Conversations with John Mayberry / Twistor Theory Survey

0:00 I'm sorry, especially when we were in the restaurant, there's particularly this very interesting point about the characterisation of normalism groups, and how it's easily sucked into the concrete view of abstract objects as things that you obtain by abstraction, when you don't... Yes, I mean, this was the point that you were making of the whole point of the maximality definition is to avoid. Besides going trafficking abstract objects. Precisely, I would say, these guys. Yes, in situative, after yes, the old-fashioned epistemological structure. It's kind of odd, you know, it's going one loop further. It's really bizarre. No, but the stuff that, and I take them that point, because you've made it very eloquently on more than one occasion, but the thing we took away from last night was this very instructive about the rigidification of the structure in the case of the automorphism group, i.e. the answer to James' wine, and I am mine as I, which is certainly Michael Popper effectively had, you know, I'll put it. But no, no, that was just... Yeah, I mean, I got, that was particularly, that was particularly important because I hadn't really got clear on the issues until you explain them, if you did not know. This is about a base and a structure, I've given an exact definition. Yes, and also your explanation of what you mean by the term oncology and what he is to say that that deserves structure and precisely I mean, it is extraordinary that somebody by much of a populous is still kind of trapped in this picture

2:30 which really simply describes that there is a natural transition of the real non-diversaries, in the sense that it's not just being in it. Well, I don't think it's as open-and-shutter case. I mean, it's exactly the problem with universals. It's what it is. Yeah, yeah. It's just much more precisely formulated in that matter. The point about non-trivial ultimately is that you can't re-getify that. I mean, to re-getify structure in the sense that the technical sense you explained to me last night has to have a trivial ultimately. No, no, that is a trivial ultimately. Yeah, okay, that's the question. Yeah. It doesn't have, I mean, the complex numbers don't have one unless you've singled out a particular element of your structure to be I. Yeah. And the point is that no element in any structure has any intrinsic properties that relate to that structure. Yeah. That's the, yeah, and that's it. Artificially, manually. Yeah. Yes, the point you made about the misleading way that we're trying to think of I and minus I would probably look at the geometric effect. I mean, everybody has ignored a pair of reels where reels are, in particular, abstract objects in this old-fashioned sense, whereas you, well, you made an analogy, didn't you, with a guy who criticized Lincoln and McClellan, and was told, yes, but I have to choose somebody. Of course, it doesn't have to be. So you always... So when Lincoln said, I have to choose somebody, you didn't, of course, have... Well, you can't have, I say, well, I've got a model of the reals in which that chair is the number pi. Well, it's quite unsuitable. And so I could won't give you an alternative. Well, anyway, anyway. But I have to have an abstract one, the abstract place, or whatever. But, I mean, the whole, it's just a confusion. Yes, yes. And it's just, as I said, in the preface of my book, a failure to understand what the axiomatic method is. Yes. The very use of large numbers of very good mathematicians, and while there is mathematics who should think about it, they still have a habit here. They still have a habit here.

5:00 No, I can see that much more clearly enough than I did before. And the very talk about positions and structures, which of course already introduces a spatial mathematical, Yeah, but there's nothing wrong with using spatial metaphors as long as they're acknowledged as a metaphor. They're a colorful but inaccurate way of speech. And you start, you'll just drive yourself nervous. You start thinking, well, what is a... It's like saying, what's a shape? The greatest solution to that is a shape of a... The shape of a triangle is the set of all similar triangles. Then people say, well, that's not what people mean by shape in the past, blah, blah, blah, it's an abstraction, blah, blah, blah. Yes, that's right. This is an Alexandrian solution to that particular Gordian God. He just lies to it. All this other stuff is just bogus and old-fashioned and out-of-date. As indeed it is. then you've got I suppose you've got a kind of psychological problem on your hand because when you've got a structure that's used all the time it's just odious well pick any particular pick a particular complete ordered feeling and we'll call that the real numbers I would refer to the numbers as real numbers blah blah blah okay you can do that or you can preface all of your propositions In any complete ordered field, if these definitions are laid out, then this equation holds. And that's all that is going on. So the problem of what? The Nassarist problem is a non-problem. Although it is preoccupied with philosophers and mathematics for 30 years. But that's just a measure of how bad the subject is. Well, I certainly recognise that a long time ago, I mean, one only has to look at the difference between the present state of the game in philosophy of physics, where philosophers of physics do spend a great deal of time talking to their colleagues in the physics department and people writing books on quantum gravity have contributions from philosophers of physics. Indeed, often

7:30 co-edited by philosophers and physics, and that stuff which is, you know, really the combination of physics, because it involves tricky conceptual issues, does come on the attention now of, you know, well, there's a philosophy of physics, and they, in terms of our respect for attention of people's physics department, this is certainly a very, very different situation from all the names of philosophy of mathematics, and their philosophy of mathematics will get here. I'm not talking about working mathematicians who effectively contribute to their central fabrication of issues that are foundational. I mean, I can't use a classroom that way. But I mean people playing this, and that's what kind of game. Of course there are classes that are real mathematics. Well, it's curious how difficult it is to say what is absolutely being novelist. That people grab, but they don't have articulation. And part of the problem is, again, a failure to define concepts. And part of the source of that, as you've had very clear, is the fact that lots of the people who are doing this stuff have not had to teach. Which does maximize dozens of examples. Yes, if you haven't teach them all, it's us, yes. They don't have science. They don't consider that they've got a wonderful product, but they don't consider that they're in the market. You know, somebody chops out books like Hale and Wright. They say, here's the theory of natural numbers. How are you going to teach that bunch of horago to a bunch of undergraduates? First of all, they won't think natural numbers need to find it. They'll have this naive idea. Yeah, that cuts you together. Okay. More. So, what do they think is being clarified? They've got to go through the whole bootstruck job about formal theories, what you mean by a sentence, what you mean by a proof, what you mean by a model, and so on, which is deep into mathematics, so it's huge. It's impossible to start there. So you have to start with ideas that they all think are simple. Now that's what Bill is trying to do in Conceptional Advantage and in the other one.

10:00 Well, the sex book of the back. Yeah, but there's an awful lot of them. If I look at those things, I mean I see calling people's attention Notions which most account the Foundations Bill calls in, these notions are quite straightforward. And, I think, he's got this business of what a function is. He thinks of function, the notion of function is quite common for people saying, but I think it's a big error. I can understand why he wants to function, but it's an error because the notion of function is deeply entangled in human practice. is set, isn't it? That's why the notion of function is unknown in classical mathematics, but the notion of set is the central notion of the original. The original notion of function And it comes out in expressions like, um, I said I'm not trying to call them as a function of temperature. What they mean is you've got quantity. Okay, now, the White Law here explains this. It's really quite intelligent. The temperature isn't, the temperature at this point, at this time, temperature is a map. That's right. I mean, you know, there's all kinds of things, but it's precisely the foundational thing then he stops over. What's the conceptual calling attention? What's important?

12:30 What intuitions have to be made precise and so on? I mean, that's a really beautiful song. Yes, well, when Mr. Pontius discusses my science, that's amazing better than that. Sometimes objective and objective. He doesn't say to him, but his idea is that he wants to make as much of this algebraic. He doesn't say that. Well, that's what he's doing. For the kind of extended modernity of algebra. That's a fact about modernity. It's just a fact. It's also a fact about the right way to do it. It's true, but it is a sort of off-right, all right, through it, through my virtual experience. All that, all that I agree with. And, uh, um, the... And I'm sure it's one of these meeting in the same direction as the specificness of the natural numbers, of, uh, when singles, more than half the natural numbers, but then when the children sight-wise at the point of the time, they also... I was looking at, I was looking at the camera or something, and, ah-ha, ah, got it, yeah. Okay, let's go. I think I'm going to be shocked to find he's still here. No, he's there. Thank you.

15:00 and various questions I'll bring up at the end, which I hope will be answered by a legal conference. I'm surprised if they all are. First of all, let me just put some standard transparency. You can see that's an older one because they're all very nutty. In twisted theory, one thinks not as space-time over here, one thinks in terms of space-time, which I should point out is three spaces in one time. I'll come to that just here shortly. And one's thinking of light rays in space-time is the first approximation to be represented by a point, And the point, Steve Haggett was pointing out earlier, that if you think of how you represent a point in this space, in terms of lat rays, you think all the lat rays are a point, and that's essentially the celestial sphere. So if you have a nice sphere, and I think you go out, if you use the missile and all that, you think about it, it would be special if you arrive in a spaceship. And the point about this sphere, from the point of a crystal sphere, is actually a human sphere in a natural sense. If you think of different observers looking out of the same sky with different velocities with the same event in space-time transformation from one observer sphere to the other. And since it's two-dimensional, one could think of that as a complex one-dimensional manifold, which is where you like to think in this theory, In other words, the reason of sphere, the simplest reason of service. And the transformations, or the morphic transformations of that sphere to itself, are precisely the non-reflective Lorentz transformations. Is that just a pun, or does that have some deep significance? And the point of this sphere is to regard that as something of deep significance. And at the space of light rays, to the first approximation anyway, you might not like to think of that as a complex space, but these things were complex sub-manipodes. The snag about that, of course, is that the space of light rays is high-dimensional, and so you haven't got much chance of making it a complex space. I can't remember.

17:30 But let me tell you, and this is basically a repeat of what I put on the other transparency. It's a form of crystal space, but it's in a sense to reduce a small part of crystal space, really. And you get a better picture by thinking of the correspondence that Steve Hagrid wrote down. is this equation here, the incidence relation, which relates the ordinary space-time coordinates in orange over here, representing which one is the space-time, to the twister coordinates, where we think of these z's as complex numbers, and if we're thinking of them the projected twister space, where the C stands for here, the twister space, we're looking really at ratios of the z's, and the incidence relation is just written down below, And as Steve pointed out, if you keep the yellow things constant and let the blue ones vary, then you see the locus over here, which represents the point over here. If you keep the blue ones constant and let the orange one vary, you see what locus over here. It's a very direct way of representing the correspondence. But we bear in mind that in order to get an actual real point over here, you have a condition that's in this matrix of omission here, a condition on the z's, which is the equation written down below, which we wrote down as well. So this equation here is your omission form on the projective complex space over here. And the point about this is the signature 2-2 emission form, and therefore it has places where it vanishes. It vanishes on a real hyperservice in this three complex dimensional space. So this is the six real dimensional space over here. And we have this five dimensional, therefore, subspace. and the lines, projected lines, or the simple stream of spheres, which lie in this five-dimensional subspace. That's one of the real points of encostal space. OK, well that's more or less already said by Steve, so I won't dwell on that. But let me point out something in the picture here, which he didn't mention, I think.

20:00 You see, light ray corresponds to a point in the subspace PN, and you might ask, how do you represent points in the complex manifold which don't lie in that subspace? Well, Steve Huggins pointed out the alpha planes and so on, I'll come to that in a minute. But there is, in a sense, a more physical correspondence, and I've put down here things with little corkscrews along them so you've got to think about these as in some sense photons which have helicity and energy and I'll come to the representation shortly which gives you this correspondence so the points in the top half of twisted space actually correspond to right-handed spinning photons they're not really don't think of them as physical photons think of them as mass as particles with a certain helicity and a certain energy And if they're left-handed, they correspond to the bottom. So you can actually represent the entire space here. You also have to consider the compactification. So there may be light rays out as infinity, so you have a particular place corresponding to infinity over here. But you can actually represent the entire twisted space, or the entire projected twisted space, in terms of physically relevant objects in space-time. With some of that, I'll come back to in a moment. I'm going to mention a few other things that you also said. Wait a minute, before getting to that, I'll mention something else. In the first place, you might say, well, that's just a mathematical transformation, so why should this tell us something different about physics? Well, of course, mathematical things can guide you in various different directions, and so we don't really know but there's also the possibility that it's something physically different and this is one of the motivations behind Trister theory, is that perhaps in some sense whatever quantum geometry is going to be, we don't really know yet but one of the sort of more forward ways of thinking about quantum geometry is to think of an ordinary space-time but where the metric somehow becomes a quantum thing and in some sense you think of it as fuzzy, it becomes an operator, which is subject to uncertainty, relations, and therefore not a well-defined thing. So the light cone dimension becomes fuzzy. And that's, if you like, a more conventional point of view about quantum geometry.

22:30 But in Twister Space, you might think, well, maybe these ideas should be applied to Twister Space rather than Spacetime. And then the kind of picture that you come up with is not that you have a space-time with black stars which are fuzzy, but in some sense the black rays are still there, but whether they meet or not is what becomes a fuzzy thing. So the fonts themselves, in some sense, which become fuzzy objects. So it leads to a rather different picture if you're thinking in terms of quantizing, where it might just lead you, and it suggests that we might actually go in a different direction a way which would naturally be taught as led, maybe in a more commonly expressed point of view with regard to quantum gravity. So that's just not yet a hit point, but maybe had relevance at some stage. I can say that Christopher Theorem hasn't got to that stage yet, but it's something I might think about where we're going. So this is just something that already we plug in and told us about, and I'll just pull that through the picture. This is the six-dimensional description. And here we have this light cone in R6 with a two-four signature, two pluses and four minuses. when you think of this parabolic section of the light cone, and Steve pointed out that there's a bit misleading, which don't have really a feature in a past cone. They're connected to another dimension. And it's also misleading because the plane U equals W, which is parallel to this one, goes to the origin, represents the rays, which corresponds to the empathification across the space. It does also show how one can think of generalizing twisted theory to high dimensions. Obviously a lot of these ideas were generalized to high dimensions. One way of thinking about twisters is that they are the spinners, or the reduced spinners really, for the autonomic group O24. So, the spinners for O24, plus the six-dimensional rotation group, pseudo-rotation group, and spinners are therefore eight-dimensional,

25:00 but the reduced spinners would be four-dimensional objects, and the whole eight-dimensional spinner would consist of two degrees, and what you have an epsilon that I find in the top of the non-government index, if you like, which serves to lower one of the indices. So you actually have an upper and a lower twister. You represent a general unreduced spin for this space here. But these ideas will continue, you know, extend to other dimensions, so that in some sense you certainly have a twister theory for arbitrary space-time dimensions. But there are certain respects in which Twister's for four dimensions, or I should say Lorentzian signature, three plus one dimensions, there's something very special about that, and I'll come to that later. And this specialness does seem to have some relevance, particular relevance, to how one might try and do physics in Twister theory. That's something I'll come to. I'll put that one over here. I think also this picture of the Klein quadriche is something that Steve had too, but I'll remind you of these things. It's a correspondence between the complex projected three stages of classical geometry. The lines are represented by single points, and Klein, I think, was thinking of, I'm probably grasping before him, was thinking about how you represent lines in free space, represented by a point in a quadric, and then you want to know how was the point represented, represented by a thing called an alpha plane. I didn't incidentally invent the term alpha-plane, beta-plane, that's been the classical geometry literature. The point goes to an alpha-plane and a plane, so it's a beta-plane. These two planes are two planes which are completely contained in this four-quadric, and if you consider this as a metric, it is the conformal metric, the light cones of which are represented by the tangents to this thing, if you think of a plane which touches this thought quadrant in five space,

27:30 then they will be metered in the light cone of the point. If you think of a metric whose light cones are given by that, then these things are self-dual or anti-self-dual two planes. The alpha planes are self-fueled and the beta planes are anti-self-fueled. Okay. Now the... and you have this incidence relation which is not just between... Well, I don't know if I'm going to get that. You can translate it backwards and forwards. Incidents basically means if things intersect in a larger dimension than they do generically. So the line here would mean when the point lies on the line, and over here it would be when that alpha plane passes through the point. And so on and so forth. So incidence here plus incidence here. Let me not say too much more about that. I'll do more of our service, please. If I'm ready, let me just put that there. Now, this notation that Stuart Ward has, I have to write the incidence relation in terms of two-spin and notation. Now, I'm going to make a point here, which is that the notation... I have to make an apology, which is that the first paper I wrote on Twister Theory, I had all my conventions wrong. The trouble is that most people usually refer me to the first paper I wrote, and when I say they're wrong, I'll explain a little bit about why they're wrong. But I didn't realise certain things which later on became clearer, and one of the things that I meant wrong, is you listen to the people calling lambda, and that thing, sometimes it's called omega, or mu or something, I think I thought it was mu if I remember. I had the index here upstairs, and the index said downstairs. And the upstairs index was chosen because I thought of it something was pointing somewhere. So that's a vector. But now I realize, later on, much later I realized, it's really a momentum thing. So it should have a downstairs index. And the other one should have an upstairs index. And this all is connected with the conformal transformation properties of these things.

30:00 So in order to understand how they behave properly under conformal transformations, It's much easier to have the indices in the arrangement that I have in here. So if you don't think about that, you might feel all sorts of ways to put the indices. And in my first paper on this subject, the list for algebra, I did have the indices the other way around, and didn't realise until later that that was not the natural place for the indices. So I'll come back to something which relates to that shortly. Also, when you take a complex conjugate, you want the thing to acquire a primed index, a dotted index, and you don't want to have to move the index at the same time. I'll say something which relates to that sort of two. But let me just indicate something about the notation. The reason for the letter pi is simply that it has to do with momentum. and p tends to yield less use of momentum, but that's not a point which I think you've got the score on. But the placing of the index is something that's inhibited, so this would be a co-conjugate, which it is. Now, also, when you take a complex conjugate, here's something where I'll let you make a remark, because I should make a remark before going back to this. It depends on the signature you're using for your space-time. And there are people who use different signatures for Twister theory, because depending on what you're interested in, you may want to use different signatures. If you're interested in pure mathematical applications for Twister theory, then very often you'd be concerned with space-time as well as it wouldn't be space-time. Spaces, four spaces with a positive density signature. Space-time, physically space-time, you don't normally expect to have one time and three space. I tend to have one plus and three minuses for various reasons, but that's not such an important point here. Much of the twist-to-string work seems to be used in a three plus two minus signature, And then, with the expectation that I'm going to do a pseudo-wip rotation or something afterwards, that that is the space. Of course, to do that, I have a lot of trouble with this, because it seems to me that much of Christopher is geared to the Lorentzian signature, very specifically.

32:30 And it seems a shame to do it in some other signature and then come back with it. Of course, one does have a, literally, nice-looking feature that the twisted space, if we can go with a real space, and that makes it easy to see what's going on, but maybe there is a price to pay for that. Let me just say something about these different signatures. The signatures relate to the type of complex conjugation operation that one has in twisted space. Now, the, often I call the physical signature, is one in which you have a Lorentzian signature on space-time, and in winter space, the complex conjugation operation sends the twister into a dual twister, so the index goes down. That's related to the fact that one has a commission form, and the commission form is the thing that we polarize with respect to in order to get the complex conjugation of the view of the operation. So, in this case, this is the case that Steve Hagit was talking about, one has the zeroes of this emission form corresponding to a real five-dimensional subspace of this complex three-space, and that's the place where this emission form vanishes. Now, if you have post-definite signature, every complex conjugation operation in twister space takes you from a twister to another twister. But this is of a particular kind where there aren't any real twisters. In fact, the complex conjugation is the probability you do it twice to get back to the minus that we started with, and it's really a plutonionic situation, and you're led to this vibration. the complex projected free space, sort of, had a bundle over the S4, and the fibers are That's the kind of picture that one has for this kind of signature, in particular, a particular thing on Hitchin, without the theory of curve-spatting of this nature in the signature. The case where you have two pluses and two minuses, this is the signature case, of course

35:00 So, this will be studied in various contexts. Denies to be a mason in the case of the curved spaces. And Whitten and others in this new twister-string theory, one tends to use the plus plus minus minus signature. And here, the structure, you have a complex conjugation which takes a twister to another twister. But this time, you do have real twisters. that can equal that, and there's the real ones, and in fact they just give you a real projected free space, which is the subspace of the complex projected free space, which is the general piece of projected foreign space. So, you can see these different situations, and the different situations show up in the different kind of complex conjugation operations. But you see, these two cases are really quite different from this, in a sense, because the conjugation operation sends a twister to a dual twister rather than another twister. Of course, in these two other cases it sends it to a twister, so that's something to bear in mind. Now, also the placing of the indices here relates particularly to this particular version the complex conjugation, if you have a different signature, this might not be the natural constraint. So in this case, as the complex conjugate is something which simply puts the prime on the upper index, it has to swap the two around, so this goes down to here, and it sends down to that one, and it takes the prime off, which is the prime index representing the complex conjugate interface, I should say that the notation here also is geared to a nice translation between space-time vectors and spinners, so if you see a pair of indices a-a-prime like that, you can automatically translate that into a small letter. You might think that the big letters also would be the

37:30 the four-dimensional ones and the small ones, the full-dimensional ones, but that's a bit messy, so I'll put it this way. But it's nice to have a translation where you can go back and forwards without having to make it explicit every time. The trouble with Greek letters is that the capital Greek letters, there aren't really very many of them which are different from the capital Roman letters. So that's a slight plug for taking these things as capital Roman letters. Certainly, if you want this to fit in with a nice two-spinner notation, it's useful to see what happens here. OK, so we have the humbles conjugation, which sends the switch into a dual twister. This is naturally something which will form a smaller product. This hand is in with that hand. It begins to be in the opposite position. So, that's the next thing, one of them is a co-object. This is a, this is a, you think of this as a two-spinner, it's a compass conjugate co-two-spinner. And then this thing becomes a co-two-spinner, and this is a compass. But, you need to see clearly what to do with it. And the alpha planes, let's see, point out, are the points which are incident with a particular twister, I'll make another comment, I think, about the complex conjugation here, because it does enable you to get a geometrical picture of what a general twister looks like. In fact, this is how the whole thing starts off. Some of you might worry about the name Twister. Where did that come from? Um, here I'll show you. You see, if you want to represent this point up here, um, how do you do it? Well, you can look to the complex conjugate. That's a complex two-plane. And that complex two-plane will intersect the, the n space, which, if you remember, represents the space of light rays in the middle there. And the complex two-plane is four-dimensional, this is five-dimensional, in a six-dimensional ambient space. They're intersecting a three-dimensional set of points here, so the intersection will be a three-parameter family of light rays. So there will be a three-parameter family of light rays representing this z-bar,

40:00 that would represent the z-bar. And what does that look like? I think the picture did appear somewhere, I think I saw it when I came in today. This is a glass filled crystal, you know. What does this mean? Well, you see, this is a family of circles and one straight line filling the whole of Euclidean free space. It's actually filling the whole of a free sphere, And these are clipped parallels, projected stereographically into these three spaces. They're oriented, so each one of these has got an arrow on it. And what does that represent? Well, it represents, remember I'm trying to represent a space of light rays. A three-parameter family of light rays in ordinary space-time. So each light ray at any one moment will have a location, position, and will be going in some direction. So you have a point and an arrow attached to that point. These circles and this one straight line fill the hole for free space, at each point you'll have a unique arrow pointing somewhere, coming along in that direction here, you see. And that's telling you one of those light waves medium. So if you want to see how this evolves in time, if you wait until the next moment, and the next moment this little particle will go on zipping off here somewhere, although it's not at all obvious staring at that picture, the whole picture moves without changing shape, with the speed of light in the opposite direction to the arrows on that line pointing downwards, so the whole thing moves a lot faster, so the whole thing goes slipping along like that. And that is the twisting object, which represents a general twister. And it was somehow playing around with this figure which got me started on the subject. This actually has some other significance which I'll mention in a moment. I should say something about this too. Complex concussive points. I think Steve did mention this. The top half of crystal space corresponds to the forward tube, or the future tube, which is, if you want your wave functions to be positive frequency, you want them to extend into this forward or future tube, And where does that correspond to, where it corresponds to the lines, which lie in the top half of the proof of space. I'll come back to that shortly in a moment too.

42:30 But all this has to do with the right signature, and this is the theory that I'm all geared to handling the appropriate space-time signatures. Now there's been a notation, I said something about that already. This is just the incidence relation written. The incidence relation is just in the yellow box here. and the Spiller notation tells you how to do it. OK, now, what physically does that twister represent? Well, you see, we have a light ray, but we want to represent the ones which don't necessarily lie on this subspace PN, this five-dimensional subspace PN. And it turns out that you can represent the general particle or system, if you like, Zero rest mass, you see, in particle physics, a zero rest mass particle has, you don't just say that the momentum is null and pointing into the future, although you do want that, if it's null and pointing into the future, that means that it's still a representation, so I've taken this A here, remember the convention is that we can change that into a capital A, A prime, and if this is a null vector, it splits into a product, and if it's pointing along the future like cone, there's a plus sign. If it's on a plus like cone, there's a minus sign. So you can always find this representation, or this momentum, if it's in the future null momentum. What about the angular momentum? Well, you can always write the angular momentum in the form given here, provided the, what's called the Pauli-Lubaschi spin vector that's this object here it's the dual of the which brought up the angular momentum with the full momentum and that's the dual of this epsilon sitting here if that Pauli-Lubaschi spin vector is proportional to the full momentum this is something that you don't expect except for a massless particle because the Pauli-Lubansky pre-vector, the spin vector, is orthogonal to the four-momentum. And so if this is an ordinary massive particle, then to be orthogonal to the four-momentum would have to be zero.

45:00 So this is a particular thing which happens when it's null, but it can be orthogonal to itself. And the Paolo-Lubansky spin vector, to be, for this to be a proper physical mass particle, it should be a multiple of a form on it. And this little s here is the real number which could be positive, negative, or zero, which is the helicity. If it's positive, that's the positive helicity particle. If it's negative, it's negative helicity particle. And what one finds is that these conditions, there are actually several of them you've got to formulate, if you're thinking in terms of ordinary space-time, four vectors of things, or four tensors, it would be a little bit of a mess to say, because you want to say, not only is this thing null and pointing into the future, but it's got to be proportional to the time of the vastness in there. And that's a nonlinear relation which you've got to express somehow. The Twister theory does all this for you automatically. You find that if and only if these conditions hold, can the angular momentum be written? The angular momentum is a skewed two tensor. The angular momentum can be written in this form here. So this is the antistelphial part of the angular momentum. This is the celestial part of the angular momentum. and you find that it's a symmetric color to the pi you already had and something else. That's the omega. And this is the complex quantity of that, so you don't need any more bits. But this defines the omega for you. So once you have pi for the momentum, the extra piece of information you need is the omega. And the omega and pi together provide you a twister and give you the physical interpretation of a twister. So that's a thing to bear in mind. There is a direct, unambiguous physical interpretation of a twister, which includes the positive and negative multiplicity, so you've got the whole picture here, and not just the space here, but you have the whole projected space. Well, in some sense it's a little bit misleading to think of projected space. So what you really have is the momentum and angular momentum structure, and there is a freedom that if you multiply, you see this, if you go one way or the other way,

47:30 if you, if you, if you note p and m, there is a freedom in pi and over. This freedom is simply an overall phase factor. So you could multiply this by e to the i theta and that by the same e to the i theta, and that doesn't change anything, but that's the only freedom you have. So that, apart from that phase freedom, the point in this twisted space, It's now not projected, so we actually need to know the scaling along with the pi, corresponds precisely to the p and the n. So we have this very nice representation of mass of particles. It's actually much neater than what you would do if you simply wrote the... if you simply wrote it all in time for the formula for the formula. What happens when you move the origin? I think I should mention that. If you move the origin, it should be very simple. The highs don't get changed, the momentum doesn't get changed. The worst you could do is just to cut the phase. You don't make it to cut the phase. You only get a set form in this very simple one here, where Q is getting to the position vector of the new origin with respect to the old origin. Now, you find that the momentum and angular momentum correctly behave, I haven't got it written down there, most of this is what I have on the other picture over there, but the momentum and angular momentum don't get changed, I say that the momentum, the full momentum doesn't get changed, but the angular momentum does get changed in a standard way, and that standard way is given by this very simple crystal transformation here. Next, I want to talk now about quantum mechanics. You see, up to this point, I've just seen some practical things, but in quantum mechanics, I saw the first quantization procedure, when we want to do the analog of what we do in ovary physics with a position of momentum. the position of momentum become canonical quantities, canonical variables. They're canonical both in the sense of the Orangian sense and in the mechanical sense. And what one does in order to get the analogue of that, or what is closely related to it in Twister theory, is to say that the canonical conjugate of a twister is actually complex conjugate. So, if Z is your twister, then Z bar is the canonical conjugate.

50:00 Now, if we're going to go to quantum mechanics, we want to have these two things now having satisfied canonical commutation rules, and the canonical commutation rules, the natural ones are here. here. So the commutator, this doesn't mean a commutator, Z, Z bar, this is the tronic delta with a minus constant over 2 pi, and the Zs commute to themselves and the Z bars commute to themselves. So you could just write this down as the most natural thing, but you find not only is that the most natural thing, but if you write it down, the commutation rules that P and Erholt or to satisfy the generators of the Conqueret group, are indeed satisfied automatically by virtue of this very simple Christer computation. It's much simpler than the standard computation rules you have to remember in the magazine. So, and also, there is no problem about factual ordering here, mainly because these things... It's really a little bit of symmetry graphish here with what we did, possibly having that quantitative here. That actually disappears. So there's no backwarding problem. The only thing you do have to worry about is that the helicity, and I didn't mention before, that the helicity thing here actually turns out to be, forget about quantization for the moment, it actually turns out to be simply the twist of norm. It's a heart of ZZ bar. Very neat. When you introduce these quantum computation rules, then you find that it's not quite that, that it's something actually more natural. It's this polarised form, that is to say symmetrical form, of ZZ bar for ZZ. It's a quarter of ZZ bar for ZZ. And this is absolutely straightforward. There's no choice about anything here once you've decided on these commutations. Okay, now what would be the analogue of simple first quantization for using twisters? It's in ordinary particle physics, or in ordinary quantum mechanics, actually, similar particles. You would say, well, you might want to use position representation, or you might want to use momentum representation. if you use position representation then the momentum becomes

52:30 differentiation of a straight position and vice versa so something similar will happen in just a theory and you can think of the z representation or the z bar representation those are two different things you might do if you take the z representation then z bar will be the differential operator now d by dz basically You want to say, of course, as with wave functions, that your wave function is a function of position or momentum. It doesn't involve both variables. So how do you say that here? Well, you want to say it's a function of either z or z bar. Well, first sight, that seems funny because z bar is determined by z, but at the second sight, you realize, of course, what we do. It's just the Kochi-Riemann equations. If you say something is independent of z bar, you simply say that f is holomorphic. So that you're led to F being holomorphic, naturally, when you're thinking about wave functions. So, a Twister wave function is a holomorphic function. Or an anti-holomorphic function if you've chosen the Z-bar. But if you haven't chosen the Z-bar, you'll probably rename it W or something, and you'll go holomorphic to W. So, Twister theory deals with holomorphic functions. Now, we also, this is the convention that Steve briefly mentioned, if you want to think about massless field equations for mass particles, then you've got these things here, they're very nice to write down in terms of Houstoners, So you have the ones with the unprimed indices representing the negative helicity particles, the ones with primed indices representing the positive helicity particles, and the wave equation is 0. And, yes, the point I should have stressed that isn't here is this formula at the bottom here. You see, if you want to know what little s is, that's the helicity operator. and recall that Z bar is now D by D Z this is basically Z D by D Z and you've got to turn these around to make them the same as that and you have a 2 coming out from that common factor so you have just basically the Euler homogeneity operator that's the Euler homogeneity operator the eigenstates of this operator are homogeneous functions

55:00 and this 2 just tells you the relationship between the homogeneity degree and the volicity. And that's just given here in the z picture which is the picture as well as your picture. If the volicity is 0, then you have homogenation 3 minus 2, and that's the case that's seen explicitly well done. And if you've got these other cases different nations. Notice that it's all lopsided. Particularly in the case of gravity. So if you've got linearized gravity, then you have plus two for the homogeneity for the negative velocity graviton, and minus six for the homogeneity for the right-handic graviton. You might not worry too much about that at this stage, but I'll come to some reason why you might also go back for the minutes. And these were things that I should This is a simple function, one over a product of two linear forms, and he had these two planes and such, such, such. That's how I'll say a little bit more about this. You see, this case is the case where the singularities lie on these two planes A and B. That's where these two factors vanish. You might imagine adding together things like this, where the A's and B's wiggle around a bit, and then you'll have a smeared-out region of a somewhat larger singularity region. You can still do your conglarynticals if they start to intersect each other. So you can still imagine doing a conglaryntical even when these regions get a bit bigger, singularity not just with the poles, but, well, they might start intersecting, or you might want to intersect, add things together with these singularity regions all over the place, and you don't really know what to do. So this is really why we do meet the derivative of something that we're more sophisticated, and again, this is something that's easy to refer to, which I will also use. One of the advantages of using Czech representation, basically using open sets and overlaps and so on, is that the functions are very much simpler. If you choose to use a D-bar description or something,

57:30 you find you are led to very complicated looking things often. And often you look at things where, in some sense, you haven't done anything. You see, you can have a nice simple function to give you something which is not so simple as a space-time field. You see, the twist of function doesn't satisfy any view equations. It's just a homogeneous polymorphic function. Whereas you solve these dual equations, putting in some arbitrary function. Whereas if you go down, if you go to a D-bar representative, then you have to solve equations again. In a certain sense, does that really work? So I think there's always trying to do with what this is, I don't want to know that, but I certainly found that the Czech description is very much similar to the work that is in practice. Well, here we have, this is again what Steve was saying, let's talk about Pt plus, one reason to talk about Pt plus, as also Steve mentioned, is that you're looking for functions which are only singular when that line moves that one, and provided you're up in the top half that's not singular, and that will give you a positive frequency field, so a wave function is something which you're really looking for here, and if you're looking at the top half of twisted space, then that will So we'll cover that with two open sets, that's what's going on here. We'll compare this picture with one over the bottom. It throws the way down on that side there. These are the singularity regions, and the open sets are these ones here. The union of them gives the top half, and the intersection of them gives you where your punch is behind. So, here we have F's behind the intersection. Now, if you start adding these things up, and of course you factor out and so on, which is a little complicated, but basically, if we're just talking about the first cosmology, which I miss here, you can express it in terms of things like this. I think I don't have a huge amount of time, but I'll put this picture on the other side. I should explain what these complicated-looking contours are down here. So you can replace your simple contra-integral, which just divides two singularities, by something which runs along over the intersections between different patches, and this is what's called a branch contour, so you can sometimes, so if you do all this in terms of contra-integrals,

1:00:00 so you talk about chief cohomology, but if you want to realize a particular element in terms of the field, then you can do a contra-integral something you can. Now, here comes the same thing in this picture. Basically the same is what I have in the bottom there. And this notation here tells you this is the first. This is the region you're talking about, top part of the trip space. This is telling you the 180 degree of the functions you're looking at. And this tells you what the first column is. Which means you're looking at overlaps between two sets. Second cohomology would be triple overlap, and so on. Sometimes it's useful, certainly to lay people, I find it useful to describe what you mean by cohomology by using a picture like this. Here we have an impossible object, but a cohomology is something which is non-local, that's the key thing. It's not telling you something about the value of a function at any point, because that function changes at any point, or in the neighborhood changes. When you trip down to a small number, you lose all your information. It's a bit like this here, you see. There's an impossibility associated with this picture. You might ask, where is it? But it's a non-orgal thing. It's not here, it's not here, it's not here, and you can certainly bear count the error, but if you could cover the rest, it's, oops, no, that's not impossible. No matter which bit you cover up, as long as you go down to the middle, you find something which is not impossible. The impossibility is in a thing as a whole. If you want to describe that mathematically, well, you can see, imagine cutting your picture up into three open sets here, what I've done, and then I look at it. Each one of these represents a supposedly consistent object, but there's an ambiguity, which is how far away is it from your eye. And if you take the logarithm of that distance, then there's an additive factor, an additive number, which is uncertain. We don't quite know what that number is. And on each one of these overlaps, you have a number, and you can write down just the things that Steve wrote down a little. And that will tell you, if there's a cohomology element there, that's a measure of the impossibility. If there isn't one, then you know the thing is possible. So I just feel this is a nice way of demonstrating that check cohomology is actually quite a nice and simple idea.

1:02:30 The only real difference here is that the distance between you and the object in your idea is just an overall number. If you think of that number as being allowed to vary as it won't go over the patches, then you have something really very close to where you're talking about. Now, there's a reason why first cohomology is something that's a special role. When I said that you might be interested in higher numbers of dimensions, indeed you might, but what you tend to find is that the commoner integrals you're led to really represent objects of higher cohomology. They're not first cohomology in the world, they could be second, third, fourth, depending on the dimension. This is really the dimension of the twisted space as opposed to the dimension of the space-time. But you find that as the dimension goes up, the cohomology also goes up. And it's a particular feature of the twister theory, the canonical twister theory, which represents the space-time that you're led to, that four-dimensional space-time, where you're led to a first cohomology. Why is first cohomology something of particular importance for physical theory? There's a good reason for that. In fact, one of the ways one might choose to explain what first homology is all about, is to think that you might have functions, you find them with overlap, or you might have vector fields. Let's think of vector fields. And I'm going to think of these vector fields as representing sliding one patch over another one. Now, if that sliding is just infinitesimal, then this represents an infinitesimal change from, say, black space to something that is infinitesimal at all. But you could think of building a genuinely curved manifold by actually sliding patches over each other by a finite amount. So basically you're exponentiating the infinitesimal sliding. And indeed, the rules, which I think are still written down at the bottom there, linearization of the rules that you would have to say when you say you're building a manifold. If you're building a manifold, you want to say that you have a transition function from one catch to the next, and you want to say that there's a consistency rule on triple overlap. Well, first of all, you say that when you go from this to this, it's an inverse of going from this to this, this and the rule of that sort. Secondly, there's a consistency rule on triple overlaps. And thirdly, there's a rule that says when you haven't done anything.

1:05:00 whether, if your manifold is being unchanged, if you just slide the whole patch to the hole. So if these arrows on the overlap can extend right over the whole patch, then you haven't done it. So, what you're doing is you're factoring out... Well, it's just the rules that I can see... Do you write the numbers? Yes, you do. That was quick. Anyway, we've done it. There's certain rules on the double overlaps. This is the linear version of what you see. Fij is minus Fgi on the double overlaps. This is a triple relationship on the triple overlaps. And you see two of them as a triple. The difference is simply expressible as in this sort of form, which is really . So, what I'm trying to say is that building a manifold is a non-linear version of first cohomology. So, what the point of view is that, look, you do things linear. You're doing linear things when you're really doing first cohomology. But if you're building manifolds or something like that, building bundles or something, then you're doing something non-linear, and this non-linear thing can reflect how fields interact with each other. So the linear thing is just how the field propagates, but then the non-linear thing is how it interacts with something else, And this is a feature of first cohomology. I don't know how you would generalize these ideas to higher cohomology, because cohomology is something which... Okay, there's a sort of nonlinear version of the first cohort of cohomology, but if it's higher cohomology, you start writing in your process. So there is something very special, apart from a number of dimensions that we've... The Twister theory leads us... Well, but if we take space-time and four dimensions, then they just appear to lead us to something with a microscope when it works for a space-time assignment. Okay, now what do you do? Can I make this more explicit? Well, this is the construction referred to as the nonlinear rabbit hole construction. I think Claude De Bruen will be talking about this in two days a minute. Yes. But he'll tell you much more about what's going on here. But just to give you a very rough picture, recall, this is something I think Steve said,

1:07:30 and I said, that the condition for null separation, of course, in the constant space, is that there are corresponding lines between the space to the descent. Now, can we generalize this in some way, so that So this is not simply CP3, it's something more complicated. Well, there's a theorem which tells you that if you're looking for a global deformation, CP3, you're out of luck. But if you're looking for a local deformation, and by local I mean it's local with respect to line, so you're taking, let's say, local in space-time. If you're looking at some region in space-time which is local, what's that mean over here? You're looking at some tubular neighborhood of the corresponding line over here. So the question is, can I deform that tubular neighbourhood of the line? So indeed you can, and the sort of way you would do it... OK, remember this is a Riemann sphere, and you can think of that Riemann sphere as separated into two overlapping, slightly extended hemispheres, and that's represented here, and that thickening of it is also correspondingly split. So you have now two overlapping, two regions which, sort of hemispheric, extended hemispherical regions which overlap over here in the middle. And then you've displaced one with respect to the other, polluted back down again, and you say that's going to be your new space. Well, the thing that used to be a straight line isn't much use to you anymore because it's broken. But there's a beautiful theorem due to Kodara, which tells you that as long as this deformation isn't too drastic, in most cases, you'll find that there is still a line here, in place of the one you had before. In fact, freedom in choosing these holomorphic curves, which are without defamation, if you think of this defamation taking place continuously, these will be the defamations of that line there. These, if you have a four-prime as a family to begin with, you'll still have a four-prime. So that means you can now go back and say, What does the space of these lines look like? Well, you have some four-manipole, and you can define the structure on it. You can say, what does the light cone look like? When you say null separation corresponds to intersecting lines here. And this gives you something with a conformal complex metric,

1:10:00 and it turns out to be a general anti-cell fuel complex space-time. So this is a way you can construct the general anticep through a complex space-time. If you want to assert that it's Ritchie-Flatt as well, then you can do that by giving this thing a bit more structure, which is actually saying it's a vibration over a CT-1. Well, anyway, this is what you find is that if you preserve this vibration appropriately, you indeed get the general antiseptic or gritty-flat complex space-time. So this is the non-linear version, recall that for spin 2 you can represent the mass of fields for each different helicity, and this is the non-linear version, if you like, of helicity minus 2, negative helicity gravitas. And in fact, it's related to such a function. You see here we have a twisted function, homogeneous degree plus 2. Yes, homogeneous degree plus 2, if you remember what that transference said, homogeneous degree plus 2 corresponds to minus 2 multiplicity. And for an infinitesimal decimation, it's just exactly the form. So this gives you a finite deformation, which actually gives you solutions to the Einstein equation, the Einstein vacuum equations, called a nonlinear gravitational construction. This could have some significance, particularly if one's interested in generalizing quantum mechanics, which I believe is the case that one will need to do at some stage when gravity enters the picture, because in some sense the wave functions have suddenly become more linear. And this is an interesting feature, which, well, one would like to extend this to the other case. You see, this is just a left-handed value one. What about right-handed one? Well, you have a minus six there, sort of a plus two. And this was the Gugliela problem that Edward Ritten referred to in his talk. It's been one of the difficulties with Twister Theory for a long time, and certainly if we get some new input from string theory, that

1:12:30 would be fantastic. So, I'm very excited by that prospect, but at least there is something one can do, let's say preach, or string theory, and it is possible to encode the information of a minus-six function in terms of deformations which actually affect the fibers. These fibers, I should say, see here's a three-space, and these fibers represent the non-projective, See, if you reach projected spots, you have a whole line for a possible non-rejective spot. And you'd have to go into that. Is that the line, or is it a sea star? It's actually a sea star, but it's even a pretty big star. The hull can be more than a point. So the deformations are a little strange. They're not simply points. You don't have to cut out those. The sea star is certainly not steep. One of these fibres is an apron of the circle. Ah, the fibre is... yes, yes, it is. In fact, it could be a bit more explicit. Well, let me... I'll practice that, but I might say this more in my last talk. You can actually be more explicit about what kind of definitions you use, but maybe I don't have time to talk about that now. Just to show that there is some serious work that I want to do about it. But it's not, I don't quite know where it lands you. This is where one's driven, but where it gets you is not totally clear yet. So let me just leave that one away. But one thing I do want to comment, which I think is a point worth making, is that, it's a fairly straightforward point, but you might think, okay, a lot of the geometry is very nice when you talk about projective crystal space, But in some sense, that's not really fully what you want to do physically. For example, even for photons, you see, if you want to describe a photon, the right-handed photons, okay, that's nice, you've got one particular bundle here, the left-handed ones, you've got another bundle. But after, what do you do if you've got a plain polarized photon? So linear superposition between a right-handed one and a left-handed one. How do you describe it intuitively? Well, you can, and this is just the sort of thing you do, I have this and down here. There are two versions of the Christa-Comper-Integral, and the one that Bersa and Steve describe is one which is done in projected space.

1:15:00 But you can do a two-dimensional Comper-Integral, which is very safe easier to write down what you're doing. And the second dimension in the contour is just a trivial one, and it just has to do with integrating around the fibre, and the complex plane, and the fact that it shows you up and down. But the thing is, it's slightly different. The thing is that for the projected case, in order to make it work at all, and make sense as a congruent rule, you have to say it's homogeneous as a zero, which is the thing you said. But, there's a different point of view you can take, which is you don't do it projectively, you do it non-projectively, so you've got those two dimensions looking at it. And then you say, well, the difference is now, there's nothing that you need to do any consistent, it's always consistent. It's just that you get zero for the homogenates as well. Now, that has an advantage, because you can represent the right and left-handed parts of the photon all together in one interval. And this is a natural thing that you'd want to do if you want, say, to describe a plain polarized photograph. You want both parts, say. And you can do the same for the linear graviton. So as long as you're looking at this two-dimensional comparable, instead of a one-dimensional one, there's no problem about representing things which are not pure orthodoxy states. So it's just a point I'm making here that one of the implications, and I'm sure Andrew, when he talks about the diagrams, will have a lot more to say about this, is that projective twister space isn't really the whole picture. We have to be thinking about the non-projective twister space in certain situations. And certainly if you want to represent non-pure multiplicity states with massive particles, you have to be prepared to do that. Right. Well, I just want to end now. Oh, hold on. Let me flash this one up because this is the water construction of the time today now. Let's look at the Yang-Mills fields. And again, you have the same problem that is a very beautiful way of the water construction, a very beautiful way of representing extra-dual Yang-Mills fields, but what about the self-dual one? Well, that's a goodly problem for Yang-Mills theory, and the hope is that these new insights from screen theory you'll tell us how to handle the both the left and right-handed politicians together. And that would be very nice, if you need to understand these things more than themselves,

1:17:30 if you're really understanding what's going on. But let me end by putting down some questions. Now, Sarah, I find it very exciting that there are... there's this new stuff coming in, and I sleep here, which may be the same as a very important new influence of the subject. But it's also a question, to me, about... OK, there are a lot of things in screen theory. I don't understand screen theory, but certainly. But there are a lot of things which seem to depend on the history of the subject. And I just want to know whether this will still be the case if we're thinking of a twist of screen theory So it's something more than something just useful for handling certain types of scattering processes. Okay, maybe that's what it is. And after all, we know Twister theory has been set up, if you like, for handling massless fields, for handling maliciously states, and that sort of thing. Maybe just using that formalism in a different context, one sees nice, simplified things coming out of it. But if, and I would like to think, there is something deeper going on, then it raises questions which I'd like to put and I'm not asking anybody necessarily to answer them here and now I'm hoping by the end of this week I'll have better understanding what the answers are. But one question is do we really need the extra dimensions this is an old question because it's so many years is there a sense in which projective species really take the place of the Kalabi-Yah spaces and What's powerful in string theory can be transported into a different interpretation, a really very different interpretation, where the space-time dimensions are in fact the four that work nicely and explicitly. The same would apply to supersymmetry. Do we really need supersymmetry? I feel less strongly about this. I've got a lot of trouble with the extra dimensions which I have for a long time. What about supersymmetry? Well, there's a question at least here. At least one of the reasons for when introducing the high dimensions and the signal symmetry and so on, it has to do with the color power, and maybe the... so I'm getting rid of the knowledge. And to what extent does that come from things like parameterization and variance? And do we really want to interest a theory? See, I have here a picture, it's not a very good one,

1:20:00 If you have these curves in projected twister space, in some sense they're naturally parametrious, because if you don't care about conforming the invariants, if you do care about conforming the invariants, they're not naturally parametrious. But if you do, if you don't care about conforming the invariants, then you've got these planes with this particular line I, which represents infinity in the twister space, and they give you a sort of two-spinner way of representing these points on the curve, which is somehow natural, and which is used in the non-linear graviton structure. So it certainly is there. Einstein's equation is not being called an invariant, as seen as S-bets. Well, there are various other points here. Do we really want the plus-plus-minus-minus signature, or should we be talking directly to the plus-plus-plus-minus-minus-minus signature? It does have implications with regard to how one might represent diagrams in Twister theory. We all know after Andrew's talk, but for the moment, in and out states, a twisted ear represented by things that you call things for two ears. There are two lines coming out, whereas in twisted-screen theory, so far, one seems to have things which basically have one ear. And that's leading in a somewhat different direction, so it would be nice to understand what's going on there. There's some point about here about the ballooning of off-shell MHV diagrams. Twister theory has a way of handling it, which would be nice to see how this relates to what people do in string theory work. Finally, one of the things that puzzled me about things like string theory initially, before I really understood that, I think. some of the things are going on, I still don't understand, I should say that. Doing things like putting Lagrangians in interest of space, it seems very odd idea, you see, because after Lagrangians have to do with the evolution field equations and so on, and in interest of space you don't have any field equations. But then, of course, what people really seem to be doing, what Ed is doing, I think, is they're looking at topological functions. Well, that's fine, because you want something where there aren't many people. That's exactly the kind of thing which Twister theory is just asking for. So, you can all even think of the Garnetons and things in Twister space provided there, in some sense, disappearing. I mean, they don't represent any actual dynamics.

1:22:30 They represent something which has a kind of topological rapport. Well, that's all I want to say for a moment. Thank you. If you consider that deformation of the non-projective circumstances, if you have all those things to minus infinity or plus infinity, you certainly will have the other one as well, that's fine. I mean, I don't quite know what to do with it, but certainly, there's nothing particular to do with 96. I mean, there is something which you can see when you start writing down the form which you make of it, there is something. The geometric road, rather, I just don't know. You're quite right, that sort of says, well, what's wrong with doing all of those things Well, you might have done it once. What I wanted to do, actually, was to write down a consistent theory with all the muscles, but I'll just mention there wasn't that wrong, but technically, if that makes sense, just as for the projected principles, you don't get much of help in inter-manifraction. And, technically, the inter-manifraction speaks in court. I see how much much in court is going to be answered. Well, I think it's worth looking at, again, probably. I mean, it's one of these things which hasn't found its right point. I mean, maybe, you know, looking at the screens may tell us that. But up to this point, okay, you can make certain kinds of deformation which correspond to a lot of other things. Now, what's the point of it, or what does it actually do for you, and what's special about Spin 2? Well, I didn't emphasize it in my lecture, but one other fact about the small g-square animal of AES-5-1-S is that it would appear that there is a theory of the fact of particle And the thought across the mind of all the bacteria is going to be something you get from the declaration of an object. There's also a variant of monochrome. You can consider it to the non-projected Christians,

1:25:00 I think those are the ones that you see. What it says is the future of the culture of the world, and you only tell when it's soft, right back to infinity, and where you are. Well, you... yes. You must know exactly what? That's fine. Well, you certainly get a difference between whether it's not soft infinity or the origins. And the capital flips over, and it certainly... Yes. But it was an uncertainty. If you don't tell where you are, that... Okay. Phase one will be positive. I see what I have there, I have a question. I was going to ask Alex, there's a great comment, you talked about how you're upstairs writing down a double comment, the attitude is together. For that, yes, if you're looking at an H, if you want to look at a place counterintuitive by sheet technology, you can say, ah-ha, H2. Because you can't take H2 or 10 to 50 by this place, then you're a light bundle up in the street and can't be added up together. It's not quite clear what's the correspondence between the two. It's quite a safe sea star because there's no... They don't have a cheap commodity. So, I'm not quite sure what it corresponds to. I mean, some of the two, which replaces that combination by cheap commodity notions, it's not quite clear to me what it is. Yes, I guess it probably is actually... Because sometimes with the counter-interviews, one runs into small talk here, what we're talking about, one becomes more of an end. But then you come back there, yes, I'm not clear what keeps you here, because what kind of keys you take to tell me what's at all, the infinities. It's not small numbers. Yes, yes. It's not easy to see. So, till you're upstairs, happy and happy. Back here, we are all good things. Right, right. No, it's hard not to do it.

1:27:30 How are you? Deadlines, right? Oh, alright. How are you? Good? Yeah, I know. Good? Yeah. Yeah. Thank you. Thank you. I'm not sure if you're a part of your paper like this. Yeah, basically... You might be, okay. No, no, no, no, no. You're not sure. I know, I'm not sure if you're going to get a sleep on me. I know, I mean, this is like a thing that's the same thing. And, uh, yeah, I, I, I, I guess I can hear you. Well, you probably got it. That's right, that's right. And I remember being fooled right there. Just, just, just, just, I'm not gonna tell you this very little one of the people who would be all talking about. Well, certainly, I remember quite well that Witten, I think you're right, I think you're

1:30:00 right. I think you're right. Well, except you could say that plus means width, and minus means width, so... But, you know, that's not... No, I mean, these things are... I think at the end of the day... I mean, I see, I don't... I never really worry about this, because... To my own thinking, geometry is by definition... In very end, under change of notation. In very end, under change of notation. That's right, it should be, and I think the way... I think the trouble is that the physicists write in a very conventional passion for the paper, and that's the issue for the price of what we're doing, so we're going to roll an action this morning to talk about everything has its own pet name, you have to ask a bit of a criminal, so and so and so, somehow you're expected to know what those things are, they're not labelled in a kind of form of the paper. Anyway, I mean, as I say, I don't feel strongly enough to really worry. Well, the truth is that you saw whether the paper is, which is not as far along as I would like, they never are. you know, at some I can't remember how I wrote my extra section, but you'll probably be right. No, you wrote it with the old-fashioned people. Can I get to catch you with scissors? Yes, absolutely.

1:32:30 Thank you.