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Of people. Well, Roger, it's very nice to talk to you in a strange interview format and someone I've talked to for who?

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42 years, a long time. So we're rehearsing all sorts of things that have come up during that period.

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And I suppose my thought goes immediately to the time.

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I mean, you're everything you've done seems to be like defeating time in one way or another, being defeated by it no more.

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Well, I do think so. I think you've defeated it more than most people could ever have.

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I don't know whether you remember my office where I used to have a clock that went backwards.

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Well, that would be that would be a good motif if we wanted to in the background, I think.

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But certainly we're thinking about the second law of thermodynamics and the mystery of time,

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direction and our consciousness and our awareness of the past.

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And so we can only talk about the past. And fortunately, it's great to.

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Disadvantage we have.

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But I was thinking perhaps he would like to say something about how your first mathematical work in the perhaps in the Cambridge,

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I'm sure it started earlier, but at Cambridge in the 1950s, it seemed to me that so much came out of that.

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And there are still some puzzles for you then, which are still very much around now.

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Well, I started off by doing algebraic geometry in Cambridge as a graduate student at St John's College,

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and I think I was misled into thinking that algebraic geometry was geometrical.

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And I very soon learnt that it was basically algebra where geometry was the thing that I enjoyed and found I could do most easily.

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So one thing I did was to develop a notation which well, Hodge was my supervisor originally, and Michael Atiyah was one of the people,

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a contemporary of mine at the time, which is rather terrifying because I thought all graduate students were like that.

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And it took me a while to learn that there was something particular about Michael.

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But I used I developed a notation initially sources to handle Hodge's lectures because he

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gave lectures on differential geometry and they had these indices all over the blackboard.

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And it was not the easiest his were not the easiest lectures to follow.

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And partly stimulated by that. I developed this notation where tenses could be represented with blobs with arms and legs,

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and you could stick them together to form contractions and so on.

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So it converted geometric algebraic problems to do with tenses into pictures, which I could understand much more easily.

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Well, that's a whole motif, which I was going to ask you about later.

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In fact, the number of ways in which you develop ways of seeing things on the page and in one's mind,

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that's quite different from the usual way of formal notation.

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But also, it just strikes me that, I mean, you didn't follow the kind of algebraic geometry of the more abstract kind.

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That's that's been so enormous since that period.

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That was I mean, there are tremendous things going on with the abstraction of mathematics.

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But you you kept to a geometric viewpoint, which must have been very unfashionable, really, in Cambridge of that period.

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I think I was very unfashionable, although if you look at my thesis, there's not a single diagram in it, but it was all done using.

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I mean, there were diagrams because I did the algebra by making these tens of pictures

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and drawing lines and having notations for Symmetries Asians and school,

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Symmetries Asians and things and how you manipulated these things.

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And although it was very algebraic what I was doing, it was done in a very geometrical way.

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And I think that's one of the big things that.

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Was important in how things developed with me was I developed this general formalism of tenses which went beyond the normal idea,

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and you could include things like negative dimensional tenses. And these turned out to have relevance to spin and quantum mechanics.

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But one of the things was I was very mystified by spinners,

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because they seem to be fractional things where you had a square root of a a vector or something like that.

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And I couldn't understand how you could do that. And Dennis Sharma, who was a great friend of mine when I was in Cambridge quite early on, he sort of.

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We made good friends and he was a cosmologist who was very much following the Cambridge line at that time,

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which was the steady state model of cosmology as Bondi and Gold and Hoyle were.

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All that is being the originators of this idea. And Denis was a strong follower of it,

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which I found very interesting and intriguing and philosophically a satisfying picture where the universe sort of was there all the time.

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It didn't have a beginning, and the expansion of the universe was compensated by a new material which was created continually.

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Which I had problems with later because it was hard to see how you could combine it with the rules of general relativity and.

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Given the choice between general relativity and the steady state model, I would go with general relativity.

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But. My friendship with Dennis was very important to me because I learned a lot of physics from him.

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You see, I was doing pure mathematics as a graduate student, but there were at least three courses of lectures I went to.

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There are a lot of pure courses I went to which were important to me.

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I remember Phillip Hall's courses shown, shown while they gave a very nice course on topology and things like that.

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And but then I also went to other things which were not really evident be anything to do with

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what my research project was one of these being a very beautifully done course by Bondy,

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Hermann Bondi on cosmology, general relativity and cosmology,

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which was done with great flair and and a wonderful course and another course, equally brilliant in a completely different way,

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was Dirac's course on quantum mechanics, which was he was all everything is very logical and very beautifully organised.

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Many of my colleagues said, Oh, well, that's just the same as this book. So I said, Well, I hadn't read his book.

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So the elegance of what he'd done came out in his lectures.

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But it was also important to me because for some reason I don't know whether Denis had been talking to him or something.

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I wasn't quite sure. But but there was a it was a course on Standard Quantum Mechanics, which was the first term.

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Then the next term was to be on quantum field theory.

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And in this course he took one week off to talk about two component spinners.

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And I had been trying to understand from reading various incomprehensible books about two components, and they made no sense to me at all.

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But these I think it was probably two lectures Dirac gave and they were just perfect.

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It became completely clear the whole subject, which is a bit ironic because people think of Dirac as.

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A four components been a man but he in fact.

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Understood not only understood about two component spinners, but he developed his the higher spin versions of his own equation using this formalism.

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And it seemed to me it was absolutely the right way to do it. They've already mentioned.

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I mean, I'm thinking that when I was at Cambridge, it was very much divided, pure and applied.

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And people hardly talk to each other at all. I mean, they in my time, they were in separate departments.

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And you had as an undergraduate, you were supposed to choose which you were and you just stuck to it.

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There was a real cultural bloc there, but you just ignored that.

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I think I ignored it, yes. Well. Well, Dennis, with all the time trying to get me interested in physics,

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he I had a conversation earlier before I went to Cambridge about the steady state,

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wonderful lectures given by Fred Hoyle, which which there was some issues which I couldn't quite make sense of.

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And, and I got talking to Dennis, who was a friend of my brother's, my brother Oliver, who was at Cambridge.

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He had a couple. Well, several years before me.

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And. So we struck up a friendship with Dennis at that point.

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So he was trying to get me to do physics all the time and get me interested in physics and maybe convert my subject to physics, which I never did,

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because there were too many and there was too much in the mathematics that I was very much involved with and interested in tensor systems in general,

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geometry, ideas, so on.

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And a lot of these ideas which I should have learned then you see one of them in particular was about shift cosmology because well,

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they used to call them stacks in those days. Theory of stacks.

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You see, I think stacks still mean something, but at that time is what became called Chief Sheaves, I suppose.

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And I was baffled by the whole thing.

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And it was only many, many years later when, when Michael Atiyah made all these things clear.

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But at the time I realised there were things that would have been very useful to me later on had I paid adequate attention.

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Then what did your brother William Hodge, think of your studying, all these different things that he does?

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You Well, I mean, I just think to many graduate students now they'd be horrified by the idea of studying or completely

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different courses not getting going on publishing the papers at the right number in the right,

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maybe in a bit different then.

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But there was also a strong a slightly strange thing about you see I started off with Hodge and well, there were two other students.

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One of them gave up quite early. Another one was Michael Hoskin, who went through and did his PhD, but then went into history of science.

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And the other one was Michael Atiyah And uh, as I say, I thought, you know, they were all like that more or less.

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And it was quite because Hodge suggested at one point, Well, if he said that I was a little unhappy with the very algebraic problem that he'd sent me.

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And so he said, Well, you might like to sit in in one of the classes of another student.

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So that was I didn't understand a single word of what was going on, but that was Michael.

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Michael C and I later became very good friends with him.

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There was another course I went to when I was at Cambridge at the same sort of time as when I went to Dirac's and bound this course.

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This was a course by a logician called Steen, and I went to that, which I also found was very influential.

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And what happened to me later on, because I learnt about Girls Theorem, I'd vaguely heard about it before and I found it rather disturbing.

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You see, I think I would have, prior to going to Cambridge, believed, you know,

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we're all computers and that's what thinking is, is computation or something.

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And mainly because I couldn't think of anything else and girl theorem I'd vaguely heard of and it was sort of touted to something which was show.

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There were things in mathematics that you couldn't prove.

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And then when I went to the Staines course, it made quite clear that although you couldn't prove them using some particular system,

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the mere fact that you trusted that system is something you could give give reliable,

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derive reliable conclusions from that mere belief in the system enabled you to transcend the system,

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and you could find statements which had to be true on the basis of your trust in the system, even though the you couldn't prove it using the system.

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So I found that very striking. Did you, even at that time,

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have some inkling that there should be some connection with the physical description of the brain and the matter generally?

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I think I did, but it wasn't very well formulated, you see, I think.

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I probably did as a result of Stephen's course, come to The View because I learned about Turing machines as well.

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That was all part of the course of Turing machines and girls there.

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And the fact therefore, because of this understanding, that seems to transcend any particular form system,

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that there must be something else going on in the brain, which is not of a computational character.

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And I probably learned from Dirac's course on quantum mechanics.

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There again is a bit of an irony because I remember the first lecture I went to, he had this little piece of chair.

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I think he broke the piece talking to or something. He's talking about superpositions, you know, in quantum mechanics.

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If you could do one thing or another, then you could have superpositions of the two.

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And so you could have a superposition of a piece of chalk over here and over here.

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And my mind wandered at that point, you see. And he I remember him saying something about energy or something, but.

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But I couldn't understand why this was an explanation of anything.

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I thought it must be because my mind had wondered at that point that I'd missed the point.

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But it worried me ever since.

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And I think I did formulate the idea that there was a big gap in our understanding of the world in quantum mechanics specifically,

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and that there probably was some link between that and and what must be going on in our conscious thinking.

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But it was pretty vague and it was only very much later when I had my radio talk with

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Marvin Minsky and Edmund Friedkin were talking from a very computational point of view,

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and I could see well from that perspective, then I see why they're taking that view.

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But it seemed to me ridiculous to extrapolate to that degree,

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and this was what made me realise that they had something to say on this subject which seemed to be different from what other people have been saying.

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So I had had the idea that in the very remote future I would write some book about trying to get people excited about mathematics and physics,

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but it didn't really have a focus. But then this thing said, Well, I shall try and describe my ideas about what's going on in the mind.

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I suppose we should just in case people may not be so familiar with the timescale we're talking about,

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because this really only what you're talking about now is the work which came out in the Emperor's New Mind.

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I've left ahead and and in fact, you started publishing on this in the mid eighties, and the book was what?

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Uh, that was probably in the eighties. Was. Yes, yes.

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And you just remind people that the cosmological picture, as you were studying it from then, is Schama.

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I mean, it was hardly how anything was known at all. And really it was just the really comparatively local expansion of the neighbouring galaxies.

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But I think people regarded cosmology as just philosophy or something.

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I mean there was no reason to believe one thing or another and it became well,

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it was the microwave background, but that was only later, that was much later.

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So when you were introduced to it, it was very it was a not exactly a clean slate because the Hubble expansion was known, but nothing like the data.

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That's right. Today. I'm afraid I am jumping around here. You're quite right.

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The but that's just interesting because you took up subjects which then would have been comparatively low proof.

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I mean relativity as the subject was not, uh, it was, I mean, Ammon Bundy put the new modern ideas.

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I think Bundy was the big influence. Yes. He gave some radio talks which were extremely good, very clear.

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And he, he certainly influenced me a lot. Yeah.

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And put the subject well it was a very physical way of talking about things, but it was extremely clear.

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I think I learnt a lot also from other colleagues,

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Felix Piranha in particular with somebody who are known to a lot of the mathematics of relativity from.

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So I think that's where we can, where you were able to put two of these otherwise completely disparate pieces, knowledge together,

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which is understanding of the null geometry and the spin of representation being relevant to general relativity.

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How that was, that was that's something which came out of the outer geometry and relativity that.

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Yes. I have a trying to think of the order in which these things.

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I was certainly I got interested in in the physics and you see Denis was very he was he knew everything that was going on in the world of physics,

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particularly cosmology and astrophysics and that kind of thing.

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But he also was interested in the foundations, and we used to he used to drive.

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We sometimes would go to Stratford and go to plays, you see. And he would drive in his fancy car, you see, at great speed.

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And as you went round the corners at this great speed he would say now that's the action of the fixed stars you see,

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because he had this is Mark's principle idea was very strong with him that somehow what

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determined the local inertia was the distant stars and galaxies with a sort of marquee,

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an idea. And if you were wrote the rotating neutrons rotating bucket,

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you see it's the reason it bulges at the edge is because the influence of the stars is sort of pulling it round, you see.

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And we used to have these discussions as we would drive to or from Stratford, and the idea would be,

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well, suppose the stars or the galaxies, we got rid of them one by one they will.

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What would happen to inertia? And so I tended to take this to an extremes where there's nothing left of the car.

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You see within. Would you feel anything? You know, you just the inertia would be, according to this view, fixed by the car itself.

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So I'd say, well, look, suppose you just had two electrons, you see.

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Then how do they know how they're spinning or one electron, you see, one electron,

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does it know which way it's spinning or if you have two or if you have several?

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And so I started thinking about individual spin systems where you had no notion of direction.

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It was only what the total spin was when they came together and you bring another one in, does the spin go up or down?

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What are the rules you see? And so I developed this idea of spin networks from basically from that idea.

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I hadn't realised that the spin networks were as early as that and therefore related to the negative dimensional tensor.

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Yes. The diagram calculus that you'd worked out. That's right. That all that all came from that very, very early.

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Yes. And then you see. The connection also between the dimensionality.

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So this was the thing that intrigued me very much was how you have with spin.

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Spin, you take an electron, a spin half particle, then it's only got two ways it can spin,

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you say, but how can you have two ways when it's got the whole sphere of directions?

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Well, that's because of quantum mechanics and spin up and spin down and all the other spins combinations of them.

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And then you see the array of the complex combinations of two states, which is really a sphere.

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And that sphere gives you the directions in space,

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whether you have an intimate relationship between the three dimensionality of space and the complex numbers of quantum mechanics.

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And so this kind of struck me as something deep in a way.

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And then also when I started thinking more about the I can't quite think of the order of this

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relativity picture where you now have the light cone and you have the directions along the light cone,

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or if you like, the sky celestial sphere and you have the different directions on the different points on the sphere,

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and then again it becomes useful to represent those as points on the Riemann sphere, the complex plane together with infinity.

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And that sphere is physically a very natural way of thinking about the directions in space.

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And this time, yes, absolutely. You go out and dark night and you see what we see the past anyway.

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You see a few a little bit of it. But open space, of course, you have a better picture.

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But somehow the thinking of Earth as a as a as the complex sphere was a very crucial way of nailing down the space time dimensionality.

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Three space in one time. And only then do you get a light cone, which is a complex structure.

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So you had Dirac's two spinners, which is completely contrary to most.

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Yeah. Most people would think that he was ready.

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The full force of the gammas and everything.

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And yet that gave you insight from from quantum mechanics into space time and relativity,

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which is, again, not the direction one would naturally think of.

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No, I just went I tended to go my own way, I think.

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I mean, I was always you see, when I was at school, I remember particularly in Canada, I was in Canada during the war years and I was extremely slow.

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And you need my mathematics papers. I wouldn't get very good marks.

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And one, in fact, once I got moved down to class because I was very bad at doing math arithmetic.

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But there was a teacher. We had a great insightful teacher, Mr. Stinnett.

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I think he was called and he realised that if I was given enough time, I might do rather well in the tests, you see.

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So he said, All right, we're going to have a test today. It's the whole.

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Usually we just this this, this period, and you're supposed to finish this, but I'm going to let you have any as long as you like.

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So I would be working away. The next period would be a play period and people would be outside enjoying themselves.

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I'd be still plugging away. Occasionally I go on into the one after that, still working away at this test, and then I would do very well.

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I would get, you know, 98% or something. And it was a huge difference.

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And I think the thing was that I was not good at remembering things you see in my tables or whatever it happened to be,

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but if I had enough information so I could work it out each time, you see.

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So I think it was something I always try to work these things out for myself.

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Of course that's not much good if you're trying to do a school exam, but later on it kind of served served its purpose.

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So I had to think these through these things on my own terms rather than learning about them from a book or whatever it was.

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So I would guess that if you put forward these, uh, I mean particle physicists of that period wouldn't have been very interested in,

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in whether it's to spinners or no or their relationship with SL to see do that.

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Uh. And this kind of thing. But. And that your direction went into into into relativity.

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But really you had a great deal that was coming from quantum mechanics as well as from classical geometry.

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And the background is that is that quantum mechanics certainly was was a big influence on them.

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I mean, I don't know exactly when I firmly thought, well,

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something's got to be done about quantum mechanics, that you have to change the rules at some level.

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I think I felt that pretty early, but I couldn't put a date to that.

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But nevertheless, the idea that that quantum features.

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I'm very fundamentally at the level of small things.

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Yes, that was what and what point also is just understanding the importance of now coldness in the light.

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Right. And conformal structure. And the metric is secondary.

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And the observation on the the moving sphere in 1959 was just such a I just a two page working out of that idea.

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But it destroys a whole lot of talk about people being squashed up when they move and relativity was that and that was that

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was all well comes in your publications but fairly it's it was before I you see I went to America this was probably in.

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I went to America in 59, I guess it was 1958.

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I went to the. A conference on General 15 gravitation and Roma near Paris.

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And that was shortly after I'd been thinking about spinners and relativity.

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Yeah. Dennis. Dennis, show him again. You see, he was very keen on getting people together who he thought might have something to say to each other.

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And I think I remember 1855 as the first general relativity conference and 58 must have been.

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The second is that I mean, it's very early days. Yes. It wasn't the first. The first one was probably Chapel Hill.

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I can't remember. There were there were two and I can't remember which was called the first one.

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I see Chapel Hill was one and Paris was the second, but it wasn't the second comparatively small body of people, relatively speaking.

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But there were a lot of people I got to know well and I knew them later on.

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Ted Newman lots of people. It was a big influence on me, but it was important to me because.

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Well, I just. Let me just backtrack a little bit.

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It would have been 57 probably. I'm not quite sure of the date, but Denis persuaded me to go down to Kings College, London,

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where David Finkelstein was giving a talk about the Schwarzschild solution and getting rid of the singularities.

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And he said, Well, that sounds interesting. I wasn't working on general relativity, really.

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Then I was thinking about these spinners and so on and.

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He gave a talk where he showed how you could extend the short term solution to within the

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horizon what we now call the horizon that people used to think of this virtual singularity.

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And he did it in one part in the future and then showed how to stick together into what was now called the classical.

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Uh, extension, and this made a big impression on me,

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but it was kind of curious because Finkelstein at that time was his main interest as general relativity,

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and mine was in spinners and playing around with things in the small and quantum mechanics and so on.

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And we, in a certain sense or combinatorial, I was doing spin that works.

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And so I explained to him about spin that works and he from then on went on and

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did combinatorial things and I picked up to general relativity after that.

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So we sort of swapped roles, but I can see how that would attach to all sorts of things it needs now coordinates to do that.

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Well it was. It was. You see. It was. The history of it was sort of like this.

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I went to the lecture and I was very impressed by how you got rid of this so called structured singularity.

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But you still had the singularity in the middle. Yes. So I thought somehow if you pushed it from one place.

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But it's still there. So I began to think, is there a general argument to show that singularities have to be?

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No. I had no mechanism, no nothing to try and tackle this problem.

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The only thing I had which I'd been studying were the spinners.

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So I thought, well, let me just see how spinners work to try and describe relativity.

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So then I did that and I looked at the Var thing and it all kind of came out so beautifully and the var curvature being totally symmetric.

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It's been there. And all this stuff was. Was. And no one else has done that.

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Felix Peron did things. Is that it was it was Lew Whitten, Edward Flynn's father, who had I didn't know about it.

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Felix Ferrante mentioned there was this paper by Lew Whitten where he had actually applied spinners and looked at the inverse.

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There were some things which weren't quite right in the paper and looked at it and corrected that and and did some other things that he had done,

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like the canonical representation into four principal directions and so on.

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And all that stuff. But somehow it all fitted together in a much more beautiful way than I had thought.

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And it was as much that as as David Finkelstein's lecture, I think, which dragged me into into studying general relativity in a serious way.

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I'd been interested in it before that. My first interest, curiously enough, my first encounter with general relativity apart from my brother Oliver.

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Vaguely describing it to me was a little book by Schrodinger Spacetime structure, which is a really nice little book.

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Apart from the last chapter. He goes on to his own funny ideas, but most of it was a very beautiful explanation of the tensor calculus and so on.

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So I learned about that even before I went to Cambridge. But then picking up things from Felix Speroni and.

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And Denis and. And so. And then I went to the Roman conference, which was in 58.

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Yes. That's where today, I think. And Denis, very generously, he was one of the he was one of the principal speakers at this meeting.

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He said, well, look, I've got an hour's talk, given that you have half my time.

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I thought that was extremely generous of him. So I gave my little talk on the on the spinners.

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I think it was an hour and a half a time where there was 40, 40 minutes and I had 20 minutes I can't remember.

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So it was a rather hurried little talk on showing how you to translate these tens of

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quantities into spinners and how beautiful it fit in with the ideas of general relativity.

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But that was actually motivated by what became the singularity theorems.

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Yes, mid-sixties. And that was before you'd actually.

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I mean, this was before we really started on publishing.

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Yes. There were certain. Certain. Well, I published the thing on the on the Spinners.

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Yes. Which is 1960. But then it was more like I went to Princeton.

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I was in two years. Well, a year and a half in Princeton and Syracuse.

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And I got influenced by John Wheeler, I think, on the idea that you have.

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Well, it may have been a little after that. Yes, I think it was that the. There was a conference in.

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In Warsaw, where I that's where I started talking about conformal infinity.

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And I think then. And in the mid sixties, early sixties.

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It became clear from the observations by Martin Schmidt that there were those were the first observations of quasars.

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And I remember Wheeler getting very excited about this and saying, look,

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this tells us there are objects which are really down to the scale of their Schwarzschild singularity.

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Before, we always used to think, oh, well, this sort of so-called singularity,

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this tiny little thing wouldn't have any real relevance to physics whatsoever.

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But here it became clear that there was something funny was going on where you really had things which varied that they

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must be sufficiently big because they were this energetic and they must be sufficiently small because they varied.

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Within weeks or days or weeks or something.

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So they can't be too big and therefore they must be of the sort of size of their Schwarzschild radius was that was what we now call a black hole.

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And the name black hole hadn't emerged at that stage.

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But Wheeler was very interested in this idea about where the singularities.

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We're generic or not. Where are you aware of the I mean, you were Oppenheimer's.

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Yes. 39. Yes. Well this was something that we made a big point.

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Yes. He was talking all about.

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So the various papers that Oppenheimer was involved in, particularly the Oppenheimer Snider paper, which was just before the war,

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where you have this collapse of a very artificial material source of dust and very artificial, and it was exactly symmetrical.

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And he had this model of it collapsing to a point, but it was regarded by many people as highly artificial.

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These idealisation wouldn't apply generally, particularly because the Russians these were malicious and Kalashnikov seemed to have proved

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that the singularities were a very special thing and they would not occur generally.

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Now I'd sort of seen a little bit about their proof, and I couldn't imagine you could really prove something like this the way they were doing it.

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So I started trying to think about this in other ways,

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geometrically kind of visualising what it would be like inside a collapsing star and trying to convincing myself

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it had to be a non-local argument that you wouldn't be able to prove anything from purely local considerations.

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And then there was this idea about. What's called a trap surface, which came about in a rather curious way.

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So when I was talking to Ivor Robinson, I was at that time I was in college in London, and I Will Robinson, who was a friend of mine.

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I learned a lot of things about spinners and deal things and so on, which became important later and twisted theory.

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And he was talking about something completely different.

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Politics, probably. And. And we came to a street and across the street and conversation stopped.

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Then. Then we got to the other side. We started talking again, you see.

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And then when he went, he went. I went home, you see, went where he was going.

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And I remember thinking at the end of this, a feeling of elation. And I couldn't pinpoint it.

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And I why am I feeling like this? So I went back to all the things I've been thinking of during the day.

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And then I remembered crossing the street. And when I was halfway across the street, a thought occurred to me.

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And this was evidently this characterisation of of a collapse, what we call a trap surface, that this characterisation,

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which was a global condition, and it tells you that this star has reached a point of no return.

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And so when that I realised that I did. I then developed a.

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Pretty well the same day, rough, roughed out proof that you had to get singularities.

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But that's misleading in the sense that the techniques were things that I had developed a bit earlier.

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Partly although never published as an argument going back to the steady state level because I was interested in steady state,

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but I also interested in general relativity and I was trying to say,

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is it possible that you can have something like steady state consistent with general relativity?

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If it was in exactly symmetrical case, you could see that big problems with energy, but if it's irregular, maybe you'll get away with it.

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But then I developed an argument with these cones and focusing and so on to realise that that wouldn't help, that you'd still be in trouble.

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I never published that. But there was an argument in which another thing in the Royal Society,

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I had to try and prove something about some topics which I waited all the time that I developed these techniques.

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I thought I was waiting a long time. But I developed the techniques which became just what were needed in the case of the collapsing.

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So those ideas in differential geometry and topology which you needed there were developed for the problem of the fifties was yes state,

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which was where she was. Plus, I mean as soon as the Big Bang was the the microwave background.

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Yeah. Yeah, it was this guy wasn't really out out of the picture. Yes.

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But it turned out just the things for black holes which were fantasy in the in the fifties.

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Yeah. But I'm very far from fantasy now.

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I mean. That's right. Well it was curious how that, how the thing develop because.

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The first I went to a lot of the early we called the Texas Conferences on Relativistic Astrophysics, and I went to the first one,

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which was a lot of the stuff about the the quasars, these, these things that my Martin Schmidt had seen and Wheeler was so excited about and so on.

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And, and Roy, uh, at that point had.

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Found the solution known as the curse solution, which is can be interpreted as a rotating black hole.

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It wasn't totally clear at that stage that you could interpret it that way, but this did become clear and knowledge of these things was important.

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And and what I did at that stage, they show that you had to get singularities under extremely general circumstances.

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Know no symmetry, assumed no particular equations of state.

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You didn't have to assume the dust that that Oppenheimer and Snyder had.

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You could have quite general material as long as you didn't violate energy, energy, positivity.

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So the obvious thing in the late sixties was to go completely into the new realm

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of general relativity that was opened up by modern astronomy and cosmology.

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And what did you do? You started thinking about elementary particle physics, which was the theory there is going along at the same time.

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Yes. Yes, it was. Well, yeah, they were definitely the same time.

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But you see, these were things that were nagging at me for a long time. I just couldn't.

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I have to give Engelbert chucking a lot of credit here because on my earlier trip to this is

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the first trip to the states where I went to first to work with John Wheeler in Princeton.

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And then I went to Syracuse. And I shared an office with Engelbert Searching,

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and he kept on talking about conformal maps and the importance of conformal

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transformations and how Maxwell's equations were variants and for what reason.

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I wasn't sure that time. But another thing he stressed was the importance in quantum field theory of the notion of positive frequency.

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And these things stuck with me. And.

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They were very important in the development of the theory.

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Partly the conformal stuff to represent radiation by squashing infinity down, making a conformal boundary to spacetime that that was one of the ideas.

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But the idea of the positive frequency was very crucial to twisted it.

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I remember I made a. I knew I wanted some kind of geometry, which was complex in some fundamental way,

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but it was really trying to describe the world as we know it as well and had to try and bring quantum theory in.

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And I made a huge table with all sorts of topics and arrows going between them and things like this.

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And, and. But the Engelbert thing about the past the frequency which.

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Nobody in in quantum field theory tends to stress that.

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At that time, it was not. I think they think of it like physics physicists generally do.

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There was a very thorough analysis. Yes. So then it's sort of trivial.

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It's just plus or minus. I think it was the combination, yes.

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Of the first analysis and the fact that if conformal, things were important.

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Further analysis is not appropriate because it's not conform invariant.

383
00:40:02,190 --> 00:40:07,070
Nevertheless, the fact that you're choosing positive frequency as opposed to negative is controlling.

384
00:40:07,950 --> 00:40:15,389
And this idea of extending. So you have your sphere again, you've got your function on the equator, a real valued function.

385
00:40:15,390 --> 00:40:17,190
That's the real numbers on the equator.

386
00:40:17,550 --> 00:40:25,590
And then if you can extend your function into the Alamo quickly into the north or the south, this gives you positive and negative frequency.

387
00:40:25,830 --> 00:40:34,170
Therefore, that's such a beautiful idea. Can that be extended in some global way to the whole of space time?

388
00:40:35,360 --> 00:40:45,849
And this was nagging me, you see. And I wanted something where see if you complex if I you you don't have splitting

389
00:40:45,850 --> 00:40:51,249
into two halves you see this Raymond sphere you complex by the circle you've

390
00:40:51,250 --> 00:40:56,300
got the hemisphere that's the real part split into two halves and take it in a

391
00:40:56,350 --> 00:40:59,400
positive in the negative frequencies or maybe I think it's negative and positive.

392
00:40:59,470 --> 00:41:06,190
Never mind. And so I kept thinking, well, what about Minkowski space where you complex if it doesn't split anything into two halves, you see.

393
00:41:07,180 --> 00:41:13,469
But then I remember. Being driven, I think was shortly after the Kennedy assassination.

394
00:41:13,470 --> 00:41:17,850
I was in America in that in in Austin, Texas.

395
00:41:18,660 --> 00:41:25,810
And the families respective families, this was the Rangers and Hotshots had gone down to San Antonio.

396
00:41:25,920 --> 00:41:30,840
I can't remember where it was exactly. And in the car back, I was pushed.

397
00:41:30,840 --> 00:41:35,070
Ashfaq was driving me and he's not very talkative. So it's a lot of silence, you see.

398
00:41:35,460 --> 00:41:39,090
And I began thinking about this thing that I. Robinson had about how you can.

399
00:41:39,930 --> 00:41:44,400
Take a light ray and somehow push it into the complex.

400
00:41:44,700 --> 00:41:49,770
And then you get these funny solutions of the Maxwell equations which are non singular and twisting.

401
00:41:50,370 --> 00:41:56,310
And so I tried to understand what was going on and then I realised that these things about the Clifford parallels.

402
00:41:57,240 --> 00:42:00,570
I can't quite remember the. I realise that the. Yes. That the.

403
00:42:01,670 --> 00:42:06,770
The solution to the Maxwell equations must have the novel directions along these Clifford parallels.

404
00:42:07,100 --> 00:42:11,360
I vaguely knew I knew about the Clifford Parallels already, but the fact that this is what you got.

405
00:42:12,330 --> 00:42:16,129
I I realise this must be what you've got this configuration.

406
00:42:16,130 --> 00:42:21,770
So the twisting of the lines around these two nested Torii.

407
00:42:22,750 --> 00:42:27,190
That configuration, which I sort of known about from the Clifford parallels.

408
00:42:27,640 --> 00:42:32,709
But when I got home I just translated all into two component spinners and it kind of dropped out.

409
00:42:32,710 --> 00:42:38,920
And that was twist the twist the theory. And and then you see you had the two the thing was split into two halves.

410
00:42:38,920 --> 00:42:43,749
Ultimately, you had the the real, the space of the light rays and then the sort of mild,

411
00:42:43,750 --> 00:42:47,260
complex ification into these two halves, the right hand ones and the left hand ones.

412
00:42:47,890 --> 00:42:52,540
And that this was the analogue of the splitting of the Raymond sphere into two halves.

413
00:42:53,200 --> 00:42:57,010
It took a long time before realising how it really was that.

414
00:42:58,900 --> 00:43:03,129
So it was because it needed the community that it's so striking.

415
00:43:03,130 --> 00:43:07,000
I mean, now that this twist of variables are used by physicists where they tend to

416
00:43:07,000 --> 00:43:11,319
call it a half of transform and think of it in a completely in any other way,

417
00:43:11,320 --> 00:43:14,680
really nothing like this geometric characterisation.

418
00:43:15,070 --> 00:43:18,309
And yet if we're going to get away from doing everything in McCloskey space,

419
00:43:18,310 --> 00:43:22,900
we really need to have some picture of what a particle is and Antiparticle is and so on,

420
00:43:22,900 --> 00:43:30,580
which doesn't depend on I mean, I don't know if you we may be coming onto this really is there are things of that period which you worried about,

421
00:43:30,580 --> 00:43:33,549
which I think is still very open. Well, it's very interesting.

422
00:43:33,550 --> 00:43:44,800
I mean, as you know, we had this group for developing ideas of twisted theory and these meetings every Friday pretty well and discussions,

423
00:43:45,490 --> 00:43:48,880
fairly broad ranging discussions on various topics.

424
00:43:49,240 --> 00:43:55,639
And then you almost single handedly. Develops these ideas of twisted diagrams.

425
00:43:55,640 --> 00:44:05,210
And and and I've always admired you how much you, you know, you felt this was a thing to do and you stuck with it.

426
00:44:05,900 --> 00:44:13,640
Well, they were your diagrams, but I kept them alive until it wasn't time with other people's developing in ways which I had conceived of it.

427
00:44:14,270 --> 00:44:19,580
I'm just thinking we I was I'm thinking this. You've always worried about what a wave function really, really is.

428
00:44:19,620 --> 00:44:25,460
Yes. And a lot of people don't worry about this. They just write down the formalism of quantum mechanics and linear and so forth.

429
00:44:25,940 --> 00:44:29,410
But you like things that we can see in a sense.

430
00:44:29,420 --> 00:44:32,130
I mean, I think seeing is is very important.

431
00:44:32,330 --> 00:44:41,780
All you do, whether it's the notation or they just the business of light or the action of consciousness and seeing the truth of the causal statement.

432
00:44:41,780 --> 00:44:47,569
I mean, this is something very important and I feel you think we we can see your two way function.

433
00:44:47,570 --> 00:44:52,940
Is is that is that what I always worried about people saying, oh, well, quantum mechanics just tell us pictures are no use anymore.

434
00:44:52,940 --> 00:44:56,270
Sort of fancy. Just calculate them. Forget about the pictures.

435
00:44:56,720 --> 00:45:03,230
But I never was happy with that. I always wanted to try and picture anything I could, certainly with spin ideas, with spin and so on,

436
00:45:03,230 --> 00:45:09,980
which seem to be very important to develop the geometrical idea as far as one could.

437
00:45:10,700 --> 00:45:14,270
But there are certain very odd things about quantum mechanics.

438
00:45:15,580 --> 00:45:22,280
Well, I think as I wrote in one of my book that Shadows of the Mind Quantum Mechanics has two kinds of mystery.

439
00:45:22,330 --> 00:45:27,400
I think people tend to confuse them. So the ones I call the Z mysteries, which are the puzzle mysteries?

440
00:45:28,940 --> 00:45:32,060
Which are things which are true of the world and baffling.

441
00:45:32,390 --> 00:45:36,500
But you can understand them. I mean, it's not quite the way we used to think.

442
00:45:36,500 --> 00:45:41,690
The world was like Spain doesn't behave like a little, you know, cricket ball or something.

443
00:45:41,690 --> 00:45:45,380
Spinning around on a well-defined axis is something much more subtle going on.

444
00:45:45,950 --> 00:45:49,310
But it can be understood and it's consistent and it makes sense.

445
00:45:49,820 --> 00:45:54,950
And it makes beautiful sense often. And there there's the X Mysteries.

446
00:45:55,580 --> 00:46:00,230
X ones were the ones which were paradoxes and the Schrödinger's cat.

447
00:46:00,260 --> 00:46:04,669
So you have quantum mechanics tells you without a very difficult experiment,

448
00:46:04,670 --> 00:46:09,740
although not so nice on the cat, you can put it into a superposition of being dead and alive.

449
00:46:10,460 --> 00:46:17,480
And so Schrödinger was basically saying, Well, look, this is what my Schrödinger Schrödinger's equation is telling you.

450
00:46:17,480 --> 00:46:21,380
You could have is a cat which is dead and alive at the same time.

451
00:46:21,710 --> 00:46:24,650
That's nonsense. You don't see cats like that.

452
00:46:25,160 --> 00:46:30,110
So although he never quite put it like that, it seemed to me he was saying, look, there's something missing.

453
00:46:30,590 --> 00:46:34,970
There's something in the theory which is not adequate.

454
00:46:35,900 --> 00:46:40,459
And Einstein felt the same way. And Dirac, surprisingly.

455
00:46:40,460 --> 00:46:45,710
Oh, yes, that's right. In that because that's one reason I was interested to hear it.

456
00:46:45,710 --> 00:46:51,200
But he was very right about it. He got more you could see on on the on the web.

457
00:46:51,200 --> 00:46:59,720
There are some lectures there. And we explicitly says this. And I have trouble finding the original quotes because I know there are some quotes where

458
00:47:00,080 --> 00:47:06,220
he quite clearly says that the theory is not what Jesus says in the Boer Einstein debates.

459
00:47:06,230 --> 00:47:11,570
He says, Well, you know, Bohr is normally thought of to have won this, but I think maybe time will tell.

460
00:47:11,610 --> 00:47:15,680
And so perhaps he had more of the right idea.

461
00:47:15,740 --> 00:47:17,630
Did you get that scepticism from him?

462
00:47:18,140 --> 00:47:28,820
No, no, that wasn't a feature of what he he was very reluctant, I think, to express his inner opinions very hard.

463
00:47:29,240 --> 00:47:35,330
I had a curious experience once I was asked by the philosophy department at Boston University,

464
00:47:35,960 --> 00:47:42,130
you see philosophers like to have a talk given by somebody and then there will be somebody to contradict yourself, as you say.

465
00:47:42,620 --> 00:47:46,510
So they asked me if I'd like to do this, you see. Who was I supposed to contradict?

466
00:47:46,520 --> 00:47:54,020
Well, actually they'd heard about Dirac, commented about how projective geometry had been useful in his thinking, you see.

467
00:47:54,030 --> 00:47:57,409
So they had him and rashly I said, Oh, well, okay, I'll make some comment.

468
00:47:57,410 --> 00:48:01,280
But you can't you can't refute that because that's what I said. I couldn't. Yes, absolutely.

469
00:48:01,280 --> 00:48:10,790
Yes. So he gave this talk. She Dirac gave it to her and it was very elegantly put directly and taught projective geometry, just on projecting.

470
00:48:10,800 --> 00:48:14,900
No, no physics, no, no influence on his own thinking or anything.

471
00:48:15,140 --> 00:48:18,620
It was just a talk on projective geometry. So I'm afraid I slightly made.

472
00:48:18,620 --> 00:48:22,730
Well, I think some of the authors were hoping you might reveal some of your inner thinking.

473
00:48:23,750 --> 00:48:28,280
And then I gave a little talk since took a leaf out of his book and gave a little talk on Twisted Theory.

474
00:48:28,560 --> 00:48:31,670
How did you know that? My version of projective geometry in physics.

475
00:48:32,390 --> 00:48:37,880
But that was slightly curious. But I projected geometry had a big influence on me cause.

476
00:48:38,360 --> 00:48:44,390
Oh, actually, that's backtracking a bit, but that really would have seemed an old fashioned subject in the real time.

477
00:48:44,450 --> 00:48:49,370
Something that Victorian. Yes. I mean that people just dropped out of the syllabus by that time.

478
00:48:49,370 --> 00:48:52,129
I think almost just the ball unit here. I just caught it.

479
00:48:52,130 --> 00:48:57,860
You see we I went to when I was at University College London, that's where I did my undergraduate work.

480
00:48:58,370 --> 00:49:03,230
There was in fact, geometry was quite a big part of the syllabus.

481
00:49:03,530 --> 00:49:12,259
You had applied mathematics, you had sort of algebra and mathematics analysis, and then algebra and geometry.

482
00:49:12,260 --> 00:49:16,640
I think it was like that, but the geometry was a significant part of that.

483
00:49:17,120 --> 00:49:23,870
And there was another guy called Rental Wren, who was a very great purist.

484
00:49:23,870 --> 00:49:28,429
He started off, you know, there were just two axioms, you know, and there's a line through any two points.

485
00:49:28,430 --> 00:49:35,560
And if there's a line through these two points and through these and it meets there that needs to meet here, how much could you prove from that?

486
00:49:35,570 --> 00:49:39,290
You see, occasionally you need another axiom a bit later on, but.

487
00:49:39,770 --> 00:49:41,480
But that kind of know everybody else.

488
00:49:41,900 --> 00:49:50,210
But I rather like that because I thought it was very nice to see these kind of very primitive ideas developing into the geometry.

489
00:49:50,390 --> 00:49:57,220
So there was some projective geometry which I learnt there, which was quite important in my own understandings.

490
00:49:58,820 --> 00:50:03,960
Now what was I going to say here? Well, you brought it back in direct context.

491
00:50:03,970 --> 00:50:07,080
Yes. I mean, I was just thinking that it did come from our own experience.

492
00:50:07,080 --> 00:50:11,210
In a way. It was quite unusual. Yes. So it was just I caught the tail end of it.

493
00:50:11,220 --> 00:50:16,620
Well, I think there was there was another gentleman who came in after,

494
00:50:16,620 --> 00:50:20,010
but then it kind of faded away and got almost removed from the syllabus completely.

495
00:50:20,190 --> 00:50:24,800
And then it swung back a bit. Yeah. But. But it was considered to be unfashionable.

496
00:50:24,810 --> 00:50:32,880
And even when you did what was called algebraic geometry, there was very little geometry in the sense of what you could actually see in it.

497
00:50:33,150 --> 00:50:37,620
So I didn't take to that too well. You I tried to put as much in as I could.

498
00:50:37,980 --> 00:50:44,070
But a lot of it was translated out when I translated my diagrams into into some incomprehensible notation,

499
00:50:44,730 --> 00:50:50,620
which is, I'm afraid, what my thesis ended up as. But the geometry was always important to me,

500
00:50:50,620 --> 00:50:59,230
but it sort of went more into the physics like geometry of quantum mechanics and relativity, general thing, that sort of thing.