Transcription
Is it possible to say something that kind of captivates, keeps being brought up by physicists? Which is gauge theory, gauge symmetry, as one of the more complicated types of symmetries? Is there—is there an easy explanation? What the heck it is? Is that something that comes up on your mind at all?
Well, I'm not a mathematical physicist, but I can say this: it is certainly true that it's been a very useful notion in physics to try to say, like, what are the symmetry groups like of the world? Like, what are the symmetries under which things don't change? Right? So we just—I think we talked a little bit earlier about—it should be a basic principle, that a theorem that's true here is also true over there. Yes, and same for a physical law, right? I mean, if gravity is like this over here, it should also be like this over there. Okay? What that's saying is we think translation in space should be a symmetry; all the laws of physics should be unchanged if the symmetry we have in mind is a very simple one, like translation. And so then, um, there becomes a question like, what are the symmetries of the actual world with its physical laws? And one way of thinking—is an oversimplification—but like one way of thinking of this big shift from uh, before Einstein to after, is that we just changed our idea about what the fundamental group of symmetries were. So that things like the Lorentz contraction, things like these bizarre relativistic phenomena—or Lorentz would have said, "Oh, to make this work, we need a thing to um, to change its shape if it's moving," yeah, nearly a speed of light. Well, under the new frame of framework, it's much better; you're like, "Oh, no, I wasn't changing its shape; you were just wrong about what counted as a symmetry." Now that we have this new group, the so-called Lorentz group, now that we understand what the symmetries really are, we see it was just an illusion that the the thing was changing its shape.
Yeah. So you can then describe the sameness of things under this weirdness. That exactly that is general relativity, for example.
Yeah, yeah. Still, um, I wish there was a simpler explanation of like exactly—I mean, you know, gauge symmetry is a pretty simple general concept about rulers being deformed. I—it's just I—I uh—I've actually just personally been on a search—not a very uh, rigorous or aggressive search—but for um, something I personally enjoy, which is taking complicated concepts and finding the sort of minimal example that I can play around with, especially programmatically.
That's great. I mean, that this is what we try to train our students to do, right? I mean, in class, this is exactly what this is like—best pedagogical practice. I do hope there's a simple explanation, especially like I've uh, in my sort of uh, drunk random walk—drunk walk, whatever that's called—uh, sometimes stumble into the world of topology and like quickly, like, you know, when you like go to a party and you realize this is not the right party for me. So whenever I go into topology, it's like so much math everywhere, I don't even know what it feels like. This is me like being a hater; is I think there's way too much math. Like they're two—the cool kids who just want to have like everything is expressed through math as because they're actually afraid to express stuff simply through language. That's—that's my hater formulation of topology, but at the same time, I'm sure that's very necessary to do sort of rigorous discussion. But I feel like—but don't you think that's what gauge symmetry is like? I mean, it's not a field, I know. Well, but it certainly seems like yes, it is like that. Okay, but my problem with topology, okay, and even like differential geom—and differential geometry, is like you're talking about beautiful things—like if they could be visualized—it's an open question if everything could be visualized—but you're talking about things that could be visually stunning, I think, but they are hidden underneath all of that math. Like if you look at the papers that are written in topology, if you look at all the discussions on Stack Exchange, they're all math-dense, math-heavy, and the only kind of visual things that emerge every once in a while is like uh, something like a Mobius strip, every once in a while some kind of uh [Music] simple visualizations. Well, there's the the vibration, there's the the Hopf vibration, or all those kinds of things that somebody—some grad student from like 20 years ago wrote a program in Fortran to visualize it, and that's it, and it's just, you know, it makes me sad because um, those are visual disciplines, just like computer vision is a visual discipline, so you can provide a lot of visual examples. I wish topology was more excited and in love with visualizing some of the ideas.
I mean, you could say that, but I would say for me a picture of the Hopf vibration does nothing for me, whereas like when you're like, "Oh, it's like about the quaternions," it's like a subgroup of the quaternions, and I'm like, "Oh, so now I see what's going on. Like, why didn't you just say that? Why were you like showing me this stupid picture instead of telling me what you were talking about?"
Oh, yeah, yeah. I'm just saying—no, but it goes back to what we were saying about teaching, that like people are different in what they'll respond to. So I think there's no—I mean, I'm very opposed to the idea that there's one right way to explain things. I think there's a huge variation in like, you know, our brains like have all these like weird like hooks and loops, and it's like very hard to know like what's going to latch on, and it's not going to be the same thing for everybody. So well, I think monoculture is bad, right? I think that's—and I think we're agreeing on that point, that like it's good that there's like a lot of different ways in and a lot of different ways to describe these ideas because different people are going to find different things illuminating. But that said, I think there's a lot to be discovered when you force little like silos of brilliant people to kind of uh, find a middle ground or like uh, aggregate or come together in a way. So there's like people that do love visual things. I mean, there's—there's a lot of disciplines, especially in computer science, that they're obsessed with visualizing—visualizing data, visualizing neural networks. I mean, neural networks themselves are fundamentally visual. There's a lot of work in computer vision that's very visual, and then coming together with some—some folks that were like deeply rigorous and are like totally lost in multi-dimensional space where it's hard to even bring them back down to 3D—they're very comfortable in this multi-dimensional space. So forcing them to kind of work together to communicate—because it's not just about public communication of ideas; it's also—I feel like when you're forced to do that public communication, like you did with your book, I think deep profound ideas can be discovered that's like applicable for research and for science. Like there's something about that simplification—or not simplification—but distillation or condensation or whatever the hell you call it—compression of ideas that somehow actually stimulates creativity, and uh, I'd be excited to see more of that in the—in—in the mathematics community.
Can you let me make a crazy metaphor? Maybe it's a little bit like the relation between prose and poetry, right? I mean, if you—you might say like, "Why do we need anything more than prose?" You're trying to convey some information, so you just like say it. Um, well, poetry does something, right? It's sort of—you might think of it as a kind of compression—of course, not all poetry is compressed, like not awesome—some of it is quite baggy—but like um, you are kind of—often it's compressed, right? A lyric poem is often sort of like a compression of what would take a long time and be complicated to explain in prose into sort of a different mode that it's going to hit in a different way. You