Theories of Everything with Curt Jaimungal - Retired Off Apple Stock Then Revolutionized Physics Living in Hawaii | Garrett Lisi

Episode Date: September 25, 2024

Garrett Lisi is a theoretical physicist known for his work on the "Exceptionally Simple Theory of Everything," which proposes that the E8 Lie group can describe the fundamental forces and particles in the universe. Lisi is also an avid surfer and has gained attention for his unconventional approach to physics outside of academia. SPONSOR (THE ECONOMIST): As a listener of TOE, you can now enjoy full digital access to The Economist. Get a 20% off discount by visiting: https://www.economist.com/toe LINKED MENTIONED: •⁠ ⁠Garrett Lisi’s TED talk on his theory of everything: https://www.youtube.com/watch?v=y-Gk_Ddhr0M •⁠ ⁠Garrett Lisi’s website: https://li.si/ •⁠ ⁠Garrett Lisi’s papers: https://li.si/Physics/CV.html •⁠ ⁠Peter Woit on TOE: https://www.youtube.com/watch?v=9z3JYb_g2Qs •⁠ ⁠Garrett Lisi’s paper on triality: https://arxiv.org/pdf/2407.02497 TOE'S TOP LINKS: - Support TOE on Patreon: https://patreon.com/curtjaimungal (early access to ad-free audio episodes!) - Listen to TOE on Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e - Become a YouTube Member Here: https://www.youtube.com/channel/UCdWIQh9DGG6uhJk8eyIFl1w/join - Join TOE's Newsletter 'TOEmail' at https://www.curtjaimungal.org SPONSORS (please check them out to support TOE): - THE ECONOMIST: As a listener of TOE, you can now enjoy full digital access to The Economist. Get a 20% off discount by visiting: https://www.economist.com/toe - INDEED: Get your jobs more visibility at https://indeed.com/theories ($75 credit to book your job visibility) - HELLOFRESH: For FREE breakfast for life go to https://www.HelloFresh.com/freetheoriesofeverything - PLANET WILD: Want to restore the planet's ecosystems and see your impact in monthly videos? The first 150 people to join Planet Wild will get the first month for free at https://planetwild.com/r/theoriesofeverything/join or use my code EVERYTHING9 later. TIMESTAMPS: 00:00 - Intro 01:24 - Garrett’s Ted Talk and Background 08:36 - Mystery of Spinors 10:13 - Garrett Poses a Physical Problem 12:22 - String Theorists 23:48 - Triality 26:13 - Garrett’s Top Points in His Talk 28:42 - “String Theory’s the Only Game in Town” 32:10 - Garrett’s Presentation Begins: Unification 38:22 - Spinors vs. Fermions 45:56 - More Particles in Higher Gauge Groups 48:42 - Threefold Symmetry 54:00- Group Theory Overview 59:37 - Quaternion Group 01:04:14 - Pin Group 01:10:25 - Spin Eigenvalues 01:14:45 - C, P, and T 01:22:12 - Biquaternionic Spinors 01:25:55 - Quaternion Triality and the CPTt Group 01:34:14 - Multi-Generational Fermion States 01:37:20 - Fermions in Exceptional Unification 01:42:12 - Exceptional Magic Square 01:44:22 - Outro / Support TOE Other Links: - Twitter: https://twitter.com/TOEwithCurt - Discord Invite: https://discord.com/invite/kBcnfNVwqs - iTunes: https://podcasts.apple.com/ca/podcast/better-left-unsaid-with-curt-jaimungal/id1521758802 - Subreddit r/TheoriesOfEverything: https://reddit.com/r/theoriesofeverything #science #sciencepodcast #physics #theoreticalphysics Learn more about your ad choices. Visit megaphone.fm/adchoices

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To my utter shock and amazement and just my whole brain just was electrified This whole algebraic structure that I had been playing with fits inside this largest simple exceptional e-algebra, E8 Complete with this triality symmetry relating the generations Most physicists believe that the key to unification lies in simplicity and overlooked symmetry. Theoretical physicist Garrett Lisey, known for his unabashedly ebullient E8 model that took Ted by storm 16 years ago, has groundbreaking new work on something called triality. A three-fold symmetry that could explain one of the most perplexing mysteries in particle physics that many don't talk about.

Why do the fundamental particles come in three generations? Interestingly, triality isn't unique to Lisi's work. The special orthogonal group SO8 also exhibits a remarkable triality relating its vector and spinner representations. In this episode, we explore the geometry of E8 and the bizarre world of bi-quaternions. We'll also discuss why Lisi's work, despite its rigor, faces skepticism from the mainstream physics community and how it challenges our understanding of the relationship between mathematics and physical reality. Garrett, Lisi, it's a long time coming.

Thank you for coming onto the podcast. Thanks for having me on, Kurt. Many know you from your TED Talk, unveiling to the public the theory of everything based on E8. So please tell us about that to set the stage for your latest research, which we're about to get into. Yeah, that was pretty remarkable. I made some very questionable life decisions early on in what one could laughingly call

a career. I was a graduate student at UC San Diego, and I was very much in love with general relativity, differential geometry, and quantum field theory. And through general relativity and differential geometry, I really got a very deep sense that our world is fundamentally geometric. But in quantum field theory, when you're first introduced to Dirac fermions,

say, which are a fundamental component of our universe, it's not introduced in a geometric way. It's introduced as a column for complex numbers. So that was very unsatisfactory to me and I figured out there had to be a geometric description for Dirac fermions the same way there's a geometric description for general relativity and the same way there's a geometric description for gauge fields as fiber bundles. And when I went to people in the department to talk to them about this, nobody was interested in the problem because everybody was going crazy about the ADS-CFT correspondence and string theory.

And no one was interested about some kid worried about, you know, how do you make spinners more geometric in the way general relativity is? No one was interested in the question, but I was. And I was very fortunate in that I was using a Next computer for my dissertation. I don't know if you're familiar at all with the history of Steve Jobs and Next up, but basically Steve Jobs got fired out of Apple and started this company called Next where he just made an amazing computer and operating system. And then Apple, meanwhile, had a horrible operating system and it was just in desperate, horrible shape. So they brought Steve Jobs back.

They basically next bought Apple for a negative amount of money. I think it was 400 million or something that Apple gave to Next. But Steve Jobs came back, put all his people in charge, and then Apple's new operating system was Next Step, the next operating system, which we now know as OS X. And this all went down, and I had been using it next, so I knew this was going to succeed.

I knew it was a great operating system, so I put all my graduate stipend and my award money and everything into Apple stock in the 90s. And but Apple didn't just do well because of Steve Jobs coming back with Next, it did well because of music. They had the iPod and then the iPhone and iTunes and the App Store and Apple just boomed in success. So when it got time to get a postdoc, I'm like, you know, I don't need to go get a postdoc. I can move to Maui and be a surf bomb and I can work on figuring out a geometric understanding of spinner fields and I can just do the research I want. So that's what I did.

It was a very strange decision because I essentially left academia even though I still have a very academic mindset. But the one question in my mind was what are spinners and fermions geometrically? Why do they exist? Why would nature have them as part of our universe? And I struggled with that. I tried all different approaches. There's this cult of geometric algebra.

I don't know if you've encountered it. Now why do you call it a cult? I call it a cult because it's a cult. You take the CL1-3 Clifford algebra and you ascribe, that can be used for spacetime to great effect. And physicists use that for, that's how Dirac formulated the Dirac equation is using this special CL1-3 Clifford algebra and its representation in terms of Clifford matrices. And it turns out Clifford algebra and its representation in terms of Clifford matrices. And it turns out Clifford algebra is really useful for doing rotations or if you're in space time, you're doing space time boosts as well as rotations, you're doing Lorentz

transformations using Clifford bivectors. But you can go over the top and formulate everything you would normally formulate with differential forms and vectors, you can do that by translating that into Clifford algebra. And there's this guy, a very wonderful gentleman I had the honor of meeting, David Hestenes, who dedicated himself, I think in the 1960s, to developing this way of describing

a vast amount of fundamental physics using Clifford algebra. He got a following and a lot of people became part of it and also wrote books on it and really good papers. So there became this whole volume of literature for how to describe fundamental physics using Clifford algebra, which they called space-time algebra, including this very unusual formulation of Dirac fermions. And since I was interested in what Dirac fermions are geometrically, it was natural to look at this literature on geometric algebra. It turns out I drank the Kool-Aid for a

while, but then I'm like, you know, some things with differential forms you can't do with Clifford algebra. Such as? Like the way integration works. When you're integrating over differential forms, that's essentially what they're for. Differential forms are things you integrate over and then use the Stokes theorem to do the integral. And then there are things you can do with vectors and forms. So a differential form is a dual space to vectors, right?

So you can do it either way. Either you can say a differential form eats a vector and gives you a scalar, or you can say a vector operates on a differential form via the interior product to give you a scalar. And those are equivalent. But if you're doing something like the geometry of fiber bundles or say the geometry of a principal fiber bundle, okay, then you have a Lie group as your fiber

over a base space. And if you want to describe how that Lie group is twisting over your base space, you do that with what's called a connection. Right. And the most natural way to describe a principal bundle connection is as a Lie algebra valued run form. Right? And there's no Clifford algebra to be seen unless your principal bundle is a spin bundle, in which case you have spin 1, three, and then Clifford algebra. So in order to build the spin algebras,

those are the same as Clifford bivector algebras. Are you aware of this? Yes. Okay, so this is how you get representation spaces of the spin group. Is you go to a Clifford algebra, you build the Clifford algebra corresponding to the spin algebra, and then the bivectors of that Clifford algebra, you build the Clifford algebra corresponding to the spin algebra,

and then the bivectors of that Clifford algebra are your spin algebra generators, and the matrix representation you're using for the Clifford algebra is the matrix representation then of your spin algebra. As far as I know, David Hestines would say that Michael Atiyah and Roger Penrose, when they talk about the mystery of the spinors. I think that's what's interesting about the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the spinors.

I think that's what's interesting about the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the spinors.

I think that's what's interesting about the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the spinors. I think that's what's interesting about the idea of the idea of the spinors. I think that's what's interesting about the idea of presumes that they can describe all of physics with this framework and that by not really looking at differential forms or vectors but incorporating that into Clifford algebra into geometric algebra language, they can still do everything. But really you lose capability when you do that. So for example, if you want to, the connection for a spin bundle is a Clifford bivector algebra valued one form and that's called the spin connection and that's a very natural thing for mapping from you know Lee

algebra so basically you feed it a vector which is whichever way you're traveling and then that transforms to how the fiber is rotating alright so and this is how you do parallel transport it It's how you do Wilson lines and those integrals. So really you need Clifford algebra and you need differential geometry. You can't just wrap up differential geometry in Clifford language and glue it together and still have the same powerful framework. There's stuff you can't do as well and there's stuff you can't think about as well. So at that point I left the Church of Geometric Algebra and I embraced Geometric Algebra just

as it's CL13, it's a Clifford Algebra, and you use it the way you would any other Clifford Algebra. So before we leave that cult, if there are some people who are David Hestene's fans, and I think there is Lassen B and someone else as well who are the current head to it. Yeah, they're brilliant people. They do great work within that that context and yeah, I have a lot of respect for them. Give a problem or a question that can't be solved or even formulated in the real geometric algebraic formulation, but it can in the differential one. D'Ram cohomology. Doesn't make a lot of sense using Clifford algebra for curved spaces, but it makes perfect sense for differential forms and arbitrary manifolds.

Well, you just gave a concept, but I mean give a physical situation or a physical problem that differential geometry is well posed to solve, but it doesn't even make sense in the geometric algebraic case. Or it can't be posed in it. So you're doing general relativity and you're doing unusual cosmological topologies, and you start doing integrals to determine topological invariance of these things, if I figure out what that is. That doesn't work well in Clifford algebra. In fact, if I recall correctly, Lazenby has a theory of general relativity that is based on flat Minkowski

spacetime with fields being on top of that spacetime using Clifford algebra language, using geometric algebra. And it works to a degree until you start to think of things that actually depend on unusual spacetime topologies and embedded topologies. So non-trivial topologies can't be described in geometric algebra or a specific type of non-trivial topology? I'd say non-trivial topologies generally, the tools aren't as powerful there.

So geometric algebra is absolutely the best at describing rotations. That's why it's so useful for describing the spin group. Because the spin group is all about space-time rotations. Apparently computer graphics artists use it as well because of this reason. Yeah, that's what it's absolutely great at is rotations. You may use it on your website, which we're going to get to later. You've just tuned into a special episode brought to you by Planet Wild, your ticket to making a real dent in the global conversation.

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Everything9 to have your first month sponsored by us. Dive into action at planetwild.com slash everything and remember it's not just a donation. It's a partnership for the planet Because there are plenty of rotations involved. Yeah, that's several layers between Okay, so you're taking us currently through your journey where you are enamored by the differential geometric aspect of spinners trying to find that connection How do you geometrize spinners or understand them in a geometric manner? You first approached your colleagues which happened to be string theorists Which is quite odd to me that they wouldn't know because in my experience the most educated people in Differential geometry and bundle theory currently in the physics scene are the string theorists. Absolutely true, but they would they would say

That fermions are anti commuting excitations of a heterotic string. So that's too algebraic for you? Well, there's a lot of structure to get there. So you have to swallow that particles are fundamentally oscillating strings. And when you do that, you know, if you associate a particle with each oscillation of a string, you get an infinite tower of particles of higher and higher, larger and larger masses, an infinite number of, but we don't see an infinite number

of different kinds of particles, right? So you don't see the tower of states. And then string theory gets more and more complicated the more and more you get into it. And okay, string theorists promise that they'd get the standard model by just finding the right compact Calabrian manifold, right? And then the standard model would just fit that and everything would be good and we'd have this wrapped up before lunch and have a good theory. That was 40 years ago or something and it never happened.

It seems like with the string theory is an amazing framework for constructing anything, but it doesn't give you a theory of everything. It gives you a theory of anything you want. And I did not buy into that elaborate structure. I took a course in introductory course in strings as a graduate student from a brilliant guy who started out in general relativity, Malcolm Perry, I think, gave a great string theory course. But there were so many jumps between the standard model,

which is very well established, to strings, to all the framework that comes along with strings. You actually made a fantastic exposition about two and a half hours long called the Iceberg of String Theory recently. What Garrett is referring to is the mathematics of string theory, the Iceberg edition, where I go into details about the math of string theory. It's on screen now and it's linked in the description. That's amazingly impressive for, if nothing else, expressing just how complicated string theory gets with what payoff. Did they deliver on giving a unified theory of physics? No, it doesn't.

It seems like they can build everything except the standard model and the decider cosmology. they can build everything except the standard model and the decider cosmology. I mean, the convolutions they have to go to to get back to that is just absolutely absurd. And so I just couldn't believe it. I think it was going to be a train wreck. And it seems like a lot of people now are sharing that view. But you have to remember, this was the 1990s. This is mid-90s, like, you know, full on string. It was in full force. There's no way I was going to get a postdoc position trying to figure out the geometry of electrons when everybody was going strings. In fact, my graduate advisor for a time was the chair of the department at UC San Diego, Roger Daschen, who was overseeing me as a dissertation I did

because I found a soliton in the Maxwell-Dirac equations that nobody thought should be there. I found it numerically. And not a lot of people were doing numerical calculations back then. But Roger died suddenly during a seminar of massive heart attack. During a seminar?

Yeah, I was there. Jeez. He was, well, I he's he's the age I am now, but he was very out of shape and Chainsmoked and you know fantastic guy, but just did not take care of himself and Had a heart attack during a seminar from choking We tried CPR, but it was there was no chance. He was just gone.

So you watched him die. He didn't just have the heart attack there and died later. No, it was, I forget who the speaker was, but I'm sure he remembers. So from that, having lost my advisor, and Roger had done research on solitons in QFT. So he was a perfect person to advise on that. But now I had no advisor,

but the UCSD had just hired Ken Intrigator, who was just full on into ADSCFT. And I went to his office to talk to him because I was at the top of my class in graduate school, maybe second out of a class of 40. And I was in his office and he's telling me about ADSCFT. And I'm like, so there's a potential correspondence between anti-de Sitter space,

which as far as we know, doesn't describe our universe and conformal field theory, which as far as we know, doesn't describe our universe. It's nice that these things exist. It's like this is correspondence, but I don't think I want to devote my physics career to studying something that might just not apply to our physical universe. I mean, I have a degree in mathematics as well as physics and I love math and I love the combination of them, but I wanted to find out what the mathematical description is of our universe and I didn't think string theory was it. So instead, I had FY money from Apple stock. So off to Maui, I went to figure out what spinners are geometrically.

I was in the geometric algebra cult for a while and then I'm like, well, that's not really working. So one thing that seemed to work really well, so if you're going to understand electrons, you have to understand how they interact with everything else. So with gravitational fields via the spin connection and the gravitational frame and also with gauge fields. Now there's a fantastic thing I'm sure you've heard of called Kaluza-Klein theory. Whereas if you're doing general relativity in four dimensions, you can just increase

the number of dimensions and compact them. Just assume they're rolled up into small distance scales. And when you do the equations of motion from general relativity, Yang-Mills, the Yang-Mills Lagrangian pops out. Well, almost every compactification in string theories of the Kaluza-Klein type. Yes. I think in that iceberg, I only covered two out of 200 or so that weren't and I forgot

what the type of compactification was. Well, one's a flux compactification, then there's a third type. Yeah. I forgot what type. So, but I'm a pretty simple minded guy. I like to stick to what I know is true about the universe and make very small steps from that. Yeah.

Okay. So I wasn't willing to accept all of the string theory framework, but I'm like, okay, start with general relativity, add a dimension, you get electromagnetism. You add, I guess we would call it a torsor, but it's just the manifold corresponding to the SU2 group or its corresponding symmetric space. So you can add a symmetric space as your compact extra dimensions. I think CP2 is the name of the one, the compact projective sphere, dimension two.

And this has the symmetries of the standard model. If you look at the killing vectors for this thing, it has SU3 for its killing vectors, and you get SU2 and U1 out. And I think you can get that most directly from a compact seven-sphere. And those are ways of doing the symmetric space. But anyway, this was how to extend general relativity,

which electrons interact with, to a higher dimensional space. Yes. And all that, the problem is, is when you try to fit fermions into that picture, like how do fermions fit in with Clus Zecline theory? They don't. Okay. The fermions you get aren't standard model fermions. The theory just doesn't work well. And Witten realized this when he was first playing with Kluzikline extensions of gravity to get the standard model. When Witten was still focused on actually getting a theory that corresponded to the standard model out of string theory, these are the sort of compactifications he played

with but he abandoned it because fermions just wouldn't work just by doing them in Kluzikline theory. It just didn't work well. So remember, you can get Yang-Mills theory as an extension of gravity, but that doesn't work for fermions. However, you can go the other way. So I'm working on myself.

I didn't have graduate students. I'm just there in Maui. I've got a huge stack of notes, but it's just me. So I'm just like, stack of notes on Clusacline theory. Okay, wipe that off the table. What happens if you start with gauge theory and the geometry of fiber bundles and you extend that to general relativity? And there are these physicists who had done this, MacDowell and Suri, who had acquired

gravity as a gauge theory of spin 1-4. How so? So if you take spin 1-4, right, it has an extra four dimensions in spin 1-3. And you take those extra four dimensions to actually be a vector that the spin 1-3 subalgebra of spin 1-4 acts on. And now you have the gravitational spin connection and the frame inside one Lie algebra. And you have that spin connection acting on the frame

the way it should as a vector. And the spin connection, that's a spin 1,3 valued 1 form. And the frame, that's a vector valued 1 form that that spin 1,3 axon. And once you have spin-1-3 and then spin-1-4 as describing gravity fully, just purely as a theory with a connection with no metric in sight, the metric emerges from the frame part of the spin-1-4 connection. It's a very unusual description of gravity, but it's very cool. And the reason it's cool

is because it says, okay, the other way of thinking about this is if you have spin one for itself as a Lie group, as a 10 dimensional Lie group, then you take a four dimensional subspace of it, such that you have, and then if you look at that entire space, that looks like the entire space of a fiber bundle with a spin one three principle fiber acting on a gravitational

frame also is part of that. And then you get this natural association between that frame for that embedded space time. It's a very pretty picture. Is this what's known as gauge gravity? Yes. Yeah. So gauge gravity is another name for it.

It's also called carton gravity is another description for it. It's also called carton gravity. There's another description of this. So basically, it's a way of formulating gravity using gauge field geometry. And because you've done it that way, it's very natural to glue on spin 10. And if you have spin 10 and spin 1, 3, and you combine those into spin 11 three, those have a lovely 64 dimensional real spinner reputation representation into which the entire standard model with spins fits.

So you get a full generation of standard model, including their spins into a 64 of spin 11 three. And that looks like what you'd call a, a, a gravagut or a theory of everything. If you're loose with your language. Okay. So what's wrong with this picture? Nothing.

There's nothing wrong with this picture. It's a great formalism. The one thing that's missing from it is there's only one generation of fermions. You have no explanation for why you have three generations. Okay. So triality. Right. So 2005 or so, I'd heard of triality. I hadn't played with it.

But I was very interested in how to get the standard model assembled into a unified geometric framework so I could understand what it is. And I applied for, this was going well enough, the structural unification using gravity as a gauge theory and acting on fermion multiplets this way, was doing well enough that this private foundation came up, the Foundational Questions Institute, founded by Max Tegmark and Anthony Aguirre, because they wanted to fund unusual foundational ideas in physics. And I'm like, I'm working on that. So I submitted a grant application and much to my surprise, I was awarded a grant from them.

I went around to conferences and I was actually getting interest in this. And at this time I was working on physics full time and I was at a friend's ski house in Tahoe when I'm like, this algebra of spin 11, 3 and a 64, it looks like it should be part of the adjoint representation of a Lie algebra because it's very cohesive, especially if you increase the number of generations to three. It starts to look like one thing and I'm like, well, what is, could this be a Lie algebra?

So I started looking at large Lie algebras and see what it could fit in. Because this is the overall strategy of the unification approach to physics is you take all our pieces of known physics and you try to embed it in ultimately one larger mathematical object. It's a new day. How can you make the most of it

with your membership rewards points? Earn points on everyday purchases. Use them for that long awaited vacation. You can earn points almost anywhere and they never expire. Treat your friends or spoil your family. Earn them on your adventure and use them how you want, when you want. That's the powerful backing of American Express. Learn more at mx.ca-yamex. Terms apply. American Express. Learn more at mx.ca slash ymx. Terms apply. And for me, I think string theory fails at this. They started out with what started out as a simple framework, but they keep adding more and more layers of complication to it until you're not doing unification anymore. String theory is no longer a unification program.

It's a toolkit for building anything. But I'm very old school. I took this old strategy of starting with what we know and extending it and embedding it in the least group. And to my utter shock and amazement and just my whole brain just was electrified, this whole algebraic structure that I had been playing with fits inside this largest simple exceptional e-algebra, E8, complete with this triality symmetry relating the generations. And that was just astounding to me. Now it turns out, as with all these unified physics theories, there are problems with it. At this point, we should probably go to the first page of my talk.

Sure. I don't know. To me, the most satisfying thing was to find exactly what I'd been looking for in graduate school, which is if fermions are geometrically part of these exceptional lead groups, then that is the ultimate geometric description of what fermions are.

Before we get to your talk, what are the three ingredients of your talk that you think the audience should know about most? Like a root system versus a carton subalgebra or something that would help the audience. So if you said these are the three ingredients that maybe people would know about who are watching, let's explain them first. My talk has a thousand threads to pull on. And what I always encourage people interested in physics to do is follow your interests. You know, figure out which aspect of anything I might say interests you and then dive deeper on it. I have a webpage that I'm actually using for this talk that has background material on any

thread you might pull in this talk. It's formatted as a wiki, so you just click on the links and follow your links as if you're on Wikipedia, except you're in my digital brain. So I've used this for over a decade now as my physics notebook, and I've found it to be wonderful for doing research and organizing research. So yeah, follow your interests. If group theory, I mean, group theory is a huge subject. So group representation theory itself is so rich. And I think it's very neglected in most undergraduate curriculums. Usually, don't even see most of it until graduate school, but it connects everything together in physics.

All the power we have in physics and all this stuff you think, oh, that's really cool. Chances are that's coming from group theory. That gets forgotten when you're doing strings, right? The strings, they use groups, but groups aren't fundamental. And if there's one thing we know from building up the success of the standard model with all of these powerful particle accelerators

and theorists working on it for their lifetimes, is that groups are fundamental to physics. Strings might not be, but we know that groups are. We know that finite groups and Lie groups, which finite groups can embed in, are absolutely fundamental to physics. Before we move on to your talk, just to defend the string theorists, they would say, look, when we say string theory is the only game in town, what we mean is it's the only finite

quantum gravity in town. So what is your response to that? They don't have a complete theory of finite gravity. So if you're doing perturbative string theory on a background, you get a linearized action for spin two particles, and they say that's gravity. Well, it's a little bit of a stretch to say, OK, it's a linearized version of gravity. We're just going to assume that the completion is Einstein-Hilbert gravity. Okay, so you don't legitimately get the curvature out of string theory.

You get a linearized version because you're doing perturbative theory in a background. They're cheats. They get there by cheating. Same thing has been said of loop qualm gravity. So what if they use string field theory? String field theory I'm less familiar with. Actually, I'm not even that familiar with string theory. Mostly tried to avoid it.

The way I've gotten so far with my research is by mostly pretending string theory doesn't exist. It's like I live in a detached bubble where string theory never happened and I'm extremely happy as a theorist. Now is that because you see it as a siren call that if you, you as Garrett, were to pay more attention to it, you'd be more and more convinced? Just like how you saw there was a riveting faction of geometric algebra, maybe you saw there was a riveting faction of string theory. The math is so beautiful, maybe the math isn't, but whatever. You think you're going to be

alert to it and you've already decided this is not a productive route. Let me not be tempted. Let me remove temptation. Like what is the reason? Because another route to take is, look, I'm in the game of toe. I'm in the game of articulating a theory of everything. And in order to do that, I need to keep track of all the players that are in the game. And so I need to know the movement of my competitors quite closely, maybe not too closely as that will take me away from developing my own. But that's the other strategy. I'm not that competitive.

I'm actually more collaborative. I just don't have many collaborators. But when I see a huge herd of researchers going in a direction, I'm contrarian in that I think that's going to be mine. They're already going to have done all the easy stuff. So I'm going to go over here, something that I think is more promising than that has been neglected. I try to research things that I have higher confidence in because they're only small extensions

of what is established by experiment, but that the rest of the community has neglected and therefore is more fruitful for me to put my attention and expertise on that. Because if I put my expertise into strength theory, I might develop a greater appreciation for it, but the chances are I'm not gonna be able to be as productive as, you know,

any one of the top people in that group of thousands of strength theorists working in that direction. I have much more potential for striking out in a weird direction on my own and being wildly successful than in just being another cog in the machine over here. Got it. It's also why I don't work for large corporations. I seem much happier and more successful on my own. And you also have that FY money.

I do. That's what lets me go on surf and paragliding trips when I want to. It's fantastic. Okay. So let's get to this talk, man. Okay. All right. Going back to what we covered, the most straightforward path forward for achieving a theory of everything that has, I think, a decent chance of success is a fairly simple-minded progression of unification

of embedding groups and their collections of representation spaces that we know of into larger groups and representation spaces. And ultimately, if you can get everything into one, you've ultimately succeeded and said nature is just one thing that symmetry breaks down to everything we see. All right. I should mention all the various criticisms of this. The most successful known example of this is the SO10 Grand Unified Theory, which was figured out in 76 or so.

And it just turns out to be a pretty wild, you know, one in a hundred coincidence that the hypercharges of the known fermion multiplets represented over here Happen to all combine successfully into a 16 dimensional spinner representation of spin 10 and And this is why the the SO 10 gut is considered so nice But a lot of people hate it Peter White hates it because you still have the complexity now of figuring out how nature does a symmetry breaking from this to this. Okay, so that says, okay, well, you have this very simple thing, but if you're starting

with a simple thing, now you have to break it to get what we know. So there are various mechanisms for that. Also there's an experimental reason. You have new gauge. Whenever you do unification, you end up with new fields that give you new interactions. And for SO10, the new interactions give you the possibility for protons to decay, which we don't have

in the standard model. So there are large experiments looking for protons decay and they've never been seen. And that hasn't ruled out SO10, but it makes it less and less likely the further and further they push the sensitivity of their detectors. Okay, there are a bunch of people like me just like holding out. It's like, ah, maybe someday a proton will decay and we'll see it, you know, but maybe it's just not true.

Maybe it's there's some other structure, maybe guts are just wrong. Well, couldn't the probability just be so low? Is there constraint on the probability for SO10? Whenever you do symmetry breaking you have all these parameters and at some point you can always add more parameters with more massive particles that make it less and less likely to decay. But the thing is, and strength theorists got themselves into this position too, they keep adding more and more levels of complication to explain why why don't you see super particles.

Supersymmetry is an intrinsic characteristic of super strength theory. And also they said supersymmetry would give you some cancellations you need for the Higgs particle to break symmetry and give masses away. It does. They thought supersymmetry would help in that. But only superpartners had a certain mass. And they did not see superpartners with those masses at the LHC.

I mean, I visited the LHC. It was like they had a giant banner across the top saying, welcome home, superparticles, waiting for them to come in. But they never showed up. They didn't show up to the party because maybe they don't exist. So, but you can't extricate supersymmetry from string theory happily.

So what they do is they just add more and more parameters to make those masses higher. So it's like, oh yeah, we'll just never see them. It's like, yeah, it keeps straining credulity when you do that. So it's just a matter of how much you're willing to believe in these theories. The third criticism of these sorts of unified theories is part of their motivation is because they're very mathematically beautiful, right?

These structures, the exceptional lead groups, when you investigate their structure, they're the most exquisitely beautiful objects in mathematics. Okay, and I'm not saying that lightly, and I'm not throwing one to whoever thinks this. They're just extraordinarily beautiful geometric objects. And the possibility that our universe is embedded and ultimately comes from the symmetry breaking

of the most beautiful object in mathematics, that's inspiring, but not for some people. If you're Sabina Hassenwalder, she says this sort of mathematical beauty is completely misleading and will never get us anywhere. It's a matter of taste. I suspect maybe Germans just don't like things that are pretty. They like things that work.

I don't know. It's a matter of taste. Well, is there anyone other than her? Yeah. Many people share that view. It's a very pragmatic view and it also goes hand-in-hand with why do you assume that there is this more beautiful structure when you don't have evidence for support it and it predicts particles you haven't seen and you don't know how to break this beautiful thing

down to what we get. Okay, those are all very reasonable concerns. But Sabina wrote a very good book on, you know, as beauty led physics astray. And I think she was mostly talking about what string theory is considered the mathematical beauty of their theories, but she was also talking about grand unified theory, including this one. So it's a valid criticism. And from a pragmatism point of view, it's valid. But from a philosophical point of view, I find it very satisfying that our universe might be special. Yeah, it's always nice to be special. Now Garrett, for the people who skipped forward, they listened in the beginning, but now they're

here and they heard you speak at length about the value of spinners and their geometry and so on. Where here in this mess of numbers and letters is a spinner. Like sure we have the word fermion. This this too. So this is a complex two dimensional spin representation space of SL2C, the special linear group two dimensional complex matrices. What is the relationship between a fermion and a spinner? Is every fermion an example of a spinner?

Do spinners have fermions in them? Is a spinner tensored with quantum numbers the same as a fermion? Tell them what is the relationship because they're used often interchangeably by physicists in the standard model as it's usually presented a Fermion which is a physical particle like an electron It corresponds to a representation space Well, let me let me add some steps a physical Fermion electron Cor electron corresponds to a field. That field is valued in a representation space and that representation space is called the spinner

representation space. And a spinner representation space is acted on by rotations in a different way than vectors are. So vectors also represent a space of rotations. Spinners are different representation space of rotations. For example, you have to rotate a spinner not 360 but 720 degrees in order to return it to its original state. In an abstract space. It's an abstract space that has a very physical implementation as electrons. Yes. Yeah, so it's more than just an abstract space.

Also, when you ask them, these things have an intrinsic angular momentum. So it's not like an electron is spinning around in a circle. It's like the electron field itself has angular momentum. Yes. Okay. Which is strange. Also, the fields themselves anti-commute, okay? Which means if you're operating with them and one goes past another, it changes sign to minus.

Uh-huh. Okay. So they're anti-commuting fields. They're anti-commuting, uh, spinner valued fields. And that, that's the best way to describe a freon. And then you go to quantum field theory and these things are quantized excitations of these fields. Yes.

Okay. And then that's how we do quantum field theory, but structurally mathematically, you can think of them as electrons correspond to states. So say you have an electron that's spinning around this way with spin up, and it's traveling along the z-axis. Then that's a spin up electron.

Let's presume for a second it's massless, then you would say this thing is entirely right-handed if it's not interacting with the Higgs. Because if you're in action with the Higgs, then electrons bounce back and forth between the right and left-handed parts. But, or say if it's, yeah, say it's a massless neutrino, and then you're talking about right-handed neutrinos,

which I also think exists. But anyway, so you have a spin, you have a direction of motion along or counter to the spin that determines whether it's right-handed or whether it's left-handed. Yes. See, this is going this way and this is going that way. So the spin direction is the same, but here my thumb's in the direction of the spin and

here my thumb is opposite the spin. So this one's right-handed and this one's left-handed. Right. All right. So spinners have this chirality aspect to them, and they have spin, spin up or spin down, which corresponds to angular momentum. But they're also complex. Your team requested a ride, but this time not from you.

It's through their Uber Teen account. It's an Uber account that allows your team to request a ride under your supervision with live trip tracking and highly rated drivers. Add your team to your Uber account today. So if you're a complex drag spinner also has a complex conjugate. Okay. And those roughly correspond to particles and antiparticles. All right.

So all this mathematical structure lives in a representation space. For spinors, that's acted on by spin 1, 3, which is identical to SL2C. Maybe I should have put SO1, 3 or something here. But that SO1, 3 in a gravitational grain unified theory combines with SO10 into SO113. And the spinors and this entire multiplet of fermions, okay, this is all electrons, neutrinos, and quarks of one generation, all fits now in a 64 spinner, real spinner, of SO113. And this is a really wonderful unification, okay. It's very succinct, and it includes gravity and gauge fields.

Now, there are a lot of people that say this sort of unification should not be allowed. And that comes down to the Coleman-Mendulo theorem, which says if you think about the S matrix for scattering in spacetime and how this works with gravity and gauge fields, then gravity and gauge fields cannot be unified into a larger and larger group. The response of that is to say if this is a unifying group, right, unifying gravity and gauge fields, you don't have an S matrix here because you don't have

space-time yet. In order to get space-time, you have to break this symmetry and out of it you get spacetime, which is this 4 here. So this 4 is the gravitational frame, which is acted on by SO and 3, and this 10 is a Higgs multiplet that's acted on by SO10. And spacetime has to do with this 4 right here, these 4 dimensions of spacetime. After the symmetry breaking happens, then you have space-time, then you have particle scattering and so forth,

and then you can apply the Kuhlman-Mendoula theorem and say, lo and behold, the gravitational and the gauge fields are not unified. That is the case over here, okay? But here, if you're thinking about a unified theory that hasn't broken yet, so it doesn't have even the existence of spacetime yet because it's all been unified, then there is no

scattering to think about. There's no, you can't apply the Cohn-Mendoula theorem because the conditions of the theorem aren't met. Mm-hmm. Okay, symmetry breaking has to happen and then the theorem applies. So this is a perfectly fine, perfectly reasonable structural unification. And it's a very pretty one. And something especially pretty about it is that this unification then fits in a specific compact real form of the E8 Lie group. The problem is there's a whole bunch of other stuff in here.

Okay. And some of the other stuff is, or is what is called mirror matter, which is, which are like the standard model for, mirror matter, which are like the standard model fermions, but it has the opposite chirality. It has the opposite handedness. And we don't see these particles in nature. So this is the criticism that Jacques Dissler and Skip Garibaldi used to say this can't work.

You can't get the standard model fermions out of E8 because you also have mirror matter. They didn't call it that and they never admitted that this does embed in E8, which it does. It was very annoying to talk with them and I mostly try to maintain my mental health by not. But that was the criticism. And this is what killed interest in E8 theory, is it has extra stuff that we don't see. So the task then is to understand. So there's specifically as well as the 64S plus, there's a 64S minus in E8. That's the mirror matter.

And we don't see those 64s. But I'm like, maybe there's a symmetry that will transform between this 64 that we know physically as one generation of fermions and that other 64 that usually you identify as mirror matter could be a transformation of one generation and then there's another 64 in SO113 or SO124 rather that could be another 64 but it's vectorial. It's not even spinorial.

Explain just a moment. You keep saying that there are more particles that are predicted when you go to a higher gauge group, but isn't it the case that in the first unification model of Casin and Condon, I believe, if I'm pronouncing their name correctly, they thought, okay, the proton and the neutron are separate particles. Well, what if their representations or members of the representation space of SU2, then the proton can be something up and the neutron can be something down. That's what they initially thought. And then anyhow, so then they're part of the same particle

with just different states of the same particle. Yeah, they could be excited states. Yeah. So when you say different particles, what exactly are you meaning? You mean different excited states. So what are the different states? So for example, and what I'm working on here is structural unification. So you can also have a philosophical unification. You can also have unification of equations of motion. For example, if you do this unification, right,

and write out these fields, if you have a field valued in this, right, it looks like this. This is the spin connection part. This is the gravitational frame, that's the four. This Higgs here is the 10, and then you have a gauge field, that's the SO10, and then you have the spinners, that's your 64.

And you just consider the spinners to be sort of a one-form, but a one-form that's perpendicular to spacetime, whereas all these others are spacetime one-forms. OK. OK. So therefore, your spinners are all anti-commute. Now, you take the curvature of this thing in the usual way,

and you get all these terms. You get the Ruhmann curvature two-form, you get an area term times the Higgs, and you get torsion, and you get the covariant derivative of your Higgs field, you get the curvature of the Yang-Mills fields, and you get what looks sort of like the Dirac derivative in curved spacetime of your spinner fields. You compute the Yang-Mills contraction of that, and you get all these nice, familiar action terms that you want for these fields, including

cosmological constant, a potential for your Higgs potential, and the Dirac action. You also get a whole lot of other stuff I didn't write. If there was a rug here, all the other terms would be under it. Got it. This proportionality says this isn't all the terms. It's just it has these. It also has other stuff.

You should just put dot, dot, dot. That's the physicist or mathematician's way of saying rug. All the other stuff, the rug. And it's under that. But this is an equation of motion unification. So I don't want to be kooky and say this is a unified field theory, but that's what it is. All right.

Now, the main outstanding mystery of this unification program is the threefold symmetry of how do you relate the generations. All right. And to make this more physical and real, I'm going to the next slide. So for every type of particle, and we're going to assume for the sake of this argument that right-handed neutrinos exist as degrees of freedom.

Okay. Fine. All right. So you have eight different kinds of particle correspond to electron neutrinos, electrons, and the up and down quarks of different colors. Okay, so there are eight. For each one of these, if you treat it as a drac fermion, there are eight basis states. All right, corresponding to different spins, left or right handedness, and particles or their antiparticles. So that's where you get the 64. That's why this is a

64 dimensional representation. There's 64 degrees of freedom. But now the weird part, this is repeated three times. Nothing in nature requires there to be three generations of fermions. Everything we see around us is first generation physics. The only thing that might not be is, you know, we have neurons coming down occasionally as cosmic rays, but they're not good for anything except maybe accelerating evolution.

The, yeah, there's nothing in physics that requires there to be second and third generation when this thing discovered the famous quote is who ordered that? Why should there be the second generation? And then again, why is there a third generation? Well, now that popped up in 76. Oh, they, they suspected that might exist. Did they? Yeah.

They suspected the third generation may exist? Well, okay. It's extrapolation. Okay. You think you have one. Oh wait, we have two. Are there more? Okay.

I see what you're saying. Okay. But it's not like anyone here thinks there's four generations, is there? There's a reason for that. So for cosmological nucleosynthesis, you know, after the big bang creates all the preponderance of elements including neutrinos, cosmology only works right if you have three generations. Okay, so that puts a limit on the number of generations and that limit is less than four.

Got it. So we're pretty sure that this is a complete description of all known fields in physics, except maybe for some dark matter here or the right-handed neutrinos. Okay, so dark matter, if it's a particle, is either massive right-handed neutrinos or it's new bosons. Okay, it's probably something. Might also be modified gravity. Sure. Chances are neither of those series are working perfectly right now. Neither only dark matter or only modified gravity work well to describe phenomena we see in the universe. So it could be either. It could be both. But anyway, the mystery I want to focus on is why The Three Generations?

And I made some progress on it, I believe, and I wrote it up and I submitted it to the archive. And because I'm not universally loved in the high energy physics theory part of the archive, this was on hold for two months, which I believe may be a new record. I'm very proud of. A new record for you or a new record for them? I think it might be a new record for the archive. I don't know of a paper that was on hold for longer than two months and still accepted. Why was that?

Why do you think? It could be a combination of you had a TED talk that was quite successful, you gained notoriety, and so now you're a combination of an outsider who gets popular by evading the ordinary rules of academia, the hoops that they have to jump through, and also at the same time you're moderately a threat. I can be considered both a crackpot and a threat, somehow in some unusual superposition. Which is it? If I'm a crackpot, this doesn't belong on the archive. It should go to Vixra.

It should just ignore them. If I'm a threat, I'm scarier. What do you do? Do you suppress it? Well, if you suppress it, you don't let it on the archive, then there might be a big blow up because you actually get some attention. Maybe I get attention because I know what I'm talking about to a larger degree than most, or maybe it's just a fluke. But anyway, they had this hot potato on their hands for two months, tossing around going, what do we do with this thing? As far as I know. And finally, since the mathematics in it,

I'm pretty sure is correct. And it's potentially a fundamental contribution. They let it on. I'm happy about that. Things probably would have gotten pretty ugly if they had just said, nope, can't post it. I'm not sure what I would have done. Wouldn't have been pretty.

But anyway, they let it on. And so now we have this paper on to my knowledge It's the only paper that's gonna mention the unification of a generational symmetry along with CP and T in the standard model So I'm happy about this. All right, so For those who are going to be learning a lot of this from the ground up I wanted to give an introduction introductory page on What group theory is. So right here where I'm circling, this is the fundamental nature of what a symmetry group is. You have an element of

the group that you can compose or make a product with another element of the group and get a third element of the group. And this is called the group product, the symmetry group product. So you can combine two symmetries to get a third symmetry. And the order matters. So this is not a commutative product, like for like 2 times 3 is 6. 3 times 2 is different, can be different in a group. So it's a non-commutative product.

It is, however, associative. So symmetry groups have to follow associativity. They also have to have an identity element, and every group element has to have an inverse. Now, this description works for finite groups. It also works if you extend it to infinite dimensional groups. So a finite group looks like a set. It's just a finite set of things.

A Lie group, which is what a infinite dimensional group is, is a, well, I shouldn't say infinite. It's a group with infinite elements that are parameterized. This is a manifold. So a Lie group is a manifold. It's a manifold with a distinguished element on it, the identity element, and therefore you can also define it as a distinguished element. You can consider group elements close to the identity,

and you call the dimensions of that Lie group the number of orthogonal elements from the identity that extend out into the Lie group. And that's called the dimension of the Lie group, and the way those elements, those directions interact is called the Lie algebra. And that's called the dimension of the Lie group. And the way those elements, those directions interact is called the Lie algebra. Now, when you want to think about these things concretely, you use a representation to represent the group elements. And usually, you use a matrix representation. So if you like playing with matrices,

you can multiply matrices. And if the matrix multiplication corresponds to the group multiplication you say that's a faithful representation. Now one of the huge advantages of differential geometry is to not use a coordinate system. When you say a matrix you're choosing a coordinate system, no? You are. By choosing a coordinate system you make computations much more concrete, but ultimately, your computations have to reproduce and match those of the group which is coordinate-free. So for example, I'll do an example on the next page. Sure.

All right. Also, since you have now these group elements represented as matrices, matrices like to act on other spaces as well. So you can have a matrix multiplying a vector. And you would say that's a representation space of the group. And there are lots of, so when you represent a group with matrices, there are lots of different matrices you could represent group elements with.

You can go higher and higher in the dimensions of your matrices. And that's called the dimension of your representation. And you want to keep... Physicists are pretty casual with calling something representation versus calling your representation space. I was about to applaud you for even making that distinction. It drives me up the wall. You wanna have in your mind the distinction. Yeah, I'm glad you're aware of this too. I don't hear it talked about enough. Just for people, just so that they know, the representation is a map

and then the representation space, so the map is from the group to the matrix, the matrices, so GLV. And then that V is the representation space. Now physicists will say a particle is an irreducible representation of so-and-so. But even there, in my understanding, it's a member of a representation space. Absolutely correct. And usually said wrong. Right. And it's also technically a basis element of a representation space. Exactly. Yep. And I'll of a representation space. Exactly.

Yep, and I'll get to those too. Wonderful. Alright, so let's go to a concrete example. As you know, on Theories of Everything, we delve into some of the most reality-spiraling concepts from theoretical physics and consciousness to AI and emerging technologies, to stay informed in an ever-evolving landscape, I see The Economist as a well-spring of insightful analysis and in-depth reporting on the various topics we explore here and beyond. The Economist's commitment to rigorous journalism means you get a clear picture of the world's most significant developments, whether it's in scientific innovation or the shifting tectonic plates of global politics,

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Hamilton, I think it was like 1860 something or anything. He's a mathematician playing around with the algebra and figured out that you could have a really nice funny algebra that might relate to rotations if you cook up these four elements that he called quaternions. Quat for four. And Quaternions are also a division algebra. Remember, there are four division algebra, the reals, which is a one-dimensional division algebra, the complex numbers, which are two-dimensional division algebra, quaternions are four-dimensional, and octonions are eight dimension. They're sort of related to groups in that they have to have an inverse. And the reals complex numbers and quaternions have an associative product, but the arctonions do not associate. Okay, so most people don't associate

with the arctonions. At least you shouldn't. All right, so back to the quaternions. You have your three basis quaternions. Sometimes they're called i, j, and k. I'm calling them e1, e2, and e3. When you multiply these things, as this is the group multiplication, you multiply two imaginary quaternions, you get the third. If you do the reverse order of their multiplication, you get minus that. Okay? So the order, they're anti-commutative. And this turns out to be great, exactly what you need we need for rotations okay and we'll get to that also so but they're

four-dimensional you also have the identity element as well as if you multiply say e1 times e1 you get minus 1 so in order for the group to close in order for all the elements to to close within the multiplication of the group it has to be an eight dimensional group. And this is called the quaternion group. This finite group is order eight. There are eight elements, pluses, minuses of these. And that's the quaternion group.

And its multiplication table is down here. So if you multiply E1 times E2, you get E3. It's a multiplication table, just like you learn elementary school for numbers. This table should actually be eight by eight. Yeah. But the multiplication times minus one is trivial since minus one commutes with everybody. Okay.

So you can fill it out. Exercise for the reader fell at this table to eight by eight with the minus one. You'll just see copies of these things. All right. The representation of these quaternions is most succinctly done using Pauli matrices. So if this is a two by two complex matrix representation of the identity, that's the representation of E1, that's E2, and that's E3. And if you do matrix multiplication, you'll see multiplying this matrix times that matrix gives you that matrix.

Just to be clear for people who aren't entirely familiar with linear algebra or group theory, they're seeing a 4x4 block above and then they're seeing a 2x2 block at the bottom. So you're saying that this E1, imagine in the block above actually contains its own 2x2 block. I can't do it with my mouse so it wouldn't show. But see how it says, you understand what I'm saying. So this is the multiplication table. It turns out the multiplication table can be related directly to the representation. Okay, but you probably don't want to think about it that way. That will swirl your head around in funny ways that you shouldn't be messed with yet.

What you want to think of is you can you can represent these group elements by matrices and if you multiply the matrices they satisfy the same multiplication table as the group elements. That's the way they do it. Also there's a vector representation of these things, two by two complex, and that's called, that's what the two was in the previous slide. So these quaternions relate to spin. And that'll be more apparent later. Is it important that the values are complex? Yeah.

Because ultimately, remember from the first slide, this is SL2C. So complex numbers are here. If you go to, you can represent these as real matrices, but then they're 4 by 4 real matrices. Now, why can't you just have a 4 by 4 real representation? Yeah, you can have a 4 by 4 real representation of quaternions.

If you do it in the spin group, I think those give you rotations of vectors. So that might be the vector representation space. So let's now go to a Lie group that has the quaternion group as a subgroup and think about actual spacetime rotations. This is going to be a little trickier, so brace yourself. All right.

So this brings it all together in one spot. You've got your spin group comes from a Clifford algebra. So if you want to represent a spin group, you need to use a Clifford algebra. OK? OK. Because ultimately, we want to see how these things act on spinors down here.

OK? Yes. And if you're going to act on spinors with a spin group, you need to use a Clifford algebra and get representative matrices for your basis elements of your Clifford algebra. These are called your gamma matrices. And this describes the preferred chiral or vial representation of the gamma matrices. You multiply these four spacetime vectors together.

So gamma zero is usually for time. That squares to plus one, the gamma one, the gamma two, gamma three, each individually square to minus one. That's your three and CL one, three. You multiply them all together, you will get what is called the pseudoscalar, your Clifford pseudoscalar, which is sort of a volume element if you're a geometric algebra guy. So like I said during our introduction, Clifford Vec... When your celebration of life is prepaid in advance, it becomes a gift from you to your family later, because no one should have to plan for a loss while they're experiencing

one. Paying in advance protects your loved ones and gives you the peace of mind you deserve. Let us help you plan every detail with professionalism and compassion. We are your local Dignity Memorial provider. Find us at DignityMemorial.ca. Vectors are really good for rotations. Clifford algebra is great for doing rotations of vectors and of spinors.

That's their reason for existing. Now, you can also use Clifford algebra to do reflections. And if you consider that rotations are essentially the combination of two reflections, you can always build any rotation as two reflections. So there's this larger group, the group of reflections of spacetime. And spin 1,3 is a subgroup of this one, and the space of reflections of spacetime is called pin 1,3. And this is a pun on the relationship

between the special orthogonal group for rotations and the orthogonal group, which also includes reflections. Okay? But here, since we have a group where we're dealing, we have a spinner representation space of it, it's a double cover of the orthogonal group and I think pin is a double cover of the orthogonal group the same way spin is a double cover of the special orthogonal group. I just realized the fact that this makes people like you and I even chuckle like just a small turtle. Yeah. It's such a nerdy joke. I know.

So I didn't come up with this joke, but I do exploit it. So, um, you can use Clifford algebra operations, including the pseudoscaler to do a reflection. So say you have a vector V and you want to reflect it through a different vector U. Okay. So basically you want to take, take this V and reflect it through a different vector u. So basically, you want to take this v and reflect it through here. So now you get a different vector over here.

So what you've done is you've taken the u component of v and reversed it to get that. And you can do this with Clifford algebra. So you get v gets the same perpendicular, this is the component that is perpendicular to u, and then you get the component parallel to u reversed. Okay? All right.

And since the spinners are essentially the square root of vectors, this reflection operation acts this way on the spinners. The same operator with this vector acts on spinners. So this pin group has a vector representation space. It also has a spinner representation space. Now, the group itself, it consists of rotations near the identity, so the normal small rotations, but it also has these large reflections.

It can have reflections in space, or it can have emmett, I'm trying to say reflections in time, but say it backwards. I can't pull that off. Sure. Anyway, these correspond to elements of the pin group corresponding to what's called parity reversal and unitary time reversal. And these are elements of the pin group. Now and you get them just as a reflection. So this is a reflection through the unit time direction.

And this is a reflection through all three spatial directions. Okay, and this is called parity reversal, which is usually given the label P, and this is unitary time reversal, which for now we'll call T. Things get more complicated when, because you can't apply unitary time reversal in quantum field theory without going to negative energy states. So we're going to have to adjust this a little bit in the next couple slides. But for now, you can just consider parity reversal and time reversal as subgroups of the pin group. And basically, the full pin group here is there are four components here corresponding

to PT and PT. Another way of saying is the full Lorentz group, including reflections, has four disconnected components. So as a group, the Lie group is not connected to the identity. Only one component is connected to the identity. And that component, I think it's usually called the special orthocrinous component of the Lorentz group. the special orthochronous component of the Lorentz group. Right. But then there are these other disconnected components corresponding to temporal and spatial reflections, parity in time.

So now, if you look at the multiplication, how these parity in time things compose when you multiply them using Clifford multiplication, they anti-commute. And in fact, they give you exactly the same finite group as the quaternion group. That's pretty neat. So you asked earlier, like, so are fermion states basis vectors in the representation space?

Mm-hmm. Not only that, spin itself as a quantum number is the eigenvalue corresponding to those eigenvectors. Right. Okay, so this is what mathematicians in representation theory use to show what are called weight diagrams and root diagrams. So basically you take your six-dimensional Lie algebra, you pull out, you say I'm going to distinguish two basis elements of this Lie algebra, you pull out, you say I'm going to distinguish two basis elements of this Lie algebra that commute and that's, if that's the most you can do, that's called a Carton sub algebra. Okay, so this J3, this corresponds

to rotations around the z-axis. This K3 corresponds to boosts along the z-axis. And since they're both along the z-axis, these commute, whereas other generators of pin 1, 3 don't. But these two commute. You could also do J1, K1, but let's do J3, K3. It's conventional. Since this, as a matrix, acts on a spinner as a column matrix, acted on by this matrix, you can just read the

eigenvalues right off the diagonal of the matrix. So this has plus or minus one half spin and this has plus or minus one half boost. So the spinner representation is just the complex four-dimensional column matrix that's acted on by this matrix. That's how the spin algebra acts on spinners. It just acts as a matrix multiplication, a matrix multiplying this vector with this unusual representation. So as a spinner, since you can just read off the eigenvalues since it's already diagonalized.

So for a spinner, you have four states. For a drag spinner, you have four states, left-handed spin have four states left-handed spin up left-handed spin down right-handed spin up and right-handed spin down okay okay and those correspond to the spin and boost numbers of these things now for vectors you also have four spin states but their spins are plus or minus one and if you act with the adjoint on the algebra itself you end up with these four plus the two in the middle with zero charge. Now, mathematicians call these things weights, physicists call them charges or spins, and essentially spin, so spin, like the spin

of an electron, is also interpretable, can be interpreted as gravitational charge okay because it's the the weight of spin 1 3 maybe the viewers can see this but currently on my screen the previous too small so what is the X and the y-axis the x-axis here is boost so that corresponds to eigenvalues of k3. The y-axis is spin. That corresponds to eigenvalues of j3. Got it. OK, and it showed explicitly here. So these are eigenvalue equations.

This is a matrix operating on a spinner. And this is the cross product of this matrix with a, in Clifford algebra vectors are also represented with matrices. So you have to do the Clifford cross product to do this product. But don't worry about it. The end result is you get a vector representation space of pin 1, 3, which you see by plotting its weights.

OK. All right. So back to the fermions, physically what they are. With these glyphs, for particles, which are triangles facing up. I've said left-handed spin up. Okay, it's got spin up. It's got this little bit up here.

Spin up right-handed particle, spin down right-handed particle, spin down left-handed particle. And the parity and time symmetries transform between them and they correspond to these eigenvectors as spinors. So just as you said these are the eigenvectors, these are the eigenvalues, this is the physical, I guess it would have to be massless fermion state corresponding to that eigenvector. Okay, it's something that's left-handed moving down. So it's moving down along the negative z-axis and spinning this way with the spinning momentum going up. So that's why it's left-handed. I'm confused because before on the previous slide there were more than just these four.

There were several others. Right. This is for spinners. The previous slide also dealt with the eigenvalues for vectors and the Lie algebra itself in the adjoint representation. I see. So I combined them all. So the next page just shows these four for a spinner. Got it.

So the next thing to deal with is, how do you relate fermions to anti-fermions? So in order to do this, you have to make your Dirac fermions complex. And then you introduce a charge conjugation symmetry operator that includes complex conjugation. So as well as this Clifford algebra operator and this complex i, you have this k here

as complex conjugation. This is why I said that the real and imaginary parts, you have complex conjugation corresponds roughly to going from particles to antiparticles. Why do you say roughly? Because you also multiply times gamma 2, which swaps, and gamma 2 swaps the chirality as well as swapping the spin. I see.

All right. So you can, there are two things you need to do. So in the future for quantum field theory, which I can show you at the end if you like pain, you want to redefine your time reflection to have not to have a unitary time conjugation symmetry, but have an anti-unitary time conjugation symmetry, which corresponds to the one we see in the physical world because it leaves positive energies as positive energies and it

just reverses motion. So instead of changing energy to minus energy, it actually reverses the motion of the field or particle. So you build that this way and you end up with this anti-unitary operator that's a combination of Clifford multiplication and complex conjugation. The parity operator, you change the same. I'm doing a slight cheat, which is physicists, particle physicists tend to like to use spin 1, 3, because it's equivalent to SL2C.

But for physically, the universe actually appears to prefer spin 3, 1. So the universe seems to have possibly a preferred negative square for time directions and a positive square for distance directions. But you can cheat just by putting these factors of i in here into your operators. This i here, this i here this i here and that i there so this this means the CPT symmetry really is a subgroup of spin of pin 3 1 not pin 1 3 but we can cheat by complexifying them and make and using pin 1 3 even though we're living in in3-1. Yeah, one of the other reasons that physicists prefer 1-3 is that that corresponds to the manifold that we live on with three spatial dimensions, no?

So are you saying that our universe somehow prefers three temporal directions? Um, no. It's arbitrary whether you choose the square of time to be positive or negative. So when you choose a Minkowski metric, remember there are two different conventions called the East Coast Convention and the West Coast Convention. The convention in which space squares to plus one for distances is typically favored by general relativists, but where space squares to minus one is typically favored by general relativists, but where space squares to minus one is typically preferred by particle physicists. And this is mostly about the standard model

in particle physics, so I'm trying to use pin 1, 3. However, for CP and T, they actually naturally live in P3, 1. And you're getting them sort of into pin 1, 3 by multiplying times i to change the signs. Sorry, it's just a matter of matching conventions. If you get into this deeply, you got to figure that out explicitly. All right, so now you combine CPT. Now this C operator commutes with P and T.

So this group of charge, parity, and time symmetries for a Dirac fermion, right, which lives not... so this whole symmetry does not live in pin 1-3 or pin 3-1. It lives in this larger group that acts on complexified Dirac fermions. So charge conjugation is outside of rotations. It lives more in the gauge field side of things. It's kind of weird. So P and T are part of spacetime.

C is not. C is particles and antiparticles, which has to do with charge. That means that's something that you have to cross with a plus or minus 1 to the previous symmetries? Yeah. So basically, the CPT group is the direct product of the PT group with charge conjugation.

So charge conjugation is this extra bit that's getting added on. Now in the standard model, for fermions, remember that charge conjugation symmetry, parity symmetry, and time symmetry are all violated. And they're violated generationally. So between the first, second, third generations, when you compute their masses, right, with the Yukawa couplings, those Yukawa couplings don't respect CP and T. They violate all of them. And they violate CP and CT and PT. They violate all of them. What they don't violate is CPT. So the combined

symmetry CPT of this three is actually a conserved symmetry of the standard model, including the Eucala couplings, but CP and T individually are not. So when we figure out a symmetry between the three generations, we want that symmetry to interact non-trivially with C, with P, and with T, but not with CPT. How does the CPT group act on a Dirac fermion? Remember before we had this square of fermion spins. Now we've added charge. Got it. So see, just all it does is it converts charge to minus charge.

It leaves the spins and the helicity unchanged. Parity leaves the spin unchanged. Time reversal changes signs of both, but leaves charge unchanged. So you could start, say, with your right-handed spin up fermion state. And using these CP&T symmetry operators, you can get to any other fermion state, any other fermion basis

state, I should say. An actual fermion state is a complex superposition of these eight. Okay, but these are the basis states. So next, the big question, how do you formulate triality to expand this cube to a larger space with the triality symmetry that is going to have to map between three cubes? Mm-hmm. with the triality symmetry that is going to have to map between three cubes. And remember, this is just a Dirac fermion. Dirac fermions, there's no, I mean, you could go from a Dirac fermion to another fermion

to a vector maybe, but that's not a natural way to think about it. But if you want to think about a trifold symmetry that goes between three things, you really want to work division algebra. So what we want to do is convert a Dirac spinner into a quaternion. And because a Dirac spinner has four complex degrees

of freedom and a quaternion only has four real degrees of freedom, we're going to actually have to use a complex quaternion. OK. All right. But then once we have quaternions, then we can use triality to map between them, and then we can get a generational symmetry. Now, why not just use two quaternion. Okay. All right. But then once we have quaternions, then we can use triality to map between them and then we can get a generational symmetry.

Now why not just use two quaternions instead of a complex one? That would be two main degrees of freedom. Oh, right. All right. All right. Beckel's seatbelts for this dictionary between Dirac spinners and complex quaternions, which are also called biquaternions. So you take a Dirac spinner, it's got left and right chiral parts, spin up and complex quaternions, which are also called biquaternions. So you take a drag spinner, it's got left and right chiral parts, spin up and spin down,

each one. You compute its charge conjugate, you get the charge conjugate spinners. And because of that W2, that lifts these up here. So now you take just the left-handed chirality of your Dirac spinner and your charge conjugate Dirac spinner. And you assemble them in this 2 by 2 complex matrix. Because this is a 2 by 2 complex matrix, now you have all the degrees of freedom of your Dirac spinner, but now you have it as a complex quaternion. Because you use the 2 by 2 matrix representation from the polymatrices for your quaternions,

and you have the complex numbers to generalize it. So here you have a matrix representation of a bi-quaternion. How are you going from Psi C to Psi Q? You pull these two degrees of freedom out and these two out, which are all four. These are just conjugated and swapped. So it's just a dictionary. You've changed from a column matrix to a two by two complex matrix and done some operations. But they're nice operations because you've left it left-handed.

You've taken only the left-handed components. And this is nice for all sorts of reasons. First of all, you have a translation now of psi of a drag spinner via this matrix representation and polymatrices into complex quaternions. Now if you do, if you use the same dictionary to go back and forth, the spin group, the generators of the Lorentz algebra, operate on this spinner just as quaternion multiplication, which is pretty wild.

And boosts, so say you want to rotate around the z-axis, you just multiply it times the e3 quaternion. That's the generator of rotations around the z-axis. K3, which is the e3 quaternion times i, gives you the generator of boosts along the z-axis. And this is how you, and this gives you directly a description of spin-1-3 as SL2C acting on these things. Okay, so this is why this is true, this correspondence. So this looks extremely familiar to me. Is this known or is this new?

So the top part in particular, is that new? This is a known result. You know, it's been through SL2C. This correspondence, it may have been done before. I don't know. I think I saw it. I think some other people have come up with it at some point, but it's really nice.

And I don't think anybody's... Certainly nobody's used it for CPT. It's been used for representation so since these things are both left-handed this operation of the spin algebra on these biquaternionic spinors is just multiplication from the left. Okay so SL2C acts as rotations in a very obvious way. I don't know the clear clearer exposition than this, believe it or not. But anyway, the real fun happens when you convert CPT using this correspondence.

CP and T correspond to these operations on your drag spinners. That corresponds to this in terms of matrix representations, which corresponds to this in terms of quaternion operations and complex operations, which corresponds to these operations, C, P, and T, as operators. You combine them, you multiply C times P times T, you get the C P T generator is minus I. So now we're in a position, now that we've converted from Dirac spinors to bi-quaternions, we can do triality. What is a triality operation in the quaternions? It's

actually pretty fun. You construct a special quaternion. This is actually called a Hurwitz integer, even though it's not an integer, it's a half integer. As a quaternion that operates on other quaternions, just do the normal quaternion multiplication. So if you take this, what we'll call a triality generator as a quaternion, and you cube it, you get the identity. So that's a good sign. But the real fun happens when you use the adjoint action of this triality on the imaginary quaternions is it cycles them. It leaves one invariant because of this, but it turns the i-quaternion, which is e1, into

e2, turns e2 into e3, and it turns e3 into e1. So you have this trifold cycling of the quaternions via this triality generator. So now we essentially have it. All we have to do is add this triality generator as T to our P and T to get a larger group, a larger finite group that includes triality. Okay, if you do it with P and T, you get this group of order 24. So it goes from the quaternionic group of order 8 to the binary tetrahedral group of order 24.

You know, you multiply it times three in there for your triality symmetry. Yes. But remember, if you look, remember T is purely quaternionic. It doesn't have a complex conjugation or an I in it. Right, well if we go back to our previous page, the C operator has an E1 in it. So this triality generator doesn't commute with C, it

doesn't commute with P, it doesn't commute with T, but it does commute with CPT. Which is exactly what we wanted. And I was very happy to find this. So the difference between this, what I see philosophically or maybe psychologically from the E8 theory that I saw from the 2007 talk I believe. In E8 it looked like you were starting from the top down thinking of what's extremely beautiful and simple and then trying to find physics from its tendrils below. But over here it looks more ad hoc like you're thinking well ad hoc in the sense that the universe tends to obey these. Okay let me hobble these all together and move upward. Okay and you're right in the sense that we're

going from the bottom up, but let me tell you the kicker. I'm not saying that as an advantage or disadvantage, by the way, it's just, it's a different way of looking and you're exactly right. It's like people find E eight to be a hard thing to swallow. It's a lot, but if you're going to extend the CPT group by a trifold symmetry, there is a unique way to do it that is not a direct product and that commutes with CPT. And this is it. OK. OK.

This, in my opinion, is huge. Because you want, for your generational symmetry between these generations, you want it not to commute with C, not to commute with P, and not to commute with T. Right, right. So it can't just be a direct product group, the finite group, that includes generational symmetry. But you do want it to commute with CPTT because that's a preserved symmetry of nature. And here it is.

This is a CPTT group. Your triality commutes with CPT. It includes this group just with PT as a subalgebra. So what we're seeing is triality doesn't have to do so much with the gauge field. It has more to do with spacetime itself, because it's wrapped up with parity and time reversal as a finite group. And when I say this is a unique way to do it,

this is the only way to have a trifold symmetry. This is the only way to extend the CPT group to a larger group with a trifold symmetry that respects CPT. How do you know it's the only way? You've shown that it's a way. Mathematicians who know a lot more finite group theory than I did said this was the unique way to do it. What you get for the CPT group, and as far as I know, nobody has named this group. It's a central product of the binary tetrahedral group and the dihedral

group. It's a finite group of order 96. So how does this act? So remember I said we have to somehow act on three cubes of fermions. How does that work? It's really freaking pretty. You get what's called the 24 cell. Okay? So the 24 cell, remember a cube has eight vertices. So the 24 cell is composed of four cubes in, is composed of three cubes in four dimensions, all linked together in a very pretty way by triality. If you go to a projective representation space

where we're dealing with just the weights of a Dirac fermion, remember you get the plus or minus 1 half everywhere for spin, plus or minus for boost, and plus or minus for charge. OK, and that's how we have the fermion cube. Now, we put this in four dimensions, and we act with CP and T, and we get the normal transformations between the fermion states of the cube. But when you include triality, this gives you two other cubes on top of the one generation cube. So if

it weren't for the fact that this way of including triality to get to this larger CPTT group as a finite group that includes triality. If it weren't for the fact that that's the unique way of including a trifold symmetry, this next thing that happens is too outlandish to swallow. Because what happens is the spins and charges of the second and third generation do not make sense as spins and charges unless you use Triality to go back and forth. Interesting. So for example using Triality, the second generation charges look like plus or minus one, which aren't the

charges of a spinner at all. It's only when you go back and forth via Triality that it maps to, okay this has exactly the same charges as the first generation. So somehow, triality is making three regions of space-time, and the second and third generation must be existing relating to the second and third versions of space-time, because they don't make sense just with respect to the first. And somehow, our space-time has to be this merging of these three copies of space-time, one for each different fermion generation. This is the only way it makes sense. Or this whole thing is just not going to make sense at all. But if it is going to make sense, that's the only way to do it. Just a moment. If I heard you correctly, you're saying that there are three copies of space-time and our universe is some merging

of the three. That's the only way this is going to work. Is there then four, you're considering ourselves a fourth or it's an intersection of those three? Has to be those three because remember we have triality, tri-triality is this trifold symmetry now between three different things and it's an essential part of a finite Lie group if we're gonna extend that Lie group if we're gonna extend this finite group to act on generations this is the only way to do it but it has

this outlandish result and the only way it makes sense is if you if you're gonna merge these three spacetimes and that's the one we live in I know it's outlandish but it's the only way. Yeah. Are you saying that the first generation lives, quote unquote, lives in space time one, the second one, second generation lives in space time two? Yes. And that somehow those three space times live on top of each other. And that's why we see the second and third generation particles at all.

But we see them with all these weird masses and mixings. Super interesting. And that merging of three spacetimes, it's similar to how a fiber bundle, you know, you have a section of a fiber bundle and you consider that as your base, perhaps your spacetime. But via a gauge transformation, you can change to a different section of the bundle. Which section are we in? Well, you have a gauge transformation between all of them. It's you merge all those sections together

and think of that as the space-time we live in. I see. So the same thing is going on here except we have a finite group, and you can't continuously vary between sections. You just have three discrete sections of this finite group. So it's, I'm not sure how to deal with this exactly mathematically and I haven't formulated it yet in a proper way, but conceptually this has to be the way it works. Now this cube or this hypercube projection or three hypercubes projected down, is this

supposed to be visually informative or is this more like a flourish to give us the idea because I'm looking at that and I can't discern What I'm supposed to derive from this. They're not they're not hypercubes. They're normal eight vertex cubes It's just now they're living in four dimensions and we're projecting them down to show the plot Is there a question that you could ask me hovering your mouse over one of the vertices and say, okay, what corresponds to the P transformation of this guy? Like something like that, something physical. Well, I didn't label them. But if you start, say, with this is a right-handed spin up electron,

where's the P transform is, oh, sorry, right-handed spin down electron. Where's the left-handed spin down electron is down here. So this is the P-transform, this red line is the P-transform. Again, I don't know if it's just my screen, but I see, is there orange and red or is there just red? No, there's red, green, blue, and then black. Okay, so anytime you see a red coming out, it's always going to be three. That's right.

It's the vertices of a cube. Now via these, this triality symmetry, this right-handed spin down electron state is related via this down here to a right-handed spin down Mee-Wan state and a right-handed spin down tau. That's a triangle. If you look at the spins, like I said, the spins only make sense for one generation. For the spins of these triality transformed generations to make sense, you'll have to transform back via triality.

Remember, this is just for a single Dirac fermion that now there are three copies of. And like I said, this is because what we're finding here is that generational triality is related to space-time and not related to grand unified theories. Okay, it doesn't come out from some unusual segmenting of a Calabi-Yah, it comes out as this symmetry that is related to spacetime. And remember, this is for one Dirac fermion.

And there's 24 here for the 3 times 8. But remember, in the standard model, there are eight of these things. So remember, at the beginning, there were like 192 fermion states total, right? So to describe all of them, there's only one way to do it. So the only way this triality symmetry merged with CP and T is going to make sense in a fully unified theory is via exceptional unification. It might be E7 and it might be E8. Both of these support

triality, but ultimately when you include all the particles, all 192 states, along with this triality symmetry, you get something in seven or eight dimensions that's going to project down to this pattern if you choose your projection nicely. Now in E8 theory the only way to get this to work is if you use a compact real version of E8 because the others won't let you simultaneously embed the weak force and space-time. I spent a lot of time trying to get that to work, it can't. So the only way to get from this to a physically realistic

theory is you're going to have to use some sort of wick rotation to go from this to a imaginary weight or something that's non-compact. And what's the problem with having real forms? The problem is we don't live in real space time. We live in Minkowski space time. We don't live in Euclidean space. We live in Minkowski space time. We don't live in Euclidean space. We live in Minkowski space locally.

Now would Peter White say, I don't know if you've looked at his latest theories. Yeah, Peter's a brilliant guy. I've tried to follow his stuff as best I can. And if he is successful in going from, you know, SO4, SO6 to SO42 with those operating on twisters, which are just two spinners. If he can wick-rotate between those two, then that may provide the path for wick-rotating between this sort of compact description and a non-compact description that includes triality. So who knows, we might

have each other's missing pieces. There aren't enough people on this planet that are looking at his theories and mine and trying to synthesize them. I don't know of anybody trying to do that at all, but that would be a fun thing to look at. Ultimately, the only lead groups that support this sort of triality as part of a finite group are the exceptional groups because they're built from quaternions and octonions and complex numbers. And they have triality as part of their core. So I've gone from the ground up here and I've tried to get you to buy it. But if you've bought it up to here, you're also going to have to accept the exceptional unification is going to be the only way to go.

Now, this may embed in some larger form. It may embed in a CACMUTI or a generalized Lie group. And you're going to have to do that anyway if you're going to strive quantum mechanics. Because so far, everything I've talked about is on the classical level. If you want to talk about quantum field, there's things we can,

but that gets much more messy. Because remember that there's a… if you remember from group theory, you go from representations that there's a correspondence for a faithful representation in terms of matrices. But for going to quantum field theory, you also have to have a faithful representation in terms of creation and annihilation operators. You're dealing with an infinite dimensional representation space. But you still have to have a faithful representation in that infinite dimensional representation space.

So there's a correspondence to C and P and T generators that operate as quantum field theory operators. And that's, it's an isomorphism because of the faithfulness of the group representation. So you can get away with doing everything on the level I have here, but you can also translate it into quantum field theory operators and operators on fermionic creation and annihilation operators, which ultimately you have to do for quantum field theory. That's wonderful, Garrett. Okay.

I think I've utterly destroyed any predicted timeline for how long this would last, but you've asked a lot of really good questions. I hope you've gotten to a fun place. There are things in here such as the unification of equations of motion like how to start with a generalized Yang-Mills action and get absolutely everything else. How does space-time embed inside a Lie group in a reasonable way? How does E8 unification work? How does E7 unification work? Is there anything to call for a and her attempt to describe things using division algebras, and how does that fit into this picture?

What's your answer to that last one? E7, complexified. So there's something called, let me show you behind the curtain here. All right, so you're actually looking at a website called differential geometry. If you go to this website, you'll see these slides as well as everything else. So there's something called the exceptional magic square for how things are built. And the for how things are built. And the Lie group E7 is made from a representation space that is the direct product of complex quaternions and octonions. So you take the quaternions,

and I already showed you how to direct fermions with these generations correspond to complex quaternions. The different particle types correspond to different octoneons, also introduces complex conjugation. And that's how you get to E7. So E7 consists of three copies of the complex quaternionic octroneons. And that's Colferase and Mia Hughes' bread and butter. That's the representation space they work with.

So if they're going to relate these things with three generations, they're going to do it with Triality. If they do it with Triality, they're going to be in Complex E7. Well, Garrett, I got to get going. And I appreciate you spending so much time with me, man. Yeah. I appreciate you spending so much time with me, man. Yeah. No, this this is a great conversation. And for anybody who wants to learn more of this

stuff, they can go play on my wiki, they can look at my papers. It's a good time. And these things go mathematically deep and are very pretty also. And I hope they turn out to be right about the universe. That's the ultimate hope. All right. I definitely got to have a one-on-one with you. We can also reference the website at the same time, but more of a podcast. And so there's a phrase that I use frequently called, just get wet. So don't try to drink from the fire hose, just get wet. And there'll be a variety of different terms that you don't understand and concepts and

structures and so on. Just pick out the ones that you're interested in and go down that rabbit hole. Exactly. Yeah. Stick your foot in the pool everywhere, but dive in where you want to. Also, thank you to our partner, The Economist. Firstly, thank you for watching. Thank you for listening.

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