The Twistor Program

0:00 The last talk is represented in marriage. I think this talk is more like the Midnight Artois. I'm not quite sure whether the topic has to do more with physics, applied mathematics, or pure mathematics. I suppose really what I want to describe is a program that I've been working on for some time. If the program is successful, then the theory is part of physics. If it's rather poorly successful, then it's part of applied mathematics. This is a complete failure. Well, I won't actually prove any theorems. I will just only, unlike the previous talk, there will be no theorems in this talk. So it will be just essentially an outline of some of this program. The idea, when I say particles in the title, what I mean by a particle is what the physicist, well, the attempt is what a physicist means by a particle, namely something like a proton or an electron or a photon or perhaps even a graviton. Now, I want to one of the points about particles is physical particles that they involve certain numbers that people don't understand that physicists empirically notice that particles have certain symmetry properties that they can be described in terms of a certain number of quantities that they call things like baryon number or charge or strangeness that. Whereas these are only empirical things, there's no kind of theoretical basis which explains how these numbers come out of some broader mathematical theory. So part of the

2:30 ideas to try and change these things into mathematics. Well, the ideas that people have talked about in theoretical physics often involve perhaps internal structures and particles might be built up out of some smaller objects, and it always has a picture, a normal sort where something, you picture things going on in space and then of course you have to apply various rules to make it consistent with quantum mechanics and various things of that sort. And it becomes very mysterious because when one talks about small particles, the classical pictures that one has tend to be rather misleading and you have to apply these rules and it's not quite clear. difficult to form a picture of what's going on. Well, part of the program that I want to describe, part of the idea of the program, involves the idea that the normal space-time descriptions should perhaps be replaced by something else, which is initially equivalent to the normal space-time equivalent, space-time descriptions, but which leads one in different directions from the way in which the So what we have is, first of all, a geometrical translation from the normal space-time to some other picture. Let me make a few headings first of all. I think what I'll do is outline broadly what I want to say, and then in a little bit more detail, and then I'll do it in more detail still, and I don't know how much I can get through, but I will just continue this way. Thirdly, we made some headings, first of all, the geometry. I want to say something about the geometrical picture. Secondly, classical particles. Thirdly, quantum mechanical particles. and fourthly, well there's a whole column which I'm not going to talk about in detail you might have many particles, these will be the descriptions of a single particle

5:00 that's actually a particle, a single singular you might talk about many particles quantum fields, or we might talk about interactions, or curved space. In fact, what I'm going to describe will be entirely in terms of flat space. The program does in fact have to do with these things. In fact, it doesn't have something to do with curved space, and there are some interesting theories that one can prove which had to do with the geometrical relations, the geometrical structures I want to indicate when one generalizes from a flat picture to a curved picture. So anyway, I won't say anything about this, but I'll try and say something about these three columns. First of all, the geometry. Well, the picture that we're going to have, we start off with the ordinary picture of a flat space-time according to special relativity. It's a four-dimensional real manifold, flat manifold with a pseudo-Euclidean metric to the signature of 1 plus 3 minus 2, so we can see the Euclidean space. And we're going to have a correspondence between that space and another space, which will be a four-complex dimensional space. Let's call this one M and this one T. Any point in this So this space here will be represented by, this will actually be essentially CD4 with some structure that you put on it. It's a complex four-dimensional vector space. Any point in this space will correspond to a linear two-dimensional subspace in this space. So here's the origin there, some linear dimensional subspace in this case, there will be only certain ones that will correspond to points.

7:30 We can also go in the other direction, a point over here will correspond to some sort of structure over here. Roughly speaking, the picture will be the point over here corresponds to which moves with the speed of light and which has intrinsic spins. So if you think of this thing in terms of a line, I tend to draw these lines to 45 degrees to indicate it's going on a screen of light. The time is going out that way, and one always scales the picture so that the velocity flies into the one. So 45 degrees means that that particle is going on a screen of light. So we imagine this thing right there, but it's also got momentum, which means it's got an energy to it, which scales it, and it also has a spin, which scales it as well, so we have to bring that picture to Vega. I'll make this more precise shortly, but at the moment let me just give a rough picture of how we translate from one to the other. The translation is not a point correspondence, but it's where points over here correspond to certain structures over here, and points over here correspond to certain structures over here. Now, notice that this is a complex space, whereas this one is real, and part of the philosophy behind the whole program is that one intends to have a different kind of marriage here, a marriage between the space-time structure and the quantum mechanical structure. Because normally in physics one has these places where two very, very basic ideas in physics. First of all, the space-time structure, where one uses real numbers to describe space-time. one has a man-code, four dimensions, it's a pure man-code, and secondly one has quantum mechanics, which is based essentially on a complex field. So we have these two basic ideas which are initially thought of as independent of one another, one could do quantum mechanics and space-times which have different numbers of dimensions than the number of dimensions that we experience.

10:00 And you need to do quantum mechanics, so you choose one or the other, you don't have to get anything mixed up. But if we are really trying to understand what's going on in nature, and these things are both features of nature, seems that one ought to try and find some closer marriage between these two ideas. And I'm not talking about quantizing general relativity, this is something perhaps more basic even than that, where one is trying to get some interrelation between the number systems which are involved here. And just as an indication that there is some relation in physics between these structures. And what I mean by that is that there is some relation between the number of dimensions of space and the fact that one uses complex numbers in quantum mechanics. One can see there's some very sort of primitive considerations. Because in quantum mechanics, you consider the simplest sort of spinning system, a spin a half particle, it's supposed to have two states of spin, you can represent by spin up or spin down, that means it's going around that way, it's going to be one double. These are the two states, two states which are supposed to span the space of possible states of spin. You can form linear combinations of these, and then the mu are supposed to be complex because that's one of the rules of quantum mechanics. And it's the ratio of lambda to mu, which you're concerned with here because you're concerned with states of proportionality. And these are complex numbers, so these ratios tend to give you the points of the sphere, the complex numbers together with infinity. and the force of the sphere gives you the different possible ways in which this arc will spin. So, for example, whatever matters together, then it gets spin in some other direction. And the fact that you have a three-dimensional space, and the possible array of directions given by a sphere is two, corresponds to the fact between these complex numbers and quantum mechanics. So when it has this kind of interrelation, I don't want to say that you couldn't do quantum mechanics in five dimensions or anything like that, but there is this relationship.

12:30 They're not really independent of one another, but some kind of things as well. And the program is to try and exploit this sort of thing, not just talking about spin directions but where one discusses the whole geometry in a way which involves complex numbers in an essential way. So one of the ideas of this sort of correspondence is that one is bringing in these complex numbers essentially into this description of the space-time and And the complex numbers of the accomplishments are really like the same numbers as the ones that come into this description. This is a bit vague, perhaps, but certainly one of the guiding principles behind the whole program. Well, I should say a bit more about this first and I'll come back to it. So, the objects which belong to this space are labeled by Lenz, and they will correspond to particles moving with the speed of light, zero mass particles. Now, this is currently a class column. I'm really now moving myself over to this column, a geometrical description one here without talking about particles at all, we can just talk about geometrical constructions. To make this correspondence more precise, I'll come back to it. But in terms of the particles, how does one describe a classical particle? Well, if we're talking about a space-time description, a particle will have some sort of a whirlline, maybe a straight line or maybe a bent line. I'm talking not about interactions, so I'm talking about three particles at the moment, so we should consider it's moving along a straight line somewhere. That line will have a certain direction in space, so it'll have a four-dimensional momentum, which tells you the direction of that line. it's also located somewhere, and it also has some sort of a spin, a particle, talking about particles which may have spin, so it's a spin as well.

15:00 If you want to represent all these things together, one of the ways of doing it is to choose some fixed origin somewhere, and think about the angular momentum about that origin. Now, if you're talking about relativity, the angular momentum is some object M, which is skew-symmetrical tensile. It has six independent components, six real independent components. The momentum is something called B, which has four real independent components, and you have a total of ten altogether. If you know the angular momentum about this point, and you know the momentum, then you can find out where the center of mass of that particle is. So this information will characterize everything about the dynamical state of that particle. It doesn't characterize other things which may be sort of internal numbers, but it doesn't tell you whether that particle is a proton or an electron or what have you. It just tells you these dynamical quantities like the momentum of the spin and where it is. We have ten numbers. If this particle happens to be a particle, a zero-less mass particle, a particle moving in the speed of light, such as a photon, then it won't have as many numbers to characterize it. The momentum has to be something which has a zero length, points along the light curve. Furthermore, spin has to be especially It's actually related to this momentum in a certain way, and in fact it has to spin out of the direction. So in fact what we have in the case of zero mass is m not equal to zero. We get to ten quantities in case m equals zero. We have a total of, well we lose one of the momentum because the mass is zero. That means that you actually have three numbers in here, and then there's the question of where it is. Three more numbers, and then the how much of spin is, which is one more number, and so we have a total of seven.

17:30 In the case of zero mass case, we have a total of seven parameters to characterize that particle. All right. Now the picture we're going to have is that the point in this space here will represent directly one of these zero-class particles. And that's seven numbers, whereas Because if we're talking about a point in here, you require eight numbers, there's one extra number, which means that there's a certain, if you go this way, there's a certain degree of freedom that you lose. Let me just take it here. This is the particle. The number of degrees of freedom of the particle. In this case it's 7 for a minus 0 particle. Whereas the dimension of the space T, and this I'm going to call Twister Space. The reason for that name may not emerge in this talk has to do with how one describes these things geometrically and one has certain twisting systems of lines over here to represent these points. This object Z will be what I call a twister and it is now, it is a vector in a four complex dimensional space. So this will have the number of degrees of freedom for the twister, this is a particle, would be eight. And this means that there's a certain freedom involved, and this is the dimension of the group, the symmetry group if you like, which is the transformations among the, in here, which leaves the description over here invariant. And so this is S1. The reason I'm putting this out like this is because this is really the first stage in the hierarchy. This is, you know, the simplest type of particle, according to the scheme, is one which is described by a single object of space, and also has more complicated ones where you might have two such objects.

20:00 And then, of course, we're going to have five dimensions here, 16, the number of degrees of freedom. So when we have two of these, then it turns out that the mass will not be zero until we're now here. And this means we're going to have 10 degrees of freedom and the group as six dimensions. I will describe this group and indicate what this group is later on. In the case of three these objects, you can go on and build them up. Take three of them. Well, you don't get particles which have any more degrees of freedom because once you've got mass not equal to zero, You still only have 10 degrees of freedom in the inanimate branches. There will always be 10, no matter how many you have, I should put the number of crystals there. One, two, three. And in this next case, the group actually has 15, sorry, this is 24 now, we've got 3 times 8, and we actually have a group of 15 rounds, it doesn't add up right, but that's because Because in this case, the group was simply transitive, the group acts on the space, but it's not simply transitive, one has actually h-dimension. You can go on like this, we've got N there, we've got 10 there, and we have 8N, and in this case it's N2N1, full of aspects. I will give you what this structure is but I really just want to outline the general program first of all it's a classical description

22:30 it's not necessarily meant to describe anything actually in nature because one has just a common description and when we go over to quantum mechanics one of the things about quantum mechanics Let's go to the ordinary description and particles in the ordinary way, where one talks about position and momentum. Now, as long as one's talking classically, these are just real quantities that you commute with one another and everything. Then when we pass the quantum mechanics, we do this mysterious thing which changes these things into operators, and they're supposed to be here with each other anymore. Convocators, one lost, usually 15 equals one, those lost the blood. Everything equal to one from the side. So we're going to have one where it's alive. And then on talks not about, one talks about the position of the momentum of an individual particle, here what one talks about is wave functions, so you have some functions of x, if you like, or it could be a function of p, and the x's and p's now become operators on this thing, where p becomes essentially p by the x's. So you throw away the p's, replace all the p's by d by the x's, and you're not going to talk about functions of one set of the variables. You couldn't throw away the l one, but I throw the x's away. When you think about d by the p's over here, you just make that choice, or you might do something more complicated, which is some mixture of the two. But over here, what we're going to do is limitate this procedure, and what we find is that the object Z, which is an eminence of C4, remember, which is supposed to describe a particle, well, instead of having MCP, what we have is Z and its complex conjugate. because these are all real quantities, so one doesn't worry about complex conjugues.

25:00 But here, what one has are two objects which are very well similar to what it seems to be, they're canonical or conjugates, and one has a commentator, like that, black, and it was an indices of these things running over four values, and from the sphere of relativism, this one, because this object belongs to the dual space. It has a certain kind of structure on C4, a Hermitian form on C4, which gives us a correspondence between the dual space and the convex conjugate. Take the convex conjugate and then that's the dual space. And I write Z bar. That's why I'm going to index this downstairs here and upstairs here, because these are things which are dual objects. And what we're going to have is I'm sorry, I'm going to change it. Zeta and Z equals chronically delta. So for each value pair of non-equity conjugate objects. The reason for this, I suppose it's a hypothesis. I'm never quite sure about this, because it's almost implied by the algebra. When you translate things from the space-time description, you get quite close to that, but it doesn't seem to be that it's exactly a logical consequence. On the other hand, it's a very natural thing to write down, so I think I don't quite care which thing is a hypothesis and which thing is a question. I don't mind which thing is, I'm not going to lie to have it. Alright, well, we have this rule which means now, when I'm talking about quantum mechanics here, that the analog of the wave function is going to be some sort of a function of, of, let's say z, let's choose that one. And whereas over here, for our wave functions we don't choose functions of z, z bar. Well ever since I was undergraduate, I was always very puzzled by this because when we did hydrodynamics people used to talk about

27:30 functions, or z functions z bar, and things like z bar, and it's always a little mystified by this, because z bar has gone to z after all, so it just means, so you think it's a function of that, and it's a function of that. So, after a while, I got used to it. Anyway, to say that aside, there's a function of z, what I mean is that the partial to be higher by z bar is 0, and if you choose to, you can write that all out in terms of real-dimensional parts, and that's a set of differential equations on the thing. I don't like to look at it that way. This thing is to mean our function which is independent z bar. In other words, it's a holo-walking function, a complex analytic function. So there's the other part of the title. Psi is to be a holo-walking function. Of course, these things here just have to be smooth things if you differentiate, they were convex functions, all right, but not homomorphic. Whereas here, we insist on the size of the homomorphic functions of z, and that's the statement of saying that we're not bringing the p's, here we're not bringing the z bars. Well, things like tricks like this are familiar with quantum mechanics and various pages, but here it's used in a sort of universal way and it's an essential part of the whole program. Alright, so we have now these functions, and these functions describe particles, but they describe particles quantum mechanically. And what one has to do is have some way of going from a function z to some sort of space-time field, which depends on position. And we want some kind of description, which goes from there and back again. See, it doesn't go back too well, but it goes this way nicely. And this description turns out to be some sort of a contra-integral. I can be more explicit if I have enough time later on. The idea of what does a contra-integral

30:00 on these functions, and again, that's crucial, these things should be homomorphic in order for the idea of contra-integration makes sense. And now we get some sort of a field on space-time, which is the standard And what you do is you insert actually a lot of the things in the table of indices, and then these indices emerge up there, and it turns out this field is some sort of a field described in a way people are more used to describing fields on field theory. satisfies certain field equations, and in fact, what you find is, at least for the case where you have one variable here, as I've written down, that you're supposed to be getting particles with zero rest mass, when we talk about fields, or when we talk about wave functions, This means that you have certain types of fields, in particular, the Maxwell field, e.g. Maxwell field, something you get. What you get, by means, this description gives you something which automatically satisfies Maxwell's equations. You start off with a function here which is quite arbitrary. So as far as the space-time description is concerned, you get something which automatically satisfies the few equations that you have been interested in. Depending upon how many things you insert here, you either get Maxwell's equations automatically satisfied, or you may get the linearized version of Einstein's equations. This is a different spin I have. Notice for equations, the spin is equal to 1. In Einstein equations, the spin is equal to 2. You can adjust that by this description. These are only these weak-field Einstein equations. They're not curved space, really. They represent the space which is flat, but you like it with the intestinal curve. So this is what this gives you. There is a more fancy prescription, which actually does something similar and produces curve spaces. But let's see this column over here,

32:30 and I won't have time to talk about that anymore. It is part of the same program. Alright, well this is, if you have one thing here, that's to do with the first row here. The analyzer for the next case, you'll get two of them. You take a function involving two variables. This function has to be homomorphic in each variable. And again, you have these prescriptions, You put a middle-up jump in there, and out you get some field in a middle-up position. And you can produce fields which are not about masses, but which satisfy massive field equations. So you can start producing fields representing into the wave functions to represent masses upon them. or you put three of them in here, or four, and things get more complicated as you go down. You become more and more alternatives to things you can insert in here. You always get out of the end perfectly decent space-time field. Now, I'm going to continue this onwards, but it seems quite possible that anyone should stop here. The idea is that you take, you look at this previous column, and you look at the group which comes in here, and perhaps I really should be in more detail, but let me just the group about this and come back to it. The group here, you look at the generators of this group and you have certain quantities which involve the z's and z-bars and the x's and x-bars and things like that. What you do is you do quantum mechanics to it and that means all those z-bars are placed by d by z's and the x's and the x's You get rid of all the quantities involved in complex quantities, and now you have a whole lot of new operators, which came from the group here, which are now just operators. And the thing is, you now subject these functions to these operators, all you do is you take the group

35:00 and you look for what are called a complete set of producing variables to construct out of the generators of the group, or as we take the end, developing out of the group, we construct a complete set of commuting operators. There's a certain amount of ambiguity in how you do this, and this is really where one has a bit of freedom in the interpretation of things, but at least a number of these things is fixed. exactly which ones you choose is a matter of decision to perhaps play around with it before knowing what's the best choice. But let me write down how many of these things are here. There are many set of commuting operators here. So again, those quantities are the things which you act on the functions f, and you ask for the functions f to be in eigenstates of these discrete state of operators. And then the eigenvalues will give you a number of numbers which will characterize that particle. what the hypothesis is. When I started at the beginning when I talked about things like barium, and strangeness, and all these things that physicists talk about, the idea is that that's what these things are. This is a hypothesis, and there's no clear contradiction with a hypothesis. As I said, there's no choice as to how you choose your operators, and it does it might be what one wants. You have actually three classes of particles, perhaps more. These ones are all masters. These ones have a concurrently simple structure. The hypothesis here is that these are perhaps what are called leptons. And these things have a somewhat more complicated structure, which you want to call hadrons. And the thing about these particles which interact strongly, leptons are particles which don't interact strongly, and then these are the ones for which the mass is zero. There's also an extra rule about this, you can see the remote cloud here, the hand-run cloud or the leptons cloud and the

37:30 rule is that anybody that belongs to one of the higher-up clouds has automatic membership one lower down. So these steps all fit in here, and these ones all fit in here too. If you read that, you know that the speed makes sense. But with that rule, it seems that one might be able to work things. And the kind of symmetries that you get, the fact that the group that actually comes in here is very close to the sort of group that physicists actually use in order to classify parameters. And so this is one of the reasons for thinking the past one has three quantities here. They have to do with what we've probably heard about quarks and things. In fact, you have three of them and you have groups which transform. Well, these things are related to that. You don't actually, they're not quarks, but they're something which gives you the same symmetry coming out of it. Anyway, so that's the general picture. Whether it will work in detail is a question for the future. and that's whether, as I say, whether this is physics or high-magnetic analysis here. Well, let me say what numbers we get here. The real number of crystals, this is an atomic calendar picture. The real number of degrees of freedom. Well now, number of degrees of freedom now means the number of complete set of computing operators that you have. Now, for a particle, for an ordinary mass of particle instead of 10, you have this 6 here. This is the, has to do with that, the set of things that compute with one another, spin component of one another. And so you can only find a subset of these things to give you the characterizing part of it. These six are part of the ten, and not the six. Yes, perhaps I should put a color, I want it to be clear if I put it here. Let's try out the quantum mechanical colors will be blue, and this will be six there. And then this would be 6 all the way down. It has to do with what we do, I'll just say briefly, you take a bit of momentum components, which is 4, and then you have a total spin, which is 1, and then you have

40:00 a spin in one particular direction, which gives you one more. And that's a lot. It would be 6. In this case, instead of 7, what one has is 4, you have essentially three components of momentum and then the velocity spinning in the direction of momentum, which is just one more. Over here, basically one has a path to number. Well, it's to do with the fact that you throw away the path to variables. So essentially it's a number of components that the function is of. So, it's a core here, it's a number complex, because that's the first number of real, imaginary parts. Here we have 8, here we have 12, etc. And here we have 4n. And then the number over here, well, from the group, the group that arises here is what one might think of as a sort of internal symmetry group. It's not quite perhaps a fair way of looking at it, but it really splits into two bits. The symmetry bit is something which really should be sitting over here. here, then just, perhaps I'm going to stop down here. What one has here is, I write 6 plus 2. So these two numbers here, the numbers that you get here are the numbers which tell you what kind of particle you've got. They don't tell you how fast it's moving and they just tell you what kind of particle it is. But in the notion of what kind of particle it is are two dynamical things. One is the actual mass and the other is the actual total spin. And that's what those two numbers are here. That leaves the six left over. I mostly want to agree are part of those six. The six left over are what's left from here. So that number plus that number has to be that number. And this number is always two. So in this case it would be 4n minus 6 plus 2. Those two here, the brackets, really belong over there, but they do come from the root. So the complete set of commuting operation, the root is that plus that, but two of them really belong over here.

42:30 And here we have 2 plus 2, and here we have actually 0 plus 1. It has a zero, and it's only a spin. Well, I'm just giving you a lot of figures there without telling you how to get them. Say a bit more about the details. Let's start off by setting a little bit of a grid. I'll sketch that, rather. I'll go back again, beginning, and say a bit more detail about each column. Alright, let's say a bit more about geometry. I want to show how we get from a point in space-time to this linear two-dimensional space over here. Normally, one represents points in space-time by a position back as its perspective. with the origin somewhere, so you would have, say, some origin here, and that would be x. You have four real components, all of x, x1, x2, x3. So that's the time component for these two space ones. What you do, first of all, is to express that as a matrix of x0 plus x1, x2 plus r, x3, x2 minus x3, x0 minus x1. So the emission matrix, we're actually going to do two in that, for various reasons. Our emission matrix provides you're talking about a real vector, and I'm going to call that x, a, a prime, representing the rows and columns of this matrix. Now over here, we're going to have the thing z, Instead, there's four components, complex, and the first two of those I'm going to label omega as an index, and the second two, pi, and this index is going downstairs, the prime

45:00 means that it belongs to the complex concrete space of this, and the fact that this is downstairs means that it's a dual space. So this belongs to the dual-complex compute space z. This is part of the notation that's supposed to keep track of the spaces. So this circuit z takes the value of one and two. One and two, yes. These are just two-dimensional modules. And what I'm going to say is that this twister z goes through the point x, if known then, the relation, the following relation holds omega equals omega pi times x times pi. Now, when I write two indices, I automatically think around this and all that. Excuse me, there's something over those things. So this is really matrix multiplication. I'm going to make it x times the column go to i. That's how it's arranged. So this is just what I've said. x, when you look for all the z's which go through x, well, that's going to be a linear space. And in fact, what you do is you let pi take all values, and then only get a six by pi. And so we're going to get a two-dimensional linear space. So this shows you how you represent points by two-dimension of these squares. Suppose not every two-dimensional in this space will in fact give us a point because there are conditions in x-intermission. If we didn't have that restriction, then apart from certain means of infinity, and I'm not going to worry about that, we would have the correspondence going the other way too. But we don't, because of the condition of the actual emission.

47:30 So we've just allowed every two-dimensional space we would get actually complex space-time from here. If we want restriction for the space to not be actually real, then we need to introduce an extra structure over here. And this is the emission structure which I mentioned. We have a correspondence between Z and its complex conjugate. here, let's say we can find Z bar to be the quantity Z. If you find the Z index, it's supposed to go down, that means it belongs to a real space, and what you do is you conjugate both of these things, which means that we have Z bar, R bar, R bar, R bar. The point about this is that we have an emission form, zz bar, on the space t. Now, it's not hard to see that if x is a mission, then, in other words, the cross comes from a real point, that this thing is necessarily equal to zero. So what we have, then, in our space T, following, we have this mission formed, which is plus, plus, minus, minus. It's not written in standard form, but if you diagonalize it, you like that. A certain space in the middle, which is zz bar 0, zz bar red to 0, and zz bar less than 0. Now those linear two spaces which lie entirely in this middle region are the ones which correspond to the points. The ones which do the other things correspond to complex points. So if you want real points, you have to make sure that thing lies in this middle region here. There needs to be some sort of reality description, otherwise you can never live in this case of course. There's a lot more I can say about the geometry here, but I think I don't really have time to go into it. It's quite interesting the kind of duality properties that you get.

50:00 If you can mention one point in the costal space here, then if we choose a point compactly on here, then we can represent that in terms of an actual straight line in costal space one of these 40 by 3 things, representing the path of a zero mass particle, with a certain momentum attached, there's a certain scaling attached to that line, and that will be represented from here, apart from the base region. Let's just tip into the point Z, Z versus Z here, will correspond uniquely to a certain line here, with a certain scaling to it, which is called the momentum p, you can like it. And there's lines of the points which that goes to. Yes, that's right. The point is x, that z goes to. In fact, what would be the lowest points on that map? If, in fact, z is up here somewhere, there just won't be any points that it goes through. Or, if you like, there may be a lot of complex points, but there won't be any real ones. One can actually obtain a real description of the points up here, and that's in terms of these twisty systems of things which I'd love to talk about if you're not going to be able to talk about anything else that I do. Okay, well I think that's all I've got to say about the jump at the moment. Let's go a little bit to the classical particle. What about the momentum and that new momentum? What one has, the momentum of a particle is described by quantity of four components. It's actually a dual object in this case, sign points. What we do is now translate that object by means of this kind of description. We actually have to be careful about machines up and down just to change signs. I won't bother about the details of it

52:30 so there might be a few old signs of things coming in. because it makes me it down, and now you have to reverse the signs of those three. So let's not worry about that. P will correspond to an object, a permission matrix again, and because this vector is null, that is to say, it's pointing on the light cone, represents a positive speed of light, this actually factorizes, so it doesn't form pi, pi a bar, that's the convex quantity of that. And that's where pi is coming in now. Before I would just say, here's what the pi means. The thing that you multiply together and it gives you the momentum vector. Now we're on the angular momentum about some fixed origin. On the angular momentum, this is taking the two indices which are skewed and real. And each one of these indices now will come to a place by a pair like that. Each vector thing will come to represent as a matrix. And so it happens. For a massive particle, you can wrap this thing out like this. I'll just write it out without an explanation. Except to say that I could write brackets around things that need to be symmetracked and add together the same thing with the other way around. Epsilon means that we've achieved a symbol and then you've got a compass. It so happens that by the introduction of one extra quantity, omega, the angular momentum has that form. This depends, crucially, upon the fact that the spin is pointing along the direction of the momentum. If you just say that the momentum points along the back, In fact, this doesn't tell you anything about where the spin points, but it has to be a fog of momentum that might be pointing in any other direction. Whereas actual particles, it so happens that the spin is always in the direction of the momentum. When I say a

55:00 fog of course is a fog of itself, so that's why. It so happens that the real massive particles have that property. Some people are mystified by that, and other people aren't. I'm not quite sure which candidate I want to. I'm certainly glad that's true because the whole formalism depends on that. That's what comes up with this. All right, so now we have omega pi, which we call z, giving rise to p and m, where p and m are the p and m for a massive particle, then there exists a z. But there exist many These are z's, maybe z, and these are z times z for any real theta, and that's the complete ambiguity. So the group that we have here, the dimension of this group, one on top of the other group here, is the group. Group one, the group of units with one by one matrices. So that's g right here. Well, let's not say more about that, but it has to do with the fact that you have one over here, too. What do I have? About two minutes or something. I could explain how this thing actually gives rise to skin. Okay, that's one of these things. The group that I'm talking about here, for n twists, is the following. Given by transformations of the following time, let's say x up to z, and then the complex conjugate let's put these things in like that and then we have some matrices that's all these things, linear it's something linear into the x and the complex conjugates

57:30 And then we have the unitary matrix sitting here, unitary n by n, if you've got any unitary here, U is the military, right there, times chronic conductor alpha-beta, and then we've got lambda times the military matrix and the thing I alpha-beta, I'll better explain that a moment, but it's bar. Lambda, bar, u, i, beta, and bar, delta, alpha, but these indices are supposed to stick onto the ones up here. This is part of the matrix containing u, Lambda, lambda is skewed, complex, and by everything. I hope I put it right to the video group. I is such that I draw 1 is equal to minus 1 more, equals 1, and the rest of the components comes to zero. The other one, the other way, is 2, 3 times 1, 2, 1, raise 0. So that's the proof, and the number of dimensions, I think you can know what my written I'm going to write that the n is 2 and minus 1, one of the real parameters. Then what we do, we want to do one kind of thing, we look at the generators of this group, we look at first of all the various infinitesimal elements

1:00:00 of the algebra, and we can express these operators are things like x alpha, for example one of them will be x alpha, we've got a z alpha in between. Things are still half of things like that. I need a combination of things like that. And now, we're also going to get things with valid formages. So, these are operators. The algebra of these matrices is the same as the algebra of the thing where you sandwich x's in here with the partial differential operators. I'm pretty honest with running through this work. So just a general idea. x's and d by the z's and x-bars and d by the z-bars and things like that and you do what I would say, which is doing quantum mechanics which is get rid of all the bar quantities and replace all the x-bars by d by the x's and all the d by the x-bars by x's and you have a set of operators which now don't involve any bar quantities at all but which satisfy exactly the same commutation rules as the ones that you have to begin with. So you start off with the algebra which is both of bar quantities, you translate this into the description and you end up with a whole lot of operators which don't involve bar quantities at all. And these operators can now be used to act on functions. And you can find a big set of new operators. And then you can ask whether a function should be... You can put in all sorts of things in here and produce a whole through the different fields. So that one function would produce many, many fields. And if you say this function must be in an eigenstate of the complete set of comedian operators,

1:02:30 then this one function will only produce one field. So the idea is that if it's not in an eigenstate, you've got a mixture between a lot of different kinds of particles. If it is in an eigenstate, then you settle down and you say, is that kind of a particle? until you get that one theologian. And the thing is that... The idea, anyway, is that the power of informalism is that you have in these functions more information than you have in the fields. You work over here rather than over here. Because if you work with the fields, all it's about as a field, you don't know what kind of a particle it is that's supposed to describe. But if you're working with a function, the information is contained in the structure of that function, so what kind of a particle it is. And one has the possibility of trying to express the differences between particles, and how to respond to this. Well, I could describe much more of the details of how this works, but you can see around time. Excuse me, what is the top function that corresponds to? Ah, yes, yes. There's one point. You were just saying... Yeah, the top function, what is that corresponding to? The velocity corresponds to the latons and the hydrony type on the cells. Oh, yes, these are... Those are the fastest particles, which might be decretors or photons, or the graviton that you are working with weak fields. if you're actually, if there is, even if you're not working with a human field, you can describe it in the same way, but the rule is a little wrong. But this would describe massive problems. But there are a number of things I should have said, let me just think one point which I searched, there are these functions, I said they're functions, they're a whole lot of functions. But they're clearly not functions defined on a whole space, because if they were, when you get a contra-integral, you just ignore. They are in fact functions that find on certain subspaces of the twisted space. So you have to cut holes in the twisted space, and you have functions defined on that, and what you're concerned with is how this can behave as you go around the holes. And that's what these contra-integrals are actually doing. Part of the most difficult problem in the whole thing is to understand the topological

1:05:00 relation between the structure of these old things. There's quite a lot of topology that's involved. In the high dimensional spaces we're concerned with and where the region of these structures are on the board and whether you can find concords which actually go around those complicated regions. It's really quite an important problem. I'm beginning to learn that we've even reached the limits of what's known Some of the questions we want answered to, in the actual way, the form of justice is really answered. Maybe there are some areas of research that would be in the region. And that's it. It's an introduction, perhaps, there. Well, Professor Mendel started by saying that if Stolt had a with physics, with bio-mathematics and geomathematics. It seems also that it has a philosophy of theology, perhaps, I'm probably able to judge this really from almost any point of view, but certainly it's a very beautiful feature of formalism and if it actually tells us how things work, Thank you. Thank you.

1:07:30 I don't know. I don't know. Oh, my God. I don't know. Thank you.