Hiley Approach to Twistors — Ideals in Algebra, Ontology of Vacuums

0:00 Thank you. We'll be right back. Oh, well, well, fine. I don't think it's served in any way. Hello, Battle. Oh, I haven't started in there.

2:30 It's Bob's downstairs. Nobody else? Keith said he was... Keith said to me he was... I spoke to him on the phone about two hours ago He said... He said it was up in the air but he was almost certainly coming, or probably coming Or probably not coming Well no, he did say no He began to understand that he was going to try and get here Owen's here and now he has disappeared Well, I mean, are you sure he's not just gone to get himself a cup of tea or something? Well, if he has, he's gone for half an hour, nearly three-fourth an hour. Well, maybe Bob knows where he is, so maybe he knows where he is. Well, I haven't seen Bob either, so... Well, he's just standing downstairs, he's just running using his mobile at this moment. Oh, right. He should be hearing him out. Yeah, I probably should have sent an email. I know our letters not well. No, no, apparently she... Well, according to Keith, she's a lot better today. She was a bit rough yesterday, but she's feeling a lot better today, nor do I, she really does need to get proper, she really does have to have herself checked out properly to find out what the source of the problem is. Oh, by the way, thanks for looking after those for me, and I'm going to get them again. Can I just... Yeah, sure, I won't interrupt. I did want to ask you a bit about this. Oh sure, I'll shut up. In fact, if you want me to go in the house, I will. Can I just sit here quietly for a minute? Just sit there. Yeah, sure. Shut up. I don't know what's happened, but there's only... I mean, where are we to go? That's why I was good here, isn't it? It seems only like yesterday. Owen was here all afternoon.

5:00 We haven't seen him around. And I said... And I said, you know, do you know who's coming? He said, well, I'm here, and I said, Keith's definitely coming. Well, and I spoke to Keith on the phone about two hours ago, and he didn't say definitely, but he said he was going to try and get here. I should have put an e-mail up and, anyway. Well, if you, if you... It's a nuisance, Bob. Well, I had a problem about getting my car back. It's not quite ready yet, so... Right. It sounded like you were having some pretty heavy household wear on the phone. Yeah, well I've also found that my current employment finishes at the end of March. Oh, I'm not very happy. Well, I thought it was going to come, but I thought it could be in May or in March. Well, you want a sort of short-term contract? Well, the company I work for is, you see, and I think they're running out of places to put them in. That's not so good. I don't like hearing those kind of things, Bob. Go away. Well, the IT world used to be really, you know, quite good. Yeah, what's happened? It's just, everything's just dropping to the floor, you know. There's just an open, there's just a... Or is it a recession or what? There's a big, in the IT, there's a really big recession, you know. Well, certainly what the financial pages are saying. And also, so many people have piled into it in the last few years, haven't they? I mean, in fact, it's over... It's over-subscribed, yeah. Like the law. If it goes to law, they're desperately sure of lawyers and solicitors and barristers. Really useful people. Really useful people. Well, you try telling that to people in the city, there are about 100 people fighting for any jobs at the gallery, so I don't know. Well, in the city, lawyers in the city, quite a bit competitive, given what wars are. Well, it probably is, because they pay a lot more than that. I've got friends who are lawyers, solicitors, and doctors, and they've all got micro-couch houses. Oh yeah, they do well. I don't know about doctors so much.

7:30 Even dentists, every space for dentists is 80,000 a year. In the south-east of England, I believe that's true. Again, mind you, they graft for it. As far as doctors are concerned, I wouldn't be surprised if they get even more to go. Medics usually do pretty well. I think they deserve it because they do have a life and death in their hands. I mean, my objection is that certainly in this society, I mean, doctors do far less well than lawyers and accountants. But in the States, they seem to be more or less on the same level financially. When I was 21, I could have been an accountant. I could have been into medicine. Well, what did you come to paraphysics for? You're a idiot. Well, I did engineering, first of all. I thought they'd always want scientists and scientists as the big Howard Wilson era. The white heat. What happened to the white heat of the technological revolution? I'm afraid it sort of got cold. I sort of grew up with the Apollo moon shot and things like that. I said no. That's the sort of time when things were happening. Well, big space. I'm going to shoot the buggers out of the sky now, Bob. I think almost any child who grew up in the 50s and 60s thought the same thing. We all had a vision of the year 2000. And quite honestly, if somebody had told me in 1962 or 1964... We were going backwards like Harold. ...that in the year 2000 there wouldn't be a large, permanent human population living on the moon, the tens of thousands, I would have startled with this plea. If in 1963, 64, so in 40 years' time, in fact, no human beings would have been on the moon for 30 years. They just would have gone once and... In real trouble? There's too many people in this world. Sorry, I don't know. Anyway, Bob, what are you going to do, Chris? Well, I was actually going to ask you, because Bob's heard it all before, so he's probably the last thing he wants to do again.

10:00 I was actually going to ask you to go over some of this stuff that you were talking about when I got here with Bob the other day. And particularly, except I haven't had a chance to look properly at my notes. Are you interested in sitting in Bob, or do you want to get on and do something? It's a target to you. The rotational orthogonal algebra and the incidence relations I hang up a bit and see how it goes I just really wanted to hear you recap in detail looking at from the point of view of the structure in the rotational orthogonal algebra and these rotor algebras the same kind of underlying the Heisenberg algebra How you see yourself as it was going one way from there, and you see Penrose as having gone the other way with the Twister programme, and the way that you're going is to combine the orthogonal and the symplectic structure. Well, trying to. Trying to, well. That's still not... To try and find ways of... And just what is the defect from the point of view of your kind of heuristic on this, you'll take on all this, with what he's done, particularly with the way that the Simpletka structure is missing in the Twister approach, and why that buggers up the super symmetry as a programme, and also how all of this fits together with these ideas of Walter Schemp and the other guy. God, this is it. Why no, it's Elsa. That's a real... No, look, we'll put it shut, because there's... If you, as it were, go over the Twister programme, the fuzzification of points in favour of Robinson Conferences in the Twister programme, and... I'm not sure I can go too far on that. Because you and David wrote a very interesting programme on the Twister way back, I believe, in the Schoenberg... That's right, that's right. Which I meant to look at, because I was thinking about all of this, It's in connection with understanding the properties of colourings of spaces. Well, what we were doing there, you see, Penrose's idea was to take the light ray as the basic structure from which you build your spaces. So that was essentially the Lewis idea of contact, but Penrose didn't have it as contact. He was just saying, I want to build a whole of my world, including points, out of light rays.

12:30 And that led him to, I don't know which way around it goes, but that leads on to the spinner. Because what you find is light rays can be described by the SL2C, essentially, the spinners of SL2C. And then he built up that big story about spinners, undotted, dotted indices, and so on. I mean, the undotted one comes about because they're contra-gredient. Yeah. Okay. It's a long time since I looked at all this stuff, but it reminds me. And also how it leads up to the flag picture. Right, so you sort of get two types of spinners present. Wait a minute, you've got upstairs, downstairs, you also get U-A dot and U-B dot. In other words, there are four groups of spinners on here. Right. Now that just gives you the mathematical structure which you play with. But the basic idea was that he wanted to, and Enoch didn't use this notation, he used Greek letters for this, but suppose you take your Ua, then that would describe a light ray, and it wouldn't be light ray localised anywhere in space at all, it would just be infinite ray. now the next thing to say was all right what what's the use of that because what we want to do really is we want to build a light cone because that's something i mean there are two ways of going one way is just to go via the light ray and then the robinson congruence on that was oh no then we've got then we've got to do something else so i say the light ray then there was another way of going and you all see these And these are really the light cones. And what the light cones are is essentially that you have Ua are the forward going light rays, and then U, I forget which one it is now, it's probably Ub dot. Which was the contra-agredient? Well, these are contra-agredient to each other, and then you've also got the complex conjugates, which are contra- Right. And I have a feeling that when you come to the Dirac-Clifford, for example, the Lorentz group divides, because you've got a four-dimensional spinner, right?

15:00 Yeah. A four-object spinner here. I think this was SL2C, zero, zero, SL2C bar, and that was the complex conjugate of that, and this corresponds to the, essentially, the back part of the light cone, and the other one corresponds to the front, okay? Now, I might not have quite right, but that's the general spirit of, okay? That certainly rings bells when I look at all this stuff, you know. Now, what Penrose does is to build these things up out of the spinners themselves. What DB and I was doing was to say, but it would be better if you could actually use the Clifford Algebra on this, and then the Clifford Algebra would mean that we've got our wave function straight away, and that this is the Dirac Spinner, and the Dirac Spinner actually describes a light gun. Okay, so I think those are similar but slightly different, coming from a slightly different perspective. Penrose, the continuous light ray, and then also taking the contigrida and forming the light cones, whereas we sort of go straight to the spinner and say, look, if we take this spinner, and then of course we're taking the middle left ideals to describe the spinner, and so we're in the algebra. Which, of course, immediately gives you the syntactic structure. No, no, no, we haven't got the syntactic structure yet. This is just the orthogonal. Oh, just the orthogonal. This is just the orthogonal. This is SO31. Because I thought you said the Clifton algebra is in there. The Clifton algebra is in there, but it's not the syntactic number. It's the orthogonal Clifton algebra. OK, got it. OK, right. Now then, the next step, and you can see it in our approach very nicely, The question would be, I'm not very interested in that. But as you said, that's already giving you the structure of the wave function. Yes, the wave function is being used to describe the light geometry. Yes, whereas in Penrose's approach, it's to be recovered from light geometry. Now, that has some advantages, and this has other advantages, and as you know, it's always a compromise in physics. Now, then, the next question that Penrose raised was, I've got a light ray here, I've got another light ray here, how can I relate these two light rays?

17:30 And so what he actually did was to say, well, what I can do is I can take the moment, I can take the angular momentum of the light ray about this point. So you introduce an X mu, the sort of the distance that the light ray is from some origin. And of course you can now begin to see that what you're going to do is you're going to ask what is the relation between the two light cores. That would be the equivalent geometrical structure. But Penrose was working in this way and he found that he got an object which was ua ua dot it's different from this one I wonder it's probably I can't remember the actual thing I mean could you forgive me for the yeah don't worry he got a pair of spinners exactly that's a pair of spinners he got a pair of spinners one spinner was a light ray and the other spinner was essentially along here so that you sort of had X mu ua dot would be the sort of the moment so you're looking for the moment about of the light ray about a with a, or about a, let's call it an origin okay And this object he showed described the X-mu and the U-mu. I think in fact what he had in here was an omega. I think it was just the downstairs one. This is why it was different. But it was a pair of spaces. Yeah, where omega was essentially x mu times this, so that it would... Oh, you'll also remember he put spin frames in here, so you don't get an index on the... But this contained information about this x mu. I think he wrote it as an omega, the actual... And it was the moment of this light ray about this. So these pair of spinners here described one light ray relative to an origin,

20:00 but everything in terms of spinners, everything in terms of light rays, so you could do everything in terms of light structures. This is very classical, isn't it? Oh yeah, this is all classical. Nothing, even what I'm doing here is classical. Oh yeah, sure. I don't like a quantum mechanics in there. No, no, no. Right, then what... It's just the way of doing the differential geometry. So then we got these objects which were called twisters, and I don't know what notation to use, I'll use the twister. A again, or A now goes through 4. I forget what his notation was. And these twisters were just pairs of these objects here. And then he showed that if ZA, ZA equals 0, the... Sorry, they're pairs of the things which are already defined in terms of the two spinners. Yes, you've got, you take these two spinners and this object here you call a twister. Right. Okay, so it's, and you'll notice it's a four-component object. Yeah. And one of the first things we said, well what is the difference between that four-component object and this four-component object in the Dirac theory? that caused a bit of a stir when he was talking to us downstairs. What a stir was a bit of a delay. I'll tell you exactly what it is in a couple of seconds. Yeah, sure. No, all I want to do, I'll tell you, in a couple of seconds, what we find here is that when this thing vanishes, we actually get the things crossing at a point, so we get a point, so we construct our points out of the intersection. And then when they don't quite cross, it then defines the Robinson congruences and you can go ahead. So you've got these things going round each other. And he was hoping that you would get rid of some of the singularities in field theory by using twisters. This was all to do with consequential infinitives. And he went to work with Aronoff and he then got into a tremendous problem and try to calculate scattering cross-sections using twisters, and it was a mess. Okay, so that's, I'm sorry, it's a very potted version, but that's essentially the structure. In general, they wouldn't cross if you had a pair of twisters. Yeah, if you had a pair, they could actually, a pair of these twisters, you know, one coming from this point, and if you like, another one from here.

22:30 This is what you're doing, and these two might not necessarily cross. Now it looks as if you started with a particular point in a space-time, but in fact, it can also be made out of, out of intersecting twisters. I mean, a lot of ways is Lorentzian barring, but it's like your X mu, is that Lorentzian barring? Yeah, still, that's X mu is still Lorentzian barring, because it's a four-vector. Yeah, it's a four-vector. That's okay. So this is all still within the, now what, sorry, within the, what, differential geometry. This is all classical differential geometry. I was going to say it was all within the Clifford Algebra still. Right. Okay, that was what I saw and I think you just put a footnote in his books. Yes, that's what I didn't pick up when I was reading the stuff at all. Did you mention that this is still a Clifford Algebra? Yeah, no, now what, the way we came at it was, well look, really Penrose is talking about light rays and one, a nice way of talking about light rays is through the hyperspirical geometry. Because as light, you know, as you've got a point here and as the light expands there is a light sphere. and the radius of that light sphere let me just draw this structure again because what we were saying was this so that when we get all these spheres touching each other put spheres coming in like this We've got two ways of looking at this. We can either say, okay, as the light goes out, then this radius expands. So that's essentially the CT. But a better way of looking at it is to think of these light cones as nested in each other and let you get a series of spheres. Right? And so you now collapse a whole series of different times into one picture. because they're all coming from different points in the space. So your hypospherical geometry can now define is another way of looking at what Penrose was doing, let me put it that way, where the light rays now are the points of intersection of these.

25:00 So rather than concentrate on the light rays, what dB, essentially dB's idea of this, was to say let's concentrate on these spheres. So if I've got a whole series of spheres meeting at a tangent, then that's the light ray, because all of these are going back in time, as it was. You've now got a different way of looking at... In other words, it's really just what Wyden's did. Newton's objects in the classical, in 17th centuries. A lot of look at the normals to the wave front, rather than to these beams on the trajectories. So immediately what you see is that... That is rather how you should think of the trajectory. under spheres going to spheres, you know, make a coordinate transformation or what have you, then you're just changing spheres into spheres, and that's a conformal group. And Penrose had already come out to say that this, now the difference between this spinner and this spinner was that this transformed under the conformal group, whereas this transformed under the Rheinsk group. but that's a story which I think comes out much better from this conformal group. But the way he introduced the conformal group, there didn't seem to be any intrinsic physical motivation in it. At least it just seemed to come out of... Well, we go to the hyperspherical, and then the hyperspherical of necessity has the conformal group. And then you say all equations of physics actually was 0 of s mass. And of course it's also very much the way that people want to think about solutions in GR anywhere. Yes, because you've got a lot of conformal symmetry. There's string theories, you're always using conformal symmetry. Right, you're using all the conformal symmetry as a string theory. And also, just going back to the highfalutin, the sexy string theory, I mean there is this view that the equations are the equations of the correct classical theory in gravity ought to be invariant under the full conformal group and not just under the general relativistic, the general covariance group. Because that will give you foliation in space time, constant mean curvature foliation which which gives you a preferred time coordinate and gets rid of all of the pathologies of closed timelight lines.

27:30 I mean, that's just a program which some people have more than others have been interested in. But again, conformal structure is obviously very, very basic. But here you'll see where it... Yeah, but to get back to the Lorentz structure, so you get the Poincare group, is that you simply fix a light cone at infinity. once you fix the light cone at infinity and always leave it there then the substructure you've got is the is the Poincaré group actually you've got translations as well because you've got a point at infinity too I've turned to you down now what happens is I say this is a potted version if I had my notes here I could give you a five hour lecture on this I'm writing this all up in a book I really would like to come and listen to the five hour lecture when you've got the time and the energy So now what we do, the next thing I did was to actually look at the conformal group and look at the generators. And what you find in there are some generators, which are essentially equivalent to translations. That is, you can show that they take like cones into like cones by moving through a distance x mu. You do it by a couple of inversions to get a translation. It's not a translation in the sense of e to the i px or anything like that that we've got in. It's really an inversion. Sometimes they're called accelerating transformations as well because it's essentially acceleration. And you can get the k-calculus the way Lou Kaufman gets his space-time structure out of contacts like this. Sorry, I don't know Kaufman's. Well, what you can do, what Bondi taught us, in fact it wasn't Bondi, it went back to Page. Do you know the Page and Adam's Electricity book? No, you wrote it, because it's not. No, I wasn't, no. there's a book that I sort of learned my electricity and the guy Page actually invented this in 1936 you can begin to see the physical motivation for all this by saying suppose I arm myself with a clock

30:00 and a torch or radar set was the way Bondi put it And with that local thing I can now construct distance, because what I do is I say I get a light signal out there and then there is a relationship with the time delay which is where the k-calculus comes out, it then strikes something and comes back again by symmetry and so on, and what Luke Calculin shows is you can actually get the Lorentz group from the k-calculus. You put in there two things, constancy, the speed of light and the principle of relativity, it doesn't matter which way you do it, and out pops the Lorentz transformation. But it's built out of light rays, you see, this is the important part. And so you can build, in other words, you can build a space-time out of this hypospherical geometry. and in fact you see what you're doing is you're actually looking at these things with moments about this axis here and I'm sure you can put that all in twist of form as well but in fact the accelerating transformation this is for constant velocity you know because then you have another chap which is moving with a constant velocity does the same thing and you've got to compare points and it gets a little bit tedious but it's the same it's the same thing now what you can also do it and this has amazed me with an accelerating observer now then what you find you get the conformal group and there are some conformal transformations which are actually called accelerating transformations but they are these inversion ones these inversion inverting So what you've got in the conformal group is you've got this structure. Now the interesting thing, now, where are these spinners? These spinners come from the Clifford group. I wish Owen was here because he was asking me about this. But it's come from the Clifford group. The Clifford group is generated by terms like e to the gamma, gamma. Pairs of gamuts generate the Clifford group, and that's a spin group. And the spinners that you get from there are these four component spinners. Now when you do the same thing for the conformal group,

32:30 and you look at the spinners of the conformal group, It's SU...SU4. The spin group of the conformal group is SU4, not SL2C. It's 2,2, but it's not right. Yeah, 2,2, OK. Well, let's put them... No, you're right, you're right. That's why I hesitated. I knew it was a 4, but I didn't know it was 2,2. That's not the first S6. I can't remember that, sorry, it's a long time ago, I've done a lot of other things since. Okay, it's SU2, too. Right, now then. SU2, too, can be represented by, actually, by an 8 by 8. The basic representation of that is actually 8 by 8. but what happens is that you can actually make a transformation of the basis so that this thing also goes 4-4 the same way as this one goes 2-2 this one goes 4-4 and what you do then is you then take the semi-spinner which has four components and that semi-spinner is the twister Right. It has exactly the right properties for the Penrose twister, but you see it comes from a very different source. And it's a source which isn't, so as it were, tied to the complex numbers. No, no, I'm glad you didn't do the complex numbers. Because this is a real Clifford algebra. And of course the reason why I got an 8x8 is because you're doing a real... The twisters definitely have four components. Yes, but it's the spinner of the SU-2-2, not the spinner of SL-2-C, in other words, not the Dirac spinner, it's a different transformation properties, and Penrose actually showed that in his paper with this, but to me it seemed to be so much easier to come through the conformal clippin algebra, look at the, you see why I concentrate all the time on the clippin groups?

35:00 And then what you can actually show, and I've got some very nice constructions, is that when you break the conformal symmetry, the elements in here actually do what Penrose says they do, i.e. you can relate light cones at different points. so there's no translation in there as such it's just taking moments so this x mu that he's got in there is not a translation but what he tries to say it's a translation is by fixing the like-coded infinity what you get I think is the Poincaré group and therefore he says right since I've got the Poincaré group it's a semi-direct product with SO31 times the translation group in four, and therefore I'll take that translation group as my translations in my space. Okay, but they are intimately tied up in the hyperspirical geometry. Yes, exactly, which doesn't come out at all. Which doesn't come out at all clearly in that approach. That's what I meant by saying This is kind of doing it in a, doubtless to an algebraist as good as Penrose, it's all very beautifully motivated. Yeah, sure. There's nothing wrong with the patterns, but you don't see it at all. You don't see this other structure coming in. And that, of course, also brings out very nicely, because it ties at the end of this picture of the hypospherical geometry, you know, the sense in which the twist is a non-local object. Yes, yeah, yeah, but there's one more thing you can gain now, you see, now we can go to Schemp and, because you did mention him. Yeah, I did ask you, didn't I? Right, and now what, what Schemp was doing was that he was essentially taking this hypospherical geometry. So this gives you the geometrical structure, and now what he's doing is on these light cones he's drawing tangent spheres in the traditional fashion this is where we're talking about how this ties up with the synthetic effectual geometry that's right now the point is this when you look at the spinner structure using the sort of thing that I did with Fabio projecting onto the plane and so on there's two ways of doing it, I don't know which way you want to do it way. In the Clifford Algebra way, if you take, now what group is it we're going to use here?

37:30 I think we're essentially using the SU2. Yeah, let's think of these now not as hypospherical geometry in four space, but let's look at the spheres in three space. So we're looking at the rotational symmetry, we're not looking at, we look at the rotation in space and then we worry about translation after translation of these spheres afterwards, ok. And if we look at SU2 what we find is that the geometrical object we're dealing with, This is the aggregate. The aggregate is always a vector plus a bivector. The aggregate. The Clifford Algebra. Normally the Clifford Algebra, if you look at the aggregates, that's the element of the Clifford Algebra. When you're doing it in terms of the SO31, what you have is the aggregate, or the general element of the Clifford Algebra is a scalar, plus a vector, plus a bivector, plus a trivector, plus a pseudoscalar. Is that right, Bob? And that exhausts the sixteen of them. And this is what? This is just the ordinary, this is the Dirac Algebra. The conformal algebra has something much bigger. I've looked at it, but it's not been written up. Now if we look at the SU2, this is the Pauli algebra. So this is Dirac. Just to put labels on them. This is the Pauli. And what you find with the Pauli is that you just get the vector plus the bivector. The scalar becomes zero and the pseudo scalar is zero for this particular construction with light rays. Remember when you've got a light ray, ua is always equal to zero. That defines the light ray for you. And so you're looking at a list, this describes the light ray. And if you notice, this is just Penrose's vector plus flag. Oh, now this is where I was going to ask you, this is where the flag picture comes in.

40:00 Yes, because what we've got is that the general element of the algebra for this structure is essentially a vector V, and then you've got a bivector. Now the question is where do you put the bivector? And what Penrose does is put the bivector as a flag. It's an orientation of the flag. but actually it's more than that Bob because you can also take now what Schemp and Binz do is to say let's take that bivector and think of it as a tangent plane in other words don't draw it as a flag but draw it as the bivector defining the tangent plane to the surface so we're doing a different construction here and if you look at my paper out much more naturally than the way Penrose does. So I'm sorry, it's just that I'm biased, because I find everything fits much more clearly here. I don't recall, from the paper with Fabio, that you're discussing the flag picture, but it's a long time since I looked at it, but I do remember very clearly you're discussing how, of course, not just the vectors and bi-vectors, and not in fact even what you have in the case of the Drac algebra, but of course, you know, all the multivetors coming out of the algebra, and the spinners that were just naturally living an object in which, as it were, all these vector space objects. This comes out by taking your aggregate to be a pair of spinners, to be a product of a pair of spinners. That's why we don't get the scalar and the pseudoscience. The term aggregates here is such a lot they're looking for. It's an element of the Clifford Algebra. It's a very old-fashioned term that people used to use for the turning of the centuries. Aggregates, because you're just summing up. You see why they're aggregates, because in terms of differential geometry, these are different objects. Yeah, yeah. But in the Clifford Algebra, they're one object. They're all on the same footing. So when you do this to it, you know, an S, S to the minus one, where that's an internal, you know, what happens is you get plus B prime plus T prime plus B prime, but what this does is it takes some of the S from here and puts it into all this, so it's a kind of fold, it connects with all the ideas of that. Yeah, it's a shuffling, and you see, and in 1890 Gibbs actually wrote a paper saying

42:30 we don't want this in physics because it's surface yeah because if you do electromagnetism you don't need you only have bivectors going to bivectors or if you do differential geometry you've only got vectors going to vectors so those are the geometrical entities he thought and therefore when you've got something which does these inner automorphisms It's an algebraic structure which can't be interpreted in his... It can't be interpreted in his... Whereas in fact it turns out that it can be. And now what happens was when Dirac came along in the 20s, he suddenly rediscovered the Clifford Algebra. And so the Clifford Algebra is basic to the Dirac theory of the Electron. Which I think is lovely. And that theory actually depends upon these unit automorphisms. In other words it is the implicate order again, if you like. All right. So there's the flag picture. Now when we come to Schemp and Bintz, then what Bintz told me was they actually consider this to be the vector and this to be the bivector. And on the bivector is a symplectic structure. Well the bivector is the kind of tangent plane. Yes, the tangent plane. And the tangent plane has got a symplectic structure on there. now you can see as Bob said it's like an angle and what is phase phase is an angle and therefore what you're doing is you're saying you've got an amplitude and you've got a phase so the structure that you're using building up from the the phase is in the V it's a relation between the V and the V and now what they showed is that this is now in the metapleptic group because you now have you have to lift so that if you get a if you get something moving this is where I'm not quite clear I remember I got a little bit confused over this last time so please forgive me I haven't sorted it out oh yes I know

45:00 And it seems like something completely different, but I have a feeling it's the same thing in the implicit in what they're doing. You can also discuss geometrical optics by means of a symplectic transformation. Yeah, you'll talk a bit about that too. Okay. Now, and this is what this guy de Goussons is doing. Yeah, only they're doing it in mechanics rather than in optics, but it's exactly the same structure as Hamilton showed us. So you get a flow defined in terms of symplectic transformation, which is a movement now. I haven't seen exactly how it's related to this hypospherical geometry, but it's there. It's got to be there because the light rays, you know, here it is, look, there's the light ray expanding. And as it's expanding it's taking this symplectic structure, this phase along with it. But because when you go into the metaple... because you're dealing with the metaplectic structure... Okay, now I've jumped ahead. I shouldn't have come here so quickly. Because now I'm going to the symplectic clifford algebra. And what they do is they build on this flag plane a symplectic clifford. I'd say I'm not clear that I've actually sorted this out in my own mind, but this is the direction they were going in. And it's certainly the direction Guisson's going in, so what you've got then is if you lift... as being in the symplectic group. Then if you lift it up into the metaplectic group, in other words you lift it up onto the fibre bundle if you like, and then on the bundle you've got the metaplectic group, then the metaplectic group for every point on the symplectic manifold you have two points for your metaplectic group. That's why you have to go around twice, because it's double covered, it's the double covering feature again. The double covering feature in the symplectic structure, in the orthogonal structure, you've

47:30 got the same thing, you've got to go around twice before you get back to where you started from. So you can look at, so that you've got two structures, this is what has not, and I have not got straight in my own mind, because what I've now done is essentially go from the structure I then look at is the fiber bundle so this is O N and then I've got a fiber bundle and this is some spin group N in the fiber bundle. Here I've got the symplectic structure and I've got a spin group and the spin group is the metaplexic group okay so I've got an exact analogy between these two and if I build this on the spheres and this on the tangent planes, so what happens is as it goes, so you can think of, you go to the next tangent space, so it's a mapping from one symplectic space to another, okay I'm treating the symplectic spaces as discrete. So as the sphere is expanding, so there is a symplectic structure unfolding on this plane. But that symplectic structure itself has a double covering, so it's got a metaplectic structure. And that metaplectic structure is carrying the phase. Right, okay. Now that's what's really going on by saying that the phase is in the B of the... in the tantrum plane, in the relationship between the tantrum plane. Now how this fits in with Penrose is I don't know. That's an interesting enough question, but I think this whole way of approaching it is just intrinsically. Yeah, from hypospherical geometry immediately say, okay let's look at the three spheres rather than the four spheres, and then let's have a look at the tangent structures on the four spheres because that's the phase of the light ray. Now this all works for Hamiltonian's quadratic maximum. Right. When you go to the cubic potentials then you're in trouble. You've got to be careful.

50:00 Now Morris reckons you can do it. But I don't think we've got the metaplectic representation which covers the cubics. But he thinks you can do it. I'd be very interested. As I say, I still want to hear you talk properly about Morris Goodson's work sometime. But what immediately grabs my attention here is the role of the spin bundle, the fibres and the double covering, because this is where it clearly connects up with this program of law here, because that's all about the role of spin-like quantities in the fibre-bundle structure and what it does to the quotients, to the quotienting of the topological equivalence relations. He actually specifically says that spin is, heuristically he's got it in the back of his mind and thinking about the way that in this entirely abstract mathematical setting the way that you would put the kind of global and local structure topological spaces that have this property of double covering together and the way that you have to sort of generalise a notion of point involved because the things are not in the case when everything is localised it will do because it will cover the point but now you've got global symmetries exactly and it also connects up with the way that you express logical properties in the technical theoretic setting. For instance, I mean, the accent of extensionality fails in these spaces precisely because you don't have the stable stability of the points function. That's always, I say, the very abstract level, just thinking about the properties of spin bundles on topological spaces. but I'm sure that there is a connection here with the metaplectic group and the physical interpretation of all this. Can I go on just here because suddenly something occurred to me while you were talking there. And that is that in the Penrose approach there is no metaplectic structure at all. Now whether there's any room for a metaplectic structure I don't know but there's no natural way of bringing it in. Whereas when you're in the hypothetical geometry, there seems to be, for light in particular, a way of bringing... And of course, we're all dealing with light, yes, there must be a way up there, but he doesn't mention it.

52:30 And the metapleptic structure so clearly connects with this kind of motivation from Hamiltonian. The Hamiltonian picture, Hamiltonian dynamics. And what you might call the kind of trying to understand ontologically what Hamiltonian dynamics is about. I think the motivation here is really beautiful. What you've got here is the fact that you've got a symplectic structure here means that you can bring in interference properties and I don't know where the interference properties come in in Penrose's approach. He doesn't put them in there. Unless he makes his twisters grow hair, but I don't see how to do them. I mean, maybe I'll have to think about it. Well, I don't know too much about the twister programme, but there have been all sorts of the twist or to use a lot of abstract machinery from sheaf theory and curve homology and to get, to get them as you put it, to grow hair so you can get an extra degree of freedom. But that I thought was motivated mainly by trying to do the sort of graviton picture in the twisted framework. But I hadn't seen any... the electromagnetic in the Twister program. Why try the gravitational? Exactly. Because the gravitation is going to do something entirely different. I haven't even begun to think about it. But you see, Shemp is beginning to think about the gravitational thing and his argument. I mean, I don't know what I told you. He almost ran me over on zebra crossing in Liege because he was talking about this thing. And he said the reason why that's why you get the bending of light. I don't know whether you've ever done the geometry of the... If you do classical bending of the light, Newtonian theory, you only get one of the Christoffel symbols. When you do the general relativistic thing you get two Christoffel symbols, and that makes the difference between the correct bending and half the bending. So the classical, this is why the bending of light by a star by an Eddington experiment was so crucial for Einstein, because you had to get the bending right, and if it was classical it would be one thing, and if it was general relativity it would be two. It's classical now, I don't think there's any bending at all.

55:00 No, there is, because you just think... Because it's the equivalence of mass and energy. A photon carries energy, and therefore it has mass. No, it's, yeah, but it's, well, you use new term in here as a term of art, it's classical. It's using the classical treating, you know, special relativity. Effective mass. Yeah, effective mass. Using it in a special relativity. I didn't know that, I think. And you can also get black holes in the classical determinism as well, because if you have a mass and you want to try and escape, you've got to shoot the thing up at a velocity greater than the speed of light. Actually I was quite surprised, when you calculate the escape velocity for the velocity c, it's exactly the radius of the black hole. It's amazing. It's ridiculously simple. And yet one thinks of the black hole as something very exotic, the way it's presented in the popular press. in fact I think the first appendix to Hawking analysis it's in there ok so now where are we so what the structure we've got here is hypospherical, conformal and then by breaking the conformal symmetry you've got these, and in the conformal symmetries you've got this covering group and then if you look at the semi-spin as they're called those semi spinners are the twisters and I think this is a much nicer way and also you can show from within the algebra that what you're doing is you're relating light cones to light cones. That X mu, is that a sort of given at the beginning and you work it pi or you actually generate it? Yes, no, you put that in, you don't generate it. You put that in, if you're generating it, wouldn't you get X mu? Yeah, but it's not, you see. The trouble is that because this is an element of the Clifford Algebra, I seem to remember that when you multiply these four times together in the supersymmetry, it actually becomes zero. It becomes zero or something. What becomes zero in supersymmetry? Well, the x mu. The x mu. If you cube it, there's nil potent to the power 4. And that's what Nick and I were always worried about, why are these boys calling this a coordinate

57:30 translation when it's nil-potent? No, no they're not, I can't remember actually what they were, I won't say no they're not elements of the Grassman, they're something different, but they're certainly nil-potent. Yes, how this connects with the Grasman algebra. Grasman algebra is very easy from here, it's clever algebras. Yes, because I can take a Grasman algebra A mu, and I can take a Grasman algebra A mu and it's dual, and if I form A mu A mu A mu A mu plus equals G mu mu, Yeah, you've got a... I can create the... Clifford Algebra. No, I take two, every Clifford Algebra is made out of two dual Grassman Algebras. Right, so that's, of course I'm sorry, that's... Yeah, it's a pair of Grassman Algebras, and they're related by this relationship, which is the duality relationship. I'm sorry I'm doing it in terms of coordinates because I'm a physicist but you can do it quite beautifully in terms of abstract symbolism and therefore and then the paper that Bob loves I keep pointing this out these things can then be used to define topological features of the space did you know that annihilation and creation operators can be used to make boundary operators and this all has a topological significance to it well that's what I really want to say well I mean it's very, well is it very simple I'm not sure but this is not what Penrose does this is what comes out of the Clifford algebra which is why I prefer Which is why I prefer the Clifid Algebra. We tried to find this the other day, didn't we, Bob? Which is the paper that you say that Bob... This one. Oh, the one that's in that... The relativistic phase space arising out of the drift. So what we...

1:00:00 Yeah, but what we do is, you see... Here's the two elements of the Clifford Algebra I was telling you there. They're just sums and differences of these elements here. Because we might. What we have to do to get this out is to go into a higher algebra. We double the Clifford Algebra in order to get this. And that's why it's a phase space. To get it working. Then you see that the interesting thing, I think, from this point, which we've never really fully exploited, is that we can now become quantum. If we look at these things, but they're annihilation and creation, so we can introduce a vacuum state, which is what we do here, a vacuum state. And what we do is that every element of the algebra is created out of this aggregate. This aggregate is now written in terms of annihilation creation operators. Operate that on a vacuum state, and it produces the Clifford algebra for you. So what you're... And you're not with all your face-based facts, aren't you? So it's now not classical anymore when you look at it this way. And these annihilation and creation operators are operating on a vacuum state, which is not in the original Clifford Algebra at all. This is a good paper, you know, nobody's ever understood what we've done, including the author, one of the authors. That's really interesting, isn't it? My main interest was trying to use what you've done then to get out the mercilessistic constant potential for the Joao. I thought I saw a way through it, but it might still be postponed. There are some nice little topological tricks you could play once you saw that structure.

1:02:30 What I've never done is to go... Well, that's one I'm really interested in understanding how this does connect up with a quantising topology. Well, it's not so much quantising topology, but rather... these things become bound, you can make boundary operators out of them and then those boundary operators are created from the vacuum in order to make them into the real world as it were you operate on the vacuum state in order to create these objects this was really what we were trying to get at when we were saying that yes all of this is classical but once you make this step yeah, and then play make these the important things there's two ways of looking at it the physicist, in a way I would like to make everything out of these pair of Grassman algears, and that's what Schoenberg did but I think probably another way of looking at it may be to say, alright, we take our classical gammas and we take our gamma plus and gamma minuses and then from that phase space we can create these things because if you double up the Clifford Algebra one of these things gives you a Clifford Algebra in space-time and the other one in phase space so once you've got a phase space it's quantum mechanical because you've got this but it's anticommutatively quantum mechanical and then you can use these on a vacuum state to create quantum objects quantum vectors, quantum bi-vectors and quantum tri-vectors because they are things which are created they are eigenvalues from the vacuum they are created from the vacuum state and I like that idea and one of the things I told I was looking at the extensons, because you can do exactly the same from the symplectic Clifford, you can use the boson annihilation creation operators. Now here you generate the symmetric algebra, here you generate the, because these are all anti-symmetric objects, so this generates all the anti-symmetric tensors, this generates

1:05:00 all the symmetric tensors. If you look at this, we haven't done this before, if you look at this acting on a vacuum state in the same way as you do this on a vacuum state, you create your geometry, your anti-symmetric tensors and your symmetric tensors from your vacuum state. I think there's something here that we haven't exploited and you can't do this coming through the penrose you have no way there's no rhyme or reason for doing this but because we've come through the geometrical structure There was a hint of this, of getting the symmetric and anti-symmetric tensors out of this, you know, the pair of grass molds, and the anti-symmetric, and this for the... And a pair of, these are symmetric algebraes. These for the symmetric. I thought, in your paper on the spinner with Fabio... Yes, but we didn't go as far as I would have liked to. No, no, no, but there was a hint of, this was certainly the way you were doing it. And that's what I like so much about that paper. I still think it's right, but we spent too long worrying about these bloody primitive impotents. I can't remember why we did that. Well, because that was how you saw the structure of a cliff in Algebra. That was what was absolutely fundamental. What I was worrying about was really Schoenberg. It's telling us that this is the origin of the coordinate systems. And you want to get everything out of the left and right ideas. As it were, the kind of unfolding. I'm not using a term that I'm saying, but the kind of unfolding of the kind of... I don't know exactly the contrast, but yes, I think there's something there in the background about Troull's theorem in algebra, which is to do with the left and right ideals. What is that? I'm trying to remember. It's to do with the... suddenly come across... It's to do with the idempotence and with the, well it's to do with the relationship between the left and right ideals. I want to know that because I haven't seen it. Where do I find it? Krull, K-R-U-L-L, Krull. It's a very fundamental theory of abstract

1:07:30 algebra, which I haven't read for about 20 years. I have to go back and look. Could you dig that out? Yeah, again, it's in Bill's book, he refers to it. Which Bill's book? The new one that hasn't come out? No, no, the one which hasn't come out yet. But it's okay. I mean, any good modern text on algebra, particularly on algebraic geometry, will tell you about Kroll's theorem. This is not a... I'll dig it out. I think the best place, probably, if you just want to say a clean statement and proof of it, is Godemmol's book on algebra. I don't remember the fat book, but it's a sort of, it's a creative text book. K-R-U-L. K-R-U-L-L, Krull's Theorem. Which is all about left and right ideals and the way they transform. Right, because that's what we're, you see, my new way of looking at, you see, I don't know, I was going to go down another avenue. It has to do with the way that you treat, in algebraic geometry, Because you see, look, what the Schrodinger equation is, is simply, and I was talking to Owen about this earlier, what I'd never, when the physicists come to the Heisenberg equation, right, we're told the Heisenberg equation of motion is more general, all it means is it's just the operators are all time dependent. So everything goes into the, except this wave function, which is now just a wave function at t0. In the Schrödinger picture, all the emphasis is on the wave function. That's where all the time dependence goes. And I've never understood why suddenly, when you go to the Heisenberg picture, you've got this stationary wave function, do all the dynamics in the algebra, and then just come at the last minute. Oh, yes, sorry, I forgot. Make an expectation value between these two. Right? That is a cruddy idea. Whereas if you come, you see, the reason why they've had to do this is because they don't recognise the need for the idipotent, and they feel that Heisenberg's algebra is a nilpotent algebra, and therefore it cannot have an idipotent, so you've got to do this. That's all bollocks.

1:10:00 So you could do everything at the level of projection operators. Yeah, that's all bollocks, you see. I maintain that the correct algebra for quantum theory, in spite of what the referees and the adjudicators say, is the Dirac algebra. Is the Dirac, yes. Maybe that's what we should call it. It also, of course, gets you away from the limitations of the Hilbert space. Yeah, absolutely. As Dirac itself long ago pointed out. Absolutely, absolutely. Why are these idiots still opposing this idea? because they don't know what you're doing well I think that's the short answer because they haven't bothered to pay attention carefully enough to what's going on here and in some cases because they've just made such a massive investment in their own we've got a bigger algebra here already what bigger than the well look when we turn these and then start talking about a vacuum state we've now introduced a projection operator in this bigger algebra so we've extended the algebra yet once once again and then we can now say well what are they you see what quantum mechanics does it takes this general a and it says I can get everything out of this provided I write my A's in my Heisenberg equation of motion in terms of in other words physics is only interested in two-sided ideals in the algebra how about that take any two-sided ideal then look at the Heisenberg equation of motion on a two-sided ideal and what you can show into two. One turns out to be just the Schrodinger equation, the other one turns out to be the complex, the conjugate Schrodinger equation. It's absolutely bloody incredible. Am I missing something? Well it's very easy, Bob. I don't understand it either. I mean, I understand what you're saying, but I couldn't quite see what it's coming from. That's what you're saying, but it's the essence of what.

1:12:30 Can I go through that, because it's fresh in my mind? Yeah, absolutely. Are you all right here? You see what we're doing, you see what we're doing. I'm very clearly now where you've come from. Are you seeing what's different between what I'm doing? I've seen very, very clearly now what is different from, and I would think, say, a great deal richer than the Penrose. Also because we have this sort of quasi-quantum stuff. I call it quasi-squat-quat because we haven't actually proved it, but there is the possibility that we use these. There's all sorts of structure in there that's very strongly suggestive of how you would get out... And therefore we've got statistical geometry coming out of this. Yeah, yeah. Is that what you said about getting an extraordinary equation? Okay, let's do it. No, let's move to that while it's fresh in your mind. The only question I've got left about the other stuff, about what you talked about on Monday and which you've just gone over again, is how this connects with the failure of the supersymmetry program. Put that aside for now. I don't... No, we'll put that aside. That's that key. One expression in mind, talk about this... But what I'm saying is that... Side-byte about... You don't need superalgebra to do what Schemp and Binsch are doing. It's just ordinary algebra. You don't want supersymmetry. Okay? Now, in here, you've probably made a superalgebra putting these two structures together, and I'm not sure whether it's trivially, non-trivially or what, because I haven't investigated that properly. And in fact, you know, that's my next thing, because I've got this beautiful manuscript from my beautiful friend Maurice. I'm looking forward to meeting it. It does sound beautiful. I read that thing out, you're dead right, about the implicate order. Have you seen this Don't get too diverted from the two-sided ideals. No, it's all part of the same thing. Get it out while it's fresh in your mind. You know, from the Heisenberg equation, I mean two-sided ideals. Well, I'll remember it too. So he sends me this book. and one of the chapters is called Metatrons and the Implicate Order.

1:15:00 And then he talks about that. Then he goes on to the Lee Trotter Formulism for flows. and then he so that's classical physics yeah that's tying it all up with the Marsden sort of that's right and then he says can we by inspection of the quantum evolution group or less more general and what he shows there is that where is the summary of what he shows there Then he comes on implicate order and phase space, then he comes on to the X and P representations, Then he comes on to the unfolded metatron. The metatron is a particle of a metaplectic group, which is what I've been calling generalised metaplectic representations, Haile's approach. And then he stabs their quantum cells. But this is this connection. When is this coming out or is it just a manuscript as yet? it's a manuscript at the moment is it possible for copy and circulate if we promise not to well as and when it comes in you may not see it obviously not for citation or anything but it would be very useful to just go and look at it but he comes back to me and he's I don't forget what happened was that the publishers of scientific war scientific would run out of IC did you know they run out of IC? no I didn't physics does And a guy rang me and said, you know, I've got this guy, Maurice de Coussons from Sweden, who's writing this book and we've had a look at it. I want to know what you think of it, you see. And I said, well, I haven't seen a book yet. But I know Maurice and I know his work and, you know, I think it's going to be good. It's going to be a good mathematical book. And he said, where does the bone theory stand amongst physicists?

1:17:30 I did mention this. I did mention this. Blah, blah, blah, blah, blah. Well, you see, the reason why he's done that, because a lot of that is about the bone mechanics. And Morris says that on the continent of Europe, in France and in Germany, the bone mechanics is accepted as a legitimate way of looking at quantum mechanics. or maybes as far as France is concerned. Just from talking to these people that they're going to analysis. The only people who don't like it are bloody physicists who've got their prejudices against it. Are you talking about physicists in France? He's probably talking about mathematicians. I think mathematicians and philosophers of physics have both taken that view. The only people who haven't... Are the only generation of physicists. Yeah, okay, certainly the origin. Well, I've said American probably wouldn't work very keen. No, I think, as a broad cultural observation, that's right. There are a lot of Americans who've worked, especially people interested in that, there's a lot of the physics who've done a lot of work on it. Anyway, look, so that's, sorry, so that book's coming out and it's bone mechanics, but it's linking it to the Lagrangian mechanics and the half forms and so on. So it's showing that it's not... That's the point of all these bright people who do all this work on Lagrangian... Well, I went to Fabio to say, look, he said we had someone, Greg, this Polish guy in South Africa. He's an expert in the garage and mechanics and stuff. I go across there, about 10 years ago now, was it? Saying to Greg, look, look, this must fit in with what these guys are doing, help. Oh, you're doing something very clever. Whereas Morris has actually taken it and taken it seriously and seen how it affects the semi-classical approach. Because the semi-classical approach is approximating to quantum mechanics. We've got an approach which gives the mechanics which is exactly quantum mechanics. And therefore there's obviously a window to look at and explore the differences and see what happens. And one of the deepest problems with standard formalism, standard approaches of formalism has always been how do you recover the past formalism without basically putting things in my hand and doing a lot of very suspect tricks, mathematically messy.

1:20:00 Yeah, the interesting thing about what he's done there, he suddenly realized, after the talk I gave about shadow manifolds, that you've also got away from the metaplectic group projecting down into P representation. And that gives you a different result from reflecting down into X representation. That's why he says what I'm doing, or what we're doing is beautiful. No, that's what he got so excited about, telling us last year. Yeah, because he, and the book actually shows, and I'm not sure whether it succeeds, but I'm looking at it now, but that's what he's saying. beautiful that, you know, the semi-classical approach that he's coming to necessitates two different pictures. And that's why he's taking the implicate order, because he suddenly realises. And he said he was going to go and talk to Korn about all this. Yes, yeah. I was very interested to tell you what the reaction is. Well, if you go, if you spend April talking to Bill Ortele, he spends April talking to Korn, the next time they get together, there's a bit of luck, they might actually bash their heads together without the ring-blad-of-the-bluff. Oh, I think so. Let's get to what you were saying about the two-sided ideals in the Heisenberg algebra where you get the Schrodinger and its complex algebra. So the idea is to say that, it's essentially a very simple question, So why isn't Schrodinger's equation in the algebra? Yeah? That's fair enough, isn't it? Why isn't the Schrodinger equation in the algebra? I'm going to get that right, you're bad. I guess we had this conversation on Monday as well, didn't we? I have asked them. I'm going to complain. It is irritating, isn't it? Doesn't it irritate you? Only when I notice it. but I'm one of those lucky people that can want to screen things out. So what we've got is... Now it just annoys me as soon as I start hearing it. I hope that's the right way round. Anyway, forgive me if it's the wrong sign. We have Heisenberg's equation in motion. So what we're saying is essentially that the dynamics in... and we don't really start there because what I want to really The dynamics is a flow of some object here which takes an element of the algebra of function of time into an element of algebra of relative time.

1:22:30 And this goes via an inner automorphism so that a prime t equals m t a t m the minus one t, okay? So that's a fundamental assumption. All my dynamics is coming out of inner automorphisms of the algebra. And in fact, we needn't even say inner automorphisms. We could say that they're outer automorphisms. But the point is, it's an automorphism in the algebra. Now, what this is then, is to assume that M is equal to E to the IHT. And that's the simplest one we can do. Where H is not a function of time. because you don't get into that mess, that's what the field series people do. Now, so the reason why we've got the I in here simply is because we want this to be unitary, or mission rather, and that's real and therefore to make it a mission. Right. Is that nonsense I was talking there? forget that, that's nothing, it doesn't matter, it doesn't matter, and then I can go to the infinitesimal and I'll write it like this. Now, Schrodinger equation works on left ideals. Well, it's dealing with left ideals, before we say it works on a bit, it's dealing with left ideals. Are we happy about that statement? The reason why I'm saying it is because when I looked at the I mean you can think, it's almost obvious isn't it, because what you want to do is you want to take some element of the algebra, you want to operate it on it from the left, different this is m prime, different there, you want that to go into the left ideal. So you want In other words, what I'm saying is, instead of having an automorphism, you want MTA to be your... So that'll take a left ideal into a left ideal. So this transformation will keep me within the ideal. and this is what Schrodinger's equation actually is.

1:25:00 Now then, I've got a general element of the algebra here, and I want to say, well, Let me consider, and this is the only assumption I think I'm going to make here, let me consider those elements which can be written in the form BC such that B is an element of the left ideal and C is an element of the right ideal. Okay? Now, anything which is made with a left and a right ideal is, in fact, a two-sided ideal. So I'm looking at two-sided ideals. I'm assuming this because I know where I'm going. because I don't want to do any old B and any old C. Now what I do is say, all right, let me put that in this equation here. So I get I dB by dt, C plus I dB, keeping the order, equals, I'll write this out, A, A, H, H, B, C, minus B, C, H. I've done nothing but just put it in. Yeah? And because A is equal to B, C, I treat this as a derivative, therefore I treat it as a linear operator. and this is just writing that out in full. Now then, what I want to do, I want to assume that there exists a b the minus one and a c the minus one, and there's no reason why in this algebra we deal with in quantum mechanics because I've already had a n the minus one here for the automorphism of the algebra so that And then let me multiply this from b-1 from the left and c-1 from the right.

1:27:30 Or I can just write it down. Multiply c so that goes out. B this side, so B goes there. Now I'm multiplying B this side. Oh, and I'm multiplying C the minus one that side. B goes there. So now what have I got? I've got b the minus 1 db by dt minus b the minus 1 hb is equal to minus i dc dt c the minus 1 minus c h c the minus 1. This holds for arbitrary B and arbitrary C. So that means this must be independent. This must be some constant. Right, so this is some constant. C holds some constant. Let me choose that constant to be zero. Why not? Yeah. and if it's not zero you can always, I don't know about that because that'll give us general, that'll give us a general mechanics which I'm not sure I want to go into at the moment. I'm sorry, what is this expression here? What here? Yeah, it's constant. Oh I see, it's not constant, I was going to put C, but I'm using C here. Yeah, some arbitrary constant alpha. Okay, Could be a complex number, could be anything. And now I'm going to make the assumption, choose alpha to be equal to zero. I mean there might be something, I don't know what it is, so I'm free to choose, and quite just as I so happen. And if I do that, if I choose equal to zero, So, then I get, oh, I've forgotten the i's here, sorry, the i's disappeared from there. i, the b by the t equals hb, and minus i, c by the t equals c h.

1:30:00 I'm sorry about that Bob, it was just, you know, it had reached the point of, you know, I was either going to have to commit suicide. Right. Now, if this, can't you see that would be the Schrodinger equation if that was psi? Okay. So how can we turn that into Schrodinger's equation? Well, now I want to say, look, I've got to be very careful, because I want A to be equal to B and C, but I want B and C specifically to be left and right ideals. Right, so this is where you're going to get the complex conduit. So why don't I just use a notation that Dirac used, and this is not arbitrary. At the very beginning you said something about left ideals. No, the wave function is a left ideal because under that operation it always goes into a wave function, it's always operating from the left. That's all I meant. I meant more in the sense that when I construct the clippin algebras and so on, you see that... The point being that all of the dynamics are in a classical way, you're getting out of the automorphisms. That's right. So now we've got this, you see, so then what we can say is, all right, let's keep the Dirac notation going. It's just a mark. It's nothing more than a mark. I could put left and right, but it's much more sexy to put a little thing in there. I'll show you the reason why it's got to be

1:32:30 there. Now then, the next step, isn't this beautiful? I'm waiting for violent objection. The next step... No, I'm just sort of seeing how this fits into what I'm interested in. I thought, you know, where you think there's a difference in the sum of... Yeah, well, it's here. I'm first of all separating them, Bob. Because yesterday with Melvin, he was trying to convince me that we could get away without separating them. We can't. Now, the next thing to know is... So, what does he think can get away without separating? It's adding and subtracting, but we can't. We got into a mess yesterday, forget that. Forget that. Think of what we're doing here. Yeah, I am, very clearly. Now, you know, all I've done is I've said, let's assume... And then if I put that in the Heisenberg equation of motion, do the b the minus one, c the minus one, And I separate out, and the second assumption I put in is that constant is equal to zero. And once I put that constant equal to zero, I get these two equations. And the reason why I put it equal to zero is because I go, aha. Now, the next step is Dirac says, I take this b, and then I define that to be B, I'll use this for the notation. Okay, now look, let me do it in modern, okay, that's a definition that Dirac has in. So let me do it slightly differently. Let me say, alright, I want to choose my B to be a function of the operator X and a function of time. Because I haven't said what their functions are, but I'm now making it even more restrictive in the sense that I want B to just be a function. Because I'm interested in Schrodinger's equation, Schrodinger's equation has to do with x. But it's an operator x, not the eigenvalue. So I've got this object here. If I use the Dirac rule, eigenvalue x there, then that's just equal to b xt. And bxt, if I write as psi xt, everybody knows what I'm talking about. Yeah? And therefore, what I do here is I put my x in here, I put my x in here,

1:35:00 and that simply becomes i the psi of x of t by dt equals h psi of x of t. Actually, hang on, it doesn't. There's an extra step I've got to put in here. Do you want me to? Yes, I'd better. Because when you get home you'll say, oi BJ. Which is the ordinary Schrodinger equation when you don't have a diagonal Hamiltonian, because you usually write this as, this thing is usually h x x prime delta x minus x prime c star of x, well actually I'll I'll put the psi in there, psi star of x too. Dagger, yeah, complex conjugate. Prime, which is then equal to h of x, psi, oh there should be prime there, psi star of x. Oh wait a minute, that's just ordinary upside. I beg your pardon, I'm just dreaming. It's an ordinary upside, like it's this one. Oh I know what's popped me off, that should be side. Because I had a C there, I thought I was dealing with the right idea when I put the complex conduit on it. Sorry. Are you happy? You know, it's this one, it's this. Yeah, thanks Bob. No, sorry, I was getting angry with the radio. And what is that? What is that? It's the Schrodinger equation. It's just the complex conjugate. You've got a minus sign here, this is going to be psi star with x and t, and this is going to be...

1:37:30 Which is what you said right at the beginning, that you were going to get the Schrodinger and the complex I have a two-sided idea. Isn't this beautiful? Why don't we do this one? Why don't we do it? See what I mean? No, I see exactly the strength of the other powerless approach. I mean, it's absolutely beautiful. It's in the algebra. It's not in any fucking Hilbert space. No, it's not in the outer space. It's not in the... It's not in all the... Yeah, and of course didn't Schrodinger, you know, bust his guts trying to understand, you know, just what the space was that these dynamics did live in. But it lives in the algebra? Yeah, exactly. Now then, the real beauty of this is... Oh, it's gone. What's that? Excuse me, get injured, mate. What's this bloody little symbol? So that's fine, okay? What? What? What? What? What? OK, very simple. How do we construct left ideals? Assume the algebra has an hidden problem. Suppose now we multiply, how do we make a left ideal out of that? We say a left ideal are all the elements which we form B epsilon and all those elements are the left ideal. I'm sorry I keep growing like this. Now, how do I make a right ideal? I say it's all the elements which are epsilon b. Okay, let me put c. Yeah? Yes, that's the conjugate, that's the definition of the star, I left it out, you can go abstract

1:40:00 and say I've got a conjugate, a star, it's a star algebra, and the star algebra always takes you to the... and this is what I want to know about the left and right ideals. Well, of course, yeah, this is all the particles theorem and how you get the... Because I have a feeling that what these Cron, no, what's this Cron and Ravelli do, yes, they actually have an operator s which takes you from a times some k nought into a star k nought, in other words this s has got something to do with relating these different left and right ideals. This is in their thermodynamics of time paper, which I got out, it's in CQG, and I got it out the other day, in fact in the University of Crosby Library, but unfortunately you didn't have a chance to copy it because they've bloody closed their photocopying machines. But this is, you know, so we're, you know, so if this is a right ideal, so we're sort of turning, and I don't quite know what the role of these things are. You see, there's got these ruddy K-naughts stuck in there again. In other words, they're not in the algebra still. No. I didn't have a chance to look at properly at the paper. It's quite a lengthy paper, and it's obviously heavy going. A lot of it is irrelevant. Yeah, and I wanted to make a copy of it, but as I said, I was hoping to go back to the knife or something. Yeah, listen. Are you sitting comfortably for this shock? Okay, so now I want to form a two-sided ideal. Well, all I'll do is I can form two-sided ideals by doing that. Or, if you like, take an element of the left ideal and multiply it by an element of the left ideal. That's what I said A was. So that's equal to B Psi long C

1:42:30 What do you think of that, Mr. Callaghan? Well, I don't know what this thing was anyway, is it, is it, is it, this is an hidden potent? This is an hidden, an, any, any hidden potent, I'm, I'm, I'm making this up. So you basically say this simply has an hidden potent? Yeah. Yeah, that's just how you should think of it in pertinences, in the switch. oh you're bringing in the Stanley, I don't know, I don't prove what it is. With no mystery. Because I start by, you know, sort of conning you and saying it's just a label. But it's more powerful than a label, it's actually an important component. And therefore all two-sided ideals now in this algebra can be written as... You're right, Mike? All two-sided ideals can be written as B, epsilon, C. I think that's right, yeah. I think, although as I say, it's a very long time since I looked at Krull's Theorem, that that more or less is Krull's Theorem. No, it's going to be more than that. Well, you're right, there is more to it, but it is about how you write left and right-sided ideals. I mean, I'm making the assumption that this is a definition of an ideal. And I'm taking that from Albert's book on simple algebra. There is a lot more than that across the room, but it's all to do with... In fact, last night, after I was talking to Mo and I went and opened... I think it was Albert's book on abstract, linear abstract or something like that. It's a lovely book. It goes into this. And I suddenly thought, oh my god, I'm dealing with two-sided ideals. So that's what quantum mechanics does, it takes two-sided ideals. and that's like taking the square root it's like taking the square root of an operator because I say here's my operator A and I've got it written as a pair and then I say I take that and then I say I take that and that's very much like taking the square root symbolically, don't take it seriously it's just like taking the square root

1:45:00 now you see the next thing we've got here is that what I don't understand is where we're going to put the measure in because what we're doing here is we're saying we're in the implicit order let's call it that is the infinite order, just to talk about it. There is no space-time in this order yet. And what we normally do, you see, we're getting space-time in when we're doing this. Yeah? So really it's through this duality operation that we're somehow putting space. So this is where the measure is going in, in some sort of process there. And you've got to be able to get what you get in the case of Hilbert's space with your L2, the square root of functions out of... I'm assuming there's a square interval here, otherwise we don't know how... They won't connect with physics. But you see where we're putting it, we're putting it in its proper place. At the end. you see I mean physicists seem to think that that is everything and it's nothing even if you look at electron going through two slits where's the bloody wave and they all admit it's something abstract they don't want to give it up I think this is beautiful. This is what Dirac was trying to do. In fact this is what Dirac was doing. But he didn't have the language left ideal and right ideal. And he didn't make any silent dance of this. No, that's what's so strange. He didn't recognise that was an independent. And yet he had such a feel for beauty in mathematics and for the algebra. Melvin was telling you, I haven't got, have you got, who's got a fourth edition of Dirac?

1:47:30 I've got a fourth edition. Yeah. Could you put it in your car one day? Yeah. Just to have a look at it, because mine's a three. Melvin said that towards the end there's some lovely quotes about him wishing he could find the correct algebra to pull, you see, and there's nothing in my edition, so it must be the fourth edition. I wanted to know more about them, just biographically, what the story is there, why he, I mean, supposedly had such a powerful mind, it's strange that he seems to have given up. It was electrical, he didn't give up, everybody else gave up, gave him up. Well, maybe that was it, he just felt he was isolated. I mean, when I talk to physicists who say, I'm talking about the standard cone, they say, yeah, yeah, we know the standard cone. Yeah, we know what that is. Everybody knows what that is. And I said, but it's not. Oh, no, we don't. But it's not that. What? What? Well, at the point, Durant didn't think it was that. He thought it was the same thing. Well, it's probably, you know, it's very brief, isn't it? Well, in the third edition it's not brief, it appears in about six or seven different places. I want to look at it again, but I'm sure there's only a little... I mean, he sort of introduces this, like, notation and then goes into something else afterwards. But can I just, oh, yeah, okay, I know he doesn't. Now then, why, to bring this out, to use this structure here, what I've really done is to call that x, because what I've picked out, or at the moment I haven't, but when I was, watch what physicists do. You see, I'm still not at the Schrödinger equation when I'm up here. What physicists do is to take this thing as a function of just the operator X. And then this thing works for X. To bring the... So that the... No, no, no, no, no, no. No, you've got it wrong, B.J. You've got it wrong. It's wrong. Sorry. Delete everything I've said. Well, hang on. It's going far far back.

1:50:00 and then forget that. Now, I'm going to tell you another thing. You were wasting a suspicion I got when you did that. No, no, no. There's more in there. Now then. At the moment, I've forgotten one step. This is not completely defined. No. Okay. And the reason why it's not completely defined, is because suppose my algebra, so that my B is, suppose my algebra is the Heisenberg algebra with X's and P's in it. My most general function I have is that. Quantum mechanics deals with, so I would like this object to result from a projection from that big object. Now the way I can do that is by further defining, so what I'm saying here is, how does x do epsilon? And the point is I don't define it. But I want to define p epsilon equals 0, and I want to define epsilon p equals 0. Then I've identified what this epsilon is in the Heisenberg algebra. It's the projector, it's the polarisation projector because what it picks out is the x subspace in my phase space the operator I'm still in the algebra it picks out my x subspace now then I can say okay I could also have another projector which I'm going to call pi, and that one I'm going to define by saying x equals 0 and pi x. This is a different projector. And this is now a different projector. Okay, so now I can put, this is my x projector and this is my p projector. Now I'm legitimate in saying this.

1:52:30 Okay. And then I can have both of these in the algebra if I want to. And then I can say, well, maybe P epsilon equals epsilon P. but you know what I'm now moving into the online lounge remember Jones polynomial where he has all these so what I'm doing here I've got I don't know whether to bring this in or not what I'm saying is that I can introduce these two projectors quite legitimately I say I still have one bit of ambiguity, what do I do with... I could! I don't know, that's an interesting question, I haven't investigated this. What would happen if I put P equals sign on P sign? Isn't that what Jones does with his... What was it? It was... I don't know what Jones does, sorry. Pi i equals, do you remember that thing that i plus one pi sine on i minus one or something? You remember what you put on? You know, in other words, the point is I've reached, I'm sorry, I've reached the end of my knowledge on this. Of course you're going to say, but, but, but, but, but, it seems as if I'm sort of becoming a little bit arbitrary in here, you know. Yeah, I'm still not sure that this is completely defined, although I understood more about you've introduced these two projections. Well I think I'm still working on this but so be patient now. I know that the enveloping Heisenberg algebra is isomorphic to the enveloping boson algebra.

1:55:00 In the enveloping boson algebra, I introduce the vacuum state. And that's defined, just one. That's why I know what I'm doing, physics. Yeah, you know you might say we don't know what you're doing because you're just bringing in arbitrary good impotence No, I'm only putting X on it because Are you happy with V? yeah you're absolutely ecstatic with me and this is these are your creation operators yes otherwise you're back again with what you were doing with your no chapter one with your uh harassment their pairs of craftsmen well except i haven't got pairs so now you're not using pairs and you're not as it were looking at topology looking at the Just doing a mechanical operation, just looking at this very simple structure. Now, I'm sorry, where do the polar coordinates come into the picture? Yeah, Red Herring. Oh, I see, that's the bit you're telling us forgets. Yeah, Red Herring. That was the bit when I said, okay, I wasn't quite sure whether that operator, you know, forget everything I've said, what the everything cover. Polar coordinates, you don't need to worry about that. All of this is independent of polar coordinates. The reason I thought, because there is, I understand it with people looking at the Hilbert space picture, the so-called relative state approach and the sort of decompositions of the basis,

1:57:30 that the polar coordinates are one, they're supposed to have a particular privileged position in allowing you to decompose the basis in the Hilbert space. Forget it, we don't need that until right at the last minute. No, no. If at all. No, there isn't even a suggestive structural analogy. It's the IS business. Yeah. But there's no even a kind of remote analogy between what they're doing and what's motivating you. Well, I don't think so. I don't think so. It's certainly not what's motivating me. We don't need that. That's just crap. Now, I obviously know and share your opinion that the whole Hilbert's base formalism, as it were, has gone... No, but I mean, you're right, because when I was doing this, I thought the polar coordinates were essential for getting those two equations out with the anti-commutator and the commutator. Right, that's what I did. You'd withdrawn that. That's not necessary. I can do it with just this thing. The other thing to notice also is that when we write things like this, sorry this is just a backtracking, when we look at these things, I hope Bob's going to drive me to the station. Otherwise I'm going. This is just the density matrix. I was going to say, where does the density matrix get into the spectra because obviously this all connects up with this. It's just the density matrix. So what we're doing is we're just looking at the density matrix. But we're not giving them that significance at this stage. We're not saying what they are or anything like that. But this might be... Sorry, we do not in physics deal with... Prigogine would love this. Yes, I was going to say. the density matrix. And I want to go back and look at this paper at Crona-Rivelli because they obviously have also got something. They've got to get the density matrix out of their approach, you know, to tie up the... Right, I start right at the beginning because I choose two-sided. Every two-sided idea is a density matrix of some kind. And hence the square root

2:00:00 Bingo! Bingo. So that ties up with what Bohm and I were doing. That's in that relative phase space paper? Yeah, it's also a generalised algebra of phase space, that's the relativistic version of it. I reckon that paper's got more in it than one of the authors realised, actually, in a generalised phase space. Well, you've also got, of course, all that beautiful stuff about the Lugil. Is that one of the Vigno-R transformations as well? Which? That's in the general... I'm not sure about that, but what I do know... You do talk about the Vigno-R transformation and the relationship between that and the load operator and the density matrix. I don't think it's in that paper, I think it's in a paper about the same time, around the mid-80s, early mid-80s. But there's one, I mean, a piece de resistance to come. Arbitrary or not. Sorry? Arbitrary or not. Can I get them all from you? Or can you get them from the back? Yeah. What is the generalisation of people in our translation? Oh, it is in there. Oh, it is in there. Let's have a look. That's a bit of a line. That's not the only place that you talk about the bit of my Alphonson Nation. No, I know, but that's the first time I've ever seen them mention it. The interesting thing is, the interesting thing is, but when I get the sums and differences, you know, the paper of Melvin's, the equations I get can be related to the thing that we are. Yeah, sure, sure. I'm sorry, you know, I'm afraid I've got a drum on my back and it's the same beast. This operation seems to produce new physics now.

2:02:30 Well, I'm not sure it's new physics, I think it's just... Out of the wave equations, you seem to get these conservation equations, and why that happens, I'm sure there's a way of showing. Well, I mean, it's certainly new and deep structural insight into the existing physics, at least. It's at least that. But seeing the Vignum-Yalphansmation, how it connects up with the density matrix. But the whole of that square root was Vignum Yr, wasn't it? You see, because the Vignum Yr we've got there is not the approximation. There is a Vignum Yr transformation. And that takes us... You see, there's a bit of a problem here, because some people, whenever you say Vignum Yr, think you're actually making an approximation. You've got a classical limit or something. It's used to know a thing, isn't it? Yeah, but that is not what we were doing. but that transformation is a specific transformation which takes you from configuration space to a phase space in the algebra and not the shadow manifold. That's what these people don't realise. These are subtle distinctions. Okay, Matt, are you ready for the final little bit? I hope so. I was talking the other day about the Heisenberg, extended Heisenberg, no, the enveloping Heisenberg algebra, and I said there were two groups of automorphisms I'm interested in. I would like to claim they're the only two, but I'm not going to be such a fool. Heisenberg group Heisenberg group is generated by E to the I AX plus BP plus CZ where Z is just the reals So that's group number one, there's a second group which is metaphylactic. And those are generated by

2:05:00 ax squared plus bp squared plus cpx minus x, or plus xp. And I think these could be complex numbers if you like. You know, we needn't have that i in there, it's just conventionally they put the i in there, So if I take V, I can do an e to the i, well let me just put alpha a dagger minus a alpha star a, e to the minus that, and that produces alpha. This is well-known, Mike, to physicists who deal with coherent states. Yeah, but this is just, if you look at this, it's just... It's the... It's equivalent to the Heisenberg group in the boson, in terms of boson algebra. And in fact if you convert this back into p's it's something like e to the ax plus bp, because a is equal to, you know, a is equal to a dagger, I don't know which way, a plus ip and a minus ip. It is the same as this, but I want to put it in A's because I want to know how A acts on B, because I know I'm in control. I'm a control figure. So this gives you the coherent state, And the interesting thing about this coherent state is that it's written as some constant term here, I can't remember, e to the a dagger zero. I think this is to the end, that's right, it's all well known Mike to coherent state

2:07:30 people. So if you look at this as a vacuum state, then it's a vacuum state which has an infinite number of particles in it. And these were the condensed states that Ramazawa talks about. So you've got a condensation going from your vacuum to to your coherent states, so new ground state if you like. Now then, what you've also got here, in coherent states is well known, is they've got operators which are alpha a squared plus beta a dagger squared b, and the same on the other side, these produce the squeeze states, well known in physics. Now, this morning I showed, like this, that v e to the q squared, oh x squared, sorry, And there might be one over the roof hanging, but we're much friends. And when I say that, what I'm saying is that if I look at this, what I've got to satisfy is this, and I've got to satisfy that x is not equal to zero, that x is not equal to zero.

2:10:00 And you can see how the x just goes through, and you write x as equal to a plus a dagger, or a minus a dagger, or whatever it is. So what you're doing is you're creating an infinite number of particles in your... So again, these are condensed states same ballpark as those context states. And then what you show, and I showed this in front of Owen this morning, this afternoon, So, that feed side equals, side feed equals, so that that is the side. What do you think of that then, Mr. Wright? And then the E that we, and you remember I had the delta the other day? I was going to say this is the, this is the transpose. The delta comes simply by putting a P in there. It's a transpose, isn't it? No, it's just, it's nothing to do, this is a different one. The P, the E and the delta come by doing this. I don't know whether it's the delta or the e, I can't remember now. Where you've got x on one side and p on the other. Right, yeah. So those are very funny things, I'm not quite sure physically what we mean. But it comes, I mean, it just comes straight away, you know, you've got this transformation, the metapocalyptic transformation in Heisenberg algebra. It comes out of the metapocalyptic group. Yeah. Or the transformation theory. Yeah. Well that's the way you construct your isomorphism, you see. And you know it's got to come from there. And that's everything done. And isn't it trivial? Why didn't Schoenberg do this in the first place? Well, they were speckling about that the other day, weren't they? But you had an answer, which was to do with why...

2:12:30 Well, he was looking for his E in a... But this is what Marco's thesis should have been about. I'm a bastard, aren't I? Well, why you say that, haven't I? Well, because there's a lot of different bits and pieces going into this. Yes, it's a sort of knowing a lot of physics to make the connections. The sort of physics I love doing, actually. Well, because I don't think I've ever got it as clear as this before. I'm sorry, but again, tell this ignoramus non-physicist a bit about the squeeze states. The squeeze states is that you want to practically think about a Gaussian. If you use a coherent state and you go to Px, your phase space, the coherent state you can show are Gaussians, and these Gaussians actually go round. Like a harmonic oscillation. Ah, I see. But they keep this spread. Because this was what Schrodinger was trying to say was the particle, because the wave packet doesn't spread. are special states for which the wave packet doesn't spread. Okay, but they're circular, right? Now then, these things actually turn them into ellipses. so that you can so this is what turns them into ellipses again? squeeze that's the squeezing you take the sphere and you squeeze it right

2:15:00 and that's why the squeeze comes in right and so what you'll know now if you remember this one was delta x delta p equals what is it h bar H-bar over two or something. It's the minimum uncertainty package. I see where the terminology comes from now and what's going on. And what you're doing is you're trying to squeeze the uncertainty in P Or x. Or x. To beat that. I mean he doesn't check, this area is always the same. And those are the squeeze states and they have certain interesting properties. You have vacuum states as well, I mean you can see this you're producing a vacuum state which has... Another interesting thing here is this squeezing in P and Q is actually obtained by taking many photons. Now I wonder whether there's a relation between... This is where you see space is being made out of all these condensed bosons. Yes, this is the so-called kind of soft photon. Ah, that's it, yeah, soft photon. So the space is not empty, it's full of these soft photons. And ultimately that's what's going to determine the gravitational field. Yeah. And they're going to be gravitons which somehow do this. Yeah, because none of this changes the geometry. Right. Now if we've got something like gravitons present, then we can start squeezing and doing all these things. I'm not sure whether it will just come out of these two. It may be that we need a bigger algebra. And this is why Harry was looking at the Einstein algebra, the Garoach. Now the only trouble with that is that's a commutative algebra. And the point is how do you generalise that commutative algebra into a long commutative algebra. The other way is the way Janus Raptis is coming in and that is to say let's take these basic idempotence and see how we can relate idempotence to idempotence and build up a structure. A network. And then see if this It's like a simplicity of decomposition, but I still like this idea of creating the geometry

2:17:30 out of what we did in that paper. That's ultimately what I wanted to do. I wanted to create quantum simplices and then have a quantum topology on those quantum simplices to produce space out of this infinite sea of soft atoms, photons, I don't know what would be. You see, having a soft photon, that would then tie up with the Clifford algebras. Because everything's made out of light rays. Now I'd say everything's made out of photons, you see. But then we've got the gravitation. The gravitation doesn't fit in there, so there's something special about gravity that doesn't. do you see the new way of I think it opens up an entirely different way of looking at these things and whether this is what Bill's got in the back of his mind I've certainly said I want to talk to him about that after listening to this I'm not too sure what's I certainly understand more now about the way you think about the structure in the ground state of that that's where our space time is going to come from now the question is whether it's a global vacuum or whether the local vacuum this is where special relativity is coming in and we've got some things to worry about there but everything starts from the vacuum that's the assumption not only the universe but everything In fact this is a nice way of looking at it, because if we say everything starts from the vacuum, then space itself must emerge from the vacuum. If the universe emerges from a vacuum fluctuation, space must emerge from the vacuum state. And therefore we've got to have a rich enough vacuum. But this is where all the topology and the structure is, you see, because once you know the vacuum state, you've then explained everything. That's the theory of everything. So if you want to think you're doing a theory of everything, you're trying to understand the vacuum state. Isn't that a ridiculous state? It's ironic, it's on the contrary.

2:20:00 Before we can understand the universe... We have to understand nothing. I think Wethead would go along with that. Yeah. Sorry, who? Wethead. Oh, yeah. No, Wethead. Redhead. Redhead, yeah. Yes, well, it's a great fan of the ground state of the vacuum. More ado about nothing. Yes, everything. And, of course, this view of the vacuum is actually a cleanup. It's deep cleanup. It contains all the degrees of freedom of all the systems. All the dynamical degrees of freedom are further than all the systems. It contains the dinosaurs and us. Well, you need to understand a great deal more, but that was hugely illuminating. No, this is a different view, but I think... Sorry, I'm still... Well, you have to be able to stand back and, you know... But at the moment, it seems to be a very, very promising way to go. I'd like to understand more about what Henness Raptus is doing there, and how his approach cracks up. Yes, I think you'll have to look hard to see the connection, but it's there. And what I'm hoping to do is just to pick his brain and pitch a few of his ideas. Well, I'm sure that'll be a two-way street. And besides, that's what physics is all about. I hope it's what all intellectual life ought to be about. But I think now I've put all this algebraic structure into a way which doesn't require Hensch and all that murderous stuff. And also you really are seeing how basic metapleptic words is and how it connects you up with it. It's getting much more clearly now what's going on in it. And that's why I want to bring out this fact that the Heisenberg... Oh, and I'll put one more isomorphism in there, the symplectic clifford algebra, I was going to say, yes. Well, you've already made that connection. I know, I know. And that really tells us this, you see. That really tells us that there is this metapleptic group. And, of course, it connects up very nicely with what you were saying about the way the metapleptic group approach? There's a much more natural way of motivating the... This is all metaplay. There's no rotation structure. Yeah, yeah. And that's why I kept saying these two things keep going apart. And the only guys who I know have brought it together in the way that I'm beginning to

2:22:30 get a glimmer of insight is Schent and Vince. And you see, this metaplectic group has got nothing to do with the Vanhove theorem, which was, you know, trying to use the metaplectic group to quantize all the Hamiltonians. This is just saying, given the Heisenberg algebra, the metaplectic group is a group of polymorphisms of the Heisenberg algebra. And therefore, it's got to be useful in physics, because we feel the Heisenberg algebra is the basic symmetry for quantum mechanics. And you see, you know, if only people had taken this seriously, they would have seen what the squeeze sticks are. Maybe these things that I'm talking about here are something, you know, particle, some kind of particle sticks, that we haven't really bothered to investigate, because we haven't another language to do it. Remind me again what the P's in the... The mountain. Yeah, so what you've got... These I think you'll find essentially... They'll be sort of X spaces. They're essentially constructed... No, I haven't quite got that. We'd better not start on it now, Basil, because you're already well after nine. Yeah, OK, let's go. No, I'm not saying because I want to kill a very, very interesting explanation, but there's already so much to chew on here. I mean, I don't know about Bob, but I think I've got at least a fortnight, no, two or three weeks' worth of chewing just on what you've printed tonight. It'd be a lot more than that, because... Well, I'm sure it'd be a great deal more than that, but just in order to formulate a few, intelligent questions. No, all I've done is just sketch. Yeah, you said this was the, yeah. Because the trouble is, you see, when I gave those series of lectures about, I've been through all of this with these guys, you know, for hours. Yeah, I'm sorry, but I'm hearing it for the first time, so, well, maybe not all of it. They don't recognise it, because you've got to have two views of all of this.

2:25:00 One is the technical details. You know, it took me five days to get this sorted out, until I suddenly realise that these are vacuum expectation values and that's what physicists use all the time. And the other is the... You're saying I'd have two views of it. The other is less of being a kind of more overarching view of where all this is coming from Well, the way that it fits together with your world picture, which, as I said, I see how very fundamental the vacuum, the structure of the vacuum is to that. I'd love to understand a bit more about... Simplicial topology is an area which clearly does bear looking at, and again, that's something Bill's very... I wish I could see where the topology comes in on this, you see, because we've got very simple topology up here. It's interesting now, with the vacuum state, isn't it? Well, yes, but the point is, what is the vacuum? Well, it's very simple, you tell us what this means. Because you've got all this high dimensional structure, which you've got to sort out as... Well, you've made the point yourself before, but effectively your vacuum is the hollow movement. Yeah, but mathematically, it's the limit point of this subject. Is that minus beta, is it? Yeah, where beta is 1 over k2, as t goes to 0. Well, that's going to take a bit of unpacking. Well, that's why I put it on the board here, because I haven't unpacked it. yeah i'm afraid i'm well it's suspicious but also um no well he made the point is you know you've got to sort out conceptually what's going on in your maths um at the same time of course as you are using it in order to to doing the mathematics and you know you know and applying of course the difficulty look at yeah well thank you very much again basil because that was an

2:27:30 absolutely bloody wonderful i can't tell you yeah it is really making sense but i think that i mean this is absolutely wonderful but what you're explaining earlier about the um the twister is beautiful too because it really makes me see how very very algebra goes and how much more general that tangent plane picture is than just the flag picture. I think that really is, that connects up with all these ideas and synthetic differential geometry. And particularly the business of the fibre bundles and the spin bundles, that's what that does to the coverings of the space. There's got to be... Well, that seems to very naturally connect up with these ideas of Bills about what he calls this cohesive hierarchy, a structure where you've got the kind of discrete and co-discrete spaces at the very lowest level, where you've got kind of a trivial action of the group. The group is just kind of trivial g equals one. It's just the way you think of the structure and character of sex in the structure and the character of a lot of the spaces. And then you've got a whole hierarchy of spaces, of structure on top of that, which degrees of connectivity of the figures, which all live in this high-dimensional structure and out of which the topological connections are really coming. And the open-sets base, the idea that Dipolo is all to do with building a structure on top of an already-given basis of open-sets, is just, doesn't go anything like deep enough. That's wrong. That's just coming out in a much deeper picture. I'm sure there's a connection there with what you're doing, you guys are doing. Yeah, don't leave anything behind this time. Yeah, I think I've got all of that. You say there's a train strike, isn't there? Well, pick up your... Yeah, sure, sure, sure. Pick up your bed and walk, yes. Well, it's up in the air. It's still up in the air. Officially, um... Officially, at this point, it, I think, is still on. They were expecting that it would be called off. But I would suggest, having told Alan and...