20: Sir Roger Penrose - Plotting the Twist of Einstein’s Legacy

Plotting the Twist of Einstein’s Legacy
Information
Guest	Sir Roger Penrose
Length	02:23:04
Release Date	24 January 2020
YouTube Date	1 March 2020
OmnyFM	Listen
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YouTube	Watch
Portal Blog	Read
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Sir Roger Penrose is arguably the most important living descendant of Albert Einstein's school of geometric physics. In this episode of The Portal, we avoid the usual questions put to Roger about quantum foundations and quantum consciousness. Instead we go back to ask about the current status of his thinking on what would have been called “Unified Field Theory” before it fell out of fashion a couple of generations ago. In particular, Roger is the dean of one of the only rival schools of thought to have survived the “String Theory wars” of the 1980s-2000s. We discuss his view of this Twistor Theory and its prospects for unification. Instead of spoon feeding the audience, however, the material is presented as it might occur between colleagues in neighboring fields so that the Portal audience might glimpse something closer to scientific communication rather than made for TV performance pedagogy. We hope you enjoy our conversation with Professor Penrose.

Eric Weinstein (right) talking with Sir Roger Penrose (left) on episode 20 of The Portal Podcast

Transcript

The Portal Group's Transcript Completion Project generates and edits transcripts for content related to Eric Weinstein and The Portal Podcast. Completed transcripts are available to read on The Portal Blog and The Portal Wiki. If you would like to contribute, you can make direct edits to the wiki, or you can contact Aardvark or Brooke on The Portal Group Discord Server.

This transcript was generated with Otter.ai by Brooke from this content's YouTube version. It was edited by Aardvark#5610. Further corrections and contributions were provided by ker(∂n)/im(∂n-1)≅πn(X), n≤dim(X)#7337.

Housekeeping and Introduction

00:00:00
Eric Weinstein: Hello, this is Eric with two pieces of housekeeping before we get to today's episode with Sir Roger Penrose. Now in the first place, we released Episode 19 on the biomedical implications of Bret's evolutionary prediction from first principles of elongated telomeres in laboratory rodents. I think it's a significant enough episode, and we've had so much feedback around it, that before we continue any kind of line of thinking surrounding that episode, we'll wait for my brother and his wife, Heather Heying, to return from the Amazon where they're currently incommunicado. So thanks for all the feedback, it's been very interesting to process.

00:00:33
The second piece of housekeeping surrounds today's episode with Roger Penrose. Now, I know what I'm supposed to do. I'm supposed to talk about quantum consciousness and The Emperor's New Mind, maybe ask Roger about the many-worlds interpretation of Quantum Mechanics, or the weirdness of quantum entanglement. I'm actually not that interested. I also don't want to go back to his earliest work on singularities and General Relativity with Stephen Hawking.

00:00:54
What I instead want to do is to remind you of what Roger is in fact famous for. He is one of the greatest geometric physicists now living. He's perhaps the best descendant of Albert Einstein currently still working in Theoretical Physics in this particular line of thought. I also think he's a great example of what the UK does well: he has a very idiosyncratic approach to trying to solve the deepest problems in Theoretical Physics called Twistor Theory. I'm not expert in it, and I can't always follow it, so if you're not following everything in today's episode, instead of deciding that the episode has somehow failed you, try to remember that people who are working in Mathematics and Theoretical Physics spend most of their time listening to colleagues completely lost as to what their colleagues are saying. So, if you start to feel that you're being left behind by some line of thinking, what we do is, in general, wait to see if another line of thinking opens up that we can try to catch. You're not going to get all of the waves, and in fact the same thing is happening to me while I'm interviewing Roger. He's not understanding everything I'm saying. I'm not understanding everything he's saying. And in fact, this is normal.

00:01:57
So what I would like to do is to instead present you guys with an idea of what science actually sounds like when people are talking from two slightly different perspectives. We spend an awful lot of time simply trying to understand each other. And if that feels a little bit uncomfortable, well, then in fact you're getting a true scientific experience, which is often very different than what you're getting when everything is prechewed and spoon-fed. Hope you enjoy it. Without further ado, Sir Roger Penrose.

00:02:29
Eric Weinstein: Hello, you've found The Portal. I'm your host, Eric Weinstein, and I'm here today with none other than Sir Roger Penrose. Roger, welcome.

00:02:37
Sir Roger Penrose: Hello, good to be here.

00:02:38
Eric Weinstein: Good to have you. I'm extremely excited about having you here. There are lots of questions that you typically get asked these days, many of them about consciousness, some of them about art objects that come out of your thinking, but I know you in a professional capacity as one of the important—most important—people at the nexus of Geometry and Physics in our time. Of course, you can't say that, you can make all sorts of faces, but I can assure you that it's true.

00:03:09
You know, there's a Leonard Cohen quote, from a song called The Future where he says, "You don't know me from the wind, you never will, you never did. But I'm the little Jew that wrote the Bible." And I have what I consider to be the bible right here, which is a book you wrote called The Road to Reality which, there's no getting away from, may be, in my opinion, the most important modern book of our time, because what it tries to do is to summarize what we know about the nature of all of this at the deepest level. And I think what I want to do is to introduce you to our audience, which has been habituated, over perhaps 16 or so interviews, not to expect to understand everything. They want to work, they want to hear conversations unlike any they've heard, and so we'll do some combination of explaining things, but [also] some combination of allowing them to look up things in their own free time, if you're game. Should we talk about The Road to Reality?

00:04:06
Sir Roger Penrose: You can talk about that if you like, yes.

00:04:09
Eric Weinstein: That'd be great. So, where are we, in the history of coming to understand what this place is in which we find ourselves, what we are made out of, and what we know about our own context?

00:04:25
Sir Roger Penrose: It's a very tough question, that. I mean, when I wrote that book, it was more or less the state of the world at the time. I now feel I should rewrite part of it because things have changed—in one important way in particular, as far as I'm concerned. Whether other people agree with me is another question. But I don't think I'm going to rewrite it because it was such an effort. And I don't think I would be likely to live long enough to do a good job out of it.

00:04:54
Eric Weinstein: Has that much really changed since you wrote the book, at a deep level?

00:04:58
Sir Roger Penrose: A lot has not changed. The thing that has changed, in my view, you see is—whether people agree with me on this is another question—is to do with Cosmology.

00:05:07
Eric Weinstein: Right.

00:05:08
Sir Roger Penrose: You see, I have a proposal, which I didn't have—I mean, it's new since the book. It's not all that new because it's about 15 years old, but it's new since I wrote that book.

00:05:19
Eric Weinstein: And in our time scales that's quite new. Now—

00:05:21
Sir Roger Penrose: That's pretty new, yes.

00:05:22
Eric Weinstein: Let's just, just to get some context. You were born in the early 1930s.

00:05:27
Sir Roger Penrose: '31, yes.

00:05:28
Eric Weinstein: Okay, you got a chance to live through, if not the original General Relativistic and Quantum revolutions, their consequences. In particular, you were able to take classes from people like Paul Dirac, who scarcely seems like a human being, sometimes more like a god.

00:05:51
Sir Roger Penrose: Oh yeah, that was an experience. Yes. When I was at Cambridge as a graduate student—You see I did my undergraduate work at London University, University College. And then I went to Cambridge as [a] graduate student, and I went to do Algebraic Geometry, so I wasn't trying to do Physics at all. And I, I'd encountered a friend of my brother's, Dennis Sciama, when I think I was at University College as an undergraduate. And he had given a series of talks on Cosmology—well it started with the Earth, and then he sort of worked his way out, and then talked about what was then referred to as the Steady-state Theory. Where the galaxies—the universe expands and expands and expands, but it doesn't change, because all the time there is new matter created—hydrogen—and the universe expands and then you get new material, and it keeps replenishing what gets lost.

00:07:00
And I thought it was quite an intriguing, I mean, Dennis was a great fan of this model, and so I was really taken by it. So that, well the story was that I was in Cambridge visiting my brother, my older brother Oliver, who did Statistical Mechanics. And he was actually much more precocious than I was, he was two years ahead. And he was, I think, finishing his research there. But I had been listening to these talks by Fred Hoyle, and he was talking, I think in his last talk, about how in the Steady-state Model, the galaxies expanded away, expanded away, and then when they reach the speed of light, they disappear. And I thought that can't be quite right, and I started drawing pictures with light cones and things like this. And I thought, well, they would fade, gradually fade, but they wouldn't just disappear.

00:07:56
And when I visited Cambridge, I was visiting my brother and we were at this—the Kingswood restaurant in Cambridge. And I said to my brother, "Well look, I don't understand what Fred was saying. It doesn't sort of make sense to me." And he said, "Well I don't know about Cosmology, but sitting over there on the table is a friend of mine. He knows all the answers to these things." And that was Dennis Sciama. And so I explained this problem I had to Dennis, and he was pretty impressed because he didn't, he said he didn't know the answer, but he would ask Fred. Fred Hoyle. And, the main thing was that when I did come up to do graduate work, in Gen—in Algebraic Geometry, Dennis decided to take me under his wing, and try to persuade me to change my subject and do Cosmology.

00:08:46
Eric Weinstein: So you were simultaneously under the great geometer Hodge as well as Dennis Sciama?

00:08:52
Sir Roger Penrose: Well, Hodge was my supervisor. See, Dennis was just a friend.

00:08:55
Eric Weinstein: I see.

00:08:56
Sir Roger Penrose: Hodge was my supervisor, originally, until he threw me out, and Todd became my supervisor. That's another little story. But Dennis just wanted to get me interested, and do working Cosmology. This was it. I never, he wanted me to change my subject. I learned an awful lot from Dennis about Physics, because Dennis sort of knew everything and everybody. And he had a real knack of getting, if he thought two people should meet each other, he got, made sure they did meet each other. In one case, it was Stephen Hawking. But, Dennis was actually—well you mentioned Dirac—Dennis was actually the last graduate, at the time he was the only graduate student of Dirac's.

00:09:47
Eric Weinstein: Is that right?

00:09:47
Sir Roger Penrose: Yes. Dennis was, was Dirac's—

00:09:49
Eric Weinstein: Dirac was famously sort of difficult. I think that, you know, in recent years, this book came out of Graham Farmelo, The Strangest Man, that puts Dirac's bizarreness, in line with—

00:10:02
Sir Roger Penrose: He was difficult to get to know. But there's a bit of an irony here. I mean, certainly, he was hard for physicists and so on to get to know him. Now there were two people—

00:10:15
Eric Weinstein: And actually, maybe if I could just say one thing to our listeners.

00:10:17
Sir Roger Penrose: Yes.

00:10:19
Eric Weinstein: In my estimation, if not yours, Dirac would be neck and neck with Einstein for the greatest of 20th century physicists.

00:10:27
Sir Roger Penrose: I think, I wouldn't be far off at that description.

00:10:31
Eric Weinstein: For some reason, his press wasn't nearly as good, maybe because of his hair. I don't know.

00:10:35
Sir Roger Penrose: Well, he didn't talk much. This is one of the problems. No, I agree. I think he was—I mean, you think about all the Quantum Mechanics people who develop that amazing subject, and Dirac was really the one who put it all in order and so on.

00:10:51
Eric Weinstein: Well his, and this gets to a very odd issue, which is that you have wielded taste and beauty as a weapon your entire life. Your drawings are among the most compelling—I remember the first time—one of the things I've done, using our friend Joe Rogan's program, is to push out discussion of the Hopf fibration, because it's the only non-trivial principal bundle that can be visually seen. And since the world seems to be about principal bundles, it's a bit odd that the general population doesn't know that stuff of which we are.

00:11:28
Sir Roger Penrose: Yes. Well the, the Hopf fibration, or the Clifford parallels, was instrumental in the subject of Twistor Theory.

00:11:37
Eric Weinstein: Well, but the first time I ever saw a diagram, it was somebody reproducing a diagram they had seen of yours. And so, the way in which you have used art and sketches was really transformative

00:11:47
Sir Roger Penrose: Yeah, but I drew it out by hand. The picture was drawn by hand. Largely, I mean, there were, I think, some circles involved which I used a compass for but basically I drew it by hand. There were two versions of it. The first one was more—I sort of threaded—the first one had more circles in it, and I thought I'd draw a little more simply the second version, but actually, I had three versions. The third version is in The Road to Reality. But I'm not sure it's the best. I think the second version perhaps is the best.

00:12:22
Eric Weinstein: So Dirac, getting back to it, had this elegance of mind that was unrelenting.

00:12:29
Sir Roger Penrose: Yes.

00:12:30
Eric Weinstein: And he famously brought in these bizarre objects with which some of us are obsessed, others of us don't understand the obsession, called spinors, which sort of are a prerequisite to getting to Twistor Theory, which you've popularized.

00:12:47
Sir Roger Penrose: Well, when I went to the... you see Dirac gave a course of lectures in Quantum Mechanics, and the first course was sort of basic Quantum Mechanics. And the second course was on Quantum Field Theory, but also spinors. And there's an interesting story about that, which I don't know the answer to. In the second course, he deviated from his normal course of lectures. Now, I understood when I talked to Graham Farmelo, who wrote this biography of Dirac, I understood from Graham Farmelo that, when I described that Dirac deviated from his normal course to give two or three lectures on two-component spinors, which for me were absolutely what I needed. You see, I'd learned from my work on Algebraic Geometry, which ended up by trying to understand tensor systems as abstract systems, and things which you can't represent in terms of components.

00:13:53
Eric Weinstein: And I should just say that in terms of these two-component spinors you're talking about, for the lay audience, all of the matter that they think about, whether it's in, bound up in electrons, or the quarks that make up protons and neutrons, if you think of these things as waves, which many people in our audience will be familiar with that concept, the question is, what are they, what medium are they waves in? And they're—the medium would be a medium of spinors, which is not something that's easy for people to understand.

00:14:22
Sir Roger Penrose: Yeah, well, it's, they're not. And certainly the formalism... You see Dennis, I told him I need to understand about spinors and particularly, two, the simplest ones are these two-component spinors. And he suggested I read this book by Corson. So I got the book by Corson, and I found it completely incomprehensible. Just, I mean, it was a fascinating book because it was very comprehensive, it described all these different spins, fields, and different things like that, and using a lot of two-component spinors, which is the right way to do it.

00:14:57
But, to introduce what these are objects were was almost incomprehensible, I found, mainly because you have these translation symbols all over the place, and they mess up the appearance of the formulae. So I just found this thing very complicated and incomprehensible.

00:15:17
But then I went to Dirac's second course. It may have been not the same year, I think he went—one year I did the first, and maybe the second course was when I was a research fellow, rather than when I was a graduate student, I can't quite remember. I think—must have been—when I was a graduate, research fellow. Anyway, this was a course on Quantum Field Theory and things like that, but he sort of deviated from his normal course, in one week, to talk about two-component spinors. And for me this was exactly what I needed. It made the whole subject clear from this complete confusion that I had before.

00:15:56
Now then, you see, many years later I talked to Graham Farmelo, and I told him the story. And he said, "That's very strange. Dirac would never deviate from his course, he just, he thought when he got his course perfect, it was perfect, he would never change." And this was true of his first year course, the shorter, the initial course, which I went to, which people often said to me, "Well, that's not such a great course, it's exactly like his book," but I hadn't read his book. So to me this was, sure the book is amazing too. But not having read the book, I found this course absolutely stunning, and it made things ab—

00:16:36
Eric Weinstein: Do you think Dirac actually understand—understood—these objects, these most mysterious of objects?

00:16:42
Sir Roger Penrose: Two-component spinors?

00:16:43
Eric Weinstein: Spinors in general. I mean, he brought them into Physics, they'd been previously found inside of Mathematics, I think by people like Killing and Lie, I'm not sure who.

00:16:53
Sir Roger Penrose: Yes.

00:16:53
Eric Weinstein: But—

00:16:54
Sir Roger Penrose: Cartan is the one.

00:16:54
Eric Weinstein: Cartan perhaps. I don't think—I mean, let me throw out a really dangerous idea. I don't think any of us understand them at all, and that part of the problem is that he understood very well what could be said about them—

00:17:09
Sir Roger Penrose: Yeah.

00:17:10
Eric Weinstein: But that, you know, I asked you before about your favorite film, you said 2001. You could make an argument that spinors are, in Mathematics and Physics, like the monolith. It's always encountered, nobody ever understands exactly what it means, but it always grabs your attention, because it seems so absolutely bizarre and highly conserved.

00:17:29
Sir Roger Penrose: Well I always like to think of things geometrically. And, [at] least for the two-component ones… You see, when you go up to higher dimensions, you still have spinors. But the spinors, the dimension of the spinors goes up exponentially. So each time—You add two to the dimension of the space, and the dimension of spinors is multiplied by two. So, they get—

00:17:51
Eric Weinstein: So [with] dimension 2D, for example, you'd get spinors of dimension 2 to the D over 2.

00:17:58
Sir Roger Penrose: That's the sort of thing, that's right. And so the... Usually one talks about the Dirac spinors, which are the four dim—the four spinors—

00:18:06
Eric Weinstein: The full, right, right.

00:18:08
Sir Roger Penrose: But they split into these two, two and two—

00:18:10
Eric Weinstein: In even dimensions.

00:18:11
Sir Roger Penrose: Yes, that's right, in even dimensions. And, I like to understand these things geometrically. So you could see what the two-component spinor represented, I had this picture of a flag. So you have the flagpole, [which] goes along the light cone. So that's a—

00:18:29
Eric Weinstein: That's the vector-like piece of it.

00:18:31
Sir Roger Penrose: It's a vector. And—

00:18:32
Eric Weinstein: And then you have an extra piece of data—

00:18:34
Sir Roger Penrose: An extra piece, which is this flag plane. And you get a pretty good geometrical understanding. The one little catch to it is that if you rotate it through 360 degrees, so you might think just to where it started, it's not the same as it was before, it's changed its sign, and then you rotate it again, so—

00:18:59
Eric Weinstein: Well that won't make any sense to anyone. But if—I mean one way of looking at that is if you have a Klein bottle—

00:19:05
Sir Roger Penrose: Yes.

00:19:05
Eric Weinstein: And for those of—some people will be listening to this on audio, some watching it in video. A Klein bottle, in a certain sense that can be made precise, has a square root that would be a torus: that is, a double cover. So it seems like a very weird thing to take a square root of a strange topological mobius-like object, but there you are.

00:19:24
Sir Roger Penrose: Yeah.

00:19:25
Eric Weinstein: So it's really the square root of the rotations that has this double effect. But we say it linguistically in a way that makes it almost impossible for anyone to understand.

00:19:35
Sir Roger Penrose: Well I think this was a mystery. I mean, I understood that a spinor was the square root of a vector, you see, and I couldn't make head or tail of that idea. And it was when I went to Dirac's course it did become clear. And he made, he gave this very impressive illustration, which I thought was due to Dirac, I learned later it was due to Hermann Weyl, that you imagine a cone, circular cone—

00:20:00
Eric Weinstein: Yes.

00:20:01
Sir Roger Penrose: Sitting in space like that, circular cross section, and another equal cone, which rolls on it. So one is fixed, and the other one rolls around on it. Now you see, when you imagine initially, the cone is almost just this little spike, and you have a tiny circle at the end. And when you roll one on the other, it's like rolling one coin on the other coin. So, and you can see when you roll one coin on another coin, it goes around twice, because it's 720 degrees as it goes around. Okay. Now, when you imagine gradually increasing the angle, the semiangle of the cone, and you do it again, you keep thinking of that motion until it becomes almost flat. And then what's the other? It's just a little wobble.

00:20:48
Eric Weinstein: Right.

00:20:49
Sir Roger Penrose: So when it becomes flat, this motion goes to nothing. So this illustrates how a rotation through four pi—

00:20:57
Eric Weinstein: Right.

00:20:58
Sir Roger Penrose: Two complete rotations, gradually can be deformed into no rotation at all. However, with a single rotation, it doesn't disappear.

00:21:07
Eric Weinstein: Well, I think with a, with a pulley system and a wheel, we don't have any trouble imagining a wheel that rotates twice as fast, half as fast, not at all hooked up to one particular crank wheel, right?

00:21:20
Sir Roger Penrose: Yes.

00:21:21
Eric Weinstein: The problem comes when that's not the generic case, the generic case is usually encountered one dimension higher, three and up has a familiar... because something called the fundamental group has a structure of Z mod two, rather than Z in dimension two. So there is something where in the place where you can see this most easily, it's slightly misleading. And then, in higher dimensions, you have to learn how to tutor your intuition, which is this problem that all of us who tried to think about higher-dimensional objects encounters, is that we have to use the visual cortex we're handed, and then we have to trick it into imagining worlds beyond where we've seen.

00:22:02
Sir Roger Penrose: But you see, Dirac had another thing that I... There's a thing called the Dirac Scissors Problem.

00:22:09
Eric Weinstein: Hmm.

00:22:09
Sir Roger Penrose: So you imagine the chair with, which has the pieces of wood going out like this—

00:22:15
Eric Weinstein: Yeah.

00:22:15
Sir Roger Penrose: And you have a pair of scissors. I think this is Dirac's joke that it was a pair of scissors, and through the, where you put your fingers, you have a piece of string which goes through this and then goes around the chair and then comes back through the other one, goes back again.

00:22:29
Eric Weinstein: Right.

00:22:29
Sir Roger Penrose: Now the problem is you take the scissors, and you rotate them through—

00:22:34
Eric Weinstein: 360. And it doesn't—

00:22:35
Sir Roger Penrose: 360 degrees, and the string's all tangled up.

00:22:37
Eric Weinstein: We can't undo that one.

00:22:37
Sir Roger Penrose: Whatever you do, you can untangle it. You're allowed to move the scissors around parallel, not rotate them, and you can move the string around it and you can undo it. But you do it twice. 720 degrees, two complete rotations. And then you find you can untangle it. So this was the Dirac Scissors Problem. And I think the joke was it's a pair of scissors, so if you get too frustrated, you just cut the string.

00:22:40
Eric Weinstein: You just cut the Gordian knot, yeah.

00:23:03
Sir Roger Penrose: And he wrote a paper explaining that, I think Max Newman—

00:23:08
Eric Weinstein: Yeah.

00:23:08
Sir Roger Penrose: Wrote a paper. Dirac did this as an illustration of how you can undo it when it's, when it's—

00:23:14
Eric Weinstein: Right.

00:23:14
Sir Roger Penrose: Four pi, 720 degrees, but to prove that you couldn't do it with this is, I think you... Max Newman had a theory—

00:23:22
Eric Weinstein: Have you seen this video called Air on the Dirac String, which illustrates this in video format?

00:23:26
Sir Roger Penrose: I haven't seen that.

00:23:27
Eric Weinstein: I would highly recommend it because it shows this off as the similarity to the belt trick, to the Philippine wineglass dance—

00:23:35
Sir Roger Penrose: Yes—

00:23:35
Eric Weinstein: All of these different versions.

00:23:37
Sir Roger Penrose: I find I could do that one actually.

00:23:39
Eric Weinstein: I had Joe Rogan try it and I think he got almost all the way around.

00:23:43
Sir Roger Penrose: Yeah, no, I've done it with a glass before, so—

00:23:45
Eric Weinstein: Yeah.

00:23:46
Sir Roger Penrose: Yes, you go like that and it comes back.

00:23:49
Eric Weinstein: Very stylish.

00:23:50
Sir Roger Penrose: Yes, so you can do two complete rotations—

00:23:54
Eric Weinstein: Two complete rotations.

00:23:54
Sir Roger Penrose: Yes.

00:23:55
Eric Weinstein: So, this is a very fundamental property of the world that is somehow not discussed. I think... I find it very interesting that people want to talk to me about the multiverse, sometimes they want to talk to me about [the] quantum measurement problem. But the idea that we are somehow based on a square root, and I would disagree with you slightly if you would permit it, that it's not just a question of the square root of the vectors, it's the square root of the algebra generated by the vectors that really the spinors are: this exterior Clifford algebra.

00:24:26
Sir Roger Penrose: Oh yes.

00:24:26
Eric Weinstein: This object has fascinated me my entire life, and it's very strange that all of, you know the stability of matter and matter's strange properties with electron shells are all coming out of this weird knot that appears everywhere in the universe, and it's not universally known that it's even there.

00:24:45
Sir Roger Penrose: Yeah, I suppose the difference between the fermions and bosons, so the particles which have a spin which is half an odd number—

00:24:56
Eric Weinstein: Right.

00:24:56
Sir Roger Penrose: Which have this curious property that you rotate them, and they get back to manage themselves. And it's crucial for matter because the Pauli Exclusion Principle depends on the Fermi Statistics, which is to do with the, this exact, this property.

00:25:15
Eric Weinstein: So without this knottedness and the scissor trick or whatever you want to call it, we wouldn't have a periodic table and chemical elements that—

00:25:23
Sir Roger Penrose: You wouldn't have anything.

00:25:24
Eric Weinstein: We wouldn't have anything.

00:25:25
Sir Roger Penrose: Yeah, you wouldn't have, you wouldn't have fermions, in other words you wouldn't have things which have an Exclusion Principle, so, and the bosons, which are the opposite, they like to be on—if you have two bosons in—you can have them in the same state, they rather like to be in the same state, so you get these things called Bose-Einstein condensates, where if you get them very cool they all flop together into the same state. But for the fermions it's completely the opposite. They hate to be in the same state, or they can't be, and this is what sort of pushes them apart. So you get the Fermi Principle... Pauli Principle

00:25:59
Eric Weinstein: So you have this, this strange thing called the Spin Statistics Theorem—

00:26:03
Sir Roger Penrose: Yes.

00:26:03
Eric Weinstein: That says that if things have a knottedness of a particular kind, then they either are highly individualistic or highly communistic, whatever you want to call it. My question would be, there's another aspect of that, that I've been very curious about, which is when we have to treat these objects quantum mechanically, and you've, of course, thought a great deal about Quantum Theory, we have two totally different prescriptions for how to make these different objects quantum mechanical, but there's a one to one correspondence between these two utterly different treatments that matter and force get quantum mechanically, it's the darndest thing.

00:26:43
Sir Roger Penrose: When you get these two kinds of particle or two kinds of atoms, the bosons and the fermions. And it has to do with the, make a complete rotation: Do they come back to themselves? Or, do they come back to minus themselves?

00:26:56
Eric Weinstein: That's the topological bit.

00:26:58
Sir Roger Penrose: Yeah.

00:26:58
Eric Weinstein: But then there's this whole thing that might go under, like, Bayesian integration, which is no integration at all. I mean, you're effectively almost lying about what you're doing to the fermions to make them look like bosons. And yet, what we, what we seem to get out of this is that nobody—I don't think anyone could have anticipated that there would be a dictionary of two totally different structures, which are—seems to be almost word for word.

00:27:29
Sir Roger Penrose: Yes, because they're not totally different in the sense that you take [a] two fermion system and you get a boson. So you, they are a part of the same world.

00:27:39
Eric Weinstein: Well they have to be related.

00:27:40
Sir Roger Penrose: Yes, that's right.

00:27:41
Eric Weinstein: Now, maybe I could ask you a little bit about that. So I want to get to Supersymmetry. But before I do—

00:27:49
Sir Roger Penrose: I see, yes. Okay. Go on.

00:27:52
Eric Weinstein: We're gonna make you work this morning, Sir.

00:27:53
Sir Roger Penrose: I can understand that.

00:27:54
Eric Weinstein: Yeah. So here's my question, am I correct that you've lived through two eras, an era of fairly rapid development in testable, fundamental Physics, coming from theory—I've tried to be very careful about setting that up so I don't walk into a trap—and a stagnant theory—era in which theoretical predictions coming at the level of fundamental theory have not been rapidly confirmed by experiment.

00:28:01
Sir Roger Penrose: You're thinking of things like String Theory?

00:28:36
Eric Weinstein: I'm thinking about a regime before the early 70s, and a regime following the early 70s.

00:28:42
Sir Roger Penrose: Well, Supersymmetry, is that what you meant?

00:28:45
Eric Weinstein: Well, it could be Grand Unified Theory, Supersymmetry, Technicolor. It could be Asymptotic Safety. It could be any one of a number of speculative theories from Loop Quantum Gravity, Regge Calculus, String Theory. It's like the kitchen sink, we've tried a million different things that don't—

00:29:05
Sir Roger Penrose: They didn't really pan out indeed.

00:29:06
Eric Weinstein: Well, it seems like, if you'll permit an American metaphor, we've been waved into third base, and we've been waiting for the signal to come home for about 50 years, and we're not even sure that anyone's still, you know, there at home plate.

00:29:21
Sir Roger Penrose: Well, you see, you might be wrong, playing the wrong game. That's the trouble

00:29:24
Eric Weinstein: You think rounders would do it?

00:29:27
Sir Roger Penrose: Well, I mean, there's a lot of intriguing ideas you mentioned. Basically, I think you were hinting at Supersymmetry as one of them, which—

00:29:36
Eric Weinstein: Well maybe I've thrown off close to 10, I [unintelligible], I could do it pretty easily.

00:29:37
Sir Roger Penrose: But I guess you had, there's nothing new about that. They were, people were playing around with knots and things—Kelvin was... the idea that knots might be—

00:29:52
Eric Weinstein: At the basis of particle identity.

00:29:55
Sir Roger Penrose: Yeah, yeah. I mean, these ideas come back again in a different form, but certainly in the, I guess the 19th century, people were playing with, well, I guess you can go back further than that... Phlogiston.

00:30:09
Eric Weinstein: Well, that's true. But I would say that Maxwell was the first great condensation of theoretical ideas, where an enormous amount of theory surrounding magnetism, electricity, visible light, invisible light—

00:30:27
Sir Roger Penrose: Well that was a huge, huge revolution.

00:30:29
Eric Weinstein: And that all of those things now can be unpacked from a single geometric equation.

00:30:36
Sir Roger Penrose: But that's the thing. I mean, people know about Galileo, they know about Newton, know about Kepler, they know about Einstein, and they also may know about the modern Quantum Field Theory: Heisenberg, Schrodinger people. How many people know about Maxwell?

00:30:56
Eric Weinstein: Not enough.

00:30:57
Sir Roger Penrose: Not enough.

00:30:58
Eric Weinstein: Although people do have Maxwell's equations tattooed on their backsides.

00:31:01
Sir Roger Penrose: Well some people do. But the general public don't know about Maxwell. But Maxwell's equations completely change our way of looking at the world. And we live off it without thinking, you know, you've got these lights here. Well, these are visible lights, so we, we know, you knew about visible light, but we didn't know anything about x-rays. X-rays, radio waves, they're all part of the same scheme. Electromagnetism, dynam—well, some of this goes back to Faraday just before Maxwell.

00:31:37
Eric Weinstein: Sure.

00:31:37
Sir Roger Penrose: So Faraday had a lot of the influential ideas. Electromagnetism, well a little bit of that was known before—Ørsted knew that if you had an electric current, then you get a magnetic field—but it was the other way around with Maxwell: now if you have a varying magnetic field, you get a current, and you combine these ideas you can make a dynamo. So these things go to Faraday, and he had sort of clues that there might be some connection with light. But he didn't have the equations.

00:32:07
Eric Weinstein: But even Max, you know, I'm very partial to this book on orchids that followed Darwin's Origin of Species.

00:32:14
Sir Roger Penrose: Oh yeah.

00:32:15
Eric Weinstein: That was the book he wrote—the title is, and I always, I love reciting it, it's On the Various Contrivances by which British and Foreign Orchids are Fertilized by Insects. And so you think, well, why would you write a damn fool book like that after Origin of Species? And the answer is he wanted to test whether he understood his own theory. And in fact, it's revealed that he didn't understand the full implications. I would say that the same thing is true of Maxwell's equations, which is, this is perhaps the best dress rehearsal for unification we've ever seen, you know, full unification, and on the other hand, it's not until the late 50s that we actually unpack the last trivial consequence of the theory with this bizarre effect of passing an electron beam around an insulated wire.

00:33:05
Sir Roger Penrose: Aharonov, yeah.

00:33:06
Eric Weinstein: Yeah, in fact we had dinner last night, we asked Yakir Aharonov if he wanted to come but he's in Israel, and he sends his regards.

00:33:14
Sir Roger Penrose: Oh, you know, send mine back. No, no, he's great fun I always—

00:33:20
Eric Weinstein: But that was a very weird thing where we learned that if you have an insulated solenoid, that the phase of the electron beam going in a circle around it would be shifted despite the fact that the electromagnetic field could be treated as zero because the electromagnetic potential, this precursor—

00:33:44
Sir Roger Penrose: Yeah.

00:33:45
Eric Weinstein: Turned out to carry the actual content, that before that it had been thought that that was just sort of a convenience product to recover electromagnetism, and it turned out that that geometric object was more important. And you know, part of the reason I bring this up is that we would have no way of visualizing this effect if it were not for your interaction with M.C. Escher.

00:34:11
Sir Roger Penrose: Oh, now you have to explain that one.

00:34:13
Eric Weinstein: Well, you know, this etching called Ascending and Descending.

00:34:18
Sir Roger Penrose: Oh yeah, sure. Yeah.

00:34:19
Eric Weinstein: Which is sometimes referred to as the Penrose stairs.

00:34:23
Sir Roger Penrose: Yes. Well, you want that story?

00:34:26
Eric Weinstein: Well, I do. But, what I was gonna say about why I'm asking for it, is that the photon is really best represented in some sense as the angles of a set of stairs like that with this very mysterious property, that what you're really talking about is what we would call horizontal subspaces, pictured as stairs, and the fact that there's a paradox of going around, and you seem to be going up all the time, but you're back to where you came is the same thing as saying I never go up and yet I come back higher or lower. And that's called holonomy. And we don't have a means of visualizing that except for like, either rock-paper-scissors or your work with Escher. Is that a fair comment?

00:35:07
Sir Roger Penrose: Well, I had, I think there's a—

00:35:09
Eric Weinstein: Is this the first time you've ever heard somebody say this?

00:35:13
Sir Roger Penrose: Well, let me... I mean, there's a quite complicated story here.

00:35:17
Eric Weinstein: All right.

00:35:17
Sir Roger Penrose: You see when I was a graduate student in Cambridge, I think it was in my second year, when the International Congress of Mathematicians took place in Amsterdam. And so I and a few friends decided we would go to this meeting, and I remember... I think I was just about to get on the bus or tram or something, and Shaun Wylie—who is a lecturer in in Algebraic Topology—he's just about to get off the bus, I was getting on, and he had this catalog in his hand of an exhibition in the Van Gogh Museum. And this was a picture... The one called Night and Day with birds flying off into the day and the night, and the birds changed into the spaces between the birds [unintelligible], and I just look at this and I think 'Oh that's amazing what is that? Where on earth did that come from?'

00:36:14
He said, "Oh, well, you'd be very interested, this is, in the Van Gogh Museum, there is this exhibition by an artist called Escher." So I'd never heard of him before. And I went to this exhibition, and I was absolutely blown away. I thought it was most amazing thing. I remember particularly one called Relativity, where people walk up the stairs and gravity directions are two different ways. And I thought this was hugely impressive. And I went away, thinking, 'Well, I'd like to do something impossible,' you see, and I didn't... See, I had an idea about an impossible structure with bridges and roads and things like that. So locally, it makes sense, but as a whole it was inconsistent. And I didn't think I'd seen anything quite like that in his exhibition.

00:37:02
So I played around with this. And then I sort of whittled it down to the triangle, which people refer to as a tribar. So it's a triangle which is locally a completely consistent picture, but as a whole, it's impossible. And I showed this to my father. And then he started drawing impossible buildings, and then he came up with this staircase. So we decided we'd like to write a paper together on this. And we had no idea what the subject was, I mean, what, who do you send a paper like this to, what journal? So he decided since he knew the editor of the British Journal of Psychology, and he thought he'd be able to get it through, we decided the subject was Psychology.

00:37:48
Of course, it's, as you say, it's not, it's more in a way Mathematics because it illustrates ideas, well of cohomology, and other things like that which I didn't quite know was illustrating at the time. But anyway we wrote this paper, and we gave some reference to Escher, I think, reference to the catalog. And my father sent a copy to a Dutch friend of his and he managed to get it to Escher. And then my father and Escher had a correspondence. So that was—

00:38:23
Eric Weinstein: This is Lionel Penrose.

00:38:24
Sir Roger Penrose: Lionel, my father Lionel Penrose, yes. But I actually visited Escher then. And, he had sent a print to my father with a dedication to it, and he gave me another. So I have in the [unintelligible]—

00:38:42
Eric Weinstein: But in some sense—

00:38:43
Sir Roger Penrose: So the Ashmolean Museum.

00:38:45
Eric Weinstein: You see, I'm very indebted to you for this reason, because when I, when I have to describe what General Relativity is—

00:38:52
Sir Roger Penrose: Yeah.

00:38:53
Eric Weinstein: And I don't wish to lie the way everyone else lies—if I'm going to lie I'm going to do it differently—I say that you have to begin with four degrees of freedom, and then you have to put rulers and protractors into that system so that you can measure length and angle. That gives rise, miraculously, to a derivative operator that measures rise over run. That rise is measured from a reference level, those reference levels don't knit together and they form Penrose stairs. And the degree of Escherness, or Penroseness, is what is measured by the curvature tensor, which breaks into three pieces, you throw one of them away called the Weyl curvature, and you readjust the proportions of the other two, and you set that equal to the amount of stuff. Now that's a very long causal chain—

00:39:41
Sir Roger Penrose: Yeah.

00:39:41
Eric Weinstein: But it is linguistically an accurate description of what General Relativity actually is.

00:39:47
Sir Roger Penrose: Yeah, well, it's illustrates that, it also illustrates cohomology, which... I was being interviewed, oh, ages ago, by... I don't know whether it's BBC, I can't remember when it was. There was an interview for some reason. They were interested in Twistor Theory, now...

00:40:02
Eric Weinstein: They think they're interested.

00:40:03
Sir Roger Penrose: Well they thought they were, I guess they'd heard that word or something. And at one point they say, well, surprisingly not at the beginning, they asked me what what good it was, you see, what can you use it for? So I said, oh, you can use it to solve Maxwell's equations, you see, that's the equations of electricity and magnetism and light, and so they got a bit interested. And they said, "Oh, how do you do that?"

"Well, it's actually involves an idea that I couldn't really explain here. It's not possible to in a sort of popular talk like this."

"No, no. What is it, exactly?"

No, no, no, I couldn't do it.

"Now what's the, what's the idea?"

"It's a thing called cohomology. No way I could explain that."

So then I went back home and I was lying in my bed and I thought, I think I can, you know, it's this impossible triangle. That's exactly an illustration of cohomology. So I went back the next day and told them, but they weren't interested. They didn't use it. I think I may have tried to explain. Yes, well you have a lo—something which is locally consistent—

00:41:01
Eric Weinstein: Right.

00:41:02
Sir Roger Penrose: But with an ambiguity about it. So here the ambiguity is you're not quite sure—you draw a picture of it, the ambiguity is that you don't know how far away it is. It could be bigger and further away or smaller and closer, and the picture is consistent. But you get an inconsistency if you go around—

00:41:19
Eric Weinstein: Right.

00:41:20
Sir Roger Penrose: And locally, because you have a freedom—

00:41:22
Eric Weinstein: Yes.

00:41:23
Sir Roger Penrose: And you misuse this freedom in a sense, so the glitch in it is this impossible structure.

00:41:28
Eric Weinstein: Well, I had this. So this is actually my son, my 14-year-old son's copy of the book.

00:41:35
Sir Roger Penrose: I see, yes.

00:41:35
Eric Weinstein: And I was having to describe this to him what cohomology was, and I said, that one-forms, which is a piece of technology in Mathematics that you can analogize to radar guns, so that while you're driving and the policeman shoots your car with a radar gun, he's measuring the component of speed in the direction of his gun.

00:41:58
Sir Roger Penrose: Yeah.

00:41:59
Eric Weinstein: And so that's something that eats the vector of speed and spits out a number. And then you could imagine a racetrack that wanted to have a circular series of radar guns to measure the speed of cars going around it. Now the question is, you also recognize that you could build a poor man's version of a speed system by heating the track to some temperature and measuring how quickly the temperature changes as the car went over it. But you can't actually have the one thing that you want, which is the series of radar guns that are always measuring the speed going around the track, because at some point, if the temperature is going down, down, down, down, down, down, down, then it's going to be 10 degrees below wherever it started, which is your paradox again.

00:42:49
Sir Roger Penrose: Yes. Well, there's a nice example somebody made, I can't remember who, where you accompany—you have a ball going up the stairs or down it, whichever it is, and you accompany that with a, a note going up or down. And you can make it sound as though it keeps on going all the way up and all the way up all the time. It's by the harmonics, you bring a new harmonic in as you go round.

00:43:12
Eric Weinstein: You're below, and it's sub-perceptual. So there's this auditory illusion that captures this—

00:43:18
Sir Roger Penrose: Yes, you have an auditory version of the same thing. And somebody had this ball bouncing around with that.

00:43:22
Eric Weinstein: That's not, but that's a bit of a cheat. You, I mean, my point would be that your Escher stairs, or your Penrose stairs, are... the cheat is that it appears to be flat. In other words, it's very easy to achieve that on a curved object, but that what you did was to create the illusion as taking place in a plane or—

00:43:45
Sir Roger Penrose: Well you can draw it in a plane—

00:43:46
Eric Weinstein: In a rectilinear system.

00:43:48
Sir Roger Penrose: You have an interpretation of a three-dimensional thing, which is an ambiguous interpretation.

00:43:52
Eric Weinstein: So you saw the movie Inception, of course, where they, they realized this actually?

00:43:56
Sir Roger Penrose: Yes, they're, they show some of that, that's right.

00:44:00
Eric Weinstein: But that effect is the soul of the Aharonov-Bohm effect, which surprised the world in the late 50s because it was discovered so late into the game.

00:44:10
Sir Roger Penrose: It is a comm—same sort of thing. That's right. Well, of course like so many things, people point out that this Oscar Reutersvärd, who is a Swedish artist who'd drawn things like this before. I think roundabout the year I was born, he had a picture which is all, with cubes going around. It wasn't exactly the same, but it was.

00:44:33
Eric Weinstein: I think I've seen these floating cu—

00:44:34
Sir Roger Penrose: The one with the cubes. Yes. And then he had versions with stairs, staircases too. But he never put any perspective in it, which seemed to me that was a something—

00:44:45
Eric Weinstein: Missed opportunity.

00:44:46
Sir Roger Penrose: Yes. Now in my triangle, I did put some perspective.

00:44:48
Eric Weinstein: Yeah.

00:44:49
Sir Roger Penrose: So it's slightly, you can see, but you can do it with a perspective and it still works.

00:44:55
Eric Weinstein: So, what I want to get at is, I think also that we have this very funny thing that happened, recently, starting from the early 70s, where we started mis-telling our own Physics history, because of the needs of the community to look like we were succeeding when we weren't, or we were succeeding at something different than we were trying to succeed at. And, in part, one of the reasons that I want to use this podcast to discuss science is to give alternate versions of what's happened. And I want to explore one or two of them with you. Now, you and I have a very funny relationship which, we don't really know each other. But you were quite close to Michael Atiyah at various points. And I was—

00:45:42
Sir Roger Penrose: Well we were graduate students together, in the same group—

00:45:44
Eric Weinstein: In the same year.

00:45:45
Sir Roger Penrose: Absolutely, the same year, yes. With the same supervisor.

00:45:47
Eric Weinstein: Incredible.

00:45:48
Sir Roger Penrose: Yes, that's right.

00:45:49
Eric Weinstein: And then you continued to cross-pollinate ideas—

00:45:51
Sir Roger Penrose: Yes.

00:45:52
Eric Weinstein: Through the years.

00:45:53
Sir Roger Penrose: Yes, absolutely, yes.

00:45:53
Eric Weinstein: Now for listeners who don't know, Michael Atiyah was one of the absolutely most dominant and generative... I don't even know what to call him, like beyond genius, a seer of type.

00:46:07
Sir Roger Penrose: But he has had, he had such a broad understanding of Mathematics. It's partly—

00:46:11
Eric Weinstein: In Geometry more generally, and Analysis, I mean, just incredible, and Algebra. I mean, he wrote a book on on Commutative Algebra. Now he had a partner for much of his career, Isadore Singer, who I was quite close to for a period of time. And Is was, again, another one of these figures that if I'd never met one, I wouldn't know that the human mind was capable of that level of repeated insight. And they came up with something called the Atiyah-Singer Index Theorem, which governs worlds in which there are no time dimensions, but only space dimensions, or no space dimensions and only time dimensions, but there's no—

00:46:28
Sir Roger Penrose: They're just equations without any differential equations.

00:46:55
Eric Weinstein: Well differential equa—if you think about differential equations as coming very often in three main fields of study, elliptic, hyperbolic, and parabolic, then the idea is that wave equations would be hyperbolic: the type that you're worried about in Physics, but things like soap films look like elliptic equations, and Atiyah and Singer had this incredible insight into the nature of elliptic equations. Do you, go ahead.

00:47:23
Sir Roger Penrose: So, no I was going to say it's an extremely general theorem, which covers, goes over all sorts of different areas of Mathematics, and has application—

00:47:32
Eric Weinstein: Well, it sort of tells you that the, the knottedness of some beautiful space that you might be exploring, like some kind of high-dimensional donut that's knotted many times around itself, whatever you want, that that topological knottedness tells you something about the kinds of waves that can dance on that space.

00:47:55
Sir Roger Penrose: Yeah. No, it's a very remarkable theorem, certainly.

00:48:00
Eric Weinstein: Does that theorem in the so-called elliptic category, world of space and no time let's say, relate strongly, in your estimation, to the most important hyperbolic equations that govern the waves that make up our physical world due to the constraints of Relativity in a world with one time and three spatial dimensions?

00:48:23
Sir Roger Penrose: Well I can say if I've used the theorem. In at least two different contexts, yes, maybe more. So, I mean, I'm not an expert in that area at all. And it was mainly when I was trying to solve a particular problem... I don't know how much detail you want to go into these things. But it had to do with how to make Twistor Theory work in curved spaces. But I ran up into a question, which had to do, it has to do with Complex Geometry.

00:48:56
So you've got Geometry in which instead of using real numbers, so you use, you think of measuring with rulers, say, and the ruler is one-dimensional. The numbers go along one dimension if you like. And complex numbers, where you have the square root of minus one incorporated into the number system, they are really two-dimensional. So the Geometry of complex numbers has twice as many as the real numbers.

00:49:26
But the Geometry of complex numbers is particularly fascinating, or the Algebra, you might say the Analysis, or whatever it is. It's particularly fascinating and I was, sort of, when I learned about this when I was an undergraduate doing Mathematics, and I thought it was incredibly beautiful. Because when you talk about real numbers you have, you can have a, say I draw a curve which is a function, so this curve has some shape. And you might want to see, well, is it a smooth curve, that means you have a tangent direction as you go around it, maybe it jumps. So it's not even continuous, or maybe it's smooth, or it maybe, you have to have a curvature of this curve, and it might not be smooth enough to have curvature. So if there's one degree of smoothness or two degrees, or you can have three degrees or four degrees, and they're all different, or an infinite number of degrees, or that you can expand your function in the power series. They're all different.

00:50:30
And then we learn about complex, you see. Oh we now do it all over again, and you've got your Analysis, or Algebra if you like, or Geometry, where the, it uses complex numbers. And then suddenly you find that if it's smooth, everything comes with it. You can differentiate as many times as you like, you can expand as a power series, and I thought that was incredibly magic. You just have to do it once, rather than all these different kinds of—

00:51:01
Eric Weinstein: Well I should just say that, that Mathematicians quite often view the complex case, the the case of complex numbers as the natural case. And the case of real numbers as artificially tortured, which is a complete reversal from how most engineers and physicists... And you have actually been quite instrumental in making the case for the fundamentally complex nature, that it's not just a convenience that we use complex numbers in Physics, but that nature appears to be essentially complex.

00:51:35
Sir Roger Penrose: I think, you see, by just hearing this nature of Complex Analysis, and how beautiful it struck me as being, and I had this sort of feeling, 'Wouldn't it be wonderful if these numbers were somehow the basis of way the physical world operates? I have no reason to think that,' and then I learned about Quantum Mechanics and I was amazed to find yeah, there they are, they're not just useful convenience. You can use them to simplify ideas in Mathematics you can, you know, might have an integral—

00:52:08
Eric Weinstein: You would have to work awfully hard to get rid of them.

00:52:10
Sir Roger Penrose: Yes, but there things you find that give you a little trick to do it. Now at least for me they come with contour integrals and they drop out in an amazing way. And I thought, well, that's a piece of magic, but it doesn't tell you anything about the world, it just tells you this is a neat way of doing things. And then I learned about Quantum Mechanics. And suddenly these numbers are right there at the basis of the whole subject. And I thought that was an amazing thing. Maybe these complex numbers are really there at the root of everything.

00:52:39
Eric Weinstein: I mean, I think you wanted to talk about twistors. And maybe I can intro that and then try to fit that into this history that I'm claiming we don't tell. Now, one of the ways of describing what Twistor Theory is, and of course it's a bit of a tall order for a podcast, is that you are replacing Einstein's spacetime with a larger structure that in some sense implies spacetime, where you can take all the data that roams around on spacetime, the waves, the force, the matter, what have you, and you can, as Mathematicians might say, pull it upstairs to this larger twistor space, where you might have a couple of extra tricks up your sleeve, because the extra space that you've created to augment spacetime with has this kind of complex number aspect baked into it.

00:53:35
Sir Roger Penrose: Yes, well it was something... Just to go back for a moment, it was to explain the Atiyah-Singer Theorem, that's why it was useful. I'll come to that in a minute because it's a very interesting story, the way these things sort of come together and take many many years sometimes before they come together. But I was really intrigued by these complex numbers. And there is, well, something, let me tell you sort of the origin of the twistor idea. I was struck by the fact that, you see people know that that when things travel with a great speed, and according to Einstein Special Relativity, they get sort of flattened in the direction of motion. Now, this is a way of talking about it and you get this Lorentz contraction as it's called.

00:54:27
Now I was playing around with Relativity and thinking about, it was this two-component spinors and thinking about how the Geometry of it worked, and I realized if you think of the sky, you see the sky is, is where... You have vectors in four dimensions. Think of a vector as something which has a magnitude and a direction to it as well. And in ordinary three-space, you've got this idea of a vector which is quite common, people know about [it]. But when you're in four dimensions, then you have space and time together. But you have particular vectors, which are called null, and these are the ones along the light cone. So these, this is, an ordinary vector might represent a velocity. So you have a particle moving along with a certain speed. And your four-dimensional vector would point along the velocity or the momentum of that particle.

00:55:28
Eric Weinstein: So, weirdly, in the spacetime metric of Einstein, these are vectors that are not zero, but if you used Einstein's special rulers and protractors, what would the length of these vectors be?

00:55:42
Sir Roger Penrose: Well they're zero.

00:55:43
Eric Weinstein: So it's a really, it's a... It's linguistically tricky to talk about these things because they're nonzero things that would be measured to be of zero length if that concept of length was in fact extended from your normal concept of length.

00:56:03
Sir Roger Penrose: Yes, the idea that something of length zero means it's, two points: the distance between them is zero, you think of them right on top of each other, or if the distance is very very small they're pretty close to each other, but in the kind of Geometry, we'll call it Minkowski Geometry, because although it's describing Einstein's Special Relativity, the Geometry was not Einstein. People often say, oh, Einstein introduced four-dimensional spacetime. That's not true. It was Minkowski. And Einstein—

00:56:33
Eric Weinstein: I'd say that this is real, that this is not just a sort of a weird artifact of the description of various processes that were being undertaken.

00:56:44
Sir Roger Penrose: Yes, well it's a kind of Geometry, and Minkowski showed that the space of Special Relativity is really four-dimensional, and it's this kind of Geometry in which you can have distances which are zero, although the points are sort of a long way away from each other. And this represents a light ray. So you have one event, say, and then the light through that event reaches another event. And when I say event, I mean, not just a spatial position, but the time as well.

00:57:17
Eric Weinstein: You mean a position in spacetime.

00:57:18
Sir Roger Penrose: In spacetime. So you need four coordinates, three space and one time coordinate. So that's what we call an event. And so you have a point, or an event, in spacetime, and imagine the particle moving with the speed of light to another such event. And the distance between those two, in this kind of Geometry that Minkowski introduced is zero. So, and he, Minkowski played around with different kinds of Geometry, and he realized that Special Relativity is really best described by this kind of—what we call Minkowskian Geometry. So you can have zero distances, and yet the points are not on top of each other.

00:57:57
Eric Weinstein: So your idea was to take all the points that are bizarrely zero distance away, and then make those the new points in a new space?

00:58:05
Sir Roger Penrose: Well it wasn't quite that. I had to come up to this slowly because it took years. But the initial idea isn't so hard to understand, really. You see, if you look out at the sky, what do you see? Well you're seeing light rays or you're seeing photons coming to your eye, which have traveled with the speed of light. So, the world line in four dimensions of that photon is tilted over at what represents the speed of light. Now, in this Minkowski Geometry, that distance, well it has a clear meaning, so let me give that. Suppose the photon is emitted at one point, one event, and received at another event. Now to that photon, the time between one and the other is zero, and that time measure is exactly the distance measure in Minkowski Geometry. Suppose the particle was not traveling the speed of light. Suppose it traveled with half the speed of light or some other speed, then its time—experience of time—is exactly the distance according to Minkowski's Geometry.

00:59:18
Sir Roger Penrose: So you say, if it travels with very, very, very great speed, suppose you travel to a planet which is four lightyears away, and you travel with, well I won't do the calculation right here, but with half the speed of light, then you would, the experience that you would, of time you experience is less than the time that somebody here on Earth would think that it took you to get there. So as you travel faster, you, your experience of the passage of time slows down in a sense, you don't think it's as long. And if you actually travel the speed of light, that experience would be zero. So this is the experience of the length of time, if you had, well you could have a very very good clock, you carry it with you, and you see how, what, how the—

01:00:04
Eric Weinstein: Clock made of pure light, it all gets pretty, pretty heady out here.

01:00:07
Sir Roger Penrose: Well, you don't make it out of—you can imagine a clock, say a nuclear clock or something, and you're not traveling with the speed of light because you can't get to the speed of light, but the time measured by that nuclear clock would be the distance in Minkowski Geometry.

01:00:21
Eric Weinstein: I should point out, just for our listeners, that even people who do this field of Differential Geometry morning, noon, and night in Math departments almost never choose to work in worlds with some temporal and some spatial dimensions, because it just, it breaks your head.

01:00:39
Sir Roger Penrose: It's a very different intuition.

01:00:42
Eric Weinstein: A very different intuition.

01:00:43
Sir Roger Penrose: And when you go back and you think about the puzzles that people had when Einstein introduced his Special and then most particularly General Relativity, they found it very puzzling. You could look at the arguments people had—

01:00:54
Eric Weinstein: Well we keep using these words like time and length and all of these things that have become... We don't recognize that in that one innocent decision to break off one degree of freedom and treat it differently, that all of our linguistic intuition goes out the window.

01:01:12
Sir Roger Penrose: You have to start all over again. Well, it was a curious experience I had, because I was giving a series of lectures in Seattle... these were the Battelle lectures given in, what was it the... I forget exactly what the dates were, maybe it's round about 1970 or something like that. And there was a collection of mathematicians and a collection physicists, John Wheeler and Cecil de Witte had organized it. It was a very interesting meeting. Well, people from both areas of expertise were brought together, and at that time, it's hard to believe now, but at that time, mathematicians and physicists were barely talking to each other. And they got me to give a series of lectures. And I—

01:01:56
Eric Weinstein: This is before Jim Simons and C.N. Yang get together in Stony Brook?

01:02:03
Sir Roger Penrose: There's a good question, when was that?

01:02:05
Eric Weinstein: That was '75, '76.

01:02:07
Sir Roger Penrose: It was before that.

01:02:08
Eric Weinstein: Okay. Wow. Okay.

01:02:09
Sir Roger Penrose: Yes, it must have been before that, is that right? I think so. Yes. I really have to... My memory of dates is not—

01:02:16
Eric Weinstein: Well if you, I know you're hot on the trail of this, but just to leaven something in, Roman Jackiw at MIT once beautifully said, and I don't think he wrote it down, he said, "We didn't understand the partnership that was possible between Mathematics and Physics, because we the physicists used to talk to the analysts." And he said, "The analysts either told us things that were absolutely trivial and irrelevant, or things that we already understood." He said, "When we talked to the geometers, we started to learn new things that we'd never considered."

01:02:52
Sir Roger Penrose: It's really, there was very much cross fertilization there. But I was gonna say, I gave these lectures at, I think it was 12 lectures, and I wasted my time on something which I won't go into, until I left myself only three lectures to describe the singularities, the black hole idea, which wasn't then the term, black hole wasn't used just then. But the collapse—

01:03:15
Eric Weinstein: It was just called the Schwarzschild singularity?

01:03:17
Sir Roger Penrose: Yes. Well, it was called singularity when it—that was thing people called it the Schwarzschild singularity, it's what we now call a horizon. And I remember in my third lecture from the end, describing the, basically what we call a black hole, I was talking about the Schwarzschild singularity. And I was explaining that, you see, it was basically to do with the zero length business. And, and Steenrod was a very distinguished mathematician, he—

01:03:45
Eric Weinstein: From Princeton, Norman Steenrod.

01:03:46
Sir Roger Penrose: Yeah, and he'd written this book on fiber bundles, which is absolutely—

01:03:50
Eric Weinstein: Impenetrable.

01:03:51
Sir Roger Penrose: Well impenetrable, but also fundamental to the subject.

01:03:55
Eric Weinstein: Yes, but it's so impenetrable that I never got to the point that you're talking about.

01:03:58
Sir Roger Penrose: But anyway, he was there at the back of the room. And I remember telling about it—And he was absolutely dumbfounded, now you see, here is somebody who's a real expert at this kind of Geometry, Riemannian Geometry, whatever you call it, where you have the notion of when the distances are small, then the points are close together. And here you have this other kind of Geometry, and the intuition you need for that Geometry was completely foreign. That's the point you were just making.

01:04:23
Eric Weinstein: Well because, we do have this weird way of talking about something that sounds like this. We might call it like Non-Hausdorff Topology, but it is a Hausdorff Topology. But it's, it's, so the problem is it's pulling apart two different notions of the word close.

01:04:40
Sir Roger Penrose: That's right, exactly. That's right. Because you think of close means a small distance. So you imagine a little tiny ball and the distance from that point is small.

01:04:49
Eric Weinstein: Well, you know, Mathematics makes you pay for every attempt to sort of intuitively encode something that isn't precise. We've been discussing the fact that this intuition is very very strange, involving how to think about spaces of the type that Einstein and Minkowski and Poincare were considering... How does that begin to lead us towards these more speculative ideas of your surrounding complex numbers and the twistor program? I don't think many people, many, many of them may have heard of it. But even in Mathematics, you have to know that you got, you were sort of seen as leading a cult. It had its own newsletter, its own bizarre drawings, it was very difficult to communicate to members of the twistor cult because they didn't speak like other people.

01:05:38
Sir Roger Penrose: Well we had this twistor newsletter which was, it started off by... just in handwriting. And it was duplicated, and then... Let's not go into that for the moment.

01:05:50
Eric Weinstein: Oh very good.

01:05:51
Sir Roger Penrose: Talk about, the basic, the origin of Twistor Theory if you like, how, where did it come from?

01:05:57
Eric Weinstein: Is this, in fact, your big bet in Physics, do you think?

01:06:01
Sir Roger Penrose: Yeah, I think so. Well, you see, it's between that and the Cosmology, but the Cosmology is a bit different because it's not such a, okay, it's a wild idea, but it's not a whole body of wild ideas, which Twistor Theory more is. But it has lots of connections with Mathematics, as pure Mathematics, and connections with Physics. Let me describe the basis of it, because I think we've got most of the things we need.

01:06:01
Sir Roger Penrose: You see, the light cone describes how, from one point, or one event in spacetime, all the different points of zero distance from it or in other words all the light rays from that point. Now, let me think of it the other way around. That is, my past light cone. So I'm sitting at a certain point in spacetime and I look out at the universe, and all the light rays that get to me at a particular instant, moment of my time, come along this past light cone. So that's, imagine this kind of stretching out into the past and getting bigger and bigger as it goes back in time. And that's all the events which are, in one moment of my time I see those events. So I see a lot of stars in the sky. Now let's suppose that, I mean, the stars in the sky look like points, you see, so that you have this sphere, the celestial sphere, which is my field of vision, if I'm imagining myself out in space—

01:07:32
Eric Weinstein: So imagine that the Earth was transparent, so you weren't occluded.

01:07:34
Sir Roger Penrose: Oh, just, let's go out into space, then I can be looking at the world all around me. Now let's imagine that another astronaut comes whizzing past me at nearly the speed of light. And just as we pass each other, he looks, he or she looks out at the sky at the same moment as I do. Now, because of a phenomenon known as aberration, the stars will be slightly... not in the same place with regard to that astronaut as me. The sky is somewhat distorted, but it's distorted in a very particular way, which is what's called conformal.

01:08:16
To say this in a simple way, suppose I happen to see a configuration of stars that happened to be on a circle. Suppose they were concyclic. And then this astronaut passing by me would also see these in a circle. Even though the transformation would not be a rotation of the spheres, the sky would be squashed up more on one end and stretched out at the other end. But the thing about that transformation, it's something which I knew about from my Complex Analysis days. Do you think of the, what's called the Riemann sphere? This is the plane of points, you see it's the complex plane, or the vessel plane: the points represent the complex numbers. So zero is in the middle if you like, and then you've got one, and then you've got minus one, and i and minus i, they're all on a circle, and you go out and infinity is way out to infinity. But the Riemann sphere folds all this up into a sphere. So infinity is now a point.

01:09:17
Eric Weinstein: So it's a little bit like if you have a, if you have a caramel coating around an apple, you're folding that disc—

01:09:23
Sir Roger Penrose: You fold it around, that's right.

01:09:25
Eric Weinstein: And at the point where the stick would go into the apple, all of the boundary of that candy would come together.

01:09:33
Sir Roger Penrose: Yes. And it's what's called a stereographic projection, you can project from the north pole, and all the other points flatten out into the plane.

01:09:41
Eric Weinstein: So you can see all the points on the sphere except for the point from which you're projecting

01:09:45
Sir Roger Penrose: Exactly. And that's called the stereographic projection. And it has this remarkable property, that it sends circles to circles. Or you could say it's conformal, that is, angles are preserved, and it's a very beautiful transformat—I used to play around with these things just for fun, often. Now, the thing is that the transformations of this sphere to itself, which preserve the angles, is also [a] transformation which is what's called analytic, or holomorphic. It's the most smooth transformation you can have—

01:10:20
Eric Weinstein: So, just the analog of smooth, but for complex objects rather than real objects, where real and complex means the types of numbers.

01:10:28
Sir Roger Penrose: Yes, that's right. So it's what smooth is in Complex Analysis. And those transformations, which send the sphere to the sphere, are exactly those in Relativity. So the different observers passing me at different speeds, looking at the same sky, the map from my sky to their skies is exactly these complex transformations of the sphere.

01:10:55
And this actually is what you exactly get when you use two-component spinors, and you see the description, when you move from one observer to another, is exactly those ones which transform the sky in this conformal way to itself. And often people find this puzzling. I find it puzzling, recently, because suppose you had a sphere which is whizzing, you know, an alien spaceship, which is a sphere, shooting past you at nearly the speed of light. Well, you see [in] the direction of motion, it will be contracted by the Lorentz contraction. So when you look at it, you should see it sort of flattened out. You don't, because a sphere goes—a circle goes to a circle, if you see it as a circle when it's not moving, you'll still see it as a circle—I mean the boundary of the thing will look like a circle when it is moving. And you work away and think about it.

01:11:44
Well, you see where the light waves go, and the front of it, and the back of it, and all that, and you see, really, you don't see the flattening, it really, it does look like a circle. Its boundary looks like a circle. So I wrote a paper on this. Almost simultaneously, there was... Somebody else wrote a paper on... mainly thinking of the small circles and spheres. But this transformation, that's really what started me off—

01:12:13
Eric Weinstein: But, if I understand correctly, and maybe I don't, we have another mutual acquaintance, or friend, Raul Bott, and he showed us that the world seems to repeat every eight dimensions in a certain way. But during the first cycle of what you might call Bott Periodicity, from zero to seven, or one to eight, depending on how you like to count, you get these things called low-dimensional coincidences. And so, that they don't recur because of your point earlier about spinors, that spinors grow exponentially, whereas vectors grow linearly. And, but during the first period, where these things are of comparable strength, you get all of these objects where, depending upon, you define in two different contexts you turn out to be the same object. Are you making use of that here?

01:13:05
Sir Roger Penrose: It is that, it's the, well the Lorentz group—

01:13:09
Eric Weinstein: Or like, you know that the rotations of space and time, which we might call SO(1,3) or SO(1,3) double cover would be equal to something else called SL(2)C, which would mention complex numbers, even though there's no complex numbers to be seen in space and time.

01:13:27
Sir Roger Penrose: Yeah, it depends on that, one of those coincidences, well it's triple coincidence, I think, you certainly get a coincidence there, which one is depending upon in this description. But the point I'm making here is that in a certain sense, Relativity is described, when you do it in the two-spinor form, which is really expressed in this fact that it's the transformation of the Riemann sphere to itself, which is a complex transformation. This is the most general transformation of the sphere to itself when you think of that sphere as a Riemann sphere, so it's a complex one-dimensional space. You might say, "Surely it's two-dimensional." Well, it's two-dimensional in real numbers, but one-dimensional in complex numbers, because the complex, each complex number carries the information of two real numbers.

01:14:17
Eric Weinstein: So for example, Mathematicians would call what most people call the complex plane, they might call it a complex line.

01:14:23
Sir Roger Penrose: It's a complex line. That's right.

01:14:24
Eric Weinstein: Yeah. And so the language, again, is intended to make things very hostile to the newbie.

01:14:30
Sir Roger Penrose: Yes, well, it's, that's true. But you have to get used to the idea that when you're thinking complex, when you think of it sort of, really, sort of concretely in real terms, that you have to double the number of real dimensions to get the number of complex dimensions.

01:14:44
Eric Weinstein: I want my audience to work, but I don't want them to feel stupid for making the mistake that every single person makes.

01:14:50
Sir Roger Penrose: You halve the number of course. So we have the complex numbers playing a fundamental role in Relativity. That's really the point we want to make. And it's the complex sphere. Sorry, the Riemann sphere, which is this one-dimensional, in complex sense, two-dimensional in the real sense, object, which is fundamental.

01:15:12
Now, this Riemann sphere appears in the most basic way in Quantum Mechanics too. You think of the, the spin. Now, that's practically the most direct comple—the most direct, Quantum Mechanical thing in a certain sense, where you see Quantum Mechanics playing a real role as Quantum Mechanics, which is hard to grasp normally, but you can see it here, the Geometry really works.

01:15:40
You see, if you have an object of spin half, that's the smallest nonzero spin you can have, such as an electron. So think of an electron, it has spin half. Now, what that means is that it's basically two states of spin, which people call spin up and spin down. Well what does that mean? Right-han—You put your thumb up like that, right-handed spin is where your fingers go, and that's, spin up means right-handed about up. Spin down is right-handed about down, or it's left-handed about up. And those are the two basic states.

01:16:16
Now what's special about up and down? Nothing. So you think of 'What about right and left, forwards, backwards?' All those are combinations of up and down. And they're combinations through these complex numbers, which lie at the basis of Quantum Mechanics, that here you can see, in a visual way, what they're doing. You see, you can say, up, down, what's left and right? Well these combinations of up and down. So you add this much of up to that much of down, and you get to the to the right, and you minus it, you get to the left, or, i times and you get to forwards or back, whatever it is. And the complex numbers come in to describe these possible directions of spin. And it's the Riemann sphere, again. So, but you are relating these complex numbers of Quantum Mechanics to the directions in space. So you have a connection between these rather abstract numbers, which are fundamental to Quantum Mechanics, and the much more concrete picture of directions in space.

01:17:20
Eric Weinstein: Well, but Roger, I think you're both... Well, let me challenge slightly, ever so slightly.

01:17:27
Sir Roger Penrose: Challenge me. Go on. Yes.

01:17:28
Eric Weinstein: What you're really talking about is a very important fork in the road for Physics: Do you wed yourself to the world that we're actually given? And you know, Mach was famous for having said this phrase, "The world is given only once." And so we happen to know that there does exist a world that appears to be well modeled by three spatial and one temporal dimension. And then the key question is, do you wish to have a more general theory, which works in all dimensions, or which works for all different divisions between how many spatial and how many temporal dimensions, and what I see you as having done, which I think is incredibly noble, brave, and scientifically valid, is to work with Mathematics that are really particularizing themselves to the world we're given rather than sort of keeping some kind of, I mean, like you're getting married to the world we live in, in a way that other people are just dating it and wishing to keep their options open.

01:18:32
Sir Roger Penrose: I think you've hit on a very crucial point. Absolutely right. I mean, for example with string theory and all that, people talk about 26 dimensions, or 10 spatial dimensions, or 11, or 12, and things like that. And, sure, the Mathematics, we've got Mathematics to handle these things, and maybe that's important to the way the world works, but I was never attracted by that for basically two reasons. One was the reason I'm just trying to describe here, and it's exactly what you're saying, that I'm looking for a way in which you find a Mathematics to describe the world, which is very particular to the dimensionality we see. So the three space dimensions and one time dimension is described in this formalism very directly. And if you're going to try and talk about other numbers of dimensions of space and time, it doesn't work.

01:19:26
Eric Weinstein: Well as much as I really like to stick it to the string theorists, that's not exactly their problem either. Because 26 is really, because it's two more than 24, and 10 is really because it's two more than 8, and in 8, you have something special called triality. And so what they were really doing was figuring out how to build different theories around different highly specific targets.

01:19:51
Sir Roger Penrose: But you see there, it's the beauty in the Mathematics, which, sure, is a good guide, but it has to be—

01:19:59
Eric Weinstein: Well they play with toy theories and they never grow up to playing with reality.

01:20:03
Sir Roger Penrose: That's the sort of thing. I mean, it's perfectly good to explore all these different things. And it's very valuable. But I'm trying to follow a route, which is viewed, I think, in many quarters as very narrow. I'm looking for a route, which is, works specifically for the number of spacetime dimensions that we have. And is, if... I mean there are aspects of Twistor Theory which do work in other dimensions, but they run out very quickly. And you can see analogs of these things, but they're kind of the—

01:20:36
Eric Weinstein: Well this is sort of a strong version of the Anthropic Principle, which is that if there weren't a beautiful Mathematics to catch you... I mean, in some sense, despite the fact that you're in your late 80s, it's like you're stage diving in a punk concert, where you're going to hope that the Mathematics catches you, because you're willing to actually marry, at a very deep level, the world that we do observe. And I find that what's very disturbing to me is that the political economy of science means that fewer people are willing to make strong speculations, strong predictions, to explore things that don't give them the flexibility in case things that don't work out to say, well, it could be like this, it could be like that. And so, in part, I see you as part of a dying breed of people who are willing to go down with a ship for the privilege of commanding it as its captain.

01:21:34
Sir Roger Penrose: Well, you can view it that way if you like. My claim is that the ship isn't actually sinking. You might think it is—

01:21:40
Eric Weinstein: No, no, no, I'm not, I'm not claiming—I think that one of the things that's happened is has been that yours has been one of the most important idiosyncratic programs, that in fact got a huge lease on life from the fact that it has positive externalities. Because it was absolutely solid Mathematics, it turned out that even if it doesn't give us a fundamental description of the world, it is at least a deep insight into how to transform one problem into another to allow solutions that wouldn't have been easily gleaned in the first, in the original formulation. Now I'm not saying that it's knocked out of the park for a fundamental theory, but I don't actually know whether... Do you believe twistors are a more fundamental description of the world?

01:22:27
Sir Roger Penrose: Well I do, yes. I mean, I don't normally say that out loud, but now you've put me in a position, yes.

01:22:34
Eric Weinstein: I think that's fucking great. I mean, in other words, it's like you have to say "This I believe", and in general people won't say it.

01:22:41
Sir Roger Penrose: Yeah, I think the thing is that I have been driven in directions, just as you're pointing out, in directions which are picking out the particular rather than the general. So, sure, you have Mathematics which, one of the huge aims in Mathematics is being more and more general, and you mentioned the Atiyah-Singer Theorem. That's a beautiful example of that, where it simply generalizes over areas which you would have never thought—

01:23:08
Eric Weinstein: But it also particularizes. So for example, it is only in low dimensions where you get to play the game with what are called deformation complexes, where the first term is the symmetries in the problem, the second term is the fields, or the waves in the problem, and the third term is the equations in the problem. And then you get to cut it off at that point, and have that be this magical concept of an elliptic complex. So for example, in dimension four, we glean something bizarre, which is that there are an infinite number of different ways to do Calculus in four-dimensional space and only one way to do it in every other dimension.

01:23:47
Sir Roger Penrose: Yes, yes. Well, there's something special there about four. Certainly, that's true. And the connections may be not that clear at the moment, but maybe we'll see that this is a—

01:23:57
Eric Weinstein: Maybe differentiable structures are a part of Physics.

01:24:00
Sir Roger Penrose: It's quite possible.

01:24:01
Eric Weinstein: But, how amazing that—You know, I'll give you another very bizarre one. I don't know whether this has ever come up. If you have two sets of symmetries known as Lie groups that act transitively on the same sphere in usual position, then either their intersection acts transitively on that sphere, or the dimension of that sphere is 15. And I believe that the intersection of the groups looks like the electro-strong group. So, it's very close to the particle spectrum of Theoretical Physics pulled out of nowhere just from talking about sphere transitive group actions.

01:24:40
Sir Roger Penrose: Well, it's clear that when, I mean in Particle Physics, I mean, I've never really been somebody who studied Particle Physics closely.

01:24:49
Eric Weinstein: Is that right? I didn't know.

01:24:50
Sir Roger Penrose: Well, I mean, in a general way I have, but I suppose I felt we may be a long way from really understanding what's going on there. I don't know. I mean, I hope, I hope that it—

01:25:02
Eric Weinstein: I didn't know that.

01:25:02
Sir Roger Penrose: Well, you know, we have, no, it's a complete... I often have different views from... people do on these things.

01:25:09
Eric Weinstein: I think we're almost at the end.

01:25:12
Sir Roger Penrose: Well, that's an interesting—

01:25:13
Eric Weinstein: So how do you come to the idea that we may be quite far?

01:25:16
Sir Roger Penrose: I'm not saying that we're necessarily far, I think it's understanding why the groups are the groups that we see. And people have different theories about these.

01:25:25
Eric Weinstein: Well let me ask you then a couple of questions.

01:25:27
Sir Roger Penrose: Go on.

01:25:28
Eric Weinstein: So, very early in this new stagnation post the Standard Model, people like Glashow and Georgi, and Pati and Salam, put forward these unifying symmetries that remain very odd, because they're so attractive and powerful, the prettiest of them being something called Spin-10, which physicists persist in calling SO(10) for reasons that escape me.

01:26:01
Sir Roger Penrose: Yeah, well, this is the one which doesn't exist, or is that not that one?

01:26:05
Eric Weinstein: Well, the original SU(5), which sits inside of Spin-10, was disproven in its most basic form, and at that point, Georgi and Glashow had been trained in the previous culture of Physics, which is that you fell on your sword when you predicted something and it wasn't true. And I think that they sort of rushed to commit ritual suicide far too quickly.

01:26:28
Sir Roger Penrose: Yeah, I just, maybe if I'd worked in the subject I'd form a clearer view. It's just that from the outside, I'm not convinced that... Clearly there are things which people have discovered which are absolutely fundamental in Particle Physics. But somehow it hasn't got to the basic level, which I feel I can see why these groups are what they are and so on. Let me not talk about it because I'm not an expert at that, and I'm only giving you an impression.

01:27:00
And I suspect there will be, maybe not too long from now, a better understanding. I'm hoping that Twistor Theory might have something to say about it. But at the moment, the area which needs to be explored here hasn't been explored. The things we did at one time... I'm sort of deviating a bit from the general trend. But there was a question of how we treat massive particles in Twistor Theory. And, naturally, Twistor Theory describes massless things. Things go along the light cone and that sort of thing.

01:27:35
Eric Weinstein: So in other words, because you privileged the light cones, then the treatment of particles that were massless got a privileged treatment.

01:27:44
Sir Roger Penrose: They have a privileged treatment. And not just that, you find transformations, there is a way of representing the Maxwell equations. This was the thing I was mentioning about the TV program. We're describing the Maxwell equations, which you get out of Twistor Theory, and it comes directly out.

01:28:03
Then what about the Dirac equation, you wanted to talk about massive particles? Well, the way it seems to lead you, is you think of the—Well you see, a massive particle has a momentum vector which is timelike, so it points within the cone. And one way you can describe a timelike one is think of two null ones, so two lightlike ones, you think of a zigzag, so it's got zig and zag. And that's one convenient way of doing it, or you might have one which is made of three, zigzagzog, something like that. And you can get the timelike line out of null ones—

01:28:40
Eric Weinstein: So you can build it up from different primitives.

01:28:42
Sir Roger Penrose: That's right. So the argument is that you have a twistor for each of these zigs and zags. And so you have, might have two of them, you might have three of them, and you see how many of them give you the same amount, and then you get these groups in Twistor Theory. And these groups look like the Particle Physics groups, so you've got SU(2) and SU(3). And the idea we had is 'Oh, well, that's the basis for these Particle Physics.'

01:29:05
Eric Weinstein: So SU(2) doesn't impress me much because it's ubiquitous, but SU(3) is a, is a very unu—so this is the group that represents the strong force that holds our quarks inside of nuclei.

01:29:18
Sir Roger Penrose: Well you see, there's a thing that SU(3) gives you, this, you can gauge it. So you have... There is a difference between the SU(2) and the SU(3), that the Quantum Chromodynamics, if you like, which is the theory which comes from gauging SU(3), is a genuine Gauge Theory. But when you try and do it for SU(2), for the lepto—for the electrons and—

01:29:44
Eric Weinstein: Weak isospin.

01:29:47
Sir Roger Penrose: The gauging it doesn't really work because you've got a, you've got a special, it's not the full group and so on, and so there's something funny about it. And there are other theories which might be a more promising way to go. Let's not go into that because this is all guessing, but the idea is that you could develop a Particle Physics using many choices—

01:30:07
Eric Weinstein: You can have it, in other words, if I'm not misunderstanding you, the idea is that the extra data... I mean we have a problem in the Standard Model, in that we have effectively an origin story with two gods: there's the god of Einstein that gives us space and time, and then there's this other god that gives us SU(3)xSU(2)xU(1), which give us the non-gravitational forces and all of these particle properties we call quantum numbers, and this has no connection to the space and time data.

01:30:41
Sir Roger Penrose: Well, that's the sort of thing, yeah, it looks as though it's quite separate. I mean, it must be tied up at some stage. But we haven't got to that. But the idea here was to try and do it via twistors. Well, I'm just trying to say that we have got very excited about this for a while, and then it was a long time ago, because when people discovered charm, I think it was charm, and then suddenly this didn't fit. And so we gave up that model.

01:31:04
Eric Weinstein: And so by charm you mean the the addition of entirely separate versions of the familiar family of matter, so that we now think we have three copies of matter, where the second two are repeated at higher mass scales.

01:31:19
Sir Roger Penrose: That sort of thing, yes. Yeah, that's right. And so people were—it didn't seem so simple at that point. So, and various things didn't seem to fit so well. But I think we should go back to that, from the insights that going from General Relativity, I mean, there's a long story which should be probably hard to describe here, but the construction... See, Twistor Theory starts off as a theory about space, flat spacetime.

01:31:04
Eric Weinstein: That's what bothers me about it.

01:31:19
Sir Roger Penrose: Exactly. And it's what bothers a lot of people when you see—

01:31:55
Eric Weinstein: I'm in good company then.

01:31:58
Sir Roger Penrose: I was at the time at the University of Texas for a year, and this Alfred Schild had put a lot of people together who were General Relativity experts, hoping that something would come out of it, I guess. And I had an office, next to Engelbert Schücking, whom I learned a lot from. And on the other side, I had an office, that was Roy Kerr's office, and Ray Sachs was a little way down. And, I have to backtrack, because the question is, where did Twistor Theory come from?

01:32:38
Now, I had lots and lots of ideas that I was trying to fit together. Part of these were trying to combine the Riemann sphere of Relativity with the Riemann sphere of Quantum Mechanics, and various other Mathematical ideas which come into Quantum Field Theory, and they were sort of floating around, and I remember drawing a big piece of paper with all these ideas, which were, roughly speaking of the nature, that the world we see is described by Real Numbers, but sort of hiding behind it is a world of complex numbers. And they somehow control this world of real numbers, so that the dynamics is somehow controlled by the way the complex numbers work. And this was the sort of vague thought I had.

01:31:58
And I couldn't think of a picture in which you added, you see, spacetime is four dimensions. And I needed to add basically one more dimension because I wanted to incorporate an idea... Again, it's difficult to describe these things on a sort of popular program. But it was an idea fundamental to Quantum Field Theory, which has to do with splitting your field amplitudes into positive and negative frequencies. And it's, Engelbert had impressed upon me that this was very fundamental to Quantum Field Theory. Most people weren't stressing it at that time. And the way to think about this is to think of the Riemann sphere again, and you have the equator of the Riemann Sphere describing the real numbers, together with infinity, and you've got this complex numbers on one side of one hemisphere and also on the other hemisphere, and the ones which are positive frequency, which is the fundamental thing for Quantum Field Theory, extend into one half. So this to me was a very beautiful way of thinking about it, rather than splitting everything into the Fourier components, and taking half of them, and that was seen to be—

01:34:30
Eric Weinstein: But you have four degrees of freedom with one extra real degree of freedom.

01:34:33
Sir Roger Penrose: I just wanted one extra dimension, like the Riemann sphere going from the halve to the whole sphere, and I wanted it to divide into two halves. And that was the picture I wanted. And you try to do it with spacetime, it doesn't work because spacetime is four-dimensional, and if you complexify it it's eight-dimensional, that doesn't divide in two, that's just something else. So I knew that wasn't right. Okay.

01:34:58
Now, I was in Austin, Texas, I had friends in Dallas. Now, this was the year in which Kennedy was assassinated. And my friends in Dallas were at a dinner, and it was the next place that Kennedy was to go to, and he was going to give a speech. And they all got worried because he didn't turn up, and they were genuinely quite right to be worried, because he'd been shot. And this was a great shock to us all. And so we decided we wanted to calm ourselves down and we went on a trip to... a trip from Austin, where I was, and Dallas where the others were, and we went off in a few cars to San Antonio and maybe to the coast, and this was to try and recover from the shock. And coming back, all the womenfolk wanted to gossip and so on, and I was with István Ozsváth, who is a nice fellow, I like him a lot. Hungarian, but he didn't speak much. So all the others went to gossip and I was sort of leftover and we the two of us went in the car driving back to Austin.

01:36:03
And so I had a nice, silent drive coming back, and I started to think about these constructions that Ivor Robinson—he was in Dallas at the time, an English fellow who lived in Dallas—and he constructed these solutions of the Maxwell equations, which had this curious twist to them. And I had understood these things, and I realized that they were described by, as you talked about, the Hopf map or the Clifford parallels, these are, you can think of a sphere in four dimensions, three-dimensional sphere in four dimensions, and you have these circles, which fill the whole space, no two intersect, and every two link. Beautiful configuration.

01:36:45
And I realized that this was the thing that geometrically described these solutions that Ivor had found, and I tried to think about this and I thought, 'Well, okay, these sort of describe...' Well, the way Ivor had to have thought about it is, think of a light ray. And then you think of all the light rays which meet that light ray. So you've got one light ray and all the other light rays which meet it. And that family of light rays, you can have Maxwell's, solutions of Maxwell's equations which point along those rays. So what he did—this was his trick—you move that light ray into the complex. So you add a complex number to the—

01:37:30
Eric Weinstein: So two extra dimensions...

01:37:32
Sir Roger Penrose: Well, it pushes the light ray into the complex. And then you can construct this twisting: you don't see the light ray anymore, it's pushed into the complex, but you've still got the complex family of light rays, which meet it in a certain sense. So I try to understand what that looks like. And I thought, this is, you're pushing something into the complex, and you describe it by means of this complicated twisting family of light rays.

01:37:56
So in the drive back, I thought, 'Well, let's count the number dimensions there are of these,' as I called them later, 'Robinson congruences,' and I was gratified, or startled, or whatever the right word is, to find that the number of dimensions of this family of light rays was six: six-dimensional family. What's the dimension of the family of light rays? Five. So the ones you actually see directly are the light rays. That's the real thing, and the thing which governing in the mysterious complex world, add one dimension, they can twist right-handed, that's one way, left-handed is the other way, divides the thing into exactly what I was looking for.

01:38:40
Eric Weinstein: Fantastic. And that additionally had this structure of three complex dimensions?

01:38:45
Sir Roger Penrose: Yes, yes. Well, I had to go back and get hold of my blackboard and try to work it out. And I, very quickly—

01:38:51
Eric Weinstein: You must have been quickly exhilarated.

01:38:53
Sir Roger Penrose: So it was a complex projective three-space. You have these two twistors, and I felt pretty chuffed with myself, [I] didn't realize what this was. You have a five-dimensional space which divides this six real-dimensional space, which is really a three complex-dimensional space into two halves.

01:39:10
Eric Weinstein: So if I'm understanding you, you would start off with a seven-dimensional sphere, you'd take an action by a circle to get the complex projective three-space, and then you could further quotient that out by two spheres to get the four-dimensional sphere?

01:39:28
Sir Roger Penrose: Well, you have to have—you can think of it as a sphere and—

01:39:33
Eric Weinstein: Maybe I'm not seeing it correctly.

01:39:35
Sir Roger Penrose: You can think of it as a sphere, and—

01:39:39
Eric Weinstein: Alright, but you've got a complex projective three-space.

01:39:41
Sir Roger Penrose: Yeah, you can think of a seven-sphere.

01:39:43
Eric Weinstein: Now, let me just tell you what I find fascinating about this story, is that you're talking about a period traveling between two cities where you realize something is the Hopf fibration—

01:39:54
Sir Roger Penrose: Well, I knew it was the Hopf fibration, but I hadn't actually thought of it—

01:39:59
Eric Weinstein: You may not know the following—

01:40:00
Sir Roger Penrose: No, go ahead. Yes, yes.

01:40:02
Eric Weinstein: Isadore Singer took the work of Jim Simons and Frank Yang, C.N. Yang, and on the trip to Oxford, where you and Michael were, he said, "Oh my god, this is the quaternionic rather than the complex Hopf fibration." He said that was the instant when he realized that the self-dual instanton equations were going to be a revolution. And so it was the exact moment of the relationship to something as nontrivial, in his case as the quaternionic rather than the complex Hopf fibration. So this is almost an exact parallel between two stories because I've never heard yours before.

01:40:42
Sir Roger Penrose: I see, that's very interesting. It also has relevance, direct relevance—

01:40:46
Eric Weinstein: Direct?

01:40:46
Sir Roger Penrose: Yes. Because, I think as you were just saying, because you think of the vector space for which the complex three-space is, and of course, that's four complex dimensions, and then that means eight real dimensions.

01:41:01
Eric Weinstein: And this is, look, I want to tie this into a bigger thread, which I think is fascinating. I am not a devotee of String Theory, nor am I of Loop Quantum Gravity. I think that most of what has been said about Supersymmetry has been overbearing and wrong.

01:41:18
Sir Roger Penrose: I completely agree with all those things you're saying, yes, yes.

01:41:20
Eric Weinstein: And I think that the intellectual carnage from these adventures in political economy, or public relations, or whatever you want to call it are not being borne by the people who benefited from them, but by those who have to clean up after.

01:41:39
Sir Roger Penrose: There's something to be said for that.

01:41:40
Eric Weinstein: Well, you don't have to say it. I can say it because I'm not inside of the university system. Now what I would claim is that while these people I think did a tremendous disservice for all of us, taking what I consider to be our most accomplished intellectual community in the history of academics, Theoretical Physics, it is not the case that these people did nothing for 45-plus years, but what they did do has never been told properly. So I claimed to you at dinner the other night, like if you just look at the role of curvature in our understanding of not only General Relativity, where it's been for over 100 years, but now in Particle Theory.

01:42:22
So we had a first revolution around the mid 1970s with what's called the Wu-Yang dictionary, where a particular geometer, who becomes the most successful hedge fund manager in human history meets arguably the most accomplished theoretical physicist, if it's not Weinberg it might be Yang in terms of what has been proven of his contributions. They have an unbelievable interaction which shows that the Classical Theory underneath Particle Physics is as or more geometric than the theory of Einstein using Steenrod's fiber bundles and Ehresmann's connections, or vector potentials or what have you. Then you have a second revolution, again involving—so that was the first one that Is Singer takes from Stony Brook to Oxford—and you have another one, which is the geometric quantization revolution with your colleague Nick Woodhouse writing the bible there, in which Heisenberg's Uncertainty Relations strangely come out of curvature rather than just being some sort of weird—

01:43:32
Sir Roger Penrose: Bundle curvature, you're looking at a connection in a bundle?

01:43:35
Eric Weinstein: Well, that there's this thing called the prequantum line bundle, where line is again one of these planes, so the terminology is all screwed up. Nevertheless, the key point is that what we had previously treated as the annoyance of the Heisenberg Uncertainty Principle became the beauty of a geometric quantum. So now you had the underlying Classical Theory is geometric, the underlying Quantum Theory is now geometric, and then again with your English group, particularly Graeme Segal is that is a real hero with Michael Atiyah pointing the way.

01:44:15
You guys figure out that this weird grab bag that was called Quantum Field Theory, which is this thing above Quantum Mechanics that is needed for if you're going to have particles that change—regimes in which the number of particles changes like something emits a photon, you need Quantum Field Theory, you can't do it in Quantum Mechanics. So that world was a grab bag that made absolutely no effing sense pedagogically to anybody coming from outside of the discipline. And what they taught us, and this is coming from the 1980s on, is that Quantum Field Theory would have been discovered by topologists and geometers, even if the physical world had never used it, because it was actually a naturally occurring augmentation of what's called Bordism Theory, which is an enhancement of what you previously referred to as Cohomology.

01:45:06
So these are three separate revolutions with people that almost nobody's ever heard of like Luis Álvarez-Gaumé, and, you know, and Dan Quillen, who I think is the world's greatest accidental quantum field theorist. For some reason, the Physics community is still telling us stories about entanglement, and about multiverses and many-worlds, and this actual thing that happened, which is as gorgeous as anything I've ever seen, has been a revolution that's been plowing through Mathematics and Physics is covered up because they want to tell a story about Quantum Gravity, which just doesn't hang together. What the F?

01:45:51
Sir Roger Penrose: Yeah, well I think—

01:45:53
Eric Weinstein: First of all, am I wildly off?

01:45:55
Sir Roger Penrose: No, I don't think you are, you see. I mean, a lot of these things I wish I knew more about, you see, for example, Quillen Theory and so on which, life is too short but, but these are things—

01:46:10
Eric Weinstein: Well this comes out of the Atiyah-Singer Theory, where he finds these determinant lines, which are coming out of non-local spectral information, and building the basis maybe for prequantum line bundles in which the functions in that world become the waves that give us the theory.

01:46:30
Sir Roger Penrose: I think the trouble here, you see, Mathematics is full of all sorts of beautiful deep theory, and most of Mathematics as it exists now, in the sense of what's written in journals and so on, has almost no bearing on the physical world. Now, you see, I feel totally convinced, and I think you're expressing something similar, that if you find the right route through this stuff, you will really find the key to what we're seeing in the physical world. Now, and we've found many such keys, and General Relativity to the Lorentzian version of Riemannian—

01:47:10
Eric Weinstein: Semi-Riemannian or pseudo-Riemannian Theory.

01:47:12
Sir Roger Penrose: Yes, that pseudo-Riemannian Geometry. So that's picked up a beautiful area of Mathematics and turned it into Physics. And then the reverse has given lot back to Mathematics, and also with Quantum Theory, clearly, and Quantum Field Theory, but I think there are things that are hiding there, which are very beautiful Mathematics and which will reveal themselves as important in the Physics, we haven't got to it yet.

01:47:43
Eric Weinstein: What do you make of the fact that we now have three separate geometries? You have Riemannian Geometry as the parent of General Relativity. You have Ehresmannian Geometry, which is based on sort of these Penrose stairs coming from Fiber Bundle Theory, which is the parent of the Maxwell Classical Theory, but also the Classical Theory that would be underneath the strong force holding protons together which want to repel and the weak force which causes beta decay, all right.

01:48:17
And then you've got this other geometric theory, which is the geometric quantum... And they're not the same Geometry. So, for example, the Geometry that the Jim Simons and C.N. Yang find has this property called gauge symmetry. You have the opportunity for gauge symmetry in the Einstein theory, but because Einstein takes curvature and uses something called Linear Algebra, to project all of the curvature information into a smaller subset, killing off something called Weyl curvature, if you gauge symmetrize and then project, it's not the same as projecting and then gauge symmetrizing. So, the opportunity to use Gauge Theory is lost by the specific genius of Einstein.

01:49:12
Sir Roger Penrose: Well you see, there's good example, because he did this amazing thing when producing General Relativity, but then in his later years, he tried to develop the theory into these unified field theories, which from a mathematical point of view was not really a very... was not likely to give much new insights. But you know, he was right to think you should find a unified scheme and so on and bring the—Well, he troubled, one trouble was he didn't really—he considered Electromagnetism, but the Particle Physics didn't play much of a role in what he was doing.

01:49:46
Eric Weinstein: Well he died before quarks were understood.

01:49:48
Sir Roger Penrose: That's true.

01:49:48
Eric Weinstein: So he never... he was innocent of SU(3) and its various sins.

01:49:53
Sir Roger Penrose: I think, the thing is, there's huge beautiful things in Mathematics and we'd like to think, and I like to think, that they do have a role in, fundamental role, roles that we've not yet discovered in operating whether... well, the way the world works is dependent on very deep Mathematics.

01:50:16
The trouble is that there's so many wrong steps, in the sense that many beautiful things in Mathematics which are guiding in certain directions, which from the point of view of Mathematics are great, and they can be generalizing ideas and, and revealing all sorts of previously unknown beauties. But the proportion of these which we find has relevance to Physics is so—is very small at the moment. Now, I think that in some ways, maybe the most powerful, the most... Sure, I mean, complex numbers and the Analysis of complex numbers is one example where one does seem to see a role in operating the way the world works, and I'm sure that we will find other things. It's just there are so many temptations and directions, which are not particularly to do with Physics.

01:51:10
Eric Weinstein: I understand that. But what I don't understand and what I'm absolutely unsympathetic with, is that we have a lot of people who once upon a time had a lot of different ideas. Now, most of the ideas at the moment, if we're brutally honest, we are so constrained by this point in our story in Theoretical Physics, that almost every new idea is dead on arrival unless you specifically keep it from predicting things that we don't see.

01:51:11
Sir Roger Penrose: Oh yeah.

01:51:12
Eric Weinstein: Right, and so, what I see is that you've got different—and this is a sociological and economic critique—is that you have a class of naughty boys, who are very badly behaved, who get to make all sorts of claims, who talk to the press incessantly over decades about all the wonderful things they're going to do, which they don't do, and then you've got another group of people whose feet are held to the fire where, the instant they start to consider something that might, for example, violate a No-go Theorem, they're roundly humiliated.

01:52:16
Now, what I see you as having done is to carve out a very unusual niche. Twistor Theory is, at a minimum, an incredibly valuable tool for generating solutions on one space from solutions on another, let's say. However, it's also somewhat tolerated within the system. It's a minority point of view, it's a minority community, but it is allowed to play a parallel game to the string world, where the string theorists have lived for years, in my estimation, on externalities. There are lots of positive externalities of having, and I do think that's the smartest community out there, I do think that in general they're smarter than the relativists, they're even smarter than most of the geometers. They're insufferable.

01:53:07
Sir Roger Penrose: Oh they're very clever people.

01:53:08
Eric Weinstein: Yeah, very clever people, very insufferable. And the problem with that community is that it's actually accomplished a great deal that isn't of a stringy nature.

01:53:23
Sir Roger Penrose: That's true, that's true.

01:53:24
Eric Weinstein: And I do think that what they've done is they've, instead of quantizing Geometry, which is what Quantum Gravity was supposed to be, it backfired and they had the Geometry geometrize the Quantum, and that's the main legacy of these people. They took off for for Paris and landed in Tokyo, which is very impressive as a feat but it wasn't what they were setting out to do.

01:53:52
Sir Roger Penrose: I think, basically, I agree with that. And certainly String Theory has had a big influence in various areas of Mathematics. But the influence directly in Physics has been pretty minimal, I think.

01:54:05
Eric Weinstein: What do you think about the legacy of something like Supersymmetry? Which is this—

01:54:08
Sir Roger Penrose: That's an interesting question.

01:54:09
Eric Weinstein: Isn't it?

01:54:10
Sir Roger Penrose: Yes. Well it's very interesting, partly from a personal point of view, because when I first heard about it, and a lot of it was on Conformal Supersymmetry, and I could see there was a lot of connection with Twistor Theory. The only thing I didn't like was you were led to these algebras which didn't commute and, well, the square of something was zero or something, whatever. I mean, they weren't the kind of Algebra that you needed in Twistor Theory, you needed Complex Analysis. But anyway, I visited Zumino at one point, and I was most intrigued because I could—

01:54:48
Eric Weinstein: This is half of the duo that came up with the sort of originally, original deep Supersymmetric model.

01:54:55
Sir Roger Penrose: Absolutely, yes. And so I thought there's enough connections here, I'd like to understand it better. And yeah, I think I understood a bit more. But one thing I remember, particularly, this is a little bit of a side point, but I was talking to him about two-component spinors, and I realized he was somebody who understood it perfectly well. And he told me this story. He said he had once written a paper in which he used two-component spinors, and it worked out very well. A few months later, Abdul Salam did the same thing, but using four-component spinors. And he said everybody referred to Abdul Salam's paper and nobody referred to his paper. And he said, well that, from then on, he said, he vowed never to write a paper using the two-spinor formalism, which I thought was pretty ironic, particularly since Dirac himself, he was, well, intriguing that a lot of people in the early days of looking at generalizing the Dirac equation, for higher spins and so on, and there were you know, Duffin Cameron were the only the ones, I forget what they were all called, different names for all the different spins. And they were all in Corson's book I remember. And Dirac wrote this, had written this, I think earlier, I can't remember the history of which is which. But I think earlier, this paper, he did a whole lot using two-spinors, and Dirac using two-spinors.

01:56:16
But you see, clearly he knew about them as far as I was concerned, because he'd lectured on them, but this is much earlier than that. He had this paper in the Royal Society describing all the different spins with two-spinors. Much more, much clearer, much more general, simple, systematic. And again, nobody seemed to refer to Dirac's paper, which is quite curious, because he, I mean, there's a huge irony there because he wrote his initial paper using these four-spinors and didn't realize until maybe pointed out by van de Landa, I have no idea where he got it from. He realized that you could write all this in two-spinors. And in some ways, it was simpler, and used this to generalize to all the spins. But for some curious reason, nobody, or very few people, seem to refer to Dirac's paper.

01:57:03
Eric Weinstein: Well you know about this famous situation where Feynman found, effectively, the path integral formalism in some paper Dirac had published in the Soviet Union. Right? And that Feynman, you know, was trembling, I think, and asked Dirac, "Do you realize when you said that these two things were analogous that they're in fact proportional?" And Dirac said "Are they?"

01:57:29
Sir Roger Penrose: You had to be careful with Dirac, because you see I had a, when I'd first written things in two-spinors and I'd written General Relativity with two-spinors, and I found certain things came out very beautifully in this thing that everybody was worrying about, calling, at that time called the Bell-Robinson tensor... And you could, it dropped out if you used two-spinors, and the principal null directions, and all sorts of things. And the thing is, you have this equation, which is the Bianchi identities written in two-spinors. You can see it's just the same equation that you write for massless fields, Maxwell's equations is the same, the neutrino if it had no mass, just the same, they're just the same, you have it in the more in, the higher the spin the more indices, but it's the same equation. And you can see it's conformally invariant, and all just the same pa—

01:58:15
Eric Weinstein: I mean, but part of the problem that you're gonna get into with all these things, is I would venture to argue that even the lowly Bianchi identity, which is at the heart of how Einstein figured out how to do his equations, to make sure that effectively his vectors pointed perpendicular to the orbits of the symmetries that he was considering, that we don't really even understand the things that are given to us for free fully.

01:58:43
There's an old paper of Jerry Kazdin, I believe, in which he actually re-deduces the Bianchi identity from much more fundamental principles. And the same thing is true, with the sudden appearance of a version of the Calculus due to Levi-Civita from merely choosing rulers and protractors. I really worry that we never actually grounded these fields properly. I don't know if you're familiar, I think it's Chekhov who said that if a gun is placed above the mantel place, in the first act, it must be fired by scene four? Well, for example, we have this thing called the torsion tensor that everybody's introduced to on the first day of Riemannian Geometry, and then they properly are encouraged to forget about it thereafter—never really seems to show up in any meaningful way anywhere.

01:59:37
Sir Roger Penrose: Well it's always a puzzle, I know. And I've never quite made up my mind about it. Let's not go into that story though because, yes, I don't use it. But I once wrote a paper...

01:59:47
Eric Weinstein: But to ask you, just in terms of the path forward... It strikes me that what we have learned about our physical world and what comes up in this book is of a very frightening nature, that Einstein's equations, when you really understand them through Hilbert's insight, come from the simplest possible thing we could minimize.

02:00:17
Sir Roger Penrose: Oh yes.

02:00:18
Eric Weinstein: Same thing for Maxwell's equations, they spread from the electromagnetism to the weak force into the strong force because it was the simplest possible thing that could be optimized.

02:00:29
Sir Roger Penrose: You're thinking of the Lagrangian, yes.

02:00:30
Eric Weinstein: The Lagrangian, I'm trying to avoid saying those words. And then Dirac's third equation, to complete this triptych, is the equation for matter which generates all of something called K-theory, which is absolutely fundamental. So I could make an excellent argument that the three major equations, to be supplemented by one for the Higgs field now that we've found it, are the simplest and best possible equations of their type, not that we've found so far, but provably so.

02:01:11
Sir Roger Penrose: Yes. I think the trouble with simplicity arguments, which I agree with, is that it's simple in one context, and what's simple in another context it may be...

02:01:19
Eric Weinstein: I quite agree. But nature has shown such, I mean, the thing that I can't get over is that her taste in Mathematics... You know, I've analogized this to raiding a jewelry store with millions of pieces, and in under half a minute finding all the best stuff.

02:01:42
Sir Roger Penrose: I wanted to finish a story there, you see, which I didn't quite finish. It relates to something you were saying earlier, which I, at one point, you see Dirac was at the same college as I was, St. John's College in Cambridge, where I was a fellow. And I happened to be sitting opposite him at one point, and I had been working on these two-spinor ways of looking at General Relativity, and so I said to him, I thought, you know, I thought something he might be interested in, could I, would he have an opportunity to talk me about it? So he reserved a room and I had a little discussion with him. And then I wrote down this equation, which is this wave equation, which represents the Bianchi identities. And I wrote this thing down and Dirac, I thought he would instantly recognize it, because it's basically the same equation that he had in his paper with all these different spins. And he asked me—I wrote down the equation, he said "Where does that equation come from?" So I said, "It comes from the Bianchi identities." And he said, "What are the Bianchi identities?"

02:02:44
Eric Weinstein: Holy cow.

02:02:46
Sir Roger Penrose: And I thought, well, he's been writing all these papers in General Relativity, and Quan—he must know perfectly well—the explanation presumably is he simply rediscovered them himself. He just didn't know they were called the Bianchi identities. I don't know, it's a very curious story.

02:03:02
Eric Weinstein: And this was in the form that the derivative of the curvature in terms of the natural derivative is equal to zero, that's...?

02:03:11
Sir Roger Penrose: You're in vacuum, say, and you take the Weyl curvature, which is all that's left of the Riemann curvature, and you write that in spinors, and it's a spinor with four indices, completely symmetrical. And then when you write the derivative, it's the derivative acting on those four things in one contraction, the derivative's got two indices, and you contract one of those, and that's the equation. That vanishes, that's the equation. Same as the Maxwell equations, same as [the] neutrino if you had one index and no mass, and it's the way I think about these things. And the conformal invariance is very crucial. That leads me to all sorts of ideas that I wouldn't have thought of otherwise. And it was clearly the sort of thing Dirac would have played with himself, because his equations, all the higher spin equations—Although, curiously in his paper, he did the massless case a different way, which I never quite understood why. But anyway, it's there clearly: things that he understood completely, and somehow maybe never connected? I don't know what it was.

02:04:17
Eric Weinstein: Do you...? Did you read his 1963 article in Scientific American where he makes a very interesting case against naive application of the scientific method?

02:04:28
Sir Roger Penrose: No, I don't... That's the Dirac in Scientific… Gosh, I should have seen that.

02:04:30
Eric Weinstein: Absolutely. And his point... He makes the point in the case of Schrodinger, and he says Schrodinger would not have been led into error if he had not been pressed for agreement with experiment, because Schrodinger then, after publishing, there was a period of time where it wasn't understood that spin somehow entered the picture, and complicated the theoretical prediction with its experimental verification. But I think secretly, he was actually talking about himself, where he had introduced the Dirac equation, there had to be positively and negatively charged particles. And at that time the electron and the proton were known, but the positron and the anti-proton were not. And so he linked those two and Heisenberg immediately dinged him and said, "Wouldn't those be of the same mass and you're obviously making an error?" and he didn't stick to his guns or have the courage of his convictions to predict the new particle.

02:05:27
Sir Roger Penrose: There is something there yeah.

02:05:28
Eric Weinstein: And I think that that 1963 paper from Scientific American is Dirac trying to give us a gift from Mount Olympus, to say stop with the incessant insistent on the naive scientific method. Give yourself more room to imagine, more room to play, more room to be wrong.

02:05:48
Sir Roger Penrose: I think that's a crucial thing, you see he—that was the thing about Dirac. He just didn't want to be wrong. Now he was very worried about saying things that were wrong, and so often he would say nothing rather than anything. So this is a big thing with him. And I think he was disturbed by, yeah, you're right. I mean, he could have predicted instantly.

02:06:09
Eric Weinstein: I think he needed more freedom, and he didn't have it, and he tried to give that freedom to others.

02:06:13
Sir Roger Penrose: Either way being too timid, yes.

02:06:15
Eric Weinstein: What—

02:06:16
Sir Roger Penrose: Depressing, yeah.

02:06:18
Eric Weinstein: Let me ask you a harder question.

02:06:20
Sir Roger Penrose: Alright.

02:06:21
Eric Weinstein: Because we haven't asked you any hard questions to begin with. You're going to be in your 90s soon. If you were to—

02:06:30
Sir Roger Penrose: I hope to make it, yes.

02:06:31
Eric Weinstein: If you were to point to younger people... There seems to be a failure to pass torches that I've noticed. And you don't seem to be the sort of person I would imagine to have that problem. Who would you be pointing to—

02:06:48
Sir Roger Penrose: A human being, you mean?

02:06:49
Eric Weinstein: Individuals—

02:06:50
Sir Roger Penrose: Ooh, that's a tricky one. I don't, I—

02:06:52
Eric Weinstein: You don't have to push anyone down, but who would you, who would you build up, who's young and vital who might be, you might say, look, if anyone's got the scent, this might be a person to look at.

02:07:06
Sir Roger Penrose: I think I'm not going to take you up on that one.

02:07:08
Eric Weinstein: I'll decline, I'll decline to push further.

02:07:11
Sir Roger Penrose: Yes. It's just that it's not so obvious. I mean, I've certainly had people, clearly good, and inspirational in ways and think of things I'd never thought of. But it's hard. You see I don't know enough people, I think. It's probably somebody I don't know, you see. Yes.

02:07:29
Eric Weinstein: Do you? Do you worry that the glory that is the Oxford school of Geometry and Physics may not continue, without—

02:07:39
Sir Roger Penrose: I suppose I do worry a bit about that, yes. I mean, there was a—

02:07:42
Eric Weinstein: It was an unbelievable nucleus of people.

02:07:44
Sir Roger Penrose: Yeah, you're absolutely right. It's very remarkable.

02:07:47
Eric Weinstein: And I worry that the UK doesn't value itself enough. I think that you guys are so idiosyncratic, and so weird, badly behaved, I don't know what to call it. But the UK has punched way above its weight by tolerating and encouraging personalities, idiosyncrasies.

02:08:06
Sir Roger Penrose: That, I think that there is a point there. I wouldn't know how to generalize across countries, because maybe... But I think I think you're right to some degree. There is tolerance of eccentricity, which is specifically a kind of tolerance, which is specifically English or British, let's just say, British. I don't know. I'm nervous about saying things like that. Because you find somebody who springs up somewhere else.

02:08:36
Eric Weinstein: Yeah, somebody else, somebody will hate you for it, but quite honestly, what are they gonna do, make you pay with your career? I don't think it's gonna happen.

02:08:41
Sir Roger Penrose: Yeah, but I think, I think what you say about the Geometry developed in Oxford is, it was pretty distinctive, yes. And then the people moved out too, I mean, Michael went back to Cambridge.

02:08:56
Eric Weinstein: Well, that's, well, but the Oxford system, I mean, I don't have to make it so peculiar to Oxford, but you know, even if I think about, like a Nigel Hitchin, or Mason, I guess, has been in that system.

02:09:11
Sir Roger Penrose: Yes. No, they're a group of very able people, clearly they're very able people, no doubt about that. Yes. But it's, I mean, you're asking me to a bigger thing then...

02:09:24
Eric Weinstein: Let me just... One final thing. Have you been to the courtyard of the Simons Center for Geometry and Physics at Stony Brook, which is tiled with Penrose tiles?

02:09:36
Sir Roger Penrose: I've not been there. I've seen photographs of the tiling, yes.

02:09:39
Eric Weinstein: May I recommend a pilgrimage? They have a Wall there as well. The so-called iconic Wall, which because Jim Simons made so much money, he was able to chisel some of the world's most important equations and principles that you and I probably think of as being the hallmarks of being alive, you know, just contact with these things. They're actually in a place that can be visited with a key. And I always think about, in a fantastic world, unlocking that Wall and seeing whether it's, in fact, a gateway to something else.

02:10:17
Sir Roger Penrose: Yeah, it's a long time since I've been there, and I haven't been there since the Wall or the pavings.

02:10:23
Eric Weinstein: I recommend it. And let me ask you one final question to close it out. We all worry, when we've gotten this far along your road to reality, if you will, that we're not going to live to see the final chapter of completion, that this is a mystery like none other that, probably in some sense, does have an explanation and a kind of end. Is that something that that occupies you? I mean, I find you absolutely vital and sharp as a tack, but like, I'm worried about this at age 54, that, what if I don't get a chance to see the end? Is that something that animates you?

02:11:08
Sir Roger Penrose: So there's a huge amount of chance involved in these things, so it's all a gamble. I think, you know, to see a real end to all this is too remote for that. But on the other hand, you see, we didn't really discuss where Twistor Theory is stuck or has stuck for 40 years, and where I think it's got somewhat unstuck.

02:11:35
Eric Weinstein: You think it will be the answer?

02:11:37
Sir Roger Penrose: Well, I'm not sure how far it'll take us. You see, that's not the question. But the main problem, as I saw it in Twistor Theory, it's a sort of rather, surprisingly, things which worked surprisingly well. And one of these is to construct solutions of the Einstein equations or the Ricci-flat four-dimensional spacetimes, which were completely generic, provided they were anti self-dual.

02:12:02
Now what that means is you've got a complex solution of Einstein's vacuum equations, which are left-handed, in some sense. Now why do we want complex solutions anyway? You know, you want the real solutions. Well you see, I thought at one point, well, one useful way of thinking about the complex ones, as these are wave functions, because wave functions are naturally complex. So I thought, well, this is a wave function, but it's a nonlinear wave function. So I called it the nonlinear graviton. And it seemed to me a big step forward in understanding how Quantum Mechanics and gravity fits together.

02:12:36
But it got stuck. It got stuck with what I call the googly problem. Now you have to... if you're not a member of the former British Empire, you probably won't know what a googly is. It's a ball bowled in the game of cricket. You see in cricket, unlike baseball, you like to spin the ball about its axis. The direction, it spins about the direction in which it's moving because it bounces, that's a key thing. So it bounces one way or the other. And to make it spin left-handed has a certain action with your hand, and there's a clever thing that people do who are really good at this who can, it looks as though they're using the same action, but it's very cleverly done that the ball spins the other way. And you throw [an] occasional one of these in, it gets the batsman completely bamboozled. And that's called the googly. So using the same action that spins the ball left-handed, you spin it right-handed.

02:13:30
So I use that term because it seemed very apt: you have the frame, the twistor framework, which naturally gives you the left-handed graviton, make it do the right-handed one, and I struggled and struggled and struggled and had all sorts of wild ideas for how to do this, and I came up with one which I was very proud of, but it needed the cosmological constant to be zero. And so I thought there is no cosmological constant. Then I had a discussion with Jerry Ostriker, who's a very distinguished astrophysicist. And I was talking about the observations that there seemed to be a exponential expansion of the universe, which seemed to indicate the presence of a positive cosmological constant. So I said, "Well, you know, surely that's not really there, it's dust or something." And he looked at me and said, "That's not the point. There are so many things in cosmology which work so much better if you put this cosmological constant in." So I had to retract my view, I threw out my construction. But it took me many years to see, when you have a cosmological constant, you can do something that didn't work without it. And this enables you to have a construction which I think solves this googly problem.

02:14:47
The trouble is, from my point of view, it translates it into Algebra rather than Geometry, although you can get back to Geometry by thinking of it as a connection on a bundle, so that is a geometrical thing. It is a good thick connection. And then you talk about this algebra, and then instead of patching spaces together to make a curved manifold, you patch the algebras together. Then they have to be noncommutative, a point that [was] made clear to me by Michael Atiyah at one point. And this is the proposal, you just construct these algebras, and they are connections on bundles, and this enables you, at least in principle, to find a generic solution of the Einstein vacuum equations with cosmological constant. When I say find, it's—

02:15:29
Eric Weinstein: So you mean it, what would be called an Einstein manifold, the Ricci scalar is constant and nonzero.

02:15:34
Sir Roger Penrose: Constant and nonzero, that's right. That's right, exactly what people call Einstein space. But this is Lorentzian, and not positive-definite. But it's not a construction that you sort of write the formulae down. It's, you construct this algebra and then you look for subalgebras in the algebra, and that's not the thing I'm good at doing, so... But still it needs to be worked out.

02:15:56
Eric Weinstein: It sounds like you need a collaborator.

02:15:57
Sir Roger Penrose: Oh, absolutely. But I got distracted by Cosmology and other things, and—

02:16:02
Eric Weinstein: Well stay away from that consciousness stuff. It'll suck all your time.

02:16:05
Sir Roger Penrose: Well, that is a problem, actually.

02:16:07
Eric Weinstein: Yeah.

02:16:09
Sir Roger Penrose: But you see, other people are doing that. I'm not really doing that. They get me to give lectures on it, and I give the same old lecture which I've given many many times—

02:16:17
Eric Weinstein: Well, I have to say that if you stick with what you've done in Physics and keep trying to push that ball forward, I can't imagine a better use of your time. You're invited any time you want to come back and return to this program. This has been an extremely heavy load for our listeners.

02:16:37
Sir Roger Penrose: Sorry about that.

02:16:37
Eric Weinstein: No, we don't apolo—We don't apologize for that. We have to start doing something different, because people are hungry to know 'What does it actually sound like to hear people talking about where things are?' rather than some spoon-fed prettified version, that's like prechewed as if it were baby food. I don't think they want that anymore.

02:16:55
Sir Roger Penrose: No, I quite agree. And I think even if you don't understand all the things, that [you] get some feeling for some of the things people are trying to do is really important, as important as the details, or perhaps more—

02:17:08
Eric Weinstein: But part of it is just respecting our listeners, they know that we don't know how to get this to them in exactly the right way, and so I think we have the best listenership of any program out there because they've been habituated to recognize that not every program and not every sentence is going to make sense. So, Roger, thank you for coming through. We'll come back anytime and we'd love to continue the conversation about twistors or anything else you'd like to talk about.

02:17:31
Sir Roger Penrose: I've really enjoyed it. Thank you very much, yes.

02:17:34
Eric Weinstein: All right. You've been through The Portal with Sir Roger Penrose, hope you've enjoyed it. Please subscribe to us wherever you listen to podcasts. And if you are the sort of person who views podcasts, navigate over to our YouTube channel. Make sure that you subscribe and click the bell so you'll be informed the next time our next episode drops. Be well.

Resources

Ascending and Descending, by M. C. Escher. Lithograph, 1960.

Penrose's Writing

Penrose's own books on Spinors and Twistor theory in the context of relativity, electromagnetism, and gravity:

Spinors and Space-Time by Roger Penrose and Wolfgang Rindler.

Penrose also has an article explaining the mathematical meaning of his tribar via the standard machinery of cohomology here. A later review of his is located here and another by Tony Philips with more calculations here

The tribar shown in pieces, embedded into three open sets. The numbered and circled subregions contain duplicate overlapping points and the rules for translating into the other open sets.

Cohomology

Cohomology of a smooth manifold can be computed by solving certain differential equations, or by combinatorially approximating the manifold with a cover as shown with the tribar. Further, it plays a necessary role in Penrose's Twistor theory. Both mathematical approaches are demonstrated in the book by Bott and Tu:

Differential Forms in Algebraic Topology by Raoul Bott and Loring Tu.

Spinors

Spinors have two main instantiations: the infinitesimal quantity usually in finite dimensions as the value of a vector field at a point, or as the vector field taken over a finite region of space(time). References will be given after the brief explanations.

Infinitesimally

For the infinitesimal quantity in finite dimensions, spinors are constructed via an algebraic means known as representation theory. The key fact that supports the construction is that the special orthogonal Lie groups of rotations [math]\displaystyle{ SO(n,\mathbb{R}) }[/math] acting on linear n-dimensional space admit double coverings [math]\displaystyle{ Spin(n) }[/math] such that two elements of the spin group correspond to a single rotation. Five algebraic structures comprise the story here: an n-dimensional real vector space, the n-choose-2 dimensional rotation group acting on it, the spin group, another vector space acted upon (the representation) by the spin group to be constructed, and the Clifford algebra of the first real vector space. To distinguish between the first vector space and the second, the elements of the former are simply referred to as vectors and the latter as spinors. This may be confusing because mathematically both are vector spaces whose elements are vectors, however physically the vectors of the first space have a more basic meaning as directions in physical coordinate space.

1)

The Clifford algebra is constructed directly from the first vector space, by formally/symbolically multiplying vectors and simplifying these expressions according to the rule: [math]\displaystyle{ v\cdot v = -q(v)1 }[/math] where q is the length/quadratic form of the vector that the rotations must preserve, or utilizing the polarization identities of quadratic forms: [math]\displaystyle{ 2q(v,w)=q(v+w)-q(v)-q(w)\rightarrow v\cdot w + w\cdot v = -2q(v,w) }[/math]. The two-input bilinear form [math]\displaystyle{ q(\cdot ,\cdot ) }[/math] is (often) concretely the dot product, and the original quadratic form is recovered by plugging in the same vector twice [math]\displaystyle{ q(v,v) }[/math]. The intermediate mathematical structure which keeps track of the formal products of the vectors is known as the tensor algebra [math]\displaystyle{ T(V) }[/math] and notably does not depend on [math]\displaystyle{ q }[/math]. It helps to interpret the meaning of the unit [math]\displaystyle{ 1 }[/math] in the relation as another formally adjoined symbol. Then, the resulting Clifford algebra with the given modified multiplication rule/relation depending on [math]\displaystyle{ q }[/math] is denoted by [math]\displaystyle{ \mathcal{Cl}(V,q)=T(V)/(v\cdot v-q(v)1) }[/math] as the quotient of [math]\displaystyle{ T(V) }[/math] by the subspace of expressions which we want to evaluate to 0. This curtails the dimension of the Clifford algebra to [math]\displaystyle{ 2^n }[/math] from infinite dimensions.

2)

The Spin group is then found within the Clifford algebra, and because the fiber of the double cover map [math]\displaystyle{ Spin(n)\rightarrow SO(n, \mathbb{R}) }[/math] is discrete, it is of the same dimension n-choose-2. This will not be constructed here, but only the following operations which distribute over sums in the Clifford algebra are needed to get the Spin group and its homomorphism to the rotation group:

a)

an involution [math]\displaystyle{ \alpha, \alpha^2=id }[/math] induced by negating the embedded vectors of the Clifford algebra: [math]\displaystyle{ \alpha (x \cdot y\cdot z) = (-x)\cdot (-y)\cdot (-z)=(-1)^3x\cdot y\cdot z }[/math]

b)

the transpose [math]\displaystyle{ (-)^t }[/math] which reverses the order of any expression, e.g. [math]\displaystyle{ (x\cdot y \cdot z)^t = z\cdot y\cdot x }[/math]

c)

the adjoint action, conjugation, by an invertible element [math]\displaystyle{ \phi }[/math] of the Clifford algebra: [math]\displaystyle{ Ad_{\phi}(x)=\phi\cdot x\cdot \phi^{-1} }[/math]

3)

Spinors require more algebra to construct in general, such as understanding the representations of Clifford algebras as algebras of matrices. In the simplest case, one can choose an orthonormal basis of [math]\displaystyle{ V: \{e_1,\cdots,e_n\} }[/math] and correspond these vectors to n [math]\displaystyle{ 2^k\times 2^k }[/math] matrices with [math]\displaystyle{ k=\lfloor n/2\rfloor }[/math] such that they obey the same relations as in the Clifford algebra: [math]\displaystyle{ \gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}= -2\eta^{\mu\nu}\mathbb{I}_{k\times k} }[/math] where [math]\displaystyle{ \mathbb{I}_{k\times k} }[/math] is the [math]\displaystyle{ k\times k }[/math] identity and [math]\displaystyle{ \eta^{\mu\nu} }[/math] is the matrix of dot products of the orthonormal basis. The diagonal of [math]\displaystyle{ \eta^{\mu\nu} }[/math] can contain negative elements as in the case of the Minkowski norm in 4d spacetime, as opposed to the Euclidean dot product where it also has the form of the identity. Then the spinors are complex vectors in [math]\displaystyle{ \Delta=\mathbb{C}^{2^k} }[/math], however an important counterexample where spinors don't have a purely complex structure is the Majorana representation which is conjectured to be values of neutrino wavefunctions.

Summary

The following diagram summarizes the relationship with the structures so far:

The hook arrows denote the constructed embeddings, the double arrow gives the spin double cover of the rotations, and the downwards vertical arrows are the representations of the two groups acting on their respective vector spaces. The lack of an arrow between the two vector spaces indicates no direct relationship between vectors and spinors, only between their groups. There is however a bilinear relationship, meaning a map which is quadratic in the components of spinors to obtain components of vectors, justifying not only that spin transformations are square roots of rotations but spinors as square roots of vectors. When n is even, the spinor space will decompose into two vector spaces: [math]\displaystyle{ \Delta = S^+ \oplus S^- }[/math]. In a chosen spin-basis these bilinear maps will come from evaluating products of spinors from the respective subspaces under the gamma matrices as quadratic forms.

In the cases of Euclidean and indefinite signatures of quadratic forms, a fixed orthonormal of [math]\displaystyle{ V }[/math] can be identified with the identity rotation such that all other bases/frame are related to it by rotations and thus identified with those rotations. Similarly for the spinors, there are "spin frames" which when choosing one to correspond to the identity, biject with the whole spin group. This theory of group representations on vector spaces and spinors in particular were first realized by mathematicians, in particular Cartan for spinors, in the study of abstract symmetries and their realizations on geometric objects. However, the next step was taken by physicists.

Finitely

Spinor-valued functions or sections of spinor bundles are the primary objects of particle physics, however they can be realized in non-quantum ways as a model for the electromagnetic field per Penrose's books. When Atiyah stated that the geometrical significance of spinors is not fully understood, it is at this level rather than the well-understood representation algebra level. Analyzing the analogue of the vectors in this scenario, they represent tangents through curves at a particular point in space/on a manifold. The quadratic form then becomes a field of its own, operating on the tangent spaces of each point of the manifold independently but in a smoothly varying way. Arc length of curves can be calculated by integrating the norm of tangent vectors to the curve along it. In general relativity, this is the metric that is a dynamical variable as the solution to Einstein's equations. In order to follow the curvature of the manifold, vector fields are only locally vector valued functions, requiring transitions between regions as in the tribar example previously. Similarly, because the spin and orthogonal groups are so strongly coupled, we can apply the same transitions to spinor-valued functions. The physical reason to do this is that coordinate changes which affect vectors, thus also affect spinor fields in the same way.

Because the Minkowski inner product can be 0, the tangent vectors for which it is form a cone which Penrose depicts as smoothly varying with spacetime

The metric directly gives a way to differentiate vector fields, or finitely comparing the values at different points via parallel translation along geodesics (curves with minimal length given an initial point and velocity). Using this, a derivative operator can be given for spinor fields. It is usually written in coordinates with the gamma matrices: [math]\displaystyle{ Ds(x)=\sum_{\mu=1}^n\gamma_{\mu}\nabla_{e_{\mu}}s(x) }[/math] where the [math]\displaystyle{ \nabla_{e_{\mu}} }[/math] are the metric-given derivatives in the direction of an element of an orthonormal basis vector at x. Their difference from the coordinate partial derivatives helps to quantify the curvature. These orthonormal bases also vary with x, making a field of frames which like before can locally be identified with an [math]\displaystyle{ O(n,\mathbb{R}) }[/math]-valued function and globally (if it exists) defines an orientation of the manifold.

Dirac first wrote down the operator in flat space with partial derivatives instead of covariant derivatives, trying to find a first-order operator and an equation:

[math]\displaystyle{ (iD-m)s(x)=0 }[/math]

whose solutions are also solutions to the second order Klein-Gordon equation

[math]\displaystyle{ (D^2+m^2)s(x)=(\Delta+m^2)s(x)=0 }[/math]

But it was Atiyah who actually named the operator, and utilized its geometric significance. On a curved manifold it does not square to the Laplacian, but differs by the scalar curvature:

[math]\displaystyle{ D^2s(x)=\Delta s(x)+\frac{1}{4}R s(x) }[/math] This is known as the Lichnerowicz formula.

Spinor References

Clearly some knowledge of linear algebra and Lie groups assists in understanding the construction and meaning of spinors. With our list in mind, other books may more directly approach the topic. Spinors are implicit/given in specific representations in the quantum mechanics and field theory books. Penrose's books being given, the following give introductions to these topics at various levels:

Use this book to learn about Clifford algebras and spinors directly, it covers the necessary prerequisite linear algebra and group theory but only briefly touches on the relation to curvature.

If following our main list here, you will encounter Clifford algebras and spin representations here.

Less general discussion of spin representations, but with focus on the low dimensional examples in quantum physics.

Immediately introduces Clifford algebras and spin representations, demanding strong linear algebra. The remainder of the book extensively introduces the theory of the Dirac operator, Atiyah-Singer Index theorem, and some assorted applications in geometry.

Notes

There were three versions. The third version is in The Road to Reality. He thinks the second version is probably the best.

MC ESCHER - Ascending and Descending (The Penrose Stairs)