Algebraic Structures in the Foundations of Physics

0:00 where do we start? And at that time, again, at Birkbeck, we had Roger Penrose. And we had some really wonderful times because Roger was also very interested in this idea of how do we start from something which doesn't take the continuum as basic. And there was the Penrose and Cronheide, really what we call it when Penrose actually developed a And he wasn't first part of the time, Zeeman actually preceded, roughly the same time, but he got his paper out quicker than Conrad and Enrox. And they were talking about causal sex, and of course you've heard about, those people have been to Raub's talks, you've heard about causal sex. And you also, you've worked on them all the time. So that was in the background. But the particular idea that really took my fancy was the spin networks. And of course you have Sorokin here, who's done a lot more on that since those early days. Whether it's going anywhere or not, I don't know. All of this was in the background. And the sort of thing, I came to this as a condensed matter physicist, so I had lattices on my mind. I was doing counting for my PhD. It's all like counting various things. Embeddings, if you make it sound good, I was embedding finite grass in regular tessellations, I was just counting one, two, three, four, yeah. And so I had that on my mind, and one idea that was floating around was that space-time was actually a discrete latitude, which I think was still perspuing, trying to see if you can get something better. But I sort of gave that up because there were some deep problems, particularly with these continuous symmetry, which seemed to be around in physics. And the problem is, this lattice structure immediately stopped everything. It seemed to be very static. And what seemed to be, to me, the sort of thing that we should be doing was we would have been thinking about something which is something in terms of activity, taking activity, taking movement, taking process as basic, not activity and process in space-time, but activity and process from which space-time can be abstracted. And this is way back in the 60s we were talking about this, and it's only just recently in the last decade that a lot of other people out there are beginning to, you know, take this idea of abstracting space and time from something deeper, and of course they get it from the non-competitive structure, which is what I'm getting at.

2:30 And you know, David Bohm created this idea of the implicate order, and the implicate order was an order which was different from the usual order which he called Cartesian order where we think about things organized particles moving in space-time and fields interacting in space-time. He wanted something beyond that in which at the primal level the process evolves in this unfolding, folding way and all this is very grandiose and very vague and people, you know, physics have tended to say, oh yes, that's all very well. interesting to discuss a few with us really interesting to discuss what he meant because I have a feeling that you know he had these nice ideas but in some sense he could not bring them forth mathematics that's not that was the problem say that's why the mathematical implementation I mean when I when I started with him way back in those heavy days you know it was my hope that we could actually create the mathematics that were coming up but it's difficult and that's a problem and so what I've been doing is sort of getting bits of mathematics that's already existing and showing how you can use that to actually articulate these ideas and just to put my final little bit of slide which I made many years ago now transparency many years ago just showing how other really good physicists and mathematicians had been looking at this question. The first one is by Einstein in 1936 where he says perhaps the success of the Heisenberg method, i.e. the algebraic method, points to a description of nature, that is to eliminate continuous functions from physics. However then we must give up in principle the space-time continuum. So in 1936 people were already saying this kind of thing. And then Eddington, who was my hero at one stage, and that is the fallacy to think of that a conception of location in space-time based on observations on the large scale still applies to the small scale. And then there's a remarkable one by Schwinger in 58, where he's looking at the renormalization problem that was facing him so that, you know, a convergent theory cannot be formulated consistent within the framework of the present space-time approach. Yeah, that was Julian Schwinger.

5:00 And of course, Penrose was the spin network, and even John Taylor, who was probably a better noire of ours. But the beautiful one I like, which sort of... Convergent theory, what was the language there with you? Which means a topology. Convergent theory. It represents a conception of space-type, which means that Zwingier had objections about the topology we applied. Yeah. Otherwise, convergence theory means topology. Yeah, yeah. But, you know, the tools weren't available at those days, and I think that's probably the problem. Well, even of the same notion of the space-type manifold, because Zwingier was something objectionable. I say the one I like is this Wheeler's idea that it's not day one geometry and day two quantum physics but it's day one the quantum principle day two geometry the only problem is what the hell is the quantum principle that's us all if we put a good question we are in a good place Okay, so that's some of the background. Now, the question about, you know, I was trying to emphasize that I feel that this idea of movement, this idea of process was the way we should, but it's a very difficult idea to get hold of. And what led me ultimately to really link to the mathematics, to the algebraic approach, was nothing to do with what was coming out of physics, but rather I started chasing the history of ideas and I went back to the I'm sorry, it's the 8080s I mean I'm not that old but went back to the 8080s to look at all the foundational work that was going on with names like Hamilton, Grassman, Clifford and people like that and I was very extraordinary, this was the title of one of Hamilton's lectures metaphysics and mathematics algebra of pure time yeah and it's not about changing things but also about the emissions okay and then grass when you say i mean math he was he was quite adamant mathematics was about thought it was not about uh material reality yeah so that you know you

7:30 It's the ordering of forms in thought. And from that came the Grassman algebra, which we now use in, I think, a very mechanical way. We lose the idea, because for Grassman, the Grassman algebra was about movement, it was about activity. When I read Clifford, again, he said, no, no, why do people make everything static in terms of little vectors drawn on boards and so on? It's actually movement is the most important thing. So it's about becoming, and this becoming is the sort of idea that we were playing with, that's why we had activity. So really Heraclitus for you. Yeah, Cline. Felix Cline. Phoenix Cline. Yeah. Phoenix Cline. He was that, was he? I didn't realise. Felix Cline. So these ideas are not new, but I just wanted to fit the picture because it's so, the approach which I think I'm coming in is very different from what is normally adopted in physics. Okay, so then the idea was how can we actually get a mathematical structure which we can say something about in this. And so the idea was simply to say, okay, suppose we have this process. We've not got this process going on against a space-time background. to have some way of making distinctions for something to happen. It's almost like going back to your early perceptions. Just as you come out of the womb, what is the first thing you do? You don't suddenly put particles everywhere and make them interact. You actually just make distinctions. You try to make distinctions of distinctions. You try to order these things. And then from later on, you then abstract the ideas of space and time and things and objects and so on. If you look at the way Piaget actually analyzed the way kids observed Those are the more primary things. So the idea was to take those primary things and see if one could actually build them up right. And that led me to actually suggest this idea of an algebraic process. Oh, I just want to say this thing. Grassman puts two objects inside a square bracket because he didn't want those two objects to be divisible, to be divided. And that's the idea of the new thought. if we've got two thoughts is the old thought separate from the new thought or the new thought

10:00 separate from the old thought and then the idea is that the new thought contains the trace of the old thought and the old thought contains the potentialities of the new thought and therefore you've got this quantum idea of the potentialities coming in all built up in the same object this bracket structure is the grassland product no it's not a grassland product it eventually becomes his his extent there is extensives it's extensives okay and then from that he then builds them up into vectors as a secondary feature can we see that the resource of means a difference a naive concept of a different yes I mean the whole distinction is essentially different but you're trying to say that you build things up from the difference. That's important for application of differential geometric concepts. In his sense, Gransman did that. What I'm trying to do is to capture that. It means a comparison. Difference means differential means comparison, anyway. How you compare that. Okay, hang on. I understand. I'm trying to do a little bit of something We try to see what he's saying in the way we look at it now, I want to say we want to try and be careful of that because, you know, the DX hook DY is a Grassman product, and what the hell has that got to do with movement from where he started from? Yes, product again, again means the basis of a difference, product, of course. these reminds me let us leave you in it which is not I was just wondering is that a commutator no no these are not come no they're not commutated they paired and then if you want to think of them mathematically they're just pairs of elements I'm giving them a meaning but they just think of them as pairs so there's sort of what I can't remember that's the sort of dual numbers where you have ordered pairs ordered pairs okay you just put them together and And you're saying, but you can't separate them, because if you separate them, you get another process. Okay. A different process. It's not the sum of the two. Yeah. Okay. No, but it's not commutating. Can we wait here? And the important one that we need to look, particularly Janice, for you, is this multiplication. And one of the things here is that my processes are directed, whereas Lou Cowen's processes are not directed.

12:30 But it actually doesn't matter because we arrive at the same for some reason or other than that, the algebra comes out to be the same. But the important thing is this product which I call the order of succession so if you have one process followed by another process then when these two elements are the same you can actually contract them and different from yours when these are different you don't contract them you just don't want them equal to zero you just leave them you just removes cracking and I cannot you if they are not equal okay but if they are equal it's a serious problem I mean like the product of two arms yes yeah but if they are not equal you do not annihilate them no as I do but you can't in the arson you keep them there as words if you like think of these think of these as letters and then you make words out of these letters right okay and then this this act this structure that here define this is actually a around groupoid would you believe? Just to make it look as if there's a real mathematics problem. Okay, now what staggered me was that I could actually get the quaternions, but I think you can get the quaternions out of almost anything, which is one of my worries, is that by these simple rules, just using that product rule, not the addition, I can actually get the quaternion structure. Can I just check, by quaternion, do you mean essentially something like a three vector with the zero conform within just a number? I mean, no, I mean the Hamilton three quaternions which all have a square of minus one. okay sorry each one of the different why you're talking about what because you you are getting I shake hey well that's not what I thought that was the conventional wire there's this time that these standard notation so what you would do I mean what I do here is the same way it's almost great yes I've Oh yeah, you can do that, you can do that, and of course Hamilton invented these fraternians in order to talk about division of vectors, right, and then you do have that sort of, yes, that's the, that's the, but what I, what I find remarkable, maybe you don't find it remarkable, is out of that simple incidence algebra, you actually produce the fraternian structure, and that's just simply, wherever you see, you see, let's have a look at this, what you do here, is these ones annihilate,

15:00 working there so just invert that with one of the axioms I had and put a minus sign there and then kill off the two p2 and you get p0 p1 so you get back inside the algebra again and you just systematically go through there and you find that you get some units now our letter has done this in a much more general way than I have it's a much better way I know there's a paper but then again she's a mathematician yeah but it doesn't matter I mean, she's done it very beautifully. Well, the difference is that you could get the whole set of people's algebra out of using the posture. Yeah, but you've done it in a more general way. This is possible, yeah. And you've got all the core patterns. No, I've got all of them out. No, I've got all of them out. We'll argue, I'll take your call. Okay? We will argue. Okay, yeah. But I say, I mean, if you want to do the Lorentz group, this is the Lorentz group in two dimensions, one time, one space, but I'm trying to make this a sketch. You can do the same. I mean, now, when you combine t's, the time arrows, the time motions, then you combine it to be minus one, not plus one. You can put a minus sign in there. And then out comes the multiplication for the, that's R11 in the Clifford Algebra notation of Portius. But you can do the Pauli, and I mean, I've got it here, but I don't want to go through it. You've got here the Pauli Algebra, and just by using those axioms that I put down there, all of this follows, so you can get the Clifford Algebra, you can get the Pauli Clifford, you can get the Dirac Clifford, and you can also get the Conformal Clifford. The covering group is SU2, but it's an 8 dimensional algebra or not. Sometimes they use the Pauli matrices as the quaternion algebra. This is the bigger algebra. And then you've got the Dirac algebra which is the 16 dimensional algebra which all physicists use. Whether they realise what it is they use, I don't know. And then finally you've got the twisters. And that came out of work that Penrose was doing with me at Birkbeck.

17:30 there when that Twister program was going. But I think the whole thing, he built everything out of spinners. And you notice I haven't mentioned the word spinner. I'm building everything out of the Clifford algebers. And I'm saying furthermore that those Clifford algebers emerge from this type of multiplication which I'm trying to argue. I mean, whether you want the philosophy or not about it is a good point. But I find it rather useful. Now, in order I also was very struck by the fact that if you took a pair of Grassman algebras, a pair of dual Grassman algebras, defined by A, and I'm putting A dagger deliberately because I want to be suggested. It didn't necessarily be the emission conjugate, but in other words, this is my current annihilation and creation of a reference. Now, they're dual in the sense that we've got this anti-commutation bracket. Now, these are commutator anti-commutation brackets, and they've got a plus on there, so it's anti-commutation. So that defines the duality for you, and then from those pairs of Clifford algebras, you can actually create all the Clifford algebras and now the Clifford algebras are essentially phase space, they're like a generalised phase space that you've got you get a double algebra and you can get back to the ordinary algebras by taking complex numbers but the point I want to just make here to emphasise is that there is a thing called a Clifford group in here and the Clifford group is the SU2 for the spin thing for the powering, it's SL2C for the Dirac Clifford and it's SU22 for the conformance of the twisters ok so that's just to note that this is all obtained by these inner automorphisms so rotations generalised rotations in these arguments come from inner automorphisms I didn't find them there. Let me put it this way. I didn't find them there in the first place. I found them afterwards.

20:00 You found out that the criminal... I had done this, yeah. You know that he's dead. I know he's dead, yeah. Well, I think last year, was it? I think something like that. Anyway, one of the reasons why I was... interesting is because if you take those clippant algebras then you can create algebraic spinners normally a spinners are a two-component object in a vector space but if you stay within the algebra the spinners come out automatically from the algebra but then they come out as algebraic elements and I think that means that you can represent them by matrices and in fact in this particular algebra R1-1, I'm only doing it just to illustrate rather than to prove it. You actually have the equivalent. So you see here's one, here's another, but as far as quantum mechanics is concerned, you say these two are equivalent, and then you use that equivalent relation to generate the Hilbert space for you. But if you come from the algebra, there's no need at all to go into Hilbert space. They're in the algebra. Sorry? They're in the algebra. You don't have to add them as an afterthought they're already there now who gave me the definition of spinners as ideals because probably no chivalry I saw it so but what this involves is essentially finding in a so what you do in this is you actually find the idempotence of the algebra. And the idempotence is going to become important in a minute because they're going to be the points of my space. But not from the clippet algebra, but from the symplectic clippet, which is something which I didn't realise. I knew it had to exist, but didn't realise that mathematicians had been slaving the way of it. Only relatively late, probably in the 70s, they started. Sorry, have you heard of the term symplectic? Yes, it's a symplectic structure, but it's actually a symplectic clifford algebra. Yeah. Okay, and that's... Krummerow's book is titled symplectic clifford algebra.

22:30 Okay, and then you see here, you've got the idea that the space-time coordinates are actually arranged in terms of that spinner which is what Penrose uses when he's building up with those structures. So that's just pointing out where that is. Just let me quickly run through. Just very quickly, this is mainly, probably people unless you know Kauffman's or the blue Kauffman's. I just want to indicate So again, here, the distinction, which is why I use distinction later on, Kaufman actually starts with Spencer's Brown law of form and uses this hook symbol to actually denote the boundary. So at this, when I mean psi and I cross, I get zero. And you can build up applying the arithmetic, as Spencer Bryant did, which has some very interesting properties. But I don't want to dwell on that. Kaufman actually replaces them by these dual numbers again, but they're not the same dual numbers of dual numbers that I'm using. He has some recursive forms of interest. Don't want to worry about that. He has a multiplication, which I call a star multiplication, connection, which is C A D B, which becomes my multiplication when B and C is equal, then essentially you've got C and C in there, and if you take C outside, you've got C times A D. So there's a kind of, I've taken the special case of what he was doing in the multiplication table. But what he does is that he actually shows that his multiplication table also brings in the the quaternions. He's actually dealing with R2, 2, 1. Now the reason why I was doing this was that the algebraic spinners in this structure actually do have a difference because if you look at this thing, what this is telling us is that if I use the annihilation operator, calling it then what it does, it annihilates the inside and takes the outside and puts it on the inside and leaves the outside zero. And if you take the creation

25:00 operator it does the opposite thing. So you can you can interpret that in terms of that and then if you really want to look at this in terms of the vacuum states, then you can think of something with the structure inside and nothing on the outside and then you annihilate that inside rather and you get something which is zero. That's just like the vacuum state and this is just like a plenum. So you've got in this structure both the vacuum state and the plenum. And in fact in here you've got the, if you introduce the vacuum projector, then and you regard AB as this, then you get exactly the modifications that you guys play around with. Okay. Now it would be nice if someone's interested to actually explore exactly what the similarities and differences are between these three structures yeah that means going into them you know being serious that could be serious I mean we can't do it at this time how does this these creation and annihilation of words how do they relate to your picture where you Yeah, what I did try to do, I've now suddenly jumped a little bit in the conceptual structure. You're right. What I originally wanted to do was to actually build the pairs of elements out of annihilation creation apparatus. You can do that but it gets very messy. So you think of them as being part of it has been the two systems but it doesn't work too well I mean you know it's it's rather artificial the interpretation here is simply that during the process you are actually annihilating the process or creating a new process it is part of the multiplication it's it's a more primitive type of process or a secondary type of process which is actually creating the processes, changing them in that way. Okay, so you've got creation of processes going on, annihilation of processes. That's something we must have otherwise we're not going to get

27:30 anything very exciting. But I say what I'm trying to do is I try to motivate it because I think this idea of keeping process is fundamental, important in what I'm doing. Otherwise, you know, you can always take everything I've done and put it into and kill all the movement. Right, now the important point where we come on to the non-commutative structures and the spaces from the non-commutative structures is from this symplectic clippard archipelage. Now you see, it occurred to me once I knew that you could get the orthogonal clippards out of anti-commuting I'll call them annihilation creation operators. Then it seemed to me natural say, well, what about the boson annihilation and creation of practice? Suppose I take a pair of them. And then you communicate the commutator relation rather than the anti-commutator relation. There's got to be something which has a structure which is analogous to the orthogonal structure. And in fact it has. But what is more important is that if you use your and annihilation is q plus ip and your creation is q minus ip standard way you've actually got the heisenberg algebra stacked inside okay so that's where the phase space that's why i said the phase space in the orthogonal clifford algebra comes in because you've now got the phase space in the symplectic clifford algebra and you've really got a one-to-one correspondence between the two and then in here you also have a clever group and this happens to be the metaplectic group and I went around even asking people like Chris Isham have you heard of the metaplectic group and he looked at me he's not even done because you know this is another double cover Janice and we look at the carton list and we find no symplectic we did not find a double cover of the symplectic groove okay but there is one and it has physical consequences just as the double covering of the spin groove that's right has this two pi four pi this one shows up in the change of phase going through an oracle through a focal point for a lens system and was actually discovered by a guy called Goyi in 1889 and somehow it's gone lost from the physics literature

30:00 I mean there are people here who might know but not in this group but in the optics group but that comes from this in all the morphism on the algebra using the the Viber invariant I'm going to exploit this idea so what have we got to summarize this this particular mathematical structure that I'm playing around with here it's essentially the enveloping algebra of the Heisenberg algebra so it's not that we don't know about it it's just that we don't call it these things Okay, so you've got an algebra, the Heisenberg algebra is generated by IQ and P. I do everything in one dimension, so I don't want to get it, I don't want to show off, right, with that commutation, usual commutation, you've got the boson algebra generated, this is a commutator, and you've got the the Bargman transformation between the enveloping Heisenberg Algebra and the symplectic Clippard Algebra. So in fact they're isomorphic structures. So really when you're dealing with a Heisenberg Algebra, or enveloping Algebra of Heisenberg Algebra, you're really dealing with a symplectic Clippard. So we've been doing it all the time, but I can't realise that we're using it. And then you get these two groups, allomorphism groups, you get the Heisenberg group, and you get the metaplectic group. And of course the immediate thing is, well, where are the symplectic spinners? How do we construct the symplectic spinners from this algebraic structure? Now, here we have a big, big problem. Because, and it held me up, this is what I put in a side and went away and did looking trajectories and simple things like that. The problem with this is, the way I construct the algebraic's left ideals, is I needed an iripotent to do it. The Heisenberg algebra of the symplectic Clippet algebra is nilpotent, and if it's a nilpotent algebra, it don't have any minimum ideals. And so this is why you get these tremendously complicated works of Dicksmere and Walter who deal with the co-adjoint orbits way of actually classifying the irreducible representations

32:30 of your symplectic structures. And they're really heady. And yet, I was coming around with something about... Ah! I'll start with non-eons. Yeah, non-eons. You've never heard of non-eons, have you? I have heard of on-eons. Non-eons. It's not my invention. It's not my invention. No, is it? Non-eons. Non-eons. Non-eons. not my invention so I'm not guilty not my invention so I'm not going to from nothing no it means nine it's a generalization of the quaternions because there are nine objects in this algebra no no no no Noctonians, no no no. Noctonians. Now why am I, just in case, why the hell is he going down this road? Because he's lost. Yes but, you said. Remember the idea is that we want to construct our manifolds, if you like, from the algebra of functions. I'm using the Gelfand idea. when it's commutative then you know I can if the algebra functions are commutative then I can actually carry in the algebra all the properties of my space I can carry the topology, I can carry the metric continuity, I can also carry the points well I won't give it I'm turning everything upside down because it's my nature it's more natural you start from an algebra and then feeling, that's my feeling, and that's what's sort of guiding me in this, yeah, your whole point of view is that one, is that one, yeah, sure, thank you, at least I've got my ideas clear across, even if there's nonsense, it's another matter, at least it's clear, so you have the maximal ideals essentially defining the points for you, okay, but you can't do structures you can't do that for non-commutative structures oh yes you can't you can't come to explode you can the way we see the way I see you just start with the algebra you you find you find the

35:00 if we have these classes are very different representations and then you take the carnalis professor he said that you can if you cannot do that that is why we say in the non-communicated we don't have a point in the in the computer we have i want to show you there's another way of doing it where you do get points yeah okay sorry this is really the punch this is really a punch line in my talk because we have to see whether this really makes sense or not all the rest was just sort of giving them mathematical background sketching it so you can see what it is I'm trying to do okay so and I've got this word shadow manifold here which I'll try to explain what that is. Right, now I can go back to the non-eons, which people found there. Toyspots. Eh? Toyspots. Yeah, I'm sorry. Yeah, it is Toyspots. Because I, you know. From the algebra, so now we've got a non-commutative algebra, a symplectic Clifford algebra the Heisenberg algebra and I want to now construct a phase space right how you know we can't do it that's okay but what do we do with the starting from the arm now that the non-ion is actually invented by Sylvester in 1884 it really shows my age doesn't it and there's actually some horrible thing Now where does it come from? It comes from, you can either call them bio-finite algebras or they're called generalized Clifford algebras. They've got various names in the literature and they're very simply defined. So as I take n equals 2, then you've got two operators u is v, and u v equals omega v, u, where omega is the two generators, but now if I got n not equal to two, but to something more general, then it's the n root of unity. And that is the whole of the algebra. Non-ion is just when n equals three. And you've got a very

37:30 simple algebra, believe me it is very simple, you can write three by three matrices, there are nine of them, you can construct the spinners, the left ideals in there, construct the right ideals in there because it has a primitive impotent. It's not a nil potent algebra, but the interesting thing is that if you let m go to infinity, it actually becomes the infinite Heisenberg algebra. Plus, plus, I need an important. Now my contention is that what we've done in quantum mechanics is that we've concentrated on the Heisenberg algebra, not realising that we really want that structure which is slightly bigger than the Heisenberg out. I'll develop this as I go along, but that's the general idea. Because when I go to the limit, these things don't vanish. They still exist. And in fact, I'll show you in just the next transparency, it actually is equivalent to the delta function. It's the algebraic equivalent to the delta function. That's what we get from these algebra as well. in some sense if you say you refine okay since you're in a way that can be well then obviously we're good we're talking with you and I are talking about the same structure and if we're coming to it from slightly different angles and you get the delta function essentially which are a determination of location yes yeah of points okay of points precisely but I say I've got this by there's finite Clifford Algebra with n not going to infinity so I'm really doing finite space times and that's why toy spaces but not space times that they're finite phase spaces it's not my face face because that's where the non-commutation is it's not in a Manning type play it's a Manning plane but it one's momentum and the other one's position. Okay. Now then, if I've got this out, the question is, what are the points? Yeah? What are the points?

40:00 We're already, we're sort of there. It's got to be the impotence. Okay? But from the interpretation point of view. So, or given an algebra, which might be the space of the other. That's right, yeah. And what we're going to find is there's going to be all sorts of spaces in there. There's not one unique space. So, and then, which is the representation theory? Yeah. Given an algebra, find the space of the other one, and then the representation theory. That's right, that's right. And then you've got your really quantum space-time. I mean, that's the grand, that's the simple. The trouble is to do it. Okay, now the points, you see, I mean, often you say pointless general relativity, I think Sam has got it in his book, yeah, chapter heading, pointless general relativity. Notice that when all I'm doing this, I have not got a space-time manifold anywhere, I've just got these multiplication rules, I've just got the addition rules, yeah, I've just got algebra I haven't got anything you can have a good algebra then you will find the space then you will find it how do you do it and then again the other through the space yeah well that's the that's the that's the story I mean I hope it's not a boring story now and here comes my philosophy in my interpretation in terms of movement you see what if everything is moving everything is action what is the point like the point well is the metamorphosis of this action well to me it's very simple it's just these these i can't these these hidden photos because what what's happening a point is something which stays itself so a point followed by a point is appointed so the movement has Well it's not permanence, it's continual repetition being the same thing. Persistence is perhaps a better term, persistence perhaps. Persistence, yeah. But it keeps transforming it into itself, and therefore it's invariant. Fine, probably there is only one, that is it. But you see, that idea was then pinched from Eddington, but he was talking about existence because he wanted to have this kind of theory where you didn't depend on space-time background. But he used groups, not algebra.

42:30 He wanted to use groups. Again, like class. Yeah, like class. Because you don't have the problem. More fundamental. But the algebra is more important for quantum mechanics than the interference problem. that's my take ok so what so what we do then is we look for the idempotence in our in our algebra and in the discrete vial algebra we have these idempotence in there and would you believe it they are actually connected by our inner automorphism again and if you look at what that transformation is than the translation operator. So you're using a translation operator to go from point to point. And you can also construct a position operator and then your position operator multiplying you by your impotent EJJ produces J. A delta Q comes in because it's a scaling function. Yeah, it tells you how far apart the points are. You can put it in there, you can put it in there. So the algebra contains, so if you like, this is the space part of my structure. So I can make points, I can make a, and when I go to the limit, when we get the delta function then we'll see that we get the continuum. but at the same time there are many different sets of idempotence, all equivalent in this function so you can then construct another set of points these set of points can be connected by translations in Q space and this is the momentum space but the way you go potence to the hour is by inner automorphism. So as you're constructing your P space, you destroy your X space. Right? And in fact, you've got an exploding transformation because every point in X space, a point in X space goes into every point in P space. So now the uncertainty principle in this view is totally different from the uncertainty principle of

45:00 disturbing something we're making the algebra if you like we're giving the algebra an ontological status via the process idea and then if we want to abstract out spaces from that we can't we can only have one space the position space or we can have the momentum space but we can't have both together okay But this is not because we're disturbing anything when we're doing our measurement, it's because the very process of nature is such that you cannot do it. Can you say that we can embed political space in a momentous space? Yes, you can do that if you, yeah. I mean, that's the whole point about the implicate-explicate order that David Bohm was so interested in. But really, the algebra is this implicate order. The way you describe the implicate order is through the algebraic structures. various, these various displays. Yeah, that's it, that's it. And then the spaces are just the projections, if you want, by choosing a particular set of components. Okay, so, do I have time just to go? Just ask me if there's a course. No. No, so can I just go on? I just want to finally connect up with the bug. The people can leave. They can leave, OK. You don't have to stay. OK, so just let me summarise this. What I'm saying is that if we all leave, there's a problem. Oh, then I'll come out with a cup of tea, no problem. OK, so where are we? I'm just repeating now in a familiar language, in classical mechanics we have P and X, all as well, and we can have a projection, so we've got dynamics present there, but when we're dealing with the quantum mechanics, you see we either diagonalise the X, in which case we have the X space, or we diagonalise the P and we have the P space, and this is all I'm doing, all the algebra is essentially the same. I hope it's a little bit. And you cannot simultaneously, they are going to. that you cannot construct these points simultaneously and in fact I should have pointed out that inner automatic automorphism that we use is just a finite

47:30 Fourier transformation which is what you expect so then it's just a Fourier transformation that we use between the innings then the question is where is the It all sounds like kinematics to you. It's totally kinematical at the moment, but the question is where is the dynamics, how can we deal with that? Now in order to do that, I still haven't explained something, I want a symplectic spinner, because you remember the Clifford algebra, if I've got the spinner, I know how to deal with the dynamics of the spin. We all know how to deal with the spin. We use the spinner, we actually use the spinner, we actually use the left ideal, or at least the equivalence class of it. So we want a spinner which is constructed from a left ideal, not the way Cos Stant and so on describe it. The dynamics is a source of your kinematics. You cannot have kinematics from one. Well, look, let's not argue about this because I'm not quite sure, you know, because I'm exploring these ideas and I'm not quite sure where the horse comes in front of the cart or behind the cart. So we have to look at the definitions to find the dynamics. Yeah, sure. I mean, sorry, I'm just giving you a scan. You know, we really ought to do much more precise to you, but if I do precise to you, and I want to just, there was a big problem for me, and I said before, and just stepped over it, this symplectic algebra has no, it's no problem, and therefore it has no, eh? Even in the future of the symplectic, you have the data, you have the start of the symplectic, and the form. The symplectic form, yeah, that's the way you do your classical, the classical, the I have a problem first of all because I can't construct these symplectic spinners as it is. However, we physicists know that there is this thing called the projector onto the vacuum and we know what its properties are. So let's add it into our algebra so that boson algebra which is 1, v, v dagger and v. With v satisfying these

50:00 relationships here and remember you annihilate, if you try and annihilate the vacuum you get zero and you just because this is a mission turn it around the other way and you can't create backwards from the vacuum if it annihilates that. This algebra is now non-nil potent and therefore you can now construct your left ideals and these left ideals are just a generalization of the wave functions that you use in the representation space but this is in the outer brain space. It's not represented in any particular representation of the X representation of the U representation. Now the problem then is we need it in that sense in the boson algebra what is the equivalent in a potent in the in the Heisenberg algebra and the way you can generate it is via doing a sandwich the V between these two metaplexic transformations. That's in a paper of Fabio Foscuro, an ideal way, way back in the paintings. And then you get the relationships between the P delta equals naught and delta P equals naught. So you have an algebra, if you use that algebra rather than just the Heisenberg algebra, then you get your you get your spinners for free and I don't know whether you guys realize it but this delta function here Dirac has it in his book quantum mechanics and it's called the standard ket or the standard bra how many people actually and I don't mean and I do not mean third edition and I don't mean I don't mean that we all know how to use that I mean how many how many people realize Dirac talks about a logic like that sorry if I'm not I don't I don't embarrass you I just want to know whether the difference between the two the difference between the two is this is a vector in an abstract space yes

52:30 This is the left ideal. If you want to put it succinctly. But he doesn't use words like left ideal. Yeah? Because he wants things, he wants everything to be operators on this But that's another story that I've only just discovered in the last couple of years actually. I kind of jumped over the table. It was incredible. Okay, so then you've got your spinners or something. just the continual limit this is taken straight from Vile this is Vile's process of going to the limit so if it's wrong I am not guilty Vile is guilty and then here you see he brings in these X's which are essentially he doesn't call them that because he's got matrix representations and they're just vectors of matrix representations but you need to go to the point is you don't have to go to matrix representation in the in the algebra you just use the algebra and then by using this limiting process which i won't go through to tell me whether it's any good or not you get this is this this is vile straight out of his book but then if i did the same thing for the impotence bump, bump, bump, get that and if I now go to just replacing the summation by an integral that's just the delta function so these are the so my delta that I've introduced in the algebra is equivalent to the limit to a delta function the limit of infinite frequency the limit of infinite so omega frequency if you want to look at it because K is, yeah it's a free if I'm even looking at it what that actually is I just want the final bit

55:00 If you see the delta arises at the limit of infinite frequency, say, frequency. Yeah, if you interpret that key as a k minus frequency. Yes, it arises. If you have an infinite power of cell localization, what is a delta? You can think of deltas as, you know, limits of Gauss. Yeah, yeah, yeah. So, yes, it's the maximum localization, maximum, physical, how can you relate to physical, you have infinite energy localization, an ideal, it's an ideal, I call it. Oh, sure. So the continuum, let us, let us, the continuum reflects that non-pragmatic arena where we can localize infinite, within. Yeah, which we don't actually have. And therefore, why do we throw the idempotent away? You see, whereas the discrete vial algebra has the idempotent in there with no messing around. So when we let N become very large, that's about the best we can do, because we don't have infinite energy. So we never actually reach the continuum. And therefore, it would be better to use the vial algebra, this discrete vial algebra, the Heisenberg algebra. I got thrown out of the room once. Now, have you noticed, not a mention of anything of the Bohm interpretation. Isn't it wonderful? I'm going to stop that now. The recent stuff all started by me asking the question, where is the quantum potential? the algebraic approach. How many people know the bug interpretation? I don't know. I mean, I do not know the intricacies. Well, I do know some. Yeah, I mean, but it comes straight out of the Schrodinger equation, just taking the, putting in the Schrodinger, let me just very quickly, you take the wave function and you write it as R, E to the R, E, S, H bar and you get the imaginary and then you separate real and imaginary parts the imaginary part just

57:30 turns out to be the conservation problem the real part is the interesting thing you get something Where Q, that's all it is, that's the whole theory in a nutshell actually, now the interpretation comes like this, if Q is equal to 0 and this is the classical action, that's just the Hamilton And if it's not zero, it's the quantum Hamilton Jacobi equation. Because all this h bar is in here, you see, perhaps I'll put it here. Can you go back, when you say you split into real and imaginary, you mean that you're using actually, you cast the differential operators in terms of like some... Not just differential. But this... So when I see this... They must be splitting into... come on my calculus no please that's all you do and now you're going to get an equation which has real terms and imaginary terms and what do you do when you've got an equation which has a real part and an imaginary part you split them off that's all you're doing and then you see this particular equation And then you build up the story about pi. My argument is very simply this. When that zero, I've got a theory which has trajectories. Suppose I make q just a little bit bigger. Just epsilon. I can still have trajectories because I've solved this thing for trajectories. At no point do I say now I must

1:00:00 Okay, so therefore if you look at quantum mechanics from this way, you can talk about trajectories. Now, don't ask me whether the particles actually follow those trajectories or not, because that's an empirical thing. And it turns out that these trajectories are nothing more than the streamlines of the probability flow. And as a condensed matter physicist, I used to use those streamlines to say that's the gave a current and, you know, treated them as if the electrons were actually following that stream. So what all the fuss is about, I don't know, the foggiest idea. Perhaps it comes from the original Heisenberg's interpretation of how much time, no trajectory is known from that. Yeah, that's where it comes from. It comes because of the prejudice in the physics. Yes, I do want to say the word, but... because it doesn't matter if my career is finished. It was finished as one time ago. And it's actually the mathematical concept. We'll look at this. The point is that if you come from the algebra... That's what I was leading to. Now I saw it. I didn't see it honestly. When I asked you to sneak into symmetric and anti-symmetric, I meant to sneak into riddle and imaginary. I had a line that's in the circle. that you're using the commutator for the usual Louville yeah this is the Louville conservation problem and this is the new I call it a new equation is it new I've never seen it written down and there is a little bit there is a little bit of a sticky problem around here all of this is in the algebra these side L's psi, r are the right items. So what I'm saying is that if I assume these two equations, because normally we say oh this is just the machine conjugate of this equation therefore it's not adding anything new. But remember Feynman actually said that the wave function is information coming from the past and the complex conjugate wave function is information coming from the future. So you're putting together these two different aspects. Again, it comes back to this idea

1:02:30 of the old thought and the new thought. So it's not coming from the future, it's actually the potentialities for the future. That was, you know, that's the sort of motivation for actually looking at this and therefore you've got these equations are not, this is not a wave function, please can I stress that, this is actually a minimum left ideal. So it's purely algebraic. If you want to make it into a wave function put a... Now then, the difference between these two gives us this, but the sum, why don't we summing. Why do we only take the sum as well? Right. We always take the difference between these two equations. We never take the sum. So what I thought was, well, let's take the sum. Then you get this thing here, which is, you can see why they never did it, because I call it an anti-derivative. Is there such a thing as an anti-derivative? Sorry? Is there? I think in super language. I've seen it somewhere but I can't remember where it was. The terminology is used for integration of an anti-derivative? When you have grassland, you have a left derivative and a right derivative. Yes, yes. I know I see it around, sorry, I'm sorry, I'm sorry, I'm sorry, I'm sorry, I'm sorry, I know I've seen it around, but if you use that, then if you find when you can write, you can't always write your left eye field in this form, RE to the IS, but you can for

1:05:00 the Heisenberg algebra. You can't for the Clifford algebra, you've got to be careful there. There's something there I haven't completely sorted out, so don't pin me in a corner. But if you use it in the Heisenberg Algebra, then it simply becomes the S by DT, so S here is an operator, here is an element of the algebra, what I've done here is I've actually projected into a Hilbert space, I'm sorry, I've projected into a space, into a Hilbert space and then it becomes then then this is well defined it's just the time derivative of the phase of the weight function but look it's actually equal to the expectation value of the anti-commitator and this in fact is just a conservation of energy you and you can do this in one thousand words as well, which is where I've spotted the connection of the lines there. Right, final transference. No, two. What does that equation look like? What does that equation look like in the X representation? It's spelled out when you've done a Hamiltonian, you know, just be harmonic oscillator, I think. Is it harmonic oscillator? Yeah, I've got a harmonic oscillator there because it balances the thing. You don't have to have harmonic oscillator. And what you find, you find, well, that equation. So this comes out as an anti-company. There's this difference between. Yeah. Yeah. It comes out of that anti-interruption. So it's hidden away in the algebra. It only appears when you project it down onto a manifold, onto a space manifold. And you've also got one on the p-manifold. And it's nice and symmetric in that case. If it's a cubic, it's hidden. We don't want to know. but the important point here is what is this this thing here P normally what

1:07:30 you do is you just identify this with P the momentum and then you can see this is the energy remember Hamilton Chakotas is minus the energy that's the momentum squared so that's kinetic energy that's potential energy that's the total energy so you've got some extra energy that's what we say but notice that ds by dx that I'm writing here as a p is not the observed momentum it's the real part of this peculiar object here, p is the operator, d by the f, minus i, d by the f. It's only the real part of it. Okay, so it's not the real momentum. The real part of it is, it's not the... Well, you know what I mean, you know, it's not the total momentum. And that's why you have to have this quantum potential in here, that's why you have to have this extra energy to conserve energy. And when I do that, equality itself is of importance, because the right member is something subtle, and you express it with the left member, which is quite algebraic. So you express analysis throughout it, the like number is ds over dx, which means you need a space, an underlying space, otherwise part of it means ds over lg. That's right. And then you express it through the other, which is important. But my take on that is that it therefore gives you the shadow manifolds. So the trajectory are in x, not p and z. Of course it is. It's no different from quantum mechanics. How can we... It's absolutely no different from quantum mechanics. But we can do other. So you express analysis to the other. So it is applicable to quantum mechanics. So you get these shadow manacles. You're constructing them. So the bone momentum is sometimes called a guidance condition, it's what you need in order to make a shallow manifold.

1:10:00 And that's why you can talk about trajectories in that situation and not get into a problem. The problem is that we try to apply classical analysis in quantum mechanical problem. There is no analysis there. It's only we hope it's algebra. So everything we can express phase or analysis in terms of algebra then we hope to be to apply something in the problem. I was hoping the sort of the activity that sort of intuition that I'm putting into this might rather than abstract algebras you know what you said there very well any questions comments comments what you have been making comments all the way through Anyway, I hope you enjoyed that. So if any of you guys write papers on the Bohm interpretation is wrong, I'll screw you. Mathematically wrong. It's exactly the same content as Bohm. Anyway. There it is.