Spin Networks, Twistor Theories, Cosmology and All That

0:00 Okay, so before we start, I just want to announce some additions to the program on Thursday afternoon. So, we write down how the reality is. So, it's added to your program. And everything else is here. The other thing to note is that we can't do with all my emails at the conference. ...so you all know why now it's still, and that is nice. OK, we're asking if you'd like us to have Roger Penrose here to stop for one next session. The title is spin network, physics of cosmology and all that. You might wonder what those topics have to do each other. And what they have to do with non-competitive geometry. I won't tell you the last because that's what I'm hoping people will tell me. But it's basically a rambling talk about things which I've done in various stages of my life, having to do with the nature of quantum space-time. So that's really what it's about, is three different angles on quantum space-time problems. I'll start with spin networks. There's nothing about spin networks. There's something I did a long time ago, when I was a graduate student, and it's been taken up by people in the loop, quantum, gravity community, but what I'm going to talk about will be the original version of spin networks and the underlying ideas. Basically it's based on discreetness and combinatorial roof, it's a completely combinatorial system, The continuum doesn't appear anywhere in the description, but it's based on the basic

2:30 rules of quantum mechanics as applied to, in fact, ordinary Euclidean geometry. And then there's Twister theory, which has started... To some degree it was the development of spin networks, or it had many motivations, And I'm going to concentrate on some slightly different motivations, but it had to do with trying to generalize spin networks from the discussion of the rotation groups in three dimensions. So one's looking at space, basically, to a space-time description, where one had the Pocahontas group and things like that, so I'll say something about that. And then finally, these are all to do with what the nature of quantum space might be. Of course, people tend to think of the relevance of this topic to quantum gravity we're talking about. And quantum gravity, one of the main places one expects quantum gravity to make its impact, is on the nature of space-time singularities. We have in the classical theory, anti-class theory, singularities in which the normal descriptions of space and time go wrong, and one of the driving forces behind trying to find a common gravity theory has always been to try and resolve this issue of the singularities and how one describes the physics which is going on. But if one really looks at the universe and tries to see what nature herself does in relation to these space-time singularities, there's some very strange things which don't seem to have much relationship to any serious version of quantum gravity that I've ever seen, although there may be some peripheral. But anyway, the main point I want to make here is time asymmetry, and what's the difference between the big band here and the various singularities, the red things here are singularities, the singularities in black holes that have a completely different character. So there seems to be some time asymmetry fundamental to whatever's going on in quantum gravity because these are places where I want to expect quantum gravity to be playing its role. I've been banging on about these things for many, many years, but I have a different take on this, which is only very recently, I talked about it, a version of it at least, just about, not quite a year ago here, at the Spitalfields conference in November, but I've been thinking about it quite a bit more since then, but what one finds is that the quantum geometry at the Big Bang,

5:00 what one doesn't really know but one can make a good case that whatever is going on there is very remarkably mild with regard to what it does to space-time and I'll say something about that informal geometry seems to be perfectly well defined it's very different from what happens at the singularities of black holes and one has to come to terms so let me leave this slide up here and say first something about It's a really three-topic feature which will require a lecture, so I'm going to give you a very rapid route to these three topics. First of all, spin networks, and as I said, this is the original space version of spin networks, that's what I've been talking about. You have these graphs where the line segments in the graph are to represent entities, or let's say a self-contained entity. You can think of it as a particle. This could seem to be a particle or it could be an atom or something that you think of as self-contained. And it has a total spin. The only thing I'm going to be interested in is its total spin. And this focus spin is nh bar over 2, so n is an integer, or it's a natural number, 0, 1, 2, etc. And there is nothing directional about this. This is 21 dm. I was interested in this. One of the original motivations for this had to do with Marx's principle. I had long discussions with Dennis Schiama when I was starting to get interested in physics in a serious way, and he was trying to impress upon me the importance of some version of Marx's principle, and so I was led to thinking what happens if there was only one electron in the universe, for instance, how does it know which way it's pointing, and that sort of question. And the idea would be that at this sort of primitive stage, you don't have any notion of direction at all. You simply know this, you're allowed to know this integer piece

7:30 of information. And the idea is to try and build up the geometry of space, simply knowing these integers and the combinatorial rules that they satisfy. So let me describe the systems in networks, and, as I say, I go through it very rapidly, without proving anything really. First of all, you have not just these line segments, but you have nodes, vertices, and I'm supposing that they're always trivalent. This is the difference between what people in the loop quantum gravity theory area, they like to have four valence vertices, then you have complications about this. You have to put in another piece of information. With the trivalent, it's completely fixed, and what one has is that the sum of these integers have to be an even number. There are two basic local rules. It has to be an even number, and secondly, there's a triangle inequality, basically satisfied by these things here. And you can consider spin networks, that is to say you have a graph which has filled up the trivalian graph with numbers attached, natural numbers attached to the edges, satisfying these rules here. Now, any closed spin network, you have a spin network to have loose ends, or you could which are closed, and if it's closed, it has a thing which I'm going to call its value. The value of an integer could be negative or positive, but it's a well-defined number. There is a question of orientation, and I fix the orientation by imagining it's drawn on the plane, and the orientation is determined by how you draw it on the plane. But the actual rendering on the plane has no real significance. It's really just a total orientation for the whole graph. It's easy as well. There's firmly the sign of this thing here. I'll tell you how to do this, how to work out V, if you know the spin network, shortly, but let me just say there is a way, for the moment, there is a way of working out this integer given the spin network, closed spin network, and the orientations given by this particular rendering on the plane.

10:00 Now, any open or closed spin network, let's say an open one, has a thing called a norm, which is obtained from the value. And the norm, I'll just give an example here, you have your spin network, I'm supposing it's open here, you draw it on the plane here, and you reflect it through a line, so you put down this mirror image here, you reflect it in the line, and join up corresponding lines, and then you take the modulus of the value of this closed graph. form is now a natural number, which is obtained from an open or closed any spin network. Now having that, you can then work out probabilities by the following rule, probability formula, And let's suppose we're given a network, which is an open one, it has a number of lines coming in, but you've got two in particular, and I'm going to consider doing an experiment, which involves combining these two, and then there are different alternatives for what the third one will be, consistent with these rules here. So let's suppose these are A and B, the numbers of the two ends that we're thinking about here. And let's suppose they come together, and we have now a third one. And so there will be various probabilities of the different possible outcomes of that consistent with the rules that I've given you there. And that is simply obtained by the norms of the following formula. If you don't know how to work out the norms, then you can work out the probabilities. It's simply the norm of the one with the C on it, times the norm of C by itself, divided by the norm of this one, the original one, times the norm of that trivalent vertex there. And that is the probability. You simply count from quantum mechanics. There's no changing the rules here. And I just, in fact, these two things here you can work out explicitly, if my memory serves me correctly, I have to be reliant on a bit here because I did this rather hastily last night, or early this morning I should say. And the norm to me is this norm of the trivalor vertex platform there.

12:30 We can work these out explicitly, but these things can be complicated. It depends how elaborate the network is here. Now, I haven't told you how to work out the value. That's the only thing, really, that's missing in this description. That's actually how to do the value, then you can work out the norm, and then you can work out the probability. Good. The value is obtained in the following way. I'm really just going to give you some examples, it's just this way around. Here we have a particular network, and I've just done it for this case and you can see what the general answer is. What you do, is you look at the numbers on these lines here, and you draw a picture where you have two strands here for that two, and three strands there for that three, and four strands there for that. And I'm dividing by this product of factorials, where there's one factorial for each of the numbers you see here. And the black line going across is an anti-symmetrizer. So, if I see a black line across two of these, it's just been straight across, mine is crossed over. And the three of them, it's just an anti-symmetry combination. And finally, when you see, for each one of these, you have to expand each of these out, and so you will have factorial n for each of these numbers here, all multiplied together. So there's going to be a lot of terms if you work this all out in a reasonably complicated graph here. There are lots of tricks for making that number easier to calculate, but I'm not going to tell you any of the truth. That's a refinement on what I'm saying here. But if you take any one of these terms in this expansion, it will consist of a number of closed loops. And you assign to each of these closed loops the number minus 2. And then there will be a power of minus 2 for each term, and there will be a lot of terms. And, in fact, you can prove it's an integer even though you've got this factorial spot.

15:00 And here's some examples here. I have a closed single line of a spin network with a number on it that's one more than a number. These things are called strand networks, so if you think of them sort of underlyingly, the spin networks and these things, and in a certain sense more primitive than these. It may look rather mysterious, all these minus twos and all that stuff. I should explain that it has, well, the way of looking at this is just slightly unconventional, which is to think of this as the orthogonal group in minus two dimensions. It's actually more the symplectic group in plus two dimensions, which I'm doing, but you have an awful nuisance of these things as signs. If you do it directly, it's SE2 you're talking about. And that should be, it's just the rotation route, basically. But you think of that as O minus 2. That's what this is doing, isn't it? I'm not going to explain where these come from specifically. I just want to give you the rules and that's sufficient to tell you how to work anything out in principle. Now, the sort of punchline of this subject is what I refer to as the Geometry Theorem. And the idea here, you see, everything is simply integers, you don't have any notion of direction. How do you get directions out of this field? Well, the idea is that if you have some complicated spin network with two, let's say, big lines big, the numbers a and b are taken to be large, so you think it can consume cricket balls or something, and they're spinning in some way, and you want to know what is the angle between the directions of spin. Well, you can do this by following experiment. Let's remove that one for a moment. Follow the experiment. I'm going to take, say, an electron on one of them, and pick it on the other one, and let's say that taking it off this one has could reduce its spin value by one, and by the stigmas on the other one, it could either increase its spin value or reduce its spin value, depending upon the angle. So if you knew what this probability was, you could work out what the angle between the axis of

17:30 rotation of those two bodies is. But that's fine, except that it's a mixture of two different things here. It doesn't actually work quite as I've said it, because you might, for example, have two things there which might be no connection whatsoever between them. And this really means that you're not talking about an angle here, you're talking about ignorance. If you think in terms of density matrices, it's really, I think, density matrices involve two kinds of probability. They involve probabilities, this would be sort of ordinary classical kind, and they call a quantum cross, but it's in them mixed together. They're all nicely combined in an appropriate way in a density matrix. But here one's really wanting to know what the quantum mechanical part is because the ignorance part is getting confused with the quantum mechanical part, and it's not really telling you an angle. So what you have to do is to try and get rid of the ignorance part in the following way. So what you do is a repeated experiment. We do it first, you have these two bodies, you take an electron off one of them, stick it on the other, these spin might go up or down, and then I repeat this, do it the second time. The thing is, if this were a question of ignorance, then the second time the result of the probability would be affected by the result of the first experiment. It's now giving us, if you don't know anything about their spins, and you do the experiment again, that's telling you something about the spins, and so the probability values But if they're just great macroscopic bodies, and I take an electron off one and put it on the other, it's not going to make any difference to the second, I mean, okay, epsilon difference, but there's an epsilon hanging in the background with all this discussion, and I'm not going to worry about that. So, that's the idea, and you say this sum has, this angle or this probability is stabilized if it really is a quantum tangent of the probability. Okay, so that defines the angles. The probabilities of these are angles and not ignorance if the probabilities in the second experiment are not affected by the result of the first experiment. Now here's the theorem. The theorem is, if these are part of the mechanical angles defined this way, then the geometry that you get of these angles is basically the geometry of directions in ordinary three-dimensional duplicate space.

20:00 So that you get, in the limits of big, complicated spin networks, the geometry of three dimensions coming out. And there are lots of puzzles about that. I've worried about this for a long time, because there's nothing directional about this originality, you see. The geometry that you get out in the end is intrinsic to the system itself. It's not as though... What would you say statement, the system is that you can embed it... Yes, that would be a... Yes, yes, that's right, yes. The angles, you can embed it. Right. That's true. That's another point which has to be made, absolutely right. But you have to consider how you build up three dimensions. Because you start with something which looks planar. You've only got real problems, real... So you can show that you cannot embed it in two space, but you can embed it in two spaces. Absolutely right. That's right. Yeah. So you generally do get the three dimensions. OK, so that's basically the original version of state network theory. Notice there's no gravity in this at all. It's just geometry and how you have systems combining with each other according to the rules of representation theory of the rotation group in three dimensions. Ultimately, you've got these directions out again. The other thing is that if we look at the combinatomic subspeeds, we cover still animation That's not because combinatomic subspeeds. That's right. But there's no... There's no direction, of course. Just the speed. That's right. There's no... See, the thing that barred me about this is it's Euclidean space, it's the directions of Euclidean space. You don't get distances. Sure. Yes, well, okay, that's really, okay, well that's about the first time. I want to move on to twisters, because one of the motivations behind twister theory was how do you do this in four dimensions? How do you gather time as well in the picture? And the sort of obvious thing to think of is how do you work with representations of the Poincaré group rather than representations of the rotation group.

22:30 But then you have this problem, you've got two problems actually. One is the non-semi-simplity of the Poincaré group, which makes the representations much more complicated. And the other, the related issue, which is the non-compactness of the Poincaré group. Now I thought about this for a while and concentrated more on, I'm worried about the non-semi-simplicity of the Wangari group and thinking that maybe one should, rather than looking at the Wangari group, one should look at the conformal group. It's a conformal group, although it's non-compact, it does have interdimensional representations and so on. Nevertheless, it's at least a semi-simple group, so that was one of the motivations underlying twisted theory. However, as far as it's gone, twisted theory has never come back to being a combinatorial structure. So this has always been a sort of conflict in my own mind. On the one hand, the feeling that one needed discrete things, that the root physics should be combinatorial discrete, and on the other hand, I've always been incredibly impressed by the power of complex analysis. And I have the feeling somehow these things may be due to each other in some way, but anyway, these are sort of ideas in the background of crystal theory, so let me say something about crystal theory. In fact, I want to give you one of the other motivations behind this theory. One of the other motivations behind this theory is non-locality. Now, people usually talk about non-locality in terms of Einstein-Badolsky-Rosa, a very distant object that can be entangled, telling us that quantum theory is in some sense non-local. But there's a much more primitive non-locality. It took me ages to realize the thing I'm going to tell you now. It's more primitive than EPR, and to encompass EPR into Tristan theory is much more difficult. I hadn't realized that the following is much more direct. Here we have a simple quantum mechanical experiment, there's a source of a quantum particle here, mirror, parabolic mirror or something, and there's a screen some more way off here. Now, the wave function spreads out here, and the nonlocality I'm referring to here is that

25:00 this wave function determines the probability of finding the particle at the screen in various places, but the normality arises because detection at some point here, suppose you find that the particle is here, that almost instantaneously tells you not to find it anywhere else. And this is a, if you thought of this as an ordinary wave in a pond or something like that, then with a local disturbance here, well then finding it here, there could be some probability of But it would be completely independent of whether you find it here or not. And so that's a very fundamental non-locality for single particle in quantum mechanics. So it's already at the one particle level one has its non-locality. Now, I'm going to come back to that. See, if it was some local phenomenon, you would have super-normal communication. Now, Twister theory is basically motivated, well, as I say, in general I can spin that There are about a dozen different motivations in it. I'm only going to refer to two which I quite often talk about. And one is the non-motality that I just said over there. And the other is monomonicity or complex structure. And one of the things that always struck me about space-time is that the string theorists of all we've taken a different philosophy here, as they seem to have, which is, my point of view is that space and time, okay, time is one dimension in space three, and this is very tied up with a homomorphic structure, if you like them, in fact, if you look out at the universe, here we have a number of stars, what you see, of course, that's a celestial sphere, or you see this sphere, and it has this particular property in relativity theory that if the transformation from one observer to another, so you can have two different people situated at this point moving with great speed with respect to each other, looking out at the same sky, and they will see a transformed sky, but this transformation is of a particular kind, it's conformal, and it's in circles to circles, which suggests that one might can't look at this as the Riemann sphere. So this is one of the basic things of twisted theory. One says, OK, you're looking at this complex geometry, and we'll look from here, and also maybe we'll have somewhat non-local characteristics. So let me just give you a

27:30 rapid picture of this subject. Here we have, coming from this now, you see the idea is think of the light rays as more primitive than the points. So here we have a light ray in space-time. I'm just talking about Mikoski space-time for the moment. And here we have a light ray represented by a point in some other space. And if you want to discuss a point in space-time, you think of the family of light rays through it, and that's this Riemann sphere in the space which I'm calling P-n, representing the light rays. Now, if this is a complex space here, of one dimension, you would like to think maybe the whole space could be a complex space. But it can't be, as it stands, because it's five-dimensional. And five is an odd number, and odd numbers can't be... odd numbers. Real dimensions can't be... That's probably an even number. Well, let me not be too disturbed by that for the moment. Just to point out these dimensions. And here we have a dimension indicated. This is four-dimensional space-time, that's right. And the light rays would do a five-dimensional family. And how do you do it? Well, I'm using these space-time coordinates, and those are complex coordinates. So let's put the twister, that's the, well, the basic formula is this incidence formula. I thought it was possible. And incidence over here means that the point lies on the line here. Incidence over here means that the agreement sphere contains the point. And the translation from one space to another is given by this installation, which can circle around the space. You can see what locus over here corresponds to one over here, and vice versa, by simply holding one or the other thing fixed in this formula. The thing at the bottom is a condition that necessarily holds in order for you to be describing a real space over here, if these red coordinates are real, then it necessarily follows this relation here. And this relation is the permission relation on the four complex coordinates here. It's also just the ratios we're interested in here.

30:00 So what we have, really, is a picture here, and I'm going to remove something so it may be a little better if you can make that one aware of things. Here's a more complete picture. We have a complex manifold here, which is a complex projective through space, and these now really are Riemann spheres, there are projective lines in this space here, but there is a subspace which describes the real geometry, as I quote, the complex geometry. If I did the complex Mikrovsky space, I would forget about the sphere, but the real geometry I need to consider this nub-manifold and its equations given on this transparency here. It's a Hermitian condition. So you have one real condition to tell you where this P in is. Signature plus plus minus minus. Now, I'm not sure whether I should have various transparencies which I can flash up at you. I'm not asking you to follow the details in the formula, that's just, I haven't got time for that, I really only want to make one point, and this one point, that's the non-locality. But what one finds is that, and don't bother about the details here, what one finds is that these ends, in this picture, these ends that I'm talking about here, can be related the angular momentum, the momentum of a massless problem. Let me be afraid of you on this slide, but let me not worry you too much. There's an explicit formula. I'm not using notation of two spinners and so on, so let me not be worried. You know about these things. You can see everything. I can't see them both on the space. I can't see them. Anyway, a little more left there. As I say, don't worry about the details. However, the following detail I would like you to worry about, especially people with non-conversative geometry, I may say that I know very little about non-conversative geometry, but if you want to talk about quantum twister theory, the crucial thing is that the twister and its complex conjugate, notice that the index is up or down

32:30 depending upon whether it's a complex conjugate or not, this means that the permission expression which I briefly wrote down on the other side, this one is being invoked when one goes to the complex conjugation, you go to the dual space, so that's what's going on here. It turns out incidentally that the helicity of the... You can relate the z's directly to the momentum and angular momentum of an object in Minkowski's space by a specific formula, and the helicity is the norm. So that you can describe things which are not necessarily light rays, but something... I'll just explain this picture I have here, and I'm jumping around a little bit. I noticed that the original picture had light rays represented as points on Pn, which you've been also told about points which are not on Pn to give the full complex space here by allowing the photon, if you like, to have felicity. So that's an extra parameter. I should say the parameter is really the energy, you fix the solicitude to be plus or minus one, plus one up here and minus one there, and then the other parameter is the energy or the wavelength, which frequency, which increases the extra dimension that's needed in order to get you up to six dimensions. But I'm afraid these things would need to be done in much more detail if I can explain properly, you know, trying to polarise and so on, but you do have these non-competitive relations, which I would be very interested to see how this develops into some form of non-competitive... But what is the intention for you to write on that specific relation? It came really from what you do in ordinary physics. You see, the motivations came from... There is a leap of logic. It's not completely forced. It's almost completely forced. But it's... It's like quantizing by...

35:00 It's related to the standard commutation rules. So you want to quantize at the volume square or something? Yeah, that's right. That's the sort of thing, that's right. There's a phase, if you like, which is not fixed, knowing that, but everything else is, and so why do anything else? But I should, just to be quite honest, there is that extra piece of information which is not forced from this, but it's over. Right, and then you want to look at the real piece, I mean, you see what you look at the equations behind the real... No, because I'm interested in, that's right, because after all, things can happen in this situation. And that fits in with the real world. But one has these connotation rules which are very basic to twist the theory. Now, I'm afraid I'll have to rattle through these things, but let me just sort of flash them out here. So if you want to describe a twist of wave function, well, since the z and z-bars are conjugate variables, a wave function ought to depend on only one or the other, but not both. Now, what does it mean for f of z not to depend on z-bar? Well, it means d by d z-bar is nought. In the Cochi-Griman equation, such as f is homomorphic. So if you're really concerned with homomorphic functions, you can talk about wave functions. And if you're interested in the helicity eigenstates, this calculation uses to tell you you're looking at homogeneous functions. So, polymorphic homogeneous means you're talking about wave functions of a particular, for a particle of a particular helicity. Okay. And then how do you, well, if you didn't know about crystal theory, you'd have to write down equations like these, for different spins, for different velocities. How do you write those things in Christa form? When you write that... I'm ready for this, so... Is that a better track track? Some people would press more...

37:30 And you can see the homogeneity degrees over here, and it's just a function, and these are the vertices in the ordinary physical physics description. But the point I'm making here is that these things are contra-integrals, and contra-integrals you can vary your contours and you can vary your functions in complicated ways. And it took me a long time to realize what was going on here. Michael Teer was a great help in creating the appropriate understanding. But what emerged is that you're really talking about cohomology. So, I have to explain why I'm looking at just the top half of the twister space, this has to do with making your wave functions at positive frequencies, so make them proper wave functions you need to incorporate the positive frequency condition in some very nicely in twisted theory by simply looking at the top half of the system. These are details which I didn't take too long to explain fully, but basically one is looking at the check of the model. That's right over here. Now, it took me an inordinate length of time to realize something very simple, which is that the non-locality that's described here is very much what's happening here. You see, Twister Theory is non-local at a sort of elementary level, because when I was talking about light rays, it says points. And light rays, after all, are not local in space-time. So there's a more subtle non-locality, which comes about, Now, because you're really talking about cohomology, you know a description of wave functions. Now, I find that there is a nice way of saying this, which you can explain to people who don't know anything about cohomology, and so let me do it, and then I want to say what's got to do with this picture here. I could use that, yes. That's why we really need three transparent AC projectors. And here, you see, there's a nice illustration of cohomology.

40:00 I thought of this when I was, a long time ago, I was involved in a television program and I was trying to explain twists and they said, well, I said, well, it's something to do with cohomology. And they said, well, how do you, what's that? There's no way you could explain that to a general audience. And I thought about it for a bit and said, well, maybe there is, you see. So, why is this cohomology? Well, you see, the cohomology I'm talking about is a precise non-local measure here of the degree of impossibility. And you might come up part of the picture, it becomes possible. So that the degree of impossibility is a non-local thing. And you can give a very appropriate check description of this thing. Do it here, you see. Here we have, OK, you break your picture up into three pieces, and each one of these represents an object. And then you have to have some patching to tell you how to put it together. And this patching is for the logarithm distance from your eyes that gives you an additive thing. And you can construct a cohomology element from these different check things here. and the non-vanishing of that cohomology element is telling you the degree of impossibility. So, honestly, it is cohomology. So this is a way of saying what it means. OK, now what's that? Let's do with this. Well, you see, you're sort of... The cohomology here is of a very nice and simple kind. It's easy to be, basically, just real numbers you're talking about. It's not very cheap, it's simple. But in complex analysis, you have to worry about something a little more subtle, but it's the same sort of idea, and you get right driven. But you see, this is where you're driven when you twist a description of a wave function. And it's a wave function of a single particle. So, here, you see, okay, the wave function here is not some local quantity, which tells you a number of different places, it's this chronological entity, in Twister states it's a chronological entity, and the measurement somehow ties into something else, and it sort of breaks it. And once you've broken it somewhere, it's gone, and so you're not going to see it anywhere So it seems to me that this notion of a chronomology element encompasses the non-locality that one sees here.

42:30 You have to tell us, in your mind, where is the measurement process? Yes, it doesn't solve that problem at all. That's something. No, no. See what I mean? I mean, it gives you an analogy to your mind, but then one will have to be more precise. Yes, yes. I mean, what I think is that you... It's to do with when you tie it into gravity, and when you have complex, you have non-linearities coming in, and the... So what is it? What you mean is that when you tie it into gravity, that would be a co-mortial direction which would take place, and which would have exactly the solar... Yes, I think that's true. Yes, that's right, yes. So somehow, when the measurement's made, you're tying it into something else, it's not a one particle problem anymore, and it's really a quantum gravity problem because there's some displacement of large objects or something involved. So that's the idea. Okay, that's all I want to say about Twister Theory. I'm going to be slugging behind. Let me say something about cosmology. You see, all these things are mathematical schemes. Okay, there's input from physics here. There's input from quantum mechanics in the calculations of spin networks, and so on. But it doesn't tell us what nature thinks about quantum gravity. There's not a huge number of clues about what nature really thinks about quantum gravity, but there are clues. And so what I want to do is to put up a picture here of standard cosmologies as we now understand them. These are the different, I put in the cosmological constant which, although people call it all sorts of strange things like dark energy, I'm just going into the old-fashioned word cosmological constant, because as far as one knows, it does fit with lots of vacations Well, the alarming thing that there is one, certainly that's not new in the sense it is in all cosmology books. It hasn't been ever since Einstein put the number in somewhat reluctantly in 1917, tried to take it back out again, but you can't do that once it has been suggested. And of course, there it is, just as well because it seems to be there. So let me say something else.

45:00 So, here I have the various different spatial geometries, positive curvature, flat, and negative curvature. It doesn't even look at the pictures. Indeed, it doesn't. But I just want to say something about the negative curvature case. We don't know which it is at the moment. There are certain people who say it's flat, but that's not much good, because if it's flat, then there's no reason why it shouldn't be one of these. So there's always a bit of error. there are some observations which seem to suggest that there might be or some interpretations on the observations which tell us that the likelihood of this magnitude but that's not important, really what I want to say what I want to do just for the moment is to remind people of pictures like this this is actually not an Escher I couldn't find my Escher just at the last minute it's a pseudo-Escher done by people at the The point I want to make is that this is the Beltrami-Fancaré so-called conformal representation of the objected plane, hyperbolic plane, and you notice that she's represented conformally nicely in the Euclidean plane. Why do you have this idea if it's in the middle, because they are not... Don't be supposed to be eyes. Don't ask me, I didn't produce that picture. Because they are not allowed, is what I've heard. The circles are allowed, so... Well, they are... Oh, I see. Well, I forget the term, the hyper-cycles, they're not... they're not... Touch. That's right. No, these are the circles. But you can have circles. I'm not very good about that. The only point I want to make here is that in representing the entire infinite geometry in the Euclidean plane, you see you've got all this crowding around the edge here, that's because that's infinity for hyperbolic geometry. Where in hyperbolic geometry, the infinity is represented very clearly and without any difficulty in the Euclidean plane as a boundary to the hyperbolic space. It's barely enough. I should also, though, make this picture more accurate, not by improving the line quality,

47:30 but by introducing singularities of black holes. You can see we have the Big Bang singularity, but we also have local collapses in the black holes, and the point that I try to show people endlessly is that the geometry is completely different here. The geometry, you could say this is a low entropy singularity, these are high entropy singularities, and the geometric nature of the difference is, I think, best described driving home to the vial curvature, and you can always, I think it's nice to tell people who don't know anything about informal geometry, what the vial curvature looks like, because it's the distortion effect due to light rays, and here we have a beautiful picture of distortion due to some object in the middle here, and the distant galaxies are stretched around in this way, just in the eye, you just look at it casually and you can see it straight is complicated and computer analyses to work out that there is indeed a tendency for the galaxies to be stretched around here. And this is a lensing effect, which is a non-conformal effect on the geometry, and that is a direct measure of the vial curvature, or the conformal curvature of space-time. Now, why do I bring the vial curvature in? Because you could ask, What is it that's special about the Big Bang? This is simply observationally. We don't have a theory for telling us, well that's something I want to say something about. We don't have a theory for telling us what's special about it. But what we do have is, you know, it's extraordinarily uniform. Which means that gravitational effects have somehow been suppressed. And the suppression of the gravitational effects in the sense of making the vile curvature very small initial singularities have the effects of, well, here, you think Big Bang singularities was enormously constrained to at least one part, what it says, to one part in at least 10 to the power, 10 to the power of 123. Actually, it's probably, it's not a lot bigger than that, but tell me how very, very precise that Big Bang was in this organization. Now, there are various ways of formulating the via curvature hypothesis. I want to concentrate on a particular way of doing this, which is being exploited particularly

50:00 by Paul Tov, which has to do with this one here, the projector. I want to tell you two tricks. One trick I've been using for many decades has to do with discussing gravitational radiation and by using a conformal rescaling. So you say the g here is the physical metric, and you say you introduce a conformal factor, the local scaling of the metric can vary from place to place, and you give you a new metric These have the same light cones in Lorentzian geometry, you can say the conformal geometry is equivalent to the light cone structure, and you can use this to make infinity finite and talk about gravitational radiation and so on. This is a nice device for discussing gravitational radiation using finite, ordinary means of of the branch of geometry. The trick here is to shrink infinity into a finite place by taking this factor only going to be zero out there. So that's the trick. Now when you've got a positive cosmological constant, it seems to have, the trick still works. In fact, in a certain sense it works better. And there are various little problems that one has, in our case, which are removed. You get other problems which are not removed, which are introduced. the fact that some of the more serious problems in this description are removed when you have a positive cosmological constant. The main difference though is that you have now a space-like boundary rather than a null one. In this picture this is a light-like boundary, whereas a positive cosmological constant you get a space-like boundary. So that's trick number one. Trick number two is the sort of inverse of that. How do you treat cosmological singularities? And as I said, Paul Cosmos formulated the Laa-Clavergine hypothesis using this trick by saying, you now infinitely expand the Big Bang to get a nice informal boundary. And that's like the picture that we have. Well, first, this is trick number one, if you like, where infinity is made finite by introducing a formal type of zero boundary.

52:30 The opposite trick is to expand out the Big Bang. There's a crucial difference between these two. One is that the trick number one is basically automatic, especially when there's a positive cosmological constant. If you've got your fields dispersing out to infinity and so on, you can more or less say that there will be a nice boundary out to infinity. So that kind of picture is relevant in the Lorentzian case. So this happens automatically. Trick number two, the second version, is a huge constraint on the nature of the Big Bang, which is basically the Valkovitch hypothesis. So trick number one works automatically, trick number two is a formulation of the Valkovitch hypothesis. So what this is saying, here's the proposal, is that the Big Bang is a constraint in this quantum graph gravity, or whatever it is, is something that tells us that, this is a proposal, quantum gravity is telling us that the initial type singularities, the ones in the past, wherever they might be, have to satisfy this condition, that they can be expanded out to be conformed to the region. I should put this transparency down here, this is just telling us It's a formal geometry, the metric up to scale changes, angles, small shapes, but not length, not preserved. And the thing is, this is nine out of the ten components of the metric, which are preserved by these formal root scalings, leaving the light cone alone. Let's say you have only one parameter, which is the big bang, you see. See, everything is regular, the idea is, with regard to conformal geometries. Conformal geometry is extendable to something behind. So you see, the mathematical tricks are to say, okay, there is some fictional space-time prior to the Big Bang, which you can extend through, but it's not supposed to be physics. This trick is, OK, there's some fictional space-time beyond infinity, but it's not supposed to be physics. The idea here is to say, oops, maybe it is physics. So now this is, OK, this is a crazy idea, and I admit it's a crazy idea. Not as crazy as many ideas I've seen in cosmology, so I can feel that I'm just fine putting it forward.

55:00 There are various people who put proposals rather like it. And then Ciano has a proposal, a little bit like what I'm saying here, for some years, and Sternhard and Turok have a proposal, which is also a bit like it. But they don't have the same philosophical underpinnings in detail. So the idea here is, okay, conformal geometry is the basic thing, and the metric is a sort of article. So, and, okay, there's two sort of physical things one says here. One is, okay, what is the remote future likely to be like in this universe? One of the motivations behind this idea is the thought of how incredibly boring the universe, as we understand it, is. Because you have this, okay, you sit around and wait for this thing to go up to infinity. It's worse than that, because what interesting things are happening in this universe? are there, and if you wait long enough, they'll disappear by hawking evaporation. So just imagine sitting around waiting for a black hole to decay. I can't do anything more boring sometimes. So, I shouldn't use emotional arguments in this discussion, but it seems to me it's all right as long as you discard them at a certain point, and we look at serious observations and so on and so forth. The emotional argument is that this is incredibly boring, and that's is that boring? Because after, who's going to be around to be bored by it? Well, the only people that are going to be around to be bored by this universe are massless radiation. Well, this is the proposal. There's some physical hypotheses which go into this. The black holes have to disappear. You have to think of some way of getting rid of the electrons mass, which is a nuisance. But let's suppose that's all right. You end up with nothing but massless radiation. And does a photon get bored? No, because a photon doesn't experience any passage infinity is no big deal to a photon, it just happens, you see, and there it is. So as far as the photon is concerned, the conformal boundary to infinity is just there. It just fits through. And so, as I said, it's an emotional argument. What about the Big Bang? Well, you see, it's another reason why you might think the conformal geometry was a really appropriate thing, and there it's quite different. There it's the energies get so large that the mass becomes irrelevant, and it's a reasonable

57:30 that the main interactions, because the mass is gone, are also conformally invariant. So the proposal is that you have conformal invariance right near the big bang, you have conformal invariance near the infinity, and this tells you that the universe doesn't really know about the metric at that stage, and so it's not an unreasonable suggestion to basically glue them together. So here's the picture I'm having, and somehow this is our universe here, there's the big Bang, there is eternity, and that's the same, because it's lost track of the metric conformally, it's the same as the Big Bang of the new universe. So you have a previous universe, and it keeps on going like this. So this is the crazy proposal. Yes, absolutely. That's why I can't have one. Yes. That's right. If you had a big crunch, this picture wouldn't work. So it's very different. Oh, it's quite different. From what people... Oh, so it's a picture in which you don't have any frame of the line. You don't have... That's right. It just... See, some people have suggested, okay, there's a big crunch and that's the new Big Bang. This is the universe expands and definitely forgets the metric. And then, therefore, it's as a conformal manifold... It doesn't change. That's right. It just fits smoothly onto the next one. Okay. Because there is no actual metric here. make mathematics work. I understand, but what about your connection between L4P and the bar curvature? That comes about because there is an infinitely scaling in the metric, but yes, there are important The bar curvature has to be zero here in order to fit for the space-time geometry. But there is an issue of where the phase space goes. And, okay, I feel two minutes. I don't know the explicit answer to this question. And, in fact, there's a lot of swallowing of phase-phase volume due to the black holes. And I think it's connected with the Hawking... The difficulty people have about the loss of information. These are two black holes spiraling into each other. And then they send out gravitational waves. Here, these gravitational waves go out. And here we have the Hawking evaporation. Does he swallow some bi-commodation? It swallows... He does, yes. Yes, yes, that's right. Somehow all the gravitational information is swallowed.

1:00:00 Okay, one needs to make sense of all that, and I agree this is a crucial point. Absolutely, yes. Absolutely, completely. Now there are a number of points which need to be sorted out in detail, and that is certainly one of them, one of the biggest ones. So I'm worried about that, and I don't know what answer to suggest. I think the one I favor most is that it is swallowed by the black holes. And that this is for quite different reasons. I am a believer that the information is lost in the Hawking evaporation process, in the final explosion. There's not much of an explosion. So then you could restore a situation which was the same as the one you started forming. Yes, yes. so that ultimately you don't lose it. But it keeps to be looked at in more detail. This is the end of the story, but it's just curious that somehow nature seems to be telling us that quantum gravity, as far as going back this way is concerned, is pretty mild. It has a conformal space, and that's nine out of the ten components of the metric, it's a perfectly smooth manifold. Okay, it's a different story in the black holes. That's a completely different story. But it's okay, we need some timely symmetry and it ties in with the dimension of the problem and various other things in physics, so it's obviously a big topic, but at least this is a scheme to play around with, and I'm leaving with a thought. And I don't know if I've left any questions, but... Question. Does that not work if you remove the big bang? Does this not work? Oh, absolutely it doesn't work. We need the big bang, yes. There needs to be a big bang constrained in this very particular way. So if you extend it, what you basically see is the next evolution of the Big Bang. You basically, the Big Bang forms a log, forms the universe. Yes? Yes. Well, it's this universe, you see. The Big Bang here. See, I tried to draw two things at once. The purple line represents infinity in this universe. And that smoothly, as a conformal manifold, continues into the Big Bang of the next universe. But if you prove the Big Bang is incorrect, if you eliminated the Big Bang, you still have the universe.

1:02:30 But it's not quite a, it's not such bad, it's a regular region from the conformed geometry point. Well, if you remove matter, you remove the electrons, and you just have radiation, they will congeal. Well, if you go back this way. If you stop it, I look at the terms where backwards. If you put this diagram upside down, you don't change it. So there has to be a time asymmetry in the whole thing. Physics has to be time asymmetric. It seems to me that's unavoidable. We're talking about that, I mean, by focusing on the mass curvature, you sign on the Vichy curvature, and in fact that's the thing which we know is not controlled by. So how does that... Well, the Vichy curvature you see, although it's infinite, you see, in fact, It's infinite with regard to the metric you use on this side. It's actually zero on this side. But that's, the conformal re-scaling that goes from one side to the other involves an infinite... Yeah? ...produces a rich code. You know, I... I would just do it with the Einstein equation in terms of the matter. Well, the proposal... I can tell you what my thinking is going, but this is a still provisional statement, is that what you want is a conformal formulation of Einstein. And I'm happy to see that in recent observations it seems that dark matter is real, and someone doesn't perhaps have to worry about modifications in Einstein's theory, which change these laws of gravity. that's rather a relief, I'm worried about it, that if there really is dark matter, okay we don't know what it is, that tells us that it's okay for Einstein's equations to be very very precisely true, and so the proposal here is okay, you can make that into a conformal theory quite easily, just introduce the scalar field, the omega is here, and putting that That scalar field equals one gives you a choice of metric, letting it vary is another field that then conformally varies. The advantage of that is you can allow it to be zero, and where it's zero, you can't then get Einstein back again, but you can't where it's non-zero. Now I need it not just to be zero, but I'd like also infinity, so it's more like a projective

1:05:00 coordinate, which allows it to be infinity as well. And so you have to have a scheme where either being zero or infinity is still okay. And that's the only place where you have to go beyond Einstein. So Einstein is precisely right here and here. But here you have to have some more equations to tell you what's going on in the description. But the conformal geometry alone gives you quite a lot of constraints which should give you observable consequences. you certainly get density preservation which are correlated in ways that people seem to see and you don't have any inflation in this model in a certain sense when the inflation took place before the Big Bang that's a bit like what Veneciana says in this version so you can have correlations between shifts and events on the Big Bang as the observations seem to indicate it's a very classical picture Some people might find disturbing. You might think the Big Bang was a good place to see where the quantum reality is in its role. It's still there because of the big, a lot of little crunches in black holes. You can't get away with doing it this way. Can we actually observe it in front of the previous one? Yes, yes. What you see here, I've tried to draw this picture. Here we have black holes. these lines may represent colliding black holes spiraling into each other they produce gravitational radiation this gravitational radiation shows up in the vial curvature here ok, it scales away to infinity but the 0-resness field is still there it has a different form factor on the other side it's the normal derivative of the vial curvature which you see on the vial curvature vial curvature is 0, but its normal derivative is not 0, that normal derivative has to carry across it's going to be smooth as a conformal manifold, that carrying across introduces density fluctuations in the mid-stage. So you should be able to see, in the detailed nature of these density fluctuations, what's been going on so far. So, I don't know how you get that information out, but in principle, in principle it's there. I just have one quick question, which is the following, you know, when one finds an action person, but one finds the Einstein action, but there is also a small, very small, very small, and I was very intrigued from the staff by the book where I got to meet the...

1:07:30 Yes, so it's conformal gravitation, basically. Yes, so it's conformal gravitation. That's an Einstein theorem. That's a huge Einstein theorem. Okay. And it only has a small, very small, additional text, which is a correction. But it's like the square of the lack of gravitation. That's interesting. That's interesting. That's an interesting idea. I think you're the last one. I love this question. So, first of all, I have a comment. As far as I remember, the question is, do I understand correctly that there exists a formulation of skin networks in terms of craft and way? I don't know. I mean, of course, these things are. I'm not quite sure how that comes in here. You see, there is a sort of, it's a trick, you just allow the, there you go. There are rational variables in the sets, because if you introduce, when I say that the orthogonal group, that's only, you're only allowed at first sight things which have an even number of indices, if you like, so they're, if you want to introduce vectorial entities, they have to be grassland variables, I think that's, this is related to what you're saying, so you do need grassland variables, if you wanted to have objects with an odd number of indices, that's defectors, trivectors, if you have an even number, that's okay, so that you get non-competitive properties there but what it's got to do with non-competitive geometry I don't know but maybe there is a connection yes yeah actually that's right yeah I think we had about it

1:10:00 so let's thank Roger again