Motivations for Twistor Theory

0:00 here with us in a relativity lunch I think you don't need introduction to this audience so here there are members of the quantum gravity group, students, PhD students and there are also master's students from our master's program well I understand I'm very glad to be here and thanks for the invitation and I understand that my introduction to Twister Theory is completely redundant because people have already been learning about it. Anyway, I will ignore that and give you my introductory discussion in any case. Excuse me. I'll just start by mentioning some of the early motivations for Twister Theory and one of them is that it's a space-time calculus specific to three space and one time dimension so it's very much not sympathetic towards theories which are extra-dimensional such as most approaches to string theory and it's not exactly aimed at being a quantum theory of gravity or anything like that It's more exploring links between quantum formalism and space-time structure, which maybe bring out connections between those subjects which won't be lying hidden in some respects. The particular connections which I focus on specifically in Twister theory have to do with the relationship between geometry on the one hand and quantum amplitudes. I've always been rather struck by the fact that if you take a spin-half mass of quantum quantum then the directions in space directly related to the amplitudes, the complex amplitudes of quantum mechanics. So these amplitudes do have some connection with geometry even though one doesn't usually emphasize that fact. So Roger, quantum theory was from the very beginning in your mind you need to start increasing stress. Quantum theory was from the very beginning. Yes, absolutely. Yeah, that's right. No, it was...

2:30 You see, quantum theory, yes, but not aiming specifically at quantum gravity or something like that. It's just, in fact, spin networks, you see, were an example of something where one is exploring a purely quantum mechanical system, but it's meant to, geometry is in some sense meant to emerge from it. So the idea is that the geometry and the quantum mechanics are interconnected in some deep way. But you're right, that was one of the very early motivations. And so here on the left hand side we have the quantum mechanical relation to the Riemann sphere. Because that's a two-state system, you have the ratio to complex numbers, and that is the, if they're not both zero, that ratio is a point on the Riemann sphere. And this Riemann sphere also has a different kind of role in the celestial sphere. The Lorentz transformations act conformally on that celestial sphere, and so it's a natural thing to think of that conformal two-sphere as being, again, an example of a Riemann sphere in a different guise. The privacy of massless particles is somehow that massless things are in some sense more fundamental than particles with mass, which in a way has become more part of, I think, the way people think of particle physics today than it had been in the early days. I don't have a date for these early thoughts, but they were arising when I was a graduate student and a research fellow in the 50s. These were in the 1950s. Quantum non-locality, this does play a role, quite a significant role, which, first of all, it was simply a geometrical rather feeble kind of non-locality. I was aware of things like EPR and so on. Of course, this was long before John Bell's emphasis non-locality that's intrinsic to quantum mechanics, quantum entanglements, but this was somewhat different. Non-locality is much closer to that that we see emerging later on. And the final thing is really complex magic. To me, this is extremely important. I think when I was a graduate student in pro-mathematics, the kind of magical features of complex analysis, complex geometry, I was absolutely hooked by.

5:00 And it seemed to me that it would be a very natural thing if somehow the magic of holomorphic functions, the fact that... I think one of the things that impressed me the most was that learning first about real functions and analysis and the fact that you could differentiate things once or twice or three times, and these were all different. Or an infinite number of times, and that was different from analytic. And then as soon as you do complex, differentiating once is the same as all, they're all the same. expanding in a power series follows immediately from being able to be complex, smooth, even to one degree so this is completely magical for me, and there are many other things which I'll learn later, so that's a kind of emotional thing that's an important one so let me just say those are initial features and these, just as I'm just stressing what I just said, here we have the Riemann sphere coming about You have, say, spin up and spin down, and you have the complex amplitudes W and Z, and the ratio of the two of them is a point on the ring of sphere, and that directly, by stereographic projection, you take the point on the complex plane, that is U is the ratio, and then you join it to the south pole, and that gives you a point on the sphere, and up the direction up from the origin to the right, setting up your frames appropriately. That is the spin direction. So you can see a deep connection, as I saw it, between the Riemann sphere of the complex numbers, the complex amplitudes of quantum mechanics, and the sphere of spatial directions, which again featured in spin networks. That's something which I was trying to explore in the early days. And now we have the other feature of the Riemann sphere, the relativistic one, which is the conformal nature of the transformations. Here we have two observers moving at great speed with respect to each other, and the sky that they see is related one to the other by a conformal map, in other words, a Mobius transformation, which is the most general transformation of the sphere as a complex one-dimensional manifold to itself, which struck me as rather remarkable, too.

7:30 And here we have basically what Twister theory is doing. We think of the light rays. I've also got a little corkscrew on that light ray because it's the light ray together with the spin of the photon, if you like. It's not meant to be a photon necessarily. It's a massless particle, and it has a spin. And this is something which could be parallel or interparallel to its motion. And the Twister picture, so if you think of all the light rays, So the point in the space-time picture looks like over here, this is the space of light rays, and of course it's all the light rays through that point, and then there's this picture here, and so that is a Riemann sphere, and the point over here is the light ray over here, and the idea is that somehow if this really is a Riemann sphere, this would want to be in some sense a complex manifold, which it can't be immediately because this is a five-dimensional space, be an even number of dimensions but to be a complex method but we'll see how that comes about later but nevertheless you still have an element of non-locality because the light ray is not localized it's spread out right from past infinity to future infinity complex that's the women's series structure over here and this a short route to twist the theory is that we use the spinners for the conformal root, which is locally isomorphic to O2.4. And that's a quick way of saying... Roger, your previous observation is that in the spheres, the directions, the Minkowski structure, the Lorentz transformation, determines a structure which is in fact a complex structure. on the light, on the directions. It's what is preserved by the So this is the most direct connection in the context. You might say it's nothing to do with it, it's just an accident. And it's conformal in all dimensions. If you take one time and n space dimensions, then you have a higher dimensional more motions of that sphere to itself but it's not complex numbers so this is

10:00 saying that there's something special about three space in one time because then and only then is the light ray space thought can you think of it as a conformal as a complex manner so it's very specific to 3d space in one time dimension which I thought was a good thing you see these days people think bad thing, because they think, well, we want to have all these possible dimensions and so on. But this is a pocket of people who don't think that. You don't think that. So I don't have to try and persuade you. No, well, I can understand Luke very much. You don't want to have other numbers of dimensions either. anyway, I was only mentioning some of the early motivations, and it's probably useful to talk about these again, and thank you for making comments about that. That's very relevant. Let me make some comment, which I always used to make in the old days. I'm not sure where it ever gets us. But it was, in a sense, the idea is that one's attitude to quantum geometry might well be rather different from, I'm not quite sure how it relates to loop variables, but certainly very different to the lots of things that people were doing in those days, where you some say that the G, the metric somehow becomes a quantum operator. And that, in a sense, gives the light cone a kind of fuzziness to it. Whereas the twister view is it's not that the light cone becomes fuzzy in some sense, but the points themselves become fuzzy. So if you say that the twister space is more fundamental than the space-time, then if there's a certain fuzziness over there, it means that the points themselves, which are these Riemann spheres, somehow become fuzzy in some sense. So this is a kind of vague point that I often used to make. The only comments that I'm going to make in this talk which are specific to maybe an approach to quantum gravity are ones which I'm going to make at the end. And these are related to things that we were talking about at the Chris Ascham meeting. I haven't gone much further than our discussion then, but I thought it was worth bringing that point up because most of the things I'm going to say are not specific to quantum gravity. there are relations between quantum mechanics in many various aspects and space-time structure but they're not really quantum gravity statements but the thing I said at the end

12:30 is something, if it has any proof to it, would be a quantum gravity type of statement so we'll come to that but that's at the end ok, so somewhat to recap what I just said here here's our basic step one here we have space time and we think of light rays or points and the twister picture sort of turns these around so your light rays are represented by points and then the points over here are represented by Riemann skills over here so that's the idea this particular version of twister space I'm calling PN there's a good reason for this the P stands for projective N stands for null. So these are what we call null twisters. And the gravitational aspects of this will be explained a bit more later on. Let me fill this picture out a bit. Trying to get it in the right place. Here we have four-dimensional spacetime. Now this now is a five-dimensional space length representing the light rays. And the algebraic way Expressing this correspondence is down here. We have space-time points with these red coordinates here and the purple ones are the twister variables. And these are complex numbers, and they're concerned with the ratios. And here we have the relationship between these two pictures as given by the incidence relation. The incidence relation is this. The first two twister components are related to the second two. The matrix here, which is an emission matrix, of the given by the space-time coordinates over here, T being, of course, time, speed of light being taken as one. And the equation of Pn, this is not the whole of the projective, complex projective space, which you would get by looking at these ratios. It's the space Pn, which is given by this emission form, whose signature is split signature, two pluses and two minuses. Of course, it's not written in a diagonal form here, but that's what it is. So this slide has in it the essentials of the standard flat space twister theory, incidents meaning in this picture that the point, which is the space-time point, lies on the line.

15:00 that's the twist-to-oreal thing, and over here it's the point, which represents the light ray now, represents on the line, which is this projective line, complex projective line, in other words, in the sphere. So that's the large complex. Basic. That's the very basic picture. get these things down. I think it's easier. Let me fill this out a little bit. Now we have the same picture all over again. And here now, I'm trying to make it a little bit more complete. This is exactly the same as what I had in the previous slide, except that I'm now trying to give, indicate an interpretation of these points, which lie not on the, this thing in the middle is PN and that is where this incidence relationship, sorry where the equation of PN is given by this is emission form and that is a subspace of the entire complex of the Z space which is simply co-ordinatized by the three independent ratios of the Zs going to tell you how to get this later the point in the top half can be physically interpreted actually a massless particle with spin, so it has a helicity, that helicity could be positive or could be negative and if it's zero then it would actually be a light ray if it's positive it would be a right handed sort of twisty thing and that's where the main twisted theory comes from the interpretation of these twisted structures, I really won't go into the theory because it's not too relevant The right-spinning particles are represented by the points here, the left points there. And it's a bit miraculous, I think, that, in fact, these things do correspond to a complex projected 3-space in a perfectly natural way. And this is the incidence that we just had before. Okay. Now let me say something which doesn't... Well, it's not going to feature too strongly in what I want to say today, perhaps to make the points I'm going to make in a minute. The lines here, see, sometimes I'll draw them as a sphere, sometimes I'll draw it as a line,

17:30 or sometimes I'll draw it as a long sausage, which is a kind of hybrid between these two pictures. But it really is a complex projective line. It is a line, so therefore I draw it as a line. But topologically, it's a sphere, to bear that in mind. So, that's the picture. what I was going to say is that the incidence relation if you take R to be real then it's a simple consequence that this Z must satisfy this relation down here just pre-multiply by Z2 bar, Z1 bar and you see this is going to be anti-emission because of the I here and that's this relation so this relation here necessarily comes about if the R's are real. So if we have a real point here, a point in Minkowski space, an actual Minkowski space, then you get this relationship. I should also say that this includes points at infinity. So this correspondence allows points at infinity on Minkowski space 2, so you have a contactification of Minkowski space included in this structure. is the same as the one in the diagram yes it's where where scribe class and scribe minus are identified ah so it's compact it's an S2 sorry what did you say you said scribe class and scribe minus are now identified so they have to be that's right they're not distinguished in this twisted description so you could say that's not physical No, we were confused about that because we were discussing that. Is that the same or is it different? There is a difference. Because it's the lycone that infinite is... this identification allows you to think of Scry as a lycone of some point. Yes, it's the lycone of the special point at infinity, which is the set. You have a point at space-like infinity, you point at future time-like and past time-like and they all fit together point. You see, the future time-like point, the space-time lies in the past cone of that. Past time-like, it's the future cone. Space-like, it's the space-time. Oh, I see. Now they're the same. They all screw together and make a point. Yes.

20:00 And physically, you are going to make a comment about... So physically, that sounds like... Of course, you have to think about what physics is doing, yes. But I'm not saying anything here. But, of course, if you were talking about a curved space, then you wouldn't necessarily be allowed to make this identification. So that's not... It's only fortuitous. It's something that people... I don't get worried about with the ADS-CFT, you see, because there you've got... There aren't people talking about the identified version or the unidentified version or what. But you've got to take a covering space. if you want to have this Minkowski cylinder if you like, which is the Einstein cylinder that is the universal cover of the compactified Minkowski space you have to make sure when you're doing how you're applying this to physics what you're doing because of course I mean if you have a free massless field with no sources are automatically identified. However, with a curious there's this Goebbin index business flip of sign or multiply by I. I shouldn't really go into all that because it's not going to play any role in what it's saying. But it's a good question. So when I say it's not the physical Nikoski space, I just mean we wouldn't necessarily want to identify square plus or square minus. Yes, they have to be identified in this twisted geometry. So it's something one has to bear in mind. We're going to use these ideas. But let me complexify, in which case also you find it all identified in this way. See, here we have... Now, Pt is the whole of projective twisted space. That's the complex projective 3-space here. We have the PT+, which is the right-handed spinning particles, the PT-, and the whole space, and the Pn as well, and they're all joined together into a complex projective 3-space, and that complex projective 3-space, this is well known to geometers in the early 20th century,

22:30 and probably a lot earlier than that too, in the 19th century, with well-known people, the right people in those days, namely that if you want to represent lines in complex projective 3-space these are points this is what people refer to as a Grassmannian these days this is a client correspondence a line in complex projective 3-space is represented by a point on a quadric in an ambient 5-space it is a 4-quadric therefore and this is the complexified Minkowski space so it's complexified as well So I'm asking now for incidents where the point need not be real at all, in which case you won't have this condition which I wrote down at the bottom. If these R's are allowed to be complex, then you don't have this relationship. And so it's just a line anywhere in the whole space here to represent this point. See, if you fix the z's, and you ask for the solutions here, in the orange coordinates, that's just a line somewhere. So if you fix your... Sorry, I'm going in a long place. I'm saying that the general point in complexified Minkowski's space now will be a general line in the space area. And that, then, is a line in complex projective-free space. It's represented by a point on a complex quadric. And if I'm not interested in reality conditions, this is in Kovsky's space, compactified, but also complexified. And then you say, OK, points here represent lines here. This is part of the Klein correspondence. What represents a point over here? Well, it's what's called an alpha plane. What represents a plane? What's what's called a beta plane? and what's that got to do with what I was just saying. You see a point with a light ray, and you see the point really would be both at once, an alpha plane and a beta plane, which pass through that. They're both in the set here. So this is this picture here. Is that right? I don't know which picture, but this one, that's right. So here we have a complex.

25:00 That's what's in the picture. This is over here. This is a complex line, and then there will be an alpha plane through it, and I'm sorry, no, I'm saying it wrong, a point of, it is the right way around. When the alpha plane and the beta plane intersect, they always intersect in a line, and that line represents the point here, so that's a line in, that's the light ray. so a light ray has an alpha plane through it and a beta plane through it I don't want to go into this particularly because I'm not going to use this I mention it also because it relates to something I want to say shortly but the other thing I should mention is this if you have two points in the Minkowski space which are null separated so if we are imaginary separated, then that corresponds to lines over here which intersect. So the Riemann spheres, or the lines in this picture here, which intersect, correspond to points which are null separated over here. So that will come back into what I want to say. But I thought I'd show you that picture even though don't worry too much if it's confusing because it's not going to play a big role in what I want to say. What will play a role, however, is it's useful as pairs of spinners, and that is, if you like, the previous picture. In the instant relation, we have these two, and we have these two, and these are the ones which I call pi, and these are the ones which I call omega. The reason for that may seem clearer in a moment. although people who use twister theory in scattering these days tend to appeal to the first paper that I wrote on this subject where I had all my conventions wrong and that's what they all use these days which I hope I corrected later on when I say wrong, the main thing that's wrong about them is I had my indices in the wrong place that is to say, well let me come to that The twister can be thought of as a pair of two spinners. Sorry, this is the same split in two spinners you do, right? That's the same problem. That's the main point. It splits into two spinners, and these have different roles to play.

27:30 One of them is essentially a momentum. You see, here we have our light ray, and that light ray has a momentum, P, and that P is null. therefore in spinner notation it splits into an outer product of a two spinner and its complex conjugate. That makes it now and future pointing. Both things we want. It's called pi and there's people who object to that because they think pi means something else in mathematics, which of course it does. And I claim that this is a different pi because it's bowtie or else italic. If it's got indices it's in italics. If it's not got indices it's boldface, whereas an ordinary pie which is is an italic sorry, it's not italic it's upright but it's not bold, so upright, not bold it's a normal pie don't worry about that don't worry about E being an electric charge if you have E to the IE you see that often in physics never mind about that I just found it was a handy notation because it meant p so p is a covector it's the momentum covector and I'm stressing that this really is a momentum and that's why its index should be downstairs and that's important for conformal transformations that's a key thing which I had wrong in my initial paper omega, that's a common letter used for angular momentum because this is more like an angular momentum thing picture here is sort of giving you a feel for it. Here we have the light ray. Pi is telling you which way it's going. Omega is roughly its moment about the origin. So roughly speaking, I'll make this a bit more explicit later on. Roughly speaking, Omega is the moment of this light ray about the origin. So it's origin dependent. If you move your origin around the omega will change, but the pi will not. So that's the role of omega and pi. To be more explicit, it's really omega pi bar and pi bar omega, which gives you the angular momentum. That will be completely explicit in the middle. But the idea is that the twister splits into those two splittings, or vile splittings, if you like.

30:00 And that's a very natural splitting. But it's not invariant under change of origin. formula in there, it is, however, splitting the pie off, it's Blunker anywhere. Okay, I'll be a bit more explicit, but I, of course, I don't know how familiar people are with two-spinner notation, but a lot of two-spinner notations is very important if you want to write down formula explicitly, so this means, of course, that you, I like abstract indices, so that saves me writing down signals all over the place, which just translate from one to the other. So a space-time index is a small letter, and that stands for this pair of capital letters. The capital letters are the two components, so four components there, two each to those. The primed index is the complex conjugate spin space. So spin space is complex conjugate, So, the outer product of those two spin spaces is essentially what a Hermitian particle is the tensor, is the space-time vector space. In components, the relationship is that. The determinant is basically the length, which is this, and that tells you you split if it's a, it has rank one if it's a null vector. The metric splits into these two epsilons, here we have the translation, and you can interpret, this is important for what I want to say later, there's a single two-spinner, and I'm doing it with an upper index, it doesn't matter too much, these are low of the epsilons corresponds to a null vector because you multiply by the complex conjugate that null vector I've drawn here, that's n but the phase see this disappears because this has e to the i theta than e to the minus i theta that phase disappears but geometrically that phase has an interpretation and here I have it if you think of this light curve I've now sliced through it so it's a Riemann sphere The point, the direction of n is a point, and the flag plane here is the tangent vector to the sphere at that point.

32:30 That's an almost complete interpretation of the two-spinner. That's to say it's this null vector together with the flag plane, the only way in which it's not completely complete. So the flag is really... Sorry? The flag is really along the light cone, it's not out of the light cone. It's within the light cone. Yeah, within the light cone. It's within the light cone. So if you want to see it, you have to chop through, slice through it, and then that, now you can visualize it properly because it's a two-sphere, that point is where the slice-through hits the null vector, and then there's a tangent direction to that sphere. So this flag plane is within the light cone. Yeah, so it goes around, and going around is multiplied by a face. The reason it's not utterly complete is that there's a sign under here. If you go all the way around once, the spinner becomes its negative, and you have to go all the way around again. But there's an extra... Yeah. There's a musical issue about the space. Yeah. Which, of course, spinners have to have that property. Yeah. You can see it very explicitly here. Bear this picture in mind, because it does have a remarkable role in what I want to say later on. It doesn't play much of a role in a lot of what I want to say. Okay, now let's be a bit more explicit about the angular momentum and all that. That's the incidence again, incidence in terms of the two spinners. Is this relation, or I may just write it omega equals i r dot pi, where r is the position vector, pi is the momentum part of the twister, omega is the angular momentum part. under shift of origin the omegas will change very simply the pi's remain alone complex conjugate of a twister you take the complex conjugate of two parts and swap them around so that the pi one now sits first and so this becomes a dual object to that and the condition that I wrote down before explicitly in terms of z's and z-bars is precisely the norm vanishing of the norm of the twister in the set. So the twister has an upper index its complex conjugate has a lower index

35:00 and is therefore a dual twister and therefore you can take the contraction of it with its complex conjugate and that is the vanishing of that is the equation PN I've just written a different notation now what does it have to do with angular momentum if you imagine you have a massless classical massless particle or massless system if you like in special relativity it'll have a four momentum which is null vector and which is future point here It has an angular momentum, which has got six independent components. And this is, from the P and the M together, you can construct the Pauli-Lubansky spin vector, which is origin independent. And the rule is that a self-respecting mass desposible should have the property that the Pauli-Lubansky spin vector is proportional to the momentum. And this factor of proportionality is the helicity. That's the thing I call it, S. Now you see that a massless particle, if you do it in terms of spinors, in terms of tensors, is a bit complicated, because you have to satisfy, okay, the momentum has to be zipped now, it has to be future pointing now, and you have to have the Pauli-Lubansky vector proportional to it. That's a bit of a mouthful, but those conditions are needed. However, suppose you have a p and an m, then you can always write it in terms of an omega according to this expression here. Or the other way around, if you've got an omega and a pi, then you can construct the p. We've had that before. But you also construct the angular momentum by taking the symmetric product of omega with pi bar and pi with omega bar multiply with a couple of epsilons to make it up to a two-index square symmetrical tensor. and then you can add these two bits to make it real and that's your MAB automatically satisfies all these conditions conversely, if you have a P and M satisfying these conditions you can construct an omega and pi out of them, the one freedom being that

37:30 multiplying by a phase so you could multiply omega and pi right the way through by a single e to the i theta this will not change any of these things but anything else will show up with P and M. The only other point I should make is that pi is not allowed to be zero. So if pi is non-zero, then it corresponds uniquely to a P and M, satisfying all the conditions that needs to be. The elicity comes out very nicely as being essentially the norm of a twist-up. I see. So now, now, that's why you were saying that one elicity and the other elicity would be true. So you now get the whole picture. So this is what it's about. But you see, there isn't really a light ray anymore. Because if you just know P and M, the condition for saying where is the light ray, it doesn't work. There is no localized light ray. It's spread out. And I don't think I brought my picture, but the name Twister came from this complicated picture picture of Clifford Parallel's projective, which I won't bring to here because it's going to play no role in what I want to say later. But if you want to understand geometrically what a twister is geometrically, it's this rather complicated structure, which is only in the limit when the S is small, if you like, that it becomes close to being localised as a library. Roger, I met you in Padova we were in a beautiful palace and there was some light coming from a window in one of these Italian palace and you started to say looking at this light you know, geometrically a twister is there was a group of people around and then you said I'll tell you another time oh dear the trouble is I suspect that I didn't bring that picture with me I can show you later on in a week it was the original thing which set me off on this line of thinking because I've come to the motivation behind that but it was thinking about that configuration why do you say that if I have P and N if I have N also I have so I know where

40:00 they're like So what do I don't know? Well, you see, if it was a massive particle, there's a condition you can say, I don't know what it is. There's a condition you say, when is your point on the light ray? Right. And that condition, when it's null, runs into a problem when there's a spin. I see. So you can't actually... I believe it, yeah. Well, if you don't know it will require a relativeistic invariance, then you can do it. but that's not very satisfying locally it's simply not localizable if you characterize your massless particle by P and M I mean you might say well also there's a light ray and I think Singh does it I'm trying to remember because Singh does talk about this in his book on special authority but I think he specifies that there is a light ray out there but if you don't say that if you say okay what's physically important is P and M it doesn't fix a library unless the helicity is zero just go through it and check but it fixes this pristine structure I can give that to you later but not now because I deliberately thinned down this lecture to what I thought was the sort of minimum but I don't think it is ok so this is classically If you like, what was the theory as well? But it seems to me it's incredibly economical. And even if you're not interested in quantum gravity or all sorts of things, if you're interested in massless particles, this is a very canonical, very tidy way of representing a massless particle. People are doing this now, the scattering. But it took people ages to pick up on this. but now something I should mention and that is infinity twisters see everything I said up to this point well that's not quite true because the P's and M's are not conformally invariant but if I want to Simone you missed the 6 I'm sorry? Simone missed the 6 when he was telling us there is a 6 number 6 Below Lambda. Below the cosmological constant. No, not you, Simone, when he was telling us.

42:30 Oh, I see. Sorry, you'll have to explain that to me later. Oh, here. Yeah, that's it. Oh, yes, you have a 6 there. And he missed it. Yeah. Well, if you're not interested in exactly what it is, yes. It's something like Lambda. Yes, no, it is Lambda over 6, yes. But infinity twisters, usually these things would be nought, you see. If there wasn't a cosmological constant, then there would be a zero sitting there. Sorry, that's the six you meant. But if there is a cosmological constant, then the infinity twisters naturally have this form, as I put them here. Lambda is the cosmological constant, and the I's, the infinity twisters, are duels to each other in this portion. And they're also complex conjugates of each other. Yeah, they're complex conjugates and also duels. And the contraction of one with the other is... So if you try and multiply them together, you'll get a lambda coming in. So if lambda is zero, they are singular objects. If lambda is non-zero, they give you actually a complex symplectic structure to twist a space, which will have an important role to play in what I want to say later. So here is this complex symplectic object, and here is the symplectic potential. And if lambda is not, it's degenerate. But now this is different from the usual one that, for instance, gives the representation of the . They're both there, yes. Now you can use them. They both have rows to play, but it's important to Barron not to confuse the two. See, one of them is holomorphic, but needs the infinity twister. The other is not holomorphic, it's real, if you like. But it's conforming Barron. They're both important, but you have to keep them separate. The other is DZ-Coffer. I haven't said it yet. DZ-Coffer. Well, I can say it, yes. It's just that you don't just think that... It's probably worth mentioning right now, actually, because I wasn't quite sure where to put this transparency in my talk.

45:00 But since it's being mentioned here, let me see. Yeah, there is a natural... symplectic structure, which is the, it's familiar, sorry, I'll probably put it, hidden it, here's the other symplectic structure which is a more familiar one probably because you have any they don't have to be non-JD6 it's just the natural symplectic structure that you get when you're p's and x's, dp dx. But if p is now, then that corresponds to the twist of it, which you can write as z dz bar. The symplectic form is this thing here. But I mention it here just so we know they're both there, and they have different blocks to do it. This has nothing to do with the metric. It's purely conformally invariant. It's a real structure. Okay? That's a second. Yeah, you should do this. It's probably worth mentioning the geometrical role of these things. In the case of the lambda equals zero, However, if you want to get the Minkowski metric, here's the quad that we had before. The I gives us a plane, which is tangent to. This is the complexified Minkowski space. This is the twister space. And the I, it's less clear what it means to metric, so it's a linear complex. which is a little confusing there it is but it's tangent to the sphere to the wait, the i is a point?

47:30 the i is a point? yeah, it's a point it's a special yeah, alright geometrically it's telling you where infinity is but it's a point in twist or space? no, it's not no, it's a line in twist or space but it's a point in compactified Minkowski space so here we have the point and this is the light curve the light curve of the point and that's the light curve infinity if you like that's why it's called infinity because it's telling you where infinity is but it also stays in real Minkowski space in what? Oh, yes, this could be a real or, sorry, it could be perfectly good, well, compactified Minkowski space. Right. Yes, it doesn't have to be complex. And it does for this picture. So it is actually, it is the point of infinity, yeah, that's what, yeah, that's right, that's the point. Right. I zero, I plus and I minus all glued together, that's right, yeah. And do you have features for the other pieces, for lambda? Yes, it's very simple. The difference is, it's just moved a little bit. So here we have, this is new. and the contraction is positive for this is a negative for any decision this is this is both this is i i it is complex but real will hold just as well except you might prefer to cut it across the top because i want there i see like infinity being Space Lake I want to see the analog of Scry If you want the Time Lake versus Space Lake Well, there you are It's just a general slice and it might be easier to think of it across here Right, right And to sit a space is normally thought of that way

50:00 the hypodoloid which people often draw for the scissor space, this would be the scribe. Now I'm cutting it horizontally in the picture. Okay. If I want to cut it for anti-de-scissor space, I would have to move my plane in. And then it would be timeline. Okay. Okay. I think that's all I wanted to say there. Yeah, this is worth seeing. See, if I have a complex point, a point in complex Mikrovsky space, there are all lots of different kinds. And these will be lines in projective twister space, and there are essentially six different varieties. It could be entirely in the top half, it could be morphed almost entirely in the top half it just touches here it could be cutting right across it it could touch at the bottom it could be entirely in the bottom or it could be entirely in here now these different cases are understood completely in terms of the imaginary part of the complex vector so you see these are represented by a complex point forget about infinity a complex finite point say And that will have an imaginary part. The imaginary part will be real Lorentz invariant, in the sense its causal structure will be real Poincare invariant. I guess Poincare plus, I'm not explaining myself well. But take the imaginary part of the position vector of that complex point. It could be entirely in the future cone, in which case it's the cast. Normal conventions are sort of backwards, but forget about it. If it's pointing into the future, the line will be entirely in the past of twisted space. If it's pointing entirely in the past, it will be the other way around. In the future, if the imaginary part is null, then you get the touching case. If it's spaced like, you get the cutting across case. you get it lying within the PNKs. This is important for studying wave functions

52:30 because this region where the imaginary part points entirely into the past is what is referred to as the forward tube or sometimes the future tube. And this is useful in quantum field theory because it expresses the condition that you're looking at positive frequency. So if you're doing quantum field theory and you want your wave function to be pose of positive frequency components, then that means that it extends homomorphically into the forward tube. Now, this was a very early motivation of Twister Theory, because I was trying to find an analogue of the picture. I guess I have a... I'm cleaning myself a a lot of orders. I'm trying to find an analogue of the picture. See, I always find it kind of awkward talking about making, if you're going to do quantum field theory, in a context so it isn't just a flat space which might involve general relativity or something and you want to say something is positive frequency to talk about it being composed of Fourier components which are positive frequency is saying something which isn't very friendly it's not friendly towards conformal invariance because conformal invariance conformal transformations mucks it up to see which other components get all messed up. However, the condition of positive frequency is explicable very nicely in terms of the Riemann sphere. You see, if you have a function which is defined for real-time variables, and then you want to say, well, that's a positive frequency function, it's a complex value function set, then it extends either into the top half or the bottom half, depending upon whether this is plus or minus. It depends on your conventions. frequency goes one way and the negative frequency goes the other way and this is a conforming invariant description it doesn't depend on actually doing the fear of Fourier decomposition you're not really interested in the individual Fourier components you're interested in the condition that it should be positive frequency as a whole and that is this very nice geometrical

55:00 description here. And I wanted a picture which did this globally for Minkowski Space. And I'd better come back to that shortly because that was very early motivation for Twister Theory which took about 10 years to be realized. But it was eventually realized. And that actually you see here just the geometry is that the forward 2 corresponds is twisted space. So in a certain sense the forward two corresponds to the top half of twisted space. There's something funny about the hosities and so on, whether it's top or the bottom half but don't worry about them here. It does work out. It's confusing. Anyway, that was an early-man duration. Somehow, twisted space did split things into two halves in this very natural way. One half you can think of as being associated with positive frequency and the other half with negative frequency. That will be made much more explicit shortly. But that's what I'd mention. Okay. Now, that's all, apart from what I was muttering just at the end, that is all to do with classical geometry, classical physics. What about quantum, just a theory? Well, sort of first quantize, just a theory. Well, the natural thing to write down is that the twisters commute among themselves, the conjugate twisters commute among themselves, but the twisters and the conjugate twisters are canonical conjugates as well as being complex conjugates. So now you're back to the conformally invariant symphletic structure that you're quantizing, not the infinity. This is the conformally invariant real complex structure, symphletic structure, that's right. has nothing to do with the infinity twist that's correct and that is here so as I said Z and Z bar are canonical conjugates as well as complex conjugates this is almost naturally comes from the ordinary P and X connotation rules historically it was very curious because I remember

57:30 a little after I was doing these things I think it was a little afterwards Wes and Zomino in their introduction of their version of supersymmetry conformal supersymmetry came across things which were in fact twisters however they had the opposite connotation rules, they had that plus sign and these were meant to be fermionic objects and people kept telling me well you should have a plus sign no no no that wouldn't work it doesn't have any sense I don't care these are fermionic because they're not supposed to be thought of that way ok they must be commutators and not anti-commutators otherwise nothing that I say from here on would make any sense and a lot of the remarkable aspects of tristan theory come about from this so I'm going to not lose it. These things, as I said, canonical conjugates. Moreover, you find that if the commutation rules here, you write these in terms of omegas and pi's, then you find that the P's and the M's, written as exactly the same expressions I had before, satisfy the commutation rules that come about from these things being generators of the Poitier group. so these are exactly the Poincaré computation Poincaré generators computation rules which one can see so the general argument is it looks a bit miraculous here is this a the analogous statement of saying that you can give the generator of SL2C from the is that these are the generators just the M Yeah, it's the Poincare generator, so P and M are generators of the full Poincare group. And the commutation rules for that simply come from these commutation rules. So if you have these, which are much simpler, you get those commutation rules. But maybe your question is at the quantum level. Maybe he wants to know if now there is a Hilbert space on which these acts as unitary operators. The generators of the Poincare. Yeah.

1:00:00 Because that's something that we, in the connection with quantum gravity, we have a slightly different representation of the quantum version of this algebra, and then the m's are act as unitary operators on the infinite dimensional representations of the Lorentz group. Yes. Well, this will be true here, but in a sense which I shall come to shortly. Yeah, I mean, there is a subtlety. But yes, that's right. Let me make another point first, though, which is s, which was the helicity, I should say, first of all, there is no factor-ordering problem here. This comes about because of these symmetry brackets, because omega and pi bar don't actually commute, but the commutator is an epsilon, and therefore that's killed by the symmetry bracket. So that, in fact, there is no ambiguity in factor-ordering problem. However, when you write s, and the s is simply a calculation from the p and the m, exactly the same calculation you find it is not ZZ-bar anymore, but the polarized form is a quarter of ZZ-bar and ZZ-bar-Z. That just comes out, so there's no choice. And here we have the commutators of what I said before, the concreting generating relationships come from the twisting relationships. And in fact, that extends if you have conformal. If these were generated as a conformal group, in fact, the seven sets are more obvious then. Okay? Now, what does this tell us if we want to construct a twister wave function? Well, you see, I like the ordinary p's and x's. You can have a wave function which was a position space or a momentum space. Choose what you like. Here, you can, say, take your z's or take your z-bars. Choose what you like. However, your wave functions shouldn't involve the canonical variables. So what does it mean to say it's a function of z and not a z-bar? Well, that just means df by the z-bar is naught. In other words, it's holomorphic. So you're looking at holomorphic functions of twisters. Now this was, to me, a very striking fact, because, as I said right at the beginning, the magic of complex numbers very much has to do with the magic of holomorphic functions.

1:02:30 So here you have homomorphic things thrown at you automatically. There is a measure on the space, on the spaceship? A measure? To define this kind of product between two of these? You can write down a scalar product, but that's something that needs a lot of... A lot of work. ...carefulness. Yes, but you certainly have to get the product. But there are a lot of subtleties involved in that, which depend on what the actual multiplicity is and it jumps when we have Which makes the generators self-adjoint They're not self-adjoint No, they're not self-adjoint Because they're complex No, no, the generators of the Pankarev Yes, yes, they would be So this will be a unitary representation It is, yes, but the question What space is it that you're talking about? And that needs some subtlety. That's right. Oh yeah. But the other point is, okay, suppose you do this, and then you say, we've got rid of our z bars because they are just d by d sets. Because I'm using the, say, the twist, I could do either way around. I could say, or the dual twister representations. Let's just for arbitrary less choose the twister ones. That means independent Z bar. That means holomorphic in Z. Now you might say, what about helicity eigenstates? Well, the helicity is just this thing I wrote to, Z Z bar plus Z bar Z, contracted. And that is slightly rearranged, this expression here. Z Z bar is the Euler homogeneity operator. so the eigenstates of this will be homogeneous functions so you're looking at functions which are homogeneous and homomorphic and those will be the eigenstates of helicity and the relationship between the helicity degree of homogeneity and the helicity the helicity is s and so 2s over h bar is this number n which should be an integer and the degree of homogeneity is minus 2 minus n If you look at the dual twisted representation,

1:05:00 I use Z bars, it's better if I rewrite it. It's W, so you don't want to see the bars. So holomorphic in Z bar means, or anti-holomorphic in Z, it's holomorphic in W. So you're now looking at holomorphic in W with a dual twisted representation, and then the helicity is minus 2 plus N, where N is the whole genesis of W. Okay. So this is all very formal. It doesn't relate to anything we might know already to do with space-time descriptions. So let me relate it. What do you do if you have a spatial description, a space-time description, I should say? Then you have, well, these are things familiar to me for a long time. Arbitrary helicity, which could be positive helicity or negative helicity. and the way it works out is the negative is the ones represented by symmetric spinners with n indices n being twice the minus twice the spin as the convention goes the field equation is this one here if it's 0 it's just the wave equation if it's positive velocity it's the same thing but with prime indices instead and if you see they have positive frequency into wave functions and the relationship should be examples given here scalar wave you have homogeneity of degree minus 2 as your twister function and that's got to be related to phi I'll tell you what's related in a moment the iraq vial massless massless neutrino equations, minus 1 or minus 3 depending on whether it's left-handed or right-handed. Maximal equations, they have a right-handed or left-handed part, and these are these. If you want to make it real, then you have to be one's complete quantity of the other. If it's a wave function, they have to be both positive frequencies. And in the case of spin 2, minus 2 or plus 2 for the helicity. You have the same thing with a couple of epsilons here, multiplying up your field, your oscillos, and you get a thing with vile tensor symmetries. So this is the linearized Einstein equations.

1:07:30 This could be your graviton linearized graviton equation. and the homogeneity degrees are you see it's curiously lopsided homogeneity degrees plus 2 or minus 6, this is the left handed gravaton, plus 2 the right handed one minus 6 of course if you use the opposite twisted invention, this would be the other way around but you have to settle on one or the other and whichever you settle on you get a horrendous This will have a key role in what I want to say, which is like in the case of Maxwell-Foughton as well. What is the difference that you're emphasizing? I'm emphasizing the fact that this is minus 6 and that's plus 2. But in the previous case it's 0 and minus 4, so in the Foughton case. That's almost as bad, yes. It's just as bad. Okay, so there's nothing special about the graviton. it's just that the discrepancies is involved so these functions can be identified as the the space of the of a particle a one particle in the space yes, I'm looking at a one particle in the space I mean it seemed to me I had to understand one particle before I could understand lots of particles whether that's right or wrong and that's what I felt. But even for one particle, it is lopsided. But you see, Twister Theory is inherently lopsided. It's got a chiral character to it, right from the start. But I mean, I used to think, okay, well, that's revealing something chiral about physics, because we know it is chiral in some strange ways which we don't understand fully. And maybe this is sort of bringing it up to the issue. But these issues have importance, for what I want to say later. Very much so. But I'm just saying this. You have to. What I'll say in a minute, this lopsidedness doesn't trouble us at all. But what will I say? Okay. I want to say what the relationship is between the twister function of homogeneity degree, whatever it happens to be,

1:10:00 and these fields. And what's the relationship? It's a complement. to understand what we're doing let's concentrate first on the wave equation so you have and you have a homogeneity minus 2 what do you do with it well you multiply it by a one form which happens to have to be plus 2 this makes the whole thing degree zero and then you put in the incidence relation here, which see, you're going to integrate out the pi dependence in this interval, that's the whole point. I'm integrating out the pi dependence and I get rid of the omega dependence by putting the incidence relation in and that gives us an x dependence. When I integrate out the pi dependence, I get an x dependence. Now you're using the global structure with the infinity twister as a sort of boundary condition for the real equation. i've written it this way the first way i wrote it down i didn't do it that way but i won't give you that because it looks more confusing it's not actually dependent on the infinity twisting i see because these uh these integration measures that you wrote down looks like the one you're right you're absolutely right it is the same thing it's the same thing but you need to think a bit about. It is actually conformally invariant. this way, right? You just box and not plus R. Yeah, you see, I'm doing two things which are not invariant. The other is this. The incident relation, which I slapped down for you, is actually if you're looking at conformal invariance it's a confusing thing because, as you can see immediately, because if x goes off to infinity what can I not? So it's not The actual representation, the actual formula I'm writing down here for convenience is using a non-conformally invariant description. I could use a conformally invariant description, which is what I initially had, but it would confuse you more, I think. Maybe. It depends what's confusing me now. but let me just say that it's not actually dependent

1:12:30 non-conformal invariant well actually these things are conformally invariant let me not go into that here it is actually conformally if you write it right but it's not so easy to write down see here you actually have the x's nicely presented to you whereas if you don't If you wanted to make this nicely conformally very low way through, it's kind of a mess. You could do it. But you know, to ask was the fact that box phi, like this, is not conformally balanced. That's why I was expecting in finding this. Oh, it is? Well, it was not plus r over 6. Oh, I see. Yeah, well, that's partly to do with the way I'm writing it. Well, this is in flat space. Oh, it is r over 6, yes. It's in there. If you actually conformally transform, yes, yes. But then you see, I'm not doing that. what you're saying is completely right it's just that my formula as written is choosing a particular conformal scale and making the Meccleston space flat to do it otherwise, which was more manifestly conformally invariant I'm not even sure I see where the R over 6 comes from but it does come out, yeah because that's what you get when you conform and transform that's right but then let me talk about other things because there's the subtleties about this to do a corner integral you've got to have singularities otherwise it's going to be trivial in fact you're going to have singularities because it's got minus 2 homogeneity degree now here's a typical case this is what we call an elementary state it's a very very special twist of function which happens to be of two linear, of a product of two linear forms. That means the singularities lie on two planes in twister space. I'm going to insist that they intersect in a line lying in the past in Pt minus. That means they're separated. The singularities are separated in Pt plus. Any line which lies in Pt plus, in other words representing a point in the forward tube, will have separated singularities and therefore there's a well-defined And that contour interval would be completely continuous as I move this line around any old how, even right as it goes into the ET. In fact, a little bit below it, I'd still be all right,

1:15:00 but I mustn't get close to that line. Now, I could have more general situations. I could add these things together and smear these things around until I get big regions. Still work, provided these big regions are separated. and then I will have a counter interval which is well defined the other point I should make is that it works also for different helicities if it's a positive helicity that means that I've got some indices here with primes on them I supply the same number of pies to give me the same number of indices and then that brings up the homogeneity with the one form here to homogeneated zero, this is now a genuine counter-integral and it works, and it automatically satisfies the field which is not hard to see here. If I want to do the negative velocity case, I put in d by the omegas, curiously related to these by the quantization rule incidentally, which I may point out. If I had pi bars, those are really d by the omegas. I should give credit to Lane Houston who first pointed out this form of expression here. When he did at first I said well what's the point, we've already got these ones but then only realising later that this was an important thing to bring these holisties into coherence with each other. we can do both and the minus the lopsidedness in the zero to minus four if you like doesn't worry anybody because you have two d by the omega's or two pi's not the size of this disappears that's fine let me just go and say what are we doing here see I worried about this for a long time I was very pleased in some respects and very worried in other respects see it's got to do with the top half of twisted space and that's giving us positive frequency automatically but it's got this funny camel hump sticking up into the top and if I add lots of things with the camel hump in different places, it's going to smear up the whole place and how do I get a nice space out of that which is Lorentz invariant or even rotation invariant because these camel humps will rotate as I rotate my frame and if I had to get the ones pointing in different directions it's going to ruin everything so I got a little puzzled by all this for a long time

1:17:30 and I have to say that it was talks with Michael Atiyah that were rather important in clarifying what was going on had to do with other things which I'll mention in a minute more specifically. This was a key realization. Okay, those are the camel hops. The function is non-singular avoiding the two camel hops. Whereas the region I'm interested in is the whole of the top half of the twister space. And the thing is, the way you think about this is that the twister function is defined on the intersection of two open sets whose union is the whole region we're interested in. And this is a little funny restate. So you can go around. Sorry? So you can go around. You can then go around if you know what you're doing. See, more generally, we may require several open sets to cover the region of interest. And if you start going around, you have to chop them up and so on. And this is the first co-homology element. was thinking about. What we're really doing is co-homology although I had learned about chief co-homology when I was a graduate student, I think I don't think I ever understood it I'm not sure. Probably held up Twister Theory or didn't I, I'm not sure Anyway, Michael, the tier came in usefully at the right moment to remind me of these things Suppose we've got the covering of the top half of Twister's Place with a whole lot of open sets, then the line will intersect these over the sets and regions and you can do your counter-intervals with bits and pieces all stuck together. This still works. The real thing you're doing is check cohomology these whole method functions. You've got a whole collection of f's instead of a single f. This is your twister function. The whole collection of f's which have the property that they have a triple relationship like this It's better to do this all very abstractly and look at refinements and so on and so forth. Stein covers. It all gets a bit fancy and then you end up with this thing being really

1:20:00 first cohomology of the top half of twister space with holomorphic functions of particular generative. And that's what we're really doing. So when... So the holes, the cohomal things, are where there are similarities in the function on the intersection. It's the overlap. See, the functions are always defined on the overlaps the open sets. And here, what is the overlap? Well, you see you've got two open sets, which each one removes one camel hump. And then the overlap, which is where both camel humps are removed. This part is clear to me. What are the regions where the function is not defined? It's just where they're... You just want to make sure that the union of all those regions covers the whole space. are defined on the overlaps between them. Right. In the previous term, I see the lack of definition of the function is just where there are the points, where the singularity is. So you don't care where your open set gives out, you see. Sure. If one of them gives out or the other one gives out, then it's okay. Sure. It depends how you... This is an area... That's in a way. This is an area where the mathematics was sitting there waiting to be used. So we were lucky. I'm not sure that's the case in what I want to say later. Maybe it is sitting out there somewhere, but I haven't found out yet. Whereas here, I didn't understand this stuff, exactly what was going on, but it was when I had these discussions with Michael and our group did, with chromology and so on and it became clear that this is what we were doing and that's see if you take a rotation which moves the camohofs around it doesn't change the chromology because the chromology doesn't care about which particular way you have of putting it into let me give you my picture which I always give at this stage tend to, which I think is rather useful

1:22:30 you see, I was once, a long time ago being interviewed by somebody trying to make a TV program and they wanted to know about Twister theory and I started telling them stuff and they said, oh yes, yes, yes, I'm not sure and then stupidly I let slip, he says, well really if you want to describe a field, it's cohomology, he said, what cohomology, what's that no, no, no, no, no no way I could possibly explain that no, no, no, we'd really like to, no, no, there's no way So I went home, and then I thought, ah, here it is. Here is a really, an honest representation of colorimology. You see, here we have something which is locally, you could make it out of wood, and you could imagine the pieces made out of wood. They never actually used this in the program, I may say. but ever since I've rather hoped that some TV program would make me you see here, what is that well that's, suppose I was going to make that thing out of wood, you see well I'd have bits and I have an instruction book which tells me ok, I glue that piece to there yes ok, and then I glue that piece to there and then I glue that piece to there and then that's my thing whoops but what I would do, you see if I was conscientious, I would say I've got this instruction, I've got that instruction and that instruction, and then work out the cohomology element. If it's zero, fine, it fits. If it's non-zero, it's a measure of the impossibility of that object. And that's completely honest. That is a... You can actually represent the impossibility of these structures as a cohomology element. So it's a precise measure of the impossibility. but it's very nice because it tells you there is a non-locality you can't put your finger on it anywhere it's no word specifically it's just a feature of the thing as a whole so here each single f of z defines a color model, right? no, yes the f of z is the whole thing the fijs are just the gluing structures there 1, 2, and f2, 3, and 3 3.1, you see. Whereas then you work out the cohomology element and you find it doesn't vanish to this object. So you're stuck. You can't make it.

1:25:00 And how bad it is, how far off it is, is given by the measure, that cohomology measure. So the cohomology element is somehow representing an inconsistency in trying to... Well, in the case of twisted space, you're trying somehow to put a homomorphic function on the whole space, in a sense. And you're not really interested in that function, you're interested in what's wrong with it. And that is... See, I like to think, although I've never been able to explore this very far, I would like to think that this is... See, right early on, I gave you some motivation for Twister Theory, non-local. And in those early days, I was aware of the EPR problems, the Einstein-Kadolsky arose, and this was before John Bell, and so the fact how this is a real problem, if you want a local description of quantum phenomena, it seemed to me that to have something which was non-local was important. If it's really going to represent some quantum structure to space-time, it's got to be non-local. Now, the non-locality that you see here is a one-particle non-locality. which is still there because you say take the photons from this thing and give you the screen suppose it was just a single photon and suppose it makes a black mark when it hits the screen and suppose it makes a black mark here just one photon then that you could say that point there has got to say okay too late I've seen it you see so the ones sitting down here are not allowed to see it because this fellow has seen it but you can't have a classical description of that because that would mean him sending a signal of this film down there faster than the speed of light so you have non-localities even with single particles and Einstein worried about these things way early on but it doesn't have the full force of the bell the non-locality which really tells you you can have a model of it and here you can have a model of these things but nevertheless it's a kind of non-locality which is going on even for single particles of particles, which is what you have for the EPR, then you need to have, if you've got

1:27:30 two particles, then it's actually second column, this is first column, where you're looking at open sets and the check description is open sets and pairs of open sets is where your functions are the second cohomology you have triples of open sets third cohomology will be quadruples of open sets and it's more complicated than how to visualize what's going on i used to set this as a challenge to my group when i was early in oxford and say well look here is a representation of first cohomology, find me a representation of second cohomology, which is nice and easy to describe. So far, nobody has succeeded in doing that. At least, I haven't seen a good description. So, second cohomology is harder to understand, but nevertheless, mathematically, you can say what it is. And for pairs of particles, it's the second cohomology. So, these things that Bell inequalities are to do with seem to be second cohomology objects. But that would be really nice explore further. All I can say is that the non-locality that was forced upon us in the twisted descriptions of master's particles well, it's a cohomology, I think. And you see that in the twisted theory, and it's also, in a certain sense, there in the physics, but I don't know how to carry that much further than just saying that. I mean, I feel it has got to be related to gravity, but that's another issue, except that I do want to say it here. There's a reason why cohomology is really rather important to a theory, and this comes about where you want to what you may call non-linearize it. The case I'm most concerned with is the gravitational And you can think of functions defined on pairs of sets. It's very much like what you have when you want to build a manifold. So if you want to build a manifold, you have transition functions or coordinate patches, and they've got to have satisfier relational triple overlaps. And this is, well, if you want to start to build a manifold,

1:30:00 In a sense, suppose you've got one where these things are just glued together and they make a flat space, and then you can put a cohomology element on, which I'm saying is now a vector field. Suppose I have a vector field on each of these things, which the triple overlaps, they satisfy a consistency condition, and I can just paint those vector fields on. Suppose I think of this paint as drying, and when the paint dries, it warps the space a bit, the space. It moves one space with respect to the other along the vector field. So it slivers, slides them along. And does that end up making you a non-trivial space? Well, it's the if it's an infinitesimal change, that is first cohomology. Another example, which is first cohomology, is if I'm trying to build a bundle. So I suppose I have a space down here, and above each one of these open sets is a trivial the bundle to the next piece or to the next piece I slide them along consistently with the group structure and I get non it's got to be consistent with the triple overlaps and I may get a non-trivial bundle this way. And that is a cohomology. That's just what it is. Now you see this was a thing which I call the non-linear graviton construction which is really what I want to talk about this was in when it was 19 very fast after that Richard Ward just a year later saw how to do a similar thing for photon, well for the electromagnetic field where you're concerned with how it interacts with charged fields or more interestingly if you like with the Yang-Mills fields where you have non-linear construction and this is I don't want to talk about it in detail here, but I should just mention it, because it had considerable importance in pure mathematical interest in Twister theory. So that was the Ward construction. But I want to be more concerned with the gravitational one, which came just a bit before the other. And that, I can describe it here, I think. applications to general relativity

1:32:30 well there are many applications in specific cases which I don't want to talk too much about but what I do want to talk about is what I call the non-linear graviton construction it's going to be local in the sense of locally in the space-time which means in twister space locally with respect to line so this could be standard projected twist of space here we have a line representing a point in cost of space a complex in the cost of space i'm talking locally so i'm looking at the tubular neighborhood at that point of that line should say here it is i've considered that as an overlap of two open sets which i then slide one respect to the other so i'm sort of exponentiating this specter field, which tells me how to move one to the other. I glue it down. That breaks my line, so that wasn't in use anymore. I use theorems. This is where my Clotier came in very useful. Theorems of Kodira, which say if this deformation is not too far, I mean it's a finite deformation, but I don't want to go too far in some technical sense, then these lines will still exist. So you can prove that although they're not straight in this since they looked over here they will still exist not only will they still exist but they form a four parameter family so that this family of lines if I represent those lines by points in a space that space will be four dimensional this is all complex I should say so this is a complex four dimensional space and now I can say I can decree that these things are null separated if these lines intersect if you remember the Dikowsky space of the lines corresponding to the null separation in the Encosca space. So you're posing a causal structure on this four-dimensional space. Yes. I guess causal is a little strong because it's complex, so it's not really a linear structure, but it's a conformal structure. I have a notion of what null separation is. Yeah, if it was real, it would give me a causal structure. But there are problems which I'll come to about making it real, what you find is that this gives you a general anti-self-dual complex

1:35:00 space-time in quotes so it's a complex space-time this anti-self-dual that means that the vial curvature is anti-self-dual the starting point what do you mean technically by sliding the angle well I can cut I can take Minkowski space Linkovsky space, I can have its twisted space, that's an complex projected 3 space, I can take a line in that, I can take a tubular neighborhood of that. Then I can say, this is the same tubular neighborhood. Now, that's a Riemann sphere. Now that Riemann sphere, I can think of as made up out of two hemispheres which overlap around an annulus. No, you slide one with the two together. And this is one, this dotted line here, this part of it represents one hemisphere, and that's the other hemisphere, and this is the overlap between the two. Then I slide where the annual region of those two hemispheres overlap, and I slide kind of a follow-up. This slide is a, it's a map from the twist of space to itself? And now you're considering generic maps, generic deformations? They have to preserve... Let me do a little bit more generally in this picture, because this picture is the case of Ritchie Flatness. If I say I want a cosmological consulate, in fact, it doesn't matter, either a cosmological consulate or not, but I do have an infinity twister. and I want to preserve that. So I'm going to slide one piece against the other, but preserving not just the complex structure. So I want to preserve the complex structure, and I want to preserve the infinity twist. I guess what I don't understand is what happened with the rest of twist of space when it was locally existing here. Well, it may... I don't say anything about it. But it's only going to give me a local thing here. Yeah, maybe this is the point. You see, it's not going to be a global space-time here. It's going to run into singularities. But locally, the space-time, in quotes,

1:37:30 I'm looking at the neighborhood of a point. Okay. And the neighborhood of that point will have a curvature. It will have a Riemannian curvature, actually. I see. So now, okay, correct. So you're forgetting what the rest. You're taking a region of space-time, and now there's a way to deform it into a general Ritchie-flat or whatever, by looking at the tube in crystal space. I didn't catch it like this. I could just say deform that tube in such a way that the complex analytic structure is preserved and the infinity to it is preserved. Then I can find, as long as the deformation is not too big, I can find a four-parameter family of these lines. Those lines naturally have a complex remaining structure, automatically Ritchie-flat. Ritchie-lambda, what you can call it, Einstein, Einstein's base. But what you can see is that the tube corresponds to regions. That's right. The tube corresponds, that's right. The tubular neighborhood of a line, I shouldn't have made that clear. It corresponds to a little ordinary neighborhood of a point. Why is it which is flat? Why is it that you stick this in? Where does it go in? I think you said something, but I missed it. No, I probably didn't say it. See, where I did it first was when the case... See, I did it first with the case where the infinity twist was singular. And then it corresponds to a projection. 0. When lambda is 0 then you, roughly speaking preserves the pi space and changes the omega space. So that's what it does. The pi space is where you project out the pi's. So this projection is saying, keep the pi's and do something with the omegas. You do something non-trivial with the omegas but you keep the pi's. And that's the same as saying, keep the infinity twister in the singular case where there is a zero case. And that's what I did originally. Somewhat later, I guess... So, infinity twister, I think of something at infinity, but asymptotic flatness or something like that. But you reach it with zero at the local... It's giving the metric everywhere. It may be sitting at infinity, but that's where the metric goes wrong, if you like.

1:40:00 I could write down an expression using the infinity twister for the distance between two points. and that expression uses essentially the infinity twist it's quite similar and that metric you get would automatically be Ritchie Ritchie flat if you use the flat well you don't do it that way it's only a few test you can write down the metric, it's slightly involved but yes you directly get the metric from the fact that preserved, and it gives you a metric. And that metric not only satisfies Ritchie's practice, it's the general solution. It's the general solution for anti-self dual, which of course is not much good if you're interested in real space-times, because anti-self dual means that the viral curvature is anti-self dual. If it's real, then the viral curvature is equal to its anti-self dual part is complex conjugalness and itself to your part, therefore the RK, which is zero and therefore if it's Ritchie flat and Vial flat, you're a bit stuck with flat space I was trying to understand because everything before was in Twister definition of Twister is very much attached to Minkowski's space-time but there is something natural about Ritchie flatness in this, even if it's something natural with Ritchie flatness, Einstein's equations Yes. I'll come to that from a different angle, if you'd like, later on. Here, you can see it in this picture, because it means that the pi space is out there. It's integrable. The pi space is global. And if you write down the commutation relations, you have commutation of derivatives, and if the pi space is global, it very quickly gives you its anti-self-deal. But also Ritchie flat. It's richly flat and anti-self-deal, but in a sense, it's a little too trivial. Well, this isn't trivial, but it's a little too trivial that you get richly flatness, because it's an integral thing. Maybe the pi space is global. And that is basically... You see, the alpha-brins are pretty green. Yeah. It's... But the space of these solutions is big, so what are the...

1:42:30 If you start from one of your signals, what are the kind of things you can do? How do you get to all the possible nearby solutions? What is the likelihood of things you can do? I'm not sure. I should have had that transparency here. I'm anticipating you might have asked questions like that. The board. I can do it on the board. See, if you have a twister function which is homogeneous of degree plus 2, and that is for the helicity minus 2, I guess. Then you can consider d by dz and d by dz here. Oh, f is the freedom now. Sorry? The function f is arbitrary now. Yes, it's a function of f, which is homogeneous. So that should correspond to a general linearized field. and I just took two patches and it was getting a general linearized field but it's also given a general non-linear field by this construction with as much freedom as you get in the FC now I see so it's ok, that was a non-linear graviton it seems to me, ok, you can think of your gravitons How do you interpret this? See, I was saying, well, I should say that the way it came about originally was by a very roundabout route involving Ted Newman. He was looking at these things he called H-spaces, which had to do with making the sheer complex and so on. I don't want to go into it here. But it became clear that that construction, once we got our conventional story, which took us probably years, when we had our convention then we would separate and we came back again then it's flipped so we kept misunderstanding each other because it looked at all the spaces you constructed the flat

1:45:00 because they were self-dual by my conventions and anti-self-dual by his conventions does this have to do with the fact that the the self-dual curvature of the anti-self-dual is zero. That's right. So these are... It's kind of striking that you get all the complete set of solutions. Of course, it's not so easy to write it down, if you want to write down ds equals, ds squared equals, because you want to know where these lines are. And that's non-trivial. In certain cases, people have done it. Oh, I see. So you know that they exist, but they have the theory. That's right. But how you match them. You should find them. It's going to be a mess. And in simple cases, yeah, a lot of people did. Paul Tom, and Richard Ward, and various people found big classes of these solutions. But I would say it's closer to being allowed to say you have an explicit formula. nevertheless you want to write down ds squared equals it's not so easy okay so that's that's the story up to about almost 40 years now here is the big stumbling block what I call the googly problem now you see probably there aren't too many cricketers here But googly is, people who belong to the old British Empire, most of them know what a googly is. It's a cricket ball which is bowled as though, looks as though it spins left-handed, whereas it actually spins right-handed. It's a clever thing, and there's a Pakistani who was very good at doing this, and you couldn't tell if the ball was going to spin the other way, you see. so it's even apt terminology because not only is this mole of photon or the graviton right handed but it's also hard to do so that was the idea so you use is there a formula you have an f plus minus 6

1:47:30 which you can put that into this formula it doesn't make much sense it spoils homogeneity and what on earth should you be doing so this is what I call the googly problem I forget what it was my term or Richard Ward so I think I originally suggested the term that he picked up so the googly problem has been with us as I say we're getting on for 40 years and I've always regarded that as the major problem not just for gravitation because remember the ward construction for the self-dual fields if you want to express interactions of any kind in physics you're pretty well stuck because you can get the left-handed pass but not the right-handed pass I've tried all sorts of ways most of them involve going off to infinity in one way or another and not terribly satisfying very fine however and this is what palatial twist of theory is about and I should explain the name and it's slightly embarrassing but let me do it anyway I'd been worrying about a year and a half ago I guess, a bit more than that I was really worrying about this because everything you do seems to force the thing to be anti-self-dual and what I tend to do is go up to infinity and then things come nicer do constructions there. I had a very what I thought was a really nice answer at least in principle it required the cosmological constant to be zero so when it turned out not to be zero I fought against that for a while but then I got persuaded it's probably not zero. And I couldn't make that thing work. Non-zero cosmological constant. But it was a bit over a year and a half ago and there are these occasions when people get together at the Back in the palace, you see, that's what the name comes from. Has the Queen told you? Sorry? Has the Queen told you how to do it? Not exactly. But as we were driving up there, I thought the Vanessa said, I've been having this problem, it's been around my head. I wonder whether Michael Atiyah will be there. You see. Well, I had a good reason for believing he might be there. So we went up to this occasion,

1:50:00 back in the palace, and we went to all these very formal things. and I saw that Michael sitting there with his wife and so I went up to him and this was in a big room with his portraits of Gainsboroughs and I think there's a Rembrandt there and all sorts of very marvellous pictures and there was Michael sitting down there and so I went up to him and I said well look, do you think it's alright if I ask you a mathematical question he says well I'm not sure about that so I asked him this question which is more or less described here it had become clear to me that the problem was patching these spaces together here if you just knew somehow the holomorphic functions but not spaces is there something you could do so I said well is there something you could do if you don't say the sheath of holomorphic functions so he said no that won't work because you just take the prime ideals or something well I sort of knew that but I really didn't want to face up to it But then he paused and said, well, that's not true if this is a non-competitive field. So I thought, my God. You've got the z's and d by the z's, which is a non-competitive field. Oh, in the quantum case. Yeah. And he said then, it's a non-competitive geometry or something like that, and this doesn't hold. it's trivial, it doesn't hold any more. And so I began thinking that it was really relieving, gave him a lift to where he wanted to be, and so I said, yeah, I think we have a non-competitive twistist algebra already waiting. See, what I rather liked about this, in a way, is you've got your zeds, and the power of twistist theory comes about because you're looking at polymorphic things. The Kodara theorem and everything works because you're looking at complex analytic structures. once you bring your z-bars in it wrecks the whole thing but if it's not z-bars but d by dz that's holomorphic so you've got holomorphic structure it's not an algebra which is the ordinary sense of sheaves of algebras but maybe it works in some sense so these things are called palatial twisters for the slightly embarrassing reason

1:52:30 that was where this thought came from and Michael was always I probably wouldn't have done this he was always very insistent so if you say anything about it you must give credit to the Queen I wasn't quite sure that she'd done that she did provide the ambiance at least that he was very insistent that the Queen should get some kind of credit for it so she can get credit for the name so this is the sort of idea it's just a bit vague here You don't really have the space down here, or unless you do it to non-competitive geometry in some sense. You've got the Heisenberg algebra of these z's and z-bars, and you've got them here, and then you patch these things together. And, okay, as special cases, you'll get both the self-do and the anti-self-do case, because you can put it like this. You see, the idea is it's a very quantum mechanical point of view. This algebra is sitting there, and as you often say in quantum mechanics, Well, okay, the z's would be one example of a complete set of commuting variables. Or the z bars, in other words, the d by z would be another example. So in a certain sense, you've got those things sitting in this general framework. So somehow you've got these complete sets of commuting variables, and they won't be necessarily just z's or z bars, but there'll be some combinations of these things, and that's the idea. And you patch them together and you try to extract the space-time line of it. Well, that's the idea. There are various things which have come from this later on, which I think are somewhat promising. Now, I can't say I know exactly how to do it, for all sorts of reasons. But there are nice suggestive features to it. At first, I didn't see how on earth you're going to express the Einstein equations, how on earth are you going to say what you mean by all this and things like that. But one important input, it seems to me, is the following. See, it's pretty vague what you're doing, because if you've got this algebra, I don't even know what the local means. I've worried about this before, because

1:55:00 it's easy enough if you've got a space and you know what local means but you've got little neighborhoods here and you might be looking at the functions to find all these little neighborhoods but if you've just got the functions and some of these things are d by dz and well for a sheaf you see in an ordinary sense you say well you have something which is holomorphic in some neighborhood and that usually means well it means you've got a power series which is convergent in some neighborhood But what do they mean by saying we've got a power series in d by dz which is convergent in some ways? I don't even know because it's also not clear whether that's what you want because things that do converge nicely still have funny properties like the exponential series, you see. Exponential function converge really nicely but then if that was d by dz is that really what you want? Because exponentials move things. If you say each of the d by dz moves along, and if your point is in some set, then it moves it to something which is outside the set. I get baffled. I'm only just saying things I don't really know what I'm talking about, you see. I'm saying I don't know exactly what one does on the analytic sense. Maybe there are probably people out there who do know. Whether they know the right things for this, I'm not sure. They know lots of things, and they tell you all the things that they know, and you're never quite sure that those are what you want. this is the idea of geometric quantization now as we saw before we have the symplectic structure on the light ray space now light ray space so long as I give a scale to the light rays I had a transparency which I probably It was the symplectic structure which is always there, but which is related to the quantization procedure, but which is not the one that's involved in the infinity crystal. never mind it is here it's the bottom

1:57:30 oh, yes, it's this no, it's not the bottom, it's the bottom it's here, it's this one what I mean to say is that if you have a space-time you've got its light rays let's suppose it's globally hyperbolic so that these light rays don't come out and meet each other in complicated ways and they are scaled by a momentum scaling. That gives me a six-dimensional, the library space is five-dimensional the momentum scaling gets a six-dimensional space, it's a symplectic manifold, and then if you've got a symplectic manifold, there is this procedure of geometric quantization which says, what do you do? First thing you do is you construct a circle bundle over the space and then you construct a connection that circle bundle. Now that circle bundle, I said it was really nice that you have this flag plane thing here, because that is a circle bundle. This is already there. You don't have to invent this. You see normal geometric quantization, you've got the symplectic manifold, and you have to conjure up this circle bundle, and you do that just saying, there shall be a circle bundle. But what I'm saying is, you don't say there shall be, it's really here already, because you've got the face of this flag plane. So you've got the seven-dimensional circle bundle, and then geometric quantization procedure gives you a connection on this space. And this connection is something which, if these were p's and x's in the ordinary syntactic kind of structure, it gives you something where the p's and x's satisfy the canonical computation rules. So the idea here is that you get if you've got z's you get z's and z-bars which will satisfy the canonical connotation rules so that's the sort of idea now yes here is the yeah that's right that this is the connected this is the symplectic structure already now you've got a symplectic potential which is this thing I might have called phi I think it is z dz bar which is equal to minus z bar dz if this is flat twisted space because these twisted sets

2:00:00 by that condition one of these has got a bar never mind so you already have your symplectic potential sitting there anyway I spent quite a little bit of time what the connection that you have to invent in the general procedure of geometric quantization or I think it's geometric pre-quantization with the general procedure you have to construct these canonical variables which have a canonical commutation relation between each other you just can see geometrically what's going on I don't think this is all very clear here but nevertheless I did convince myself that I knew this connection was geometrically and basically it's the following suppose you've got light ray space and I can think of this as just ordinary flat space for the moment and I've got not just flat space, I'll take a moment in time that means I've got each light ray has got a position and it's going somewhere it's got a direction now if I look at the ones that are neighboring to that there are those which i call abreast those are the ones where this one form vanishes and the ones which are non-abreast are sort of ahead of it or behind it and the head of it or behind it comes what that one form is doing and the two form which is comes from taking the of that has to do with the curl of these things so you can sort of understand geometrically what's going on. And the, if you want to construct the connection that you get from here, it's basically, you use this one form as the, I'm not sure whether it's this, it's this one here. Yes, you use that one form and you make the connection out of that one form. So you've let me not go into the details partly because I don't quite know what I'm doing and I can see that there is that kind of freedom I can understand geometrically what this connection is how much it helps me I'm not really sure I'm going to give up very soon because the point comes where I need a bit more

2:02:30 help to know what I'm doing speaking I'm doing what this picture says and I am roughly speaking saying that I need to know this connection which has to be basically what d by dz means initially and so if I if I try to match the light ray space here, which is what I will try to do the light ray space is at least geometrically clear what that means but the light ray space since I have the geometric quantization I've got the roughly speaking what d by d z means geometrically I think I'm going to just throw you slightly disconnected thoughts here is the other part of it which needs to be tied in with it there is a thing called local twisted transport which has been around for quite a while it's an instance of If I think of omega and pi locally, so I've got a point in a space-time, okay, now I'm back to an ordinary real Lorentzian space-time, four dimensions, at each point I choose an omega and a pi. Now, first of all, I can change a metric that I have on that space by a conformal factor. this changes the epsilons by the square root of the factor that we have here and the omegas and pi's if I change my metric change according to this where this epsilon thing is simply the root of the log of the conformal factor so this is conformally invariant notion and then I have a transport if I have a curve in space time I can move these omegas and pi's around by means of this of transport. Now, if the space-time is conformally flat, I get my old twister space. This is integrable, I go around a loop, I'm back to where I was before, and the twisters are defined below me. If it's not conformally flat, then when I go around a loop, I may get the thing not to agree exactly as I go around the loop. Now, there's a remarkable thing which I've known for a long time and I can never think of anything to use it for. But the idea here is You see, suppose this curve is a null geodesic.

2:05:00 Then along that null geodesic, I will have a flat twister space. So for every null geodesic in the spacetime, not only do I have a null geodesic with its pi associated with it, but I've got an entire flat twister space there. As I move that null geodesic around, I'll have other flat twister spaces. and somehow that flat twister space will be where I know what d by dz means. So I can not only carry up omega pi along here, but I can carry d by d pi and d by d omega along here. And it's the same as how I would carry omega bar and pi bar along here. So it really, as far as more single light rays concerned, this quantization thing makes sense. now here's the point that I knew before but I didn't know what to do with it that is if I have an infinity twister I don't think I only knew it when it was a zero cosmological constant but it can have a cosmological constant if you like if I have an infinity twister in the local twister space which is defined by the expressions I had before in the twister space then a necessary and sufficient condition that that shall be globally defined is the vacuum equations. Now, so the vacuum equations, if this is a lambda, if the infinity twister has a lambda in it, then this is Einstein vacuum equations with that lambda. If lambda is zero, this is now a singular infinity twister and is still constant. So the condition that I have The Einstein equations in this framework is that I have a constant infinity twister. And that is the vacuum equations. It's sort of strange how simple it is, because I was wondering quite a long time how on Earth, not only would you make this work to construct any kind of space-time, how will you express the Einstein equations? Well, it's that in this algebra that you have here, you have an infinity twister, which is globally defined in the algebra. And if that matches from side to side, that will give you the vacuum equations. Now, I think that's correct. The only thing is I'm not quite sure what I'm doing. So what does it mean to say

2:07:30 that the algebra is somehow defined? I don't know if I'm pointing at the wrong point. how do you know that that algebra is see what I think you do is you do know for a real space time, you've got your light ray space and that light ray space because of the geometric quantization stuff does at least know what d by dz means as long as you don't go too far into the see if you're just staying along pn roughly speaking but you're not moving part of twister space where you more than say one derivative into that that's free so the idea is that you keep that free and you join these together in such a way that it matches and the idea would be that if it globally matches all the way around then you have something like a Kadara theorem which tells you the four parameter family of things where it's consistent all the way around So that's the sort of picture we have. Now, as I say, I don't know how to make it work in detail, but the general idea is, roughly speaking, that we need a generalization of these Skadori theorems, which is, I was hoping Michael Atier would give me, but I didn't quite know exactly what I wanted at that stage. I'm still not quite sure I know what I wanted. But the idea is that it has to be the algebra that is defined by the lightweight space by the geometric quantization rule and then you forget about where the lightweight space is you've just got your Twister algebra and then you match that Twister algebra all the way around and if that matches all the way around and I say the algebra of the Twister-Heisenberg algebra if it matches all the way around the consistency of that all the way around will give you a four-parameter family of these sort of pseudo-lines points in your space. I wish I could say more about that. Just to understand that better, the obituary of these points are now somehow non-commutative, right? Well, I guess there would be something like a commutative subspace or something. It's where the set of the commuting variables extends globally over the whole region. I don't know, you see. I don't quite know how to answer your

2:10:00 question, because I don't quite know what I'm doing. But the line is not a line but is it your logic that okay the the the oromorphic for parameter family is lost if space-time is not is not complex and self-dual but maybe I can now use something that is normal or thick but think of it has some Isenberg operation operative sense so the normal market part of it is more like operators yes and then I can try to still associate a unique complex structure to also space things which are not I don't necessarily give you a complex structure of the space-time I mean you might raise issues of to what extent this will work if you've got a long analytic space which I don't know I mean, these things arose a long time ago also with the edge spaces that Ted Newman was talking about. If it's not analytic, can you do these constructions? Right, exactly. Or even if it is analytic, is it analytic enough in the sense that you've got enough of a complexification that these surfaces that you need will only exist? So there were awkward things we had even there, which is much worse here, I'm sure. But I think if you restrict your attention probably to, for the moment, to analytic spaces, which is what I would do, in the hope that we wouldn't be restricted to that algorithm. But I think you're right, that this could be a big problem, or does it solve itself in some way? Does it mean that your light ray space, even if it's not polymorphic, has enough holomorphism? Maybe I completely misunderstood, but it seems that you're saying, that you're suggesting that one can go beyond the self-duality by introducing some non-commutativity. Exactly. Okay. So let me try to rephrase in maybe all question terms. One reason to ask the self-duality is so that alpha planes nicely exist. So maybe now you could say, okay, I now try to think of the alpha planes as being fuzzy somehow because they're built from some non-commutative structure and then even

2:12:30 if the antiseptic dual part of I does not vanish I may still be able to define the such an object yes I I haven't thought of it that way but it may be that would be working yes that that's it's somehow fuzzy I guess fuzzy in the sense that people think of quantum mechanical things is fuzzy you don't have you don't have position and momentum defined anymore at the same time so you don't z's and z-bars defined in the same time, but in a fuzzy sense, they can still exist. Yeah, I would think that's true, but you have to be careful about in what sense they do exist, otherwise you're not going to be able to use your Podaro-type theorems, even if we had them. So I think it's got to be something which is honestly holomorphic. It won't be the space-time, probably. To what extent... You see... See, if you think of the light ray space, I get confused about this too, but you see, locally, and what I mean by locally is in the neighborhood of a light ray. If you have a light ray, and you've got all the light rays nearby it, okay, if you're asking for too much, say what the shear is and so on, or what the divergence of the shear is, you're lost, you see, because that's too much structure. I want to have something which doesn't have that much structure, so that I can have floppiness enough when I glue the two halves together. say that my library space doesn't have much structure. If all I've got is the symplectic structure, then maybe that's all I've got. But I need also something from the d by dzs. You see, if it's just that symplectic structure, it's too floppy. I think it's too floppy. Even if I know it globally, you can see it's too floppy for various reasons. So it's got to have a bit more to it than that. Now what is the bit more that it's got? I think it's a bit like, although this is wrong, but it's a bit like since you've got a symplectic manifold and you want to know, is it the cotangent bundle of a space? Well, you've got your p's and dx's and things. You say, well, you know where p goes to zero. If it is a symplectic, If it is the cotangent bundle of an honest manifold, you know where p goes to zero. p goes to zero, and there's your manifold.

2:15:00 But if you only know the symplectic manifold, and you don't know where it goes to zero, you let parts be cut away from it, then you don't know. You can't reconstruct the manifold. There are many ways you could do it. It's like taking different polarizations or giving different manifolds. That's a bit like that here, I think. So you've got your d by dzs. exactly what z was you could take it to zero or even if you knew what what am i trying to say yeah suppose you had the complexification where you had z and z bar as independent variables you could see where z goes to zero keeping your z bar which now called zero because the w is in the w is still there even though the z bar is going to zero you can do that in the complex because they're now not complex conjugates in each other so i can keep one zero and keep the other one alive. And that gives a structure of a global kind which could contain the information you want. Now I'm not sure if it's that. But it's a bit like that. So it's a bit like knowing that it's a sequence symplectic manifold which is a cotangent model of a manifold because you know where P goes to zero. But if you don't know that part of the space, you haven't information. It's got much more freedom. There's something like that going on. And to know the d by dz, if you like, is like knowing that kind of thing. But it's too vague in my head to be able to say something. In the previous picture, somehow an intuitive intuition that one could maybe make, or I don't know, is that one is using this concept, I mean, twisters are associated with now, now, now, and one reconstructs solutions with non-linear gravitons, which are interpreted as spacetimes. Yes, well, there you need to hold alpha sign, yes, yes. But here you've quantized this twister state, so the output will not be a classical thing anymore. I guess it's like, the analogy could still hold? Like you're sort of like, you're testing the geometry now with quantum stuff. In a sense. Yeah, you see, well, there's an interesting question, perhaps I should come to the end and then I finish my talk, but there's one more thing I want to say, you see.

2:17:30 and I saw him to Carlo in London at the Chris Isha meeting and I was saying this isn't quantum gravity now it's not quantum gravity because there is no h-bar no there's no there's no Planck Planck length in it there's no Planck length in it and then I thought well you can put one in This is wilder, but maybe it's the right thing to do. Yeah, this was maybe. See, up to this point, I would expect it's maybe more general than classical, but it's not quantum gravity. It may be a framework for quantum things, because you're certainly taking ideas from quantum mechanics, very much. You're saying, like with the position of momentum, you're saying you don't know either of them exactly, but you've got some canonical variables, make one more clearly defined, and then the other one gets fuzzier. But you have some structure there, which is there in the algebra. And there's something similar here. That you've got a structure in the algebra, which you want to say is there all the time, and even for a real non-conformally flat space-time, this sort of idea would be saying, Okay, you've got a space-time. It's a bit like going back to the experience in space-time. We tried to say how you define a space-time or a manifold. Well, you can define it in terms of the functions on that manifold. If you know your algebra functions, then you know where the points are. And that's the same question that came up with Michael here. Now here, the idea is you don't have a global twist of space. you have a global twister heisenberg algebra and that's the idea that that really is global you have got in some sense and i don't quite know what the sense is a heisenberg algebra which is global to the space-time and which from which you could extract the real space-time from and which if you've got a light ray you can focus it down onto that light ray and it gives you the local twister space. But that global twister space, as you go from light ray to another light ray, it gets shifted around in the Heisenberg algebra,

2:20:00 so what used to be your z's get mixed in with it. But, nevertheless, that Heisenberg algebra would be global to the whole space-time. And knowing it, you could reconstruct the space-time. That would be the idea. Yes. But it's still classical space-time. So it's not quantum gravity. However, now this is simple. This may be, you can maybe shoot me down in five minutes, or maybe it's an idea worth saying. I'm not sure. See, at this stage, there is not a proposal for quantum gravity. And there is no, if only for the reason that the Planck length or time doesn't appear. So here is the crazy proposal. Remember that I had the commutation rules for z and z-bars, but the other two I wrote down as simply commuting the z's with themselves and the z-bars with themselves. I remember in the very early stages, many years ago, when I was worrying about this operation, and I had thought about putting an infinity twister here, but then I realized it didn't do any good because I only knew about infinity twisters which were singular, not the cosmological consonants. And I think I realized that by redefining the z's you could simply eliminate this part. It was trivial. Whereas here I don't think that works. Because if you have a cosmological constant, then you could say that the connotator of the z's is some tiny number. When I say epsilon, epsilon bar extremely small, compared with the cosmological constant. This is what the i's, the lambda is hiding in the i's. But i, alpha, b, and i, multiply the two together to get the cosmological constant. But the thing you want here is in some sense absolutely absurdly tiny by comparison to that. And maybe that contains information of a, it's a bit of a quantum quantum space-time, where we're on the small scale, we've got something which is non-commutative. And that isn't too unlike what people do when they try to do non-commutative geometry. There are various people who try to use

2:22:30 non-commutative geometry ideas in that the position coordinates don't commute. Now, this isn't saying exactly that. It's saying the twisters don't commute with themselves, not just with their complex complex. All I've done is to play around with these commutation rules for a bit to see whether I'd run into a ridiculous contradiction. And I was not successful or unsuccessful, I'm not sure what the right word is. I'm not clever enough to see whether there was a contradiction in that. It didn't obviously lead to a contradiction. I do have a curious feature here, though. That is that if the zeds... I'm a bit worried about this because the zeds contain this flag plane. information. Now the flag plane, it's got some geometry to it. So if I change my definition of z by multiplying by e to the i theta all the way through, and e to the minus is i theta from z bars, does it change anything? Well, it changes the face of this epsilon. I don't know quite what that means. It needs more thinking to that. But I don't see quite anything wrong with what i've written there there may be something wrong with it and as i say it's only a crazy thought which was what when carla was talking to me in london a few weeks ago i reminded myself of having tried this when this was a singular and it didn't do anything when it's non-singular i think that could give you something new now is that a quantum gravity theory? I have no idea. But it certainly has a better chance. Because it's, you have a role for the blank scale, epsilon big blank scale entity. But that's another step ahead of many of the others. But it does mean that if these things commute, I'm not doing quantum gravity than I would expect to be. So I would expect, okay, it's doing physics where quantum aspects of it would be, would find a natural home. So I think that is true. So things where ordinary quantum ideas find this a geometrical role. I have no clear idea about that. It's like quantum fields on a classical mind.

2:25:00 Well, something, yeah. I don't know about that. On a classical mind. I hate to say what's going on. Not in the sense that it's intermediate before getting to quantum gravity. No, I would think, you see, I haven't talked about the Ward construction. But the Ward construction, also, these ideas should be relevant to. because you're stuck there again with Yang-Mills fields which are anti-self-dual. And if you want to describe ordinary particle physics, that's no use. You've got to have the entire gauge of the fields in there. They can't just be anti-self-dual. But this, if you extend these ideas to the bundle, in the way, the manner of the water construction, I can't see why you shouldn't include honest quantum fields ideas from particle physics. So, I suppose what excites me about it is, although I don't quite know what I'm doing, it does give hope to extend twister theory, as we've known it, from the straitjacket that it's been in for all these years. It's been stuck with the googly problem. I mean, there are ways of bringing, there are things called ambitwisters, you see, where you can bring the twisters and the dual twisters together that somehow we've lost the things which are nice about it, how the wave functions come out as very natural structures. Of course, there's another problem, which is mass. Now, this doesn't address that as a stance. How do you deal with mass? There have been ideas about that, but nothing I've said today has any clear relationship to that. So that's another issue. But I regard the Googling problem as a much more basic and fundamental obstruction to progress and to a security area. And if that can be surmounted by something like this, and if it can be made manageable, maybe somebody will produce the Schwarzschild solution with even simple examples, which are obviously not antiseptic, or can you see whether you can get them out of this kind of structural construction? It would be interesting. Can we finally move to the questions, actually? I have a very simple technical question yeah go ahead because now you have this relation that is equal to I

2:27:30 the infinite twister but I wanted this to be related to the previous proposal in which you got zero there so eventually if you take the cosmological constant equal to zero you should find back the other case but I cannot see it from the definition of I well I suppose you still have an epsilon there well the idea would be if epsilon is very very small you might get away with treating these variables as commuting yes technically they don't commute so you'd say well it's got to make its mark somewhere but you see if this is a Planck scale thing which is what I would say ok there's got to be something like a factor of 10 to 120 or something between these things because in terms the cosmological constant is set equal to, well I like to put it equal to 3 for various reasons. If I make it equal to 3, these are my cosmological units, it's not the normal Planck units because I'm now not saying the gravitational constant is anything that special. But then if lambda is 3, this epsilon would have to be some very tiny number. So the epsilon is some number that you introduce by hand in this case. It's a mission cosmological constant Yes, in relation to the cosmological constant it would be very small. So it would be something like 10 to the minus 120. I don't know if it's that or if it's 10 to the minus 60 or 10 to the minus 240 or something. I haven't thought it through. But it's some tiny relation to the cosmological constant. That would be the idea. now I have no I have no idea whether this works in any sense what you do with it or anything I'm just throwing this out so as to agree with the title of the call can we move to the formal session the question session thank you for this, is that a good point? applause Can I ask you a sort of motivational question? Because we've got a lot of your motivation. I suppose question session means I have to sit down. Yeah. Have you eaten? Can I eat? Oh, I see. I don't know. You tell me what the rule is.

2:30:00 You don't want to be responsible for you starving. No, it's okay. Yes, go ahead. It's sort of, I half see, I half don't see. You're getting up, that's more or less what I have. Go on, yes, yes, yes. If I have a scalar, a free scalar field, there is a relation between solving the classical equation of motion and constructing the one particle here in the space. What is the relation for you? You have two motivations here One is essentially solving the Einstein equations But then you have a side Motivation which is understanding Something about quantum gravity maybe Yes but that's only the last slide So do you think that the two Have to be related? Are related? Should be related? Might be related? I think it should be, yeah. It should be. Because, I mean, I said... In relation to the earlier question, that if it does, when you can ignore epsilon, give you solutions, classical solutions, the Einstein equations, that would be the approximation in which the classical solutions make sense. You want to say there's a better level at which you need to go beyond that, and that would be when you take terms in epsilon seriously. maybe you could do some perturbation series in epsilon I have no idea I mean a lot of the theorems probably depend on epsilon being exactly zero and if they weren't zero you might have to worry about I mean I'm talking about theorems that I don't even have but suppose there is a Kodara theorem which I think is quite plausible because to make things homomorphic all the way around in some sense is a very big constraint analytic complex analytic. And I would be altogether surprised if that didn't work. But it was, if you had an exact theorem, you see that exact theorem would probably not extend when you've got this epsilon in there. But when you were thinking about formulating the Googly problem, what you had in mind is the structure of the solution of Einstein equations. So I miss what you said, Simon. When you were formulating the Googly problem, What you had in mind is mostly the structural solution of the Einstein equation.

2:32:30 But I'm... You see, I think you're right in suggesting, I think, that I'm being a bit... I'm thinking about two things. I'm trying to do two things, opposite things at the same time. Yeah, which might be good, but you're doing two things. Yes, I think that's true. but you see the original anti-self-dual that non-linear graviton stuff came from classical general relativity I mean it's been lost somewhat and the people don't worry about these problems so much but Ted Newman was trying to have a notion, I think it was very specifically, a notion of angular momentum in general relativity I mean you go out to asymptotic now infinity, and you've got the nice Sachs-Mondi-Sachs formulae, mass loss, and you've got momentum, as Sachs explained very nicely. But if you talk about angular momentum, there are these horrible problems about when you're doing a super translation or when aren't you... And then Ted had this way of introducing complex sigma, which meant that you looked as though you were... The problem was when can you find a good cut, you see. You have light cones going out, and do they know when they come from a point? But that point is not a real point in space time. It's something which looks asymptotically like a point. So you could talk about angular momentum about a point, because you need to have the notion of a point to see what you mean by angular momentum. So he said, well, what are these good cuts, as he was saying? And the good cut has to be one which looks, in a certain sense, Minkowski's base. And that would be kill off the shear. You can't kill the whole shear off, but you can kill say the self-deal shear or the anti-selfdeal shear. Choose one of them, kill that one off, and you find a four-parameter family. It's a space. And that is the construction from which this came. So I thought about this. I don't understand what he's doing. Where are the equations for these things? No, no, there aren't many equations. It's a globally... globally defined. My gosh, yes. So these lines in my curved twister space are defined globally. It's the same story, yeah.

2:35:00 Yes, yes. So that, my God, yes. I sort of knew enough about these things already to think that was likely. That that would actually, that global condition would work. And it was then I talked to various people, Singer, and he told me what to do, do these, solve these equations, and that was too much for me. So then I talked to Michael Atiyah, and he said, no, no, it's this, and then he went off He said, yes, well, it's because of this. Take this special case and just did some geometry and said, yes, the number is four, you see. Four dimensions. I see. No, that was... So I thought I'd try them again later on, you see. But somehow now you hope that this opening up of the Planck length a little bit could help you to solve the second half of the problem. Could help you solve it? problem, the googly problem, the missing part. Well, that's the hope. Without the blank link. Without the blank link, with an external commutativity. With the extra non-commutativity. Well, yeah. Well, I think it's going to make everything harder, is my general feeling, like you normally would hear, gravity problems. I think it's going to make it harder, but maybe not. I think it's the first check to see whether there is something inconsistent. Maybe if you play around with these commutation rules and go around in some big thing, you'll see that it's going to spoil the other one. Maybe. I wouldn't swear to it making proper algebraic sense. But maybe. Actually, it reminds me also of things that people were doing at Penn State a long, long time ago when I paid no attention because in those days there was no cosmological constant. I'm talking about this, maybe it's a different subject altogether. What was it, these non-competitive? Chen Simons. The Chen Simons, these quantum groups maybe? Quantum groups, yes, quantum groups. Well, quantum groups, you have the two scales, right? The big scale and the large scale. Yes, here there's two scales. Right, so that's what... That's beautiful of the mathematics of the quantum group. In a sense, you have a natural home for putting a small, minimal Planck scale

2:37:30 and a big, maximal cosmological scale. Yes, but I was just thinking about these quantum groups where you seem to have something which is identified as a cosmological constant. And I remember thinking, oh, I'm not interested in that because the cosmological constant is zero, isn't it? But now I don't say that anymore. So I can go back and see we're talking about does that make any is that it connected with these non-competitive geometries or is that something a different subject yes this was to do with these lines across one way or the other you have the uh at the time that's the way lee was thinking about it that's right that's right the difference between the going down that's yeah that's right I don't know if it's got anything to do with this or is it might be has that to do with deforming the communication algebra of the B algebra of the group there is a relationship between these twisters and the conformal group via this sympathetic structure as Simone explained to us can I ask a question to clarify so of course the old formalism is very much chiral so there is built in this strong requirement of self-duality of the wild tensor in order to reproduce a space-time point out of twister space and it is nicely related to holomorphicity and it seems to me that you are trying to hope that the notion of holomorphicity could carry through also for real space-times So you're willing to give that up then? I certainly would not restrict to spacetimes which are anything other than analytic. You see, analytic, real analytic, you can certainly complexify and talk about holomorphic

2:40:00 things, as with Ted's H-space constructions. Oh, that needs a little more, because you needed it to be holomorphic in a big enough neighborhood that you could actually find your solutions. So it might be something like that. Now, I don't want to be stuck there for life, but nevertheless, temporarily, I would think I would be happy enough if you look at real analytic spacetimes. Okay, no, but I meant holomorphism in Twister's space. So, somehow, at some point, you would like to impose an equation that tells you something like self-dual-weil is equal to the complex conjugate of anti-self-dual-weil. And I would like to, what intuition makes you think that you can still do this, preserving some... No, I think the way to think about it, perhaps separate the problem into two parts, you see. One of them is to forget about the Einstein equations, forget about the infinity tensor structure. Just look for the light-ray structure, you've got the symplectic structure, the real symplectic structure and all that. Now, let's make it analytic, real analytic. So if I've got this light-ray space and then I can imagine doing the geometric quantization procedure, which I think just amounts to modifying your derivative by putting that one form into it. And that will give you the curvature when you go around. And incidentally, the charge, you see, it's like an electromagnetic connection. But the connection only applies when you've got a charged field. And what is the charge? the helicity, because it's the eigenvalue of the homogeneity operator. So you can see, I'm not exactly sure whether it's that or that momentum one is pushed up by two or something, but it's basically that. And you can sort of see that the, it probably has to be real analytic to get away see it's something like having a twister space where you've got

2:42:30 P end because that's the lightweight space and you want to try and push it up into the top half or the bottom half you've got the derivatives you've got the quantization working kind of tangentially within the P-N space. And that's given to you from the light ray space. There are different ways of extending it into the top half of the bottom half. Those different ways are the sort of freedom you need. Because you need a bit of freedom. You're going to say, OK, I've got my geometry here, my non-comitative algebra here, and one patch. I've got the non-comitative algebra in the other patch. And I want to patch them together. And I try and do it here. it. And that freedom is essential, because I say, okay, that freedom has got to be consistent all the way around, and if it matches all the way around, then I have a global structure. And that global structure, the idea would be, fixes me to a four-parameter family, and that would be the space-time. But it's got to be holomorphic. Maybe that just means that the D by D Z's exist, you see. The D by D's, I'm not sure quite what it means. Well, the D by D Z's can be iterated to a higher order. You don't have just first derivatives which you want to have higher order derivatives. I mean, that's the... They say it's analytic. It means only one derivative. I don't know, you see. I'm confused at that stage because I don't really know what mathematically the structure is that's working with you. But the general feeling is that you'll have if you go to your D by DZs and you just postulate that those D by DZs continue in some way maybe locally to the top of the bottom half and that continuation has to be consistent as you go around, because there's lots of ways of doing it here lots of ways of doing it here but those, if I go all the way around the patching, will only be consistent for a certain small family and that sort of small family will be the ones I'm looking for Okay, so The reason why I was asking this, whether you were willing to get this up, is because what came up at the beginning of the discussion is that in loop quantum gravity, we use the twisters

2:45:00 and we quantize them in a slightly different way in the sense that we represent the same commutation relation as we have there, but consistently with the fact that the theory is chiral and does treat omega and pi differently, we look for representations as functions not olomorphic functions of the twister but rather as generic complex functions of just one of the two twisters so you see it's like we do a real polarization instead of a complex polarization and this is so let's say for functions of omega and omega bar or pi and pi bar instead of omega and pi bar and this is some nice features because these functions immediately carry a unitary representation of the Lorentz group with a very simple scalar product, no poc hammer contours, no complicated subtleties like that. So that is one thing that is nice. The thing, of course, is that the theory is piecewise flat. So we are more like in a reggae framework, in which we try to recover curvature by gluing together flat things. this part of the program is quite different. Well, it's interesting historically because when I first went to Princeton, I met Reggie, you see. You know he died the day before yesterday. Reggie died? Yeah, the day before yesterday. He wasn't too well for a long time. He had a illness. He had a what? He was sick. He was pretty sick and becoming weaker week or so. No, I hadn't heard that, I see. Yesterday or the day before, the day before yesterday. I saw him last time, probably a few years ago. But he'd been ill for quite a long time. Yeah, he had disappeared. I hadn't seen him for a while. Yeah, but this was, I first went to Princeton, and I remember Wheeler, Todd, son Wheeler, and he said, Well, let me introduce you to Reggie and Dickie. Now, people are on very nice first-name terms of people. Not just first-name, but very sort of friendly colloquial first-names. Yes, so I met Reggie there, and he was doing the Bones. You see, he was very famous at that time, of course, for his Reggie's trajectory,

2:47:30 Reggie Poles and all that stuff. And that's what all the excitement was. and the excitement he was interested in in putting these flat triangles together but that did influence me because I think in some strange way when I first was thinking of Twisters when I was in the University of Texas and there was a hotbed of relativists who were assembled there by Alfred Schild and I thought how am I going to get any of these people interested in what I'm doing because it's all flat space Engelbert Schuching, he was always interested in these things, but I don't think I got anybody else interested in it. But my sort of feeling was well, maybe... A lot of flat spaces make... Yes, the flat spaces, by a Regi sort of procedure, would make you a curved space. But that's what Simone is doing. In a sense, that's what happened, exactly. What Simone is doing is a lot of twisters. That's right. It's one sort of one twister space attached to each... Attached to a couple, right? well it's exactly it is that in a sense yes it is that but not quite in the original sense i think i always always worried that reggie's things only approximately satisfied the vacuum equations and he just put a lagrangian you know and said well you minimize this lagrangian and whatever it was and that okay they never have you quite satisfied the vacuum equations approximately and then I tried to think about things which were like Regi but they were stuck together on hyperservices so these were these flat plane wave, impulsive waves and that had a big influence on this stuff the twist-to-quantization rule came from that so I mean there was a chain of thoughts which influenced each other just to finish this whole discussion with the final question so you do not believe that a real section in twisted space as opposed to a homomorphic section could play any role this is really what I wanted to get to even if we know that we have to impose an equation that is like psi left

2:50:00 equal psi right bar It's very much like a reality condition. You still think that it would be described by some olomorphic functions in Twister P's and not some... No, you see, you have a side of side tilde. I'm not a bar with a twister bar. And those are independent. And then, okay, you want a real one, you make them the same. And you want a complex conduit to be done. And you think that this can be done with an olomorphic description? Yeah. I'm not saying it can be done. You're hoping. I see no obstruction to being able to do it. Because there's another instance of... So, in a sense, you build the curved structure, which, for the moment, let's say, is just self-dual. Out of, basically, C2, out of the chiral part, the left-handed sector, let's say, of your spinner. So, there's an epsilon AB that plays a sort of strong role in building up. on the structure, and at least one of them you need still to be fixed. Well, of course, when there is no cosmological constant... Yeah, sure, let's discuss. This goes on the C2s. Yeah, let's discuss just for the sake of simplicity. Incidentally, Richard Ward was the one who worked out the details of that. I mean, it was fairly clear what you had to do, but he wrote a paper on that. But I don't think we looked at it this way. Well, not here, but this is one way I try to think of it I mean, maybe I can't remember, but it was clear that you had a symplectic type structure which is not degenerate as you have in the in the Ritchie-Flatt case so it's Ritchie-Landac constant case Because there's a, how can I say a toy mod, or a simpler version of this nice non-linear graviton construction the same idea of building up an a priori complex metric out of just flat C2 properties, which is this Urbanka matrix that in luquanto gravity we use all the time, at least in the covariant constructions. I don't know if you remember this, Plebansky and Urbanka, this work here, right? Right, and there too, you can build a self-dual or an anti-self-dual complex metric to begin with and then you have the issue of reality conditions

2:52:30 and one way to get rid of them is to actually instead of working with complex B fields is to sort of taking the real or imaginary parts of this Urbanka metric and so again to work with real fields does this work for special cases is that it? no no no this you can do in general I have to remind you to what that's doing I mean this is I know there was a plebansky way of looking at this So, I don't know, the example also for us working in Leuquantogadi and spin forms has taught us that simpler reality conditions are maybe the way to go, so that's why I was kind of making this question of whether, although the olomorphic structure is certainly really beautiful, whether there could be actually a role for, since you were talking also about geometric quantization, right? maybe a real polarization could at some point be tried the one that gives race itself dual structures for sure I was building on this analogy with the Plebansky metric and the fact that you want to impose something like psi equal psi bar to say that maybe you know well yeah functions instead of functions of z, I don't know, functions of part of z, or something like this. And, in a sense, this is not too, too far from what we are trying in look quantum gravity, by using these functions of omega and omega It's a polarisation that is... I see what you mean, I think, yes. But it's a bit like... I suppose I worry that kind of you're losing the holomorphic... The power of the holomorphicity, which is what gives you uniqueness. Certainly, it's a But how that would work here, I don't know. It's very powerful, but very heavy, too, because it seems that it's blocking us from discussing. I understand, sure. But most of it, you don't have to do anything in practice, which is likely to be true. but yeah I mean there might be something like that because you see in a sense if this framework works you would say well within that you have both the self-dual

2:55:00 and anti-self-dual constructions because you say in a certain sense you have a complete set of commuting variables which are the z's or the z bars which are the d-mending sets but then if you're going to have a mixture which is going to give you a real solution, then your variables have to be even-handed with respect to the z's and the d-value z's. So there will be some real structure coming. Okay, I see. Some structure, yes. I suppose I'm trying to think that slightly differently, but I think you're right what you're saying. There would be... Certainly the polarizations, or whatever the right word is, the right word here. That is, in a sense. It's going to be something intermediate. Right, maybe at the crossing section, at this overlap region, there will be a request that mixes the... Well, there's probably some global freedom, which is now you can mix all the way around. I don't know. Yeah, no. If I knew more clearly what was going on, maybe I could ask you a question. But in a certain I think of the light ray space as keeping you close to the real structure. See, because the light rays Well, again there's this thing about ambitwisters. People say well, there's one way of doing this with ambitwisters. That's the complex analogy of these things. In a sense, this is not really doing that. I'm saying you take the light ray space and then I'm allowing myself to broaden out and round it where I forget where the light rays are but I use the light rays as as to say don't go don't wander too far from that what does it mean to be a small local deformation look that you're close to the light wave space in some sense you say okay I'm going to forget exactly where the light waves are and I look at just where the z's and d by the z's that I get out of that geometric cross-ident quantization procedure and then I retain that and match that all the way around I don't really know what I'm talking about, I'm just saying things which I feel that they've got that kind of structure, how far off that is, I don't know. I have just a question that I tried to make before, but it wasn't.

2:57:30 So, you showed us the generators of the Lawrence group before, I mean the Boncari group. So, if I use this algebra there with zeros, with epsilons equals zeros, then I can show, one can show that the commutators reproduce the Poincaré algebra. There's a good question. What do you get? No, I mean, I thought. Isn't, I mean, you have... Oh, I'll say it again. Oh, if epsilon is zero. Yeah, if epsilon is zero. Yeah. And actually you can get the whole of the conformal group. Now, if you put epsilon different from zero, what do you get? I mean, you can get something. You certainly get something. I haven't a focused idea. So that is what smells like a quantum group. Quantum group is something like that. So you have this Poincaré group, and now there is epsilon all over the place. It's not only the Poincaré group. But maybe it looks like what people do with groups when they make them quantum. when they, when they, I mean, quantum groups are not groups anymore, but they resemble groups in that, that the algebra, when there is a parameter, when you take it to zero, they just get the, the group, the algebra. So what, these algebras don't close, is that, or what, something like that, they just grow. Yeah, in a sense, they don't close, because it's a universal enveloping algebra, so in a sense, it doesn't. I've never played with these things. There is a geometrical picture, which is related to quantum group, which is related to phase sphere, which is related. Let me start from physics, from geometry. Imagine the Planck scale, physically, which is sort of a minimal scale. Yeah. And, you know, cosmological is a constant in the universe. Yeah. Which, the Euclidean universe would be the size, and the Lorenzo universe is sort of the horizon, the distance of the horizon. Yeah. the smallest thing in terms of angle the angular the minimal angular resolution under which you can see a thing is a plank thing that the cosmological distance I see what you mean okay okay yeah yeah which means that no angle smaller than that makes any sense because it's nothing you see smaller than that So, on the two-sphere, on the two-dimensional, the smallest resolution you can have is something.

3:00:00 Which means that if you think of expanding functions on a sphere in spherical harmonics, you have a maximum j. Maximum spin, a maximum. And in fact, that's what the representation of quantum s2 has. gets a maximum j which is the smallest angular and if so on your somehow on your remote spheres that represent there should be a minimal resolution so there will be some non-computativity in the geometry that describes this is even more vague Maybe you can read this fuzziness between two twisters, between the two light lines that could be exactly the fuzziness in the angle that we are talking about here. I remember in the state people were talking about the non-competitive sphere or something. Is that the same thing? Yeah. The non-competitive sphere is a very simple object. Imagine you describe the sphere in terms of the functions of it. Yeah. and now you take all these functions and you expand this very good money can you cut it up to maximum yeah yeah so you're describing something which you don't see this details under a certain scale yeah and there's a large mathematics of doing this property and that's what they call it it's like you have two different non-commutativity quantum gravity because one is given by h bar and the fact that for instance we have the h bar in the minimal area of quantum gravity and there is another non-commutativity that comes from the cosmological constant so if you take the cosmological constant to zero you get rid of this non-commutativity. And it seems to me that the non-commutativity of non-commutative geometry is this second one. Yes, I agree. So that will be the true questions. Yeah, that seems to be right. It's curious that it involves big and small at the same time, isn't it? Because you're looking at the cosmological concepts, because that's the I, alpha beta contains that. And you're looking at the Planck scale, because the epsilon contains that.

3:02:30 But somehow, But in pure quantum gravity, H-bar is not really there, right? So you only have two-dimensional, so one-dimensional is... But you have a big number, probably. Yeah, you have one big number, that's right. You have one big number in the theory, this is the... But when you said that... So you have the infinite twister that tells you where infinity is, so in fact it has the cosmological constant there, like telling about this maximum distance it's the radius of the horizon of the hyperboloid this is a hyperboloid yeah well lambda lambda over 3 or something that's right so but that's over 6 it happens to be roughly right about now too but that's a coincidence that's probably that's one of the people coincidence is this lambda is this really the cosmological constant or the square root of the cosmological constant well just because we happen to be living in a round at the time when the cosmological constant is starting to dominate so people say oh isn't this an amazing coincidence well I'm not sure it's amazing I think it's no more or no less amazing, which may be still amazing, than the other Dirac. It's a Dirac coincidence. Dirac used to worry, the original worry he had was that one of these big numbers was the age of the universe. And how can the age of the universe be a number? Because as you wait, the number gets bigger. So he argued that the other constants were actually changing. But then, that was then explained, or actually I don't know when it was explained, because Dickie already had an explanation, and then Brandon Carter, that if you look at how the constants of nature determine the scale of time of a main sequence star, you find that they do and it was to do with the relation

3:05:00 between the electric and the gravitational attraction in the hydrogen atom which one was tentative before so they do tune to so that the age of the universe is what it is yes, you said that if you happen to be the kind of being that likes to live around the main sequence star and you then look out at the universe and if you are a physicist or an astronomer where are you supposed to live here? You see, goodness me, you say, this is an amazing coincidence. This number is in Planck units. I think Dirac was saying in electron units or something, it was 10 to the 40 or something. Isn't that amazing? So he said, well, that's ridiculous. It couldn't be 10 to the 40. Numbers that big don't come into physics, you see. So it must be, it's not a constant. And so therefore, all the other numbers which were 10 to the 40 must also be not constants. Well, that turned out to be wrong. but it was at the time a reasonable suggestion but then Dickie I guess it's not so surprising because people if we happen to be the kind of being that comes about from a planet going around the main sequence star then when we are in the middle of our existence the whole evolution we look out and we see this coincidence because the star the lifetime of the star is governed by by these numbers and the ratios. So there you have an explanation. It's one of the two uses of the anthropic principle which makes any sense as far as I'm concerned. Yes, exactly. It's the proper use of the anthropic principle. Yes, it's proper. I agree. Which one is the other one? The other one is the Hoyle coincidence. Which some people say, oh, well, it was not really anthropic. Well, you see, I think it was, because Hoyle was really interesting. this was the energy levels of carbon and he puzzled about building up elements and he didn't believe in the Big Bang so he said the only way you could have building up elements was in stars and he got stuck at carbon I think it was carbon and oxygen and there was some imbalance, how do you get a comparable amount of carbon and some people say well that's nothing to do with anthropic because it's not that much carbon anyway so it's got to be there somehow but Hoyer was interested in it for an anthropic reason and I think that's really true

3:07:30 from other things I've heard him say so I think he was interested in that question an anthropic argument saying well we exist, we need carbon where did it come from and it could only come because of this energy level which he predicted I found it. It was really there. These coincidence things, I found it doubtful. I mean, given the Carter answer was good, but still, I found it doubtful for another reason. Because if you think about the past, you find it surprising if you assume equiprobability in a logarithmic scale. But if you throw a dice and plop the outcome in logarithmic scale, of course you find something. So assuming equal probability of something plotted in a logarithmic stage, it's just wrong to start with. In other words, along the light of the universe, say, from now to double the light of the universe, we're at a completely random point. Why should we at the very beginning? Now, the second reason is, yeah, what about the future? Because the future then is very long. but if you think about the future which is very long I mean unless something happens the future is too long it's infinite so how do you put a probability on infinite so in any case just do something it's very dicey I should make clear that I'm not a supporter of the anthropic arguments I know but I think there are things to think about I think the Hoyle one and the and the Carter-Dickey one and the Carter-Dickey one is really an answer I mean, that's nothing I think that's all right no problem with that one well, Carter-Observicio, the weakest version there are things we observe which are about the universe which show that we're not in a random point of the universe right, if you measure the density around us the mean density of the universe is not is this a strange coincidence no, I don't think anybody should say that so there's a mistake in using the cosmological principle too strongly by saying that absolutely anything we measure

3:10:00 we should be in a totally random place no, I think you're right it might be that certain regions, for example in the void, you might say why don't we live in the void if there's nothing Good, so now in the next two hours, Roger penguins will talk about consciousness Nobody warned me about that Am I? Oh yeah, we had some Don't run away, it's chocolate or at least leave the key thank you sorry for exploiting you well I'm here I'm here come here I just wanted to make some oh sorry applause applause applause Thank you.