Conversations

0:00 The physics hasn't got much shadier sense, or a lot of it hasn't, I suppose.

2:30 In order to solve some difficulties. As far as publicity goes. Yes, yes, you have to say it's not a radio, isn't it, the character? I have to say it's very good at self-promotion. But actually, just coming up the stairs, I thought, well, gee, if that's really true, then maybe the redshift doesn't exist after all. Well, you know, Vigier, an old French friend of mine, has long said that the redshift's got nothing to do with the expansion of the universe. And so on and so forth, due to the finite mass of the photon, you see. So if the photon has a finite mass, then the speed is not going to be c, and all sorts of things could happen. So that's been going for a long time, actually. But I have a feeling this guy says it's a function of time, isn't it? Yeah, I haven't looked at metadata or stuff. I didn't look at it seriously. I did see that it did make it look very, very messy. Well, I think it was just that it was.

5:00 This alleged first 10 to the minus whatever. Yes, that's right. That's where they hide everything. Then it cooled down, you see, and became... Well, as long as you can do it in the first nanosecond of the big bang, within the first nanosecond, you can do anything you like. Later on, you can run into trouble. You didn't happen to bring along, or you didn't, we don't actually have a way of downloading that paper, do we? Oh, that's right, I was going to, sorry, I discovered something. No, it was just that I thought there would be interest in that, I meant to bring one along this morning. To be read, either outside work or to be read, please be patient with me. No, it's one I have read. It's not in the other one. Madday's work. That's the one, yes, that's the one. When was this published? That's a very good question. Well since it might do a paper that appeared in the AMS, well it's you, so in that case you have seen it, right, it is the same one, yeah, it is the same one. I just found a rather interesting, amusing paper, a very long one, I think. A last page of the article. It's the last page of the article. I don't know what this is, but it's something else. Strokes of submarines, gravitation, the things people put on there, this is a paper, content 2 megabytes, which has gotten, I think, most of us taken up in this picture. What were you saying this morning about the Cartier paper, or what we got started on in the Cartier paper?

7:30 Non-commentative geometry, has it been demonstrated that there is any such thing? I would be very interested if you could just run past me again the point you were making about how Grothendieck in his construction, as it, the needle, I want to say, rather the body, actually could be subsumed with the logic, the structure of sub-effects, and by looking at the ring classifier as the central. Of central interest, but the point is that in that article Cartier says something about how Tierney and I put logic into Topo's theory. Actually, I wrote to Cartier that Burgundy didn't really need logic in the following sense that taking logic in the narrow sense, it's about Sub-objects, you see, relations, everything is included into relations and the properties and relations and then logical operators are applied to those which are ultimately, well, the names of the sub-objects are mere statements, you see, so you're actually just journaling statements, whereas there's the geometrical concept of direct limit and inverse limit and Grotendieck, you'd be interested in which kinds of theories. These structures can be classified by toposes. These are clearly those which can be defined using direct limits and finite inverse limits. So my point was that he didn't need to encode the definitions of integral domain or local range or something like that into logical language. He could see directly. Well, of course it's closed onto direct limits and finite inverse limits and therefore it's classified. I lamented to Cartier that this kind of intuition had not been systematically developed and wrote in the cadet file. In a way, it's part of the Ratzmann idea of dealing with the geometrical objects directly as much as possible. He coordinates and statements. I mean, that's not directly related, I guess, to the non-geometrical business.

10:00 Well, you know, I'm not very keen on the idea of non-commutative geometry. No, well, you see, I mean, it's, I don't know if I'm getting conservative in my old age, but it's just I never have seen it. Well, the basic thing is, what are the morphisms of the spaces? You can't have geometry, and we know for a long time that when you talk about one space, there are other spaces that come up, you know, the Cartesian product of the space with itself, and subspaces, and spaces of maps, and all these. Other spaces, or even doing the idea of a curve in a space is a map from a curve-like space into your space, right? So you need to have a category of spaces in some way, or you don't really have a geometry helping you relate these different things. So I asked Kanye, what are the morphisms of these non-communic spaces? He didn't really know, so he gave me one answer, which was obviously inadequate. I came back and said, well, on the other hand, it's this other thing, it's a completely different thing. Well, no, that's a little too general. So then he came up with the thing, well, it's the second thing, but it's sort of like the first thing. So in spite of, you know, having won this huge Nobel Prize and been working on this thing for decades or more, two decades at least, he still doesn't know what morphisms are, or spaces. Where is the geometry? These people are just sort of formally, by analogy, noting that some structures and analogs can be made without any real physical formal analogies. The notion of vector bundle, you see, for example. Over commutative space is well known. It's a well better place to space by a non-commutative algebra, so it's considered modular over that algebra. Okay, well, that's been known for a long time that there are certain analogies, but to call it geometry... Okay, so it's really, you use the word geometry that you're using. Yeah. I should imagine, or certainly I use it in a much looser sense, that it's really the non-commutative algebra that I feel.

12:30 Of course. Well, this is the thing that they are constantly using to draw people into believing that they're doing something worthwhile. Of course, everybody knows that they're not really algebras and quantum mechanics. But there have always been non-community algebras... Yes, indeed, this is one of the points that came to stress, that the non-communitivity is already there in the classical form. Yeah, I mean, in fact, Leibniz's rule, the basic rule of differentiation, that's the source of the so-called Heisenberg relation. It's essentially the same thing. I mean, one can consider it more abstractly, but the concrete representations that one uses... There are always some now in terms of differentiation. Yeah, but my question would be, does it have to be, or are there more general structures which I would want to break away from that? That's really what's interesting to me, because it always seems to me from a physicist's point of view that we have a very rich structure in which representing it in a Hilbert space seems to be cracking a... There are a lot of people with a sledgehammer and it misses the key algebraic structure. I'm just wondering whether I've missed something because maybe it's been down this road where I'm trying to oversimplify everything. I mean, as Mike has heard me talk many times, it's almost trivial what I do, in a sense, and it's certainly, well, I just noticed, the reason I just came in here, I picked up a paper on, I'm trying to get back to this modular spaces and modular, you know, Tomita theory, which Carlton thinks a lot of, bringing in thermal time into quantum physics, and...

15:00 At the moment it just feels like maybe some sort of formal device in which they've noticed some connections, but there doesn't seem to be any physical connection between these. These connections don't seem to engender any physics from the intuitive point of view. You can see the formal point of view, yes, that would be interesting, but trying to see where the physics comes in. And I was just trying to look at that and... There, my trouble is that a lot of it is done with very using one-woman algebras and the abstract forms of one-woman algebras. I'm not up to scratch on that because it seems to, I mean, I'm getting things out of just looking at the algebras and sort of mimicking what I can do with a clip of algebra, doing it on the symplectic algebra. And this then brings me very close to what Gallup found did. I mean, he was using maximal ideals to describe the points of the space, and so when you go to the non-commutative algebra, you try to do the same thing. I don't know whether you would criticize these, because you can get many ideals, minimal left ideals, minimal right ideals, and sometimes they are called points, you know, and in fact... What I found was that I could build up these points in a very simple way, but to do that I have to know all about the ideal structure, I have to introduce impotence, and this became particularly serious in the symplectic algebras, because I think essentially Heisenberg algebra is a symplectic algebra, so that one really should be studying the symplectic algebras much more seriously than they, here again, not much more seriously, because the mathematicians have done it. But it seems to be very different from what you do. And it seemed to me that because of the symmetry between the bosonic and filamionic cases that you ought to be able to do what you do for the symplectic, for the orthogonal algebra, you should be able to do the same for the symplectic algebra. And many years ago I kept looking and thinking, why don't these mathematicians talk about symplectic algebra?

17:30 I told you about symplectic, I'll stop on that, I'll clip it out, and that's standard, and the great joy about that is everything is finite, so even people like me can just wrap matrices together and get a feel for what's going on. And when you get the symplectic structure, first of all, you've got an infinite bunch of algebra, an infinite bunch of representation. The thing that bothered me for a long was that I couldn't find any impotence, because if you look at it from the... But that was the thing that struck me. If they're as similar as they look on the surface, where the hell are the idempotents? I came across some work by Mario Schoenberg, who was a Brazilian mathematician. I don't know where he came from. He's a friend of David Bohm's. And he had introduced an idempotent as it wore out of the blue. He did it for the algebra, and it wasn't difficult to connect his... It was very easy, well, very easy to find out what that was. The difficulty was curly P, and I'm afraid I drove people like Mike and Matt and said, God, I wish I understood what this curly P was! Well, like a couple of years ago, you think we got the answer. I like Mike, but I don't think we got the answer. But it took some doing. But the point is that in order to construct your symplectic Clifford algebra,

20:00 It doesn't come out of the algebra itself, and I was always worried, I called it an extended Heisenberg algebra, and there were certain rules that it had to obey, and finally, you can see it. For me, of course, being a physicist is to actually try and identify what this thing is. What's it supposed to do for you? It generates the symplectic spinors. And therefore now I've got an analogue between orthogonal spin and we can actually do experiments to show its double covering nature. 2 pi 4 pi, all that stuff. But then what I wanted to do was build a symplectic spinner accessible to physicists that actually would do this double, would show that there's a double covering. And that that double covering is very important even in symplectic structure. ...produce some physical consequences, and in fact it does, but they've been discovered a long time ago, but not by this... And not in context of quantum theory, right? And not in context of quantum theory, right? Like in classical optics. Optics. And this is tied... the double covering of this inflected group is essentially... a subgroup of it is the metaplectic group. You've probably heard about the metaplectic group. I've heard of it, but I've never studied it. I don't want to be working away, but the idea being that one thing... Can I go on talking like this? Sure, yeah. It's all right with you, isn't it? Don't mind my interrupting. No, please do. That's what we're here for, yeah. Please do. No, only... It might be helpful just to mention the motivation in classical optics, the... Okay, yeah. Sorry, I've come off on the expression for the... Yes, the Hamiltonian, what do you call it? Sorry, I've forgotten the term, where you've got the focus of... Focal points. You're going through the focus. Yes, but there's a specific term for it, isn't there? The focal. Never mind. It doesn't matter anyway. No, well that was just, I mean, it was extraordinary the way the bits and pieces fell in place,

22:30 because I was doing this sort of abstractly looking at this from a physicist's point of view. Finding that there should be a double cover and then hunting through the physics literature to see whether there had ever been anything. Where does the double cover come from? How do you see it in real life? Well, there was an experiment done way back in 18... 1890 something. It was 1890. It was an 1890s. 1890. Goy. Goy. This is the Goy phase. The Goy phase, you see, which had been in optics, which Schrodinger, not Schrodinger, Sommerfeld had actually got in his book, but he got it from the differential equation. It was sort of a typical physicist's getting at it without seeing what the mathematical structure was and it was just popping out because there was discontinuity in the phase, essentially. Well, and the discontinuity can be seen because if you, which is equivalent to the 2 pi 4 pi business, and that is if you have a concave mirror and a plane mirror and you set up interference pattern between these two, okay, so you've got some beam of light coming in here and you've got an interference pattern set up here. As you go through the focal point, You get a sudden transition from the bright fringes on one side become dark fringes, and the dark fringes become bright fringes. In other words, you can actually see a discontinuity in the phase, and that discontinuity in the phase actually comes out of the metaplectic structure, out of this double cover of the synaptic. So that's the sort of thing I was looking for, but... The question was to act to make the analogous structure in the symplectic clifford algebra or to construct a symplectic clifford algebra where you could actually see the analog between these two. Now I came across the work of Costat. Costant. Costant, yes. French. The French mathematician. Was he French?

25:00 I thought he was. He did a lot of work in France because he and Sarai were the people who did most of the early work in geometric context. I'm not a symplectic speeder, but he did it in terms of fiber bundle structure, and I didn't want to have a geometric, I wanted an algebraic approach rather than a geometric approach, and I sort of got stuck for a long time, until I actually eventually found out what this, what this impotent was, and it's very strange. Let's see, how can I tell this story without boring... If I work, let me go straight to the answer. If I work with the boson annihilation creation operators to represent my symplectic structure, then that turns out to be nothing more or less than a projector onto the vacuum state. Now what we physicists do is we take the algebra defined in terms of the annihilation creation operators of the bosonic type. And then we sort of find that something goes wrong with what we're doing, and we just add this ad hoc out of the blue, as it were, not realizing what it's doing, but we just do that. And that is, in fact, what this curly P is, if you look at that algebra. But the question was, what is the equivalent if you're using the PQ algebra? Position of Momentum, who uses the phase space algebra. And there, the clue came by actually looking at Dirac's work. Now, you know Dirac has this notation. I don't think mathematicians really know what the hell it's all about, the Braquette notation. No, we don't. He introduces something which is like that which he calls a standard ket. Without the line in front of it. And he also introduces a standard bra. And it was sitting there annoying me for a long time wondering what the hell it was. And then one day the penny dropped. If I put these two things together it was simply what I got.

27:30 So that what Dirac had smuggled in without realizing it was a primitive idempotent. And it really is that primitive idempotent which enables him to do the rest of the story. So his algebra, my claim is, that he's using is actually the extended Heisenberg algebra. And that's the way he's able to generate, that's why he can use these bras and kets. Because these are just essentially left ideals and the right ideals. Because once you introduce an idempotent, you can then generate the right ideals, and then you've got a symplectic clip of algebra, exactly the same, or analogous structure to the orthogonal, the complementary, with covering groups, SU2 in the case of SO3, and the covering groups of the metaplexic. So that really enabled me to say, right, now I've really got the... Well, and then, just after I, well, actually, I knew about him before Cromerol. You remember Cromerol? He's actually written a book eventually after I discovered, well, I think the impetus for me pushing this even further was his book in which he described, which is called Orthogonal and Symplectic Clifford Algebras. Okay, in other words, he... Yes, I think I have looked at that book. Yeah. And in fact, I spoke to him many years ago before the book came out, because he was down at a conference at Kent. He didn't introduce an idempotent. He introduced what he called a formal device. He didn't call it an idempotent. I think he just called it omega star or something. He just imported in from outside again.

30:00 Without any, because I did actually ask him why he did it, and he sort of got a little bit niggled with me and said, well, I could have called it beta or gamma, you know, as if I was worrying about the name rather than the actual, why he was found it necessary to introduce it. And in his book, he eventually put it that it is a formal analogy with the idempotent. In other words, he didn't take the extra step and say it was the idempotent from which you can construct this. I just want to take this one stage further now, because this enables us to do quantum mechanics purely algebraically, and I've shown that this thing relates to the vacuum expectation values in the GNS construct. Without having to bring in Hilbert space. Without having to bring in Hilbert space. Now I think Hilbert space... Well, Dirac himself, his approach was basically algebraic. That's right, absolutely, yeah. I mean, all of this is inspired by Dirac. But the other thing, of course, that has always puzzled me, made me curious and frustrated me, was the Bohm interpretation of quantum mechanics, which I have contributed a little bit to by exploring this magical quantum potential, trying to see what it is. And because he was trying to, in fact, derive the Schrödinger equation, he doesn't succeed in doing it, but there's a sort of a strange, if I had my book where you see all the coffee stains on it, because I felt this was the clue to the connection between what he is doing and what, you know, the quantum potential. Bill, shall I go through the quantum potential with you? Just very briefly, and then I'll come back here, and then I hope you'll see the geometry, you know, the non-commentary geometry.

32:30 Just a few months ago, I mentioned this to you, actually. I don't often go to the physics department at Buffalo, but I saw this notice of a talk that sounded very interesting by someone called Li from Buffalo. I don't know if you know that name. I know A-Li, but I don't know where he is. There are a lot of Li's around us. Yeah. Sofus Li, oh sorry. That's no longer with us. I don't think he's with us, is he? This is Chinese, alright. Well, we think this is the Li of Li and Yang. I knew he was part of the New York University system, but I didn't even know that he was part of it. We were terribly fragmented during that time. Anyway, the point was that he felt that he had pushed this much further, basically by splitting the wave function into amplitude and amplitude to search for the propolar decomposition of the wave function. If you only set the initial condition, then the whole thing would just be an ordinary differential equation. Yeah, well, I mean, that's essentially the idea. I mean, there's nothing... In a way, the Bohm theory is very trivial. But in another way, it's very profound from a philosophical point of view, because it opens up the possibility of an ontological description rather than... There is no reality and all that, but far is from that. Nevertheless, it is still very strange. And really I think the way I presented it to Henry Stafford, I'm discussing it with him on an email at the moment, my motivation has been why the hell does it work? You have the office to say it can't work. And I know it can work, because I've worked with it, and I know all the tricks of the trade, and there's no way of showing it's going to be different from standard quantum mechanics, because it uses the same formalism, the probabilities are assumed to be the same, if you haven't got any new formalism and you haven't got any new expression of the probabilities, your probabilities are going to come out the same, whether it's an orthodox theory or not. So, in one way, it seems to be a waste of time.

35:00 Because you're not getting any new predictions, this is what a physicist would say, but in another way it's opening up a different way of looking at all of this abstract symbol manipulation, no connection to classical domain. And so what is it? I mean it's very, very naive, which is probably what Lee was saying, and therefore it's a differential equation and you just work it out. So you start essentially by, I'll put h bar equal to one, by putting that in the Schrodinger equation. And then you get two equations, and then you've got a complex equation, you separate out the real and imaginary part. The imaginary part just gives you conservation of probability. If you still assume that r squared is equal to probability, you find a continuity equation for p. Okay? I won't put that on because it's trivial, it's not very exciting. The real part, however, becomes, and I'll do just one particle because that's good enough. You get one term which looks like that, where this Q is this rather curious, I'll put the H-bar in there, you get this rather curious additional term. It's not put in, it just comes out. Now, the thing that's noticed is that if this was the classical action and that was equal to zero, that would just be the Hamilton-Jacobi equation. So that clearly, as we know from the turn of the century mathematics, this just gives us an ensemble of trajectories. P is equal to grad s, if that's the classical action. So now what Bohm said was just simply this. Suppose we keep this condition when this is not the classical action but the phase of the wave function. In other words, let us identify that P with this.

37:30 There's no reason, you know, where it's infinitesimal, it's still a trajectory. As it gets a bit bigger, it's still a trajectory, but not quite the same trajectory. You're right. Yeah, and nowhere in that letting Q become large does anything strange happen. It's just a differential equation. Yeah. Okay? So you get actually a deterministic trajectory. So you can get out trajectories. Right. Okay? And they're deterministic in the sense that given the initial condition, you're trivial. So now then the question is, but what about the uncertainty principle? That's the first thing that worries everybody, because if we attribute, sorry, I should say, so what we do is we assume that the particle has a position and a momentum, just as in classical physics, and the momentum is always given by that. Then you've got the trajectory. Now then. The problem with most people is they say, but surely the uncertainty principle tells us you cannot have X and P. No phase space in quantum mechanics. No phase space in quantum mechanics. And that has always been the problem. Where is... Heisenberg's... Everybody's trying to get phase space when they're doing condensed matter physics. Yeah. So what this really is, is Bohm is constructing a phase space because this P here is not the observed P. Okay, now you can see that in the following sense that suppose you've got something, a plane wave, coming in. Well, we know the particles have a momentum p, which is the eigenvalue of the plane wave. And then you say, all right, now if I want to try and find out where the particle is, what do I do? I actually put in a slip. But we know what happens when you put in a slip. There is a new polar potential generated. And that then makes the trajectory to this, and therefore, as you're trying to measure x with a given p, you're changing the p, so you're sort of participating in the process during measurement. So measurement now is not something where you're just standing back and saying, I'm not influencing the system at all. I'm actually changing the whole thing. That's why there is no difference between orthodox quantum theory and this quantum theory. Because an attempt to measure that trajectory will immediately produce this effect and therefore destroy the trajectory.

40:00 But nevertheless, now then, the other thing to notice is that this is just the probability column. It's just related to the probability column. So, it's essentially in quantum mechanics, and most people in condensed matter physics say If I want, if I've got a beam of electrons coming in here and I want to find what the current is, I just put an E in front of this. So in other words, they're implicitly assuming that the particles are moving. Okay, but they don't, but you don't have to say they're moving in all trajectories, but nevertheless, you know, it's so close that... Okay, so, so that, you've got that thing. And then of course when you look at this, this queue has some really glorious features. I don't know, have you seen some of these pictures? I mean, for the two-slit experiment, it's always worth speaking about scenics. I haven't been in this talk for a long time. For example, suppose you take the two slits. Now, we're always told that you cannot have trajectories and get interference. That's not true. This, I mean, this is done using the Feynman propagator technique and so on, so this is kosher quantum mechanics, there's no nonsense going in there, and you can see what happens, that the curves, the particles follow one of these trajectories near the screen, so you can see why it builds up individually, because it depends upon the contingent condition, initial condition, and you can't control the initial condition because of this. So you cannot put any particle on a given trajectory, but nevertheless if you just spurt a lot in and it happens to sit on one of these trajectories, then it will follow them. And then why is it not, why are these bent in the middle here when there's nothing there? Because this is just the two slits and this is the screen, there's nothing in between. So you then calculate this thing and that's what it looks like. There are the slits in the background. And here is the, the way it's moving along here is no potential, therefore it moves straight and then goes across. So this potential is somehow introduced by the experimental setting. Yes, it's an environmental, we call it a morphogenetic, it's like a morphogenetic field in the biological sense, that it's set up by the surroundings.

42:30 So the particle is moving along and it's being fed information about the surroundings. And we've even called it an information potential, you know, trying to argue that in quantum mechanics it's not merely a Efficient causality, but there's a formative causality, there's a morphogenetic field playing, in which information itself is playing an important role. I mean, that's just a given, I don't know, that's a speculation. That's obviously a great deal more speculative. That's much more speculative. There's nothing speculative about these curves and so on because they come out of the mathematics. The speculation is, you know, what is the meaning of this, from the philosophical point of view. The crucial point is that somehow the evolution of probability is, as in classical statistical mechanics, actually induced by a flow on the phase space itself, which, however, involves... I wish I had remembered better what Lee was saying. It was quite impressive to me at the time. Everything is deterministic, including the evolution of Q itself, so that the only thing that really needs to be specified beyond the, you can hypothesize the initial momentum and position and so forth, is what is the initial value of this Q field, you see, and he said, well that's, I asked him, how do you figure out what that is, he said, well that's, everyone knows how to do that. It depends on the pyramidal setup, so the two splits, two splits there's one thing and for another setup there's another thing, and so you put that in, so that's the, where the active measuring enters the mathematics is in the specification of the initial conditions for the Q, the other stuff is sort of classical. And of course I've been interested in seeing whether one can from outside influence the Q. So that one's feeding in outside information, so if you've really got an information exchange going on there and so on, you know, altering the size of the slits or doing something simple like that, and all that's possible, but that's not the reason why you might be interested in this, but that's what you can do, and everything works this way, you know, because every time we did this, it started in the 70s, people would sort of look at us in disbelief and say, oh, well, that's obvious you can do this.

45:00 But what about so-and-so? They'd do another one of the classic paradoxes, and then you'd solve that one, and then they'd say, oh yes, that's interesting, but what about, yeah. And in the end, there was a- The etiology is stronger than the science. Absolutely, absolutely. And then John Taylor down at King's, they'd get a scathing report for David Bohm's, it was the wholeness in the implicate order, which was obviously too much for John Taylor. And he had a go at this and said, yeah, but they can do it with particles, but we know physics is really field theory, you see. Well, I'd just written a paper, which had just been published with David on field theory, showing how you could do exactly the same with field theory. Oh, yeah. So I posted it to him. He immediately wrote back, oh, my God, he said, I didn't realize you could do that, and then started asking me questions. Because he was interested in gravity, you see, and quantum gravity. One of the reasons why I got into this was that I thought, here we've got a system where we've got geodesics, if we put it in this form, will it enable us to answer questions about quantum gravity, where we know general relativity is all about geophysics. I mean, we failed on that, but I mean, that was the reason for me becoming interested in this. Okay, so the real question, and I think a lot of people still... They try to, the ideology really is so strong that I'm still getting papers saying that this is wrong. And look, the mathematics is all there. P is equal, probability is r squared. Q, these are just auxiliary variables. They're not even observables in that sense. So you can never prove it wrong, experimentally or anyway. You can say I don't like it, but that will then show your prejudices. But I have noticed that there are a few more people saying why does Q exist? Well, there's a lot more that I mean people want to say at some point about the way this connects with the phase-space picture particularly with the Louisville I'm just coming up to that, this is the next one because it's tied up with the Yeah, it's tied up with the

47:30 And what is that lovely expression of Marius de Goosen's, the symplectic camel? Yeah, that comes from Gromov's theorem, you know Gromov's theorem, which is a topological theorem which actually Ed Witten puts in a sequence, you know topological field theory that Witten's been talking about just recently, he put Donaldson's modular spaces, which a lot of people are probably familiar with, but he's got a sequence, I can't remember the two middle Donaldson's in four dimensions and Gromov is in two dimensions. The Chang-Sans index comes out. It's a sort of a generalization from two dimensions to three dimensions to four dimensions. I don't remember who the mid guys are at three dimensions. So there's a series of important topological considerations. And Gromov, I was introduced to by Maurice de Goussons, who's a French mathematician who worked with Jean Leray. He's a Ph.D. student of Jean Leray. And I came across it when, quite extraordinarily, because I went in to talk about what I'm going to talk about in a second to Shelley Goldstein, who has been supportive. How can we be supportive of the Bohm? He calls it Bohmian mechanics. Except the whole point is it's not. I don't think it's mechanics. It looks as if it might be mechanics, but when you look at the quantum potential, it has all sorts of strange properties. It has strange properties in the sense that because you've got this factor in the bottom here, it means that it doesn't matter how big the amplitude of this wave is, the guiding wave if you want to call it that. It doesn't matter how big the amplitude is, the quantum potential could still be quite small, and it doesn't matter how small r is, conversely, this could be quite large. So this would explain why it could be non-local in the sense that it would spread over, because it doesn't have to have a massive amplitude which is decreasing with distance. It's independent of the amplitude so that, you know, even though the waves are very far apart, you could still have quite a significant contribution. And the non-locality in quantum mechanics, the EPR paradox, all comes out of this.

50:00 And you can show that it is non-local. In fact, that's where John Bell got his question from. Are all theories of the type that Bohm has produced non-local? And out came his inequalities to say, he said yes. Okay, so it really was the... The non-locality is no longer a kind of piece of mystery mongering. I mean, you're actually seeing here how it's... Well, everybody is using non-locality now, which is... You know, I used to have to defend it very carefully and sort of poke my head around the door and say, don't mind me talking about non-locality to you. And now, I think it was about 12 years into... Preaching about this that a couple of students were getting restless in the front and said look get on with it We know that we know quantum mechanics is not local. So it's this Perception has suddenly changed from non-locality. They're idiots To non-locality is obvious, which is quite Which is quite extraordinary Okay, now maybe they've just become less critical I think there are a lot of experiments now which show that the Bell inequalities are violated and people cannot explain them locally. There are people looking for loopholes in the Bell inequality. I've got a paper now which I'm refereeing. And true enough that if this assumption is made, Bell's inequality doesn't hold. But the point is produce a quantum mechanics which has this assumption in it. And that's what reading is doing. Then we'll take notice. That's that's a that's an aside yeah let's let's go to grandma uh okay you really want me to go that way okay well okay like bill just can i just say i want you to go i'm sure you i want you to go okay the theme is going to be face space yeah i cannot have a manifold with a well-defined p and q So I've got a problem. How do I construct my fate spaces? I always thought it would bring back to something much simpler. It always seemed to me that the notion of state has to involve some aspect of becoming. And traditionally it's with momentum. But for some reason, the fact that quantum mechanics got formulated in terms of complex ways...

52:30 ...wasn't already a big mystery because Penrose is a little bit complex, right? Yeah, yeah. ...which is... I don't agree with... No, but no, it's a simple matter that you've got to have this, you know, some kind of aspect of becoming in the States. Absolutely. This could be... What do you think Basil tells you about his take on the twister? It could be encoded as... You see, let me tell the story as it came to me, because I think it's much clearer. So we're worrying about where this quantum potential count comes from. The next thing I was worrying about was... Oh yes, I know. Now that comes out just by writing down a Schrodinger equation and putting psi equals r e to the i s. Trivial. Mathematically trivial. But we are told that the Heisenberg picture, i.e. the matrices, is equivalent to the Schrodinger picture. So the question then was, where the hell is the quantum potential in the Heisenberg picture? So that really was, in fact, that was Melvin Brown's thesis problem. He came to me, I want to do a thesis with that. I said, all right, there's the thesis. It wasn't quite that, actually. What I was worrying about was that if you have this equation, the S equation, we know in classical physics, in other words, when the Q is not there, this other equation is symplectic invariant. You know, it's a symplectomorphism. If we bring the Q in, do we break the symplectic symmetry on that? Yeah? Because now we've got something else in there which is not in classical physics, yet we know it's got to be symplectomorphism of some kind, because Heisenberg's equation is essentially a symplectic equation. So that was the puzzle as well.

55:00 But I say, for me, the real question was, how do you get the quantum potential in the Heisenberg equation? Now all the classic approaches to the Heisenberg mechanics is where you take the time dependence out of the wave function and put it into the operators, but you're always left and then you get the Heisenberg equation of motion of any operator. You've essentially got that equation for the Heisenberg equation of motion. This is Bill Levier. This is Arleta, who's one of the students now. She's a lecturer here. She's a mathematician. I'm just talking to Bill about all the stuff I've talked to you about, so it's going to be very boring. I'm sorry for the static. Are you teaching tonight? Yeah. Okay, but we're not disturbing you if we go and talk? No, no, no, I mean, I'm sorry to say. And then, of course, what you always put in the wave function at time t equals naught, as you're always looking in the back, you say, okay, that's it. Where is the phase development? That was my question, because where the hell is the phase development? Right, so then, it was at this time that this symbol came in.

57:30 And what Dirac was saying was that I really want something... Which I want to talk about where I prevent anything multiplying it from, in this case, the right. In other words, his wave function was something, he wanted this thing to be a function of the operators. So he's saying that it doesn't matter. You see, here's the argument essentially like this. If I've got that's equal to f of a psi of a. I can put this inside the bracket there, if I've got an operator here. And then he argues, well, I don't know what this is anyway, so why don't I make this some function of A? Then I can take care of the eigenvalues by bringing my A eigenket to sandwich it on this side. So really, I don't really need that bar in there at all. So I can keep multiplying on here and building up more and more complicated elements of the matrix. If you take this as the... And we have to do that too. But if you think of this as an impotent, then what you're doing is just generating the left. Okay, so you've got a way of generating ideals. And then you've also got a way of generating right ideals because there is the standard bra as well. So that when you're dealing with something like this, I'll call that sine. This essentially is the density matrix in a pure state. You can make it a mixture if you want to, but just think of it as a pure state at the moment. So if you're dealing with the algebra, what you should be dealing with is the density. But the trouble with the density operator is that it doesn't have a phase in it. Okay, so what you really should be dealing with is looking at the left ideals and the right ideals separately. Okay, and then the phase is going to come in from there somehow.

1:00:00 How does it come in from there? Well, the argument has to be that you need both the Schrödinger equation And the dual of the Schrödinger equation, normally they're treated as just being, if you've got one, you don't need the other, you need both, because if you think of it more generally, why should the left ideal be simply dagger the right ideal, emission conjugate of the right ideal, what should it be? That's something special, so suppose we treat these two things differently, then the Schrödinger equation will be playing one game on it and the... Conjuncturing the equation that we do, but it's the two together that determines how the thing unfolds. Now how do I see that? I see that because of something that Feynman told me. Feynman's, in his, in his ninety-four hours, he didn't teach me, I mean I just read it in his paper, in his 1948 thesis it was, and that was when he was thinking about quantum mechanics before the path interval but the beginning of the path, and he said, suppose we think of The wave function as describing information coming from the past. Suppose we describe as information coming from the future. Remember his other idea of . Then what we should be looking at is the action where these two meet each other. So the intersection. And what he then does is he actually obtains Schrödinger's equation. By putting these two things apart, in other words, by taking psi x1, I think it's better if we use this, psi t1, t1 is that point there, and thinking about psi t2, and then going to the limit as t1 goes to t2.

1:02:30 What he did, then, was to say, what about this? My thesis is that the algebra is describing the process, picking up on Rasmussen, picking up on Heisenberg, Paddington, the right thing, because the algebra is describing the process, or something like that. Therefore, if I am processing them, I would be interested, far more in this, in applying two points to them, and not applying two points to either. This is the Schrodinger equation. That's what I'm going to show you the Schrodinger equation. This one is the Schrodinger equation. So let's treat these two things as different. Because I've got different times here, phases, look at these as elements of the algebra. A and B. Left ideals and right ideals. How do you break elements? And I can do that because I can generate my left ideal. I'm not sure whether they need to be pinned down. This is a treatment of left ideals. Normally they're the irreducible representations of some groups, so they're going to be minimal left ideals.

1:05:00 Okay, they're going to generate the state. But I'm not. At this stage, I don't think you need to pin it down. Maybe you do, but I don't feel it's necessary to worry about that. Left ideal is... it could be a maximal left ideal. Then it would be close to the maximal ideals that Gerolf Hand had, but that's jumping ahead a bit. If there is a mathematical problem, then let me know about it. Let me tell you the story. So what I then do is I take the Schrödinger equation and I take its dual. And I simply add and subtract those two, okay? So I'm going to get... Now, this is going to work on one, of course, and this is going to work on two, so that my adding and subtracting is... I want to keep these two... I want to go to the limit afterwards, and what I find from here is that... I won't go through all the steps because I'll probably get them wrong. I find I get the Luthier equation, so I get an equation which is i d rho by dt... Equals, I'm not sure if there's plus or no, with a commutator, because I'm adding and subtracting, so I'm going to get commutators and anti-commutators. This is the standard Libyan equation where I've written rho is equal to, now, if I write these, not the l's, but the psi, I've got l is equal to psi times the idempotent, if I write this in a polar decomposition, If I polar decompose that function, so I write it in terms of, I'm sorry about the r, but amplitude and e to the is, I then find that this, I get a second equation, which is rho ds, or this is the limit, sorry I've taken the limit as t1 goes to t2, and I take the limit here, plus, if I've got the sign wrong...

1:07:30 So now what I get is two equations, not one equation. Normally most physicists have said, this is all you need, the Livian equation, when you're doing statistical physics. I say, no. When I first got this out, I thought I was doing something. I haven't seen it for a hundred years. And what I've done is so naive that it's going to have to be somewhere in the literature or else something's wrong with it. The first thing I did was to check. But this was gauge invariant because I've got a phase operating and it turns out it's gauge invariant and the Bovenhoff effect just falls in one line, the Berry phase falls out in two lines and so on. It comes straight out of this equation with no messing. When I was in... For the S equation... This is an equation I have never seen written down anywhere in the literature. So the S equation depends on rho, but the rho equation doesn't depend on... No, they depend on... they're coupled again because... Because the, well you see, let me, let me just, as long as I'm sort of confident that this was all right, these are all operators, okay, I'll put hats on, no representations in the Hilbert space, no quantum potential. Quantum potential arises from this equation when I find a representation for it. In other words, I put the equation with x's on either side of it, and then what I get, in other words, this equation contains the quantum potential.

1:10:00 But it only contains it when I project it into a subspace and I take a representation. When I was in Urbana just before Christmas, I was talking to John Barker who's been playing around with the Bohm interpretations. He's a nanotechnologist and he's interested. The great thing about the Bohm interpretation is you have statistics. You can talk about individual trajectories and everything. You know, printed circuits and all that jazz. He was analyzing them in terms of the Wigner distribution, which is the classic way of doing it, and he was doing it in the bone and showing that the bone method gives you a much better handle on how to deal with these things. Anyway, we were slapping each other on the back, and so I had the courage to say, look, I've got this equation. And he says, oh, we know about it. Not in this form, he said, but we know that is not the complete story. And he gave me a reference to, what was his name, Carruthers and Zachariason. And they sort of use it, but in a very sloppy way. They realize that this is not complete because it doesn't, when they're trying to find the eigenvalues of the density operator in this equation, they find... It doesn't give you a complete solution, but you have to introduce the anti-commutator in order to treat it. And if you use the commutator and the anti-commutator together, you get a complete specification of the eigenvalues of the density matrix in the system.

1:12:30 So I'm going to patent it. Patent it? Well, if they apply the nanotechnology as well, patent it. Yeah, but he's been doing the work. He can patent it. I mean, all right. Anyway, it then turns out that Farley, David Farley... You've got patent equations. You've got patent equations. And then, when I put this on Los Alamos site, David Farley actually said, I think it's got something to do with the cosine bracket in the Wigner-Moyal. That's right, yes. It's the Wigner-Moyal distribution. Because the Wigner-Moyal distribution produces a star product, which is actually a cosine plus a sine product. And this is the sign for it. So, you know, it's all fitting together. Now then, then my conclusion might be, yes, well that's what I'm trying to, and in fact in the average world I've got this whole story about T1 and T2 about being and becoming moments. I've got an idea of a moment where the time is ambiguous. And it's when the time is ambiguous that movement takes place. Okay? So it's not, Zeno's paradox disappears in this, in this one. And I think, did I, did Keith publish it in Empire? After All the Coming. Did Keith publish that one? I think so. I haven't seen this year's Empire. No, I haven't seen this last year's. Oh, I haven't seen it. Because I did write one. No, I didn't come last year, I came to the one, the talk you gave this year on the Hamiltonian, the Jacobinian. That was, I was very pleased with that. No, no, my take on it. Okay, so we've got this equation. We've seen the quantum potential, we now know where it's coming from. It's coming from the algebraic structure, essentially, when you project. So now we've got a picture that... And of course you're absolutely right to pick up on the underlying philosophical motivation about this. So you've got your algebraic quantum mechanics. No reference to space, position.

1:15:00 Time is still in there, I haven't tackled that part. Keep time as it is, classical, but no position anywhere. There's no positions in this at all. So now then, what I say is that when I project, I can either project my phase space into x, this x is going to be what I observe when I measure x. But then, in order to make this into a phase space, I have to have convert, which was equal to grand, into the p-iron space. x equal to, can I remember what it is? Minus the p s for the phase. In other words, I get an x which I have to identify with minus the derivative of p. And so I can now get, so I can get a phase space for every projection. Now my claim is that this is, if I was playing in the Gelfand spirit or a Cohn spirit, then I would say that I have shadow manifolds. What we are doing in physics where we see the uncertainty principle, I apparently call this that I can either look at this through the x-eye or the p-eye, and of course you can look at it through many other eyes as well, so that Bohm looked as if he was doing something special with the x, but you can also do, so that puts back the symmetry in the Bohm approach that Heisenberg so objected to, and a lot of people object to. What I've done is to restore the symmetry in the mathematics of the Bohm. And now we can decide, do we take the x-representation? What is the philosophical or physics reason for taking the x-representation as opposed to the p-representation?

1:17:30 And so then you've got a sort of a debate of, well, all measurements are done in x-space, and therefore that's what makes x special. But I mean, that's... it would be nice to know... because if you look at the... the whole idea of this was, the algebra described the infinite order. Did you... have you ever come across Bohm's ideas of the... Okay, his idea was that, take an analogy. He had a brilliant example. Suppose you get two cylinders and you put in a spot of dye. You've got glycerine between the two cylinders. You know the old viscometers that used to do that? Yeah, yeah, yeah. And you can see through them, they're made of perspex. You put a spot of dye in there. And if you turn the thing several times, the spot disappears. If you turn it back again... This is what reappears. So the argument is the following that nature is such that there is an implicit order even though you can't actually see the structures that are there they're implicit overall structure of the process movement whatever you want to call it but then you can unfold them you can bring them out and one way we do that is by putting the experiment in its place and that's when you explicate a certain order. So then, philosophically, what we would say is that's Bohm's Implicit Order and these are Bohm's Explicit Orders and you cannot display Explicit Orders simultaneously because if you put two spots, one spot in, then another spot, you cannot display both spots at the same time because of the structure of the process.

1:20:00 He's going to, that would be a, yeah, it's somehow the stability, I think, bringing the stable structure in, has got something to do with it. And that breaks the symmetry. So I've got the symplectic symmetry here, but then I have to break the symplectic symmetry when it comes into the world of matter. And the breaking of the symplectic symmetry is sort of captured by that Q in some way. But you see why I sort of picked up on the on the non-commutative geometry which I know you're having the shadow manifolds now somebody I think Hiller, Michael Hiller, has done a bit on this and he talks about shadow manifolds although I read a paper that David and I wrote many years ago and we talked about shadows there so it's obviously been in the back of my head for a while but I think this this opens up a whole new way of looking at what Bowling actually putting the emphasis on the process putting the emphasis on the algebra And the non-commutivity is in p and x, you see. It's still different from the orthogonal, although the mathematics is the same between the orthogonal, triplet, and the symphatic, but you're seeing it in a different way. There's nothing that we aren't getting from the anticommutative. It might be useful to say something about the way that you tied in these ideas of David's about the enfolding activity with the disinterpretation of the spinner.

1:22:30 In terms of the correlation between the left and right ideals, rather than, you know that structure that you did with Fabio, where you put out the bivectors and the... I don't know why I did that. What would it help? I'm not sure. I'm not sure I like it. Well, I haven't tied this up with Gromov yet. No, no, I was going to say, is this still... We're talking about the topology as well. Anyway, that's that part. Shall I give you another story? Yeah, well, take a while for this one. I'm sorry, I've been through it and just sketching it, but this, from my point of view, is very nice because it's emphasizing the algebra and there's then just one more step and I'm into the GNS construction and so on. Changes all this lot into a buyout. When I come to field theory, we change it all into a buyout. Well, this is just simple, straightforward algebra, but you have to double-check. I don't remember how to do that because that's what we're doing in the network stuff. Going through Lou Kauffman's book on... I started in the icing model a few years ago, a nice lumber crunching, and Lou has shown how the icing model, I'm not sure it's only Lou, but Lou's book has shown, Lou Kauffman's book is lots of physics, has shown beautifully how you relate what I was doing in the early days with Hoppe algebras. The algebraic structure lying behind my county was essentially... You also get the grammatical polynomial out, don't you, and the state-lead-template algebra? Well, that's counting again. For me, the most important thing is how you can get it all out of the template in the algebra.

1:25:00 In other words, you have some configurational program which you can then put in algebraic form, and from the algebras you get out the configurations. The structure is very similar to what I'm doing here. You're not some form of vacuum expectation values here. And this could be thought of essentially as an academic lecture. Yeah, that's right, that's right. So there's some mathematical principle here which is not just what I'm doing here, but actually is quite extensive across a whole couple of fields of activity. And they seem to be totally different. The really staggering thing... Is that the configuration work that we've been going through counting, just counting spin up and spin down on lattice, just on a finite graph, you just take a singular finite graph like that, and you count the number of configurations first turning one spin up, with a rule that you get an energy contribution with when the two ends of the bonds have got the same spin orientation, because you get a partition function. Right, what you can do is you can get that partition function out of an algebra and that algebra uses exactly the techniques that I'm using up here so somehow this process of taking vacuum expectation values is working out the contribution of some algebraic process to the amplitude of the total process. It's giving you some numerical value. So if you can think of these processes as actually executing knots of some kind, just then you're looking at those knotting variance, and that's really topological field theory. I mean, this is a long way from topology, but the point is it's the same mathematical structure, it appears to be. Because we even got idempotence, didn't we? Yeah, yeah, yeah. First of all, you have to go to the chromatic, you have to go to what Lou calls the universe.

1:27:30 Yeah, that's right, yeah. So you've got a, this thing can be, this thing can, if you go to the, what I call the dual of it, Lefschitz dual, you know, where you take a simplex and you... Yeah, yeah. And then make a structure out of that, that's essentially, I mean there's different ways of doing it. That's right, yes. And this thing then becomes... Yes, in fact that's a little illustration of a slightly more sophisticated, and you get the Wiedermeister moves out of this. That square lattice can then becomes this knot and Lu then shows how you decompose this knot to get exactly the same partition function out of it, but then you can decompose this using the algebraic rules and get this out of there, and the algebraic rules are just like quantum expectation values, in other words you sandwich hidden potents on either side of your analysis. It's extraordinary, wasn't it? Do you hold this paper by law? Because he's exactly like you. Of course. I mean, it's in his book. This is another paper, but it's in his book. Yeah, that's exactly... Well, you missed my talk. If you'd come yesterday... Well, but I was reading it yesterday, so I didn't think you'd see. I was thinking... Yeah, that's exactly what I'm doing. Exactly. Exactly. Except I'm just doing a simple... Yes, you're just doing it. I'm just doing a whole checkerboard. And writing from calculators. But essentially you're right. It all connects up with the housing model. You're just calculating partition functions. It's none of my... This is Luke Cartman. Everything. Yeah, he writes it's a mystery. I mean, why? Because there's von Neumann algebra there. Yeah, and then the next step, I haven't got it right, the von Neumann algebra is exactly the same algebra as you're using here. Yeah, he says it's a mystery. Well, is that strictly correct? I thought that in fact the Jones algebra was actually a sub-algebra there. No, it's not the Von Neumann, it's the way you stack Von Neumann algebras together. Oh yeah, I agree. That's right. I'm sorry I do not know enough about Von Neumann algebras. I want to understand what Jones did there. Because I'm building everything up out of idempotence.

1:30:00 And do you know what those idempotence are, by the way? They're just delta functions. They're the algebraic equivalent of a delta. When I saw Bob getting it and then I got my points out, cock-a-doodle-doo. So this is, anyway, that's the... One would have thought that we would have jumped Hilbert space long ago because... I don't know, why do they keep... I have not been able to. So he was... There are again these strange ideological fixations that people have, unfortunately. I mean, an extreme example of that was when I was in Prague. Physicists at Charles University had asked me to, I was in Prague for some time, and physicists had asked me to come and talk to them about what I did with the bone interpretation, and I just went through, I knew there was trouble. You could sort of feel it in the atmosphere. So I actually bent over backwards to say, well, all I did was this, and I found this, and look, isn't this interesting? No, I've no idea what ideology I am. I must speak to you about this. I could not do this because I'd never mentioned the word Hilbert space. I'd never mentioned the word Hilbert space at all. I went red in the face, he was shouting. Even when I went down in the lift afterwards with him, he trusted me. And I say, I wasn't provocative. I can be provocative when I want to be, but I wasn't. I was bending over backwards not to be provocative, because I knew there was trouble coming. That's one of the things I was hoping to hear Bill talk a bit about, as and when you feel the sport knows you, about your own views about Hilbert space and also nuclear spaces and the way that the construction actually fits into a categorical setting.

1:32:30 Well, it doesn't, of course, fit very well in the case of Hilbert space. Can I just do one more bit of the story? Okay with me, if Bill's up for it. Actually, I want to say, I don't quite understand what you're saying about ideals, because, you see, Gelfand's original work, according to Grothendieck's reworking of it, it's not about ideals, you see, it's about homomorphisms. If you have a homomorphism into complex numbers, or the real numbers, or fields, you know, mathematical fields, then the kernel of the homomorphism is indeed an ideal. But you often have to consider several different fields and morphisms between them. So, in other words, the basic idea of a point, algebraic point, is that you have variable quantities of various kinds. Particularly, you have constant ones. And so what a point is, it's a way of evaluating variable quantities and producing a constant value. So it's only in a certain special situation that those kind of homomorphisms can be equivalently interpreted in terms of ideals. So again, you see, again, this is a whole ideological commitment. Everybody thinks they have to talk about ideals because Galifian talked about ideals. This was why I raised the question of that Cartier paper with you this morning in Paddington. Did you see what I posted? No, I didn't see that. That's a couple of days ago. The point is that Brodkenbeek already, you know, in the 1960s came up with this idea of skiing. He and Doudanet latched onto this idea. You see, again, there's...

1:35:00 So there's a lot of writing about schemes in terms of prime ideals and so on, which is an incredibly complex, you have to master a whole, well, that's not just a question of mastering, you have to imagine carrying along an immense mathematical baggage just to understand the definition of scheme according to that original, well, in 1973 already, so 30 years ago. Grotendieck gave a lecture, happened to be at Buffalo, and I happened to be there when I heard it. I wasn't working at Buffalo. So he said basically that should all be overthrown, and a much simpler definition of scheme should be adopted. And in particular, this was based on the fact that points are homomorphic, of a special kind, perhaps, in that they have the very flavor of reducing more variable to less variable, let's say. There are a lot of people who tend to say, well, okay, we want an algebraic algebras. Let's mimic the idea of the spectrum of an algebra by calling it legal ideals. But it's purely a formal analogy of the particular formulation that was given to the book. If you take a more invariant formulation, then the analogy would be something quite different. And of course, that was the point that you were making to Kahn when you had that exchange with him, and particularly when you got to the point that it doesn't respect the Morita equivalence in his construction. And there's no control over the way that one passes from the variable to the constant in his construction, because of the angle of all this machinery. I'm coming into it very, very, at least mathematically, very naively. David Bohm had been prattling on about algebras and process, and Penrose was around.

1:37:30 and Penrose was doing his twisters as that was here in this department this is where Penrose began his career before he was before he went to higher places sir roger if i got into a taxi in virginia i said i'm not sure it's just africa I'm not sure I should come and sit next to Hugh, Roger, because you've got a lighter. He actually said, he said, my wife and I spent several weeks wondering whether we should accept it or not. So it obviously wasn't. And then he said, well, I thought I could do my, I thought I could do science some good if I became. So anyway, sorry. Very nice guy, Roger. He was prattling all about, so I had algebras coming out of one ear, I had spinors coming out of the other ear, into, rather, and I then discovered Clifford, and then to my amazement, I discovered that the spinner was actually just an element of the algebra, okay, but now the spinors are what we have been dealing with as elements of a Hilbert space, and for me, to regard the... Weigh function as in fact part of the algebra was absolutely crucial to the whole because now we have one structure in which we have both operators and state there's no difference between the two and therefore putting together with a sort of philosophical ideal that we had that everything was coming out of wholeness and so on so wholeness was always in the background. Bohr was kept telling us quantum mechanics is about our inability to separate the object that we're investigating from the means that we are investigating with and so that whole thing quantum mechanics is essentially about wholeness was there. Whereas I always felt the algebra of operators and then Hilbert spaces not being only something you know that you're one structure and multiplying it by another structure and getting your answers out.

1:40:00 And naively I thought... You certainly don't have that kind of categorical distinction between the notion of state and the transformation of state and the dynamics in that picture. Right, that's right. And then the idea was, well, why not put it all together? And then I saw you could do it with the Clifford algebra. And I can construct the ideals in there. And the ideals were the wave functions, were equivalent to the wave functions. This is where I come from. Ideals are equivalent to... The left ideals, the minimum left ideals, are essentially the wave functions. That's why I went to this left and right ideal story here. Because I'm trying to get everything in the algebra. And because, you know, that... For me it was a sort of a... My God! So the spinors are actually in the algebra. They're not something that comes as an afterthought. The way we're presented in physics, okay? So that I see it as one part of an algebra and then I say to myself, why can't I do that for the Heisenberg algebra? You see the logic, I mean, sorry, my line of reasoning, it's not a deep line of reasoning in a mathematical sense. Well, you may be able to biographical statement, yeah. Okay, but it's because, it's my obsession with the spinner, if you like. And it's seeing the symmetry, the analogy between the two. One based on anti-commutators, the other one's on commutators. The anti-commutators give me the Clifford algebra. I mean, if I take the fermionic annihilation and creation operator and I make, add them together, I construct the elements of the Clifford algebra. In other words, I can make a geometric algebra out of fermions. Being a physicist, I say, well, why can't I make bosons? In the same way, with a boson annihilation operator, I should get something which is equivalent to the Heisenberg algorithm. In other words, it's a symplectic structure lurking in there. And why can't I do the same as I do... Most people say, oh, well, one's anti-competitor, the other's commentator. I don't accept that.

1:42:30 Yeah, yeah, yeah, yeah. That's simply a formal expression part of the story. But deeper down, there's a different aspect. The fact that the fermions were coming out and giving me a geometric algebra, as Clifford called it, then seeing the bosons coming out and giving you, hopefully, another geometric algebra, then you have to put the two together. That's where the Penrose twister was coming in, but you see he doesn't put them together out of these two different structures, he puts them all in. The twisters come out of the orthogonal. In other words, the displacement. And then he thought he'd done it, but that's what I thought was wrong about the twister. It doesn't get the momentum side of things, it doesn't bring in momentum space, and that's where the commutation is. And of course building this assumption about complex magnitudes that for some mystical reason when they're in the position space picture we have to go to the complex one. Whereas in fact there's a quite different way of doing twisters. Which, well, I touched on it earlier on, which is to look at them as elements of spherical hyperbolic geometry. That's right. The Leigh's hyperspherical geometry. That gives us a much more different algebra is coming from a symmetric bilinear form. That's right. It's sort of a superstructure on the idea of a metric. But I want to look at it, I was taught by Penrose to look at it the other way. Namely, spin networks to the things you start with. There's a quantum spin network and then you extract out your your quadratic invariant as a consequence of this basic... Well, fine, fine. That's my... Yeah, the origin of the form, but it's... I certainly, I'm well aware that, because I've often had it pointed out to me, that that's the way the mathematicians think about it. And I'm just saying that if you read the way Clifford formulated it... It was all about rotors and motors and so on. In other words, it's all about activity. It's all about action. I actually went back and, well, I didn't go back and see it, but you know, Clifford was at UC, just across the road, so we have a collection of Clifford's works here, you see, and because it was on the shelf in front of me, being a lazy guy, if it hadn't been in front of me, I wouldn't have read it.

1:45:00 And I was absolutely fascinated. I was fascinated by going back and reading Hamilton as well, you know, the algebra of pure time. Have you seen that? You've probably never seen that. No, I've seen that, yes. I mean, I've seen it. And the really staggering thing for me was realizing that, I think it was Grassman who said that mathematics is about thought, not about material process. Hmm, well, that was a very interesting conversation this morning, Ralph. I'm afraid you couldn't have said. I hope we're not actually going to come to blows, gentlemen, because I was going to take you out to the pub shortly. Here I have to say, I do part company with you. I agree with you in assigning great importance to Grassman. Listen to what I'm saying. I'm saying that for me, it was a tremendous shock. ...to think that was possible, you know, and it opened up a... it got me completely away from my practical physics, as it were. Thinking about all mathematics was a material process. Well, I'm not so sure that Grassman, or the use made of Grassman on that occasion was a contribution. But I do think you're absolutely right. So, for him, geometry is not part of mathematics. Rather, geometry is the application of formal science to real science, because whatever your geometry is, it's motivated by the real world. And so it's imported into when you apply formal methods to it. So, he separated. Many people have made this claim. People are trying to discredit him from all sides, you see. Yeah, I'm not trying to discredit him, I'm trying to say... Oh, no, I read your great admirer. It was just published, or just written, I should say, a lengthy review of the English translation of Grassman's. Okay, I haven't read... I read just snippets, not whatever I read. Take what I want to hear, not the rest.

1:47:30 Well, unfortunately, an awful lot of, and that is, well, until this volume was published, there was no complete presentation of Grassman's work in English, am I right? No, no, you're absolutely right, because I remember I was down with Clive Kilmister, and he thought it was a good idea if a few of us got together and talked about concepts of space and time. And he talked about relativistic space and time. He wanted me to talk about the sort of pre-space, the process space that I was talking about. He had Richard Serabje, who was a very competent philosopher, talking about Greeks' notions of space and time. He was a very Aristotle scholar, a very fine scholar. And there was a Japanese lady, which seems rather... Anyway, she was talking about Leibniz. Well, it was Hideyoshi Guru, wasn't it? I think it was. I mean, I don't know if she was any good or not. Well, if it was Hideyoshi Guru, she's a very fine philosopher, yes. If it was her. I wasn't there, so I'm not sure. Anyway, it was a Japanese lady. An American student came up to me after I said, well, have you read Grassman, he said. And I said, well I would love to read Grassman, but the trouble is my German is pathetic. And he said to me, well you don't have to read Grassman. There's a guy called Lewis who's actually trans... Oh God! No, no, no, this is where we can build... I'm sorry, Bill. No, no, no, we're here too. This is a very useful exchange. It's amazing how things come around. This is just in the circuit of one short day. Bill was just talking when we were at Paddington Station just after Bill came from the airport this morning before we went off to Quaker House. This is exactly what we were talking about. Yeah, so Lewis is... The thesis was that Grossmann's philosophy of mathematics and geometry came from, derived from the theology of Schleiermacher, one of the founders of the Prussian church. However, this analysis of Grasper has been thoroughly refuted by your truery. I wasn't taking any of that. I was just taking the snippets that he had translated of Grasper's original work. Well, it's not the time to take snippets.

1:50:00 But the only time I've been doing this for a hundred years, more than a hundred years, whereas my colleague Shanyuel and I... Simply reading Grassman from the point of view of finding the mathematics, we found mathematics there that nobody knows about. Absolutely. We found some absolutely wonderful mathematics. In fact, I must recommend to you, and Pierre Cartier also recommended it to the seminar in Paris. The paper which Bill wrote about five years ago now, in the 150th anniversary of Grassman's publication. Most of the talks were philosophical, historical. I have the mathematical paper, which is entitled Gassman's Dialectics. It's basically mathematical. There's one or two other mathematical papers that I mentioned, but Schubring himself analyzed the historical aspect refuting these. But there was a book by Forder called The Theory of Extensives. Do you know where? Well, no, I haven't looked at it. I've seen that name on my list of things to look at. I came across that. It was a very fine class book. I was at Forder. H.G. Forder. I'm just trying to think. Sorry, it's all autobiographical, unfortunately. I'm trying to think why I got... I think it was really Clifford that... Well, of course, Gifford starts out by saying Grassmann's algebra is going to be used for more things than people saw it. It could be that that's where I picked... Around 1850 or so, right? Yeah, it could be that's where I picked up the looking at Grassmann. There's some snippets, and I wanted to know more about Grassmann. I picked his Ashdemon era up and looked him up. And then I came across Fowler's book, which I went through, actually, and was quite interested in what he was doing. Didn't make that much sense of it, but it just... well, I was very young then. I found it quite interesting, but never was able to make really good sense of it. There were some things there which were a little bit helpful in what I was doing.

1:52:30 I made some notes on it, you know, I should work through some of these. Put it in the file somewhere and not come back to it again but it probably left some impression but it was really it was really Clifford though that got me on this motors and rotors and so on as activity and then he comes back and criticizes Gibbs because Gibbs had a paper in Nature in 1890 something where he said we don't want this Clifford nonsense. We only have vectors going to vectors, bivectors going to bivectors, and so on, and this whole idea of a little bit of vector going to bivectors, you know, the transformation of aggregates, he said, we don't want this at all, and then wrote his book on vector calculus, totally ignoring Clifford. And Grassman. And Grassman. And then suddenly you find that Dirac rediscovers Clifford. And nobody realizes it. This is the extraordinary, even now, people, certainly in the physics community, don't realize that, and they're still, instead of using as, Clifford was very adamant that he wanted to look at algebra as A follows B follows C, as a process, not as transformation of things. It's very much a process. But without things, you can't have processes. Yeah, that's a moot point. Well, I think it's rather basic, actually. You might start disagreeing. I will, it's a good thing. It's a creative disagreement. I have to say, I completely concur with Bill. And this, of course, is the case of understanding what category theory is really telling us, that there isn't, as it were, a prior... You have to have the identity maps and categories, really. And people have tried to eliminate them, and they've always... I think it also connects with what you were making about the need to study more business between spaces spaces in general but there's not just one space

1:55:00 There's other reasoning that applies to specific universes, like three-dimensional space, and there's other reasoning that applies to four-dimensional space, and there's relation between the two, but there's, you know, in other words, the unity, what was Engels, Engels said the unity of matter is in its materiality and not anything more, in other words, that somehow you, yes, there's a unity, but it's within that the specific, the specific comes down to. Many, many limited things that relate to each other. But also, of course, studying them in the course of their transformation. Well, yes, you see, I was going to throw in a Maoist idea there about the bleeding aspect of a contradiction, perhaps, which I think is really a very crude attempt to deal with the asymmetry that contradictions have. Somehow becoming is more important, more interesting, blah, blah, blah, than being. But that doesn't mean that you don't have both of them. No, no, I would say you have both of them. You might be taught by David Bowie that you have both of them. You have the discrete and the continuous. Not everything should be discrete or everything should be continuous. Yes, right, right. But you have an interrelation. Well, it's the same way as when Gibbs is wrong because, again, there are these theological things that are looked around that seem to be stronger than mathematics. You can meet people today who will tell you, well, quaternions are outmoded. You can't really use quaternions. They don't know why, though. No, no, no. Of course, different algebras are just an extension of the idea. Well, I mean, they are indeed appropriate when you have symmetric bilinear forms and so forth, and often you have asymmetric ones and anti-symmetric ones, and so, you know, these concepts have a definite role and a definite context where they make sense, and they relate to other things, but it's trying to get everything in one and say, well, this is, you know, I'm a Lutheran, you see, and that's it. And of course in logic this results in this extraordinary view that there's just a single... It's a tremendous distortion by Frege, which we're still, in fact, probably in my lifetime we won't succeed in rooting out this.

1:57:30 It's been a distortion in the heart of mathematics for the first part of a hundred years, although suddenly it's now increasingly being sorted out. Or another aspect of it is, you see, Frege actually states, It's interesting because he criticizes Dedekind and Schroeder, who were great mathematicians and logicians, as being mere technicians because, this is a letter to Husserl, actually, that he said, well, these guys are mere technicians, they don't understand philosophy in the profound way that we do, and they're not precise the way we are, and so forth and so on. But then he goes on to make this incredibly precise, imprecise argument. Concepts are properties. No argument really is given, but concepts are replaced by properties. Properties are what? You have, you know, properties make sense within a given universe of discourse. So if everything is a property there must be just one universe of discourse and therefore you have to invent these screwy things like the singleton operation and so forth. I mean it's of course because he wanted to get numbers out of extensions of concepts. I'm trying to build arithmetic, which obviously is a precise and profound part of our thought on this cloudy metaphysics that you have to have. On the other hand, I was just arguing with Colin a couple of days ago. He wants structuralism. Mathematics is structuralism. I've heard that one before. I've been on that bandwagon for some time. I've been trying to wean him off it. I think it's more true than the idea of thinking about properties, but basically you need both, so that a concept can be analyzed in various ways. That's right. So a given analysis of a concept will involve structure and properties of that structure. The same concept could be analyzed a different way, different structures, correspondingly different properties, and so to identify the mathematical objects or the concepts with structures per se is, again, is a prejudging issue and inaccurate.

2:00:00 Some things are more preferred than others. It's nice if you can put it in a situation where you can find reasons why it should be preferred. Exactly, exactly. The thing that's always worried me about your stress on structuring activity, as activity process based interpretation of algebraic structures, the thing that's always worried me about that... I mean, it's not that there isn't a very important aspect, that it's not a useful heuristic context in which to think about certain structures and algebra, but the idea that there is just this one single a priori, if you like, ontological source of the meaning of constructions... And that this is more fundamental than, as it were, any other aspect of structuring mathematics. That's the thing which worries me about... I did ask David Bohm if the whole movement was gone. Oh, yes, that's precisely because it leads to that kind of speculation that I've always been a little bit worried about. I mean it comes out of Whitehead and this whole idea of process, processes being the fundamental category, and I think that the whole idea of the one fundamental category, in the metaphysical sense of it, is itself an illusion. There is this immensely complex structure of the real world which we shall, over the centuries of development of science, gradually develop the tools, the tools that we objectively need to penetrate it and to characterize different portions of it and the way that those different portions of it, the main concepts fit together. Classical idealist philosophies, they talk about, there's understanding, you see, which mere mortal scientists have, and then there's reason, or something equivalent, you see, which is a qualitatively higher thing, which only they have, you see. Sometimes labeled transcendent for a reason. You know, there are different labels for it, but you see, what I claim is that really understanding is all there is. That we understand more and more, we understand relationships. More and more and more, but this is the only way that knowledge can progress.

2:02:30 I completely agree with that person, Colin, in this respect. I'll just say that very quickly to Bill. Although, no, Colin's a great guy. You know, wonderful stuff. But that book that he's just published, which he's presumably sent to you to draft the manuscript on, in which he puts in this plug for the structuralist position, how structural mathematics works. Again, I'm always worried about that, that there is this... It's a little bit like Russell's mutual monism, the idea that there is this one specification of structure as such, this fundamental category out of which everything else comes, within which everything else fits, and it's just pure abstract structure, which is in some sense ontologically at a higher level than anything which it would make sense to speak of as the structure of. I'm a physicist and everything comes from the vacuum state. The different kinds of structures that start in different branches of mathematics and in physics are all supposed to fit within. This is what I don't like about that. What I see as the kind of objective idealist tendency in Collins' version of structuralism. But there is a problem. I mean, I did write that paper on the vacuum of Holland. Well, no, you did, and it was an interesting paper. I know we've argued about it. We absolutely disagree with the kind of philosophy in it. But there's some very nice insights into the algebraic structures. Yeah, but don't forget, one of the things that we are cursed with in quantum mechanics The quantum formula is what I call the Cartesian category, which is the category about continuity, about particle identity, and so on. Well, I'm not so sure that that is a fundamentally mistaken idea, but... But it will fit! I mean, how do I understand the theorems of Cauchon and Specker and Gleason?

2:05:00 Well, one of the ways that you understand the appearance... The apparent non-locality is perhaps precisely through the construction that you were discussing earlier on, the way that these apparently non-local effects show up, and if the quantum potential does have that characteristic, DR over R, there's no reason why it shouldn't be very small. And still have large amplitude effects. Yeah, but you're still denying our argument, which is still denying one of the features, I think, that is in Cartesian, and that is absolute locality. What do you feel? Do you feel there's such a thing as absolute locality? No, I'm not sure about it. Because, you know, one of the things that, you know, one of the things that I was just reading was Haag's idea that really locality in the physics community The reaction against locality is incredible. I mean, I'd like to look at locality as a relationship, which means you can look at it in different ways, that you can look at it a-locally, even. But most physicists feel very uncomfortable. They want locality to be somehow absolute, in the sense that the whole space-time structure is and has a local relationship. Differential equations have gotten us a hell of a long way in understanding the structure of the world. I think that's probably one thing I'm happy with. Without contact action, without differential equations, it's difficult. I must admit I have nightmares at times whether when I go to the algebraic structure, whether I've thrown the baby out with the bathwater as well, you see. But that's where I was sort of, this is where this notion that I had the notion of that the primitive idempotence are actually the points in this structure and the reason being, I mean I picked this up from Eddington so you can blame, let's blame him, and that was that the great thing about the idempotence was that it either existed or it doesn't exist. But it doesn't, it's not the sense that it exists as an entity that is static and is always there. It's rather an entity which keeps turning into itself.

2:07:30 So in the process... Or into something else. No, I mean, I'm thinking more about the things which... Well, then of course it would be. Then it just keeps turning into itself. But my, all my emphasis is always... But is it when it's abstracted from this kind of structuring activity? Yes, yeah. It's a kind of stable form within. I keep finding it a little bit out of the way all the time. I mean, for example, an arbitrary second-order differential equation is really a good one. Yeah, yeah. On the space of possible second-order infinitesimal paths. See, in other words... A particular law will say, okay, if you try to go this way, it'll say, no, you have to go that way. But you can contemplate an arbitrary continuation and project it back onto the actual one. So the whole specificity of the law is contained in that input. Right, right. I keep somehow coming back to input formulations. That was a second-order path, you see, not a first-order path. I don't think I would have emphasized it in potency so much if it hadn't been for the fact that it was what I wanted to do the simplex experience with. And that's the reason why I went for it. It was something. And then it turns out to be the delta function, which I think is rather... Now that is extremely interesting, because I mean, obviously Bill's right. I mean, delta functions just don't live in Hilbert space. No, they don't live in Hilbert space. And they're not intensive quantities at all. No, I mean, the delta function is simply an evaluation of the point. In fact, it's not a function. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. It's a measure. Because it is the thing which colludes together to do like a punch to quantum mechanics. I think what you've done, a bit of what you've done is showing how the phase space, you can get the phase space structure back into. In fact, you have to in order to make sense of the theory. And when you do this, it does show clearly, particularly when you connect up with the Wigner-Moyle distribution, that there is this quantum potential term, which has incidentally been known about for a long time, but not under the name of the quantum potential. In fact, one of the interesting things Basil turned out was that the Moyle in a paper is 1949.

2:10:00 Actually, Isolated did effectively produce this equation that I'll offer. This equation? Yeah, what was that? Moyau. Moyau. Wigner-Moyau. He was trying to approach quantum mechanics from a purely statistical point of view. Moments, second-order moments, and third-order moments, and so on. And this equation came out of a second-order moment. Yeah. Well, that's Nielsen's idea. Nielsen, yes. Nielsen rediscovered it. Yeah, but it was in fact rediscovered by a number of people. It's just that I didn't realize, and I've been reading the paper for a long time now, I didn't realize he had the quantum potential there. But it's quite clear if you read the paper. I've actually spotted the equation. You can see exactly where it goes. And he's also got that condition, this condition there as well, the first mark. But he's got both the bone conditions. From a purely statistical theory. And that's sort of reassuring in a sense that, you know, it's not something completely arbitrary. It's something that's in the structure of the whole thing. Yeah. Well, it's interesting because I thought for a long time that the main difference between quantum mechanics and statistical mechanics was that the evolution of probability was not induced by a point transformation. If you put in the... Yes, then it is. Except for the fact that this is a time-dependent equation, is that right? Yeah, yeah, yeah. Because I'm coming from the Schrodinger equation, where q has time. Yeah, so the q will have a time, which is, actually I raised that point. We use this argument in the ballet. The ballet is a very, the quantum potential is very much like the ballet dance, in the sense that when ballet dancers coordinate their movements, they don't push each other around. That's the efficient course.

2:12:30 But rather there is a coordinated movement, and that coordinated movement actually develops in time, and so the quantum potential also develops in time. So you've got a sort of a ballet unfolding. Yes, that's enough. Well, thank you. You said you raised this question with me, I was going to say. Was it Lee that you were mentioning to me when we were talking in Nancy that first evening, that Lee had mentioned to you that he now no longer thought that the plant constant was a fundamental constant of nature, or was that somebody else that you had been speaking to? That's not a new idea either. No, but it's interesting when an eminent physicist produces a reason for... He might be desperate. Well, I mean, Dyson actually was worried about what happens if the fundamental constants vary with time. I mean, that's been a very... No, that's been around in Dirac, obviously, played with that idea, and other people, but... I think Willow actually put it much more clearly and says, if you're in an evolutionary situation, what do you mean by law? Of course, the law there before the evolution, i.e. some sort of plank, idealistic, platonic idealistic form. Did the laws evolve? I think that's a confused idea. I think it's a question of order. In other words, I would say laws, we take laws as if they've got given, you see. That's the way we interpret them. And of course, for a lot of physics, I mean, if you violate the conservation of energy, you know, you better watch out because... But that doesn't mean to say it's absolute. No, I mean, it's a reflection of the boundary conditions of the initial conditions. It just works in this local region of the universe. Well, I have a very naive question about this non-locality business. I mean, these kind of calculations, they are essentially about a particular that shoots off toward the star seriously. But now, what about the fact that everybody else is shooting at the star?

2:15:00 It's again a question that in fact the equations are about a definite system, but in fact there are a lot of systems that are interacting, so will this photon really make any difference when it reaches the star Sirius? It seems like it won't. It won't make any difference to the star Sirius. The extraordinary thing about the whole process is that, first of all you can't send signals. And yet, if you look at it our way, you are transforming active information faster than speed of light, because the particle is behaving here in a way which is being changed by something you do here. But as soon as it hits... So there are similar velocities, but at the same time there is some kind of influence. It's very puzzling, it's very puzzling. I've not run the proper measure of which can be controlled. If you look at it this way... Then you could say that in the algebra that locality is not important and if you've got if you've got these relationships which you could project out of whatever it is that's going on. Then locality is not the most important thing anymore. It's the coupling that goes on in the algebraic structure, in the infinite order, I want to say. You sure as hell better be able to recover locality for all practical purposes. That's the problem. That's the problem. That's the £64,000 or whatever million pound problem. Given the sum of our experience of the world is of local interaction. Yeah, sure. That's the problem, you see. That's what we have not solved, and I don't think any of this is solved. What it's done is to say, yes, if you look at it this way and we analyze it in our traditional ways, we find this Q is a non-local quantity, but then because of the participatory nature of this Q, you can't actually use it to signal independent information, and yet there is something because we teleport. It's claimed. I'm not sure how. I mean, I'm not up in that literature, but... I mean, I had teleportation in 1976, but I was too stupid to see what it was.

2:17:30 Yeah, it's the choice of name that is off-putting. But you had these effects that they now use and... Yeah, I had all the effects. I tried to talk to David Bowie about them and say, look, this is very important. I'm not quite sure. Let's talk about it. You know, just let it go. Good old David. Well now of course it's a whole industry. Now it's a whole industry, yeah. Which has the National Security Agency throwing billions at it and putting these cryonic machines underneath Lake Geneva. But you see, if you look at it from this way it's not surprising because... The reason why, you see, if you analyze it this way, then it looks as if there's some influence traveling, but in order to unlock that influence, you've also got to send a classical signal. In other words, you can't do anything with the damn thing if you don't send the classical signal. What does the classical signal do? The classical signal sets up, tells you what this background is for the appropriate cue. Okay so that there's a very very interesting relation between text and context here and this i mean do you get you do you get anything like that in category theory where you're you're sort of you've got to try and create your text and your context together yeah and i i just don't know how to get my hands i keep asking people telling talking people about it but there's nothing we do in our world that gives us text and context coming out it's a sort of a dialectic Is there a well-developed theory about how you combine two systems described by this type of equation into one bigger system? In other words, interactions between them. No, that was what I was trying to develop, actually, in this idea of fusion. We've got in our book fusion and fission. Now, that's not nuclear fission and nuclear fusion. This is the idea of entanglement and non-entanglement. So what is the process that entangles things, and what is the process that disentangles them?

2:20:00 And I don't think we've got a full story on that yet. Now, one thing I kept going around asking people... The trouble is, in the mainstream theory, there are all of these programs, like the decoherence interpretation, which, of course, I think is... Garbage. It doesn't actually answer any principles. It assumes the classical world. Or even worse, the many words interpretation. No, there's something much deeper involved. The interesting thing about the decoherence interpretation, of course, is that people are now abusing the... But it's not only decoherence. You don't have to form coherence. Look, one of the things that I was very puzzled about was, because we were doing an experiment, one of the first experiments we were doing in our laboratory downstairs actually on this. Yeah, actually, before Aspect, 10, 15 years before Aspect. And in fact, Aspect took some ideas from us this year and never acknowledged it, because we were the first, we, it had nothing to do with me, Dave Butt and another couple of colleagues, experimental colleagues, listened to what we said, which made a change, I was quite surprised. But what they did was to actually... Create their entangled pairs by positronium annihilation in copper 64. Copper 64, positron comes out, forms positronium, out comes a couple of photons in an entangled state. And our idea was that, and this was one of the first ones, the second one, because we had some results from an Italian who said that the thing was changed and we were checking it. Your detectors further and further apart before you detect the photons so that you rule out any possibility of space-like connection, sorry, non-space, you've got, if you think about it as a light curve, if you can do, here's the photons coming out, do your experiments here so that, and have a resolution time so small That within that resolution time there is no chance for a signal to go across there.

2:22:30 And we were the first ones to check that it was space-like separated. The second point was that when you've got photons coming out, of course, or any quantum particle is a wave packet, and therefore you've got to make jolly sure that this wave packet overlap is negligible because we know that the covalent bond These overlap with wave functions and that's one of the strongest bonds you could possibly get. So just in case there's something there that we don't know about, you've got to make sure that the wave packets were far enough apart. Now in our case, the wave packets were probably about half a centimeter. The resolving time was 5 nanoseconds, roughly this. The path difference, the light path in 5 nanoseconds was 30 centimetres, and we had the 6 metres apart, and with the, what did I say, the half, which was about half a centimetre, we measured 6 metres, these were separated by 6 metres, and in fact as we pulled, I couldn't do it absolutely, but as we pulled it out you could see no change whatsoever. We were hoping for some sort of dramatic spontaneous localisation. You know, some effect would take place which would decouple spontaneously. No such deal. We have on the record 23 meters. Now it's 40 kilometers. So we were testing that. Then the point was, because the positronium was forming here, the entanglement was local. And I kept trying to think, could the entanglement be non-local? You know, I think this is probably one of the things you had in mind, that could you, well I'm not sure, if you've got two things out here, can you make them entangled? Yeah. Okay. And the answer is you can. And the way you do it is that you have an entangled pair which is separating that way, you have another entangled pair which is separating that way, and then by doing an appropriate measurement on these two here,

2:25:00 You can actually entangle those. So you can actually connect the photons when one's on Mars and one's on Jupiter by doing something locally. You turn them into an entangled state. Now the teleportation, the teleportation comes about if you think of the quantum potential as active information, then what you can show is... ...that you change the active information on the system at the other side. You don't teleport particles. Of course. That's absolutely clear. There's no question of teleporting matter. No, matter. So if you wanted to beat me up, Scotty, what you'd have to do is you'd have to have a whole set of molecules that is Scotty, and another set of molecules which is potentially Scotty. And then if you can... Thank you for the problem of the script. If you can program the information of the sun into the waveforms, then you can create Scotland. I saw a movie called Braveheart, where they were slicing up the Scots. No, no, I don't know. You're pretty good. Okay, now this picture, again, raises this question I was trying to get to before. So we here on Earth are trying to connect up these two. Thank you for watching. And the thing about this is that the entangled states are very fragile. Yeah. Very fragile. Yeah. Now, right after we'd done our thing, there was a paper came... That's why they have to use these supercool.

2:27:30 No, no. I thought when you were doing these experiments, they had to use that. No, no, no. We have nothing. No supercool. No, I mean to preserve the entangled. I'm not confused. I thought there was no business about these... You don't have to have them... I mean, you want to keep them away... I suppose you're right, but you want to keep them in free space. You don't want them to collide with anything. Because if they collide with something, you immediately break it. But you see, the interesting thing was that... Just let me tell you what I mean about the fragility. Otherwise the dick would hit. Holt. I think his name was Holt. One of the first experiments after ours, but before Aspect, reported that he found the Bell inequalities were satisfied. Quantum mechanics is wrong. Okay, nobody believed him. But then he got a very good experimentalist in that field. He was a research student. You don't trust research. Then he got a very good experimentalist to help him called Pipkin. Then Clauser, who is an even better experimentalist, he repeated it using the same source and the same set-up, but a different transition at the end of the same difference. That was the point at which, historically, the physics community really started sitting up and taking much more of it. They didn't take much notice of it, I speak curiously. Not immediately, I guess. Aspect made a big name for himself by not doing his experiment for about five or six years. He put a proposal in Phys. Reb, which everybody was talking about. But it took him five or six years to actually complete the experiment. And his experiments were the best, you know, they're sort of at the end of the line, because if you work long enough... Everybody else has got bugs in their experiments, so by the time you come to do your experiment, you've learned from other people, and that's where you're bound to get a better result at.

2:30:00 But Clauser repeated the whole... They started fiddling about, and one of the things that optical... These are gamma rays we were using. These guys are using optical photons. And the reason why they didn't do this was because their paths, the coherence length of one of these photons was about 12 meters. Metres, they're metres. And also they're resolving time, they only use piles of plates, so God knows what they're resolving time. So it was not clear that they were space-like separated, it was not clear that they were weight-packet separated. But Clauser was fiddling around, and what they had was, I said they're optical, they had lenses. You know, remember your practical physics, when you put a lens, you always put it in a clamp? Yes. And he was just loosening the clamp. We measured it and found that the values were violated. In other words, the stress caused by screwing up the lenses, you know, pressure, you make it birefringent. And that's sufficient to destroy. Well, you've got to say on the lines just how fragile the effects were. That's what I wanted to get across to people. That's the sort of thing you've got to watch out for. It doesn't matter if the lens is not stressed, you don't break the coherence. But if you've got any sort of stressing in the lens, then you immediately break the coherence. So if anybody's fiddling about with that, you just break the coherence and that's it. So there's no danger of... It's when someone of the same sits here and sends out all these, all these photons and then does a measurement and correlates them all, I don't know what would happen then. And that's an interesting thing to consider, I mean what would one...

2:32:30 Can I just go back to, right to the beginning of what we're talking about and to go back to classical optics and the effect we've always faced. Can you remind me a little bit about the, in Morris's book, about the Maslow index? Oh no, that's due to the double cover. That's how it connects with the double cover. So I think that might be of interest to Bill to see this. Because here you've got a very geometric structure. It's coming out of a nice piece of... I think it's the same thing. Yeah, I think it is, but he's using slightly different language. It connects with the way that this metaphylactic group enters the... The idea is this, that we've got our classical physics on our face. Is that why we point just how much of quantum theory is already there in the classical? Okay, let's do it. First of all... Essentially all of it. Let's take a... Let's... We're playing with classical physics. Classical physics gives us a symplectomorphism defined by some Hamiltonian, and it gives you the flow. Now one of the interesting features that this is a sort of a side comment of Gromov's theorem. One of the interesting features about Gromov's theorem is the following. Normally when we look at a region in phase space, and it's got a certain volume, then we can use, we argue in the elementary physics book, that if we make a general transformation, we can actually, which is a symplectic transformation. We can smear this thing out as long as the volumes remain constant, and therefore the idea is that you can actually put any distribution through the eye of the needle. That is not so, if you have a transformation which keeps the action, if you take every process as a classical action, and if you have a transformation which leaves the action invariant, and Hamiltonian flow will do that, whether it's a complete symplectomorphism I'm not sure, I'll just have to check it out.

2:35:00 What you find is, what Romoff showed was that there is a topological invariant even in the classical space, which says that you cannot deform a volume p to below a certain volume defined in terms of the action, the classical action. You can take x and y and you can deform that to a smaller volume, so you can thread that through a needle. And this is known as the symplectic camel. The complementary variables, X, P, X, has got a limit on it if you keep the action invariant. Now of course in classical physics you can make that action go to zero and therefore you come back again and put it through the eye of the needle. But you have to change the action. And the reason why you can't change the action in quantum mechanics is because... You have a topological feature which stops you from compressing that to below a certain value. So if you've got a cylinder, you can't always put a general blob. And this bound depends on the particular law of motion? Yeah, but it's the classical laws of motion. But there's a slight restriction, I think, which is to do with keeping the... The only thing I can understand to tell people about it is that there is a... The transformation keeps the action constant, as nearly all our processes have got in classical physics as far as I'm concerned, the way Morris puts it. But the action is just x times p, right? That's right. And therefore, if you keep that constant, I mean, it's deeper than I've given them. It's deeper if you've gone through the theorem step by step myself, which is probably something I should do.

2:37:30 I did try to go through Morris's chapter on that. We've spent a couple of years looking at it. It's not a very clear chapter, but it is unclear that there is this abstraction. There is an abstraction. There is an abstraction. I understand the abstraction is that the action has to be invariant in some way, otherwise you're not going to be sure. The mathematical writings often use the word action to mean various things. I'm using the word action. I'm not sure Gromov used the word action. He uses the word capacity and then defines capacity in a rather abstract way. I'm sorry, Bill, I can't, you know, I can't really... It's a capacity of what? It's a particular... It's a volume. The capacity is obviously a volume of some kind. Yeah, but... And what restricts the volume? The only thing I can think of that restricts the volume is the action. In this one eventual case, the volume is the same thing as the classical action. Yes, but if you've got a complicated, you know, if you've got a many-body system. How's that? I think that's where it probably gets its more generality. Sorry, you didn't ask me to say anything. No, it's okay, I was just interested in... The point is, please, just to put in note, that there is a topological... Looks as if the trace of the uncertainty principle is already in classical physics. That's the thing that really struck me. The Planck's constant is already there. There is something that if there was a Planck's constant there, then you certainly couldn't thread them together. I'm sorry, I expressed it very badly. Because I can certainly, I can make the action zero. You can make it interesting. Then it works very, then I can get it through. The action and the energy are sort of independent variables. Well, you have the action angle variables, don't you? They come into it, and there's certain transformations.

2:40:00 Oh, I'm sorry, I, I, I've heard of this eye-of-the-needle business, but I didn't. I haven't helped you. No, but thanks for reminding me. I always meant to look more closely. Yeah. But my son, by the way, is working on it. In fact, he may be, you know, he's going to be defending his thesis this month at the University of Arizona. To do it? No, no, my older son, John, and part of his idea is quantum mechanical effects and RF oscillators, a couple of systems which satisfy the van der Pol equation, which I saw the van der Pol equation contains the quantization of the action. This is just one of his conclusions. I was just trying to think how one could see that, I mean, because there's... I think the suggestion that it has to treat the energy and the action as dependent, you know, possibly, yeah. Morris's main theme is the following, that... This is this guy who's LeRoy's student, by the way. LeRoy's student, is the following, that because this is a symplectic structure, and as we've already, as I've already indicated, there is a covering space to this structure. Now this covering space can be, it can cover it an infinite number of times.