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?
42 years, a long time. So we're rehearsing all sorts of things that have come up during that period.
And I suppose my thought goes immediately to the time.
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.
Well, I do think so. I think you've defeated it more than most people could ever have.
I don't know whether you remember my office where I used to have a clock that went backwards.
Well, that would be that would be a good motif if we wanted to in the background, I think.
But certainly we're thinking about the second law of thermodynamics and the mystery of time,
direction and our consciousness and our awareness of the past.
And so we can only talk about the past. And fortunately, it's great to.
Disadvantage we have.
But I was thinking perhaps he would like to say something about how your first mathematical work in the perhaps in the Cambridge,
I'm sure it started earlier, but at Cambridge in the 1950s, it seemed to me that so much came out of that.
And there are still some puzzles for you then, which are still very much around now.
Well, I started off by doing algebraic geometry in Cambridge as a graduate student at St John's College,
and I think I was misled into thinking that algebraic geometry was geometrical.
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.
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,
a contemporary of mine at the time, which is rather terrifying because I thought all graduate students were like that.
And it took me a while to learn that there was something particular about Michael.
But I used I developed a notation initially sources to handle Hodge's lectures because he
gave lectures on differential geometry and they had these indices all over the blackboard.
And it was not the easiest his were not the easiest lectures to follow.
And partly stimulated by that. I developed this notation where tenses could be represented with blobs with arms and legs,
and you could stick them together to form contractions and so on.
So it converted geometric algebraic problems to do with tenses into pictures, which I could understand much more easily.
Well, that's a whole motif, which I was going to ask you about later.
In fact, the number of ways in which you develop ways of seeing things on the page and in one's mind,
that's quite different from the usual way of formal notation.
But also, it just strikes me that, I mean, you didn't follow the kind of algebraic geometry of the more abstract kind.
That's that's been so enormous since that period.
That was I mean, there are tremendous things going on with the abstraction of mathematics.
But you you kept to a geometric viewpoint, which must have been very unfashionable, really, in Cambridge of that period.
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.
I mean, there were diagrams because I did the algebra by making these tens of pictures
and drawing lines and having notations for Symmetries Asians and school,
Symmetries Asians and things and how you manipulated these things.
And although it was very algebraic what I was doing, it was done in a very geometrical way.
And I think that's one of the big things that.
Was important in how things developed with me was I developed this general formalism of tenses which went beyond the normal idea,
and you could include things like negative dimensional tenses. And these turned out to have relevance to spin and quantum mechanics.
But one of the things was I was very mystified by spinners,
because they seem to be fractional things where you had a square root of a a vector or something like that.
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.
We made good friends and he was a cosmologist who was very much following the Cambridge line at that time,
which was the steady state model of cosmology as Bondi and Gold and Hoyle were.
All that is being the originators of this idea. And Denis was a strong follower of it,
which I found very interesting and intriguing and philosophically a satisfying picture where the universe sort of was there all the time.
It didn't have a beginning, and the expansion of the universe was compensated by a new material which was created continually.
Which I had problems with later because it was hard to see how you could combine it with the rules of general relativity and.
Given the choice between general relativity and the steady state model, I would go with general relativity.
But. My friendship with Dennis was very important to me because I learned a lot of physics from him.
You see, I was doing pure mathematics as a graduate student, but there were at least three courses of lectures I went to.
There are a lot of pure courses I went to which were important to me.
I remember Phillip Hall's courses shown, shown while they gave a very nice course on topology and things like that.
And but then I also went to other things which were not really evident be anything to do with
what my research project was one of these being a very beautifully done course by Bondy,
Hermann Bondi on cosmology, general relativity and cosmology,
which was done with great flair and and a wonderful course and another course, equally brilliant in a completely different way,
was Dirac's course on quantum mechanics, which was he was all everything is very logical and very beautifully organised.
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.
So the elegance of what he'd done came out in his lectures.
But it was also important to me because for some reason I don't know whether Denis had been talking to him or something.
I wasn't quite sure. But but there was a it was a course on Standard Quantum Mechanics, which was the first term.
Then the next term was to be on quantum field theory.
And in this course he took one week off to talk about two component spinners.
And I had been trying to understand from reading various incomprehensible books about two components, and they made no sense to me at all.
But these I think it was probably two lectures Dirac gave and they were just perfect.
It became completely clear the whole subject, which is a bit ironic because people think of Dirac as.
A four components been a man but he in fact.
Understood not only understood about two component spinners, but he developed his the higher spin versions of his own equation using this formalism.
And it seemed to me it was absolutely the right way to do it. They've already mentioned.
I mean, I'm thinking that when I was at Cambridge, it was very much divided, pure and applied.
And people hardly talk to each other at all. I mean, they in my time, they were in separate departments.
And you had as an undergraduate, you were supposed to choose which you were and you just stuck to it.
There was a real cultural bloc there, but you just ignored that.
I think I ignored it, yes. Well. Well, Dennis, with all the time trying to get me interested in physics,
he I had a conversation earlier before I went to Cambridge about the steady state,
wonderful lectures given by Fred Hoyle, which which there was some issues which I couldn't quite make sense of.
And, and I got talking to Dennis, who was a friend of my brother's, my brother Oliver, who was at Cambridge.
He had a couple. Well, several years before me.
And. So we struck up a friendship with Dennis at that point.
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,
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,
geometry, ideas, so on.
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,
they used to call them stacks in those days. Theory of stacks.
You see, I think stacks still mean something, but at that time is what became called Chief Sheaves, I suppose.
And I was baffled by the whole thing.
And it was only many, many years later when, when Michael Atiyah made all these things clear.
But at the time I realised there were things that would have been very useful to me later on had I paid adequate attention.
Then what did your brother William Hodge, think of your studying, all these different things that he does?
You Well, I mean, I just think to many graduate students now they'd be horrified by the idea of studying or completely
different courses not getting going on publishing the papers at the right number in the right,
maybe in a bit different then.
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.
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.
And the other one was Michael Atiyah And uh, as I say, I thought, you know, they were all like that more or less.
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.
And so he said, Well, you might like to sit in in one of the classes of another student.
So that was I didn't understand a single word of what was going on, but that was Michael.
Michael C and I later became very good friends with him.
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.
This was a course by a logician called Steen, and I went to that, which I also found was very influential.
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.
You see, I think I would have, prior to going to Cambridge, believed, you know,
we're all computers and that's what thinking is, is computation or something.
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.
There were things in mathematics that you couldn't prove.
And then when I went to the Staines course, it made quite clear that although you couldn't prove them using some particular system,
the mere fact that you trusted that system is something you could give give reliable,
derive reliable conclusions from that mere belief in the system enabled you to transcend the system,
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.
So I found that very striking. Did you, even at that time,
have some inkling that there should be some connection with the physical description of the brain and the matter generally?
I think I did, but it wasn't very well formulated, you see, I think.
I probably did as a result of Stephen's course, come to The View because I learned about Turing machines as well.
That was all part of the course of Turing machines and girls there.
And the fact therefore, because of this understanding, that seems to transcend any particular form system,
that there must be something else going on in the brain, which is not of a computational character.
And I probably learned from Dirac's course on quantum mechanics.
There again is a bit of an irony because I remember the first lecture I went to, he had this little piece of chair.
I think he broke the piece talking to or something. He's talking about superpositions, you know, in quantum mechanics.
If you could do one thing or another, then you could have superpositions of the two.
And so you could have a superposition of a piece of chalk over here and over here.
And my mind wandered at that point, you see. And he I remember him saying something about energy or something, but.
But I couldn't understand why this was an explanation of anything.
I thought it must be because my mind had wondered at that point that I'd missed the point.
But it worried me ever since.
And I think I did formulate the idea that there was a big gap in our understanding of the world in quantum mechanics specifically,
and that there probably was some link between that and and what must be going on in our conscious thinking.
But it was pretty vague and it was only very much later when I had my radio talk with
Marvin Minsky and Edmund Friedkin were talking from a very computational point of view,
and I could see well from that perspective, then I see why they're taking that view.
But it seemed to me ridiculous to extrapolate to that degree,
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.
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,
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.
I suppose we should just in case people may not be so familiar with the timescale we're talking about,
because this really only what you're talking about now is the work which came out in the Emperor's New Mind.
I've left ahead and and in fact, you started publishing on this in the mid eighties, and the book was what?
Uh, that was probably in the eighties. Was. Yes, yes.
And you just remind people that the cosmological picture, as you were studying it from then, is Schama.
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.
But I think people regarded cosmology as just philosophy or something.
I mean there was no reason to believe one thing or another and it became well,
it was the microwave background, but that was only later, that was much later.
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.
That's right. Today. I'm afraid I am jumping around here. You're quite right.
The but that's just interesting because you took up subjects which then would have been comparatively low proof.
I mean relativity as the subject was not, uh, it was, I mean, Ammon Bundy put the new modern ideas.
I think Bundy was the big influence. Yes. He gave some radio talks which were extremely good, very clear.
And he, he certainly influenced me a lot. Yeah.
And put the subject well it was a very physical way of talking about things, but it was extremely clear.
I think I learnt a lot also from other colleagues,
Felix Piranha in particular with somebody who are known to a lot of the mathematics of relativity from.
So I think that's where we can, where you were able to put two of these otherwise completely disparate pieces, knowledge together,
which is understanding of the null geometry and the spin of representation being relevant to general relativity.
How that was, that was that's something which came out of the outer geometry and relativity that.
Yes. I have a trying to think of the order in which these things.
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,
particularly cosmology and astrophysics and that kind of thing.
But he also was interested in the foundations, and we used to he used to drive.
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.
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,
because he had this is Mark's principle idea was very strong with him that somehow what
determined the local inertia was the distant stars and galaxies with a sort of marquee,
an idea. And if you were wrote the rotating neutrons rotating bucket,
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.
And we used to have these discussions as we would drive to or from Stratford, and the idea would be,
well, suppose the stars or the galaxies, we got rid of them one by one they will.
What would happen to inertia? And so I tended to take this to an extremes where there's nothing left of the car.
You see within. Would you feel anything? You know, you just the inertia would be, according to this view, fixed by the car itself.
So I'd say, well, look, suppose you just had two electrons, you see.
Then how do they know how they're spinning or one electron, you see, one electron,
does it know which way it's spinning or if you have two or if you have several?
And so I started thinking about individual spin systems where you had no notion of direction.
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?
What are the rules you see? And so I developed this idea of spin networks from basically from that idea.
I hadn't realised that the spin networks were as early as that and therefore related to the negative dimensional tensor.
Yes. The diagram calculus that you'd worked out. That's right. That all that all came from that very, very early.
Yes. And then you see. The connection also between the dimensionality.
So this was the thing that intrigued me very much was how you have with spin.
Spin, you take an electron, a spin half particle, then it's only got two ways it can spin,
you say, but how can you have two ways when it's got the whole sphere of directions?
Well, that's because of quantum mechanics and spin up and spin down and all the other spins combinations of them.
And then you see the array of the complex combinations of two states, which is really a sphere.
And that sphere gives you the directions in space,
whether you have an intimate relationship between the three dimensionality of space and the complex numbers of quantum mechanics.
And so this kind of struck me as something deep in a way.
And then also when I started thinking more about the I can't quite think of the order of this
relativity picture where you now have the light cone and you have the directions along the light cone,
or if you like, the sky celestial sphere and you have the different directions on the different points on the sphere,
and then again it becomes useful to represent those as points on the Riemann sphere, the complex plane together with infinity.
And that sphere is physically a very natural way of thinking about the directions in space.
And this time, yes, absolutely. You go out and dark night and you see what we see the past anyway.
You see a few a little bit of it. But open space, of course, you have a better picture.
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.
Three space in one time. And only then do you get a light cone, which is a complex structure.
So you had Dirac's two spinners, which is completely contrary to most.
Yeah. Most people would think that he was ready.
The full force of the gammas and everything.
And yet that gave you insight from from quantum mechanics into space time and relativity,
which is, again, not the direction one would naturally think of.
No, I just went I tended to go my own way, I think.
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.
And you need my mathematics papers. I wouldn't get very good marks.
And one, in fact, once I got moved down to class because I was very bad at doing math arithmetic.
But there was a teacher. We had a great insightful teacher, Mr. Stinnett.
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.
So he said, All right, we're going to have a test today. It's the whole.
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.
So I would be working away. The next period would be a play period and people would be outside enjoying themselves.
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.
I would get, you know, 98% or something. And it was a huge difference.
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,
but if I had enough information so I could work it out each time, you see.
So I think it was something I always try to work these things out for myself.
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.
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.
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,
in whether it's to spinners or no or their relationship with SL to see do that.
Uh. And this kind of thing. But. And that your direction went into into into relativity.
But really you had a great deal that was coming from quantum mechanics as well as from classical geometry.
And the background is that is that quantum mechanics certainly was was a big influence on them.
I mean, I don't know exactly when I firmly thought, well,
something's got to be done about quantum mechanics, that you have to change the rules at some level.
I think I felt that pretty early, but I couldn't put a date to that.
But nevertheless, the idea that that quantum features.
I'm very fundamentally at the level of small things.
Yes, that was what and what point also is just understanding the importance of now coldness in the light.
Right. And conformal structure. And the metric is secondary.
And the observation on the the moving sphere in 1959 was just such a I just a two page working out of that idea.
But it destroys a whole lot of talk about people being squashed up when they move and relativity was that and that was that
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.
I went to America in 59, I guess it was 1958.
I went to the. A conference on General 15 gravitation and Roma near Paris.
And that was shortly after I'd been thinking about spinners and relativity.
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.
And I think I remember 1855 as the first general relativity conference and 58 must have been.
The second is that I mean, it's very early days. Yes. It wasn't the first. The first one was probably Chapel Hill.
I can't remember. There were there were two and I can't remember which was called the first one.
I see Chapel Hill was one and Paris was the second, but it wasn't the second comparatively small body of people, relatively speaking.
But there were a lot of people I got to know well and I knew them later on.
Ted Newman lots of people. It was a big influence on me, but it was important to me because.
Well, I just. Let me just backtrack a little bit.
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,
where David Finkelstein was giving a talk about the Schwarzschild solution and getting rid of the singularities.
And he said, Well, that sounds interesting. I wasn't working on general relativity, really.
Then I was thinking about these spinners and so on and.
He gave a talk where he showed how you could extend the short term solution to within the
horizon what we now call the horizon that people used to think of this virtual singularity.
And he did it in one part in the future and then showed how to stick together into what was now called the classical.
Uh, extension, and this made a big impression on me,
but it was kind of curious because Finkelstein at that time was his main interest as general relativity,
and mine was in spinners and playing around with things in the small and quantum mechanics and so on.
And we, in a certain sense or combinatorial, I was doing spin that works.
And so I explained to him about spin that works and he from then on went on and
did combinatorial things and I picked up to general relativity after that.
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.
Well it was. It was. You see. It was. The history of it was sort of like this.
I went to the lecture and I was very impressed by how you got rid of this so called structured singularity.
But you still had the singularity in the middle. Yes. So I thought somehow if you pushed it from one place.
But it's still there. So I began to think, is there a general argument to show that singularities have to be?
No. I had no mechanism, no nothing to try and tackle this problem.
The only thing I had which I'd been studying were the spinners.
So I thought, well, let me just see how spinners work to try and describe relativity.
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.
It's been there. And all this stuff was. Was. And no one else has done that.
Felix Peron did things. Is that it was it was Lew Whitten, Edward Flynn's father, who had I didn't know about it.
Felix Ferrante mentioned there was this paper by Lew Whitten where he had actually applied spinners and looked at the inverse.
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,
like the canonical representation into four principal directions and so on.
And all that stuff. But somehow it all fitted together in a much more beautiful way than I had thought.
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.
I'd been interested in it before that. My first interest, curiously enough, my first encounter with general relativity apart from my brother Oliver.
Vaguely describing it to me was a little book by Schrodinger Spacetime structure, which is a really nice little book.
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.
So I learned about that even before I went to Cambridge. But then picking up things from Felix Speroni and.
And Denis and. And so. And then I went to the Roman conference, which was in 58.
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.
He said, well, look, I've got an hour's talk, given that you have half my time.
I thought that was extremely generous of him. So I gave my little talk on the on the spinners.
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.
So it was a rather hurried little talk on showing how you to translate these tens of
quantities into spinners and how beautiful it fit in with the ideas of general relativity.
But that was actually motivated by what became the singularity theorems.
Yes, mid-sixties. And that was before you'd actually.
I mean, this was before we really started on publishing.
Yes. There were certain. Certain. Well, I published the thing on the on the Spinners.
Yes. Which is 1960. But then it was more like I went to Princeton.
I was in two years. Well, a year and a half in Princeton and Syracuse.
And I got influenced by John Wheeler, I think, on the idea that you have.
Well, it may have been a little after that. Yes, I think it was that the. There was a conference in.
In Warsaw, where I that's where I started talking about conformal infinity.
And I think then. And in the mid sixties, early sixties.
It became clear from the observations by Martin Schmidt that there were those were the first observations of quasars.
And I remember Wheeler getting very excited about this and saying, look,
this tells us there are objects which are really down to the scale of their Schwarzschild singularity.
Before, we always used to think, oh, well, this sort of so-called singularity,
this tiny little thing wouldn't have any real relevance to physics whatsoever.
But here it became clear that there was something funny was going on where you really had things which varied that they
must be sufficiently big because they were this energetic and they must be sufficiently small because they varied.
Within weeks or days or weeks or something.
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.
And the name black hole hadn't emerged at that stage.
But Wheeler was very interested in this idea about where the singularities.
We're generic or not. Where are you aware of the I mean, you were Oppenheimer's.
Yes. 39. Yes. Well this was something that we made a big point.
Yes. He was talking all about.
So the various papers that Oppenheimer was involved in, particularly the Oppenheimer Snider paper, which was just before the war,
where you have this collapse of a very artificial material source of dust and very artificial, and it was exactly symmetrical.
And he had this model of it collapsing to a point, but it was regarded by many people as highly artificial.
These idealisation wouldn't apply generally, particularly because the Russians these were malicious and Kalashnikov seemed to have proved
that the singularities were a very special thing and they would not occur generally.
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.
So I started trying to think about this in other ways,
geometrically kind of visualising what it would be like inside a collapsing star and trying to convincing myself
it had to be a non-local argument that you wouldn't be able to prove anything from purely local considerations.
And then there was this idea about. What's called a trap surface, which came about in a rather curious way.
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.
I learned a lot of things about spinners and deal things and so on, which became important later and twisted theory.
And he was talking about something completely different.
Politics, probably. And. And we came to a street and across the street and conversation stopped.
Then. Then we got to the other side. We started talking again, you see.
And then when he went, he went. I went home, you see, went where he was going.
And I remember thinking at the end of this, a feeling of elation. And I couldn't pinpoint it.
And I why am I feeling like this? So I went back to all the things I've been thinking of during the day.
And then I remembered crossing the street. And when I was halfway across the street, a thought occurred to me.
And this was evidently this characterisation of of a collapse, what we call a trap surface, that this characterisation,
which was a global condition, and it tells you that this star has reached a point of no return.
And so when that I realised that I did. I then developed a.
Pretty well the same day, rough, roughed out proof that you had to get singularities.
But that's misleading in the sense that the techniques were things that I had developed a bit earlier.
Partly although never published as an argument going back to the steady state level because I was interested in steady state,
but I also interested in general relativity and I was trying to say,
is it possible that you can have something like steady state consistent with general relativity?
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.
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.
I never published that. But there was an argument in which another thing in the Royal Society,
I had to try and prove something about some topics which I waited all the time that I developed these techniques.
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.
So those ideas in differential geometry and topology which you needed there were developed for the problem of the fifties was yes state,
which was where she was. Plus, I mean as soon as the Big Bang was the the microwave background.
Yeah. Yeah, it was this guy wasn't really out out of the picture. Yes.
But it turned out just the things for black holes which were fantasy in the in the fifties.
Yeah. But I'm very far from fantasy now.
I mean. That's right. Well it was curious how that, how the thing develop because.
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,
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.
And, and Roy, uh, at that point had.
Found the solution known as the curse solution, which is can be interpreted as a rotating black hole.
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.
And and what I did at that stage, they show that you had to get singularities under extremely general circumstances.
Know no symmetry, assumed no particular equations of state.
You didn't have to assume the dust that that Oppenheimer and Snyder had.
You could have quite general material as long as you didn't violate energy, energy, positivity.
So the obvious thing in the late sixties was to go completely into the new realm
of general relativity that was opened up by modern astronomy and cosmology.
And what did you do? You started thinking about elementary particle physics, which was the theory there is going along at the same time.
Yes. Yes, it was. Well, yeah, they were definitely the same time.
But you see, these were things that were nagging at me for a long time. I just couldn't.
I have to give Engelbert chucking a lot of credit here because on my earlier trip to this is
the first trip to the states where I went to first to work with John Wheeler in Princeton.
And then I went to Syracuse. And I shared an office with Engelbert Searching,
and he kept on talking about conformal maps and the importance of conformal
transformations and how Maxwell's equations were variants and for what reason.
I wasn't sure that time. But another thing he stressed was the importance in quantum field theory of the notion of positive frequency.
And these things stuck with me. And.
They were very important in the development of the theory.
Partly the conformal stuff to represent radiation by squashing infinity down, making a conformal boundary to spacetime that that was one of the ideas.
But the idea of the positive frequency was very crucial to twisted it.
I remember I made a. I knew I wanted some kind of geometry, which was complex in some fundamental way,
but it was really trying to describe the world as we know it as well and had to try and bring quantum theory in.
And I made a huge table with all sorts of topics and arrows going between them and things like this.
And, and. But the Engelbert thing about the past the frequency which.
Nobody in in quantum field theory tends to stress that.
At that time, it was not. I think they think of it like physics physicists generally do.
There was a very thorough analysis. Yes. So then it's sort of trivial.
It's just plus or minus. I think it was the combination, yes.
Of the first analysis and the fact that if conformal, things were important.
Further analysis is not appropriate because it's not conform invariant.
Nevertheless, the fact that you're choosing positive frequency as opposed to negative is controlling.
And this idea of extending. So you have your sphere again, you've got your function on the equator, a real valued function.
That's the real numbers on the equator.
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.
Therefore, that's such a beautiful idea. Can that be extended in some global way to the whole of space time?
And this was nagging me, you see. And I wanted something where see if you complex if I you you don't have splitting
into two halves you see this Raymond sphere you complex by the circle you've
got the hemisphere that's the real part split into two halves and take it in a
positive in the negative frequencies or maybe I think it's negative and positive.
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.
But then I remember. Being driven, I think was shortly after the Kennedy assassination.
I was in America in that in in Austin, Texas.
And the families respective families, this was the Rangers and Hotshots had gone down to San Antonio.
I can't remember where it was exactly. And in the car back, I was pushed.
Ashfaq was driving me and he's not very talkative. So it's a lot of silence, you see.
And I began thinking about this thing that I. Robinson had about how you can.
Take a light ray and somehow push it into the complex.
And then you get these funny solutions of the Maxwell equations which are non singular and twisting.
And so I tried to understand what was going on and then I realised that these things about the Clifford parallels.
I can't quite remember the. I realise that the. Yes. That the.
The solution to the Maxwell equations must have the novel directions along these Clifford parallels.
I vaguely knew I knew about the Clifford Parallels already, but the fact that this is what you got.
I I realise this must be what you've got this configuration.
So the twisting of the lines around these two nested Torii.
That configuration, which I sort of known about from the Clifford parallels.
But when I got home I just translated all into two component spinners and it kind of dropped out.
And that was twist the twist the theory. And and then you see you had the two the thing was split into two halves.
Ultimately, you had the the real, the space of the light rays and then the sort of mild,
complex ification into these two halves, the right hand ones and the left hand ones.
And that this was the analogue of the splitting of the Raymond sphere into two halves.
It took a long time before realising how it really was that.
So it was because it needed the community that it's so striking.
I mean, now that this twist of variables are used by physicists where they tend to
call it a half of transform and think of it in a completely in any other way,
really nothing like this geometric characterisation.
And yet if we're going to get away from doing everything in McCloskey space,
we really need to have some picture of what a particle is and Antiparticle is and so on,
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,
which I think is still very open. Well, it's very interesting.
I mean, as you know, we had this group for developing ideas of twisted theory and these meetings every Friday pretty well and discussions,
fairly broad ranging discussions on various topics.
And then you almost single handedly. Develops these ideas of twisted diagrams.
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.
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.
I'm just thinking we I was I'm thinking this. You've always worried about what a wave function really, really is.
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.
But you like things that we can see in a sense.
I mean, I think seeing is is very important.
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.
I mean, this is something very important and I feel you think we we can see your two way function.
Is is that is that what I always worried about people saying, oh, well, quantum mechanics just tell us pictures are no use anymore.
Sort of fancy. Just calculate them. Forget about the pictures.
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,
which seem to be very important to develop the geometrical idea as far as one could.
But there are certain very odd things about quantum mechanics.
Well, I think as I wrote in one of my book that Shadows of the Mind Quantum Mechanics has two kinds of mystery.
I think people tend to confuse them. So the ones I call the Z mysteries, which are the puzzle mysteries?
Which are things which are true of the world and baffling.
But you can understand them. I mean, it's not quite the way we used to think.
The world was like Spain doesn't behave like a little, you know, cricket ball or something.
Spinning around on a well-defined axis is something much more subtle going on.
But it can be understood and it's consistent and it makes sense.
And it makes beautiful sense often. And there there's the X Mysteries.
X ones were the ones which were paradoxes and the Schrödinger's cat.
So you have quantum mechanics tells you without a very difficult experiment,
although not so nice on the cat, you can put it into a superposition of being dead and alive.
And so Schrödinger was basically saying, Well, look, this is what my Schrödinger Schrödinger's equation is telling you.
You could have is a cat which is dead and alive at the same time.
That's nonsense. You don't see cats like that.
So although he never quite put it like that, it seemed to me he was saying, look, there's something missing.
There's something in the theory which is not adequate.
And Einstein felt the same way. And Dirac, surprisingly.
Oh, yes, that's right. In that because that's one reason I was interested to hear it.
But he was very right about it. He got more you could see on on the on the web.
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
he quite clearly says that the theory is not what Jesus says in the Boer Einstein debates.
He says, Well, you know, Bohr is normally thought of to have won this, but I think maybe time will tell.
And so perhaps he had more of the right idea.
Did you get that scepticism from him?
No, no, that wasn't a feature of what he he was very reluctant, I think, to express his inner opinions very hard.
I had a curious experience once I was asked by the philosophy department at Boston University,
you see philosophers like to have a talk given by somebody and then there will be somebody to contradict yourself, as you say.
So they asked me if I'd like to do this, you see. Who was I supposed to contradict?
Well, actually they'd heard about Dirac, commented about how projective geometry had been useful in his thinking, you see.
So they had him and rashly I said, Oh, well, okay, I'll make some comment.
But you can't you can't refute that because that's what I said. I couldn't. Yes, absolutely.
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.
No, no physics, no, no influence on his own thinking or anything.
It was just a talk on projective geometry. So I'm afraid I slightly made.
Well, I think some of the authors were hoping you might reveal some of your inner thinking.
And then I gave a little talk since took a leaf out of his book and gave a little talk on Twisted Theory.
How did you know that? My version of projective geometry in physics.
But that was slightly curious. But I projected geometry had a big influence on me cause.
Oh, actually, that's backtracking a bit, but that really would have seemed an old fashioned subject in the real time.
Something that Victorian. Yes. I mean that people just dropped out of the syllabus by that time.
I think almost just the ball unit here. I just caught it.
You see we I went to when I was at University College London, that's where I did my undergraduate work.
There was in fact, geometry was quite a big part of the syllabus.
You had applied mathematics, you had sort of algebra and mathematics analysis, and then algebra and geometry.
I think it was like that, but the geometry was a significant part of that.
And there was another guy called Rental Wren, who was a very great purist.
He started off, you know, there were just two axioms, you know, and there's a line through any two points.
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?
You see, occasionally you need another axiom a bit later on, but.
But that kind of know everybody else.
But I rather like that because I thought it was very nice to see these kind of very primitive ideas developing into the geometry.
So there was some projective geometry which I learnt there, which was quite important in my own understandings.
Now what was I going to say here? Well, you brought it back in direct context.
Yes. I mean, I was just thinking that it did come from our own experience.
In a way. It was quite unusual. Yes. So it was just I caught the tail end of it.
Well, I think there was there was another gentleman who came in after,
but then it kind of faded away and got almost removed from the syllabus completely.
And then it swung back a bit. Yeah. But. But it was considered to be unfashionable.
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.
So I didn't take to that too well. You I tried to put as much in as I could.
But a lot of it was translated out when I translated my diagrams into into some incomprehensible notation,
which is, I'm afraid, what my thesis ended up as. But the geometry was always important to me,
but it sort of went more into the physics like geometry of quantum mechanics and relativity, general thing, that sort of thing.