, which are just fields defined on superspace.
This construction, in Eric's mind, was erroneously applied to Minkowski spacetime, rather than the space of connections. To Eric, this explains why there's been so much time and money and effort wasted on finding superpartner particles on spacetime, because superpartners exist in connection space. The decomposition into three pieces occurs prior to the kernel operator, each 16-dimensional.
And now the first generation is the regular spinner-valued function, so the zero forms. And the second and third come from the decomposition of the one forms on the spinner field. And those break up into gamma trace and gamma trace list parts. The Higgs field comes from the vertical component of the gauge potential, so the variational pi.
And when you pull it back to your base space, you get three parts. One that looks like a gauge potential downstairs, so that is to say an ad-valued one-form, interpretable as such on X4, the base space. Additionally, you get a spin-zero piece downstairs, which gets interpreted as the Higgs field in Eric's framework. And you get a remainder piece that has not been identified or not seen and doesn't figure into our experimental observations in the physical universe so far.
Just remember, there either is right now or is going to be an accompanying PDF to this Geometric Unity iceberg, so if you'd like more notes such as expansions on these topics and proofs that I wasn't able to get to in this iceberg, then subscribe to my sub-stack as that's where I'll publish it. It's curtjamungle.org or C-U-R-T-J-A-I-M-U-N-G-A-L.org. Open Questions
Soon I'm going to do a full recap of the whole 30 steps of geometric unity except in a flyby overview just to hammer the point home further and show you that all of this initially abstruse math is now understandable by you. It's actually in part to congratulate you. At first you heard this foreign language and now you're able to be somewhat conversant in it. You can order dinner. You can go back to your hotel. You can pick up a date. I'm taken by the way, but
Thank you. Now first, let me talk about what open questions I have.
One of my questions is, what is the phenomenology of the theoretically predicted but currently unobserved decoupled sectors? I put them in quotations because I've heard Eric call these dark sectors, but I find that terminology to be too evocative of dark matter or dark energy, so let's just call them decoupled sectors. Would gravitational interactions at high energies enable us to have direct or indirect observations of these? How?
Okay that's one question. My next is about how does geometric unity account for the matter anti-matter symmetry slash asymmetry. Is this observed asymmetry explained by disconnected chiral sectors where the anti-matter is hidden so it's a decoupled chiral sector or is the asymmetry still an initial condition? And another question I have is given that Eric
has squeezed plenty out of these numerical coincidences, particularly about spinner structures like SL10 and SU5, their connections to the Einstein field equations, the metric in four dimensions. I believe Wilczek actually remarked about this. So given that, and there's also the coincidences of the observed generations of fermions being precisely 16 fields, etc., how far do these numerical coincidences go?
What other numerical coincidences are meaningful? So what about Dirac's large number hypothesis? What is a red herring versus a smoking gun? Also, you should know that I'm going to be speaking with Eric directly on the podcast in the next couple of weeks. So if you have questions, leave them below and feel free to subscribe for that conversation as we'll go further in depth into geometric unity.
At this point, I'd like to emphasize that I don't want you or other people to think that there's something negative given that they're open questions, as every single theory has open questions. For instance, here are some of my notes about open questions in string theory, in loop quantum gravity, and in asymptotic safety. Eric's theory is a tour de force, and unless you have an understanding of physics, it's difficult to fully appreciate how many pieces there are in this one theory.
Despite the open questions that I mentioned, which this theory has been generated by a single person in isolation, now believe it or not, what I've shown you for the past three hours or so still leaves maybe 30 to 40% of GU unexplored.
Just so you know, if you have any questions, which you likely do, that's all right. I'm going to have a large solo podcast with Eric just on Geometric Unity. It will be unlike any other podcast because we'll delve into the particularities of it, especially now that they've been explored in a fair amount of detail. Therefore, feel free to subscribe to be notified of this upcoming podcast with Eric Weinstein.
Now, let me give a recap of the 30 steps of geometric unity for the interested viewer who cares about the steps, and this is a treat because you'll be surprised how much of this finally makes sense. Maybe not all of it, but initially maybe 5% made sense. So number one, you begin with a smooth four manifold, say X4. You then construct its metric bundle Y14, where each fiber is the symmetric two tensors on X4, and that happens to be 14-dimensional.
Number two, you construct the frame bundle with the structure group GL4, then you lift to its double cover to enable finite dimensional spinner representations. Next, number three, you define observation maps, which are actually sections or local sections of this bundle.
Next, you construct the tangent bundle of the metric bundle, and you also construct its dual, so its cotangent bundle. Next, you split the tangent into its vertical bundle, which encodes the metric variations, and the horizontal part, which is just directions along the base space.
Next, you use a certain construction which Eric happens to call the Zorro construction. It's a way of getting a section from the base space to the metric bundle. So you choose a section, you're choosing a different slice of this whole metric bundle. How do you get from there to a connection on the metric bundle? That's what this construction is about. Next, you define the chimeric bundle, which is the vertical space direct sum the dual of the horizontal. Next, step eight, you introduce the Frobenius inner product.
and that acts on symmetric matrices to metrize the vertical fibers. Next, number nine, you decompose the symmetric zero two tensors into a trace and traceless part, and you choose a signature. This is one of the only parts in geometric unity where you make an actual choice that wasn't there implicit in the structure. In this case, you choose four comma six, and that's for the vertical part, and you choose one comma three for the H part, the horizontal part.
Number 10, you construct spinner bundles, but you do so on the chimeric bundle and you use that exponential property as follows on screen. Next, you identify the structure group of spin 7,7 who has a real spinner representation of
two to the power of seven so 128 dimensional and that splits because you can divide that into two into two chiral 64 dimensional real spaces but you have to complexify now number 12 from the principal bundle with the structure group of U of the unitary matrices 64 comma 64 via the spinner representation here then you define an inhomogeneous gauge group where the first part are actual bona fide gauge transformations
And the second are just gauge potentials. You just semi-direct product them together. Number 14, you specify a right action on the affine space of connections.
And you do so with the following somewhat hairy formula, but not too hairy. And this formula has group associativity to it. Number 15, you define the augmented torsion tensor. This is the gauge-equivariant analog of Einstein's contraction operator, and this addresses Einstein's quote-unquote gauge problem, or more accurately, it addresses Weinstein's rendition of Einstein's gauge problem. Number 16, you introduce the Schiap operator. That acts on two forms.
And it's defined on screen here that's the generalization of Einstein's contraction number seventeen you formulate a first order action so this is the action of geometric unity as far as i can tell now eighteen you vary the action to derive field equations and this is what gives you something that's analogous to Einstein's field equations.
And also Yang Mills. Number 19, you introduce Fermion fields as a spinner-valued form. So there's firstly a zero form, which is just a regular spin field, and then there's a spinner one form here. And that gives a Dirac-Rarita-Schwinger operator coupling both sonic and fermionic sectors. And that's why Eric sometimes refers to this as supersymmetry.
I'm not a fan of that term because supersymmetry means something in particular, but Eric is taking the analogy here of interchanging fermionic and bosonic degrees of freedom as supersymmetry instead of the supersymmetry algebra. Number 20, construct the deformation complex here. This has a cohomology which classifies gauge-inequivalent linearized fluctuations about a solution. Now number 21, you see that there's a seesaw structure inside this Dirac-Rarita-Schwinger complex.
And that's what explains mass hierarchies, mixing between light and heavy spinners. Number 22, you can do a reduction. So you go from spin 7 comma 7 to that spin 1 comma 3, and you cross that with spin 6 comma 4. You can do some further reductions, as I've described earlier on, and that recovers the standard model gauge group. Number 23, you decompose the diracuarita-shringer complex, and again, like I said, there's a zero form and there's a one form,
The one form splits into two different parts, a gamma traceless part and a gamma trace part. One of those is the second generation and the other one is the third generation, whereas the zero form is just the first generation.
Number 24. There's a Higgs field. Where is this Higgs field? Well, when you take the variational pi, the one form on the principal bundle, it has a scalar part and that corresponds to the Higgs field upon pullback. Number 25 is that the symmetric 0-2 tensors themselves, which have a trace and a traceless component, there's a trace component here which gets identified with the Higgs.
and the traceless part becomes identified with a spin-to-graviton-like field. Number 26, you can derive a quartic potential when you do some expansions, and that was covered earlier, from the Yang-Mills term A wedge A squared. And that actually gets you a Mexican hat potential for the Higgs field.
Number 27, you show that there's a minimal coupling in the Dirac operator, so on screen here, and that gives you Yukawa interactions when the A, the connection here, includes Higgs components. Number 28,
There's a correspondence between the Higgs and the Yang-Mills sector because you decompose this variational pi and that gives you this unified origin of the Higgs and Yang-Mills in this geometric structure of the metric bundle. This isn't even a step, it's more like a realization. Number 29, you incorporate a quadratic term into the field equations and this gives full gauge covariance analogous to Yang-Mills self-interactions. And finally 30,
You can decompose that gauge potential like we talked about under the pullback, and its constant component produces an effective term. This then gets identified with the cosmological constant. And that's all of fundamental physics.
Alright, thank you for coming along with me on this journey through fundamental physics and its potential unification. Thank you to Eric Weinstein for providing this theory and giving me plenty to chew on over the past few months. Thank you to all the video editors who helped edit this together, even though all of this probably look like and sound like hieroglyphics to you, or at least look like hieroglyphics don't sound like anything.
If you're interested in more icebergs like this, you should know I have a string theory iceberg where I outline the math at the graduate level for string theory. I also have one where I cover different theories of consciousness. Soon I'll be doing interpretations of quantum mechanics as well as an iceberg on algebraic geometry, so feel free to subscribe. Again, thank you to Eric Weinstein. It's an avant-garde and creative theory.
Kurt here, several months later. This has been so long in the making, geez, you have no idea. Anyhow, I wanted to say that I mean what I just said. I may have said this before in the iceberg and if I haven't, I should have because it bears repeating, I haven't seen a theory like this come from any single individual ever. Not one that's this fleshed out or has this amount of unexampled connections within itself as well as to what's known as the
I've received several messages, emails, and comments from professors saying that they recommend theories of everything to their students and that's fantastic.
If you're a professor or a lecturer and there's a particular standout episode that your students can benefit from, please do share. And as always, feel free to contact me. New update. Started a sub stack. Writings on there are currently about language and ill-defined concepts as well as some other mathematical details.
Much more being written there. This is content that isn't anywhere else. It's not on theories of everything. It's not on Patreon. Also, full transcripts will be placed there at some point in the future. Several people ask me, hey, Kurt, you've spoken to so many people in the fields of theoretical physics, philosophy and consciousness. What are your thoughts? While I remain impartial in interviews, this substack is a way to peer into my present deliberations on these topics. Also,
Thank you to our partner, The Economist.
Firstly, thank you for watching, thank you for listening. If you haven't subscribed or clicked that like button, now is the time to do so. Why? Because each subscribe, each like helps YouTube push this content to more people like yourself, plus it helps out Kurt directly, aka me. I also found out last year that external links count plenty toward the algorithm, which means that whenever you share on Twitter, say on Facebook or even on Reddit, etc,
it shows youtube hey people are talking about this content outside of youtube which in turn greatly aids the distribution on youtube thirdly you should know this podcast is on itunes it's on spotify it's on all of the audio platforms all you have to do is type in theories of everything and you'll find it personally i gained from re-watching lectures and podcasts
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"text": " Welcome to the iceberg of geometric unity, a comprehensive and technical edition. This iceberg format is one that will guide you through the intricacies of this theory of everything, beginning with foundational concepts and then advancing into the more sophisticated hinterlands. In this special episode, we rigorously explore Eric Weinstein's geometric unity, moving beyond metaphorical explanations to engage directly with the mathematical underpinnings of the theory."
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"text": " You can't explain what a Dirac operator is to them outside of making a TikTok video that gives the impression of knowing without actually understanding. My name's Kurt Jaimungal, and on Theories of Everything, I use my background in mathematical physics from the University of Toronto to explore the unification of gravity with the Standard Model, and I've also become interested in fundamental laws in general as they relate to explanations of some of the largest philosophical questions we have, such as what is consciousness and how does it arise?"
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"text": " In other words, it's a peregrination into the all-encompassing nature of the universe. Today we'll cover the abstruse math of bundle theory, of index theory, of course the standard model with general relativity. Just so you know, this episode took a combined 250 hours across three different editors and several rewrites on my part. It's on par with the most labor that's gone into any single theories of everything video, comparable to the iceberg of string theory, and that's saying something."
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"text": " I'll explain GU in four words, in 30 words, and then in 20,000 words. In four words, Einstein knows petit salam. Not terribly informative to people without a physics background. However, you can see my notes here on this substack on eschewing simplistic explanations, as you only get to choose two of these three, simplicity, accuracy, and succinctness."
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"text": " after trace reversing the Frobenius metric on the fibers. Again, that's a handful, that's actually only 28 words, and that will make sense to someone who has a differential geometric background, but maybe not to you yet. Now the 20,000-word explanation is the rest of this iceberg. One problem is that there are three legs to this mathematical physics stool, geometry, algebra, and analysis, or in other words, calculus. The issue with quantum field theory is that it developed in an unbalanced manner."
},
{
"end_time": 385.435,
"index": 17,
"start_time": 376.493,
"text": " It's predominantly analysis, and we discovered super late that we'd been neglecting certain topological, geometric and algebraic aspects."
},
{
"end_time": 409.838,
"index": 18,
"start_time": 385.862,
"text": " Starting in the 1970s with Jim Simons, for the last 50 years, people like Witten, Seagull, Quillen, Singer, Attia, Hitchens, Donaldson, Dan Freed, C.N. Yang, and Alvarez-Gomay have been making quantum field theory more geometric. When you're taught quantum field theory in graduate school, it's generally from the point of view of effectively a generalization of multivariate calculus."
},
{
"end_time": 427.517,
"index": 19,
"start_time": 409.838,
"text": " But this tool of analysis is too crude of a tool to bear the responsibility of advancing fundamental physics. Now I'll give some simple examples. How do you avoid issues with pseudotensors by ensuring that physical quantities are tensorial and coordinate independent?"
},
{
"end_time": 448.456,
"index": 20,
"start_time": 428.08,
"text": " 2. How do you ensure that the time evolution of a quantum state preserves the non-negativity of the probability density during propagation? 3. How can you tell if you have an anomaly in quantum field theory? 4. How do non-local spectral contributions arise in ostensibly local theories?"
},
{
"end_time": 466.681,
"index": 21,
"start_time": 448.626,
"text": " And number five, of course, how do you formulate quantum theory on curved spaces? None of these considerations that I've just mentioned are natural in an analytic framework and analysis, but geometric principles ensure that all of these conditions are met. I won't say the answers now, but I will address them later in the iceberg."
},
{
"end_time": 484.599,
"index": 22,
"start_time": 466.681,
"text": " Needless to say, it's exactly the same issue where phenomenon that are difficult to prove regarding convergence and analyticity in the real case become completely obvious or even trivial when you extend to the complex case. For now, let's look at the current state of fundamental physics."
},
{
"end_time": 514.633,
"index": 23,
"start_time": 485.111,
"text": " Hi, everyone. Hope you're enjoying today's episode. If you're hungry for deeper dives into physics, AI, consciousness, philosophy, along with my personal reflections, you'll find it all on my sub stack. Subscribers get first access to new episodes, new posts as well, behind the scenes insights, and the chance to be a part of a thriving community of like minded pilgrimers. By joining, you'll directly be supporting my work and helping keep these conversations at the cutting edge. So click the link on screen here."
},
{
"end_time": 526.305,
"index": 24,
"start_time": 514.633,
"text": " Hit subscribe and let's keep pushing the boundaries of knowledge together. Thank you and enjoy the show. Just so you know, if you're listening, it's C-U-R-T-J-A-I-M-U-N-G-A-L dot org, KurtJaimengel dot org."
},
{
"end_time": 556.323,
"index": 25,
"start_time": 527.756,
"text": " What's on screen are terms which are the ingredients of modern physics in the same way that you have flour, sugar, eggs, butter, and milk to form a cake. The following are ingredients of the universe as we know it. The universe combines these all, but we don't know how. And further, we don't know of a source that could give rise to all of these. Is there something like the four simple base pairs of DNA that give rise to the protein bonanza that we call life?"
},
{
"end_time": 568.012,
"index": 26,
"start_time": 556.8,
"text": " Let me go through these one by one. Firstly, we have Einstein. Now specifically, this term represents the Ritchie curvature tensor defined as the contraction of the Riemann tensor."
},
{
"end_time": 594.206,
"index": 27,
"start_time": 568.268,
"text": " Most people don't realize that this is quite odd, as the Riemann tensor has its anti-symmetry in the last two factors coming from it being a two-form, whereas the first two factors have anti-symmetry coming from the Lie algebra, which is actually after you pull down the i with the metric. More on this later, but intuitively, the Ricci scalar measures how volume changes in a curved spacetime, capturing the gravitational effects of matter."
},
{
"end_time": 621.561,
"index": 28,
"start_time": 594.923,
"text": " Next is Yang-Mills-Maxwell, and this term is the Yang-Mills action for gauge fields, where F A is the field strength, also known as the curvature, associated with the gauge connection A, given by this formula here. And this inner product is something called the killing form on the Lie algebra of the gauge group. Intuitively, this term represents the energy stored in the gauge fields. The next term describes the action for fermionic fields,"
},
{
"end_time": 649.855,
"index": 29,
"start_time": 621.852,
"text": " psi and psi bar, where psi bar is the Dirac adjoint of psi, and this D with a slash is the Dirac operator coupled to the gauge connection. The gamma matrices here satisfy the Clifford algebra, and this guy describes how particles like electrons and quarks move and interact with gauge fields within spacetime. Confusingly, this psi here isn't the same psi as a wave function. This is just one of the many ambiguities in physics. You're just going to have to get used to it."
},
{
"end_time": 673.746,
"index": 30,
"start_time": 650.299,
"text": " The next is Higgs and this governs the dynamics of the Higgs field and through the potential here it gives mass to gauge bosons like the W and Z ones via spontaneous symmetry breaking when this Higgs field acquires a vacuum expectation value. And this next factor here is when the Higgs field acquires a VEV of vacuum expectation value, this time giving rise to mass for fermions."
},
{
"end_time": 691.698,
"index": 31,
"start_time": 674.241,
"text": " which is in proportion to the strength of the Yukawa coupling. At this point, don't worry if you're confused. Again, this video is meant to be watched and rewatched. And furthermore, this is standard physics, so nothing here is specific to geometric unity. All of this will make sense and I'll explain and re-explain. Just keep watching."
},
{
"end_time": 711.288,
"index": 32,
"start_time": 691.698,
"text": " Now this next term, spin 1,3, I could have also said SL2C, technically, which is the double cover of the proper orthochronous Lorentz group SO plus 1 comma 3, which is actually necessary for representing spinner fields, and anyhow, this is what allows us to correctly describe particles with half-integer spins."
},
{
"end_time": 727.551,
"index": 33,
"start_time": 711.732,
"text": " Then there's this, which you hear of plenty, SU3 x SU2 x U1. This product group is the gauge symmetry group of the standard model, where roughly speaking SU3 corresponds to the strong force, also known as quantum chromodynamics,"
},
{
"end_time": 752.159,
"index": 34,
"start_time": 727.824,
"text": " and SU2 corresponds to the weak force, technically isospin, and U1 corresponds to electromagnetism, though technically hypercharge. Next, we have the family quantum numbers. These will be outlined later in this talk, but intuitively speaking, this space encodes all the intrinsic properties that distinguish different types of particles within a particular generation. Now next,"
},
{
"end_time": 779.241,
"index": 35,
"start_time": 752.159,
"text": " Speaking of these generations, this denotes that there are three generations of matter. Now, some people call them three families. I've also heard it called three flavors of matter, confusingly again. This field here, psi, can be decomposed as follows. That's what this symbol here means, which represents these three generations or families or flavors or whatever you call it. This accounts for the existence of particles like the electron, the muon and the tau."
},
{
"end_time": 801.476,
"index": 36,
"start_time": 779.548,
"text": " which are all similar in behavior, but they differ in mass. And lastly, please don't make me pronounce these, the CKM matrix explains why quarks can change types or flavors during weak decays leading to phenomenon like the decay of a strange quark into an up quark. And the PMNS does something similar, but for neutrino mixing or oscillations as they're sometimes called."
},
{
"end_time": 830.111,
"index": 37,
"start_time": 802.978,
"text": " There are two key equations which can be derived from these terms, but I'll include them on screen anyhow for completeness. So one is the Einstein field equation and the other is the Yang-Mills equation. Since the Einstein tensor on the left-hand side can be thought of as an operator that acts on the Riemann tensor, you can rewrite it into this form. Now I'm going to write both of these equations suggestively as Weinstein suggests in a suggestive manner."
},
{
"end_time": 858.951,
"index": 38,
"start_time": 831.203,
"text": " If you squint, you'll see an analogy here between Yang-Mills and Einstein. Both theories involve operations acting on curvature tensors to map them back into field variables and sources. However, one huge difference is how these operators interact with symmetries. In Yang-Mills theory, gauge covariance is preserved under gauge transformations due to the structure of the covariant derivative and field strength. In contrast,"
},
{
"end_time": 884.684,
"index": 39,
"start_time": 859.206,
"text": " The linear contraction operator, which I've denoted P, which is used to form the Einstein tensor, denoted G, in general relativity, doesn't commute with gauge transformations. I'll leave my proof of this on screen for those who don't believe me. You can just pause. But also, there are show notes in the description with a full breakdown of everything in this video. This, by the way, is partly what Eric means when he talks about the twin origin problems. But more on that later."
},
{
"end_time": 912.398,
"index": 40,
"start_time": 884.889,
"text": " In Yang-Mills theory, this involves Lie algebra-valued one-forms, which are also known as the gauge potentials or connections, while in general relativity, it involves symmetric rank two tensors, or the metric tensor as you may know it. What other analogies exist? And how can we make these analogies not only precise, but derived? Furthermore, how can this be done with the smallest set of simple assumptions possible?"
},
{
"end_time": 930.64,
"index": 41,
"start_time": 913.097,
"text": " That's what Geometric Unity aims to achieve. Layer 2. Geometric Unity begins with a four-dimensional manifold, X4, and then it constructs a bundle of metrics called Y14."
},
{
"end_time": 943.848,
"index": 42,
"start_time": 931.203,
"text": " This 14-dimensional space comes about naturally. How? Well, at each point in the original manifold, you can assign a symmetric bilinear form, which is a metric of 10 independent components,"
},
{
"end_time": 966.834,
"index": 43,
"start_time": 944.087,
"text": " Plus, of course, the four dimensions of the base space to account for each point in the manifold. Rather than choosing a metric which is what's ordinarily done, GU instead works with the space of all possible metrics simultaneously point-wise. This is extremely important because traditionally, spinners require a metric for their definition, and this creates a chicken-and-the-egg problem with quantum gravity."
},
{
"end_time": 991.937,
"index": 44,
"start_time": 967.295,
"text": " How is it that matter can exist for mionic matter when a metric isn't well defined? We'll speak more about this later. The key innovation is what Erich calls the chimeric bundle over the space, constructed as follows, where V is the vertical bundle, so changes in the metric at a point, and H star is the dual of the horizontal bundle, so movement in the base space. The vertical space inherits a natural metric"
},
{
"end_time": 1018.575,
"index": 45,
"start_time": 991.937,
"text": " via something called the Frobenius inner product, which is a fancy word for this formula on screen here. Again, all of this will be explained with examples. Now, what about that horizontal space? Well, it gets its structure from a connection choice. This allows us to define spinors without first choosing a metric on X4. Now, the structure group of GU is precipitated from spin 7, 7 or spin 5, 9."
},
{
"end_time": 1037.278,
"index": 46,
"start_time": 1018.575,
"text": " And this signature, by the way, is because we have this decomposition here between the vertical and horizontal components. This leads to a complex spinner representation. This dimension 64, by the way, comes about because the spinner representation of spin 7 comma 7 has dimension 2 to the 7, which is 128."
},
{
"end_time": 1064.514,
"index": 47,
"start_time": 1037.278,
"text": " and that splits into two 64-dimensional pieces. Note, these vial spinners are not of definite signature. Importantly, they are two vial spinners of split signature, 32, 32, and then, of course, another 32, 32. Eric then introduces something called the inhomogeneous gauge group, and that combines gauge transformations, which are H, with so-called translations in the space of connections. That's just an analogy, and we call that N."
},
{
"end_time": 1089.394,
"index": 48,
"start_time": 1064.514,
"text": " This structure parallels how the Poincare group combines Lorentz transformations with spacetime ones. But this time we're in the context of gauge theory. Later on, Eric then introduces something called the augmented torsion tensor. Again, this is plenty of jargon and new terminology. It does need to be introduced because when you name something, you can then use that concept on its own rather than having to construct it from scratch every single time."
},
{
"end_time": 1119.206,
"index": 49,
"start_time": 1089.394,
"text": " Now this augmented torsion tensor, which again is on screen but will be explained more later, is what combines aspects of gravity and maintains that gauge covariance that we talked about before. This is what resolves Einstein's quote-unquote twin origins problem, or more specifically, Eric's coining of the twin origins problem. Now where does the standard model's gauge group SU3 cross SU2 cross SU1 come about? Well, what Eric does is instead of using a compact group, he uses a larger non-compact one"
},
{
"end_time": 1149.343,
"index": 50,
"start_time": 1119.497,
"text": " which has as a maximal compact subgroup the standard model. In other words, the gauge group of the standard model comes about not by an arbitrary choice, certainly not three arbitrary choices, but rather as a necessary consequence of the geometry itself. The three generations of matter come about from decomposing spinner-valued forms on this larger space Y14, with something special happening for the two generations, and then the third is more like a remnant. So the first two come about from what's on screen here,"
},
{
"end_time": 1167.705,
"index": 51,
"start_time": 1149.582,
"text": " a function, a spinner-valued field, so spinners, and then a spinner-valued one-form. However, the third generation is actually a Rarita-Schwinger field in the decomposition of zeta. And all of this comes about from minimal assumptions on the original manifold x4."
},
{
"end_time": 1192.125,
"index": 52,
"start_time": 1168.012,
"text": " Again, Eric's theory, geometric unity, makes unification not by adding new structures, but via recognizing that the ingredients of modern physics, remember that Einstein equation, Yang-Mills theory, fermions, Higgs, et cetera, they all come about from geometry and moreover from X for itself. Just a moment. Don't go anywhere. Hey, I see you inching away."
},
{
"end_time": 1215.862,
"index": 53,
"start_time": 1192.602,
"text": " Don't be like the economy, instead read the economist. I thought all the economist was was something that CEOs read to stay up to date on world trends. And that's true, but that's not only true. What I found more than useful for myself personally is their coverage of math, physics, philosophy, and AI, especially how something is perceived by other countries and how it may impact markets."
},
{
"end_time": 1239.838,
"index": 54,
"start_time": 1215.862,
"text": " For instance, the economist had an interview with some of the people behind deep seek the week deep seek was launched. No one else had that. Another example is the economist has this fantastic article on the recent dark energy data, which surpasses even scientific Americans coverage. In my opinion, they also have the chart of everything. It's like the chart version of this channel. It's something which is a pleasure to scroll through and learn from."
},
{
"end_time": 1262.227,
"index": 55,
"start_time": 1239.838,
"text": " Links to all of these will be in the description, of course. Now, the economist's commitment to rigorous journalism means that you get a clear picture of the world's most significant developments. I am personally interested in the more scientific ones, like this one on extending life via mitochondrial transplants, which creates actually a new field of medicine, something that would make Michael Levin proud. The economist also covers culture,"
},
{
"end_time": 1282.039,
"index": 56,
"start_time": 1262.227,
"text": " finance and economics, business, international affairs, Britain, Europe, the Middle East, Africa, China, Asia, the Americas, and of course, the USA. Whether it's the latest in scientific innovation or the shifting landscape of global politics, The Economist provides comprehensive coverage and it goes far beyond just headlines."
},
{
"end_time": 1306.732,
"index": 57,
"start_time": 1282.039,
"text": " Look, if you're passionate about expanding your knowledge and gaining a new understanding, a deeper one of the forces that shape our world, then I highly recommend subscribing to The Economist. I subscribe to them and it's an investment into my, into your intellectual growth. It's one that you won't regret. As a listener of this podcast, you'll get a special 20% off discount. Now you can enjoy The Economist and all it has to offer."
},
{
"end_time": 1332.602,
"index": 58,
"start_time": 1306.971,
"text": " Let's begin with what's familiar, that is a four-dimensional manifold."
},
{
"end_time": 1359.411,
"index": 59,
"start_time": 1332.79,
"text": " This is that tiny input. It's the priming of the pump necessary to construct the rest of geometric unity. There's no metric needed. There's no connection. There's no additional structure yet other than being spin or orientable or connected, et cetera. But there are generalizations of GU without these facets. I personally find it easier to assume these. However, instead of working with X4 directly, Eric constructs what's called the observer's."
},
{
"end_time": 1387.892,
"index": 60,
"start_time": 1359.821,
"text": " Now this observer is actually a triple, even though before I denoted it simply as Y14, it's technically the base space X4, the total space Y14, and then the map pi that projects down between them. So recall, at every point in the manifold, there's a fiber, and this consists of all possible metrics that one could have had at that point. Now let's be precise, what the heck is meant when I say all possible metric tensors at this point?"
},
{
"end_time": 1416.374,
"index": 61,
"start_time": 1388.251,
"text": " Okay, at any point x in x4, a metric tensor is a symmetric non-degenerate bilinear form g sub x, say, which takes in two vectors at that same point and then outputs linearly something from the real numbers. The space of such metrics forms a ten-dimensional manifold because a symmetric four by four matrix has ten independent components and then you get the extra four because of the base space. So what's different about this construction?"
},
{
"end_time": 1432.841,
"index": 62,
"start_time": 1416.374,
"text": " Rather than fixing a metric which is what we ordinarily do in general relativity we consider all possible metrics simultaneously and then this allows one to study how would physics look if you didn't make any specific choice of a metric eric maneuver which i haven't seen done before."
},
{
"end_time": 1462.09,
"index": 63,
"start_time": 1433.166,
"text": " is instead of quantizing gravity directly on the base space X4, which as I'm sure you know, has problems with renormalization. Instead, Eric works on this larger space Y14 and then he pulls back information to X4 via what he calls observation maps. So I should spell out what a pullback is and what an observation map is. Eric, confusingly to me initially, calls the local sections of this bundle observations. So that's all it is. It's nothing more."
},
{
"end_time": 1486.135,
"index": 64,
"start_time": 1462.278,
"text": " Then you look at a local patch of the manifold and you think, okay, what am I smoothly going to assign as a metric from the larger space Y14? Okay, let's clarify what's meant by pullback. Anytime you have a map, let's say something that goes from A to B, and then you have some structure on B in that target space, whether it's a differential form, a tensor field or a function, a pullback is the corresponding structure"
},
{
"end_time": 1503.422,
"index": 65,
"start_time": 1486.391,
"text": " Step 2. The frame bundle and its double cover. Okay, let's construct one of the other essentials. The frame bundle over our base space."
},
{
"end_time": 1526.425,
"index": 66,
"start_time": 1503.763,
"text": " At each point in our base space for this frame bundle, the fibers consist of all possible frames or bases for the tangent space. The group structure here is GL4R. Nothing here is unique to GU. This is standard in differential geometry. But what exactly is a frame? Well, again, at each point, a frame is an ordered basis for the tangent space, and I'll give you some examples."
},
{
"end_time": 1554.206,
"index": 67,
"start_time": 1526.732,
"text": " Here is the standard basis that you know probably by x, y, z, and t in some coordinate system, and then another frame could look like so. In differential geometry, by the way, your bases are vectors, which are differential operators. These frames are related to one another via an element of GL4R. In other words, if you had a base and another base, there exists a transformation between them, which is a member of GL4 in order to get you from one to the other. But here we hit our first snag."
},
{
"end_time": 1561.476,
"index": 68,
"start_time": 1554.701,
"text": " GL4 does not have a finite dimensional spinner representation. Why?"
},
{
"end_time": 1589.65,
"index": 69,
"start_time": 1561.664,
"text": " If you take an element of it, say A, and you square it and you get negative i, any finite dimensional representation A would need to satisfy that A also squares to negative i, which is impossible over the reals. This is an extremely important point, so it deserves some elaboration. Let's consider this matrix, which satisfies that condition of A squared equals minus i. The eigenvalues of this matrix would also need to be square roots of minus 1, which don't exist."
},
{
"end_time": 1605.282,
"index": 70,
"start_time": 1589.65,
"text": " So to fix this, we need to do something called passing to the double cover. The map is on screen here. And then we do something called the lifted frame bundle, which has the structure group of this double cover of GL4R, also known as the meta linear group."
},
{
"end_time": 1622.995,
"index": 71,
"start_time": 1606.92,
"text": " Step 3. We construct observation maps. Now what exactly do we mean by observation map? I understand this is plenty of new terminology as we have an observer so now we have observation maps and you may have heard of the Shiab operator or the Chimeric bundle."
},
{
"end_time": 1652.483,
"index": 72,
"start_time": 1623.473,
"text": " Eric is evidently an enormous fan of these neologisms, and this is just something we'll have to get used to. There is a method to the madness though, because when we give a name to something, it means that we can then pick it out again and reference it without having to construct it anew each time. So back to the question, what are observation maps? These are local sections, so some map from a neighborhood of a point X going into the larger space Y, but why do we call these observations?"
},
{
"end_time": 1680.06,
"index": 73,
"start_time": 1652.483,
"text": " More on this later, but for now, you can think of it as how one measures geometry. Each of these IOTAs, each of these maps, picks out a specific metric at each point in its domain. For any such of these maps, we get a pullback map that brings geometric data from y back down to x4. Specifically speaking, if you have any tensor, say omega, on y, then you can pull back the corresponding tensor on x4."
},
{
"end_time": 1709.855,
"index": 74,
"start_time": 1680.452,
"text": " Think of this like a probe into the space of all possible metrics. The spin 1 comma 3 bundle comes about as follows. At each point little y in the larger space big Y, we have a metric g little y on x4. This metric determines a principal s o 1 comma 3 bundle of orthonormal frames. The insight from Eric is that this bundle naturally admits a canonical double cover by a spin 1 comma 3 bundle by this construction on screen here."
},
{
"end_time": 1734.548,
"index": 75,
"start_time": 1710.316,
"text": " The other insight from Eric is that this bundle has a 14 dimensional matrix representation broken up into the 10 vertical and then four horizontal actually asterisk on that dual horizontal and that is the bundle that we're calling C the chimeric bundle. Now this C has a metric on it and part of it is actually canonically isomorphic to the original 14 dimensional tangent bundle."
},
{
"end_time": 1764.462,
"index": 76,
"start_time": 1734.889,
"text": " The other part requires some extra structure. By the way, when someone says a bundle admits something, they use this word admit, what's meant is that there then exists something. In this case, when we say that a bundle admits a double cover, we mean there exists a bundle map that's locally two to one. More precisely, for any point in here, there are exactly two points in here that map from here to here. And this mapping here preserves the bundle structure."
},
{
"end_time": 1790.077,
"index": 77,
"start_time": 1764.667,
"text": " This is analogous to how the complex function, say z which maps to z squared, gives a 2 to 1 map from the circle to itself. Why do we need this double cover? Because SO1,3 doesn't admit spinner representations. More precisely, there's no finite dimensional representations of SO1,3 that, when restricted to rotations, gives the spin-half representations of SO3."
},
{
"end_time": 1816.288,
"index": 78,
"start_time": 1790.486,
"text": " However, spin 1,3 does admit such representations, and this is how Eric will eventually describe fermions. Step 4. We construct the tangent bundle and its dual. At each point in the larger observer's capital Y, we have a tangent space which has the same amount of dimensions, namely 14. And again, 10 of them come from the symmetric matrices at each point, which represent all the possible metrics"
},
{
"end_time": 1833.763,
"index": 79,
"start_time": 1816.715,
"text": " And we have the other four dimensions coming from the base space below. The dual space consists of linear functionals from the tangent space to whichever is your underlying field, namely the reals in this context. To be specific, at a particular point in the observer's Y, it can be decomposed"
},
{
"end_time": 1849.735,
"index": 80,
"start_time": 1834.002,
"text": " as the base space X and then the metric G. Specifically speaking, you can see this isomorphism on screen here, where this guy represents the symmetric zero comma two tensors. And recall that in four dimensions, symmetric matrices have 10 independent components."
},
{
"end_time": 1871.681,
"index": 81,
"start_time": 1849.974,
"text": " Now you may ask, why do we need both the tangent space and its dual tangent space? The answer lies in how Eric defines field contents later. Some fields will naturally live in one bundle and others in its dual. In fact, we'll see that without a metric on why these bundles are not naturally isomorphic. And this is what leads Eric to later construct the Chimeric bundle."
},
{
"end_time": 1897.79,
"index": 82,
"start_time": 1871.681,
"text": " I'll be using this phrase field content so i should define it in physics when people talk about field content we just mean the collection of fields that appear in the theory so these could be a scalar field like the Higgs for instance could also be a vector field like the electromagnetic potential it could also be tensor fields like the metric all the previous examples were tensor fields as well or spinner fields like electrons."
},
{
"end_time": 1918.626,
"index": 83,
"start_time": 1898.063,
"text": " The word content just means collection or set. It's like the complete list of ingredients in our theory."
},
{
"end_time": 1942.773,
"index": 84,
"start_time": 1919.121,
"text": " Jokes aside, Verizon has the most ways to save on phones and plans where you can get a single line with everything you need. So bring in your bill to your local Miami Verizon store today and we'll give you a better deal. Rankings based on root metrics, root score, report data to 1-H-2025, your results may vary. Must provide a post-paid consumer mobile bill dated within the past 45 days. Bill must be in the same name as the person reviewing the deal. Additional terms apply. This Marshawn beast mode lynch. Prize pick is making sports season even more fun. On prize picks whether you're"
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{
"end_time": 1959.872,
"index": 85,
"start_time": 1943.166,
"text": " Football fan, a basketball fan, it always feels good to be ranked. Right now, new users get $50 instantly in lineups when you play your first $5. The app is simple to use. Pick two or more players. Pick more or less on their stat projections."
},
{
"end_time": 1975.23,
"index": 86,
"start_time": 1959.872,
"text": " Anything from touchdown to threes and if you're right you can win big. Mix and match players from any sport on ProgePix, America's number one daily fantasy sports app. ProgePix is available in 40 plus states including California, Texas,"
},
{
"end_time": 1996.869,
"index": 87,
"start_time": 1975.486,
"text": " Florida and Georgia. Most importantly, all the transactions on the app are fast, safe and secure. Download the PricePix app today and use code Spotify to get $50 in lineups after you play your first $5 lineup. That's code Spotify to get $50 in lineups after you play your first $5 lineup. PricePix. It's good to be right. Must be present in certain states. Visit PricePix.com for restrictions and details."
},
{
"end_time": 2015.674,
"index": 88,
"start_time": 1997.5,
"text": " Step 5. The vertical and horizontal bundles over Y. At each point in the observer's Y, we have a splitting of the tangent space, decomposed as follows. The V is the vertical space, which is tangent to the fiber, and the H is the horizontal space, which is non-canonical, meaning one needs to make a choice."
},
{
"end_time": 2030.196,
"index": 89,
"start_time": 2015.674,
"text": " Precisely speaking, the vertical bundle is defined as follows. To visualize this, you can think of the vertical subspace at y as the space of all possible velocities for the changing metric at x while staying in the same fiber over the x."
},
{
"end_time": 2046.288,
"index": 90,
"start_time": 2030.452,
"text": " In other words, if h of t is a path in this metric space where h of zero is g, your initial g, then taking the derivative with respect to time and setting it to zero is an element of the vertical tangent space. This is all standard in differential geometry."
},
{
"end_time": 2071.715,
"index": 91,
"start_time": 2047.398,
"text": " Step 6. The Zorro construction is named for its zigzag pattern and this is what provides a canonical manner of defining the horizontal distribution. Now when I say canonical here, what I mean or what is generally meant by canonical is that there's a natural choice or natural option that doesn't depend on arbitrary decisions. It's similar to how when you have a vector space, there is no canonical basis."
},
{
"end_time": 2088.422,
"index": 92,
"start_time": 2072.142,
"text": " However, you do have a canonical dual space. Here, the Zorah construction gives us a natural method to split T y, T y being the tangent space of y or the observer's or the metric bundle, in other words, without making loathsome arbitrary choices."
},
{
"end_time": 2103.558,
"index": 93,
"start_time": 2088.422,
"text": " Here's how it works. You'll see this construction here, which looks like the backward Z of Zorro, which is why Eric denoted it the Zorro construction. And this funny symbol here is a Gimel, which is a Hebrew letter. And then this symbol here, which looks like an N is an Aleph."
},
{
"end_time": 2121.715,
"index": 94,
"start_time": 2103.558,
"text": " Note, I'm unfamiliar with using Hebrew letters in math unless it's Aleph for cardinality. Now these two symbols on screen here look almost identical to me, which are Gimel and Beth respectively. You may see me confuse these symbols throughout, but don't worry because anytime they're referenced, they mean the same thing."
},
{
"end_time": 2148.439,
"index": 95,
"start_time": 2122.073,
"text": " namely, a section of the metric bundle. Note that the augmented torsion tensor is now called the displacement torsion tensor by Eric. I've also heard him call it distortion, but I'm going to continue to call it the augmented torsion tensor for the remainder of this iceberg. To me, I wouldn't use these Hebrew letters, and in fact, I'd change this gimbal to the iota from before. The only difference is that the gimbal is a global section, whereas the iota from before is a local one."
},
{
"end_time": 2175.572,
"index": 96,
"start_time": 2148.729,
"text": " Now the Aleph represents the Levi-Chevita connection, G sub-Aleph is the induced metric on Y, and A sub-Aleph is the resulting connection on Y. However, I should point out that Eric uses this Gimmel symbol and the lowercase g to prevent confusion that was coming about from calling two different metrics on two different spaces both by the traditional G. Thus I understand that the Gimmel and Aleph aren't there arbitrarily, as the way Eric sees it,"
},
{
"end_time": 2201.698,
"index": 97,
"start_time": 2175.572,
"text": " All of the drama takes place on this larger observers. The reason Eric has this induction from the Zorro construction is that he wants the freedom to later not have a metric on the base space when not observing the system, for instance. And this is Eric's move to make the quantum metric make sense later. Let's break this down step by step. We start with a metric, which is a choice of a global section on X four akin to iota from before."
},
{
"end_time": 2227.585,
"index": 98,
"start_time": 2202.415,
"text": " This determines a unique Levy-Chevita connection, which Eric denotes as Aleph on X4, and this comes about by the fundamental theorem of Riemannian geometry, nothing not standard here. Next, this Aleph then introduces a metric on the larger space Y, through the Frobenius inner product on symmetric matrices. And finally, the G of Aleph determines its own Levy-Chevita connection, A sub Aleph on Y."
},
{
"end_time": 2233.626,
"index": 99,
"start_time": 2227.841,
"text": " This process gives one a canonical manner of splitting Ty, so the tangent space."
},
{
"end_time": 2263.899,
"index": 100,
"start_time": 2233.899,
"text": " into the"
},
{
"end_time": 2265.213,
"index": 101,
"start_time": 2263.899,
"text": " From X for."
},
{
"end_time": 2294.804,
"index": 102,
"start_time": 2265.469,
"text": " to y, the larger space, without making choices. The horizontal subspace is precisely the space of such lifted vectors. When I say lift, by the way, what I mean is we take something that's defined on a lower space and we find a corresponding object to it in a higher space such that it projects downward to give us what we started with. More precisely, if this here is a fiber bundle and we have a vector that's a tangent vector at say b, the base space,"
},
{
"end_time": 2320.708,
"index": 103,
"start_time": 2294.804,
"text": " Then a lift of the vector from the base space is another vector in this larger space such that when you project down, you get the same vector. Now the horizontal distribution or the choice of horizontal subspace gives us an approach to choose such lifts. Step seven, the chimeric bundle. You'll notice we're introducing plenty of terminology. There's chimeric bundle. There's the observers. This isn't just jargon for jargon sake."
},
{
"end_time": 2346.647,
"index": 104,
"start_time": 2321.101,
"text": " Instead, we're actually enhancing the clarity because we're avoiding repetitive exposition, having to define these over and over. These terms will become familiar as we proceed and recall this entire iceberg is meant to be watched and rewatched where you learn something new every single time. Alright, let's define this beast. First of all, notice that if we take Y14 as its own bundle, the Observer as its own entire bundle,"
},
{
"end_time": 2363.2,
"index": 105,
"start_time": 2346.647,
"text": " and we take the tangent space at a particular point in it, it can always be decomposed as follows, a vertical component and a horizontal one. You know from differential geometry, a choice of a horizontal subspace is a choice, that is the same as a connection"
},
{
"end_time": 2386.032,
"index": 106,
"start_time": 2363.473,
"text": " which then becomes something like curvature. However, in GU, you're always trying to minimize the amount of choices you make. You can even think of summing up GU as, if you cannot have one, then you must have them all. Anyhow, let's take a look at that Frobenius inner product that we referenced before, and let me just give you an example of two matrices. So, this isn't actually from the bundle, it's just what I could write on screen."
},
{
"end_time": 2411.186,
"index": 107,
"start_time": 2386.357,
"text": " Let's imagine you have 1, 2, 2, 3, 0, 1, 1, minus 1. Well, you can just do the math and compute its Frobenius inner product, and it works out to minus 1. Now, this chimeric bundle differs in the second component, its H-dual. It's not the horizontal bundle, but the dual of it. This asymmetry will turn out to be required for the theory's ability to unify gravity with gauge theory."
},
{
"end_time": 2437.5,
"index": 108,
"start_time": 2412.824,
"text": " Hi, Kurt here. If you're enjoying this conversation, please take a second to like and to share this video with someone who may appreciate it. It actually makes a difference in getting these ideas out there. Subscribe, of course. Thank you. Step 8. The Frobenius inner product. At each point in the large space Y, we need to define an inner product on the chimeric bundle."
},
{
"end_time": 2463.063,
"index": 109,
"start_time": 2437.841,
"text": " So how do we do this without already having a metric on Y? The way Eric goes about doing this is by noticing that V inherits a natural metric via the Frobenius inner product. Again, for symmetric matrices A and B, the Frobenius inner product is defined as follows on screen. Also notice that you can decompose the trace and the traceless parts of a symmetric two tensor. The trace part has dimension one,"
},
{
"end_time": 2480.572,
"index": 110,
"start_time": 2463.063,
"text": " and the traceless part has all the other dimensions. To see why this decomposition is reasonable and valid, consider that for any symmetric matrix A, you can always write it as follows, where you have a traceless part and a traceless part. The traceless part has a signature, in this case, 3,6."
},
{
"end_time": 2508.268,
"index": 111,
"start_time": 2480.879,
"text": " whereas the trace itself contributes either a 1 comma 0 or a 0 comma 1, depending on a choice. Here is where you make a specific choice. So we can either choose 4 comma 6 or 3 comma 7. For geometric unity, Eric chooses 4 comma 6 for reasons that will become clear later, though there are generalizations of geometric unity with other choices. Step 9, choosing a signature."
},
{
"end_time": 2518.541,
"index": 112,
"start_time": 2508.66,
"text": " Why is it that we have to be so careful about this signature? The answer is representation theory. The signature determines which spinner representations are possible."
},
{
"end_time": 2545.589,
"index": 113,
"start_time": 2518.933,
"text": " With our choice of 4,6 for the vertical space, and 1,3 for the horizontal space from spacetime, we get a total signature of 7,7. Note, GU could have had a signature of 5,9, and I believe Eric isn't sure which of these is the sector of our universe, in his theory. But for the remainder of this iceberg, we're going to select spin 7,7. But why these particular signatures, Kurt, you may ask?"
},
{
"end_time": 2551.323,
"index": 114,
"start_time": 2545.589,
"text": " Now the magic lies in representation theory, the representation theory of spin seven comma seven."
},
{
"end_time": 2579.735,
"index": 115,
"start_time": 2551.63,
"text": " When we have a metric of a signature, an arbitrary one, P, 6, the real spinner representation has dimensions 2 raised to the floor of P plus Q over 2. Now for 7, 7, it gives us 2 to the power of 7. This signature is essential because if you take the dimension of a spinner bundle, it equals this formula on screen here where you get 2 raised to some floor function. In this case, it becomes 2 raised to 7, which equals 128."
},
{
"end_time": 2606.51,
"index": 116,
"start_time": 2579.735,
"text": " And this 128 splits not into C64 plus C64, but into the equally split signature that we talked about before, C32, 32, and then another C32, 32. Remember, these are vile spinners of split signature. Again, the same is also true for a spin 5, 9 bundle matching what's required for the standard model. Again, more later, this is somewhat of a flyby overview."
},
{
"end_time": 2625.367,
"index": 117,
"start_time": 2607.637,
"text": " Step 10, defining spinors without a metric. Here's where everything so far comes together. The spinner bundle on the chimeric bundle decomposes as follows. Now what's so special about this decomposition? Well, it's the exponential property of spinors at work."
},
{
"end_time": 2641.049,
"index": 118,
"start_time": 2625.845,
"text": " For any direct sum of vectors, v, say, direct sum with w, as long as they have metrics, we have the following. Think of it like if you have a particle that can move in two independent directions, its quantum states multiply rather than adding."
},
{
"end_time": 2662.654,
"index": 119,
"start_time": 2641.681,
"text": " This is the quote-unquote exponential property of spinors that Eric mentions. When you pull this back via an observation map, IOTA, you get the following decomposition into tensor products. This decomposition eventuates in both spacetime spinors and internal quantum numbers exactly what's required for the standard model fermions."
},
{
"end_time": 2691.425,
"index": 120,
"start_time": 2662.654,
"text": " Now this alchemy happens because the vertical part V contributes internal symmetries, whereas the horizontal part, or more specifically the dual to the horizontal part, gives us spacetime properties. When one pulls this back to X4, the spinners decompose perfectly to give both the spacetime transformations of the particles and their internal quantum numbers like color and isospin. The implication? Eric has now constructed spinners without choosing a metric on spacetime."
},
{
"end_time": 2719.48,
"index": 121,
"start_time": 2691.749,
"text": " Instead, they're fomented from the geometry of the observers. This resolves a long-standing chicken-and-egg problem in quantum gravity, which is how can matter exist between measurements if a metric is required for that matter to be defined? The answer, according to Eric, is that matter lives in the observers, namely that larger y-space where spinners exist prior to any choice of a metric on spacetime. Step 11, the structure group."
},
{
"end_time": 2748.882,
"index": 122,
"start_time": 2720.486,
"text": " The structure group of our theory originates from the spinner representations of spin 7,7. Why these dimensions? Recall, 7,7 comes about from combining 4,6 with 1,3. And you can see that one is vertical and then the latter is horizontal. Let's pause and ask, why do these signatures even matter? Again, the 4,6 signature comes from the Frobenius inner product on symmetric matrices, while the 1,3 comes from spacetime."
},
{
"end_time": 2774.957,
"index": 123,
"start_time": 2749.07,
"text": " It's actually here that we make a choice. Because we could have had anything that's summed to 4, and we're just choosing 1, 3. This is one of the only places in this entire theory that I can see a choice being made. Now you may be wondering, why can't we just use any group here? And the answer lies in representation theory. We would like a group whose representations can accommodate both gauge fields and fermions."
},
{
"end_time": 2802.142,
"index": 124,
"start_time": 2775.401,
"text": " The spinner representations of spin 7,7 do exactly that. Why do we need a structure group at all? When we started, we had a base space, an X4 manifold, and then we immediately got the frame bundle with the structure group GL4. But now we're working on Y14, and we require a group that preserves the structure that we've built. The signature 7,7 is not arbitrary. It comes from the natural metric structure on the chimeric bundle."
},
{
"end_time": 2829.275,
"index": 125,
"start_time": 2802.637,
"text": " Now you may be wondering, why can't we just use any old group here? We need a group whose representations can accommodate both gauge fields and fermions. The spinner representations of spin 7,7 do precisely that. So why 7,7 and not say 14? The key is that we would like to preserve the signature that comes about naturally from the vertical and horizontal decomposition. But here's the rub."
},
{
"end_time": 2850.196,
"index": 126,
"start_time": 2829.565,
"text": " The Frobenius inner product, which by the way, Eric sometimes calls the Frobenius metric, but I'm going to continue to call it the inner product, is given by this formula on screen here in components. When you have 4x4 symmetric matrices, this naturally gives a 4,6 signature because the space of symmetric matrices compose into what's called a trace."
},
{
"end_time": 2862.193,
"index": 127,
"start_time": 2850.435,
"text": " and a traceless part."
},
{
"end_time": 2889.957,
"index": 128,
"start_time": 2862.432,
"text": " it naturally gives a 3,7. And this isn't often remarked on. Seeing this 4,6 signature here is extremely subtle because it requires remembering that you can do something called flipping the trace, which is what Eric says, and technically that's a trace reversal. And that's, by the way, what Einstein did when he realized that his equation couldn't just be the Ricci tensor equal to the stress energy tensor. Instead, you require the minus r over 2 correction."
},
{
"end_time": 2919.019,
"index": 129,
"start_time": 2889.957,
"text": " So let's take a look here. Recall the trace is a single number, which is why you can represent it by something of dimension one, which is just the real numbers here. And then the rest becomes the traceless component. And then you wonder, well, what's a representation space of seven comma seven, and it's U 64 comma 64. And that 64 again comes from half of two to the power of seven. Thus, we preserve the Z two grading on the spinners. Step 12, the principal bundle construction."
},
{
"end_time": 2936.425,
"index": 130,
"start_time": 2919.991,
"text": " Let's pause again. We've constructed this elaborate chimeric bundle with its spinner representations, but how does one actually implement gauge theory here? You may think, well, let's just use this spin 7 comma 7 directly. However, there is a subtler approach that Eric takes."
},
{
"end_time": 2956.783,
"index": 131,
"start_time": 2936.647,
"text": " His idea is that spin 7 comma 7 acts on that 128 dimensional complex vector space via its spinner representation. This space splits into two 64 dimensional pieces as we've said before and we wonder why is it we're using this unitary 64 comma 64 rather than just U of 128."
},
{
"end_time": 2972.09,
"index": 132,
"start_time": 2956.783,
"text": " It's because the spinner representation preserves a metric of signature 64,64. This brings us to our principal bundle on screen here, where we finally have a gauge group or a structure group, namely U64,64."
},
{
"end_time": 3000.247,
"index": 133,
"start_time": 2972.09,
"text": " And then we have what's called an associated bundle. This is from taking the frame bundle of the chimeric bundle and then doing what's called a lift to its double cover. We then use that row representation to convert the spin 7 comma 7 transformations into U 64 comma 64 transformations. Step 13, the inhomogeneous gauge group. OK, let's carefully build up our gauge structure. First, what do we mean by gauge group?"
},
{
"end_time": 3030.555,
"index": 134,
"start_time": 3000.589,
"text": " In physics, gauge groups represent redundancies in the descriptions of nature. In other words, they're different mathematical descriptions of the same underlying physical reality, which is unobservable. So for instance, you can measure your height in inches, you can measure it in centimeters, you can measure it in meters. It doesn't change your height. Those are just different representations of your height. Now let's think about electromagnetism. You can add a gradient to the vector potential without changing the physics. That's a gauge transformation."
},
{
"end_time": 3043.387,
"index": 135,
"start_time": 3030.555,
"text": " But there's something deeper going on here that we're going to explore. So let's clarify an important distinction in gauge theory. There are two related but distinct concepts that are often confused and so I'd like to spell it out."
},
{
"end_time": 3066.869,
"index": 136,
"start_time": 3043.712,
"text": " There's a choice of connection and then there's gauge transformations. The choice of connection, let's call it a one-form A on a principal G bundle, which has a total space of P going down to M, is a Lie algebra-valued one-form satisfying certain properties. The space of all of these connections, calligraphic A, is an affine space. Here's what we mean by affine space, by the way."
},
{
"end_time": 3094.718,
"index": 137,
"start_time": 3066.869,
"text": " A vector space has an origin. Some people like to say a vector space has a preferred origin. Actually, anything that has an origin is not an affine space, so you don't even need to put the word preferred there. Okay, now what about gauge transformations? Gauge transformations are bundle automorphisms that preserve the fiber structure. Again, they're not just bundle automorphisms, which are often said, they have to preserve the fiber structure. They form a group, calligraphic H,"
},
{
"end_time": 3116.834,
"index": 138,
"start_time": 3094.718,
"text": " acting on connections via what we see here. Now let's unpack this. The first term here you can think of as a rotation of a sort of the connection and the second term is a correction to the connection that you need in order to preserve the transformational properties. In physics language this ensures that the quote-unquote covariant derivative transforms properly."
},
{
"end_time": 3134.053,
"index": 139,
"start_time": 3117.261,
"text": " For a concrete example, let's just take the non-abelian gauge theory case of QCD. If this A mu here is a gluon field and this g of x here is some varying spacetime dependent element of SU3 and that SU3 comes from the SU3 cross SU2 cross SU1."
},
{
"end_time": 3163.422,
"index": 140,
"start_time": 3134.053,
"text": " Then you have this situation over here being satisfied. And importantly, this describes the same physics, which is why people call it a redundancy. So the choice of connection is not redundant, but the gauge transformation is. And that's something that's quite confusing when you first learn about it. Now here, this calligraphic H, by the way, is the smooth sections of the associated bundle P sub H with the adjoint action of H, where H, again, not calligraphic,"
},
{
"end_time": 3192.841,
"index": 141,
"start_time": 3163.422,
"text": " is U of 64 comma 64. In fact, I'm going to stop just calling it H because that's confusing. I'm going to say U 64 comma 64 from now on. In simpler terms, we have a principal U 64 comma 64 bundle and these are our gauge transformations of it. Again, I can be even more precise. Let's say we have a Lie group G and we have its adjoint representation where G goes into the automorphisms on the Lie algebra, which acts on the Lie algebra by conjugation."
},
{
"end_time": 3222.961,
"index": 142,
"start_time": 3193.183,
"text": " Then the adjoint bundle with the lowercase a is the vector bundle associated to the principal bundle P subscript U 6464 via this representation. The inhomogenous gauge group is then this calligraphic G which has the calligraphic H semi-direct producted with this calligraphic scripted N. Now this is like translations in the space of connections but not in a trivial manner. There's a multiplication rule here."
},
{
"end_time": 3241.425,
"index": 143,
"start_time": 3223.166,
"text": " And this is what shows how gauge transformations act on the translation part. It seems like Eric is creating the structure to parallel Poincare's group's combination of Lorentz transformations with spacetime translations, but this time in the context of gauge theory rather than spacetime symmetry."
},
{
"end_time": 3259.77,
"index": 144,
"start_time": 3242.09,
"text": " It should be noted here that this is quite a large move. We're neither working on a space of metrics like Einstein, nor on a space of connections like Yang Mills or Yang Mills Maxwell as Eric says. Instead, what I'll do is I'll overlay a dictionary here that you can take a screenshot of."
},
{
"end_time": 3282.705,
"index": 145,
"start_time": 3259.77,
"text": " But recall, there are further notes in the substack at c-u-r-t-j-a-i-m-u-n-g-a-l dot org, curtjymongle dot org, myname dot org. So you can sign up there if you'd like the PDF. This is, as far as I can tell, Eric's interpretation of Einstein's unified field. Unified means algebraic in the eyes of G-U. Step 14. Defining a right action on connections."
},
{
"end_time": 3306.715,
"index": 146,
"start_time": 3283.507,
"text": " Now we need to understand how our gauge group acts on connections. For any connection A in this calligraphic A and any element lowercase g in this calligraphic g, we define the following. And you may ask, Kurt, are you going to read that? No, it's tedious. Just look, take a screenshot. Why this complicated formula? Well, let's be like Kurtis Blow and break it down. This term here is the familiar gauge transformation."
},
{
"end_time": 3333.66,
"index": 147,
"start_time": 3307.244,
"text": " This next term represents translations in the space of connections, and this term here is a correction term to ensure the consistency. This right action was constructed to satisfy this specific property, making calligraphic A into a right calligraphic G space. So what happens when we try to combine gauge theory with gravity?"
},
{
"end_time": 3348.626,
"index": 148,
"start_time": 3333.985,
"text": " Well, there's several problems, but an immediate one is that gauge transformations don't play well with Einstein's way of contracting indices. However, what if we could find a quantity that transforms correctly under both?"
},
{
"end_time": 3374.445,
"index": 149,
"start_time": 3349.019,
"text": " That's what Eric has found with the augmented torsion tensor, defined on screen as follows. And now this variational pi, at first I thought this was an omega, it's technically a variational pi, is a member of the adjoint valued one-forms on the larger space y. And that's going to be the gauge potential. Whereas this variation on epsilon here, belonging to calligraphic H, is a gauge transformation."
},
{
"end_time": 3404.121,
"index": 150,
"start_time": 3375.009,
"text": " Now the key property is that under gauge transformations, we have this formula on screen here. So let's break this down piece by piece. What is firstly this variational pi? Well, it's an adjoint valued one form, as I said before, which means it takes in vectors from T Y and it returns elements of the Lie algebra H of the gauge group H, which again is U 64 comma 64. And now this other term here looks complicated."
},
{
"end_time": 3415.572,
"index": 151,
"start_time": 3404.343,
"text": " However, it's just the gauge transformation of the base connection A0. By the way, when I use the word variational pi, it's not in reference to anything about variational calculus."
},
{
"end_time": 3445.077,
"index": 152,
"start_time": 3415.896,
"text": " It's instead because Eric's notation is to use this symbol right here. And in LaTeX, it's written with a slash and then var pi. It's a variation of pi in the same way that there's a variational phi. So var phi or var phi as some people call it. Just letting you know because you'll be hearing me use this term variational pi and upon rewatching this iceberg for maybe the fourth, fifth time, too many times to count, I realized that this can be confusing. Now let me make a concrete example."
},
{
"end_time": 3470.111,
"index": 153,
"start_time": 3445.077,
"text": " Let's say we have a U1 gauge theory and our variational epsilon is e to the i theta, then what we have is this on screen here. This is exactly the gauge covariant combination that appears in electromagnetism. Now this is quite interesting because it combines aspects of both gravity, so torsion, and gauge theory, so covariant derivatives, while maintaining gauge covariance."
},
{
"end_time": 3494.889,
"index": 154,
"start_time": 3470.418,
"text": " The way that I see it is it's like finding a method to make Einstein's gravitational theory speak the language of Yang-Mills theory. How do we generalize Einstein's contraction of the Riemann tensor in a gauge covariant manner? The answer lies in what Eric calls the Shiab operator that's spelled S-H-I-A-B."
},
{
"end_time": 3515.401,
"index": 155,
"start_time": 3495.145,
"text": " Now for a gauge covariant two-form C, which some people call cassis, but I'm just going to say C, and it's not the letter C, it's this symbol on screen. You have this formula here, where the Shiab operator acts on this two-form and gives you a Ritchie-like term, which we'll explain more later, and then a scalar curvature-like term. Again, more will be explained later."
},
{
"end_time": 3544.565,
"index": 156,
"start_time": 3515.828,
"text": " Note, the circle with a dot in the center is my notation for the shiab operator. Eric actually writes two concentric circles with a dot, but I wasn't able to get this to consistently render in my workflow, so just note that whenever you see this shiab online outside of this video, it will likely have two circles and a dot. Let's further break it down, piece by piece. First, what is this operator doing? It's taking a gauge covariant 2 form, and then it's returning another differential form"
},
{
"end_time": 3573.933,
"index": 157,
"start_time": 3544.787,
"text": " that transforms this time properly under gauge transformations. So why do we need such an operator? Think about Einstein's theory. When you contract the Riemann tensor to get the Ricci tensor, you're using the metric to raise and to lower indices. However, this operation doesn't respect gauge symmetries. It treats all copies of two forms in the same way. The Shiab operator fixes this by incorporating the gauge transformation epsilon explicitly."
},
{
"end_time": 3599.616,
"index": 158,
"start_time": 3574.36,
"text": " Here's how it works. These forms phi, phi 1, phi 2, for instance, are invariant under the action of spin 7, 7. When one conjugates them by an epsilon, we get objects that transform covariantly under gauge transformations. And now you also may ask, hey, Kurt, why these particular combinations of wedge products and Hodge stars? And again, the answer lies in representation theory, just as the Einstein tensor"
},
{
"end_time": 3615.538,
"index": 159,
"start_time": 3599.838,
"text": " For a concrete example, consider the case of U1 gauge theory. Here, the C would be the electromagnetic field strength."
},
{
"end_time": 3643.063,
"index": 160,
"start_time": 3615.776,
"text": " The general non-abelian case is more subtle, but the principle is analogous. Eric is building an operator that combines the metric and gauge structures consistently. The action principle in GU takes a form reminiscent of both Einstein-Hilbert and Chern-Simons."
},
{
"end_time": 3658.029,
"index": 161,
"start_time": 3643.319,
"text": " Now you may look at this and say, this looks nothing like Einstein, it looks nothing like Chern-Simons. What's remarkable about this action? First, notice that it's first order in its derivatives, like Chern-Simons theory, but unlike the second order of Einstein-Hilbert action."
},
{
"end_time": 3684.701,
"index": 162,
"start_time": 3658.302,
"text": " Also notice that the field variables are omega, which actually comprise this epsilon and then this variational pi here, where epsilon is a gauge transformation and pi is a gauge potential. The first term combines the augmented torsion tensor with the curvature through the Shiab operator. This generalizes both the Einstein-Hilbert term and the Yang-Mills term. The second term is like a mass term for the torsion with coupling constant kappa."
},
{
"end_time": 3703.916,
"index": 163,
"start_time": 3684.957,
"text": " Remember, in vanilla gauge theory, one can't put the gauge potential directly in the action because it's not gauge covariant. So in general relativity, you can't put the connection directly in the action because it's not diffeomorphism invariant. But here, the augmented torsion gives us a covariant object we can use directly."
},
{
"end_time": 3730.299,
"index": 164,
"start_time": 3704.309,
"text": " Note you also have to recall that every element omega that comes from the inhomogeneous gauge group actually produces two connections. One is A and another is B. The difference between these two is called T. Also you should note that at this point the theory is purely bosonic. The fermions haven't come about yet. This reminds me of how string theory was initially bosonic prior to being fermionic or having both. Step 18. Field equations."
},
{
"end_time": 3756.067,
"index": 165,
"start_time": 3731.22,
"text": " From our action principle, we derive the field equations through a variational principle. The result is deceptively simple. It's on screen here, so what is going on? The Shiab operator acts on the curvature here, much like Einstein's contraction acts on the Riemann tensor. The primary difference is that this operation preserves gauge covariance. The augmented torsion tensor term T omega enters with a coupling constant kappa."
},
{
"end_time": 3780.572,
"index": 166,
"start_time": 3756.425,
"text": " This E term here is essentially something that helps the theory's consistency as it makes the equation properly reflect the variational principle from which it's derived. I think of it like an error term. Eric demands this because it's necessary for recovering both Einstein's equations and Yang-Mills theory in the appropriate limits. Step 19. Fermions and supersymmetry. Where do fermions enter?"
},
{
"end_time": 3799.906,
"index": 167,
"start_time": 3780.794,
"text": " Recall that fermions act as quote-unquote square roots of gauge potentials. For spinner-valued forms, which we see here as a zero-valued spinner form and a one-valued spinner form, we get a Dirac-like operator. Now this is fascinating because traditional supersymmetry relates bosons and fermions through spacetime translations"
},
{
"end_time": 3827.722,
"index": 168,
"start_time": 3800.23,
"text": " Here we're seeing a different sort of supersymmetry based on the affine space of connections. The operator here combines aspects of the Dirac operator with our gauge structure. The upper left block involves the Shiab operator acting on the derivative of a spinner-valued one-form. This is like the square root of the Yang-Mills operator. The off-diagonal blocks couple scalar spinners to vector spinners similar to how supersymmetry transforms fermions"
},
{
"end_time": 3854.548,
"index": 169,
"start_time": 3827.978,
"text": " into bosons and vice versa. When one decomposes the spinner representations under spin 7 comma 7, one finds the following formula. Now this is how the three generations of fermions come about. Notice that upper index of three there. The first two generations come from a spinner-valued zero form and a spinner-valued one form directly, whereas the third generation comes about from Rarita-Schwinger fields in the decomposition of zeta."
},
{
"end_time": 3884.07,
"index": 170,
"start_time": 3855.64,
"text": " Step 20, the deformation complex. How do we study small perturbations around solutions? We require a complex. Now this complex is what's necessary for understanding the physical content or the field content of the theory. The first map here encodes infinitesimal gauge transformations. It tells us how the fields change under small symmetry transformations. The second map gives us the linearized field equations. It tells us how the field propagates."
},
{
"end_time": 3908.422,
"index": 171,
"start_time": 3884.565,
"text": " The cohomology of this complex, which is the kernel module of the images, describes the physical degrees of freedom. At the first level, we have the first homology, and it gives us the gauge-inequivalent perturbations. This is analogous to how in electromagnetism, two gauge potentials differing by a gradient describe the same physics. Explicitly, these operations take the form on screen here."
},
{
"end_time": 3936.288,
"index": 172,
"start_time": 3908.422,
"text": " And we also have the d squared equals zero property that makes this a complex, ensuring the gauge invariance of the linearized theory. Note that the second term is actually slightly more complicated, but I will cover that later and or in the PDF notes on my sub stack and or with the upcoming podcast with Eric Weinstein himself. Step 21, the seesaw mechanism. Now, here's where it gets fascinating if it wasn't already."
},
{
"end_time": 3960.93,
"index": 173,
"start_time": 3936.613,
"text": " Our Dirac-Rarita-Schwinger complex leads us to an operator of the form, which is on screen here, that we've talked about before. Now why is this interesting at all? It's because this structure mirrors the neutrino-seasaw mechanism. So the seesaw mechanism explains neutrino masses through the mixing between light and heavy states. Here, we're mixing between different spinaural sectors."
},
{
"end_time": 3979.855,
"index": 174,
"start_time": 3961.254,
"text": " The zero block in the lower right corner is actually essential because this is what allows for the hierarchy between different types of fermions and this potentially explains why we see three generations of matter with such different masses."
},
{
"end_time": 4004.224,
"index": 175,
"start_time": 3980.572,
"text": " I debated whether this section should go earlier in the script or later just because it's representation theory and there's nothing specific to GU here. However, I placed it here due to its length. The reduction of spin 7 comma 7 to the standard model gauge group follows a path through intermediate subgroups. So let's go over how does this reduction work. Let's peel back the layers."
},
{
"end_time": 4029.36,
"index": 176,
"start_time": 4004.548,
"text": " Firstly, you have a spin seven comma seven acting on a 128 dimensional space of spinners that we've talked about ad nauseum and it splits into positive and negative chirality parts. Now here's where we get something different. You can reduce this to a maximal compact subgroup spin seven cross spin seven. Why this particular reduction of all the reductions we could make? Well,"
},
{
"end_time": 4058.148,
"index": 177,
"start_time": 4029.701,
"text": " Experimentalists haven't ever observed non-compact internal symmetries in particle physics. Indeed, there are compelling theoretical reasons why physicists don't consider non-compact groups. So for instance, the famous or infamous Coleman-Mendula theorem, which essentially states that the symmetries of the S matrix, which describes particle interactions, must be a direct product of the Poincare group and an internal symmetry group, though there are some assumptions here."
},
{
"end_time": 4080.111,
"index": 178,
"start_time": 4058.695,
"text": " This internal group must be compact for unitary representations, which means you need this for consistent quantum theory. But why must it be compact? Well, it boils down to these two reasons, unitarity that we mentioned already, and then positive energy. Non-compact groups often lead to these theories with negative energy states, which are physically problematic."
},
{
"end_time": 4107.483,
"index": 179,
"start_time": 4080.538,
"text": " We'll discuss the reasons why Eric's model circumvents these objections later. For now, let's break this down into the standard model's gauge group step by step. Again, the complete structure group reduction path begins with U64 comma 64. In low gravity and in Eric's model, this decouples into two vial halves, bringing us to spin 7 comma 7. This contains a spin 1 comma 3 cross spin 6 comma 4,"
},
{
"end_time": 4127.398,
"index": 180,
"start_time": 4107.483,
"text": " where the first term, the spin 1,3 represents spacetime, or specifically the spacetime symmetries. Then you have to notice that the spin 6,4 part has spin 6 cross spin 4 as its unique maximal compact subgroup. And then this gives us SU4 cross SU2 cross SU2"
},
{
"end_time": 4155.469,
"index": 181,
"start_time": 4127.398,
"text": " via an isomorphism, which is precisely the Pati-Salam model. This answers a constitutional question. What is the maximal compact subgroup of the fiber structure group of our observable universe? The final reduction down to SU3 cross SU2 cross U1, our standard model, comes about when the metrics carry an additional special unitary structure. Keep in mind that much of the above is somewhat standard in GUT circles, so Grand Unified Theory circles."
},
{
"end_time": 4164.531,
"index": 182,
"start_time": 4155.469,
"text": " Eric, though, follows a different path by using the non-compact group SU3,2, which is a real form of SL5,C."
},
{
"end_time": 4192.91,
"index": 183,
"start_time": 4165.026,
"text": " The standard model gauge group comes about now as the maximal compact subgroup of SU3,2. The specific real form corresponds to the A4-Dinckin diagram and this distinction is what allows Eric to resolve the proton decay controversy and other issues that plague 1970s grand unified theory schemes and approaches or what have you because they used real forms that in Eric's eyes were incorrect. So how do we get this SU3,2 while spin 7,7"
},
{
"end_time": 4214.633,
"index": 184,
"start_time": 4192.91,
"text": " has a spacetime split given on screen here. One of those, the 6, 4, has SU3, 2 as a complex structure inside. We can then take the maximal compact subgroup of that, which is taking the special part of U3 cross U2. And we can further make an isomorphism of that to the standard model."
},
{
"end_time": 4242.739,
"index": 185,
"start_time": 4214.957,
"text": " Keep in mind that each step involves symmetry breaking, which in physics corresponds to the vacuum state not respecting the full symmetry that used to be there in the Lagrangian. This breaking can be spontaneous, dynamical or explicit, put it into the Lagrangian by hand. Now, in this section, there are a plethora of subgroups being taken. So one question naturally comes up, which is, can we just take any subgroup willy nilly in physics? And the short answer is no."
},
{
"end_time": 4270.913,
"index": 186,
"start_time": 4243.251,
"text": " Each time you take a subgroup, you're saying that the symmetry is broken and thus you're introducing new physics. So for instance, the breaking from SU4 to SU3 cross SU1 in step four corresponds to the separation of leptons and quarks. This is both a boon and a curse. Since there's new physics, it means you're deviating from what's standard. However, it also means that there are predictions which can be falsified. Step 23."
},
{
"end_time": 4293.882,
"index": 187,
"start_time": 4271.408,
"text": " Again, we're going to make this painfully clear. The Dirac-Rarita-Schwinger complex on Y14 gives us this generalization of the Diram complex, which is what we need to deal with the spinner-valued forms. On screen here, you take a one-form with the Shiab operator, technically the Hodge star of the Shiab operator and then some other differential,"
},
{
"end_time": 4322.688,
"index": 188,
"start_time": 4294.138,
"text": " which takes you from a one-form to one dimension less than the manifold, so 13 in this case. Note, you may be wondering why you haven't seen this complex before, and that's because as far as I can tell, it's a novel Dirac-Rarita-Schwinger-like complex introduced by Eric. This is brand new in the physics literature. This complex yields three distinct sets of fermions. How? Well, the scalar spinner here, new, gives the first generation."
},
{
"end_time": 4350.145,
"index": 189,
"start_time": 4323.08,
"text": " The zeta vector spinner here splits into two parts, a gamma traceless part, which gives the second generation, and a gamma trace part. And that's what gives the third generation. So the first generation again, this space is the scalar valued spinners on Y14. And when you pull it back to X4, these correspond to the familiar first generation fermions like the electron, the electron neutrino and up and the down quark."
},
{
"end_time": 4377.995,
"index": 190,
"start_time": 4350.52,
"text": " The second and third come from zeta, which is a spinner value to one form. The decomposition of this is where it gets interesting. The first term here on the left side of the direct sum gives what Eric considers to be the second generation. And the last term, which is on the right hand side here of the direct sum, is where the third generation comes from. These decompose to give us distinct generations because of how these spaces transform"
},
{
"end_time": 4407.637,
"index": 191,
"start_time": 4378.251,
"text": " Step 24. Higgs from Yang-Mills. So where's the Higgs field at? Well, it's lurking in the gauge potential variational pi here. Here's how. Firstly, we decompose this form as follows. The second term here on the right-hand side contains the Higgs field."
},
{
"end_time": 4434.855,
"index": 192,
"start_time": 4407.91,
"text": " because symmetric two tensors decompose, remember, into a trace and a traceless part that we talked about ad nauseum again. Let's unpack this further. The gauge potential is a one-form on the Y14 space, which is valued in the adjoint bundle of the principal bundle P sub G. When we decompose the one-forms on this space, we're essentially splitting it into parts that live on X"
},
{
"end_time": 4464.053,
"index": 193,
"start_time": 4435.367,
"text": " and parts that live in the vertical direction in the fiber. The key term is this one. This is the space of symmetric two tensors on X4, and it contains scalar fields from the perspective of X4. Among these scalar fields is our Higgs field, according to Eric. But what's the justification for even calling it the Higgs field? Well, it behaves like a Higgs field because of how it transforms under gauge transformations and a few morphisms of X4."
},
{
"end_time": 4490.742,
"index": 194,
"start_time": 4464.565,
"text": " The trace component R in the decomposition transforms as a scalar under diffeomorphisms just like the Higgs field should. Moreover, under gauge transformations of the structure group G, this component transforms in the adjoint representation exactly how one would expect the Higgs to transform in gauge theories. Step 25 Trace and Traceless Contributions"
},
{
"end_time": 4518.763,
"index": 195,
"start_time": 4490.998,
"text": " The decomposition of the symmetric tensors into trace and traceless parts sounds like some mathematical pedantry, but it's not. Why? Consider the symmetric tensor aij, which you can write as follows, where you decompose it into trace and traceless parts, done this many times. The traceless part gives spin two contributions, and the trace part gives spin zero contributions. This mirrors exactly what happens with gravitons and the Higgs."
},
{
"end_time": 4548.592,
"index": 196,
"start_time": 4519.155,
"text": " Now we've discussed the trace and traceless decompositions before in the context of the Frobenius inner product, but let's go further. So we have this gravitational sector here where the traceless part is on screen and it corresponds to the spin two field in general relativity. This represents gravitational waves or gravitons in quantum theory. So why is it spin two? Because it's a symmetric traceless rank two tensor, which transforms under the spin two representation of the Lorentz group."
},
{
"end_time": 4575.367,
"index": 197,
"start_time": 4549.189,
"text": " Now the scalar sector here is what engenders the Higgs field, in geometric unity at least. This decomposition has these as a consequence. It suggests that the graviton which is spin 2 and the Higgs field which is spin 0 are intimately related. Interestingly, it's called geometric unity for a reason because these two are different aspects of the same geometric object on Y14."
},
{
"end_time": 4600.623,
"index": 198,
"start_time": 4576.664,
"text": " Step 26 Natural Quartic Potential Why does the Higgs field need that particular Mexican hat potential? Is it possible to get it to emerge in something that resembles something natural? Well, the Yang-Mills action contains terms like the following, where you take the normed squared and you get quartic terms. This matches the structure of the Higgs potential."
},
{
"end_time": 4626.561,
"index": 199,
"start_time": 4601.118,
"text": " Is it possible that the emergence of the Higgs potential comes from the Yang-Mills structure? Let's continue to explore this. In standard Yang-Mills theory, we have this term here, and that represents the self-interaction of gauge fields. However, in GU, remember that A contains components that we identify as the Higgs field. Let's write this out explicitly. Here, phi represents Higgs-like components."
},
{
"end_time": 4649.019,
"index": 200,
"start_time": 4626.8,
"text": " Now then, we expand the a wedge a squared term to get terms like the following. Notice this last term here. This looks precisely like a quartic term in phi. Is this the origin of the famous Mexican hat potential? Eric says, of course bro. You may wonder, where's the negative mass that gives the potential its characteristic shape?"
},
{
"end_time": 4675.282,
"index": 201,
"start_time": 4649.718,
"text": " Now this comes from the coupling between phi and the other components of A, specifically from terms like ADX wedge phi comma ADX wedge phi. This geometric unity specific derivation is to me phenomenal because it shows that the Higgs potential, far from being an ad hoc addition to the standard model, emanates inevitably from the geometry of gauge fields."
},
{
"end_time": 4704.906,
"index": 202,
"start_time": 4676.561,
"text": " Step 27, Yukawa as minimal coupling. The traditional Yukawa coupling also looks ad hoc, at least to me and to most other physicists and mathematicians. But in GU, it comes about inevitably. This time, it stems from viewing the Yukawa coupling as minimal coupling. Minimal coupling, by the way, to a mathematician just means a gauge covariant derivative. This A mu term, when it contains the Higgs component, gives the Yukawa interaction."
},
{
"end_time": 4732.346,
"index": 203,
"start_time": 4705.299,
"text": " This is the reinterpretation of the Yukawa coupling, and it's a prime example of how geometric unity unifies seemingly disparate aspects of particle physics. To be specific, in the Standard Model, the Yukawa coupling is introduced by hand to give fermions mass. In contrast to geometric unity, this Yukawa coupling comes about from the geometry. Here's how. Recall that in GU, the Higgs field, phi, is part of the gauge potential A."
},
{
"end_time": 4759.548,
"index": 204,
"start_time": 4732.739,
"text": " This Dirac operator coupled to A is D slash A here on screen. Expanding this you get this plus phi mu where this new A with a little tilde on top are the usual gauge fields and this Higgs is the component and we get this full formula on screen here. This last term is precisely the Yukawa coupling. Step 28. The correspondence between the Higgs and the Yang-Mills sectors."
},
{
"end_time": 4786.544,
"index": 205,
"start_time": 4760.213,
"text": " First, let's write out the Yang-Mills Dirac action in the language of geometric unity. Here this alpha is our gauge potential and f sub a is the field strength as usual and we also have some left and right-handed fermions. The covariant derivative acts as follows. Now compare this to the Higgs-Yukawa action. Here this capital phi is traditionally viewed as a scalar field which is valued in a representation of the gauge group."
},
{
"end_time": 4816.323,
"index": 206,
"start_time": 4786.988,
"text": " but in geometric unity it comes about as a component of this variational pi under the decomposition of one forms when pulled back to x4. This correspondence is surprising because we have kinetic terms like d sub a alpha squared and d sub a phi squared and they match and we also have this quartic term which correspond to one another. We also have this quadratic coupling which parallels this. We also have the fermion couplings and it takes on analogous forms"
},
{
"end_time": 4843.933,
"index": 207,
"start_time": 4816.613,
"text": " But how can a gauge field component act like a scalar field? Just remember, there either is right now or is going to be an accompanying PDF to this Geometric Unity iceberg. So if you'd like more notes, such as expansions on these topics and proofs that I wasn't able to get to in this iceberg, then subscribe to my sub stack as that's where I'll publish it. It's curtjamungle.org or C-U-R-T-J-A-I-M-U-N-G-A-L dot org."
},
{
"end_time": 4867.739,
"index": 208,
"start_time": 4845.111,
"text": " Step 29, the missing quadratic term. The Einstein field equations have traditionally been written on screen here with g mu nu plus the cosmological constant and we set that all as something proportional to the stress energy tensor. However, in GU, one needs to include a quadratic term in the field equations. So you get a modified equation which takes the form on screen here."
},
{
"end_time": 4897.432,
"index": 209,
"start_time": 4867.944,
"text": " This term here with the T omega comma T omega is the self-interaction of the augmented torsion, similar to how the Yang-Mills field strength contains A comma A. And it means that the theory maintains gauge covariance while it preserves Einstein's intuition about geometric contraction. Think about it. In Yang-Mills theory, the field strength is F sub A, which equals DA plus A comma A, and it needs a quadratic term to be covariant, namely that last part."
},
{
"end_time": 4925.828,
"index": 210,
"start_time": 4897.875,
"text": " Similarly, our augmented torsion needs its quadratic interactions to maintain gauge covariance while allowing for some Einstein-like contraction. Step 30, the emergence of the cosmological constant and CKM matrix. Lastly, both the cosmological constant and the CKM matrix come about from components of the gauge potential, variational pi here. How does that work?"
},
{
"end_time": 4947.5,
"index": 211,
"start_time": 4925.998,
"text": " The gauge potential decomposes as follows. You see that it splits into these components based on how they transform under the structure group when pulled back via an observation, which recall is iota. So the first part is what gives the standard model gauge fields. The second is what gives the Higgs field as we've discussed earlier."
},
{
"end_time": 4977.875,
"index": 212,
"start_time": 4947.892,
"text": " The third contributes a constant term into the Einstein equations, and the fourth determines mixing between generations. When one pulls this back to x4 via an observation, the component variational pi sub lambda gives the cosmological constant term in Einstein's equations, and then this variational pi sub ckm is what gives the mixing angles between the quark generations. Seemingly disparate physical phenomenon like dark energy, quark mixing,"
},
{
"end_time": 5003.968,
"index": 213,
"start_time": 4978.353,
"text": " All derived from the geometry of the observers. Ordinarily, we think of these as independent parameters that we need to add in by hand. However, in Geometric Unity, they're intrinsic parts of the geometric structure. Just so you know, if you have any questions, which you likely do, that's all right, I'm going to have a large solo podcast with Eric just on Geometric Unity. It will be unlike any other podcast because we'll delve into the particularities of it."
},
{
"end_time": 5032.073,
"index": 214,
"start_time": 5004.275,
"text": " Layer 3 Welcome to Layer 3 of the Geometric Unity iceberg. Let's recap what's been done so far. In Layer 1, we gave a brief overview of Geometric Unity, as well as the universe as we know it. In Layer 2, just now,"
},
{
"end_time": 5060.828,
"index": 215,
"start_time": 5032.483,
"text": " We went over geometric unity in four minutes and then I gave the longer one hour or so version of it as well. It's been quite a journey and now in layer three, here's where it all starts to come together. Again, if you haven't understood much of this so far, then that's entirely fine because this iceberg is designed to be watched and rewatched where you glean something new every single time, not only from geometric unity, but perhaps from physics and math as well."
},
{
"end_time": 5084.411,
"index": 216,
"start_time": 5060.828,
"text": " I've worked on this like i would work on a documentary with hundreds of hours put in also note that i will be recapitulating often so that your hand is held metaphorically so. Unless you're into that now as i mentioned in layer one we talked about the universe as it's currently known and we went over symbols and equations that are at the heart of our models of the physical universe."
},
{
"end_time": 5101.852,
"index": 217,
"start_time": 5084.582,
"text": " They were the Richie Scaler, which is at the heart of the Einstein-Hilbert action, as well as for other actions, there's the Yang-Mills Maxwell one, there's the Dirac one, there's the Higgs one, there's also the Yukawa coupling, there's also the Lorentz group, or the double cover of it, namely spin one comma three,"
},
{
"end_time": 5123.575,
"index": 218,
"start_time": 5101.852,
"text": " There's the internal gauge group of the standard model, so SU3 across SU2 across SU1. There's also the family of quantum numbers. There are also the three generations of matter or families of matter or flavors of matter. I've heard some people call it. There's also the CKM matrix and the PMNS matrix for quark and neutrino mixing respectively."
},
{
"end_time": 5153.285,
"index": 219,
"start_time": 5123.575,
"text": " Now there are also these which are alternative writings of what I've just mentioned. So there's the Einstein field equations, there's the Dirac equation, the Klein-Gordon equation, the Yang-Mills equation, and the Higgs field equation. While I've reviewed the different steps of geometric unity as I see it, the question is, well, how do we integrate these concepts that I've just mentioned within geometric unity? Exactly. Now, GU's explanation for these equations and concepts encompasses essentially everything in modern physics."
},
{
"end_time": 5159.667,
"index": 220,
"start_time": 5153.49,
"text": " Einstein-Hilbert action in GU"
},
{
"end_time": 5183.985,
"index": 221,
"start_time": 5160.503,
"text": " In standard GR, one writes the Einstein-Hilbert action as follows, where the Ricci tensor is computed by contracting the Riemann curvature tensor with the inverse metric, and then you further do a trace. In geometric unity, however, there's no single metric that's privileged, as you know, and instead, Eric begins with the curvature of a distinguished connection, A0, which is defined on the frame bundle over the manifold lifted to the double cover,"
},
{
"end_time": 5196.425,
"index": 222,
"start_time": 5184.155,
"text": " so that spinners can exist. Now this A0 is obtained via the Zoro construction that we talked about earlier and more precisely one starts with the frame bundle of X4 lifts it to its double cover"
},
{
"end_time": 5219.753,
"index": 223,
"start_time": 5196.698,
"text": " Notice I use the term lifted here, and I do so in the sense of a standard bundle theoretic lift."
},
{
"end_time": 5240.486,
"index": 224,
"start_time": 5220.247,
"text": " That is to say, given a projection from F tilde to F, we can look at a connection which is A0 on the regular F, and that can be lifted, quote-unquote, to F tilde by composing with the covering map. Precisely this means what's on screen here, which is a 2 to 1 covering, and A0 is a connection on the base F,"
},
{
"end_time": 5259.172,
"index": 225,
"start_time": 5240.486,
"text": " which is then lifted to an A tilde on F tilde, and it satisfies this equation here. In other words, for any vector in the tangent bundle of F tilde, you can define a connection by just pushing that vector forward. This allows for the proper transformation properties that Eric wants for spinors."
},
{
"end_time": 5287.415,
"index": 226,
"start_time": 5259.172,
"text": " As I'm reviewing this iceberg several weeks later, recall this is many months, many, many months in the making, I'm realizing that I sometimes use A0 and I sometimes use A aleph. You may be wondering what is the relationship between these. These are the same. My mistake is that in ordinary math notation, one would use A0, but in GU, because Eric is referencing something specific with the aleph, then the A sub aleph is used and it refers to the choice made in the Zoro construction."
},
{
"end_time": 5314.48,
"index": 227,
"start_time": 5287.415,
"text": " Now, geometric unity reinterprets this contraction process as coming from an algebraic operator, which compresses the full curvature two form on Y, the metric bundle, the observer's, into a symmetric two tensor. You'll notice that the domain of P E consists of a tensor product with two fundamentally different factors. So this first component is a differential form, which is intrinsically tied to the manifold Y itself. The second factor here has a Lie algebraic character."
},
{
"end_time": 5331.152,
"index": 228,
"start_time": 5314.77,
"text": " In Einstein, it's SO1,3, the Lie algebra. So what's interesting is that this P-E operator contracts mathematically distinct structures using a metric on y. So the operator P sub e, it contracts these two using the metric on y"
},
{
"end_time": 5359.582,
"index": 229,
"start_time": 5331.152,
"text": " much in the same spirit as how Einstein in his original approach obtained the Ricci tensor by contracting the indices of the Riemann tensor like we talked about just a minute ago. Now, where did these two forms come from? Well, one comes directly from the curvature of the distinguished connection, namely what's on screen here, while the other is an internal two-form from the contraction mechanism itself induced by the metric on y. Thus, both copies of the two-form are essentially the same curvature data, however,"
},
{
"end_time": 5389.206,
"index": 230,
"start_time": 5359.787,
"text": " Eric regards them as having different transformation behaviors under the gauge group. This is reminiscent, again, to how Einstein in his approach also had two appearances of two forms in the Riemann tensor treated identically by the metric contraction. In fact, one way that I think of this algebraic operator is that it encapsulates the same idea as forming inner products via the Frobenius metric. If you take two symmetric matrices and multiply them together using the natural inner product on screen here, you obtain a contraction that produces a scalar."
},
{
"end_time": 5403.729,
"index": 231,
"start_time": 5389.206,
"text": " Here, however, this algebraic operator acts extended bilinearly to the tensor product to yield an element of the symmetric 0,2 tensors, thus compressing the curvature information into a form that mimics the Einstein tensor."
},
{
"end_time": 5431.493,
"index": 232,
"start_time": 5404.121,
"text": " What you're doing is you're allowing an algebraic operator, which is defined to act on a curvature tensor, and then it's supposed to return something from the parameter space of field content. In Yang-Mills, this involves a Lie algebra-valued one-form, while in general relativity, it involves symmetric two tensors. After this algebraic operator on y, we then pull back the result along the section, which was that observation map, so that"
},
{
"end_time": 5461.254,
"index": 233,
"start_time": 5431.681,
"text": " the effective Einstein tensor on X4 is given by this formula on screen here. So you begin again with FA0, which is a two-form. The contraction operator is defined locally such that we have this formula on screen here so that we can then pull back with an observation and we get the formula on screen here. And that's where the R comes from in geometric unity, re-derived from a gauge covariate contraction on Y."
},
{
"end_time": 5469.804,
"index": 234,
"start_time": 5462.841,
"text": " the Yang-Mills-Maxwell term. Next, we will derive the Yang-Mills-Maxwell term within geometric unity."
},
{
"end_time": 5498.336,
"index": 235,
"start_time": 5469.974,
"text": " In conventional settings on a fixed spacetime x4, the Yang-Mills action is written as follows, where the fA here is the curvature as usual of the connection A on a principal G bundle, and the inner product denotes the invariant bilinear form, also known as the killing form, on the Lie algebra G. Editors note, in a local trivialization, that is, on a patch of the base space where the bundle is trivial, each connection is represented by a Lie algebra-valued one-form A,"
},
{
"end_time": 5523.063,
"index": 236,
"start_time": 5498.643,
"text": " Its curvature is the following, but when you're comparing two different connections, say A and B, then you often write it as follows. The connection B here can be viewed as a reference or a base connection. Now when you set B to equal zero, that is the trivial connection, you then recover the usual single connection formula. This will become important later in the Higgs sector portion of this geometric unity iceberg."
},
{
"end_time": 5533.251,
"index": 237,
"start_time": 5523.473,
"text": " Eric considers the principal bundle here with the structure group of U64,64 coming from that spinner representation of spin 7,7."
},
{
"end_time": 5556.63,
"index": 238,
"start_time": 5533.524,
"text": " The gauge field is defined on y as a curvature 2 form, with the transformation property as follows. Now recall that the key step is that this inner product here, which we use to contract F A, is actually induced from the metric on the larger space y, which is in turn built from the vertical Frobenius metric on the symmetric 2 tensors from the base manifold."
},
{
"end_time": 5586.288,
"index": 239,
"start_time": 5556.63,
"text": " and the horizontal lift via the Zoro construction. Explicitly for local coordinates on y, we can define the following slightly hairy formula at least to look at on screen. This ensures that the contraction is performed in a manner that's compatible with the dynamical nature of y. Now in our current context, A represents the full gauge field on y and A0 denotes that distinguished background connection which is chosen by the Zoro construction. Therefore, you should actually understand that this connection"
},
{
"end_time": 5604.684,
"index": 240,
"start_time": 5586.288,
"text": " is supposed to be expressed as A equals A zero plus alpha, where alpha is like a correction or a fluctuation. In contrast, this F A zero that appears in some derivations is simply the curvature of the background connection alone. Our full curvature, F A, is thus containing extra pieces."
},
{
"end_time": 5627.585,
"index": 241,
"start_time": 5604.684,
"text": " and then the contraction via the Shiab operator is engineered to handle these extra terms. Moreover, the domain of this operator is the space of Lie algebra value two forms on y, and its target space is typically the space of scalar fields or symmetric two tensors, for instance, because it compresses the two forms into a lower degree object using the"
},
{
"end_time": 5658.865,
"index": 242,
"start_time": 5630.742,
"text": " But under the gauge transformations H, for instance, we have the following, which preserves gauge covariance. Technically, it's the below, but the above is much more succinct. Now, you may be confused. Look, is P E an example of the Shiab operator or vice versa or nothing? Now, the answer is that the P E in Einstein's theory is an algebraic contraction that maps two of something. This is a two form and then some Lie algebra data. Their domains and target are different."
},
{
"end_time": 5683.575,
"index": 243,
"start_time": 5658.865,
"text": " Also, you'll see that this lambda with a bullet is what represents, in Eric's theory, the U64 comma 64 Lie algebra structure, while the Shiab operator maps these two forms, which locally involve both a two-form part and a Lie algebra data part, to a one-form or another suitable counterpart. The idea here is that the Shiab operator is a generalization of Einstein's contraction operator."
},
{
"end_time": 5711.954,
"index": 244,
"start_time": 5683.797,
"text": " except Eric constructs it to be compatible with both the base manifold and the fiber gauge structure. At this point, I should point out that the function space for the bosonic theory is the inhomogeneous gauge group. It has two components. One looks like the gauge potential and functions like a connection. The other piece does the symmetry work. It allows the chiab to be defined without destroying equivariance in contraction. Why don't we just do a step-by-step derivation? So step A, let's say,"
},
{
"end_time": 5740.93,
"index": 245,
"start_time": 5711.954,
"text": " you get that distinguished connection from the Zoro construction. Step B, you define the gauge potential correction. Step C, you compute the curvature. Step D, you then apply the Shiab operator. Now you'll notice that this form is analogous to the standard Yang-Mills action, except contracted by the Shiab operator here and replacing the usual inner product contraction on FA. Now one has to show that in a suitable limit after pulling back to X4,"
},
{
"end_time": 5770.811,
"index": 246,
"start_time": 5741.152,
"text": " that this actually reduces to the familiar expression. For an explicit demonstration, you would take a local section from x4 to y, then you would note that the induced metric on x4 satisfies that the metric is a pulled back metric and that the Shiab operator on the curvature when pulled back is the same as the inner product of the curvature with itself. That completes our derivation of the Yang-Mills-Maxwell term in GU."
},
{
"end_time": 5800.572,
"index": 247,
"start_time": 5771.596,
"text": " Eric derives the Dirac action in the framework of geometric unity, solving the previously unsolved problem of defining spinors without a prior metric. Now, in conventional formulations, the Dirac action is given as follows, with the Dirac operator also written on screen here. And the gamma matrices, of course, satisfy that Clifford algebra so equals twice the metric. Even though many people think Clifford algebras don't rely on a metric, there is an implicit pre-assigned metric, and that's the dilemma."
},
{
"end_time": 5821.749,
"index": 248,
"start_time": 5800.947,
"text": " The very existence of spinors demand a metric. However, one would like to work in a framework where the metric is allowed to be dynamic, allowed to vary. Now, geometric unity sidesteps this by constructing the chimeric bundle, which recall is the vertical component, direct summed with the dual of the horizontal."
},
{
"end_time": 5851.203,
"index": 249,
"start_time": 5822.108,
"text": " Eric then uses the exponential property of spinors, which you can recall if you have different vector spaces V and W direct summed together, that their spinner bundle is isomorphic to the tensor of the individual components. Now in our case, because our components are V and the dual of H, we have the following. The prowess of this construction is that it's like liberating. It's like liberating the definition of fermions from an a priori choice of a metric."
},
{
"end_time": 5879.804,
"index": 250,
"start_time": 5851.493,
"text": " Instead, the metrics are actually determined by the point that they are in Y. This means that the Dirac operator can be defined on Y, the larger space, rather than X4, the base space. Concretely speaking, you construct a Dirac operator acting on the sections of the spinner bundle by creating something familiar here, where the nabla is of course the covariant derivative associated with the distinguished connection A0, which is from the Zoro construction."
},
{
"end_time": 5905.742,
"index": 251,
"start_time": 5879.804,
"text": " The gammas are the gamma matrices or more specifically the corresponding gamma matrices to the 7,7 signature determined by the vertical Frobenius metric on V and the induced metric on the horizontal space or the dual of the horizontal space H star. The I of course runs over the entire indices of 14 dimensions and the alpha represents additional potential coming from the inhomogeneous gauge group structure. We're going to talk more about that."
},
{
"end_time": 5929.087,
"index": 252,
"start_time": 5905.742,
"text": " Now one uses these observation maps to pull back the spinner fields from the larger space y to x4, and this is done using the standard pullback of vector bundles on screen here. You'll notice that this pullback yields a decomposition, and what happens is that the v part becomes the internal quantum numbers, whereas the h part becomes the spacetime Lorentz group spin 1 comma 3."
},
{
"end_time": 5956.749,
"index": 253,
"start_time": 5929.241,
"text": " Now, to spell it out in detail, if you look at the chimeric bundle, and let's just pick a single point Y, then we have the following, and we then construct the spinner bundle as follows, again at a single point, and we take the pullback, which here I'll just show on screen every element. By this construction, this last factor here is equivalent to the conventional spinner bundle on X4, since H star evaluated at the pullback corresponds via the Zorro construction"
},
{
"end_time": 5974.002,
"index": 254,
"start_time": 5957.039,
"text": " to the cotangent space on the base manifold, which we denote by that S and then the slash and spacetime. Meanwhile, the other factor here with the V, that's where the internal symmetry space comes from, so that's why I placed a S with a slash through it and said internal as an indice."
},
{
"end_time": 5998.046,
"index": 255,
"start_time": 5974.309,
"text": " whose dimension encodes the family quantum numbers, thus that decomposition that you see above. Finally, in GU, the action, or the Dirac action to be specific, is defined as follows, where that field there is a section of the chimeric spinner bundle, and mu is the natural measure on y. What I mean by natural is that the measure actually comes from the induced metric on y itself."
},
{
"end_time": 6025.026,
"index": 256,
"start_time": 5998.046,
"text": " specifically the square root of the absolute value of the determinant of G with the 14 dimensions dy. This measure has the required properties, that is, it's locally given by the Lebesgue measure in suitable coordinates and it transforms appropriately under coordinate changes so that the integrals such as the Dirac action are well defined and gauge invariant. The Higgs sector, the derivation of the Higgs kinetic and potential terms."
},
{
"end_time": 6049.411,
"index": 257,
"start_time": 6025.486,
"text": " In geometric unity, the origin of the Higgs sector is the gauge potential written on screen here with this variational pi, and the origin of the Higgs sector is when we decompose this according to the splitting of the observers. Now here, let me clarify what this gauge potential means. In GU, this gauge potential variational pi here is a one-form defined on the larger observers"
},
{
"end_time": 6077.688,
"index": 258,
"start_time": 6049.599,
"text": " whose domain is the tangent space of Y, and whose target is the Lie algebra of the gauge group, specifically lowercase u, 64, 64. You can think of this as a rule that goes from here to here, that tells you how to differentiate sections of the associated bundle. There's nothing miraculous about this specific one-form. It's the canonical gauge potential coming from the inhomogeneous gauge group structure on Y."
},
{
"end_time": 6102.875,
"index": 259,
"start_time": 6078.183,
"text": " In other words, any connection on the principal bundle Y provides an adjoint-valued one-form, but this variational pi is our chosen representative that encodes fluctuations over the distinguished background connection. Now, the gauge potential will decompose when pulled back to X4 via a local section. See it decomposing as follows with alpha and phi."
},
{
"end_time": 6117.978,
"index": 260,
"start_time": 6103.404,
"text": " where the alpha represents the Lie algebra valued one form component corresponding to the Yang-Mills fields and the phi is a scalar field or at least appears as one coming from additional degrees of freedom in the vertical direction."
},
{
"end_time": 6143.643,
"index": 261,
"start_time": 6118.336,
"text": " When one pulls back a one-form, variational pi from y, via the section, we then decompose, or Eric then decomposes the tangent vector into horizontal and vertical components. The horizontal part gives a one-form on the base manifold, since it involves directions along the base, and we denote that by alpha. In contrast, the vertical part is associated with the variations along the fiber."
},
{
"end_time": 6161.237,
"index": 262,
"start_time": 6143.951,
"text": " So that is the variations of the metric at a fixed point, which when pulled back effectively lose their quote-unquote direction along x4, becoming functions. That is to say, they become zero forms valued in the symmetric two tensors, which we denote by phi."
},
{
"end_time": 6185.879,
"index": 263,
"start_time": 6161.237,
"text": " Editors note, in a more fine-grained analysis, and this is as Eric points out, you have to decompose both the one-form part, the horizontal versus the vertical, and the adjoint part, which is the pure horizontal, pure vertical and mixed. And concretely, the pulled back one-form is written as a direct sum of six different pieces. Each of the summands here are different sectors in 4D."
},
{
"end_time": 6209.155,
"index": 264,
"start_time": 6186.067,
"text": " So some give a spin-1 field, like the usual gauge bosons, and others give a spin-0 field, like the Higgs scalar, and it just depends on how it transforms. This is a refinement about the splitting of what I've just showed of alpha plus phi, but my initial wave writing it was much more compact. As you can see, it's quite tortuous otherwise to show every pairing of horizontal versus vertical in both the one form and the adjoint representation."
},
{
"end_time": 6236.374,
"index": 265,
"start_time": 6209.548,
"text": " Note that the vertical component corresponds to the infinitesimal changes in the metric at a fixed point. Given that they're symmetric matrices, they contain both a trace and a traceless part. In our construction, or more specifically in Eric's construction, we use a contraction, so the trace, to isolate the scalar degree of freedom, which is then identified with the Higgs field. To derive the Higgs Lagrangian,"
},
{
"end_time": 6257.381,
"index": 266,
"start_time": 6236.698,
"text": " Eric starts with the Yang-Mills action on Y as follows, where the curvature is here. When we decompose A as A0 plus variational pi, with A0 or A0 being that distinguished connection induced by the Levi-Chevita connection on the base manifold, the field strength splits into parts involving our gauge connection."
},
{
"end_time": 6286.988,
"index": 267,
"start_time": 6257.841,
"text": " In particular, the terms that are quadratic in the gauge connection lead upon pullback to terms like D phi squared and phi wedge phi squared. Now remember, A zero is that classical background connection that encodes the standard geometric data, while the variational pi, the full gauge potential, represents the fluctuations. Now by writing A equals A naught plus variational pi, you separate this unchanging classical piece"
},
{
"end_time": 6310.077,
"index": 268,
"start_time": 6287.278,
"text": " To explain this clearly, we begin with a gauge connection that gets decomposed as follows. Then you insert that into the curvature and that leads to this formula on screen here."
},
{
"end_time": 6336.101,
"index": 269,
"start_time": 6310.776,
"text": " When you then pull back this variational pi via the local observations, its vertical component yields phi. The derivative then contains a term that projects onto the derivative of phi, giving a kinetic term here. Meanwhile, the gauge potential wedged with itself when restricted to the vertical directions and then contracted via the Frobenius inner product provides a quartic self-interaction term"
},
{
"end_time": 6347.807,
"index": 270,
"start_time": 6336.323,
"text": " of the form on screen here. In GU's language, after the appropriate contractions on the metric on X4, these pieces match the standard Higgs kinetic term"
},
{
"end_time": 6373.148,
"index": 271,
"start_time": 6348.012,
"text": " and the Mexican hat potential, typically written as follows. What I find remarkable, personally, is that the gauge potential, when viewed through the lens of the observer and decomposed via the pullback, bifurcates into these two sectors. One is that alpha, which reproduces conventional Yang-Mills, as we'll see, and the other is the phi, which embodies the scalar fluctuations of the metric, which then become the Higgs field."
},
{
"end_time": 6400.009,
"index": 272,
"start_time": 6373.677,
"text": " In this way, GU unifies the disparate sectors of gauge theory and spontaneous symmetry breaking within a geometric framework, all following from the metric. Now, every step as follows. Let me just repeat that as you start with the full gauge potential. You then split it into horizontal and vertical components using iota to pull it back so that the vertical part, which carries the metric fluctuation degrees of freedom, provide a natural candidate for the Higgs field."
},
{
"end_time": 6428.882,
"index": 273,
"start_time": 6400.469,
"text": " The kinetic term is then given by the norm squared here, while the quartic self-interaction is from the contraction, and both are computed via the Frobenius inner product on the space of symmetric 0,2 tensors. Yukawa coupling, the derivation via minimal coupling. The Yukawa coupling term that gives mass to fermions come out as part of the minimal coupling of the Dirac operator on the spinner bundle to the gauge potential."
},
{
"end_time": 6456.834,
"index": 274,
"start_time": 6429.428,
"text": " Start with the Dirac action on the Observer. So we have this action here, and of course this D slash is as follows, which is the Dirac operator coupled to the full gauge connection A, and psi is a section of the Chimeric Spinner Bundle, S of C, or S slash of C, actually. Now, the minimal coupling is another way of saying that the derivative in the Dirac operator is replaced by a covariant derivative."
},
{
"end_time": 6472.159,
"index": 275,
"start_time": 6457.227,
"text": " Inserting the decomposition of A we saw just a minute ago, we obtain the following. In standard physics, the term here, phi psi, isn't actually present in the definition of a derivative because the Higgs field is taken to be a separate independent scalar."
},
{
"end_time": 6494.48,
"index": 276,
"start_time": 6472.602,
"text": " However, Eric's identification is that the Higgs field originates from a particular component of that variational pi. Now in practice, one arranges the theory so that the coupling of phi to the fermions takes the form on screen here, where y is the Yukawa coupling constant. Let's clarify the connection between this guy and this guy."
},
{
"end_time": 6516.954,
"index": 277,
"start_time": 6495.179,
"text": " Now in GU, the gauge connection again decomposes as follows, so that the minimal coupling replaces the ordinary derivative as follows, and next one decomposes the chimeric spinner into its left and right components. This is so that the Dirac operator then splits into off-diagonal blocks as follows."
},
{
"end_time": 6545.862,
"index": 278,
"start_time": 6517.483,
"text": " Here, the splitting into the left and the right-handed parts is fomented from the Z2 grading of the chimeric spinner bundle S of C as constructed via that exponential property we talked about before ad nauseum. In more detail, once a spinner bundle is defined on a space with a metric of indefinite signature, that is 7,7, the associated Clifford algebra admits a decomposition into even and odd parts. This decomposition"
},
{
"end_time": 6570.247,
"index": 279,
"start_time": 6546.084,
"text": " allows one to identify chiral or vial sub-bundles, usually denoted by S slash L and S slash R. Some people put a plus and a minus instead. Thus, psi is a section of the chimeric spin bundle which decomposes into its chiral components. Here the operator D minus A and D plus A incorporate the corresponding parts of the connection."
},
{
"end_time": 6591.357,
"index": 280,
"start_time": 6572.739,
"text": " The Higgs field contribution appears precisely in the following blocks, where the first part, the A0 parts, denote the chiral pieces of the Dirac operator associated with the background connection A0, and phi, after some projection or contraction, behaves like a scalar."
},
{
"end_time": 6619.445,
"index": 281,
"start_time": 6591.971,
"text": " Inserting these into the Dirac action yield the following. Now these two terms here combine after appropriate normalization, absorbing constants into the Yukawa coupling to yield the Yukawa action as follows. Thus, geometric unity derives the Yukawa action from the original Dirac action by expanding the covariant derivative to include the gauge potential's extra component phi, which is associated with the Higgs field."
},
{
"end_time": 6634.991,
"index": 282,
"start_time": 6620.128,
"text": " Now this derivation is entirely quote-unquote minimal once one posits that the gauge potential contains both the conventional gauge fields and an extra component phi, the Lorentz group and the standard model gauge group."
},
{
"end_time": 6657.978,
"index": 283,
"start_time": 6635.776,
"text": " The Chimeric bundle is defined as the vertical component plus the dual of the horizontal component. With the overall signature of 7,7, now the relevant real spin group is spin 7,7, however, Eric works with complex Dirac spinners, thus the natural symmetry group is U64,64. The real spinner representation of spin 7,7"
},
{
"end_time": 6688.2,
"index": 284,
"start_time": 6658.2,
"text": " has of real dimension 128. However, once you complexify and then you split into chiral components, it decomposes into two 64 dimensional complex of vial representations. In this complex setting, the indefinite unitary group U 64 comma 64 is what preserves the Hermitian form. And this is what acts as the full symmetry group of the spinner bundle, even though the spin seven comma seven governs the underlying real Clifford structure at signature seven comma seven."
},
{
"end_time": 6716.664,
"index": 285,
"start_time": 6688.2,
"text": " All of this is as far as I understand it, and I could be incorrect, so I'm just telling you what I've understood. Now, to recover spacetime versus internal degrees of freedom, one uses the observation map, written on screen here, and the differential of this, or the push forward of this, maps from the tangent space on the base space to the upper space, so that the image is isomorphic to a four-dimensional subspace, the dual of H."
},
{
"end_time": 6747.005,
"index": 286,
"start_time": 6717.312,
"text": " while the complement V of dimension 10 corresponds to the vertical variations. Now this splitting is what is allowing this reduction of the structure group. So we start with spin 7 comma 7, that then gets reduced to spin 1 comma 3 cross spin 6 comma 4. Now this spin 1 comma 3 acts as the spacetime horizontal part and it's isomorphic to the double cover SL2C of the Lorentz group."
},
{
"end_time": 6766.203,
"index": 287,
"start_time": 6747.466,
"text": " the remaining factor 6,4 acts on the vertical 10-dimensional space. Now since 10 equals 2 mod 4, this is just a result, that means that the vertical space admits a complex structure, and you can reduce 6,4 in at least two ways."
},
{
"end_time": 6794.292,
"index": 288,
"start_time": 6766.561,
"text": " So one approach is to reduce it to a non-compact real form, SU3,2. Alternatively, you can reduce spin 6,4 to its maximal compact subgroup. And then you'll find spin 6 cross spin 4. Now there are some isomorphic coincidences where spin 6 is SU4 and spin 4 is two copies of SU2, and then that is precisely the Patiss-Lomb group."
},
{
"end_time": 6824.411,
"index": 289,
"start_time": 6795.128,
"text": " If you know anything about grand unified theories, this is quite remarkable and should be surprising. At least it was to me. Now, to obtain the standard model gauge group, we then do a further reduction. And from here, this is standard in GUT folklore. So, provided you choose a correct embedding of the residual U1 factor as the difference between the two SU2 factors or equivalently by requiring certain trace conditions on the symmetric tensors,"
},
{
"end_time": 6847.875,
"index": 290,
"start_time": 6824.701,
"text": " Then the physical gauge group comes about. So you take U3 cross U2 and demand that the determinant equals 1 and then you get something that's isomorphic to the standard model gauge group SU3 cross SU2 cross U1 up to discrete identifications. Note that the left-hand side is the maximal compact subgroup of SU3 comma 2."
},
{
"end_time": 6872.978,
"index": 291,
"start_time": 6848.319,
"text": " Now, this identification is enforced by the requirement that the internal quantum numbers, which ultimately turn out to be 16-dimensional for a single family, and we'll see more about that soon, they match the observed hypercharge and color assignments of elementary particles. So, let me just sum up. First, we start with the full symmetry group, spin 7, 7, on the chimeric bundle."
},
{
"end_time": 6900.384,
"index": 292,
"start_time": 6873.507,
"text": " Then, by application of the observation map, we split the geometry into a four-dimensional spacetime with the spin 1, 3 symmetry, and then a 10-dimensional vertical part which has a different structure group, spin 6, 4, which then can get reduced to SU4 across SU2, SU2, and that further gets reduced to the standard model. Now, GU is broke with"
},
{
"end_time": 6928.49,
"index": 293,
"start_time": 6900.503,
"text": " many alternate paths like a labyrinthine roguelike video game as far as I can see. So one path that I mentioned is that pati salam first which is spin six comma four reducing to pati salam providing an immediate quark lepton unification picture. Now the other path is the one that goes through what I mentioned before where you take u three cross u two and you take the special part or you enforce speciality which means the determinant equals one"
},
{
"end_time": 6954.787,
"index": 294,
"start_time": 6928.49,
"text": " And then you recover up to discrete factors, the standard model gauge group. Neither of these, as far as I can tell, is more fundamental. They're just complementary ways to see how the indefinite group spin 7,7 breaks into the spin 1,3 spacetime group and the standard model gauge group. The family of quantum numbers. In conventional SL10 grand unification,"
},
{
"end_time": 6972.193,
"index": 295,
"start_time": 6955.094,
"text": " The fundamental spinner representation is 16-dimensional. Here, however, in GU, the 16 is reinterpreted as coming from the structure of the metric bundle. So let's recall that the chimeric bundle has that 7,7 signature."
},
{
"end_time": 7002.278,
"index": 296,
"start_time": 6972.654,
"text": " In Clifford algebra theory, for a space of signature PQ, I'm being general now, the real spinner dimension is given by 2 to the P plus Q over 2, and you have to take the floor function of that. Now for 7 comma 7, that works out to 2 to the power of 7, which is 128. This 128 dimensional complex representation on Dirac spinners naturally splits by chirality into two 64 dimensional Weyl spinners. By the way, you may wonder,"
},
{
"end_time": 7027.381,
"index": 297,
"start_time": 7002.739,
"text": " Why is this with a W pronounced vial? It's just German. As I mentioned a couple minutes ago, the properties of the Clifford algebra are such that there's a Z2 grading into positive and negative chirality parts, each of that 64 dimension. In technical terms, consider that for Clifford algebra CLPQ, the real spinner representation, if it exists, has this dimension here."
},
{
"end_time": 7045.674,
"index": 298,
"start_time": 7027.722,
"text": " and it decomposes as follows. Notice that this time I'm calling it S plus and then S minus. This operator is unique up to scaling so that the splitting is canonical. Now this action is defined by the standard representation induced by the Clifford algebra multiplication on this space."
},
{
"end_time": 7075.657,
"index": 299,
"start_time": 7046.305,
"text": " In our context, the full chimeric bundle's spinner representation later will be pulled back and partially reduced through projection onto the internal degrees of freedom to yield a 16-dimensional space corresponding to exactly the family of quantum numbers. Let me explain this reduction in more detail. Here, think of the spin bundle on the vertical part as encoding the internal degrees of freedom associated with the variations of the metric, a kind of quote-unquote gauge content."
},
{
"end_time": 7101.971,
"index": 300,
"start_time": 7076.152,
"text": " and then the spin bundle associated with the dual horizontal as encoding space-time properties. When one performs an observation and one pulls back via this observation back onto your base manifold, then you get the decomposition on screen as follows. The careful matching of these components with the quantum numbers like weak hypercharge and isospin forces the internal part to be 16-dimensional."
},
{
"end_time": 7129.172,
"index": 301,
"start_time": 7102.483,
"text": " Now, this matching process exploits the fact that for any given element, say k of u1, one can assign it a weight. And the sum of these weights in a chiral multiplet typically adds up to produce the correct quantization conditions. Now, for instance, representing all of these components as little quote unquote bit strings, so not in the string theory sense, but in the computer science sense, as in conventional SU-5 language,"
},
{
"end_time": 7149.206,
"index": 302,
"start_time": 7129.735,
"text": " You'll find that the total number of independent states is exactly 64. You can actually prove this to yourself by looking at the exterior algebra of C5, which has dimensions 2 to the power 5, which is 32, and if particles split into particles and then antiparticles, then you get 16, one for each chirality."
},
{
"end_time": 7173.217,
"index": 303,
"start_time": 7149.616,
"text": " However, GU doesn't just postulate SU5, instead it recovers the same number from its more fundamental chimeric construction. Explaining the Three Generations of Matter Some of you may know that in Hodge theory you obtain cohomological information from harmonic forms, but this requires a metric."
},
{
"end_time": 7201.425,
"index": 304,
"start_time": 7173.592,
"text": " However, there's also the Dirac theory. Now, in Durham theory, there's the metric independence. So, where is the Durham version of the Dirac operator? What I mean to say is that Nigel Hitchens showed that the dimension of the space of harmonic forms changes with respect to variations in the metric. It's only the difference between these dimensions that's topologically determined by the A-roof genus. Now, unlike harmonic forms,"
},
{
"end_time": 7227.381,
"index": 305,
"start_time": 7201.749,
"text": " The individual kernels don't behave consistently. In Hodge theory, regardless of the metric chosen, the dimension of the kernel of the Laplacian remains constant and it's topologically determined. So this then raises the question, can you make Dirac theory resemble Diram theory? The way that I see it is that in answering this question, Eric derives the three generations of matter."
},
{
"end_time": 7253.797,
"index": 306,
"start_time": 7227.637,
"text": " To explain the three generations, we studied the Dirac-Rarita-Schwinger complex on the full metric bundle Y14, also known as the Observer, so I know I keep saying this over and over. By the way, the way I'm calling it a complex may give you the false impression that this complex exists in the literature, but as far as I can tell, this is a novel construction by Eric himself. Instead of merely considering spinner-valued zero forms, which are just regular spinner fields,"
},
{
"end_time": 7283.916,
"index": 307,
"start_time": 7254.104,
"text": " Eric assembles the combined complex as follows and then he defines a Dirac operator or a Dirac-like operator because it's not exactly Dirac where here the D with the subscript A with the further subscript omega is a gauge covariant exterior derivative involving the full connection of A and then the D with the star is the adjoint constructed with the Hodge star and then this little dot circle guy is our Shiab operator that everyone knows and loves"
},
{
"end_time": 7310.964,
"index": 308,
"start_time": 7284.531,
"text": " It ensures that the quote unquote projection is gauge invariant. I call it a contraction. I've heard Eric repeatedly call it a projection. Also, this operator itself is another novel and central contribution of geometric unity. You can tell that by the presence of the Shiab operator, but I'm just hammering the point home anyhow. What does it mean for this operator to quote unquote mix different spinner valued forms? Okay, let's take a look at the block matrix form."
},
{
"end_time": 7338.814,
"index": 309,
"start_time": 7311.732,
"text": " The off-diagonal operators couple zero forms and one forms. Consequently, any solution, like let's just say there's a psi from the kernel of this operator, it will take the form of these two guys here with one guy coming from a spinner-valued zero form and the other guy coming from a spinner-valued one form. Now, a detailed index theoretic analysis, which uses the Atiya-Singer index theorem adapted to this context,"
},
{
"end_time": 7369.019,
"index": 310,
"start_time": 7339.104,
"text": " shows that the solution space, so the kernel of this, splits into three gauge inequivalent sectors. Note that I do need to be careful as the T.S. Singer index theorem only applies to Euclidean signature, thus more work needs to be done here, which I've glossed over. In simple terms, the first generation comes directly from the zero-form spinors, and I'll just rewrite that here with psi 1. Now the second part is where it's interesting, because the second part is actually a one-form,"
},
{
"end_time": 7396.92,
"index": 311,
"start_time": 7369.394,
"text": " and this decomposes under the action of the Clifford algebra into two components. So you'll see here there's one that's traceless or gamma traceless, then there's another gamma trace part. The gamma traceless piece is gotten to by contracting with the gamma matrices and requiring that this contraction vanish, yielding the second generation, according to Eric. And the complement is the gamma trace part, which produces the third generation."
},
{
"end_time": 7418.882,
"index": 312,
"start_time": 7397.312,
"text": " Also known as the imposter generation by Eric. Those are Eric's words. What you may not know is why he calls it an imposter. So he calls it an imposter in GU because unlike the first two generations in Eric's framework, the third one has different unification properties as one goes up toward U64 comma 64."
},
{
"end_time": 7438.524,
"index": 313,
"start_time": 7419.428,
"text": " Now, I should emphasize that each of these components is isomorphic to the 16-dimensional representation predicted by grand unification schemes. That is, isomorphic at the level of reduction to spin 6 cross spin 4 representations. But how do we know these pieces are exactly 16-dimensional?"
},
{
"end_time": 7458.763,
"index": 314,
"start_time": 7438.712,
"text": " Recall that the internal part of the Chimeric Spinner Bundle here comes from the vertical bundle of metrics, and by representation theory of the Clifford algebra for signature 7,7, you get that the full spinner representation's dimension is 128, and then you just divide by 2 to get the 264 dimensional Vial representations."
},
{
"end_time": 7477.483,
"index": 315,
"start_time": 7459.428,
"text": " Now you have to then pull this back, and when you do, the horizontal, so spacetime, component factors out as the standard 4-dimensional spinner representation of spin 1, 3, leaving out a factor of 4, which you then just divide 64 by to get 16."
},
{
"end_time": 7502.568,
"index": 316,
"start_time": 7478.046,
"text": " Thus, you get the full matter field on spacetime as follows. Now, not only does this yield the correct field content for a single generation in conventional SL10, but you get the other three. You get this splitting of the three generations. Again, this is as far as my reading of Eric's theory goes. Now, why does this Rarita-Dirac-Schwinger complex give three dimensions instead of, let's say, two or four?"
},
{
"end_time": 7522.022,
"index": 317,
"start_time": 7502.841,
"text": " Well, let's just take a precise look at the structure of the differential complex and its interaction with the Clifford algebra. The zero form part here contributes one block. The one form part, through the action of the gamma matrices, splits into a part that vanishes upon contraction, so the gamma traceless part,"
},
{
"end_time": 7537.176,
"index": 318,
"start_time": 7522.21,
"text": " and a complementary gamma trace part. An index theory argument, which I'll leave as a PDF, shows that the net index forces the total kernel to split into three parts, and then you use spectral theory computation to confirm that this has 16 dimensions."
},
{
"end_time": 7562.415,
"index": 319,
"start_time": 7537.722,
"text": " You can show step by step that this Dirac-like operator, when acting on the 0 form plus the 1 form part, is Fredholm, and that its index, which is the difference between the dimensions of its kernel and co-kernel, is topologically determined by the Arouf genus of the observers. Now I should repeat that this Dirac-like operator is novel to GU, so perhaps we should call it the GU fermionic operator."
},
{
"end_time": 7587.756,
"index": 320,
"start_time": 7562.79,
"text": " Now let me just recapitulate this all. The operator structure on the complex here implies that its kernel splits as follows, with each of these guys here individually being 16-dimensional. By the way, to clarify the distinction between operators and complexes in the context of Dirac, Rowida, Schwinger, etc., and these quote-unquote Dirac-Rowida-Schwinger types, formalisms, and so on, let me just go through this one by one."
},
{
"end_time": 7617.108,
"index": 321,
"start_time": 7588.114,
"text": " First, let's take a look at the Dirac operator on n-dimensional spin manifold. Now, this is standard. Nothing here I'm saying is new. We have a spinner bundle S with sections as follows, and the Dirac operator is an endomorphism on this space. And the gammas here are the regular gamma matrices, and that's the spin connection. In even dimensions, S splits into its chirality parts, its chiral parts, and you have this as follows, where it's not exactly an endomorphism. You map instead the left to the right and the right to the left."
},
{
"end_time": 7626.135,
"index": 322,
"start_time": 7617.892,
"text": " Now D alone is just an operator, a single first order map actually, and one often embeds it into a Dirac complex."
},
{
"end_time": 7655.572,
"index": 323,
"start_time": 7626.834,
"text": " So you'll see this on screen here. Complexes just mean that you have many vector spaces and you tie them together by differential maps that eventually terminate to zero. And then there's some other conditions such that you want subsequent compositions to equal zero. And this is what allows people to study the cohomological properties and the index. Now, if none of those words meant anything, it doesn't actually matter. Next is the Rarita-Schwinger operator, R, which deals with spin three half fields."
},
{
"end_time": 7675.981,
"index": 324,
"start_time": 7655.572,
"text": " And these are usually realized as vector spinners. Concretely, you can write it as follows, plus some other possible constraints. These are rarely encountered in standard physics because there isn't any known Rorita-Schwinger fundamental particle. But this too, anyhow, can be placed into a complex, as follows here, where you see this sequence here. And again, you have these consecutive maps that go to zero."
},
{
"end_time": 7703.217,
"index": 325,
"start_time": 7676.391,
"text": " actually if you pause here this should confuse you because the one form spinner part is not spin three halves the reason is that the one form spinner part it's that spinner trace sector that's the ordinary spinners and only the spinner traceless part has the spin three halves and it's that part that gets mapped to the two forms of the spinner bundle this two form of the spinner bundle contains a part that looks like pure spinners so"
},
{
"end_time": 7722.363,
"index": 326,
"start_time": 7703.217,
"text": " the zero forms valued in spinners then it has a piece that looks like a rarita swinger spin three half part however it has another piece that we haven't seen yet and that part looks like pure two forms valued in spinners and these will vanish under any contraction"
},
{
"end_time": 7752.363,
"index": 327,
"start_time": 7722.585,
"text": " in Eric's framework. So this becomes analogous to scalar, traceless Ritchie and vial sectors of the curvature. See now in geometric unity, you actually merge the spin half and the spin three half fields into a single structure. And it's that that gives this Dirac Rarita Schwinger type operator here, where the Zeta is what contains the Rarita Schwinger part. It's just that it also has a trace piece that has ordinary spinners as well. And the new is the Dirac part."
},
{
"end_time": 7776.698,
"index": 328,
"start_time": 7752.961,
"text": " This block matrix operator generalizes both Dirac and Rarita Schwinger simultaneously. Why? Because it mixes the zero-form or the one-form spinner fields. It's neither purely Dirac nor is it purely Rarita. Hence, I've heard Eric call it a Dirac-Rarita-Schwinger-type system. The CKM and PMS Mixing Matrices"
},
{
"end_time": 7807.227,
"index": 329,
"start_time": 7777.824,
"text": " In the standard model, the CKM matrix comes about when you consider flavor mixing of quarks, also known as generation mixing, which happens under the weak interaction. Now in conventional math terms, the CKM matrix is a unitary three by three matrix, and it comes about because the weak interaction eigenstates, so that is the states that participate in the charged weak currents, don't align with the mass eigenstates, that is the states with definite mass."
},
{
"end_time": 7837.039,
"index": 330,
"start_time": 7807.517,
"text": " So I actually talked about this here in this podcast with Lawrence Krauss if you're interested. Link is on screen and in the description. Let's say U, C and T are the weak eigenstates and I'm just putting a subscript of L here because we're dealing with left chirality. Now these are weak eigenstates of quote-unquote up-type and now we have the DSB for the down-type quarks. Then the physical mass eigenstate down quarks are in a linear combination of these via the CKM matrix."
},
{
"end_time": 7865.23,
"index": 331,
"start_time": 7837.261,
"text": " Mathematically, we express it as follows. Where the primed guys here are the mass eigenstates and the DSB unprimed are the weak eigenstates. Now the question is whether this has an interpretation or an explanation or a derivation in geometric unity. Let's proceed by recalling again that GU constructs this unified field associated with the gauge interactions from an inhomogeneous gauge group. And it's on screen here with the variational pi."
},
{
"end_time": 7879.445,
"index": 332,
"start_time": 7865.879,
"text": " This then gets pulled back via the observation and after we perform this pullback, the spinner representation gets decomposed into those three generations that we talked about a few minutes ago. Concretely, we can write it as follows."
},
{
"end_time": 7899.292,
"index": 333,
"start_time": 7879.821,
"text": " Now what about this indicates mixing well because as you move in the internal space determined by the fourteen dimensional construction the observation construction the representation of the unified spinner field isn't fixed in a way that preserves the labels of the individual generations."
},
{
"end_time": 7922.244,
"index": 334,
"start_time": 7899.77,
"text": " So in other words, that internal bundle that we talked about before, which is the spin bundle of the vertical part, has extra substructure. If you change trivializations or perform gauge transformations that act non-trivially on these three families, then the mass term acquires off-diagonal contributions, generating a non-diagonal mass matrix MAB."
},
{
"end_time": 7941.578,
"index": 335,
"start_time": 7922.551,
"text": " In standard physics, you usually allow MAB to be non-diagonal by fiat, meaning that you just impose it. However, from the GU standpoint, this non-diagonally is viewed as a consequence of leftover gauge freedom, the residual transformations that don't block diagonalize the three different families."
},
{
"end_time": 7957.807,
"index": 336,
"start_time": 7942.346,
"text": " So let's say if you write the following, then you have a Yukawa coupling here, which is the capital Phi, and the diagonalization of M A B is gotten to by finding two unitary matrices that separate the uptype and the downtype quark sectors."
},
{
"end_time": 7979.582,
"index": 337,
"start_time": 7958.097,
"text": " The physical misalignment between these two guys here, these two block diagonalizations, that is what gives a unitary matrix the CKM matrix. So geometrically, you can see the gauge freedom in the internal 16 dimensional representation spaces permitting these off diagonal transformations among, say, Psi 1, Psi 2, Psi 3."
},
{
"end_time": 8007.022,
"index": 338,
"start_time": 7980.026,
"text": " The way that I understand it is that suppose you write a gauge transformation as epsilon, like a three by three array of blocks. Let's say epsilon a b. Each block acts between the spaces corresponding to the different generations. In the absence of any of these off diagonal blocks, the three families are unmixed. They do not mix. However, if some of these guys with a not equal to b, so the off diagonal parts are non zero,"
},
{
"end_time": 8020.572,
"index": 339,
"start_time": 8007.329,
"text": " in which the mass matrix appears block diagonal can be altered producing mixing. So we have under a gauge transformation epsilon the following here and similarly for say psi 2 and psi 3."
},
{
"end_time": 8043.746,
"index": 340,
"start_time": 8021.067,
"text": " Because weak interaction eigenstates differ from the mass eigenstates by these transformations, the resulting mixing is codified in the CKM matrix. You can view the CKM matrix as the relative rotation required to diagonalize the up and the down type quark mass matrices so that we have this equation here as the overall residual rotation."
},
{
"end_time": 8064.991,
"index": 341,
"start_time": 8044.07,
"text": " So why would this be unitary? Well, again, the way that I understand this is that because the gauge transformations in geometric unity, as in typical quantum gauge theories, are unitary at the relevant stage, the resulting mixing matrix must itself be unitary, preserving probability. That's the advantage or the main want of unitarity."
},
{
"end_time": 8078.148,
"index": 342,
"start_time": 8065.52,
"text": " In effect, you no longer impose by hand that psi phi psi can be off diagonal in generation space. It's forced by the leftover gauge transformations that don't preserve the decomposition here."
},
{
"end_time": 8106.203,
"index": 343,
"start_time": 8079.241,
"text": " The GU derivation of neutrino mixing is essentially a copy and paste of the quark sector derivation just applied to leptons, particularly the neutrino mass matrix. The only difference is that quarks and leptons reside in different parts of the 16-dimensional multiplets. Now, because of this, the unitary matrix that diagonalizes the quark mass matrix is called the CKM matrix, while the one that diagonalizes the neutrino mass matrix is called the PMNS matrix."
},
{
"end_time": 8130.794,
"index": 344,
"start_time": 8107.517,
"text": " The Dirac equation in geometric unity. Let's begin by recalling the conventional Dirac equation, written as follows, where psi is a spinner field defined on a four-dimensional spacetime. Here, the gamma matrices satisfy the Clifford relation, and that's what ensures compatibility with the spacetime metric. See, there's always an implicit metric used to define spinors."
},
{
"end_time": 8159.138,
"index": 345,
"start_time": 8131.186,
"text": " Next, any other connection A on this bundle can be expressed in the affine space of connections as A0 plus an alpha. Again, this alpha is an adjoint valued one-form, representing fluctuations. In conventional Dirac theory, you usually couple a spinner, psi, to a connection A with the derivative, so the Dirac operator here. However, in GU, the Dirac operator must be defined on the chimeric-spinner bundle."
},
{
"end_time": 8183.609,
"index": 346,
"start_time": 8159.497,
"text": " And it takes the form of this calligraphic D, where the omega, the subscript, has two components, an epsilon and the variational pi. And that represents the overall gauge data, where we have a gauge transformation, which is epsilon, and a gauge potential, which is that variational pi. And precisely, we define or Eric defines this calligraphic D operator as follows."
},
{
"end_time": 8208.985,
"index": 347,
"start_time": 8183.899,
"text": " You can clearly see that this acts on a two-component object. Which one? Well, let's just write it as follows, zeta and nu. The top one, zeta, is a one-form and the bottom one is a zero-form, so just regular spinners. Here, the operator is the shiab, which is designed or engineered to perform that generalization of a contraction that Einstein used when forming the Einstein tensor. But at least this time, it's gauge covariant."
},
{
"end_time": 8221.698,
"index": 348,
"start_time": 8209.821,
"text": " Now notice that D, the exterior covariant derivative, defined relative to the connection A omega, itself is built from that distinguished A zero and a fluctuation, so the variational pi."
},
{
"end_time": 8250.162,
"index": 349,
"start_time": 8222.193,
"text": " Now see this decomposition into zeta and nu. It seems like it's different than the original decomposition into this tensor, the spinner bundle of H-dual. It actually is consistent because the decomposition into zero and one forms come about because of the exterior algebra structure on y and it later plays a role in distinguishing the different generations. Recall just a couple minutes ago, maybe 10 or 20 minutes ago at most, I talked about the calligraphic d's"
},
{
"end_time": 8278.046,
"index": 350,
"start_time": 8250.35,
"text": " kernel decomposition into subspaces of the first and the second and third generation. Now when someone says that fermions are quote-unquote square roots of gauge potential, someone like Eric for instance, this is because the Dirac operator is a linear first-order differential operator whose square yields the Laplacian up to curvature terms and it's exactly like taking the square root of the Klein-Gordon operator which leads to the Dirac operator. This isn't language that I use personally."
},
{
"end_time": 8293.541,
"index": 351,
"start_time": 8278.507,
"text": " In GU, the Dirac operator, so the calligraphic D, plays a dual role. It not only governs the dynamics of the matter fields for a given connection, but it also encodes via the square, the meshing of gravitational and gauge degrees of freedom,"
},
{
"end_time": 8320.896,
"index": 352,
"start_time": 8294.121,
"text": " Let's now derive the gauge covariant form of the Dirac operator in GU step by step. So firstly, we start with the chimeric spinner, which is the decomposition of zeta and nu, the one form and the zero form. Perspectively, you can think of them as a vector-valued spinner versus just a regular spinner. Step two, you introduce that distinguished connection A0 on the principal bundle, and you can get that however you like with an observation or what have you."
},
{
"end_time": 8340.776,
"index": 353,
"start_time": 8321.288,
"text": " Step 3, you define the covariant differential coupled to a connection A omega, acting on differential forms valued in the spinner bundle, and the operator satisfies what you see here, which is just you move it up a form. This obeys the Leibniz rule, and when I say you move it up a form, I mean if you are a P-form, you become a P plus 1 form."
},
{
"end_time": 8361.34,
"index": 354,
"start_time": 8341.493,
"text": " Step 4 is you construct the adjoint with the Hodgstar operator, which is actually defined because you do have a metric now. The Hodgstar requires a metric, or at least it requires a volume form, which usually comes from a metric, which we do have here on Y. Step 5, you introduce the Shiab operator, which is a contraction map."
},
{
"end_time": 8374.428,
"index": 355,
"start_time": 8361.698,
"text": " And it's constructed to mimic the contraction of the Riemann curvature tensor into the Einstein tensor while maintaining gauge covariance. And when I say the SIAP, I mean Eric's SIAP because this is something Eric is introducing."
},
{
"end_time": 8403.268,
"index": 356,
"start_time": 8374.65,
"text": " and it's not a standard term in the physics or math literature. Neither is the following. Step 6, you assemble the Dirac-Rarita-Schwinger operator as follows. Also note that this could be called the GU-Fermionic operator. Step 7, you verify that the composition of the lower and upper blocks satisfy that the square is approximately the following, analogous to the standard property that we have with the regular Dirac operator, Modulo Curvature Corrections."
},
{
"end_time": 8433.2,
"index": 357,
"start_time": 8403.268,
"text": " Step 8, you notice that under gauge transformations of the entire operator that it transforms covariantly. In other words, if you take a capital Psi transformed by an element of the inhomogenous gauge group, calligraphic G, then the calligraphic D, capital Psi, transforms in the same way, preserving the physical content. Step 9, you conclude that the Dirac equation in GU may be written as follows, which covers both the propagation of the fermionic fields"
},
{
"end_time": 8462.585,
"index": 358,
"start_time": 8433.2,
"text": " Many of the ideas I'm about to lay out are inspired by geometric unity. However, I wasn't able to find specific source notes on it, so this is my best attempt to piece things together. It may not align with how Eric sees it. Let's now turn our attention to the Klein-Gordon equation, which is the conventional way that we talk about spin zero fields."
},
{
"end_time": 8484.65,
"index": 359,
"start_time": 8463.114,
"text": " In standard quantum field theory, the Klein-Gordon equation is written as follows, where the square is that double Laplacian, also known as the de Lambertian operator, and phi is the scalar field. This equation is inherently second order in its derivatives, and it comes about from the Euler-Lagrange equation, which is on screen here, and of course we're up to a potential term."
},
{
"end_time": 8514.326,
"index": 360,
"start_time": 8485.316,
"text": " So how does this come about in geometric unity? Well first, remember that we decompose the symmetric 0,2 tensors as follows with a trace component and a traceless one, and the one-dimensional R subspace, so the trace subspace, is going to be the Higgs-like scalar field. Now in GU, the gauge potential A with a subscript omega includes both the usual Yang-Mills vector components and the scalar components coming from the vertical decomposition."
},
{
"end_time": 8540.265,
"index": 361,
"start_time": 8514.701,
"text": " When we form the kinetic term, which is the square of taking the derivative of phi, it comes about from the Yang-Mills-Lagrangian term on screen here after decomposing the curvature into the parts that are quote-unquote purely horizontal and those that are mixed or vertical. More concretely, begin with the gauge curvature here. This is standard. Now note that A decomposes into the following."
},
{
"end_time": 8565.52,
"index": 362,
"start_time": 8540.538,
"text": " Now, the first part represents the conventional gauge fields, like I mentioned, Yang-Mills, and the second part is going to be something like the Higgs component. So, then you expand this expression and you obtain cross terms. Cross terms that are linear with respect to the Higgs, so the variational pi subscript H, and quadratic in A. You'll get terms like the following. That's actually quark tick in the variational pi."
},
{
"end_time": 8585.674,
"index": 363,
"start_time": 8565.998,
"text": " Now this quartic term then plays the role of the Mexican hat potential in the Higgs sector, typically written as follows. Keep in mind that in Eric's framework, the structure and normalization of this potential are not arbitrary. They instead come from these contraction rules. When one projects the curvature, so F A,"
},
{
"end_time": 8614.189,
"index": 364,
"start_time": 8586.067,
"text": " onto the vertical subspace of the chimeric bundle. To be precise, we can define an algebraic operator as follows, which compresses the curvature tensor in a gauge covariant fashion. When applied to the curvature, the operator, P e in this case, yields a symmetric two tensor whose trace reproduces the scalar curvature relevant to gravitational dynamics. And then there's some deviation, and that deviation from the idealized form"
},
{
"end_time": 8634.258,
"index": 365,
"start_time": 8614.445,
"text": " is what encodes the self-interaction of the Higgs. You can vary the action with respectify and it leads to the following equation, which one recognizes as the Klein-Gordon equation except with a nonlinear potential. To say this differently, the same contraction mechanism that transforms the full gauge curvature"
},
{
"end_time": 8661.578,
"index": 366,
"start_time": 8634.565,
"text": " into a form suitable for gravity also when applied to the vertical fluctuations yields the kinetic and the potential via the quartic self-interaction and these terms are characteristic of the Klein-Gordon equation for a scalar field. Again, much of this is based on my own reading of Eric's notes and discussions about this but this is my current understanding so I'll share it with you."
},
{
"end_time": 8690.094,
"index": 367,
"start_time": 8661.971,
"text": " This is about Einstein's field equations. So in standard form, they read as follows, g plus the lambda with the lowercase g equals the kappa times t. It's quite familiar to most of you. Now in geometric unity, you still want to write some Einstein-like tensor, capital G, but defined on the 14-dimensional observer's or the metric bundle. Instead of simply writing the following, Eric introduces a projection operator, which I'm calling P"
},
{
"end_time": 8708.916,
"index": 368,
"start_time": 8694.923,
"text": " Erick adjusts the vertical Frobenius metric via trace reversal."
},
{
"end_time": 8734.138,
"index": 369,
"start_time": 8709.138,
"text": " is treated so as to preserve the covariance. Now the result is a modified Einstein tensor g, satisfying g plus lambda lowercase g equals kappa t, where the t comes about from the matter fields that are on y except when pulled back via an observation. Now something I was confused about is what is the relationship between this p and the Shiab operator."
},
{
"end_time": 8751.561,
"index": 370,
"start_time": 8734.428,
"text": " Well, the Shiab operator is a more general mechanism for contracting curvature in a gauge covariant manner. In GU, one usually denotes the Shiab operator that mixes the torsion type terms with the curvature to project out a Ritchie-like combination. Thus, this P operator"
},
{
"end_time": 8781.476,
"index": 371,
"start_time": 8751.732,
"text": " can be seen as an Einstein-specific restriction of the broader Shiab framework, focusing on producing the usual Ricci tensor minus half the Ricci scalar g structure, except while remaining compatible with the inhomogeneous gauge group. Consequently, the final Einstein equation on y, the larger space, is not, like I mentioned, a naive contraction. Instead, it's from the Shiab operator or a specific instance of it to get this Einstein-type decomposition."
},
{
"end_time": 8797.432,
"index": 372,
"start_time": 8781.903,
"text": " This modified scheme is what recovers a gauge covariant version of G, the Einstein tensor, plus a cosmological term equated to a derived stress energy tensor. The Yang-Mills equation in geometric unity."
},
{
"end_time": 8824.189,
"index": 373,
"start_time": 8798.183,
"text": " Finally, let's examine the Yang-Mills equation in the context of geometric unity, beginning with the classical expression of dF equals J, where F is the curvature of the gauge connection A and we're on a principal G bundle over spacetime in the standard formulation. Now again in standard Yang-Mills theory, we're in four dimensions and the connection A is a Lie algebra G, so a lowercase g algebra valued one form,"
},
{
"end_time": 8847.517,
"index": 374,
"start_time": 8824.445,
"text": " and its curvature is given as we've seen here ad nauseum, where the wedge product incorporates both the exterior derivative and the Lie algebraic structure. The gauge covariant derivative acts on sections of the adjoint bundle as follows. Now under a gauge transformation, the connection and curvature transform like so, which guarantees that the Df transforms homogeneously and makes it already gauge invariant."
},
{
"end_time": 8864.292,
"index": 375,
"start_time": 8848.148,
"text": " In Geometric Unity, however, Eric sees the Yang-Mills as recast, except on the observers, of course, where the gauge fields aren't extensions of some external field, but a component of the Chimeric bundle associated with the space of metrics."
},
{
"end_time": 8882.978,
"index": 376,
"start_time": 8864.292,
"text": " More precisely, the connection on the principal bundle over y is written as follows, where we have, like I mentioned, the distinguished levy-chevita connection coming about from the Zoro construction, and the variational pi, which is just a one-form that is Lie algebra-valued, representing gauge potential fluctuations."
},
{
"end_time": 8909.855,
"index": 377,
"start_time": 8882.978,
"text": " The Yang-Mills curvature in Eric's framework is then given as follows, which is a Lie algebra value 2 form, so there's nothing new here, except that in GU, Eric's trying to solve the problem that the usual contraction of the curvature tensor that appears in Einstein's equations don't straightforwardly commute with gauge transformations as we discussed earlier, and something similar occurs in Yang-Mills theory if one tries to insert the gauge potential directly into the action. To remedy this,"
},
{
"end_time": 8936.886,
"index": 378,
"start_time": 8910.111,
"text": " Eric treats the gauge potential as living in an affine space, which is modeled on the one-forms, and defines its action indirectly via the curvature, which is manifestly gauge covariant. Now, the Yang-Mills action in GU is defined as follows with the inner product defined by the killing form on the Lie algebra and the volume being the volume form on Y, which is induced by the Chimeric metric. This affine space that I mentioned"
},
{
"end_time": 8966.237,
"index": 379,
"start_time": 8937.142,
"text": " The structure guarantees that every term, especially the curvature, transforms correctly under the inhomogeneous gauge group, and I'll just write it here again for reminder. In other words, while the final expression mirrors the familiar Yang-Mills action, its derivation is fully rooted in GU's framework, so Eric ensures gauge covariance and a natural incorporation of the horizontal and vertical degrees of freedom. When you try and vary that action with respect to a sub-omega,"
},
{
"end_time": 8985.742,
"index": 380,
"start_time": 8966.442,
"text": " The Euler-Lagrange equation gives a familiar Yang-Mills equation form, where the D with the star is the adjoint of the covariant derivative, and J will represent the current from derived matter fields. Now there's a subtlety here in that GU with the curvature has to further be processed by the Shiab operator."
},
{
"end_time": 9005.06,
"index": 381,
"start_time": 8986.101,
"text": " For clarity, the Shiap operator is defined as an algebraic contraction map designed to replace what I've called the naive metric contraction present in Einstein's quote-unquote projection, although I see it as a contraction. Concretely, let's say we have a two-form Kazai. The Shiap operator combines other forms,"
},
{
"end_time": 9021.578,
"index": 382,
"start_time": 9005.333,
"text": " phi 1 and phi 2 which are built from the geometric data of y via the construction as follows where the star is the hodge star operator associated with the induced metric on y and the conjugation by epsilon is what ensures gauge covariance."
},
{
"end_time": 9050.282,
"index": 383,
"start_time": 9022.022,
"text": " In effect, this operator is what compresses the two-form curvature into the lower degree structure analogous to a symmetric two tensor in a way that commutes with the residual gauge transformations. This construction is baroque and carefully done. It's necessary because Einstein's usual contraction doesn't commute with gauge transformations, leading to inconsistencies when unifying gravity with gauge theory. It's another reason why Eric calls it the ship in the bottle"
},
{
"end_time": 9074.019,
"index": 384,
"start_time": 9050.52,
"text": " When I was telling him, hey, isn't this just conjugation, which is just something viewed from another perspective? And it's from here that you can see why the ship in the bottle reference is more appropriate because of, like I mentioned, it's quite baroque and carefully constructed. My current understanding is that the effective Yang-Mills equation in GU takes the form as follows. I'm writing it like this."
},
{
"end_time": 9103.882,
"index": 385,
"start_time": 9074.275,
"text": " But also know that the following equations are already gauge invariant by construction, and that's why they don't require the same specialized operator. There will be an accompanying PDF to this, so if you would like more notes, such as expansions on these topics and proofs that I wasn't able to get to in this iceberg, then subscribe to the substag, as that's where I will publish it. And that's it. That's all of physics explained in geometric unity."
},
{
"end_time": 9131.305,
"index": 386,
"start_time": 9104.428,
"text": " You can breathe now. The rest of this iceberg will just be reprises of the derivations and the concepts that you've already encountered. So congratulations. Before we move on to the next and final layer, layer four, which will be summaries and open questions, I want to cover some notes on the simplicity of the equations that we've just outlined that underlie theoretical physics. The Dirac equation, for instance, it's the quote unquote simplest of its kind."
},
{
"end_time": 9134.701,
"index": 387,
"start_time": 9131.698,
"text": " Why? Well, in part, because it generates k-theory."
},
{
"end_time": 9163.319,
"index": 388,
"start_time": 9135.06,
"text": " So, what is K-theory? K-theory is about the differences between vector bundles. So, sure we can add vector bundles, but then the question is, what does it mean for one vector bundle to be subtracted by another? In order to make vector bundles into a complete abelian group, you require a way of talking about their quote-unquote formal differences. This is done by Grothendieck, called the Grothendieck completion of the monoid of isomorphism classes of vector bundles under direct sum."
},
{
"end_time": 9187.756,
"index": 389,
"start_time": 9163.712,
"text": " It's quite a hairy term, but the point is that you have something called the index, which generates the k-theory classification of vector bundles through something called the Atiya-Singer index theorem, which we've talked about, and the Dirac operator is the simplest operator in k-theory. Well, now how about the Yang-Mills equation? Well, it's the simplest geometric equation involving the curvature of a connection."
},
{
"end_time": 9203.763,
"index": 390,
"start_time": 9187.875,
"text": " What about the Einstein field equation? Well, it's simple in the sense that it's derived from the simplest Lagrangian possible. It's built not even from the curvature tensor, but just the scalar curvature. And also the Klein-Gordon is seen as one of the simplest equations, if not the simplest of its class."
},
{
"end_time": 9226.186,
"index": 391,
"start_time": 9204.019,
"text": " Paradoxically though, the Klein-Gordon equation, it appears to be the most geometric one with its metric hat potential, is actually the least intrinsically geometric of the four equations that I've just mentioned. Differential geometers often overlook the Higgs sector. Note, what I mean is that the Mexican hat is easily seen to be visually geometric, but not easily seen to be differential geometric."
},
{
"end_time": 9256.288,
"index": 392,
"start_time": 9226.544,
"text": " However, to Eric, the Higgs sector comes about as a vertical component of a connection form. So in GU, when decomposing the gauge potential, the Higgs field appears naturally, and the Yukawa coupling is just minimal coupling. Now conventional field theories introduce various vector bundles that are seemingly disconnected structures from the derived space-time geometry. In general relativity, symmetric zero-two tensors decompose, like we mentioned, into the trace and the traceless component, with the trace component"
},
{
"end_time": 9276.715,
"index": 393,
"start_time": 9256.63,
"text": " Carrying an eric's formulation to spin zero so eric's innovation is that he lifts the frame bundle to the double cover with the gl double cover also known as the metal linear group and this structure may be new to most of you even though you're familiar with physics and it's because mathematicians tend to avoid the double cover as there's no"
},
{
"end_time": 9297.602,
"index": 394,
"start_time": 9276.988,
"text": " Finite dimensional representation that can do the job of representing spinners. A fundamental group's obstruction exists because the gl4,r is real. Now, GU overcomes this by quickly passing to the spin subgroup and moving to the observer's rather than working on the base x4 directly."
},
{
"end_time": 9310.52,
"index": 395,
"start_time": 9297.91,
"text": " This then is topological rather than metric dependent. Now the connection between these spaces allow the vertical components of the connection to appear as the Higgs field when pulled back to space-time."
},
{
"end_time": 9328.012,
"index": 396,
"start_time": 9310.845,
"text": " Eric constructs these chimeric versions of spinner representations in indefinite signature space with the spin 7,7 that we mentioned before. And then you reduce it through its maximal compact reduction in one of the paths that I mentioned. And that's how he unifies previously disconnected geometric structures."
},
{
"end_time": 9352.517,
"index": 397,
"start_time": 9328.387,
"text": " Just so you know, if you have any questions, which you likely do, that's all right. I'm going to have a large solo podcast with Eric just on geometric unity. It will be unlike any other podcast because we'll delve into the particularities of it, especially now that they've been explored in a fair amount of detail. Therefore, feel free to subscribe to be notified of this upcoming podcast with Eric Weinstein. Layer four."
},
{
"end_time": 9381.34,
"index": 398,
"start_time": 9353.899,
"text": " Finally, we're in the deepest layer, which is actually the most accessible now that you've had the previous two or three hours of background. Man, you don't see this, but this iceberg took hundreds of hours of work across 10 months on and off. I hope you enjoy it. The feeling that I get from Geometric Unity, I figured out how to make it into an analogy, is that firstly, if you take a look at the standard model, it resembles a jigsaw puzzle."
},
{
"end_time": 9402.312,
"index": 399,
"start_time": 9381.664,
"text": " It's broke, the edges aren't clean, there are little protrusions. It's unclear where these pieces came from or what the source of the jagged edges, the spikiness, the messiness is. What is clear is that it fits together and works somehow, except there's this other piece beside it, which is a pristine disc, also known as gravity."
},
{
"end_time": 9413.302,
"index": 400,
"start_time": 9402.637,
"text": " And it's unclear how to combine these, they look like they're different elements. One is like that colorful, baroque object that I talked about, the jigsaw, and then the other is the gravity, the nice porcelain disc."
},
{
"end_time": 9439.275,
"index": 401,
"start_time": 9413.763,
"text": " What geometric unity looks or feels like to me is if you start with this disk, generalize it, you get, say, a sphere, a perfect pearl. Then, if you allow this pearl to drop on the floor, it will shatter into different pieces, but what's remarkable is that some of these pieces exactly outline the aforementioned messy jigsaw puzzle."
},
{
"end_time": 9460.981,
"index": 402,
"start_time": 9439.275,
"text": " Then you wonder, what are the odds? Look, I was trying to combine these two different pieces, the jigsaw puzzle and this disc, and I couldn't make it work. But actually, they're not just meant to naively be combined. Instead, they all fall out of the same structure. And you don't need to do any work to get it to do so. You just let it drop on the floor and examine the pieces."
},
{
"end_time": 9486.459,
"index": 403,
"start_time": 9461.271,
"text": " Well, that's the feeling of geometric unity, at least to me. Now enough about feelings. Let's get to this layer. Right now, I'd like to give you an explanation at four different levels. So one is the explain like I'm five level, even though I have my issues with that. And you can read this sub stack here, where I go into detail about how misleading and foolish this enterprise of hey, if you can't explain it to a five year old, you don't understand it is."
},
{
"end_time": 9502.824,
"index": 404,
"start_time": 9486.852,
"text": " after the ELI 5 you'll get the ELI U so that is explain it like I'm an undergrad and then after it's explained at the undergrad level I'll explain it at the graduate level and then I'll explain it at the PhD level then we're going to get to open questions and then there's one more treat"
},
{
"end_time": 9532.824,
"index": 405,
"start_time": 9503.285,
"text": " Hi, everyone. Hope you're enjoying today's episode. If you're hungry for deeper dives into physics, AI, consciousness, philosophy, along with my personal reflections, you'll find it all on my sub stack. Subscribers get first access to new episodes, new posts as well, behind the scenes insights and the chance to be a part of a thriving community of like minded pilgrimers. By joining, you'll directly be supporting my work and helping keep these conversations at the cutting edge. So click the link on screen here."
},
{
"end_time": 9548.695,
"index": 406,
"start_time": 9532.824,
"text": " Explaining geometric unity to a five-year-old"
},
{
"end_time": 9571.459,
"index": 407,
"start_time": 9550.247,
"text": " In physics today, we have these two primary theories that don't get along well. One is about gravity, so general relativity, and one is also about particles, so it's the standard model. The universe seems governed by particles, but it also seems to be governed by gravity, since you're made up of both particles and you stick to the ground, and you orbit the sun, etc."
},
{
"end_time": 9588.285,
"index": 408,
"start_time": 9571.459,
"text": " However both of these theories even though they describe the universe they don't combine well together and the prefix of universe is unique which means one so is there a unification that combines these that's what geometric unity attempts the key insight from eric weinstein."
},
{
"end_time": 9608.882,
"index": 409,
"start_time": 9588.336,
"text": " is to take a look at this 4D space time and instead of putting a metric on it, so it's actually not even a space time, it's just a 4D space, you think of what are all the possible Einstein theories that can be placed on this? Mathematicians sometimes call this a moduli space, but technically in this case, it's a 14 dimensional manifold."
},
{
"end_time": 9630.572,
"index": 410,
"start_time": 9608.882,
"text": " What Eric calls the observers now in this higher dimensional space forces and even the three generations of matter that we observe in the standard model they aren't added in by hand instead there in gendered from the geometry of this fourteen dimensional space itself. Explain geometric unity like i'm an undergrad in math or physics."
},
{
"end_time": 9646.152,
"index": 411,
"start_time": 9631.323,
"text": " Start with a four manifold. You then think about how do I make it geometric since currently it's topological. Now geometric in this case means metrical, so adding a metric. However, you want to be general. You want to think about all metrics."
},
{
"end_time": 9667.807,
"index": 412,
"start_time": 9646.425,
"text": " One way of thinking about this is by attaching all possible metrics at a single point and that is technically called the metric bundle. You then think about if this metric bundle itself carries a metric that would be meta and it turns out it doesn't but there's a way that you can metricize part of it."
},
{
"end_time": 9684.77,
"index": 413,
"start_time": 9667.807,
"text": " namely the vertical parts of the tangent space. You do so with a certain type of metric called a Frobenius metric. The name doesn't matter. It's actually a somewhat natural metric and there are two choices here. There's a regular Frobenius metric and then there's something called a"
},
{
"end_time": 9704.497,
"index": 414,
"start_time": 9684.77,
"text": " Trace reversed one or just the reversed one doesn't matter they're both choices the reversed one is preferred for various reasons from there you can then think about what is the signature because you have a metric now and now that you have the signature you can think about the spin group of the signature so spin seven comma seven."
},
{
"end_time": 9727.568,
"index": 415,
"start_time": 9704.735,
"text": " And now that you have a spin group, you can think about what does this spin group act on, and it turns out one of the spaces it can act on is a real dimensional space except of 128 dimensions, so R to the 128. Now since chiral fermions are complex chiral, and since real vector spaces can always be complexified, Eric complexifies here."
},
{
"end_time": 9756.561,
"index": 416,
"start_time": 9727.841,
"text": " and that also splits into two copies of C32, 32. Note, I'm not saying anything special like a Newlander, Nuremberg integrability condition or that there's the vanishing of a certain nine house tensor. We're not dealing with complex manifolds. I'm just making the observation that anytime you have copies of R, you can complexify it. That ability is there in the same way that when you have a manifold, you get with it a tangent space."
},
{
"end_time": 9774.036,
"index": 417,
"start_time": 9756.561,
"text": " You don't have to provide anything special to the manifold, it just comes with a tangent space. We also know that a manifold has a cotangent space, just like I said it has a tangent space. However, there is not an isomorphism between the tangent and the cotangent without a metric."
},
{
"end_time": 9791.647,
"index": 418,
"start_time": 9774.036,
"text": " Or without additional structure like a symplectic form a metric in this case for the base manifold would be a choice of a section of the metric bundle you can think about why that is and you'll be able to convince yourself that is an equivalent notion so let's assume that you make that selection of a section."
},
{
"end_time": 9821.186,
"index": 419,
"start_time": 9792.09,
"text": " Now although we won't fix what type of section, we'll just say that you choose some section. At that point, you can make an isomorphism between the tangent and the cotangent. And from there, given that your tangent space of the full metric bundle will split into vertical and horizontal parts, you can think of the dual of the horizontal now. I should say at this point that there's plenty of twists and turns, like there's so much to remember, but the point is that you start from something simple and you look at what structure is contained within."
},
{
"end_time": 9835.828,
"index": 420,
"start_time": 9821.527,
"text": " In many ways you can think of the claim of geometric unity as the claim of what we think of as simple actually has extreme complexity inside it and furthermore a subset of that complexity matches the standard model almost verbatim."
},
{
"end_time": 9861.118,
"index": 421,
"start_time": 9837.005,
"text": " okay so now we keep moving around in our little space of complexity and we use a result from differential geometry about spinner bundles which is that if you have a direct sum b has bundles and you take the spinner bundle of that it's the same as the spinner bundle of a tensor the spinner bundle of b you can use this along with the isomorphism provided before to form spinners and now you wonder well"
},
{
"end_time": 9878.148,
"index": 422,
"start_time": 9861.118,
"text": " Okay these spinners are built on the metric bundle and we live in a base space or at least we supposedly do so what do these spinners look like from the perspective of the base space now the question about perspective from the base space is the same as a pullback operation."
},
{
"end_time": 9895.538,
"index": 423,
"start_time": 9878.148,
"text": " In this instance so that's what we do when we're allowed this pullback operation because we've already chosen a section of the bundle once this is pulled back we then get what matches the standard model spinners as for the gauge group this one you can see because there's a reduction from spin seven comma seven."
},
{
"end_time": 9910.93,
"index": 424,
"start_time": 9895.913,
"text": " down to spin 1 comma 3 cross spin 6 comma 4, and from there, after moving to the maximal compact subgroup, the second factor goes down to spin 6 cross spin 4, which is Pati Salam's grand unified theory."
},
{
"end_time": 9932.534,
"index": 425,
"start_time": 9910.93,
"text": " In this way, the quantum numbers and the standard model gauge group actually have different origins and the quantum numbers are fixed while the gauge group could have been different or been broken down differently, theoretically, at least in geometric unity. Now as for the Higgs and other parts of the standard model, those can be seen as different connections on the full metric bundle, but also pulled back."
},
{
"end_time": 9949.019,
"index": 426,
"start_time": 9932.534,
"text": " In some ways you can think of this as wondering about how we have these two different theories one that describes particles and one that describes gravity gravity is like a simple dove in that it's beautiful and innocent and the standard model is like this torturous snake."
},
{
"end_time": 9977.483,
"index": 427,
"start_time": 9949.445,
"text": " in that the standard model is effective but it's convoluted. People have been trying to put these together for decades and what Eric has found is that if you look at gravity itself and generalize it, the particles come out. You can visualize the four manifold as a chiapet which grows fibers naturally and that living on the blades of grass are the different particles we're looking for. Be as innocent as doves and as wise as serpents."
},
{
"end_time": 9981.237,
"index": 428,
"start_time": 9978.285,
"text": " Explaining Geometric Unity to a Graduate Student"
},
{
"end_time": 10011.63,
"index": 429,
"start_time": 9982.261,
"text": " Firstly, we have to think about what's the goal. So to explain to a graduate student, I'm going to use Eric's frame of mind. What's the goal of Eric? Eric is thinking, how do I resolve the chicken and egg problem of quantum gravity? That is, how can matter, which is described by spinors, exist between metric measurements if you require a metric in order to define the spinors? The answer, according to geometric unity, is that matter lives in the observer's in this 14 dimensional metric bundle."
},
{
"end_time": 10032.142,
"index": 430,
"start_time": 10011.63,
"text": " and spinners can exist there even though you're not making a canonical choice of space-time metric. So you start with a four manifold x four, GU then constructs a metric bundle which is on screen and at each point you have the decomposition in the tangent space as follows with the vertical space representing the metric variations."
},
{
"end_time": 10062.295,
"index": 431,
"start_time": 10032.295,
"text": " Eric then uses a certain construction, where if you choose a section of this metric bundle, it then gives you a connection, the Levi-Chevita connection. Now that metric on y induces a connection itself. And this is what a choice of a horizontal subspace means, a connection. Now the Chimeric bundle combines the vertical space, which has the signature 4,6, with the dual of the horizontal space, so the signature 1,3, and that gives the total signature of 7,7."
},
{
"end_time": 10085.811,
"index": 432,
"start_time": 10062.295,
"text": " I should note once more that spinners are derived on this chimeric bundle, which has a metric without requiring a connection. Now because this chimeric bundle is isomorphic to both tangent and cotangent bundles of Y, you get spinner bundles to exist prior to choosing a metric on the base space X, or even a connection on Y. But once you do select a connection, you become canonically isomorphic."
},
{
"end_time": 10103.353,
"index": 433,
"start_time": 10085.811,
"text": " This is that Zorro construction and it's a constitutional mechanism that powers geometric unity. The spinner bundle on the space then decomposes as follows. There is something else called the augmented torsion tensor and that is supposed to resolve. Again, we have to think what is Eric thinking is the problem? What is the goal?"
},
{
"end_time": 10128.882,
"index": 434,
"start_time": 10103.353,
"text": " Well, Eric's thinking I'm looking for a map from the inhomogeneous gauge group to the space of connections, which is equivariant under this right action and equivariant on the left hand side as well. This is what becomes dark energy. Eric sees another problem which he calls the gauge incompatibility problem. What that is is that in Einstein's formulation, the Riemann curvature tensor is viewed as a two-form taking values in a two-form."
},
{
"end_time": 10152.449,
"index": 435,
"start_time": 10129.292,
"text": " So essentially you have two copies of a two-form, one from the connection and then one from the metric structure. When you contract this tensor to produce the Einstein tensor, you're treating both copies symmetrically. However, that makes it not transform properly under gauge transformations. This mismatch in the origins of the curvature components is what Weinstein, Eric Weinstein, refers to as the twin origins problem."
},
{
"end_time": 10171.561,
"index": 436,
"start_time": 10152.722,
"text": " So this augmented torsion tensor is one way of solving the problem with gravity that is not gauge covariant, but gauge theory is gauge covariant. So how do you make gravity gauge covariant? Eric then introduces a Shiab operator, which generalizes that contraction I referenced about Einstein in a gauge covariant manner."
},
{
"end_time": 10187.5,
"index": 437,
"start_time": 10171.903,
"text": " The formula is on screen here. And then there's a Dirac-Rarita-Schwinger complex on this larger space. This generalizes the Diram complex, twisted by spinors in dimension three. Now this complex gives three distinct sets of fermions."
},
{
"end_time": 10217.654,
"index": 438,
"start_time": 10187.739,
"text": " The scalar spinner, which gives the first generation, and then there's a vector spinner, which splits into two parts, a gamma trace part, which gives the second generation, and a gamma traceless part, which gives the third. Each of these is 16-dimensional, and that matches the standard model's Fermion structure. In particular, it matches spin 6, 4, which is an alternative real form of spin 10 complexified, and that's closely related to the well-known SO10 real form from grand unification theory."
},
{
"end_time": 10237.619,
"index": 439,
"start_time": 10218.609,
"text": " Hi, Kurt here. If you're enjoying this conversation, please take a second to like and to share this video with someone who may appreciate it. It actually makes a difference in getting these ideas out there. Subscribe, of course. Thank you."
},
{
"end_time": 10254.599,
"index": 440,
"start_time": 10239.224,
"text": " How about rather than quantizing gravity directly on X4, which gives renormalization problems, how about we work with fields on Y14, that metric bundle. Then from there, we're going to pull it back to X4 via sections of the metric bundle."
},
{
"end_time": 10276.51,
"index": 441,
"start_time": 10254.599,
"text": " Geometric Unity's foundation is in constructing principal bundles over this Y14, so principal bundles on principal bundles, this time with groups spin1,3 and spin7,7 and u64,64 as the structure groups, and we have a chain of inclusions on screen here. There's also something called the inhomogeneous gauge group, which is on screen here."
},
{
"end_time": 10305.691,
"index": 442,
"start_time": 10276.988,
"text": " where the first factor is a genuine gauge transformation and the second is actually just gauge potentials. Now this inhomogeneous gauge group calligraphic G combines the gauge transformations with the translation quote unquote in the space of connections with a certain product structure given on screen as follows. Now let's say for any connection and any element inside this inhomogeneous gauge group we also have a right action that right action"
},
{
"end_time": 10334.94,
"index": 443,
"start_time": 10306.067,
"text": " is defined as follows. The augmented torsion tensor, that is what gives a gauge covariant object that transforms correctly under both gauge and diffeomorphism symmetries, resolving that gauge incompatibility problem of the Riemann tensor. The Shiab operator generalizes Einstein's contraction in a gauge covariant manner, and I'll put the formula on screen here again. The action principle, because we all want to know what is the action, what's the Lagrangian,"
},
{
"end_time": 10353.37,
"index": 444,
"start_time": 10335.128,
"text": " That takes the form of a first order equation reminiscent or it rhymes with Einstein-Hilbert and even Chern-Simons theories and that's on screen here. The field equations are derived from this action. The term involving the torsion T is what ensures the Euler-Lagrange current remains exact."
},
{
"end_time": 10374.753,
"index": 445,
"start_time": 10353.609,
"text": " This exactness property is what allows Eric the geometric formulation of the field equations. There's also a quadratic term, and that maintains the gauge covariance. The Dirac-Rarita-Schwinger operator takes the form on screen."
},
{
"end_time": 10392.398,
"index": 446,
"start_time": 10375.111,
"text": " and it acts on regular spinners plus spinner-valued spinners. This operator on this space here comes from Eric thinking differently about supersymmetry. There is something pioneered in the late 70s by Salam and Strotti about constructing superfields, which are just fields defined on superspace."
},
{
"end_time": 10416.92,
"index": 447,
"start_time": 10392.398,
"text": " This construction, in Eric's mind, was erroneously applied to Minkowski spacetime, rather than the space of connections. To Eric, this explains why there's been so much time and money and effort wasted on finding superpartner particles on spacetime, because superpartners exist in connection space. The decomposition into three pieces occurs prior to the kernel operator, each 16-dimensional."
},
{
"end_time": 10435.759,
"index": 448,
"start_time": 10416.92,
"text": " And now the first generation is the regular spinner-valued function, so the zero forms. And the second and third come from the decomposition of the one forms on the spinner field. And those break up into gamma trace and gamma trace list parts. The Higgs field comes from the vertical component of the gauge potential, so the variational pi."
},
{
"end_time": 10461.578,
"index": 449,
"start_time": 10436.203,
"text": " And when you pull it back to your base space, you get three parts. One that looks like a gauge potential downstairs, so that is to say an ad-valued one-form, interpretable as such on X4, the base space. Additionally, you get a spin-zero piece downstairs, which gets interpreted as the Higgs field in Eric's framework. And you get a remainder piece that has not been identified or not seen and doesn't figure into our experimental observations in the physical universe so far."
},
{
"end_time": 10487.295,
"index": 450,
"start_time": 10461.971,
"text": " Just remember, there either is right now or is going to be an accompanying PDF to this Geometric Unity iceberg, so if you'd like more notes such as expansions on these topics and proofs that I wasn't able to get to in this iceberg, then subscribe to my sub-stack as that's where I'll publish it. It's curtjamungle.org or C-U-R-T-J-A-I-M-U-N-G-A-L.org. Open Questions"
},
{
"end_time": 10518.183,
"index": 451,
"start_time": 10489.104,
"text": " Soon I'm going to do a full recap of the whole 30 steps of geometric unity except in a flyby overview just to hammer the point home further and show you that all of this initially abstruse math is now understandable by you. It's actually in part to congratulate you. At first you heard this foreign language and now you're able to be somewhat conversant in it. You can order dinner. You can go back to your hotel. You can pick up a date. I'm taken by the way, but"
},
{
"end_time": 10523.37,
"index": 452,
"start_time": 10519.019,
"text": " Thank you. Now first, let me talk about what open questions I have."
},
{
"end_time": 10551.34,
"index": 453,
"start_time": 10524.121,
"text": " One of my questions is, what is the phenomenology of the theoretically predicted but currently unobserved decoupled sectors? I put them in quotations because I've heard Eric call these dark sectors, but I find that terminology to be too evocative of dark matter or dark energy, so let's just call them decoupled sectors. Would gravitational interactions at high energies enable us to have direct or indirect observations of these? How?"
},
{
"end_time": 10575.828,
"index": 454,
"start_time": 10551.34,
"text": " Okay that's one question. My next is about how does geometric unity account for the matter anti-matter symmetry slash asymmetry. Is this observed asymmetry explained by disconnected chiral sectors where the anti-matter is hidden so it's a decoupled chiral sector or is the asymmetry still an initial condition? And another question I have is given that Eric"
},
{
"end_time": 10601.749,
"index": 455,
"start_time": 10576.101,
"text": " has squeezed plenty out of these numerical coincidences, particularly about spinner structures like SL10 and SU5, their connections to the Einstein field equations, the metric in four dimensions. I believe Wilczek actually remarked about this. So given that, and there's also the coincidences of the observed generations of fermions being precisely 16 fields, etc., how far do these numerical coincidences go?"
},
{
"end_time": 10623.558,
"index": 456,
"start_time": 10602.295,
"text": " What other numerical coincidences are meaningful? So what about Dirac's large number hypothesis? What is a red herring versus a smoking gun? Also, you should know that I'm going to be speaking with Eric directly on the podcast in the next couple of weeks. So if you have questions, leave them below and feel free to subscribe for that conversation as we'll go further in depth into geometric unity."
},
{
"end_time": 10653.08,
"index": 457,
"start_time": 10624.599,
"text": " At this point, I'd like to emphasize that I don't want you or other people to think that there's something negative given that they're open questions, as every single theory has open questions. For instance, here are some of my notes about open questions in string theory, in loop quantum gravity, and in asymptotic safety. Eric's theory is a tour de force, and unless you have an understanding of physics, it's difficult to fully appreciate how many pieces there are in this one theory."
},
{
"end_time": 10672.056,
"index": 458,
"start_time": 10653.541,
"text": " Despite the open questions that I mentioned, which this theory has been generated by a single person in isolation, now believe it or not, what I've shown you for the past three hours or so still leaves maybe 30 to 40% of GU unexplored."
},
{
"end_time": 10694.275,
"index": 459,
"start_time": 10672.517,
"text": " Just so you know, if you have any questions, which you likely do, that's all right. I'm going to have a large solo podcast with Eric just on Geometric Unity. It will be unlike any other podcast because we'll delve into the particularities of it, especially now that they've been explored in a fair amount of detail. Therefore, feel free to subscribe to be notified of this upcoming podcast with Eric Weinstein."
},
{
"end_time": 10722.978,
"index": 460,
"start_time": 10694.275,
"text": " Now, let me give a recap of the 30 steps of geometric unity for the interested viewer who cares about the steps, and this is a treat because you'll be surprised how much of this finally makes sense. Maybe not all of it, but initially maybe 5% made sense. So number one, you begin with a smooth four manifold, say X4. You then construct its metric bundle Y14, where each fiber is the symmetric two tensors on X4, and that happens to be 14-dimensional."
},
{
"end_time": 10739.275,
"index": 461,
"start_time": 10722.978,
"text": " Number two, you construct the frame bundle with the structure group GL4, then you lift to its double cover to enable finite dimensional spinner representations. Next, number three, you define observation maps, which are actually sections or local sections of this bundle."
},
{
"end_time": 10756.323,
"index": 462,
"start_time": 10739.599,
"text": " Next, you construct the tangent bundle of the metric bundle, and you also construct its dual, so its cotangent bundle. Next, you split the tangent into its vertical bundle, which encodes the metric variations, and the horizontal part, which is just directions along the base space."
},
{
"end_time": 10785.64,
"index": 463,
"start_time": 10756.766,
"text": " Next, you use a certain construction which Eric happens to call the Zorro construction. It's a way of getting a section from the base space to the metric bundle. So you choose a section, you're choosing a different slice of this whole metric bundle. How do you get from there to a connection on the metric bundle? That's what this construction is about. Next, you define the chimeric bundle, which is the vertical space direct sum the dual of the horizontal. Next, step eight, you introduce the Frobenius inner product."
},
{
"end_time": 10813.626,
"index": 464,
"start_time": 10785.896,
"text": " and that acts on symmetric matrices to metrize the vertical fibers. Next, number nine, you decompose the symmetric zero two tensors into a trace and traceless part, and you choose a signature. This is one of the only parts in geometric unity where you make an actual choice that wasn't there implicit in the structure. In this case, you choose four comma six, and that's for the vertical part, and you choose one comma three for the H part, the horizontal part."
},
{
"end_time": 10833.268,
"index": 465,
"start_time": 10813.626,
"text": " Number 10, you construct spinner bundles, but you do so on the chimeric bundle and you use that exponential property as follows on screen. Next, you identify the structure group of spin 7,7 who has a real spinner representation of"
},
{
"end_time": 10860.657,
"index": 466,
"start_time": 10833.268,
"text": " two to the power of seven so 128 dimensional and that splits because you can divide that into two into two chiral 64 dimensional real spaces but you have to complexify now number 12 from the principal bundle with the structure group of U of the unitary matrices 64 comma 64 via the spinner representation here then you define an inhomogeneous gauge group where the first part are actual bona fide gauge transformations"
},
{
"end_time": 10870.589,
"index": 467,
"start_time": 10860.657,
"text": " And the second are just gauge potentials. You just semi-direct product them together. Number 14, you specify a right action on the affine space of connections."
},
{
"end_time": 10896.954,
"index": 468,
"start_time": 10870.947,
"text": " And you do so with the following somewhat hairy formula, but not too hairy. And this formula has group associativity to it. Number 15, you define the augmented torsion tensor. This is the gauge-equivariant analog of Einstein's contraction operator, and this addresses Einstein's quote-unquote gauge problem, or more accurately, it addresses Weinstein's rendition of Einstein's gauge problem. Number 16, you introduce the Schiap operator. That acts on two forms."
},
{
"end_time": 10918.558,
"index": 469,
"start_time": 10896.954,
"text": " And it's defined on screen here that's the generalization of Einstein's contraction number seventeen you formulate a first order action so this is the action of geometric unity as far as i can tell now eighteen you vary the action to derive field equations and this is what gives you something that's analogous to Einstein's field equations."
},
{
"end_time": 10941.152,
"index": 470,
"start_time": 10918.558,
"text": " And also Yang Mills. Number 19, you introduce Fermion fields as a spinner-valued form. So there's firstly a zero form, which is just a regular spin field, and then there's a spinner one form here. And that gives a Dirac-Rarita-Schwinger operator coupling both sonic and fermionic sectors. And that's why Eric sometimes refers to this as supersymmetry."
},
{
"end_time": 10971.305,
"index": 471,
"start_time": 10941.34,
"text": " I'm not a fan of that term because supersymmetry means something in particular, but Eric is taking the analogy here of interchanging fermionic and bosonic degrees of freedom as supersymmetry instead of the supersymmetry algebra. Number 20, construct the deformation complex here. This has a cohomology which classifies gauge-inequivalent linearized fluctuations about a solution. Now number 21, you see that there's a seesaw structure inside this Dirac-Rarita-Schwinger complex."
},
{
"end_time": 11001.101,
"index": 472,
"start_time": 10971.852,
"text": " And that's what explains mass hierarchies, mixing between light and heavy spinners. Number 22, you can do a reduction. So you go from spin 7 comma 7 to that spin 1 comma 3, and you cross that with spin 6 comma 4. You can do some further reductions, as I've described earlier on, and that recovers the standard model gauge group. Number 23, you decompose the diracuarita-shringer complex, and again, like I said, there's a zero form and there's a one form,"
},
{
"end_time": 11015.009,
"index": 473,
"start_time": 11001.169,
"text": " The one form splits into two different parts, a gamma traceless part and a gamma trace part. One of those is the second generation and the other one is the third generation, whereas the zero form is just the first generation."
},
{
"end_time": 11039.616,
"index": 474,
"start_time": 11015.179,
"text": " Number 24. There's a Higgs field. Where is this Higgs field? Well, when you take the variational pi, the one form on the principal bundle, it has a scalar part and that corresponds to the Higgs field upon pullback. Number 25 is that the symmetric 0-2 tensors themselves, which have a trace and a traceless component, there's a trace component here which gets identified with the Higgs."
},
{
"end_time": 11058.831,
"index": 475,
"start_time": 11039.616,
"text": " and the traceless part becomes identified with a spin-to-graviton-like field. Number 26, you can derive a quartic potential when you do some expansions, and that was covered earlier, from the Yang-Mills term A wedge A squared. And that actually gets you a Mexican hat potential for the Higgs field."
},
{
"end_time": 11073.848,
"index": 476,
"start_time": 11059.155,
"text": " Number 27, you show that there's a minimal coupling in the Dirac operator, so on screen here, and that gives you Yukawa interactions when the A, the connection here, includes Higgs components. Number 28,"
},
{
"end_time": 11102.005,
"index": 477,
"start_time": 11074.189,
"text": " There's a correspondence between the Higgs and the Yang-Mills sector because you decompose this variational pi and that gives you this unified origin of the Higgs and Yang-Mills in this geometric structure of the metric bundle. This isn't even a step, it's more like a realization. Number 29, you incorporate a quadratic term into the field equations and this gives full gauge covariance analogous to Yang-Mills self-interactions. And finally 30,"
},
{
"end_time": 11118.848,
"index": 478,
"start_time": 11102.329,
"text": " You can decompose that gauge potential like we talked about under the pullback, and its constant component produces an effective term. This then gets identified with the cosmological constant. And that's all of fundamental physics."
},
{
"end_time": 11143.592,
"index": 479,
"start_time": 11120.316,
"text": " Alright, thank you for coming along with me on this journey through fundamental physics and its potential unification. Thank you to Eric Weinstein for providing this theory and giving me plenty to chew on over the past few months. Thank you to all the video editors who helped edit this together, even though all of this probably look like and sound like hieroglyphics to you, or at least look like hieroglyphics don't sound like anything."
},
{
"end_time": 11167.568,
"index": 480,
"start_time": 11143.592,
"text": " If you're interested in more icebergs like this, you should know I have a string theory iceberg where I outline the math at the graduate level for string theory. I also have one where I cover different theories of consciousness. Soon I'll be doing interpretations of quantum mechanics as well as an iceberg on algebraic geometry, so feel free to subscribe. Again, thank you to Eric Weinstein. It's an avant-garde and creative theory."
},
{
"end_time": 11194.241,
"index": 481,
"start_time": 11168.2,
"text": " Kurt here, several months later. This has been so long in the making, geez, you have no idea. Anyhow, I wanted to say that I mean what I just said. I may have said this before in the iceberg and if I haven't, I should have because it bears repeating, I haven't seen a theory like this come from any single individual ever. Not one that's this fleshed out or has this amount of unexampled connections within itself as well as to what's known as the"
},
{
"end_time": 11220.503,
"index": 482,
"start_time": 11194.241,
"text": " I've received several messages, emails, and comments from professors saying that they recommend theories of everything to their students and that's fantastic."
},
{
"end_time": 11238.08,
"index": 483,
"start_time": 11220.759,
"text": " If you're a professor or a lecturer and there's a particular standout episode that your students can benefit from, please do share. And as always, feel free to contact me. New update. Started a sub stack. Writings on there are currently about language and ill-defined concepts as well as some other mathematical details."
},
{
"end_time": 11266.425,
"index": 484,
"start_time": 11238.268,
"text": " Much more being written there. This is content that isn't anywhere else. It's not on theories of everything. It's not on Patreon. Also, full transcripts will be placed there at some point in the future. Several people ask me, hey, Kurt, you've spoken to so many people in the fields of theoretical physics, philosophy and consciousness. What are your thoughts? While I remain impartial in interviews, this substack is a way to peer into my present deliberations on these topics. Also,"
},
{
"end_time": 11269.121,
"index": 485,
"start_time": 11266.63,
"text": " Thank you to our partner, The Economist."
},
{
"end_time": 11298.78,
"index": 486,
"start_time": 11271.391,
"text": " Firstly, thank you for watching, thank you for listening. If you haven't subscribed or clicked that like button, now is the time to do so. Why? Because each subscribe, each like helps YouTube push this content to more people like yourself, plus it helps out Kurt directly, aka me. I also found out last year that external links count plenty toward the algorithm, which means that whenever you share on Twitter, say on Facebook or even on Reddit, etc,"
},
{
"end_time": 11319.787,
"index": 487,
"start_time": 11298.78,
"text": " it shows youtube hey people are talking about this content outside of youtube which in turn greatly aids the distribution on youtube thirdly you should know this podcast is on itunes it's on spotify it's on all of the audio platforms all you have to do is type in theories of everything and you'll find it personally i gained from re-watching lectures and podcasts"
},
{
"end_time": 11339.718,
"index": 488,
"start_time": 11319.787,
"text": " I also read in the comments"
},
{
"end_time": 11363.166,
"index": 489,
"start_time": 11339.718,
"text": " and donating with whatever you like there's also paypal there's also crypto there's also just joining on youtube again keep in mind it's support from the sponsors and you that allow me to work on toe full time you also get early access to ad free episodes whether it's audio or video it's audio in the case of patreon video in the case of youtube for instance this episode that you're listening to right now was released a few days earlier"
},
{
"end_time": 11369.787,
"index": 490,
"start_time": 11363.336,
"text": " Every dollar helps far more than you think. Either way, your viewership is generosity enough. Thank you so much."
}
]
}No transcript available.