Noncommutative Algebraic Structures — Talk 1 of 3

0:00 It really means all the form, knotted up, twisted up. It's the old Greek word. I can't remember the precise meaning of it. The trouble is, the way we use it is not the same way as Lewis, Noss, and things like that. The symplex is a symplectic Greek word. Oh, when you think of complex, there's complex in the simplex. Yeah, you think that simplex meant, but it doesn't. It's not even spelled the same way. We've got balance here, um, different algebras, orthogonal and simplex explicit algebras. Yeah, but what Lou was talking about was really the orthogonal reproduction. But that might happen. All his stuff is . And Lou was saying, how do we do the translations? And the kinematics, in fact. And that's where the sympathetic group comes in. But everything Lou was doing was . I know he had time going up, but if he wasn't really doing the Even though he's using bras and cats. problem okay which is about rotations right and the general direction of calculus is to stay at the point and look at the directions okay but what i was trying to do is say all right now what happens if we move over to here and look at the world from this point of view or move over here and look at the world from this point and how do you put all that up Do you think that's big enough room? I can see it. I can see it. I can see it. I don't need to... Oh, I have to sit from here. This is the back row. I don't need to come any further back and lower. Okay. Look at this. How are you? No, they'll be okay. Looks like perimeter room. It's huge. Maybe you've a lot of people on this corner. I don't know what they're talking about. Oh, look at that. I can't give any thought to that, have you, Baz?

2:30 I'm going back to your old stuff again, I'm afraid. Well, I don't have to be afraid, but not your new stuff, your old stuff. I wanted to show you what I was doing and related in the context of what you were doing. Well that's easy enough to set up. Put it out the window, do they manage to get into that bar? It's only within 10 minutes. That didn't look like it did. Was the coffee bar open? No, I mean, I think it's open. It is open, it's open. It's open. It's open. It's open. I've sorted it now. I was a bit stressed. What I wanted to do was space time, I wanted to come up here. I started with Roger, you see. I was in the 6th office. Roger was in the 5th office. I was in the 6th office. I was in the 6th office. And we interacted with him a bit. And then he would just come. That was before I was famous. Before he came out at night, you know, Sir Roger. That's it, Sir Roger. And just after he got it, we were at a conference together in Radford, Virginia, and a taxi was sent up to us to pick us up from the hotel. I said, sorry Roger, I'm not sure I can sit in the same taxi as a night in Burrell. I did, yeah. I mean, if anybody is stupid enough to offer me a knighthood, I would say stuff it would. Yeah, I'm afraid I'm very anti. Do you want to become a knight? I do not want to become a knight. A little bit. Sir Hullery. No, it's a little bit. People would say I love Sir Hullery. Yeah, I don't know. Fortunately, I'm never going to be in a position of having to refuse. Oh, thank you. No, no, he didn't like that. I was reading somewhere in the middle of April. There was an online society and they had everyone to join and they were like, what? Well, yeah, he has to pretend that the people in these societies spend all their time wasting their time figuring out who to admit. Yeah, who is the next one to come in? Who would you like to be in here?

5:00 Yeah, right, right. But academics is based on that. That's what academics is about. Really? Okay. Well, I mean, that's just not my story. Why a person doesn't appear? But you see, well, you know how to clone. I've met a couple of colleagues, so I've got a clone in the background. I'm a wandering minstrel and we sing today. I've never had that at the beginning. I've heard that no one had two things impacted the games. I think that was it. This person is in a fight back position and that's what I'm hearing, so therefore you shouldn't hire me. That was the person. No, no, no. This one I'm slapsing on, she couldn't possibly be for this. It's incredible. The one that I sat in on, that the woman who mentioned it was physics down, saying, I got the white woman appointed and not the black Canadian. In public, there was only a few of us in there, but that's what she said. Well, she was following after Margaret Thatcher died, right, about dealing with universities. Yeah, absolutely, absolutely. She was a veteran, yeah. That's right. Even though she was especially a socialist. And then I realise... Was there a chemistry degree? No, that was Hatcher, that was Hatcher. I was talking about the Baroness Blackstone, who actually... I'm sorry, so everyone seems to talk about this as well, but I think you don't talk about this as well. Well, I've got on to it for some reason or other, because of Penrose's anointment. Oh, yes, yes. It's all Penrose's anointment. So how many scientists have been made, how many mathematicians and physicists have been made? It's because we have Penrose, Tia, Penrose, Berry, Hawking was offered that he refused to give him the O.M. Oh my goodness me, my god, Stephen. Well, he already had the O.M. as much. Michael has the O.M. as well. Yeah, yeah, yeah. The Order of Mary.

7:30 It's the only one that Queen actually gives her soul. So that, of course, is the... Because they do tend to be allowed to very distinguished people. Do not believe in... No, I wouldn't mind the money. What? I wouldn't mind the money. The money's alright, but I wouldn't refuse money. On the subject of money, since they're in the bar now, I don't know if I have time to explain to you. Dimitri is putting up a prize of a million dollars for this resulting film's geometry that he wants to choose. And I've been over in Oxford trying to persuade Roger to become chairman of the prize committee, but the problem is that Roger thinks that the thing is too imposed. And I'm a bit upset because Dmitry did flag the proposal very clearly, you know, basis for discussion, not any kind of final draft, please give me your reactions and then we'll go back and talk about it. Um, he, because he doesn't know these guys, you know, he's not a rat, he's got a member of the Russian business, and of course we need to talk about elsewhere. Um, I'm hoping, I might be able to participate in talking around, at least talking around and offering some friendly advice as to how to be drafted. One of the things I have to say to Dimitri is that, frankly, to just as much in terms of raising the profile of the subject and getting people interested in the offer of $50,000 and put the rest of the money into establishing a couple of research fellowships. Is this a one-on problem? It's not well known. This is his research group. I don't agree very much of people offering huge amounts of money for NAF problems to begin with. And I certainly would not approve, in principle, of offering a lot of money for a problem that isn't well known. Yes, I agree with you. So I think what I need to explain is it. But bear in mind, the country is kind of wrong. They did very isolated for a long time. I mean, that's what I need to say. You know, at least I need to say it very different. Because I'm in negotiations with him, but no talking to him for the work I've been in this place with his needs. But I did basically need to try and bring me down a little bit. There's another difficulty here. If it's the reason why I say it should be well known, it's not for the prestige of it.

10:00 But you'd better be down on the shore that it's really hard. Or else it's a plot. Yeah. You announce it and somebody shawls in and that's the end of it. And what happens, don't you? Well, I don't know much about Prince of Dramatry, having been thanks to the genius of many series knowledge of Prince of Dramatry. But on the other hand, I spoke to two people who were in Prince of Dramatry's research group when I was in Russia. And it's certainly a difficult problem. It's not actually a question of trying to prove. What they want is they want to produce a generalization of the GR equations in what's technically called the Burr-Hard-Moore, which is defined in something new type of product. I'm sorry, I'm trying to figure out more. What is this? I don't know what you're interested in. And the reason we look at how space is actually a lot of now, I don't know if there's a different terms of logic. No, I don't know what I'm saying. I don't know what I'm saying. Well, if I understand rightly, it's a little bit like what I'm saying. If you condense it down by doing that map of that puncture, you know, onto a vector-spur, you can't get to it, it's different from the top. I'll tell you how many of you create a vector-spatial, which is illegal for the type of technique to do, but I'm using it at this point, and it's still to promote a string network, so it can be much better. You see a big-offering, a string network, which is the sample with nothing at all. Maybe you just come in a tutorial and get straight up and out of it. That would be a virtuous ambition. And if you want to try and do something from that point of view, you want to know a lot of exercises such for which you can discuss in terms of the remit to the CELSO and that particular area. Then you'll be able to do some technique and allow your dressers to know what they are on the pilot's level of respect. And this is one particular technique offered. I think it isn't really that, because that's for all the students well. But I really want to try. The other thing is that they've got a happy example. They're not having a university. But it's a big block. It's the state of the technical university, which is the library. And so you can add, and all the places you might have no need there at all.

12:30 We've got a general relativity conference in July, and other people as well, of course. What got me attracted to it in the first place? That's after the icing there. The icing was a bit different. I think it's a co-operative problem. Many of you are talking about it. My thesis was System Mechanics of Interacting Systems. In fact, the art scene was a very simple, primitive model which goes from... I'm certainly not going to mention it from where it goes to be based on it. Right. And, um, I like that, so I don't know about some of this. Did that in my thesis, and I thought my theory would help. Yeah. But I don't know what was written on the notes and it can't be anything in the sections. and that would be a few, so it's a band of bands. But it's always, you know, these things are always in the back of your life. Now, I get this analogy very much, actually, and I kept reading the book off of the shelf, and I suddenly saw that he was pulling the same mathematical structure that I was using for all of these, to be awesome, to be able to leave them. So I then, this is why the algebra is attempting to be algebra, he just very quickly glossed through. And I was able to relate what he was doing to what I did in my thesis. Thank you. Thank you. Very good. Not a hint to me. Didn't have to tell you what that was different. And then, the fact that the only problem is, how is it going to run the numbers? And I've been doing it very creditably on the record. I'm not going to run the numbers. I had a question of what, how is it going to do that? Yeah? That was a question I had. Still, I'm still thinking about why. Why should I run the numbers that I need to do that?

15:00 Are you going to introduce me, Mike? Yes, I am. We still have one person to come, and the good news is that Mark St. Johnson has got the flight and is on his way. He should be here about 5 o'clock at the time, which is when you and I are going to have to leave to fix the break here. I want to hear what Boontree has to say. We'll never miss that, that's why I'm getting anxious. No, no, absolutely not. That's why... I'm not anxious about this, right? I'm not anxious about this, right? No, exactly. We were just saying, Sergey, that since we are running a bit late, we all very much want to hear Demetrius talk this afternoon. And Debatt and I have to go and get Van Eystern on the stage and we'll have to leave quite a bit before. Could we just shorten the lunch hour and start again? What's the time now? Well, if you can finish, if you can keep this down for an hour, if we can start, we can start when we just talk about 2.30, and we should have time to get it all in. Okay, I think we won't wait for everybody else, we'll ask that all to start now. Now, Dinosaur Heide is known for some of you and not others. I would like to say that he and I have become very good friends over the last five or six years. He did his PhD in London, originally, was it actually at Berkeley? No, I did King's. King's College, London, where he worked on some people, Morris Law Kings. I'm so old. And he then became the collaborator of David Bohm, David Bohm, at Birkbeck College, and together they called the many papers of extremely striking and original foundational nature in quantum theory, and eventually a book, The Undivided Universe, which I hardly recommend to anybody who hasn't read it. And since David Burns, by the time I get in 1992, Vassal has continued his work, he was in fact David's successor and chair at Perfect College, and he has continued that program

17:30 that he and David have worked on for many years, particularly in the direction that he's going to talk about now, the direction of exploring the algebraic structures underlying the phase space in problem systems and getting the conceptual understanding of what those other great instructors might be telling us about quantum reality. So, over to you. Thank you, Mike. Yes. Some of these transparencies were a talk I gave up for just last week and I hope that it's appropriate here to give an overview of what it is that I've been trying to do here. Just to put your historical background My thesis was in many-body systems, cooperative phenomena, but I then changed over to quantum mechanics when I started working the day before. And one of the interesting features was my introduction was really through quantum mechanics rather than quantum theory. Because he showed us in that very nice book that we had to do and then we'd leave algebra how to do the partition function, the icing, and so on, which is what I've been working on. And I've been doing it in a numerical fashion without any real sophistication. And then Lou's book took all my attention to the fact that you could do the same problem from the algebraic point of view. And the interesting thing was the algebraic techniques in that work on the temporary deep seemed to be very familiar with what I was doing in Parmajans. That's point number one. is that I also worked with Roger Penrose, well worked with, we collaborated, we had seminars together in the 60s. And that's just when he was developing a spin network, which is what Luke is talking about. And I was puzzled, there were certain things that I didn't understand from the way he was approaching, and I got interested in Clifford Algebras that was motivated more from a philosophical point of view than from a mathematical point of view. On the philosophical point of view, I actually think of David Bowen, how do we understand quantum mechanics? What are the processes underlying the formism? We know what formism does, we know how to work it, we know how to get results out. What does it mean, you know, the nature of physical process?

20:00 What is the nature of quantum processes? Can we find an ontology behind that? Now, the most people who know me, I've got this horrible reputation that a guy they think wants to go back to classical physics that I haven't grown up yet. This is totally wrong. the stuff that appeared in the Undivided Universe was not talked about with David Bowman and myself for 10 years in fact his hidden variable paper we did not discover, I hadn't even read it for 10 years and then a student came to me and said well look what's wrong with this paper and of course I didn't know how about that and then the student said you haven't read it so I then had to read the paper and then side of the story developed. But this was really where I was going right at the beginning. Okay? Now, I'm coming up today by saying, you know, why noncommutative geometry? Well, of course, in the... superficially, it's obvious, because we have the noncommutation relation between the operators x and p, position momentum. Therefore, we can't, if we're using eigenvalues construct a phase space, or a classical physics moves in a phase space, where do we find phase space? So if we have no phase space, what do we do? And this was the whole point about the enormous theorems, which were thrown out of time and time again, that you cannot put it in a phase space. But we always, the physicists anyway, and maybe the philosophers always try to look at it from a classical perspective, and now it's completely possible why it doesn't fit in. That was a tradition when I grew up. It may be different, but that was certainly the future. There were lots of arguments about the relevance of this. I accept that absolutely from all that theory. I'm not violating any of it. And then you've got the uncertainty principle is putting hugely, again, to physicists as if it's because we're stupid. It's not, I believe it's something intrinsic in nature that allows us to simultaneously specify position in the world. And this comes back to the algebra approach. Now, I found myself just recently using a number of techniques that appears in Hart's local quantum physics.

22:30 But Hart's quantum physics is that the algebra is on space-time. In other words, assume our priority of space-time exists, and then Roger doesn't know. So this is why they have a net of algebra. But being imposed by Roger, can we start with something deeper? In other words, let us not start with the space-time structure, let us not start with the continuum, let us not start with the space-time manifold, and our priority of the space-time manifold. Is there something deeper? That was the program. Okay, now, that problem seems to be orthogonal to what I would make my name, you see, which is what David Bohm made, isn't he? And that was that if you take the work that we did in the Undivided Universe, which went back to Bohm's 1952 paper, we can construct trajectories in a basis. The question then is, why are we violating the computation of elections? Because what Robert's answer is, we're not using the eigenvalues of the operators. It's the simple answer. But then the question is, why is the relation of these objects to the eigenvalues? And that comes later. And you can see here, the sort of thing that John Bell came up with. these are the same, without basis, the same man. In 1952 I saw the impossible day. And most physicists believe that this is nonsense. Is that a reference to Dom's model? Yes, absolutely. The 52, the original B-52, remember the B-52 bombers? Well that's Dom 52. Another question, one of the questions that's been behind my research, particularly since this day you died, is what is the exact relation of this to this non-compensative structure? I'll do that later on. Today what I want to do is, just to show you the beautiful things that came off the computer, just in case people haven't seen them, what is the Bohm-Mod? Bohm-Mod just takes the Schrodinger equation and writes sine equals r e to the i s, substitute in Schrodinger's equation

25:00 and then just concentrate on the real part the imaginary part of just conservation of physics and this is the real part of the problem of the Schrodinger equation under that form of neutral position and for people who have no classical physics if that term wasn't there, forget about that that's just classical physics that's the Hamilton-Jacoby equation classical physics and all astronomers and physicists knew that at the turn of the 1900s. it's got a new feature and it's a very strange looking function. R is the amplitude of the wave function and S is the phase of the wave function. And if we assume that this carries through that in classical is action but in quantum is the phase. Carry that relation through by writing that as a phase Then what you've got is a termistic equation which will give you a trajectory. And this then is some sort of fancy potential, the quantum potential. Now here's the two-slit experiment and what? Conventional physics tell us, you cannot talk about particles following trajectories because of this problem. If it goes through here, how does it know the other one's open? because it goes to different points on the screen when they're both open and when only one's open it will do something else. Okay, now this one says, yes it's alright, you've got all these, these are possible trajectories. And it depends upon the initial condition, where the electron, whatever it is you're feeling, hits the screen. And it will end up in here. So this explains why we see spots. It also explains why they're organized like that, because there's a quantum potential present. And this is the constant potential. Here are the steps. And this is the screen in the front here. And then as the particles come down these plateaus, that's where the things are straight lined. And as soon as it hits one of these dips in the potential, rate change potential force. Bang. Beep. Okay, so there is a causal explanation of quantum phenomena. and then the barrier near the barrier particles coming through I'm sorry, times in this direction it comes up to the barrier like that and we're then watching what happens there's a Gaussian packet here I think this is about the central

27:30 I can't see it properly I think the central of the wave packet Gaussian is there and we put a lot of trajectories in there because we just wanted to see what happened Okay, so this explains why the particles go through. And this quantum potential, as it were, either speeds up the particle, there's no violation of energy conservation here. There's a swapping of energy between the quantum potential and the kinetic energy. So we introduced a new, this quantum mechanics seems to introduce a new quality of energy, quantum potential energy. and a lot of people don't like this because and that kind of potential is where the non-locality resides if you look at the potential you find it is non-local and it was John Bell please can I make this point it was John Bell reading Bones 52 paper that made him ask the question which eventually led him to the inequalities people have often said to me well what Bones here is dangerous opened up the whole field of non-locality by making it very clear. Ok, because in 1935 and 1936 there was a big debate about the squadron non-locality. Nothing happened then, right through the 40s, 50s, until 52. In fact, until 1964 when Bell chopped Boone's paper and said are all explanations which give properties to the particle do they have this non-local effect? And the answer is yes because there are equalities, now you've got experimental tests and we're aware. Okay? So our P and X here are B-of-Halls as John Brown called them, they are carried by the particles. Okay, so I'm just summarizing here. The non-low path, and you see, there's this amplitude underneath, so it's not an ordinary wave. I mean, if you're swimming under the sea, and the waves are very large, the amplitude, you go, wow. But here, you can have very large, even though the amplitude is very small. Sorry, yes, that's right, even if the amplitude is very small, you still have a very strong course. Or you can have a very strong course. And that, if you like, is a kind of a heuristic explanation of why non-locality comes in,

30:00 because usually energy spreads from the amplitude going down, but it doesn't depend upon the amplitude. So, the question really was, how do we understand quantum non-locality, and more recently, how do we understand telecomptation in a theory which starts with locality as absolute? we have to plaster it on, this is why people don't like plaster, we have to plaster it on afterwards but for me it was much more, it was the Penrose feature and it's really the failure to quantise the improvisational field that I feel we should, that really should focus our attention on the fundamental space time, that's where I came in with this story and therefore should we look at the hour Now, I'm very glad to find that I'm not alone on this, because even Hamilton, long before quantum mechanics, said that in algebraic relations are between successive states of some changing thing or thought. This is Hamilton talking about process relations. And he had this beautiful task, the algebra of pure time. But then Einstein, he said, perhaps the success of Heisenberg's method points to a purely algebraic description of nature, then however we must give up in principle the space-time continuum. And then finally his correct poet John Weaver, who put it so succinctly, it's not day one geometry and day two quantum physics, but rather day one quantum principle and day two geometry. So how do you extract from quantum principle, from quantum process, or something? How do you extract this? And this, by the way, in a philosophical debate, is about bone indicated by the infinite order. But I probably won't spend too much time to work for that position here, but that's where the infinite order is essentially coming from. The algebra of infinite order we have to get space-time as an explicit order coming out of the algebra. Okay, now how we can do this, well, if I go back to the Gell-Pan construction, and the

32:30 traditional way for physicists anyway is to start with a topological space or a metric space, and then construct. Collegative algebra functions on that space, that's classical. But then Gell-Pan, you can take, you can use it with algebra, and you can abstract topology and therefore you can get points, which essentially are the maximal ideals, and you can get the points and the topology on the mountain unit. We can go away. But can you do the same with a non-community structure? What does that mean? You can actually extract. You mean extract? So the question is, can we do it for a non-commitary structure? This is where the problem starts, as you know in Paris, looking at it from different points of view. Now, we have no maximum two-sided ideals, so they're not behaving like Geltman points, and there appears to be no underlying space at all. But then I read Eddington, and he said elements of existence are not hating metaphysical elements of reality. existence is represented by an important element in the object because that's the whole point that's right is he thinking that way I just can't remember but certainly and I thought that immediately you know, something which stands but I don't want it to be stationed I want it to be something to be continuously going into itself and so an idempotent is the obvious thing for a point and the point is we've got idempotents now we've got either left ideals or we've got right ideals and so on so we're in that discussion and that's why I say that they're static, these points are not static now can we order these items so the really one thing is can we order the

35:00 potence into a topological set of sections of the structure. I can't resist making an easy remark. When you're looking at the temperament legal to writing potence, then you see that it's not identity. What happens when you square it is you get a little topological thing which collapses backwards. That's it. That's what struck me when I was reading that. It's a lovely diagrammatic And the fact that from Neumann algebra, Jones' work is actually saying, we're ordering these components. I still want to keep everything finite, but we can go on to type 2 and type 3 later, but let's get the structure on type 1. Okay? And then my philosophy behind this is that we want not to think in terms of particles and fields and interaction on a space-time nanomot. But rather we want to think about becoming itself. But not to think about particles, I mean... Particles are physical, but physical cells. They are extracted. They are almost going back to the vortex models. They are structures of this underlying non-community structure, and they are the relative invariant, quasi-stable structures in this process. So you don't start with particles, rather particles emerge from this structure. That's the idea. And this is what I call abstraction, in that sense, with extraction. Now, of course, the big problem, how do we put it in that box? But there's the philosophical background. Now, just to sort of emphasize this common existence. Now, if we had a commutative structure, that E squared equals D and the Eddington idea, then the thing either exists or it doesn't. And that's the answer. So you have a whole series of things which exist, but they don't exist. Fine. But now, what happens? Why don't your opponents don't commute with each other? Because then you can either talk about the existence of A, and you can say nothing about B, whether it exists or not, or vice versa.

37:30 So as you, within this structure, you've got things existing and not existing, and then go into an ambiguous state of neither existing nor non-existent, neither existing or non-existent would be. But how productivity amounts, if existence is about, I don't know what that, right? Right, so now we're going to have a new way of thinking about Adam Poppins, which cannot be defined when certain Adam Poppins are being. What do you want word N to mean? Sorry? What do you want word N to mean? What I want them to mean, let me clarify that. And would be like classical physics. Okay? If I know about A, if I know whether A book exists or A doesn't exist, I can say nothing about B. Because I know, taking my mechanics seriously, and it's only the values that have physical meaning. So I've got a sort of situation where it's neither A nor B nor not B. It's something else. It's in the implicate order, which be the way. So it doesn't have a manifest form. Okay? But this is the spirit of the day. So you're saying you would know about that you can sound tenuously diatomal. And this is a fundamental principle in making the nature of quantum processes. It's not because we're hand-visted, but the processes of nature themselves do not allow those things to communicate. Okay? And then you've got this idea, and this is where the phase first comes in, you can either diagonalise the X-matrix, or you can diagonalise the P-matrix. It's a shadow manifold, the guys are going to try to do all the shadow manifold. I would like you to abstract apologetic trauma, algebraic trauma, sort of subject that's like pointless approach. Yeah, but then you're in the university and you just apply it to point A. He's a such one, huh? You can give it that to your practice. Yeah, but I won't then say that we really can't talk in physics without having points in some way. relate it to the point of class. I don't want to do something totally abstract.

40:00 I think he's moving towards the point of class. Yeah, but that's what I would want to do, but I'm trying... You see, but my class has always criticised me, because I don't start my lunas in a way of starting with nothing and then putting small actions. I'm sort of reaching ahead to see where we've got to go, and then hoping it'll tell us something You're putting out a notion of, because of the underlying non-communativity, you're having more than one space associated with a given algebraic structure, depending on what it is you want to observe. It makes a lot of sense, right? Just like there's more than one eigen-space, why not have more than one top-watches? Absolutely, and somehow they live together in this implicate order. See, we haven't got a way of talking about it, except from these two words, intricate order and extra order. So I bet you might expect things like curvature coming out because of the fact that people have the structure of the observance. That's precisely the conversation. That's precisely the conversation. You're forcing it into a rope space. Anyway, I'll come to that. This is the spirit of the book. Okay, and then, the sort of thing that drove me away from the, you probably know this, I apologize if you do, drove me away from thinking of former characters talking about objects. The English I'm going to call cricket. Okay? And to play cricket, you need red balls. But I'm living in a world where I have to have two pairs of glasses. One pair of glasses sees colour, and another pair of glasses sees shapes. And I cannot see shapes and colors simultaneously. So this is my simple idea of non-cognitivity. So then what happens is, I collect all these red balls, I put my colored glasses on and I got all my red balls, sorry, I got red objects, very much, red objects. and then I put my shaped spectacles on and I pick out the red spheres and then I triumphantly say I can now play cricket because I have red spheres

42:30 but then you will immediately say ah, put back on your color glasses and what you find is half of those balls have turned blue so it's not that you're not looking at the properties somehow, the actual process of looking at them changes their quality through the design and the game. Yes, yes. As if cricket wasn't difficult enough for you guys to understand. So, and then of course, that comes to your point, with the necocubius, you know, the way my hair's standing on there, because the intricate order is that if you have a structure, it's very much like explication in perception, and you can see where I'm coming from, but So this I had before, but I saw you're going to do it, I'm sorry, because this is part of the implicate aspect of the rule. So you have one common structure, but there are two ladies in that picture. The Afroclassic picture. My worry has always been that I go straight to the old lady. I think there's something wrong with me. but this is the idea that you have one implicit in this structure are these two possibilities and you cannot see both of them yes, she has a huge number but I eventually see that she's not looking at me the other way, she's looking away OK, that's the problem. OK, now I want to try and motivate you know, I want to try and motivate the way actually I think this should be somewhere or somewhere else. Can we have a metaphor for trying to understand this, not just perception, but actually a mechanical metaphor? David Bohm was wasting his time watching television one day, when he saw this apparatus display. You've got a perspex, an anti-cylinder and an inner-cylinder, and in here you've got githril.

45:00 And then you've got a spot of dye in the githril. And we actually constructed this. Then what you do is you wind it around a few times and it disappears. Obviously. The staggering part is that if you unwind it, it reappears. The physics is linear flow. It's just linear flow. But the point is, it looks as if you lost all the structure, but you haven't. It's implicit in the glycerin. That was the whole idea of this, to illustrate that. Now then, if you put one drop in, turn it, another drop in, turn it, a different place, then you unturn it. It looks as if there's a... Instead of having a bubble chain, a track of a particle charging through, series of condensations where the bubbles can fall. So thinking that there's a particle going through there, could be lost. And so the idea here is that this is a justified with a process philosophy and the invariant. So it makes itself manifest and so on. Okay, so that's just a metaphor. Now, the amazing thing is that Lou thinks it's hard I got Heisenberg's equation of motion out of this. That's good. And in fact, it's Schreiber's equation. Would you believe it? Almost. What I say is, here is the dot, the line which is in explicit order at one time. It gets folded into the glycerin. I have an M1 process on it. It gets unfolded and produces a new explicit order. assume there is what I call the continuity of form. In other words, this opening process, this collapsing process, can be equated. And this is pure. So I can write it like this, and you can see now this is an automorphism. So all movements now are automorphisms, which now ties in with the algebra. And if you make these assumptions that m1 equals m2 equals and that m is equal to the exponential of i h, double h, h bar is equal to 1 in this.

47:30 And then just, like, you immediately get, I think there's a correction in motion. And then what you can do is you can actually... I just want to comment that your picture would be better if the arrows are running the opposite direction on the M2. On the M2? Yeah. Because both of them are opening up. But this is just a way of thinking. I know. But that is precise. That is exactly precise. Because it's M2. Okay. And what you do, so like that? The last one, I thought it was true like that. It's a good time. Okay. So, is my mind right? I go this way. It sounds like it's back again, but... No, I didn't do that. And what would be, like, population-like flags on that song? Right. No, that's not... If you explain the eye, then you really explain it. No, I'm not explaining anything. Yeah. Please, don't get me wrong. We are. That's the thing. I am using a metaphor, and if I do these things, then look, okay, I'm not going to get any with you, never mind, I've got plenty of time, I'm going to bore you later on as well, but then if you take that element, and it's an element of the algebra, I'm going to tell you, Now, I can always make my elements in my algebra by not applying a left ideal but applying a right ideal. And we put that in the Jones polynomial in the story there. And then if you split these off, you see you get essentially two equations, which are two Schrodinger equations, one for the left ideal and one for the right ideal. And if you think about these as bras and kets, that's the bra equation and that's the kets, Can you put this around the derivative of the style? If you do, you have the rack equation. Yes, you can. You can do that. That's right. What can you start with when you're on the rack? But then you get into the same problems that Finkelstein and Craig got into. You can do that.

50:00 But then you've got to think about going up and down a lot of rays and so on. You're right, but let's hold it. The point about attuneness is very important when I come to explain the... Oh yes, okay, so that's the solution. And I was just sort of wandering around, wondering what the hell we can do, until Clive Kumister organized a set of seminars at King's College, when he said, look, why don't we get together, I will do space and time from a relativistic point of view, you do space and time from the point of view that you're trying to struggle with. This is way back in the 70s. We had Richard Surabjai, brilliant philosopher, you know him. He came and gave us a brilliant lecture on the Greek's He actually refers to me in his book, doesn't he? And then we had some Japanese lady named I'm sorry I've forgotten. And she was doing Lightman's fire. So we had this big, right back in the same beginning, And then an American came up to me and said, well, I mentioned Grassman, I think it was in German, you know, because I was doing Euro, I don't know how you pronounce it, but something like that. I get a pronunciation when I go to Germany and laugh at me. This is a very famous book where Grassman algebra, as we now teach them, originated. Okay? But when I read the work, Grassman did not come through a Cartesian network. He says mathematics is about relations of thought in thought. Remember Hamilton said the same thing. And my thinking, okay, here we are, look, these thoughts are not in space and time. But there's an order in relation, so let's go on. So it's really getting, I'm sorry, getting myself kind of encouraged because I'm a very timid person here. Now, okay, how do you do that?

52:30 Well, if you've got an old thought and a new thought, can we distinguish it? Can we separate it? And Grassman's idea was no. because the new thought contained the trace of the old thought and the old thought contained the potentiality of the new thought. And so if you put this image and now interpret it in that way you begin to see why you don't want to cut the tree. And what he did was that his grassland calculus came out of manipulating symbols of this type. And then I see where you come in. And you could think of the process as essentially light-light connections. Because I've always been fascinated that even if you could sit on a photo. This is like Hamilton, a succession. A succession, yes. Okay? And now I think Hamilton's got a word here that says this and that. Exactly. It's all the same game. And I say that's where I got my ideas from. And then when I saw you had done something similar. I don't know if it's the same, but it's something similar. And in fact, I even set up a... these are the rules that Alessa was using, by the way. I don't know if you saw her work. She's done a much better job on this. Because I was just motivating it. And you'll see that the new calculus comes in. I set up these actions, some are important and some are not, because I'm not going to go into the details. So I've got this process, this undivided process between two logically distinct points in the arm. And I don't think I distinguish between the two because if I cut them, it's like a north and south pole. You know, you've got a magnet. You've got a north pole and a south pole. If you cut the magnet in half, you've still got a south pole and a north pole. It's not what comes. Not what comes from this. So this is the strength of the process. I have my processes directed, I don't think Lou does. He's got shift operators and things like that to generalize it. But then the key thing was this. I want a modification so that when these two things are the same, I can, as it were, contract to the end point.

55:00 and I have a plus and minus, I started with a plus but then I realised if I've got a minus in there, I could actually get the quaternions out of there, and if I put the minus side I can get the line screen out of there I'm talking about two dimensions in that and then there was also addition in there, which I'm not going to use here and there's also a little bit of refinement that I thought I think I put her credit somewhere else, but she's actually done much better math on her job, I think. I just know how much, actually it's a rather problem, it's motivation. But when you say there is two, say, what I should say events, right? Yeah, right, but he hates, to a physicist, his events in space-time. Oh, okay, but anyway, it's sounding like two parts of a process. But why, what does that mean, which you would allow to say that, over the river, that two parts of course are somehow weak, stronger than, say, the particles in space? Everything, yeah, well, and... Strong, yeah, but it's different. Not like it's stronger. It's different, but it's different, not like it's stronger. No, it's different. It's different. Okay, so the best way of looking at me, these things are like ripples on the top of the surface of water, and there is this deeper connection underneath. And we're trying to get at that structure. Okay? And then we say, Lou introduced this modification rule. Lou's work does not have all these actions in it. The point one is I just want to look at the modification because I'm interested in groups. It's an intended work. Sorry? It's an intended work. Of course, of course. But you see, my surprise was that you were getting, from here, the contenders and the logarithmic. I was getting it, and therefore I read it, but why is he getting it, and why am I getting it? And then, if you make this special assumption that when v is equal to c, and they commutative elements inside, you can pull it out of the common one equal to plus or minus one. So it's a special case of... Well, you see, the star is like the addition, and in fact, I also had addition, right? Yes, I'm not going to add addition, or the case is just like...

57:30 But the star is not the product, the product with the bees, the last thing. Oh, it's not this product. No, okay, but I'm not inviting it from... This is likeness, really, the order of succession, one after the other, after the other. Right. I didn't have it inside the album. Even if you go out to B to C, then this star product is not the same as number 3. Obviously not. The product number 3 is a different product. But there is a similarity. Would you agree that there is a similarity? But do you get something like total order here when you talk about success? I would probably say no. I'm talking about the algebra used here. Actually, I tried once a thing, which I just call like intentional steps. And you just go, because it's a little better answer. Just make it up to the ground. Right, right, right. And then there was also Yanis Rattis and Zapatra. Do you know Zapatra? Have you ever come across Zapatra? He's Russian. Right. Zapatra. Zapatra. Zapatra. Zapatra. Zapatra. Zapatra. Zapatra. But I just pulled these together to show up. Because at first I thought I was anyone who was doing this. And there was obviously someone who was doing it. And you see, this multiplication tree is like a contraction in tensors. Absolutely. That's what motivated me. That's what I'm thinking about when I'm doing the area I was about to do. Is that the element AD itself can be processed as a temporal process. And I wasn't thinking of making temporal processes out of them. thinking of the A-B as some distinction that's made in the space where the label is A and the other label is B. And the order is just the difference of the sides. So all these things fit together in the multiple interpretations that are possible for an order. Because in this input you just have an ordering notion. But the ordering notion can come out in the

1:00:00 because it's temporal, or spatial, or some kind of mixture. That's it. That's it. That's exactly the problem. This leads me directly to Clifford Altebrus. Because when I read Clifford, Clifford, did you know Clifford was talking about processes? Not what we teach now as Clifford Altebrus. He would talk about those rotors and motors, right? So they were really, and he said the worst thing that happened was Gibbs, I'm sorry to come back to America, Gibbs, and Gibbs wrote a paper in Nature saying that we did not want clever articles, nonsense, because of, I didn't know that Gibbs did. Yeah, yeah, yeah. There was so much, um, of course, the Giggs, the Sergians, and the Cliftonauts, and so on. Well, I can give you a reference, if you want, in Nature, 1887, or something. You know, Herbstein is also a great movie of, uh... Okay, okay. Well, I have the Giggs, you see, because he was dealing with electro-management. And his argument was, well, vectors always go into vectors, boi vectors to boi vectors, they're all separate. But the Clifford Algebra, we have a little bit of bi-vectagons to the back set, and a little bit of the back set. You're scrambling them all up, because they're in automobiles. And they didn't want that. And then the amazing thing, was his period again, Clifford Algebra, was when Dirac did his equation. The Dirac equation used his Clifford Algebra. But everybody has forgotten the original source of technology, and therefore gets into this sort of a manipulative character, which doesn't have too much reason to put it to me. Okay, so then, what, you've got from these two basic movements, I've got, these are space-like movements, I'm sort of building little things in here, I'm slipping things in, you've got to watch me very carefully, If I test that too, I wish I knew how to get dimensionality out of something primitive. How to get dimensions of your space out of something primitive. Now when I was doing icing model stuff, you remember the critical exponents? they only depend upon dimensions.

1:02:30 Just like what I was doing was counting the number of random walks which excluded, you mustn't cross yourself. Two-dimension, three-dimension. Yeah, yeah, yeah. And just by counting, I could tell whether I was in a two-dimensional space or a three-dimensional problem. One dimension, two dimension, three dimension. Once I got past three dimensions, all the same. So this is a place where the four-dimensional problem that Tumitri was talking about doesn't seem to be there. But that's a topological way of defining dimensionality, simply by counting. Amazing. OK, so that means if you give me two movements, which are going to be my following movements, then I will get the quaternions out of that, with the multiplication rule, and if you give me one space-like point and one time-like point, that means I choose the minus sign, where T's are together. So what you want, you see, is you want to introduce a process which turns P1 into P2. So you make it the same variety, the two numbers in it, and this is actually what's called the inversely in Grassman's work. And so then, so you've got this. Here, look, we've got these two together, plus. So that's equal to that. These two are, we've got to turn that around first to hit the minus, and then the two P's that opt us together, and you get minus P. Okay? And then you get this one, which you then have to, this way, P naught's it together, and you've got a minus because you've turned it around, so the thing closes so the multiplication closes and here is the multiplication table and these are some more difficult two things now with the two dimensions of the right screw all you do is when you've got two T's together same rules for the B's when you've got two T's together you change the sign and you get the

1:05:00 You get the Lorentz group. And in fact, my claim is you generate all in this way. That's what those other actions were intended to produce. Now see, the other way of going, where you're thinking of distinctions, and then you say you're thinking of evaluating the size of the distinctions, and then you'll say, well, if the product of the two things is the same, then you pull out a Lorentz transformation. I get to the different question how is that related to this construction? Interesting. Very close. And yet different. There are similarities and yet there are differences and the differences might give us some new insights. Anyway, my story I don't think I've got a slide that I really wanted to see but I hope I can get it before you leave, where I get a grasp of elements in my clippet algebra and show them that it's taking the inside to the outside and the outside to the inside and destroying things, it's very nice. But I might ask you to do this talk first to get the motivation for it. What I'm really doing is, because I've got the clippet algebras there, what I'm really doing is I'm discussing the light cone structure. So when you talk about the Penrose spin network giving you the directional calculus, I have the directional calculus built in the algebra using algebraic ideals rather than vectors in your spin space. Everything is in the algebra. That was my aim. Now, the question about translation is where the twister comes in is it actually relates light cones at different positions. They're still stationary but they're in different positions. And the interesting thing is that the, and again I've got more details of this, I just want to give a sketch at the moment. you've got a relation between the light cones at the point one, they're labeled with one, and the light cones at two, and there is this relationship, which is essentially the kind of a twisty relationship, only done in a very different way from the way Roger does it, I think. Concentrating on the curve, no.

1:07:30 Okay, so what I'm saying here is, I've got to get back, how am I dragging the points in some way? It's principle to have only one ideal in the left picture, right, in the left picture. Sorry? It's principle to have only this kind of ideal in the picture to the left. Here? Yes, but then you've just got the forward light cone. You have to put the other spinner in, the correct spinner, to get the past light cone. And then we have the dramatic relation between them. So I think that I have sort of sneaked in the points there because of the intersection of light rays, but then that's panoramic's idea anyway. But I notice, first of all, because I've forgotten the puzzle about spinners, because they seem to be taught to us in the visit as an arm before them. We've got all the angle of momentum vectors and then suddenly says, but look, there's a two-pole representation. Why is it there? In the clever algebra, it is already in the algebra. the spinner is there it's a given, you don't have to put it in afterwards it's part of the clipper group as they call it, it's the spin representation which is the clipper group ok, so then my idea was the following, here's the key to construct algebraic spinners I have left ideals and you always make left ideals by taking a primitive component and multiplying it by the algebraic left so here is my point and saying that impotents are like points I'm actually building the point in and the ideal is the like way and that's exactly what what one thinks that's exactly what one gets now, my crazy way of doing things is equivalent It's a standard way of doing it, where you've got a clipped algebra of four elements. You find the primitive impotence, you form the ideals, and then what I'm talking about is this object, you've got two of them.

1:10:00 In the ordinary physics, these are two treated as equivalent ideals, and we map them onto health. Okay, and then we've got this structure. And here you'll notice that we've got the minimum in a potent, the in a potent is done in light red coordinates. And then very quickly, do you guys know about the K-calculus, where you actually explore, I'll just do it very quickly, where you actually get a radar gun direct it out to a point and it's reflected back again. And by assuming there is a delay in the signal and assuming special relativity, a relative motion you can actually abstract out the Lorentz transformation. So you're exploring space with radar signals clockwork and radar signals. In terms of the light ray coordinates so that you get this particular relationship here. And this is where the k is. It's got the speed of the observer built into it. And then, I don't really want to... Also, by going through this calculus, looking at another observer, you can actually get out of the Lorentz transformation, or the Lorentz boosts. I'm trying that quickly, but I just want to say you can do it, because time's running, you can do it. And if you want to see it, it's in New's book, Notes, and Physics, or Physics and... Which one is it? No, it's not. Of course, the K-character list is to the Herman Bond. No, it's not. It is? Who's it to? Page. Page. You remember Page and Adams? Pardon? Page and Adams, the electric... Oh, okay. The Herman Bonding and popular audience? Okay. It goes back to 1936. I've got the original paper. What I'm doing here is pointing out that the very simple calculus of a comma b times c comma b is a b times c b. Yeah. Multiplied of coordinates gives you exactly the right algebra for handling this. Yeah.

1:12:30 So I'm going to do it. I'm not, I'm not kidding. I mean, I taught the K characters before I saw your work. Because I saw the K-characulars being taught on British television on the horizon programme by Bondi himself, way back in the 60s. And I thought, wow, isn't this beautiful? And I used to teach, and I also taught David Bowen K-characulars. Did you know you can get the control room out of the K-characulars as well? Yeah. Page number. So who's Page? Do you know that classical electricity? I think it's the same page on that one, I think it's the same name. I'm sorry, I can't remember what his first name is. But I actually got the zip, and in fact I saw it when I was preparing for this tour at Oxford. I got it out hoping I could quickly reconstruct the conformal group, but I just ran out of time. You know, like you run out of time as well. I will do that later on. Okay. Yeah, and then the other thing I want to say is that here, the light-coding objects are interpoblants. So we've got our light rays described by Spencer Penrach said that these are our interpoblants. And the source of the light-coding is the primitive interpoblants. So I've got these in impotence, but now, of course, I want to know how to order them. I'm not just emphasising it's impotent. OK, this is a clever answer, but the question is where is one? That is a nice slide. Oh, sorry. I've got so much to do. Because where I've got is the whole, let me, like, I have got the whole of the Klippan algebra, the orthogonal Klippan algebra graph, in this way I'm looking at it. Now we've got it formally, as everybody knows, but I've also got it this way, showing the process philosophy lying, possibly lying behind it. OK? Now, the point is, you know, where is cosmic? See, we've got the orthogonal structure, but that's just classical space-time. Where is the cosmic act?

1:15:00 Where is the expected orders? Where is this? OK? That's the problem. And this is just to emphasize, not only do you get the bone trajectories in a face space, but you also get a bigger distribution which is in a face space. So the face space is not unique to bone. And there's an interesting, can I do a slide? Yes, why not? A very dear friend of mine, John Larker, said he works in nanotechnology and he works with the Vigna distribution. Now, there are problems with the Vigna distribution because this distribution becomes negative and therefore it's not a probability distribution. But nevertheless, you can actually reconstruct the whole of quantum mechanics by averaging over that object there. And we can... you know, the whole quantum mechanics this way. Now I'm slightly thinking, and one of the things is what is the relation between these two? I think I'll discuss this when Luke's here, because this is his, that's why I was quickly drawing over it. But I looked at it, and I'll show how they're written. These two are actually related to each other. So when John Parker was saying, when I do the Vigna interpretation everybody listens to me, but as soon as I mention Boehm's name, Boehm, they all live the wrong. Well, not quite, but you know, mentally they use the right. And yet they are both different aspects of the same theory. And I will show how, maybe, if I get time to try anything I want to do, OK? Look at that. So the aim is now, where's this non-commitivity coming in? OK, there's non-commitivity in capital. Algebras, but I haven't really shown the quantum aspects of that non-commitivity. And then, have you come across this level? I'm sorry there's a private seminar, it's a nice small group. I can get a bigger distribution from the k-calculus with one difference. My point is, but my point is moving.

1:17:30 So we're doing radar, really, real radar. These are aeroplanes, and therefore we need phase And if you take the autocorrelation function of these two, the outgoing signal and the backcoming signal, you find you get exactly the bignard distribution. You get exactly the bignard distribution. And the bignard distribution happens to be an expectation value over the height of the group. and therefore if we're going to seacorn mechanics we have to go to the Heisenberg group and the Heisenberg group group of all morphisms symplectic structure so we're now moving from the orthogonal structure to the symplectic structure and many years ago I was talking to Clyde I pointed out to him for reasons which I'm not going to go into that there must I won't give you the reason why I said that, but from what I was doing it was obvious. And I went round asking my physics colleagues, is there going to be a double covering of the symplectic groove? Yeah, because the spin is the double cover of the orthogonal groove. So where is the double... Luke, do you know what a double covering would be? This is not a test, it's just for information. Don't be embarrassed if you don't know it, you know it. Do you know what the double covering of the symplectic group is? That's the one that keeps invariant the Hamilton's equations of notions. Classical transformations. What's the definition of? Uh, I don't have it here, um, x1, p2, minus x2, p1 is invariant. What's p1? Oh, position 1 into. No, I mean, like, your quantum number keeps invariant x1 squared plus x2 squared plus x2 squared plus x2 squared. And this one, this one. X1 to xn is the coordinates in Rn. Yeah, this one keeps invariant cells in phase space. It deforms the cells, but keeps the body.

1:20:00 I guess I need a definition that's more... Okay, in order to think about what the covering... Okay, can we leave that a bit? Double covering. Sorry? You heard the double covering? Double covering. This is my trick that I didn't do yesterday. Double covering. There's a difference between 2 pi and 4 pi luxations. Have you seen this trick? 2, 5 and 6 I've got a thing but if I do 4, 5 I hope I write out 4, 5 it's not quite I didn't quite do 4, 5 it comes out ok so that's a demonstration of the double cup I'm sorry I didn't do it properly there's a double cover of the symplectic group and it's called the metaplectic group very few businesses knew about it but there's not more people who know about it now and here it is the clipper group is the metaplectic group and the generators are x squared, p squared, xp plus px and that is a very fascinating group which Maurice de Cosson knows a lot about, and I've learned a lot from him on that metaphorically. Now, the problem with the Heisenberg algebra is that it's new potent, and this means that there are no impotents in the Heisenberg algebra. And therefore you're forced to use very complicated of either Dixmere or Shem, Walter Shem. It really is hearing about that. I mean, I don't know, maybe, do you know it? But it says it's a representation of the lead, no problem lead groups. You're assuming that there's a final order to, I mean, that x half to the n is equal to zero to some n, is that what you're saying? No, I'm just taking, I'm just taking the, if I use this object here, what do you mean by I have one? I have one, it means that, if you take the coffee table and you use that as your product, and you do it again, it vanishes.

1:22:30 It's almost, I can follow what they call almost, I don't know what I'm talking about, but it's almost cognitive. Because if you take the explanation, it's always constant. You're telling me that if I take, if I have some elements, say, and I'm given the best x commentator piece of a lot, and then if I take, oh, well, right. X, P with P, it's obviously zero. Yeah, I see what you're saying. Okay. Yeah, because it goes down the stairs. That's right. That's right. Yeah, I know. At some point, the product vanishes. Yeah, vanishes. And then there's another theorem that says that if you've got an algebra like that, it has no inconvenience. Which, again, I think for us very trivially, let the company know very well. And of course, now I'm in problem, you see, because the whole model, I was in a mess for a long time with this. Where are the hidden photons that correspond to my weight functions? Because in the clever algebra, I've got the weight functions in the other spinners in the algebra. So where? Where are they? I think it's like bashing my head against a brick wall. And then, we discovered that there was a generalization of the Quartanians that had been done by Sylvester in 18... I'm sorry, I'm a historian, I'm sorry. In 18... I still haven't got the reference, where is he? No, I haven't got the reference, but... Sylvester in 1884 or something. You guys know, I think he's a very good mathematician, so I can say something. Now, look at the quaternions this way. You can define it as E1 E2 equals omega E2 E1, where omega is the square root of unity. OK? And the quaternions. But what happens if you say omega is the cube root of unity, which is why I asked you the question over at all, about roots of unity, which is the

1:25:00 rest of the audience. Then what I got thinking about, why are you? Yeah, why are you, every time I look at your work, it's, it comes out. Okay. And these are the lines, they are just nine of them, whatever I call them, the identity. And that was the generalization of the Clifford Algebra. And you can actually generalize it even more by just taking the efforts. And now the interesting thing about this is that if you let pain go to infinity, you recapture an extended version of the Heyselberg Algebra. So that now means that this algebra, which is not the extended Heisenberg algebra, is not the impotent, it's specially extended, and therefore that contains impotent. But I don't have to prove that, I just want to show you how you do it in finite, because Because if you go back to the, if you go back to the, uh, the layout, these, you can very quickly show this has inverts here. It's a final clue, there's no problem. You can actually construct them. So you know that they've got construction. So you've got points in this space. Okay, and then the interesting thing, one of the things I was looking at was, So let me take the final play space, I can put points in and see what it looks like. Can you describe that as standardizing word algebra in and of itself? No, because I've got a whole paper which if you come to my talk in Anthem you will see the display. Especially for you. Is it in your notes somewhere? Yes, yes. I mean I have this device that's just going to capture all these notes and electronics. Okay, okay, okay. The question we just think about like fixed point theorem, right? Continuum transformation, is there a proper way to reverse the theorem? If we get something like points, could we go to it? Go to the continuum? That's what VAR actually did. These hidden elements are actually delta. They turn out to be delta functions in the continuum. They are algebraic equivalents on the delta function.

1:27:30 And what is the dimensionality of the delta? Not the odds, it's a non-dimensional. Non-generatorism you find representations of. The dimension is none. And then, if you add the n-group of unity, then you have a dimensional n, so you've got a whole succession of higher and higher algebra, or bigger and bigger algebra. And the limit that then goes to infinity is the height of the algebra. Okay, now, since this is a finite space, space has got very big components. Here they are, I've constructed them in terms of the two basic operations, u and v. This is, if you like, another way of talking about it, this is the discrete vial algebra. Often when you're doing quantum mechanics you talk about the vial algebra. and it actually is in bulk of the theory groups. So I know you're telling us that the actual I am going to use the screen action is but then the algebraic equivalence of that So you see what I've got, I've now got a translation operator which actually becomes the momentum of the limit, but I can generate by points. So this obviously must be related to your way of doing the points in a discrete structure. And I can also label these points, and the points are just labeled by left ideals or by minimum. I can label them with left ideals, and that's just the Ket X, the equivalent of the Ket X, the infinity continuous limit. Okay, so, and then, I think they're really hovering now, they're going to start. No, no, no, no, no, no. Okay. Now, what we've also got in this algebra is we can find another set of impotents. And they're actually equivalent to the momentum space, and they're generated by this object here. So I've got another set of points, but they're created that the points of p-space and the points of x-space

1:30:00 are related by the wrong body. And David and I used to call this our exploding transformation. So the points in x-space are exploding from all the points in x-space and vice versa. There's no way that it corresponds between them. And these are my position of momentum in an actual mathematical structure. So what I've actually constructed here is two discrete explanations. And there are many of them. And they are related by momentum. can't be good, but these are examples, real-life toy examples of a shadow robot, or which will become shadow robot. Then I've got the continuum limit here, and this continuum limit is lifted directly from biology.