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[https://en.wikipedia.org/wiki/Roger_Penrose Sir Roger Penrose] is arguably the most important living descendant of [https://en.wikipedia.org/wiki/Albert_Einstein Albert Einstein's] school of geometric physics. In this episode of [[The Portal Podcast|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 [https://en.wikipedia.org/wiki/Twistor_theory Twistor Theory] and its prospects for unification. Instead of spoon feeding the audience, however, the material is presented as it might occur between colleagues in neighboring fields so that the Portal audience might glimpse something closer to scientific communication rather than made for TV performance pedagogy. We hope you enjoy our conversation with Professor Penrose. | |||
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[[File:ThePortal-Ep20 RogerPenrose-EricWeinstein.png|600px|thumb|Eric Weinstein (right) talking with Sir Roger Penrose (left) on episode 20 of The Portal Podcast]] | [[File:ThePortal-Ep20 RogerPenrose-EricWeinstein.png|600px|thumb|Eric Weinstein (right) talking with Sir Roger Penrose (left) on episode 20 of The Portal Podcast]] | ||
== Transcript == | == Transcript == | ||
{{transcript blurb | |||
|bloglink=https://theportal.group/20-roger-penrose-plotting-the-twist-of-einsteins-legacy/ | |||
|ai=[https://otter.ai/ Otter.ai] | |||
|source=[https://www.youtube.com/watch?v=mg93Dm-vYc8 YouTube] | |||
|madeby=Brooke | |||
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|editors=Aardvark#5610 | |||
|furthercontributors=ker(∂n)/im(∂n-1)≅πn(X), n≤dim(X)#7337 | |||
}} | |||
=== Housekeeping and Introduction === | |||
00:00:00<br> | 00:00:00<br> | ||
'''Eric Weinstein:''' Hello, this is Eric with two pieces of housekeeping before we get to today's episode with Sir Roger Penrose. Now in the first place, we released Episode 19 on the biomedical implications of Bret's evolutionary prediction from first principles of elongated telomeres in laboratory rodents. I think it's a significant enough episode, and we've had so much feedback around it, that before we continue any kind of line of thinking surrounding that episode, we'll wait for my brother and his wife, Heather Heying, to return from the Amazon where they're currently incommunicado. So thanks for all the feedback, it's been very interesting to process. | '''Eric Weinstein:''' Hello, this is Eric with two pieces of housekeeping before we get to today's episode with Sir Roger Penrose. Now in the first place, we released Episode 19 on the biomedical implications of Bret's evolutionary prediction from first principles of elongated telomeres in laboratory rodents. I think it's a significant enough episode, and we've had so much feedback around it, that before we continue any kind of line of thinking surrounding that episode, we'll wait for my brother and his wife, Heather Heying, to return from the Amazon where they're currently incommunicado. So thanks for all the feedback, it's been very interesting to process. | ||
00:00:33<br> | 00:00:33<br> | ||
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. | 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 ''[https://en.wikipedia.org/wiki/The_Emperor%27s_New_Mind The Emperor's New Mind]'', maybe ask Roger about the [https://en.wikipedia.org/wiki/Many-worlds_interpretation many-worlds interpretation of Quantum Mechanics], or the weirdness of [https://en.wikipedia.org/wiki/Quantum_entanglement quantum entanglement]. I'm actually not that interested. I also don't want to go back to his earliest work on [https://en.wikipedia.org/wiki/Penrose–Hawking_singularity_theorems singularities and General Relativity] with [https://en.wikipedia.org/wiki/Stephen_Hawking Stephen Hawking]. | ||
00:00:54<br> | 00:00:54<br> | ||
What I instead want to do is to remind you of what Roger is in fact famous for. He is one of the greatest geometric physicists now living. He's perhaps the best descendant of Albert Einstein currently still working in Theoretical Physics in this particular line of thought. I also think he's a great example of what the UK does well: he has a very idiosyncratic approach to trying to solve the deepest problems in Theoretical Physics called Twistor Theory. I'm not expert in it, and I can't always follow it, so if you're not following everything in today's episode, instead of deciding that the episode has somehow failed you, try to remember that people who are working in Mathematics and Theoretical Physics spend most of their time listening to colleagues completely lost as to what their colleagues are saying. So, if you start to feel that you're being left behind by some line of thinking, what we do is, in general, wait to see if another line of thinking opens up that we can try to catch. You're not going to get all of the waves, and in fact the same thing is happening to me while I'm interviewing Roger. He's not understanding everything I'm saying. I'm not understanding everything he's saying. And in fact, this is normal. | What I instead want to do is to remind you of what Roger is in fact famous for. He is one of the greatest [https://en.wikipedia.org/wiki/Geometry#Physics geometric physicists] now living. He's perhaps the best descendant of Albert Einstein currently still working in [https://en.wikipedia.org/wiki/Theoretical_physics Theoretical Physics] in this particular line of thought. I also think he's a great example of what the UK does well: he has a very idiosyncratic approach to trying to solve the deepest problems in Theoretical Physics called Twistor Theory. I'm not expert in it, and I can't always follow it, so if you're not following everything in today's episode, instead of deciding that the episode has somehow failed you, try to remember that people who are working in Mathematics and Theoretical Physics spend most of their time listening to colleagues completely lost as to what their colleagues are saying. So, if you start to feel that you're being left behind by some line of thinking, what we do is, in general, wait to see if another line of thinking opens up that we can try to catch. You're not going to get all of the waves, and in fact the same thing is happening to me while I'm interviewing Roger. He's not understanding everything I'm saying. I'm not understanding everything he's saying. And in fact, this is normal. | ||
00:01:57<br> | 00:01:57<br> | ||
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00:03:09<br> | 00:03:09<br> | ||
You know, there's a Leonard Cohen quote, from a song called The Future where he says, "You don't know me from the wind, you never will, you never did. But I'm the little Jew that wrote the Bible." And I have what I consider to be the bible right here, which is a book you wrote called The Road to Reality which, there's no getting away from, may be, in my opinion, the most important modern book of our time, because what it tries to do is to summarize what we know about the nature of all of this at the deepest level. And I think what I want to do is to introduce you to our audience, which has been habituated, over perhaps 16 or so interviews, not to expect to understand everything. They want to work, they want to hear conversations unlike any they've heard, and so we'll do some combination of explaining things, but [also] some combination of allowing them to look up things in their own free time, if you're game. Should we talk about The Road to Reality? | You know, there's a Leonard Cohen quote, from a song called [https://www.youtube.com/watch?v=AKwr3DDvFpw 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 [https://www.amazon.com/Road-Reality-Complete-Guide-Universe/dp/0679776311 The Road to Reality] which, there's no getting away from, may be, in my opinion, the most important modern book of our time, because what it tries to do is to summarize what we know about the nature of all of this at the deepest level. And I think what I want to do is to introduce you to our audience, which has been habituated, over perhaps 16 or so interviews, not to expect to understand everything. They want to work, they want to hear conversations unlike any they've heard, and so we'll do some combination of explaining things, but [also] some combination of allowing them to look up things in their own free time, if you're game. Should we talk about The Road to Reality? | ||
00:04:06<br> | 00:04:06<br> | ||
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00:05:08<br> | 00:05:08<br> | ||
'''Sir Roger Penrose:''' You see, I have a proposal, which I didn't have—I mean, it's new since the book. It's not all that new because it's about 15 years old, but it's new since I wrote that book. | '''Sir Roger Penrose:''' You see, I have [https://physicsworld.com/a/new-evidence-for-cyclic-universe-claimed-by-roger-penrose-and-colleagues/ a proposal], which I didn't have—I mean, it's new since the book. It's not all that new because it's about 15 years old, but it's new since I wrote that book. | ||
00:05:19<br> | 00:05:19<br> | ||
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00:05:28<br> | 00:05:28<br> | ||
'''Eric Weinstein:''' Okay, you got a chance to live through, if not the original General Relativistic and Quantum revolutions, their consequences. In particular, you were able to take classes from people like Paul Dirac, who scarcely seems like a human being, sometimes more like a god. | '''Eric Weinstein:''' Okay, you got a chance to live through, if not the original General Relativistic and Quantum revolutions, their consequences. In particular, you were able to take classes from people like [https://en.wikipedia.org/wiki/Paul_Dirac Paul Dirac], who scarcely seems like a human being, sometimes more like a god. | ||
00:05:51<br> | 00:05:51<br> | ||
'''Sir Roger Penrose:''' Oh yeah, that was an experience. Yes. When I was at Cambridge as a graduate student—You see I did my undergraduate work at London University, University College. And then I went to Cambridge as [a] graduate student, and I went to do Algebraic Geometry, so I wasn't trying to do Physics at all. And I, I'd encountered a friend of my brother's, Dennis Sciama, when I think I was at University College as an undergraduate. And he had given a series of talks on Cosmology—well it started with the Earth, and then he sort of worked his way out, and then talked about what was then referred to as the Steady-state Theory. Where the galaxies—the universe expands and expands and expands, but it doesn't change, because all the time there is new matter created—hydrogen—and the universe expands and then you get new material, and it keeps replenishing what gets lost. | '''Sir Roger Penrose:''' Oh yeah, that was an experience. Yes. When I was at [https://en.wikipedia.org/wiki/University_of_Cambridge Cambridge] as a graduate student—You see I did my undergraduate work at [https://en.wikipedia.org/wiki/University_College_London London University, University College]. And then I went to Cambridge as [a] graduate student, and I went to do [https://mathworld.wolfram.com/AlgebraicGeometry.html Algebraic Geometry], so I wasn't trying to do Physics at all. And I, I'd encountered a friend of my brother's, [https://en.wikipedia.org/wiki/Dennis_W._Sciama Dennis Sciama], when I think I was at University College as an undergraduate. And he had given a series of talks on Cosmology—well it started with the Earth, and then he sort of worked his way out, and then talked about what was then referred to as the [https://en.wikipedia.org/wiki/Steady-state_model Steady-state Theory]. Where the galaxies—the universe expands and expands and expands, but it doesn't change, because all the time there is new matter created—hydrogen—and the universe expands and then you get new material, and it keeps replenishing what gets lost. | ||
00:07:00<br> | 00:07:00<br> | ||
And I thought it was quite an intriguing, I mean, Dennis was a great fan of this model, and so I was really taken by it. So that, well the story was that I was in Cambridge visiting my brother, my older brother Oliver, who did Statistical Mechanics. And he was actually much more precocious than I was, he was two years ahead. And he was, I think, finishing his research there. But I had been listening to these talks by Fred Hoyle, and he was talking, I think in his last talk, about how in the Steady-state Model, the galaxies expanded away, expanded away, and then when they reach the speed of light, they disappear. And I thought that can't be quite right, and I started drawing pictures with light cones and things like this. And I thought, well, they would fade, gradually fade, but they wouldn't just disappear. | And I thought it was quite an intriguing, I mean, Dennis was a great fan of this model, and so I was really taken by it. So that, well the story was that I was in Cambridge visiting my brother, my older brother [https://en.wikipedia.org/wiki/Oliver_Penrose Oliver], who did [https://en.wikipedia.org/wiki/Statistical_mechanics 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 [https://en.wikipedia.org/wiki/Fred_Hoyle 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 [https://en.wikipedia.org/wiki/Light_cone light cones] and things like this. And I thought, well, they would fade, gradually fade, but they wouldn't just disappear. | ||
00:07:56<br> | 00:07:56<br> | ||
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00:08:46<br> | 00:08:46<br> | ||
'''Eric Weinstein:''' So you were simultaneously under the great geometer Hodge | '''Eric Weinstein:''' So you were simultaneously under the great geometer Hodge as well as Dennis Sciama? | ||
00:08:52<br> | 00:08:52<br> | ||
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00:08:56<br> | 00:08:56<br> | ||
'''Sir Roger Penrose:''' Hodge was my supervisor, originally, until he threw me out, and Todd became my supervisor. That's another little story. But Dennis just wanted to get me interested, and do working Cosmology. This was it. I never, he wanted me to change my subject. I learned an awful lot from Dennis about Physics, because Dennis sort of knew everything and everybody. And he had a real knack of getting, if he thought two people should meet each other, he got, made sure they did meet each other. In one case, it was Stephen Hawking. But, Dennis was actually—well you mentioned Dirac—Dennis was actually the last graduate, at the time he was the only graduate student of Dirac's. | '''Sir Roger Penrose:''' Hodge was my supervisor, originally, until he threw me out, and [https://en.wikipedia.org/wiki/J._A._Todd Todd] became my supervisor. That's another little story. But Dennis just wanted to get me interested, and do working Cosmology. This was it. I never, he wanted me to change my subject. I learned an awful lot from Dennis about Physics, because Dennis sort of knew everything and everybody. And he had a real knack of getting, if he thought two people should meet each other, he got, made sure they did meet each other. In one case, it was Stephen Hawking. But, Dennis was actually—well you mentioned Dirac—Dennis was actually the last graduate, at the time he was the only graduate student of Dirac's. | ||
00:09:47<br> | 00:09:47<br> | ||
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00:09:49<br> | 00:09:49<br> | ||
'''Eric Weinstein:''' Dirac was famously sort of difficult. I think that, you know, in recent years, this book came out of Graham Farmelo, The Strangest Man, that puts Dirac's bizarreness, in line with— | '''Eric Weinstein:''' Dirac was famously sort of difficult. I think that, you know, in recent years, this book came out of [https://en.wikipedia.org/wiki/Graham_Farmelo Graham Farmelo], ''[https://www.amazon.com/Strangest-Man-Hidden-Dirac-Mystic/dp/0465022103 The Strangest Man]'', that puts Dirac's bizarreness, in line with— | ||
00:10:02<br> | 00:10:02<br> | ||
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00:10:51<br> | 00:10:51<br> | ||
'''Eric Weinstein:''' Well his, and this gets to a very odd issue, which is that you have wielded taste and beauty as a weapon your entire life. Your drawings are among the most compelling—I remember the first time—one of the things I've done, using our friend Joe Rogan's program, is to push out discussion of the Hopf fibration, because it's the only non-trivial principal bundle that can be visually seen. And since the world seems to be about principal bundles, it's a bit odd that the general population doesn't know that stuff of which we are. | '''Eric Weinstein:''' Well his, and this gets to a very odd issue, which is that you have wielded taste and beauty as a weapon your entire life. Your drawings are among the most compelling—I remember the first time—one of the things I've done, using our friend Joe Rogan's program, is to push out discussion of the [https://nilesjohnson.net/hopf.html Hopf fibration], because it's the only non-trivial principal bundle that can be visually seen. And since the world seems to be about principal bundles, it's a bit odd that the general population doesn't know that stuff of which we are. | ||
00:11:28<br> | 00:11:28<br> | ||
'''Sir Roger Penrose:''' Yes. Well the, the Hopf fibration, or the Clifford parallels, was instrumental in the subject of Twistor Theory. | '''Sir Roger Penrose:''' Yes. Well the, the Hopf fibration, or the [https://en.wikipedia.org/wiki/Clifford_parallel Clifford parallels], was instrumental in the subject of Twistor Theory. | ||
00:11:37<br> | 00:11:37<br> | ||
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00:12:47<br> | 00:12:47<br> | ||
'''Sir Roger Penrose:''' Well, when I went to the... you see Dirac gave a course of lectures in Quantum Mechanics, and the first course was sort of basic Quantum Mechanics. And the second course was on Quantum Field Theory, but also spinors. And there's an interesting story about that, which I don't know the answer to. In the second course, he deviated from his normal course of lectures. Now, I understood when I talked to Graham Farmelo, who wrote this biography of Dirac, I understood from Graham Farmelo that, when I described that Dirac deviated from his normal course to give two or three lectures on two-component spinors, which for me were absolutely what I needed. You see, I'd learned from my work on Algebraic Geometry, which ended up by trying to understand tensor systems as abstract systems, and things which you can't represent in terms of components. | '''Sir Roger Penrose:''' Well, when I went to the... you see Dirac gave a course of lectures in [https://en.wikipedia.org/wiki/Quantum_mechanics Quantum Mechanics], and the first course was sort of basic Quantum Mechanics. And the second course was on [https://en.wikipedia.org/wiki/Quantum_field_theory 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 [https://en.wikipedia.org/wiki/Spinor#Component_spinors two-component spinors], which for me were absolutely what I needed. You see, I'd learned from my work on Algebraic Geometry, which ended up by trying to understand tensor systems as abstract systems, and things which you can't represent in terms of components. | ||
00:13:53<br> | 00:13:53<br> | ||
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00:16:43<br> | 00:16:43<br> | ||
'''Eric Weinstein:''' Spinors in general. I mean, he brought them into Physics, they'd been previously found inside of Mathematics, I think by people like Killing and Lie, I'm not sure who. | '''Eric Weinstein:''' Spinors in general. I mean, he brought them into Physics, they'd been previously found inside of Mathematics, I think by people like [https://en.wikipedia.org/wiki/Wilhelm_Killing Killing] and [https://en.wikipedia.org/wiki/Sophus_Lie Lie], I'm not sure who. | ||
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'''Sir Roger Penrose:''' Cartan is the one. | '''Sir Roger Penrose:''' [https://en.wikipedia.org/wiki/%C3%89lie_Cartan Cartan] is the one. | ||
00:16:54<br> | 00:16:54<br> | ||
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00:17:10<br> | 00:17:10<br> | ||
'''Eric Weinstein:''' But that, you know, I asked you before about your favorite film, you said 2001. You could make an argument that spinors are, in Mathematics and Physics, like the monolith. It's always encountered, nobody ever understands exactly what it means, but it always grabs your attention, because it seems so absolutely bizarre and highly conserved. | '''Eric Weinstein:''' But that, you know, I asked you before about your favorite film, you said [https://en.wikipedia.org/wiki/2001:_A_Space_Odyssey_(film) 2001]. You could make an argument that spinors are, in Mathematics and Physics, like the monolith. It's always encountered, nobody ever understands exactly what it means, but it always grabs your attention, because it seems so absolutely bizarre and highly conserved. | ||
00:17:29<br> | 00:17:29<br> | ||
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00:18:59<br> | 00:18:59<br> | ||
'''Eric Weinstein:''' Well that won't make any sense to anyone. But if—I mean one way of looking at that is if you have a Klein | '''Eric Weinstein:''' Well that won't make any sense to anyone. But if—I mean one way of looking at that is if you have a [https://en.wikipedia.org/wiki/Klein_bottle Klein bottle]— | ||
00:19:05<br> | 00:19:05<br> | ||
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00:19:05<br> | 00:19:05<br> | ||
'''Eric Weinstein:''' And for those of—some people will be listening to this on audio, some watching it in video. A Klein bottle, in a certain sense that can be made precise, has a square root that would be a torus: that is a double cover. So it seems like a very weird thing to take a square root of a strange topological mobius-like object, but there you are. | '''Eric Weinstein:''' And for those of—some people will be listening to this on audio, some watching it in video. A Klein bottle, in a certain sense that can be made precise, has a square root that would be a torus: that is, a [https://en.wikipedia.org/wiki/Double_cover 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. | ||
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'''Sir Roger Penrose:''' Well I think this was a mystery. I mean, I understood that a spinor was the square root of a vector, you see, and I couldn't make head or tail of that idea. And it was when I went to Dirac's course it did become clear. And he made, he gave this very impressive illustration, which I thought was due to Dirac, I learned later it was due to Hermann Weyl, that you imagine a cone, circular cone— | '''Sir Roger Penrose:''' Well I think this was a mystery. I mean, I understood that a spinor was the square root of a vector, you see, and I couldn't make head or tail of that idea. And it was when I went to Dirac's course it did become clear. And he made, he gave this very impressive illustration, which I thought was due to Dirac, I learned later it was due to [https://en.wikipedia.org/wiki/Hermann_Weyl Hermann Weyl], that you imagine a [https://en.wikipedia.org/wiki/Cone cone], circular cone— | ||
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00:21:07<br> | 00:21:07<br> | ||
'''Eric Weinstein:''' Well, I think with a, with a pulley system and a wheel, we don't have any trouble imagining a wheel that rotates twice as fast, half as fast, not at all hooked up to one particular crank wheel, right? | '''Eric Weinstein:''' Well, I think with a, with a [https://en.wikipedia.org/wiki/Pulley 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? | ||
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00:23:22<br> | 00:23:22<br> | ||
'''Eric Weinstein:''' Have you seen this video called Air on the Dirac String, which illustrates this in video format? | '''Eric Weinstein:''' Have you seen this video called [https://www.youtube.com/watch?v=CYBqIRM8GiY Air on the Dirac String], which illustrates this in video format? | ||
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00:23:27<br> | 00:23:27<br> | ||
'''Eric Weinstein:''' I would highly recommend it because it shows this off as the similarity to the belt trick, to the Philippine wineglass dance— | '''Eric Weinstein:''' I would highly recommend it because it shows this off as the similarity to the [https://www.youtube.com/watch?v=JaIR-cWk_-o belt trick], to the Philippine wineglass dance— | ||
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00:24:45<br> | 00:24:45<br> | ||
'''Sir Roger Penrose:''' Yeah, I suppose the difference between the fermions and bosons, so the particles which have a spin which is half an odd number— | '''Sir Roger Penrose:''' Yeah, I suppose the difference between the [https://simple.wikipedia.org/wiki/Fermion fermions] and [https://en.wikipedia.org/wiki/Boson bosons], so the particles which have a spin which is half an odd number— | ||
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00:28:45<br> | 00:28:45<br> | ||
'''Eric Weinstein:''' Well, it could be Grand Unified Theory, Supersymmetry, Technicolor. It could be Asymptotic Safety. It could be any one of a number of speculative theories from Loop Quantum Gravity, | '''Eric Weinstein:''' Well, it could be Grand Unified Theory, Supersymmetry, Technicolor. It could be Asymptotic Safety. It could be any one of a number of speculative theories from Loop Quantum Gravity, Regge Calculus, String Theory. It's like the kitchen sink, we've tried a million different things that don't— | ||
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'''Sir Roger Penrose:''' Yeah, yeah. I mean, these ideas come back again in a different form, but certainly in the, I guess the 19th century, people were playing with, well, I guess you can go back further than that... Phlogiston. | '''Sir Roger Penrose:''' Yeah, yeah. I mean, these ideas come back again in a different form, but certainly in the, I guess the 19th century, people were playing with, well, I guess you can go back further than that... [https://en.wikipedia.org/wiki/Phlogiston_theory Phlogiston]. | ||
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'''Sir Roger Penrose:''' Well some people do. But the general public don't know about Maxwell. But Maxwell's equations completely change our way of looking at the world. And we live off it without thinking, you know, you've got these lights here. Well, these are visible lights, so we, we know, you knew about visible light, but we didn't know anything about x-rays. X-rays, radio waves, they're all part of the same scheme. Electromagnetism, dynam—well, some of this goes back to Faraday just before Maxwell. | '''Sir Roger Penrose:''' Well some people do. But the general public don't know about Maxwell. But Maxwell's equations completely change our way of looking at the world. And we live off it without thinking, you know, you've got these lights here. Well, these are visible lights, so we, we know, you knew about visible light, but we didn't know anything about x-rays. X-rays, radio waves, they're all part of the same scheme. Electromagnetism, dynam—well, some of this goes back to [https://en.wikipedia.org/wiki/Michael_Faraday Faraday] just before Maxwell. | ||
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'''Eric Weinstein:''' But even Max, you know, I'm very partial to this book on orchids that followed Darwin's Origin of Species. | '''Eric Weinstein:''' But even Max, you know, I'm very partial to this book on orchids that followed [https://en.wikipedia.org/wiki/Charles_Darwin Darwin's] ''[https://en.wikipedia.org/wiki/On_the_Origin_of_Species Origin of Species]''. | ||
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'''Eric Weinstein:''' That was the book he wrote—the title is, and I always, I love reciting it, it's On the Various Contrivances by which British and Foreign Orchids are Fertilized by Insects. And so you think, well, why would you write a damn fool book like that after Origin of Species? And the answer is he wanted to test whether he understood his own theory. And in fact, it's revealed that he didn't understand the full implications. I would say that the same thing is true of Maxwell's equations, which is, this is perhaps the best dress rehearsal for unification we've ever seen, you know, full unification, and on the other hand, it's not until the late 50s that we actually unpack the last trivial consequence of the theory with this bizarre effect of passing an electron beam around an insulated wire. | '''Eric Weinstein:''' That was the book he wrote—the title is, and I always, I love reciting it, it's ''[https://en.wikipedia.org/wiki/Fertilisation_of_Orchids 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. | ||
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'''Eric Weinstein:''' Yeah, in fact we had dinner last night, we asked Yakir Aharonov if he wanted to come but he's in Israel, and he sends his regards. | '''Eric Weinstein:''' Yeah, in fact we had dinner last night, we asked [https://en.wikipedia.org/wiki/Yakir_Aharonov Yakir Aharonov] if he wanted to come but he's in Israel, and he sends his regards. | ||
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'''Eric Weinstein:''' Well, you know, this etching called Ascending and Descending. | '''Eric Weinstein:''' Well, you know, this etching called ''[https://en.wikipedia.org/wiki/Ascending_and_Descending Ascending and Descending]''. | ||
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'''Sir Roger Penrose:''' You see when I was a graduate student in Cambridge, I think it was in my second year, when the International Congress of Mathematicians took place in Amsterdam. And so I and a few friends decided we would go to this meeting, and I remember... I think I was just about to get on the bus or tram or something, and | '''Sir Roger Penrose:''' You see when I was a graduate student in Cambridge, I think it was in my second year, when the International Congress of Mathematicians took place in Amsterdam. And so I and a few friends decided we would go to this meeting, and I remember... I think I was just about to get on the bus or tram or something, and Shaun Wylie—who is a lecturer in in Algebraic Topology—he's just about to get off the bus, I was getting on, and he had this catalog in his hand of an exhibition in the Van Gogh Museum. And this was a picture... The one called Night and Day with birds flying off into the day and the night, and the birds changed into the spaces between the birds [unintelligible], and I just look at this and I think 'Oh that's amazing what is that? Where on earth did that come from?' | ||
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So I played around with this. And then I sort of whittled it down to the triangle, which people refer to as a tribar. So it's a triangle which is locally a completely consistent picture, but as a whole, it's impossible. And I showed this to my father. And then he started drawing impossible buildings, and then he came up with this staircase. So we decided we'd like to write a paper together on this. And we had no idea what the subject was, I mean, what, who do you send a paper like this to, what journal? So he decided since he knew the editor of the British Journal of Psychology, and he thought he'd be able to get it through, we decided the subject was Psychology. | So I played around with this. And then I sort of whittled it down to the triangle, which people refer to as a [https://en.wikipedia.org/wiki/Penrose_triangle tribar]. So it's a triangle which is locally a completely consistent picture, but as a whole, it's impossible. And I showed this to my father. And then he started drawing impossible buildings, and then he came up with this staircase. So we decided we'd like to write a paper together on this. And we had no idea what the subject was, I mean, what, who do you send a paper like this to, what journal? So he decided since he knew the editor of the British Journal of Psychology, and he thought he'd be able to get it through, we decided the subject was Psychology. | ||
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'''Eric Weinstein:''' So you saw the movie Inception, of course, where they, they realized this actually? | '''Eric Weinstein:''' So you saw the movie ''[https://en.wikipedia.org/wiki/Inception Inception]'', of course, where they, they realized this actually? | ||
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'''Eric Weinstein:''' But that effect is the soul of the Aharonov-Bohm effect, which surprised the world in the late 50s because it was discovered so late into the game. | '''Eric Weinstein:''' But that effect is the soul of the [https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect Aharonov-Bohm effect], which surprised the world in the late 50s because it was discovered so late into the game. | ||
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'''Sir Roger Penrose:''' It is a comm—same sort of thing. That's right. Well, of course like so many things, people point out that this Oscar Reutersvärd, who is a Swedish artist who'd drawn things like this before. I think roundabout the year I was born, he had a picture which is all, with cubes going around. It wasn't exactly the same, but it was. | '''Sir Roger Penrose:''' It is a comm—same sort of thing. That's right. Well, of course like so many things, people point out that this [https://en.wikipedia.org/wiki/Oscar_Reutersv%C3%A4rd 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. | ||
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'''Eric Weinstein:''' So, what I want to get at is, I think also that we have this very funny thing that happened, recently, starting from the early 70s, where we started mis-telling our own Physics history, because of the needs of the community to look like we were succeeding when we weren't, or we were succeeding at something different than we were trying to succeed at. And, in part, one of the reasons that I want to use this podcast to discuss science is to give alternate versions of what's happened. And I want to explore one or two of them with you. Now, you and I have a very funny relationship which, we don't really know each other. But you were quite close to Michael Atiyah at various points. And I was— | '''Eric Weinstein:''' So, what I want to get at is, I think also that we have this very funny thing that happened, recently, starting from the early 70s, where we started mis-telling our own Physics history, because of the needs of the community to look like we were succeeding when we weren't, or we were succeeding at something different than we were trying to succeed at. And, in part, one of the reasons that I want to use this podcast to discuss science is to give alternate versions of what's happened. And I want to explore one or two of them with you. Now, you and I have a very funny relationship which, we don't really know each other. But you were quite close to [https://en.wikipedia.org/wiki/Michael_Atiyah Michael Atiyah] at various points. And I was— | ||
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'''Eric Weinstein:''' In Geometry more generally, and Analysis, I mean, just incredible, and Algebra. I mean, he wrote a book on on Commutative Algebra. Now he had a partner for much of his career, Isadore Singer, who I was quite close to for a period of time. And Is was, again, another one of these figures that if I'd never met one, I wouldn't know that the human mind was capable of that level of repeated insight. And they came up with something called the Atiyah-Singer Index Theorem, which governs worlds in which there are no time dimensions, but only space dimensions, or no space dimensions and only time dimensions, but there's no— | '''Eric Weinstein:''' In Geometry more generally, and Analysis, I mean, just incredible, and Algebra. I mean, he wrote a book on on Commutative Algebra. Now he had a partner for much of his career, [https://en.wikipedia.org/wiki/Isadore_Singer Isadore Singer], who I was quite close to for a period of time. And Is was, again, another one of these figures that if I'd never met one, I wouldn't know that the human mind was capable of that level of repeated insight. And they came up with something called the [https://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem 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— | ||
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'''Sir Roger Penrose:''' Well I can say if I've used the theorem. In at least two different contexts, yes, maybe more. So, I mean, I'm not an expert in that area at all. And it was mainly when I was trying to solve a particular problem... I don't know how much detail you want to go into these things. But it had to do with how to make Twistor Theory work in curved spaces. But I ran up into a question, which had to do, it has to do with Complex Geometry. | '''Sir Roger Penrose:''' Well I can say if I've used the theorem. In at least two different contexts, yes, maybe more. So, I mean, I'm not an expert in that area at all. And it was mainly when I was trying to solve a particular problem... I don't know how much detail you want to go into these things. But it had to do with how to make Twistor Theory work in curved spaces. But I ran up into a question, which had to do, it has to do with [https://en.wikipedia.org/wiki/Complex_geometry Complex Geometry]. | ||
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'''Eric Weinstein:''' Well if you, I know you're hot on the trail of this, but just to leaven something in, Roman Jackiw at MIT once beautifully said, and I don't think he wrote it down, he said, "We didn't understand the partnership that was possible between Mathematics and Physics, because we the physicists used to talk to the analysts." And he said, "The analysts either told us things that were absolutely trivial and irrelevant, or things that we already understood." He said, "When we talked to the geometers, we started to learn new things that we'd never considered." | '''Eric Weinstein:''' Well if you, I know you're hot on the trail of this, but just to leaven something in, [https://en.wikipedia.org/wiki/Roman_Jackiw Roman Jackiw] at MIT once beautifully said, and I don't think he wrote it down, he said, "We didn't understand the partnership that was possible between Mathematics and Physics, because we the physicists used to talk to the analysts." And he said, "The analysts either told us things that were absolutely trivial and irrelevant, or things that we already understood." He said, "When we talked to the geometers, we started to learn new things that we'd never considered." | ||
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'''Eric Weinstein:''' It was just called the Schwarzschild singularity? | '''Eric Weinstein:''' It was just called the [https://en.wikipedia.org/wiki/Schwarzschild_metric Schwarzschild singularity]? | ||
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'''Eric Weinstein:''' From Princeton, Norman Steenrod. | '''Eric Weinstein:''' From Princeton, [https://en.wikipedia.org/wiki/Norman_Steenrod Norman Steenrod]. | ||
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To say this in a simple way, suppose I happen to see a configuration of stars that happened to be on a circle. Suppose they were concyclic. And then this astronaut passing by me would also see these in a circle. Even though the transformation would not be a rotation of the spheres, the sky would be squashed up more on one end and stretched out at the other end. But the thing about that transformation, it's something which I knew about from my Complex Analysis days. Do you think of the, what's called the Riemann sphere? This is the plane of points, you see it's the complex plane, or the vessel plane: the points represent the complex numbers. So zero is in the middle if you like, and then you've got one, and then you've got minus one, and i and minus i, they're all on a circle, and you go out and infinity is way out to infinity. But the Riemann sphere folds all this up into a sphere. So infinity is now a point. | 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 [https://en.wikipedia.org/wiki/Riemann_sphere Riemann sphere]? This is the plane of points, you see it's the complex plane, or the vessel plane: the points represent the complex numbers. So zero is in the middle if you like, and then you've got one, and then you've got minus one, and i and minus i, they're all on a circle, and you go out and infinity is way out to infinity. But the Riemann sphere folds all this up into a sphere. So infinity is now a point. | ||
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'''Eric Weinstein:''' But, if I understand correctly, and maybe I don't, we have another mutual acquaintance, or friend, Raul Bott, and he showed us that the world seems to repeat every eight dimensions in a certain way. But during the first cycle of what you might call Bott Periodicity, from zero to seven, or one to eight, depending on how you like to count, you get these things called low-dimensional coincidences. And so, that they don't recur because of your point earlier about spinors, that spinors grow exponentially, whereas vectors grow linearly. And, but during the first period, where these things are of comparable strength, you get all of these objects where, depending upon, you define in two different contexts you turn out to be the same object. Are you making use of that here? | '''Eric Weinstein:''' But, if I understand correctly, and maybe I don't, we have another mutual acquaintance, or friend, [https://en.wikipedia.org/wiki/Raoul_Bott 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 [https://en.wikipedia.org/wiki/Bott_periodicity_theorem Bott Periodicity], from zero to seven, or one to eight, depending on how you like to count, you get these things called low-dimensional coincidences. And so, that they don't recur because of your point earlier about spinors, that spinors grow exponentially, whereas vectors grow linearly. And, but during the first period, where these things are of comparable strength, you get all of these objects where, depending upon, you define in two different contexts you turn out to be the same object. Are you making use of that here? | ||
01:13:05<br> | 01:13:05<br> | ||
'''Sir Roger Penrose:''' It is that, it's the, well the Lorentz | '''Sir Roger Penrose:''' It is that, it's the, well the [https://en.wikipedia.org/wiki/Lorentz_group Lorentz group]— | ||
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'''Eric Weinstein:''' What you're really talking about is a very important fork in the road for Physics: Do you wed yourself to the world that we're actually given? And you know, Mach was famous for having said this phrase, "The world is given only once." And so we happen to know that there does exist a world that appears to be well modeled by three spatial and one temporal dimension. And then the key question is, do you wish to have a more general theory, which works in all dimensions, or which works for all different divisions between how many spatial and how many temporal dimensions, and what I see you as having done, which I think is incredibly noble, brave, and scientifically valid, is to work with Mathematics that are really particularizing themselves to the world we're given rather than sort of keeping some kind of, I mean, like you're getting married to the world we live in, in a way that other people are just dating it and wishing to keep their options open. | '''Eric Weinstein:''' What you're really talking about is a very important fork in the road for Physics: Do you wed yourself to the world that we're actually given? And you know, [https://en.wikipedia.org/wiki/Ernst_Mach 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. | ||
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'''Eric Weinstein:''' So, very early in this new stagnation post the Standard Model, people like Glashow and Georgi, and Pati and | '''Eric Weinstein:''' So, very early in this new stagnation post the Standard Model, people like [https://en.wikipedia.org/wiki/Sheldon_Lee_Glashow Glashow] and [https://en.wikipedia.org/wiki/Howard_Georgi Georgi], and [https://en.wikipedia.org/wiki/Jogesh_Pati Pati] and [https://en.wikipedia.org/wiki/Abdus_Salam Salam], put forward these unifying symmetries that remain very odd, because they're so attractive and powerful, the prettiest of them being something called Spin-10, which physicists persist in calling SO(10) for reasons that escape me. | ||
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'''Sir Roger Penrose:''' I was at the time at the University of Texas for a year, and this Alfred Schild had put a lot of people together who were General Relativity experts, hoping that something would come out of it, I guess. And I had an office, next to Engelbert Schücking, whom I learned a lot from. And on the other side, I had an office, that was Roy Kerr's office, and Ray Sachs was a little way down. And, I have to backtrack, because the question is, where did Twistor Theory come from? | '''Sir Roger Penrose:''' I was at the time at the University of Texas for a year, and this Alfred Schild had put a lot of people together who were General Relativity experts, hoping that something would come out of it, I guess. And I had an office, next to [https://en.wikipedia.org/wiki/Engelbert_Sch%C3%BCcking Engelbert Schücking], whom I learned a lot from. And on the other side, I had an office, that was [https://en.wikipedia.org/wiki/Roy_Kerr Roy Kerr's] office, and [https://en.wikipedia.org/wiki/Rainer_K._Sachs Ray Sachs] was a little way down. And, I have to backtrack, because the question is, where did Twistor Theory come from? | ||
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And so I had a nice, silent drive coming back, and I started to think about these constructions that Ivor | And so I had a nice, silent drive coming back, and I started to think about these constructions that [https://en.wikipedia.org/wiki/Ivor_Robinson_(physicist) 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. | ||
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'''Eric Weinstein:''' And this is, look, I want to tie this into a bigger thread, which I think is fascinating. I am not a devotee of String Theory, nor am I of Loop Quantum Gravity. I think that most of what has been said about Supersymmetry has been overbearing and wrong. | '''Eric Weinstein:''' And this is, look, I want to tie this into a bigger thread, which I think is fascinating. I am not a devotee of [https://en.wikipedia.org/wiki/String_theory String Theory], nor am I of [https://en.wikipedia.org/wiki/Loop_quantum_gravity Loop Quantum Gravity]. I think that most of what has been said about Supersymmetry has been overbearing and wrong. | ||
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So we had a first revolution around the mid 1970s with what's called the Wu-Yang dictionary, where a particular geometer, who becomes the most successful hedge fund manager in human history meets arguably the most accomplished theoretical physicist, if it's not Weinberg it might be Yang in terms of what has been proven of his contributions. They have an unbelievable interaction which shows that the Classical Theory underneath Particle Physics is as or more geometric than the theory of Einstein using Steenrod's fiber bundles and Ehresmann's connections, or vector potentials or what have you. Then you have a second revolution, again involving—so that was the first one that Is Singer takes from Stony Brook to Oxford—and you have another one, which is the geometric quantization revolution with your colleague Nick Woodhouse writing the bible there, in which Heisenberg's Uncertainty Relations strangely come out of curvature rather than just being some sort of weird— | So we had a first revolution around the mid 1970s with what's called the Wu-Yang dictionary, where a [https://en.wikipedia.org/wiki/Jim_Simons_(mathematician) 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 [https://en.wikipedia.org/wiki/Yang_Chen-Ning 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 [https://en.wikipedia.org/wiki/Charles_Ehresmann Ehresmann's] connections, or vector potentials or what have you. Then you have a second revolution, again involving—so that was the first one that Is Singer takes from Stony Brook to Oxford—and you have another one, which is the [https://en.wikipedia.org/wiki/Geometric_quantization geometric quantization] revolution with your colleague [https://en.wikipedia.org/wiki/Nick_Woodhouse 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— | ||
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You guys figure out that this weird grab bag that was called Quantum Field Theory, which is this thing above Quantum Mechanics that is needed for if you're going to have particles that change—regimes in which the number of particles changes like something emits a photon, you need Quantum Field Theory, you can't do it in Quantum Mechanics. So that world was a grab bag that made absolutely no effing sense pedagogically to anybody coming from outside of the discipline. And what they taught us, and this is coming from the 1980s on, is that Quantum Field Theory would have been discovered by topologists and geometers, even if the physical world had never used it, because it was actually a naturally occurring augmentation of what's called Bordism Theory, which is an enhancement of what you previously referred to as | You guys figure out that this weird grab bag that was called Quantum Field Theory, which is this thing above Quantum Mechanics that is needed for if you're going to have particles that change—regimes in which the number of particles changes like something emits a photon, you need Quantum Field Theory, you can't do it in Quantum Mechanics. So that world was a grab bag that made absolutely no effing sense pedagogically to anybody coming from outside of the discipline. And what they taught us, and this is coming from the 1980s on, is that Quantum Field Theory would have been discovered by topologists and geometers, even if the physical world had never used it, because it was actually a naturally occurring augmentation of what's called Bordism Theory, which is an enhancement of what you previously referred to as Cohomology. | ||
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'''Sir Roger Penrose:''' Yes. Well it's very interesting, partly from a personal point of view, because when I first heard about it, and a lot of it was on Conformal Supersymmetry, and I could see there was a lot of connection with Twistor Theory. The only thing I didn't like was you were led to these algebras which didn't commute and, well, the square of something was zero or something, whatever. I mean, they weren't the kind of Algebra that you needed in Twistor Theory, you needed Complex Analysis. But anyway, I visited Zumino at one point, and I was most intrigued because I could— | '''Sir Roger Penrose:''' Yes. Well it's very interesting, partly from a personal point of view, because when I first heard about it, and a lot of it was on Conformal Supersymmetry, and I could see there was a lot of connection with Twistor Theory. The only thing I didn't like was you were led to these algebras which didn't commute and, well, the square of something was zero or something, whatever. I mean, they weren't the kind of Algebra that you needed in Twistor Theory, you needed Complex Analysis. But anyway, I visited [https://en.wikipedia.org/wiki/Bruno_Zumino Zumino] at one point, and I was most intrigued because I could— | ||
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'''Sir Roger Penrose:''' You're thinking of the Lagrangian, yes. | '''Sir Roger Penrose:''' You're thinking of the [https://en.wikipedia.org/wiki/Lagrangian Lagrangian], yes. | ||
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'''Eric Weinstein:''' Well, that's, well, but the Oxford system, I mean, I don't have to make it so peculiar to Oxford, but you know, even if I think about, like a Nigel Hitchin, or Mason, I guess, has been in that system. | '''Eric Weinstein:''' Well, that's, well, but the Oxford system, I mean, I don't have to make it so peculiar to Oxford, but you know, even if I think about, like a [https://en.wikipedia.org/wiki/Nigel_Hitchin Nigel Hitchin], or Mason, I guess, has been in that system. | ||
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'''Eric Weinstein:''' Let me just... One final thing. Have you been to the courtyard of the Simons Center for Geometry and Physics at Stony Brook, which is tiled with Penrose tiles? | '''Eric Weinstein:''' Let me just... One final thing. Have you been to the courtyard of the [https://en.wikipedia.org/wiki/Simons_Center_for_Geometry_and_Physics Simons Center for Geometry and Physics] at Stony Brook, which is tiled with Penrose tiles? | ||
02:09:36<br> | 02:09:36<br> | ||
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'''Eric Weinstein:''' All right. You've been through The Portal with Sir Roger Penrose, hope you've enjoyed it. Please subscribe to us wherever you listen to podcasts. And if you are the sort of person who views podcasts, navigate over to our YouTube channel. Make sure that you subscribe and click the bell so you'll be informed the next time our next episode drops. Be well. | '''Eric Weinstein:''' All right. You've been through The Portal with Sir Roger Penrose, hope you've enjoyed it. Please subscribe to us wherever you listen to podcasts. And if you are the sort of person who views podcasts, navigate over to our YouTube channel. Make sure that you subscribe and click the bell so you'll be informed the next time our next episode drops. Be well. | ||
== Resources == | == Resources == | ||
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[[File:Ascending and Descending.jpg|thumb|Ascending and Descending, by M. C. Escher. Lithograph, 1960.]] | [[File:Ascending and Descending.jpg|thumb|Ascending and Descending, by M. C. Escher. Lithograph, 1960.]] | ||
</div> | </div> | ||
=== Penrose's Writing=== | |||
Penrose's own books on Spinors and Twistor theory in the context of relativity, electromagnetism, and gravity: | |||
{{BookListing | |||
| cover = Penrose Spinors and Space-Time cover.jpg | |||
| link = Spinors and Space-Time (Book) | |||
| title = | |||
| desc = Spinors and Space-Time by Roger Penrose and Wolfgang Rindler. | |||
}} | |||
</div> | |||
Penrose also has an article explaining the mathematical meaning of his tribar via the standard machinery of cohomology [https://www.iri.upc.edu/people/ros/StructuralTopology/ST17/st17-05-a2-ocr.pdf here]. A later review of his is located [https://personal.math.ubc.ca/~liam/Courses/2022/Math527/tribar.pdf here] and another by Tony Philips with more calculations [https://www.ams.org/publicoutreach/feature-column/fc-2014-10 here] | |||
[[File:Penrosetribar.png|thumb|The tribar shown in pieces, embedded into three open sets. The numbered and circled subregions contain duplicate overlapping points and the rules for translating into the other open sets.]] | |||
== | === Cohomology === | ||
Cohomology of a smooth manifold can be computed by solving certain differential equations, or by combinatorially approximating the manifold with a cover as shown with the tribar. Further, it plays a necessary role in Penrose's Twistor theory. Both mathematical approaches are demonstrated in the book by Bott and Tu: | |||
{{BookListing | |||
| cover = Bott and Tu Differential Forms in Algebraic Topology.jpg | |||
| link = Differential Forms in Algebraic Topology (Book) | |||
| title = | |||
| desc = Differential Forms in Algebraic Topology by Raoul Bott and Loring Tu. | |||
}} | |||
=== Spinors === | |||
Spinors have two main instantiations: the infinitesimal quantity usually in finite dimensions as the value of a vector field at a point, or as the vector field taken over a finite region of space(time). References will be given after the brief explanations. | |||
</div> | </div> | ||
< | ; Infinitesimally | ||
For the infinitesimal quantity in finite dimensions, spinors are constructed via an algebraic means known as representation theory. The key fact that supports the construction is that the special orthogonal Lie groups of rotations <math> SO(n,\mathbb{R}) </math> acting on linear n-dimensional space admit double coverings <math> Spin(n) </math> such that two elements of the spin group correspond to a single rotation. Five algebraic structures comprise the story here: an n-dimensional real vector space, the n-choose-2 dimensional rotation group acting on it, the spin group, another vector space acted upon (the representation) by the spin group to be constructed, and the Clifford algebra of the first real vector space. To distinguish between the first vector space and the second, the elements of the former are simply referred to as vectors and the latter as spinors. This may be confusing because mathematically both are vector spaces whose elements are vectors, however physically the vectors of the first space have a more basic meaning as directions in physical coordinate space. | |||
</div> | </div> | ||
:;1) | |||
== | ::The Clifford algebra is constructed directly from the first vector space, by formally/symbolically multiplying vectors and simplifying these expressions according to the rule: <math> v\cdot v = -q(v)1 </math> where q is the length/quadratic form of the vector that the rotations must preserve, or utilizing the polarization identities of quadratic forms: <math> 2q(v,w)=q(v+w)-q(v)-q(w)\rightarrow v\cdot w + w\cdot v = -2q(v,w) </math>. The two-input bilinear form <math> q(\cdot ,\cdot ) </math> is (often) concretely the dot product, and the original quadratic form is recovered by plugging in the same vector twice <math> q(v,v) </math>. The intermediate mathematical structure which keeps track of the formal products of the vectors is known as the tensor algebra <math> T(V)</math> and notably does not depend on <math> q </math>. It helps to interpret the meaning of the unit <math> 1 </math> in the relation as another formally adjoined symbol. Then, the resulting Clifford algebra with the given modified multiplication rule/relation depending on <math> q </math> is denoted by <math> \mathcal{Cl}(V,q)=T(V)/(v\cdot v-q(v)1) </math> as the quotient of <math> T(V) </math> by the subspace of expressions which we want to evaluate to 0. This curtails the dimension of the Clifford algebra to <math> 2^n </math> from infinite dimensions. | ||
< | |||
</div> | </div> | ||
:;2) | |||
< | ::The Spin group is then found within the Clifford algebra, and because the fiber of the double cover map <math> Spin(n)\rightarrow SO(n, \mathbb{R}) </math> is discrete, it is of the same dimension n-choose-2. This will not be constructed here, but only the following operations which distribute over sums in the Clifford algebra are needed to get the Spin group and its homomorphism to the rotation group: | ||
::;a) | |||
:::an involution <math> \alpha, \alpha^2=id </math> induced by negating the embedded vectors of the Clifford algebra: <math> \alpha (x \cdot y\cdot z) = (-x)\cdot (-y)\cdot (-z)=(-1)^3x\cdot y\cdot z </math> | |||
::;b) | |||
:::the transpose <math> (-)^t </math> which reverses the order of any expression, e.g. <math> (x\cdot y \cdot z)^t = z\cdot y\cdot x </math> | |||
::;c) | |||
:::the adjoint action, conjugation, by an invertible element <math> \phi </math> of the Clifford algebra: <math> Ad_{\phi}(x)=\phi\cdot x\cdot \phi^{-1} </math> | |||
:;3) | |||
::Spinors require more algebra to construct in general, such as understanding the representations of Clifford algebras as algebras of matrices. In the simplest case, one can choose an orthonormal basis of <math>V: \{e_1,\cdots,e_n\} </math> and correspond these vectors to n <math> 2^k\times 2^k </math> matrices with <math> k=\lfloor n/2\rfloor </math> such that they obey the same relations as in the Clifford algebra: <math> \gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}= -2\eta^{\mu\nu}\mathbb{I}_{k\times k} </math> where <math> \mathbb{I}_{k\times k} </math> is the <math> k\times k </math> identity and <math> \eta^{\mu\nu} </math> is the matrix of dot products of the orthonormal basis. The diagonal of <math> \eta^{\mu\nu} </math> can contain negative elements as in the case of the Minkowski norm in 4d spacetime, as opposed to the Euclidean dot product where it also has the form of the identity. Then the spinors are complex vectors in <math> \Delta=\mathbb{C}^{2^k} </math>, however an important counterexample where spinors don't have a purely complex structure is the Majorana representation which is conjectured to be values of neutrino wavefunctions. | |||
</div> | </div> | ||
:;Summary | |||
::The following diagram summarizes the relationship with the structures so far: | |||
[[File: | [[File:Spinor_construction.png|frameless|center]] | ||
</div> | </div> | ||
::The hook arrows denote the constructed embeddings, the double arrow gives the spin double cover of the rotations, and the downwards vertical arrows are the representations of the two groups acting on their respective vector spaces. The lack of an arrow between the two vector spaces indicates no direct relationship between vectors and spinors, only between their groups. There is however a bilinear relationship, meaning a map which is quadratic in the components of spinors to obtain components of vectors, justifying not only that spin transformations are square roots of rotations but spinors as square roots of vectors. When n is even, the spinor space will decompose into two vector spaces: <math> \Delta = S^+ \oplus S^- </math>. In a chosen spin-basis these bilinear maps will come from evaluating products of spinors from the respective subspaces under the gamma matrices as quadratic forms. | |||
< | |||
</div> | </div> | ||
::In the cases of Euclidean and indefinite signatures of quadratic forms, a fixed orthonormal of <math> V </math> can be identified with the identity rotation such that all other bases/frame are related to it by rotations and thus identified with those rotations. Similarly for the spinors, there are "spin frames" which when choosing one to correspond to the identity, biject with the whole spin group. This theory of group representations on vector spaces and spinors in particular were first realized by mathematicians, in particular [https://www.google.com/books/edition/The_Theory_of_Spinors/f-_DAgAAQBAJ?hl=en&gbpv=1&printsec=frontcover Cartan] for spinors, in the study of abstract symmetries and their realizations on geometric objects. However, the next step was taken by physicists. | |||
< | |||
https:// | |||
</div> | </div> | ||
;Finitely | |||
Spinor-valued functions or sections of spinor bundles are the primary objects of particle physics, however they can be realized in non-quantum ways as a model for the electromagnetic field per Penrose's books. When Atiyah stated that the geometrical significance of spinors is not fully understood, it is at this level rather than the well-understood representation algebra level. Analyzing the analogue of the vectors in this scenario, they represent tangents through curves at a particular point in space/on a manifold. The quadratic form then becomes a field of its own, operating on the tangent spaces of each point of the manifold independently but in a smoothly varying way. Arc length of curves can be calculated by integrating the norm of tangent vectors to the curve along it. In general relativity, this is the metric that is a dynamical variable as the solution to Einstein's equations. In order to follow the curvature of the manifold, vector fields are only locally vector valued functions, requiring transitions between regions as in the tribar example previously. Similarly, because the spin and orthogonal groups are so strongly coupled, we can apply the same transitions to spinor-valued functions. The physical reason to do this is that coordinate changes which affect vectors, thus also affect spinor fields in the same way. | |||
[[File:Penroseblackhole.jpg|thumb|Because the Minkowski inner product can be 0, the tangent vectors for which it is form a cone which Penrose depicts as smoothly varying with spacetime]] | |||
</div> | </div> | ||
The metric directly gives a way to differentiate vector fields, or finitely comparing the values at different points via parallel translation along geodesics (curves with minimal length given an initial point and velocity). Using this, a derivative operator can be given for spinor fields. It is usually written in coordinates with the gamma matrices: | |||
< | <math> Ds(x)=\sum_{\mu=1}^n\gamma_{\mu}\nabla_{e_{\mu}}s(x) </math> where the <math> \nabla_{e_{\mu}} </math> are the metric-given derivatives in the direction of an element of an orthonormal basis vector at x. Their difference from the coordinate partial derivatives helps to quantify the curvature. These orthonormal bases also vary with x, making a field of frames which like before can locally be identified with an <math> O(n,\mathbb{R}) </math>-valued function and globally (if it exists) defines an orientation of the manifold. | ||
</div> | </div> | ||
Dirac first wrote down the operator in flat space with partial derivatives instead of covariant derivatives, trying to find a first-order operator and an equation: | |||
</div> | </div> | ||
<math> (iD-m)s(x)=0 </math> | |||
< | |||
</div> | </div> | ||
whose solutions are also solutions to the second order Klein-Gordon equation | |||
</div> | </div> | ||
<math> (D^2+m^2)s(x)=(\Delta+m^2)s(x)=0 </math> | |||
< | |||
</div> | </div> | ||
But it was Atiyah who actually named the operator, and utilized its geometric significance. On a curved manifold it does not square to the Laplacian, but differs by the scalar curvature: | |||
</div> | </div> | ||
<math> D^2s(x)=\Delta s(x)+\frac{1}{4}R s(x) </math> This is known as the Lichnerowicz formula. | |||
<div | === Spinor References === | ||
[https:// | Clearly some knowledge of linear algebra and Lie groups assists in understanding the construction and meaning of spinors. With our [https://theportal.wiki/wiki/Read list] in mind, other books may more directly approach the topic. Spinors are implicit/given in specific representations in the quantum mechanics and field theory books. | ||
Penrose's books being given, the following give introductions to these topics at various levels: | |||
<div class="flex-container" style="clear: both;"> | |||
{{BookListing | |||
| cover = Garling_Clifford_Algbras.jpg | |||
| link = Clifford Algebras: An Introduction (Book) | |||
| title = | |||
| desc = Use this book to learn about Clifford algebras and spinors directly, it covers the necessary prerequisite linear algebra and group theory but only briefly touches on the relation to curvature. | |||
}} | |||
{{BookListing | |||
| cover = Fulton-Harris Representation Theory cover.jpg | |||
| link = Representation Theory (Book) | |||
| title = | |||
| desc = If following our main list [https://theportal.wiki/wiki/Read here], you will encounter Clifford algebras and spin representations here. | |||
}} | |||
{{BookListing | |||
| cover = Woit Quantum Theory, Groups and Representations.png | |||
| link = Quantum Theory, Groups and Representations (Book) | |||
| title = | |||
| desc = Less general discussion of spin representations, but with focus on the low dimensional examples in quantum physics. | |||
}} | |||
{{BookListing | |||
| cover = Lawson Spin Geometry cover.jpg | |||
| link = Spin Geometry (Book) | |||
| title = | |||
| desc = Immediately introduces Clifford algebras and spin representations, demanding strong linear algebra. The remainder of the book extensively introduces the theory of the Dirac operator, Atiyah-Singer Index theorem, and some assorted applications in geometry. | |||
}} | |||
</div> | </div> | ||
<div data-type=" | == Notes == | ||
<div data-type="note" data-timestamp="00:14:00"> | |||
There were three versions. The third version is in The Road to Reality. He thinks the second version is probably the best. | |||
</div> | </div> | ||
<div data-type="note" data-timestamp="00:38:00"> | |||
<div data-type="note" data-timestamp=" | MC ESCHER - Ascending and Descending (The Penrose Stairs) | ||
</div> | </div> | ||
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