Roger Penrose is arguably the most important living descendent 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 Prof. Penrose.
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 incomunicado. So thanks for all the feedback, it’s been very interesting to process. 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. What I instead want to do is to remind you 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. 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.
Hello, you found the Portal. I’m your host, Eric Weinstein, and I’m here today with none other than Sir Roger Penrose. Roger, welcome.
Sir Roger Penrose: Hello, good to be here.
EW: 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. 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 maybe, 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 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?
SRP: You can talk about that, should I?
EW: It’ll be great. So, where, 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?
SRP: It’s a very tough question there. 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.
EW: Has that much really changed since you wrote the book, at a deep level?
SRP: 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.
SRP: 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.
EW: And in our time scales that’s quite new. Now–
SRP: That’s pretty new, yes.
EW: Let’s just, just to get some context. You were born in the early 1930s.
SRP: 31, yes.
EW: 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.
SRP: Oh yeah, that was an experience. Yes.
When I was at Cambridge as a graduate student, so I did my undergraduate work at London University, University College. And then I went to Cambridge as 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 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. And I thought it was quite an intriguing, I mean, I, 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, they would fade gradually fade, but they wouldn’t just disappear. 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.
EW: So you were simultaneously under the great geometer Hodge, as well as Dennis Sciama?
SRP: Well, Hodge was my supervisor. See, Dennis was just a friend.
EW: I see.
SRP: Hodge was my supervisor, originally, until he threw me out, and Todd became my supervisor. That’s another little story. But, 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–when you mentioned Dirac–Dennis was actually the last graduate, at the time he was the only graduate student of Dirac’s.
EW: Is that right?
SRP: Yes. Then this was, was Dirac–
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–
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.
EW: And actually, maybe if I could just say one thing to our listeners.
I think, I wouldn’t be far off at that description.
EW: In my estimation, if not yours, Dirac would be neck and neck with Einstein for the greatest of 20th century physicists.
EW: For some reason, his press wasn’t nearly as good, maybe because of his hair. I don’t know.
SRP: 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.
EW: 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 principle bundle that can be visually seen. And since the world seems to be about principle bundles, it’s a bit odd that the general population doesn’t know that stuff of which we are.
SRP: Yes. Well the, the Hopf fibration, or the Clifford parallels, was instrumental in the subject of Twistor theory.
EW: 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
SRP: 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 was had more circles in it, and I thought I’d draw the most 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.
EW: So Dirac, getting back to it, had this elegance of mind that was unrelenting.
EW: 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.
SRP: 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 geometry, which ended up by trying to understand tensor systems as abstract systems and things which you can’t represent in terms of components.
EW: 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.
SRP: 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. 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 formula. So I just found this thing very complicated and incomprehensible. 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 it 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. 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, and 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 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 that–
EW: Do you think Dirac actually understand–understood–these objects, these most mysterious of objects?
SRP: Two-component spinors?
EW: 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 Li, I’m not sure who.
SRP: Cartan is the one.
EW: 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 was is that he understood very well what could be said about them.
EW: But that, you know, I asked you before, about like 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.
SRP: Well, I always like to think of things geometrically. And, least for the two-component ones… You see, when you go up to higher dimensions, you still have spinors. About 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–
EW: Dimension 2D, for example, you’d get spinors of dimension, 2 to the D over 2.
SRP: 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–
EW: The full, right, right.
SRP: But they split into these two, two and two–
EW: In even dimensions.
SRP: 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, of a flag. So you have the flagpole, [which] goes along the light cone. So that’s a–
EW: That’s the vector-like piece of it.
SRP: It’s a vector. And–
EW: And then you have an extra piece of data–
SRP: 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, it’s changed its sign, and then you rotate it again, so–
EW: 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.
EW: For those of–some people 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 Taurus: 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.
EW: 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.
SRP: I think this is 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–
SRP: It’s a 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 semi angle of the cone, and you do it again, you keep thinking about motion until it becomes almost flat. And then what’s the other, it’s just a little wobble.
SRP: So when it becomes flat, this motion goes to nothing. So this illustrates how a rotation through four pi–
SRP: Two complete rotations, gradually can be deformed into no rotation at all.
However, with a single rotation, it doesn’t disappear.
EW: 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?
EW: 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.
SRP: But you see, Dirac had another thing that I… There’s a thing called the Dirac scissors problem.
SRP: So you imagine the chair with, which has the pieces of wood going out like this–
SRP: And you have a pair of scissors. I think this is Dirac’s joke that it was a pair of scissors, and through the, the way 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.
SRP: Now the problem is you take the scissors, and you rotate them through–
SRP: 360 degrees, and the string’s all tangled up.
EW: We can’t undo that one.
SRP: 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, it’s 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.
EW: You just cut the Gordian knot, yeah.
SRP: And he wrote a paper explaining that, I think Max Newman–
SRP: –Wrote a paper. Dirac did this as an illustration of how you can undo it when it’s, when it’s–
SRP: Four pi, 720 degrees, that to prove that you couldn’t do it with this is, I think you… Max Newman had a theory–
EW: Have you seen this video called Air on the Dirac String, which illustrates this in video format?
SRP: I haven’t seen that.
EW: I would highly recommend it because it shows this off as the similarity to the belt trick, the Philippine wineglass dance.
EW: All of these different versions.
SRP: I find I could do that one actually.
EW: I had Joe Rogan try it and I think he got almost all the way around.
SRP: Yeah, no, I’ve done it with a glass before, so–
SRP: Yes, you go like that and it comes back.
EW: Very stylish.
Yes, you can do two complete rotations–
EW: Two complete rotations.
EW: 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, it’s not just a question of the square root of the, of the vectors. It’s the square root of the, the algebra generated by the vectors that really the the spinors are: this exterior Clifford algebra.
SRP: Oh, yeah.
EW: 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.
SRP: Yeah, I suppose the difference between the fermions and bosons, so the particles which have a spin which is half an odd number–
SRP: Which 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, the Fermi statistics, which is to do with the, this exact, this property.
EW: 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–
SRP: You wouldn’t have anything.
EW: We wouldn’t have anything.
SRP: 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, 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 the 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, where they can’t be, and this is what sort of pushes them apart. So you get to the Fermi principle… Pauli principle
EW: So you have this, this strange thing called the spin statistics theorem–
EW: 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.
SRP: 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?
EW: That’s the topological bit.
EW: 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.
SRP: 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.
EW: Well they have to be related.
SRP: Yes, that’s right.
EW: Now, maybe I could ask you a little bit about that. So I want to get to supersymmetry. But before I do–
SRP: I see, yes. Okay. Go on.
EW: We’re gonna make you work this morning, Sir.
SRP: I can understand that.
EW: 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.
SRP: You’re thinking of things like string theory?
EW: I’m thinking about a regime before the early 70s, and a regime following the early 70s.
SRP: Well, supersymmetry, is that what you meant?
EW: 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, Reggie calculus, string theory. Like the kitchen sink, we’ve tried a million different things that don’t–
SRP: They didn’t really pan out indeed.
EW: 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.
SRP: Well, you see, you might be wrong, playing the wrong game. That’s the trouble
EW: You think rounders would do it?
SRP: 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–
EW: Well maybe I’ve thrown off close to 10, I [unintelligible]. I could do it pretty easily.
SRP: But I guess you had, there’s nothing new about that. They were, the people were playing around with knots and things, I’m Kelvin was the idea that knots might be
EW: At the basis of particle identity.
SRP: 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.
EW: 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–
SRP: That was a huge, huge revolution.
EW: And that all of those things now can be unpacked from a single geometric equation.
SRP: That, 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?
EW: Not enough.
SRP: Not enough.
EW: Although people do have Maxwell’s equations tattooed on their backsides.
SRP: Well some people do. 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.
SRP: So Faraday had a lot of the influential ideas, electromagnetism, well there was 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 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.
But even Max, you know, I’m very partial to this book on orchids that followed Darwin’s Origin of Species.
EW: 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.
SRP: I’ve heard enough. Yeah.
EW: Yeah, in fact, we had dinner last night, we, we asked Yakir Aharonov if he wanted to come but he’s in Israel, and he sends his regards.
SRP: Oh, you know, send mine back.
Oh, no, he’s great fun I always–
EW: 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–
EW: Turned out to carry the actual content, that before that it had been thought that that was just a sort of a convenience product to recover electromagnetism and it turned out that that geometric object was more important. And you know, in part 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 MC Escher.
SRP: And now you have to explain that one.
EW: Well, you know, this etching called ascending and descending.
SRP: Oh, yeah, sure. Yeah.
EW: Which is sometimes referred to as the Penrose stairs.
SRP: Yes. Well, you want that story?
EW: 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?
SRP: Well, I had, I think there’s a–
EW: Is this the first time you’ve ever heard somebody say this?
SRP: Well, let me. I mean, there’s a quite complicated story here.
EW: All right.
SRP: 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 Sean Wiley–who is a lecturer in in algebraic topology–he’s just about to get off the bus, I was getting on, and here 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?’ He said ‘Oh, well as you’d be very interested, this is this, 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 is 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. So I played around with this. And then I sort of whittled it down to the triangle, which people refer to as a tri-bar. 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. 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–
EW: This is Lionel Penrose.
SRP: 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]
EW: But in some sense–
SRP: So the Ashmolean Museum
EW: Because I’m very indebted to you for this reason, because when I, when I have to describe what general relativity is–
EW: 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 Echerness 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–
EW: But it is linguistically an accurate description of what general relativity actually is.
SRP: 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…
EW: They think they’re interested
SRP: 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’t, 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–
SRP: 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–
SRP: And locally, because you have a freedom–
SRP: And you misuse this freedom in a sense, so the glitch in it is this impossible structure.
EW: Well, I had this. So this is actually my son, my 14 year old son’s copy of the book.
SRP: I see, yes.
EW: And I was having to describe this to him called 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 the radar gun, he’s measuring the component of speed in the direction of his gun.
EW: And so that’s something that eats the vector of speed and spits out a number. And then you could imagine it 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, 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.
SRP: 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. By the harmonics, you bring a new harmonic in as you go round.
EW: You’re below, and it’s sub-perceptual. So there’s this auditory illusion that captures this–
SRP: Yes, you have an arbitrary version of the same thing. And so you had this ball bouncing around with that.
EW: 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–
SRP: Well you can draw it in a plane–
EW: In a rectilinear system.
SRP: You have an interpretation of a three dimensional thing, which, which is an ambiguous interpretation.
EW: And so you saw the movie Inception, of course, where they, they realize this actually?
SRP: Yes, they’re, they show some of that, that’s right.
EW: 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.
SRP: 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.
EW: I think I’ve seen these floating cu–
SRP: 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–
EW: Missed opportunity.
SRP: Yes. Now in my triangle, I did put some perspective.
SRP: So it’s slightly, you can see, but you can do it with a perspective and it still works.
EW: 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–
SRP: Well we were graduate students together, in the same group–
EW: In the same year.
SRP: Absolutely, the same year, yes. With the same supervisor.
SRP: Yes, that’s right.
EW: And then you continued to cross pollinate ideas–
EW: Through the years.
EW: Now for 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.
SRP: But he has had, she had such a broad understanding of mathematics. It’s partly
EW: in geometry more generally and analysis. I mean, just incredible and algebra. He wrote a book on on commutative algebra. The now he had a partner for much of his career Isidore 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
SRP: Equations without any differential equations.
EW: What differential equation if you think about differential equations as coming very often in yhree 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
SRP: 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
EW: 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.
SRP: Yeah. No, it’s a very remarkable theorem, certainly.
EW: 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?
SRP: Well I can say if I’ve used the theorem, in at least two different continent 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. 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. 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 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. 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, 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 it was incredibly magic. You just have to do it once, rather than all these different kinds–
EW: 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 the convenience that we use complex numbers in physics, but that nature appears to be essentially complex.
SRP: 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 that 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–
EW: You would have to work awfully hard to get rid of them.
SRP: 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 us 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.
EW: 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 space time with a larger structure that in some sense implies space time, where you can take all the data that roams around on space time, 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 space time with has this kind of complex number aspect baked into it.
SRP: Yes, it was something just to go back for a moment to explain the Atiyah-Singer Theorem, and 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. Now I was playing around with relativity and thinking about, it was this two components 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 or something which has it has a magnitude and the direction to it as well. And into ordinary three space, you’ve got this idea of a vector which is quite common people know about. 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.
EW: So, weirdly, in the space time 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?
SRP: Well they’re zero.
EW: 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.
SRP: 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 space time. That’s not true. It was Minkowski. And Einstein–
EW: 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.
SRP: 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 subtle. 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 special position, but the time as well. So you mean a position in space time, space time. 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 space time, and measuring 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 Minkowski geometry. So you can have zero distances, and yet the points are not on top of each other.
EW: So your idea was to take all the points points that are bizarrely zero distance away, and then make those the new points in a new space.
SRP: 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 let me give that. The, suppose the, 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 it’s time–experience of time–is exactly the distance according to Minkowski geometry. So you say, if it travels with very, very, very great speed, suppose you traveled to a planet, which is four light years 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–
EW: Clock made of pure light, it all gets pretty, pretty heavy out here.
SRP: Well, you don’t make enough–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.
EW: 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.
SRP: It’s a very different different intuition
EW: A very different intuition.
SRP: 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–
EW: 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.
SRP: 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–
EW: This is before Jim Simons and C.N. Yang get together in Stony Brook?
SRP: There’s a good question, when was that?
EW: That was 75, 76.
SRP: It was before that.
EW: Okay. Wow.
SRP: Yes, it must have been before that, is that right? I think so. Yes.
I really have to… My memory of dates is not
EW: 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 used, 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.
SRP: 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 the term, black hole wasn’t used just then. But the collapse–
EW: It was just called the Schwarzschild singularity?
SRP: Well, it was called singularity when it–that was thing people call 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–
EW: From Princeton, Norman Steenrod
SRP: Yeah, and he written this book on fiber bundles, which is absolutely–
SRP: Well impenetrable, but also fundamental to the subject.
EW: Yes, but it’s so impenetrable that I never got to the point that you’re talking about.
SRP: But anyway, he was there at the back of the room. And I remember telling– 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.
EW: 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.
SRP: 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.
EW: 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 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.
SRP: 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.
EW: Oh very good.
SRP: Talk about, the basic, the origin of Twistor theory if you like, how, where did it come from?
EW: Is this, in fact, your big bet in physics, do you think?
SRP: 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. You see, the light cone describes how, from one point, or one event in space time, 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 space time 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 measuring myself out, and so–
EW: So imagine that the Earth was transparent, so you weren’t occluded.
SRP: 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. 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 plain. 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 the 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.
EW: 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–
SRP: You fold it around, that’s right.
EW: And at the point where the stick would go into the apple, all of the boundary of that candy would come together.
SRP: 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.
EW: So you can see all the points on the sphere except for the point from which you’re projecting
SRP: 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, it’s also [a] transformation which is what’s called analytic, or holomorphic. It’s, it’s the most smooth transformation you can have–
EW: So, just the analog of smooth, but for complex objects rather than real objects, where real and complex means the types of numbers.
SRP: 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. 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. 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, mainly thinking of the small circles and spheres. But this transformation, that’s really what started me off–
EW: 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?
SRP: It is that, it’s the, well the Lorentz group–
EW: 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 SL2C, which would mention complex numbers, even though there’s no complex numbers to be seen in space and time.
SRP: 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 numbers, carries the information of two real numbers.
EW: So for example, mathematicians would call what most people call the complex plane, they might call it a complex line.
SRP: It’s a complex line. That’s right.
EW: Yeah. And so the language, again, is intended to make things very hostile to the newbie.
SRP: 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.
EW: I want my audience to work, but I don’t want them to feel stupid for making the mistake that every single person makes.
SRP: 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. So, the Riemann sphere, which is this one dimensional, in complex sense, two dimensional in the real sense, object, which is fundamental. 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. 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 that. And those are the two basic states. Now what’s special about up and down? Nothing. So you think of what, about right, 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, whichever 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 were 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.
EW: Well, but Roger, I think you’re both… Well, let me challenge slightly, ever so slightly.
SRP: Challenge me. Go on. Yes.
EW: 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.
SRP: 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.
EW: 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.
SRP: But you see there, it’s the beauty in the mathematics, which, sure, is a good guide, but it has to be–
EW: Well they play with toy theories and they never grow up to playing with reality.
SRP: 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 space time 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–
EW: 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.
SRP: 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–
EW: 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?
SRP: Well I do, yes. I mean, I don’t normally say that out loud, but now you’ve put me in a position, yes.
EW: 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.
SRP: Yeah, I think the thing is that I have been driven in directions, as 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’ve never thought–
EW: 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.
SRP: 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–
EW: Maybe differentiable structures are a part of physics.
SRP: It’s quite possible.
EW: 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.
SRP: Well, it’s clear that when, I mean in particle physics, I mean, I’ve never really been somebody who studied particle physics closely.
EW: Is that right? I didn’t know.
SRP: 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 –
EW: I didn’t know that.
SRP: -was, well, you know, we have, no, it’s a complete… I often have different views from … people do on these things.
EW: I think we’re almost at the end.
SRP: Well, that’s an interesting–
EW: So how do you come to the idea that we may be quite far?
SRP: 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.
EW: Well let me ask you then a couple of questions.
SRP: Go on.
So, very early in this new stagnation post the Standard Model, people like Glasgow and Georgia, and Pati and Salaam, 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.
Yeah, well this is the one which doesn’t exist, or is that not that one?
EW: Well, the original SU(5), which sits inside of spin 10 was disproven in its most basic form, and at that point, GA and Glasgow 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.
SRP: 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. 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.
EW: So in other words, because you privileged the light cones, then the treatment of particles that were massless got a privileged treatment.
SRP: They have a privileged treatment. And not just that you find transformations, there is a way of representing the Maxwell equations. This is 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. Then what about the Dirac equation, you want to talk about massive particles? Well, the way it seems to lead you, as 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–
EW: So you can build it up from different primitives.
SRP: 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.’
EW: 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 a nuclei
SRP: 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–
EW: Weak isospin.
SRP: 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–
EW: You could 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 gives 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.
SRP: 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.
EW: 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.
SRP: 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 space time.
That’s what bothers me about it.
Exactly. And it’s what bothers a lot of people when you see–
EW: I’m in good company then.
SRP: 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? 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. And I couldn’t think of a picture in which you added, you see, space time 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 weird way to think about this is to think of the Romans sphere again, and you have the equator of the Riemann sphere describing the real numbers together with an 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–
EW: But you have four degrees of freedom with one extra real degree of freedom.
SRP: I just wanted one extra dimension, like the Riemann sphere going from the half to the whole sphere, and I wanted it to divide it in two halves. And that was the picture I wanted. And you tried to do it with space time, it doesn’t work because first space time 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. 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 pitch to Ashford, 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. 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. 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–
EW: So two extra dimensions…
SRP: 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. So in the drive back, I thought, well, let’s count the number dimensions there are of these, as I call 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.
EW: Fantastic. And that additionally had this structure of three complex dimensions?
SRP: Yes, yes. Well, I had to go back and get hold of my blackboard and try to work it out. And I, very quickly–
EW: You must have been quickly exhilarated.
SRP: 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.
EW: 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?
SRP: Well, you have to have– you can think of it as a sphere and–
EW: Maybe I’m not seeing it correctly.
SRP: You can think of it as a sphere, and–
EW: Alright, but you’ve got a complex projective three space.
SRP: Yeah, you can think of a seven sphere.
EW: 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–
SRP: Well, I knew it was the Hopf fibration, but I hadn’t actually thought of it–
EW: You may not know the following–
SRP: No, go ahead. Yes. Yes.
EW: Isidore 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 non trivial, 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.
SRP: That’s very interesting. It also has relevance, direct relevance–
SRP: 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.
EW: 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.
SRP: I completely agree with all those things you say.
EW: 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.
SRP: There’s something to be said for that.
EW: 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. 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 Iz 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–
SRP: Bundle curvature, you’re looking at a connection in a bundle?
EW: Well, that there’s this thing called the pre quantum 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’s is that is a real hero with Michael Atiyah pointing pointing the way. 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 what’s called Bordism theory, which is an enhancement of what you previously referred to as cohomology. 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?
SRP: Yeah, well I think–
EW: First of all, am I wildly off?
SRP: 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–
EW: These come 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 pre quantum line bundles in which the functions in that world become the waves that give us the theory.
SRP: 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, and a 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, we’ve found many such keys, and general relativity to the Lorentzian version of Riemannian–
EW: Semi-Riemannian or pseudo-Riemannian theory.
SRP: Yes, that pseudo-Riemannian geometry. So that’s picked up a beautiful area of mathematics and turned it into physics. And 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.
EW: 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. 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.
SRP: 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.
EW: Well he died before quarks were understood
SRP: That’s true.
EW: So he never… he was innocent of SU(3) and its various sins.
SRP: 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 roles that we’ve not yet discovered in operating whether… well, the way the world works is dependent on very deep mathematics. 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– it’s 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.
EW: 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.
SRP: Oh yeah.
EW: 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. 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.
SRP: Oh they’re very clever people.
EW: 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.
SRP: That’s true, that’s true.
EW: 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.
SRP: 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.
EW: What do you think about the legacy of something like supersymmetry? Which is this–?
SRP: That’s an interesting question,
EW: Isn’t it?
SRP: 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–
EW: This is half of the duo that came up with the sort of originally, original deep supersymmetric model.
SRP: 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 they 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 cost and 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. 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 use this to generalize to all the spins. But for some curious reason, nobody Very few people seem to refer to direct paper.
EW: So 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?’
SRP: You had to be careful with Dirac, because 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, right. And you could dropped out of use two spinors and the principal now directions and all sorts of things. And the thing is, you have this equation, which is the big identities written into spinors. You can see it’s just the same equation that you write for masters fields. Maxwell’s equations is the thing neutrino if it had no mass, just to say, 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, or just as part of the problem that you’re gonna get into with all these things, as 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, yes, to make sure that
EW: 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. There’s an old paper of Jerry Kazdin, I believe, in which he actually reduces 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 levy civita From merely choosing rulers and protractors Yeah, I really worry that we never actually grounded these fields properly. I don’t know if you’re familiar. I think it’s checkoff who said that if a if a gun is placed above the mantel place. In the first act, it must be fired by seen for. Well, for example, we have this thing called the torsion tensor gun that everybody’s introduced to in the first day of Romani in geometry, and then they properly are encouraged to forget about thereafter never really seems to show up in any meaningful way anywhere.
SRP: Well it’s 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…
EW: 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.
SRP: Oh, yes.
EW: 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.
SRP: You’re thinking of the Lagrangian, yes.
EW: 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.
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…
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.
SRP: 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 opportunities 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?
EW: Holy cow.
SRP: And I thought, well, he’s been writing all these papers in general relativity, and quan– he must know perfectly what–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.
EW: And this was in the form that the derivative of the curvature in terms of the natural derivative is equal to zero, that’s…?
SRP: 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, the same as neutrino if you had one index and no mess, 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.
EW: 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?
SRP: No, I don’t. That’s the Dirac in Saturday. Absolutely. And his point should have seen that
EW: 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. 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.
SRP: There is something there yeah.
EW: 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.
SRP: 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.
EW: I think he needed more freedom, and he didn’t have it, and he tried to give that freedom to others.
SRP: Either way being too timid, yes.
SRP: Depressing, yeah.
EW: Let me ask you a harder question.
EW: 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–
SRP: I hope to make it, yes.
EW: 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–
SRP: A human being, you mean?
SRP: Ooh, that’s a tricky one. I don’t, I–
EW: 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.
SRP: I think I’m not going to take you up on that one.
EW: I’ll decline, I’ll decline to push further.
SRP: 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.
EW: Do you? Do you worry that the glory that is the Oxford school of geometry and physics may not continue, without–
SRP: Well I do worry a bit about that, yes. I mean, there was–
It was an unbelievable nucleus of people.
You’re absolutely right. It’s very remarkable.
EW: 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 is playing way above its weight by tolerating and encouraging personalities, idiosyncrasies.
SRP: That, I think that there is a point there. I, 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, 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.
EW: 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.
SRP: 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.
EW: 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.
SRP: Yes. No, they’re a group of very able people clearly, they’re very great people. No doubt about that. Yes. But it’s, I mean, you’re asking me to a bigger thing then.
EW: Let me just… One final thing.
Have you been to the courtyard of the Simmons Center for Geometry and Physics at Stony Brook, which is tiled with Penrose tiles?
SRP: I’ve not been there. I’ve seen photographs of the tiling. Yes,
EW: 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.
SRP: Yeah, it’s a long time since I’ve been there, and I haven’t been there since the wall or the pavings.
EW: I recommend it.
SRP: Yeah. Now I’d like to do that.
EW: 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?
SRP: 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.
EW: You think it will be the answer?
SRP: 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 space times, which were completely generic, provided they were anti self-dual. 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. 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, 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 is, 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 a 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. 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 a 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. The trouble is, from my point of view, it translates 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 non commutative. The point that 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–
EW: So you mean it, what would be called an Einstein manifold, the Ricci scalar is constant and nonzero.
SRP: 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 this formula down. It’s, you construct this algebra and then you look for sub algebras in the algebra, and that’s not the thing I’m good at doing, so. But still it needs to be worked out.
EW: It sounds like you need a collaborator.
SRP: Oh, absolutely. But I got this attracted by cosmology and other things, and
EW: Well stay away from that consciousness stuff. It’ll suck all your time.
SRP: Well, that is a problem, actually.
SRP: 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–
EW: Well, I have to say that if, 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.
SRP: Sorry about that.
EW: 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 pre chewed as if it were baby food. I don’t think they want that anymore.
SRP: No, I, I quite agree. And I think even if you don’t understand all the things that–they get some feeling for some of the things people are trying to do is, is really important, as important as the details, or perhaps more,
EW: 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.
SRP: I’ve really enjoyed it. Thank you very much, yes.
EW: 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.