A Physicist's Perspective on Noncommutative Geometry — Algebraic Approach to Quantum Pre-Geometry

0:00 I can just show that he is not completely practical. In the course of that, I think I've seen a way I've been part of my head to help in the same experience. Have you seen this paper by Kosh Abishar? He's done it. He's just, um, just in the last few weeks, he's done a test of experiment and showed us. I think as much as you can show it at all. He does very definitely. It's about time to push, because it's been a mission. No, no, no, no. There's one office in it. Look under, just go and Google. Oh, good. Good, good, good, good, good. OK, so, may I start? I'm pleased to introduce Buzin Hailey, he's going to talk about non-community geometry in algebraic quantum mechanics, yeah? Yeah, yeah, sorry, that's you. A physicist's perspective. Did I not put the correct title on? I don't know. Yeah, well thank you Bob for inviting me here to explain these ideas that I've been playing around with on a lot for quite some time now. Now, the original work actually started way back in the late 60s when Roger Penrose was at Birkbeck College with us, and we had some seminars and discussions together, and he was developing spin networks at that stage, that's when the spin network idea was born. and I looked at that work that he was doing but came at it more from the Clifford Algebra point of view and what I'd like to do today is just to talk a little bit about those ideas and tell you some of the things that I've been doing just recently on this. Okay so why non-creative geometry? Well I suppose the would be that if you look at quantum mechanics you have a non-commutative structure and what

2:30 physicists would like to do is to find some sort of phase space where they can describe what is actually going on with the particle etc. But of course if you assume that all of it physical variables have got the eigenvalues of operators your non-commutation means that you cannot simultaneously specify x and p and therefore you should not be able to make a phase space. And the great struggle to try and make a phase space of course has a long history and there are these famous no-go theorems with Neumann, Gleason, Cushing and Specker which also that you you've got problems if you try to just talk about the eigenvalues of the operators. so really all we have is we have the non-cognitive algebra we have some kind of probabilities and physicists usually reconcile that by saying well the reason why we can't have a phase space is because of the uncertainty principle that is when we try to measure things we disturb etc etc if we just concentrate on the algebras But then we have a more general approach to quantum mechanics, which is particularly described by Haag in his book on local quantum physics. But what he does there is he puts a net of algebras on space-time. I want to take it a little bit deeper and ask the question, you know, do we have to assume space-time to begin with, or can we actually abstract space-time from the processes themselves? And this is, as I say, going back to Rodgers, Penrose's idea of the spin networks where he was trying to start with something which was more quantum mechanical and abstract the classical space-time out of it. Now, some of you know in the audience there that I have worked a lot on the Bohm interpretation and therefore it might be puzzling because in that interpretation you seem to be able to construct a face space and you have trajectories in that face space and even in John Bell when he saw Bohm's original paper in 1952 actually gasped, I saw the impossible done the question that's always intrigued me is how is it possible that we can have a face space description

5:00 the points that were made in the previous slide. Now what you find, I don't know how many people are familiar with this, but the way you get the trajectories is almost ridiculously simple. That was comments, I think, by Einstein, was Bohm got it too cheap. And all you need to do to see how it works is take the Schrodinger equation, let your wave function be written as Re to the Is, and look at the real part of the Schrodinger equation. and the real part of the Schrodinger equation is at the top. Do I have a pointer or anything? There's a laser in the... In here? How do you switch it on? Now what's happening? I think you may have to put the light on here. Sorry? You need to put that up there. If someone help me out, please. Sorry about that, a typical theoretical physicist. Up there. I think I'll use my fingers to point in the future. What did I do? It was effective. Is it direct or is it...? No, it's on the screen. It's a good one. All I did was press that green. Yeah, that will switch it off. Never mind, Basil We know what they say about a lousy dress rehearsal It's where it says laser Oh, there it was So I'm talking about this equation here which is just the real part of the Schrodinger equation and you'll notice that if in fact you knock that Q out, that is just the classical Hamilton-Chicolby equation.

7:30 So the idea is to say this, well, suppose that Q was actually zero, then we know we have classical physics, and we know we have trajectories. And so if that Q was just a little bit epsilon, slightly greater than zero, you would still have trajectories, but they would be slightly altered. and if you keep letting Q grow then you would still have trajectories because at no point is there a critical value where suddenly you go from trajectories to no trajectories and if you do that then what you can do is you can say alright let's assume that we take over the classical expression for the momentum and then you can solve that equation for the two-slit experiment and then you can actually see the trajectories forming the fringe pattern and the reason why the fringe pattern is formed is because of this glorious quantum potential here which is just working out the value of Q and you can see here that along the plateaus the particles move in straight lines and as soon as they hit one of these dips you actually get a change of potential which gives rise to a force and then flick over. And here's the barrier for those people to see the way. We had a big industry going at this in the 70s where we were applying the quantum potential to all sorts of things. Okay, so then you could say, well, why are you going to non-comitant in your sense if you've got all that beautiful explanation in terms of trajectories in the phase space? Well, it really depends on this. it hinges on what on earth is this quantum potential. And a lot of people have said they don't have anything to do with it, mainly because they don't understand where it comes from and what it is and why it should be there and so on and so forth. Now it's that quantum potential which makes the total difference between classical physics and quantum physics. As you can see it's a very strange looking potential. If you think of the wave function real field if you like. Normally a force depends upon the amplitude. Here we've

10:00 got the amplitude for the potential in the denominator which means that the force doesn't depend upon the amplitude of the wave. Very strange. But that dependence goes partly to explaining this quantum non-locality. The quantum potential contains the quantum non-locality. One way of looking at quantum non-locality is to say if we assume these properties then we've actually explained quantum non-locality and we've also explained quantum teleportation etc. But that leaves this horrible queue, what is it, why is it there? And that really is the motivation for me going in that direction but there's a deeper problem which I've always been interested in, which we're all interested in, is how do we find a quantum gravity? Now here's some of the clues in this non-locality, will it help us to answer some of the problems in quantum gravity? But we know that this quantizing the gravitational field is going to raise deep questions about the nature of space-time. The gravitational tells us about the potential gravitational field, tells us about the metric structures of space and time, we have a quantized gravitational field, the field is going to fluctuate, your space-time is going to fluctuate and the question is what the hell do we mean by fluctuating space-time. Maybe we should start somewhere deeper and see if we can actually understand space-time from some other deeper process. I say concentrate on the algebras and I'm in good company, thank goodness. Hamilton, long before relativity or quantum mechanics. This 1976 is the book was published by these guys, not when Hamilton published it. It goes back to 1860s, I think it is. The algebraic relationships between successive states of some changing thing or thought. He thought algebra, look at the title, algebra of pure time. But even Einstein said perhaps the success of Heisenberg's method points to a purely algebraic description of nature, that is to the elimination of continuous functions from physics, then however we must give up in principle the space-time continuum. And then we have a beautiful neat way of

12:30 putting it it's not day one geometry day two quantum physics but rather day one the quantum principle and day two geometry. In other words we abstract space-time from the quantum principle. But of course the question is what the devil is this quantum principle. Okay let's let's start and try and take this algebra seriously. Now if we start with the algebra, can we say anything about the properties of space-time? I want to take the example of Galfand's construction. The traditional way of doing physics is you take a space-time structure, you take a topological space, you have a metric space, and then you a commutative algebra of functions on that space. But Gelfand shows you can do it the other way round, that given a commutative algebra, then you can abstract the topological properties and metrical properties from this algebra. And in fact the idea of points, they turn out to be two-sided maximal ideals, i.e. where the functions actually cross is essentially a way of defining a point. And that was very much the idea of Roger Penrose with his light rays, taking light rays as the basic thing, finding the points where the light rays actually intersect, and that's where you get the twisters coming when they don't quite intersect. Now the question, can you do this construction for a non-cometive structure? And the answer is no. There are no two-sided maximal ideals and therefore if we take the idea of points being maximal two-sided ideals, then there are no points and therefore there's no underlying space-type. So what can you do? Now, there's a clue from Eddington actually, believe it or not. What would be a point? Eddington was more interested in groups rather than in algebras. And he made the following argument, elements of existence are not hazy metaphysical elements of reality. Rather, existence is represented by an idempotent element of the algebra. It either exists or it doesn't exist. In other words, idempotents, p squared equals p,

15:00 either have eigenvalues 0 or 1, therefore they either exist or they don't exist. So maybe we should regard points as idempotents in the algebra. In other words, a point is not something which stays like that, but it constantly repeats into itself. So there's a sort of process idea lying behind this. static but persistent. Now if we were dealing with a commutative structure, very simple, things either exist or they don't exist. But when you come to a non-commutative structure and you've got your rhythm patterns not commuting with each other, then you can have the existence of A or non-existence of A or you can have the existence of B, non-existence of B. When you've got the existence here, you can't talk about A and vice versa. So you never have A and B together. There's change in our view, if you believe Eddington. Change in our view, what do we mean by existence in this case? If we apply this idea to points, we have either space-time points or momentum points. So we look here at the uncertainty principle not coming from us disturbing things when we're measuring them, but there is something which makes it impossible to have these two aspects together okay so you have this this is sort of an image so what you have underlying the algebra these things called shadow manifolds so you either have the p space shadow manifold or the x space as a shadow manifold so you'll never have one unique manifold underlying it but you have these and David Bowen called these the explicit orders in the structure of implicit orders, but that's a philosophical idea which we need to worry about here. Okay, so what I'm saying is we need a radically new way of talking about space and time. And it's not particles and fields and interaction on space-time, but I want to suggest we should be looking at process. That is, what is, is the process of becoming itself. This is the idea that we're trying to get the mathematics and space-time itself is obstructed from this process. So how do we settle the mathematics? That's the question. Well, I was giving a seminar on this many, many years ago,

17:30 saying I don't know where to start. And my attention was drawn to Grassman, in which he actually, this is the Ausdung der, in the turn of the last century, but right now. Mathematics is about thought, not about material reality. And that really shook me, because I'd been a physicist and I thought mathematics was about material reality. But Grassman, no. And he invented, remember, the Grassman algebra. And he invented it through the methods I'm talking about there, which is quite revealing. mathematics is a relationship of forming thought and not a relationship of the content of thought. Now thoughts for me, the important point is thoughts are not located in space and time so is this the clue that I'm waiting for? If you think about thoughts what happens, how do you, you've got a new thought and an old thought, can you really make a sharp separation between the old thought and the new thought. Grassman's idea was no, the new thought contains the trace of the old thought, and the old thought contains the potentiality of the new thought. So you should not make a cut between these two thoughts, it's just one process. If you make a cut, it's a different process. And he actually wrote his symbols of his algebra in terms of pairs of labels. Now you can think of this, and Clifford also, as we get to Clifford algebras, so this is thought as a movement rather than something static. It's static when you write it down but the idea behind it is that it's a movement. And one way one can look at it is to say well really a light ray is of this nature, because if we were to sit on a photon, then I know there are problems, we can't but let's do this exercise, if we were to sit on a photon, then there would be no time difference between emission and absorption and there would be no distance between the source and the sink So maybe we should try to build up an algebra with a composition like this

20:00 Okay, so I'm now pressing the laser to change this. So what are the rules of this algebra of process? And I developed the following set of rules independently of anybody else, I have to say that, because I discovered later on that essentially the same set of rules, particularly concentrated on the multiplication rule, appeared in Kaufman's iterant algebra. Not quite the same, you say, but if I put that B, and I get B there, the B can come out, so I'm taking a special case where B is plus or minus 1. Now, I also give some philosophical arguments. This is the order of succession, and this is the order of coexistence. Now, what I'm going to show is that you can actually get algebras out of this kind of modification scheme. Luke Kaufman also gets exactly the same results out of his scheme. And also I discovered not so long ago talking to Yanis Raptis that in his causal sets which we were talking about at lunch, he has exactly the same modification rules and the same structure in there. oh this is okay so now how do we get it okay now my claim is that if I just take two basic movements and the basic movements are sort of if you want to think of them as in the plane and perpendicular to the plane but I haven't got an angle in there for a moment I just want to say these are too independent and I can get the quaternions out of my with just those two objects and I can get the Renns transformations out of these two rules and you'll notice that in that there's a plus and minus here and that plus and minus tells me whether I'm a space-like or a time-like object, whether I've got space or time in there and that's something I don't understand I just put it in and it works for some reason like that. Okay so now how do we get the quaternions out of this? Well we take the two objects and then we want to introduce another process, a versor as they called it in the olden days, which is just a, versor just means to turn, it's a turner if you like.

22:30 so I want to go from P1 to P2 I describe it like that and I simply use these rules of whenever I turn them around I put a minor sign in and whenever I see two of these things sitting together I just contract them P0 P2 and just from those simple rules I find I get some two-sided, three two-sided units I get a multiplication table like that and these are isomorphic And that's just the Clifford Algebra H. I do the same in the, for Lorentz, I'm only doing this in two, you can generalise it but I'm not going to spend time generalising it. You can do the Lorentz group in two dimensions, now your versa is changing between P and T and you find you get the Clifford Algebra R1, if you think of this as a metric, my metric is plus one minus one. So I've got my Clifford algebra. Now what you can do is you can really nicely polish this up and this is what Arletta Griffo has done quite beautifully, almost put me to shame with the mathematics, but she's done it very beautifully, where you can get all of this in a very coherent form. You can generalise it, Paraly-Clifford, Dirac-Clifford and the conformal Clifford, and it's the conformal Clifford that contains the twisters. We're talking about twisters at lunchtime, and that's where the twisters come from. Now the conventional way of looking at these Clifford algebras is to look at the spinner structure as such, and then what we're doing in the Lorentz geometry is we're looking at the like-kind structure. The Dirac spinner, all it and the past. And the twister is the thing which enables you to relate one point in your space with one light corollic to another point in space with another light corollic. And the twister is really just a way of specifying the distance between these two points. Wait a minute, it looks as if I've smuggled in with that diagram because it's the conventional diagram. I've smuggled in the putting the RF space down a manifold. And the question is where are the

25:00 impotence, where are the primitive impotence in particular. Now the clue is that those light cones are essentially described by spins, the twisters are described by pairs of spins as Penrose's show. The question is then where are the impotence when the impotence come out when you don't terms of vector spinners or objects of column object with two entries but you think of them in terms of left ideals. If you think of them in terms of left ideals and the way you generate left ideal is take any element of the algebra and multiply by the left on the impotent and you stay in that ideal and your spinners now become left ideals. This is a generalization. What we have done in quantum physics is we take these ideals, we find their equivalent ideals and then we project it onto the Hilbert space. I say let's not do that now, let's keep it open for the moment. But what I want to concentrate on is that we've actually got inimpotence present. And these inimpotence are my points. So how do I look at that if I go oh this is just just to connect up with the more standard way of doing algebraic spinners here's your basis you can generate your left ideals here is your impotent here's your other impotent and here is your left ideal one here is your left ideal two and for people who just want an idea what you do when you go to the spinner structure is just knock that off and Just take that common object. Okay, now the other thing I want to just point out is that if you take the left ideal and the right ideal and multiply them together, you actually get all the properties of the coordinates coming out of them. You're not sticking them in. They are actually coming out of the coefficients in front of these unimpotents. What I'm trying to say here is that the structure that we normally associate with the spacetime is actually encoded in the algebra itself

27:30 Now I just want to very quickly go through this is Kauffman calculus which is using pairs of objects This is my A and this is my B, or T1 and T2. And it's showing that if you use the Bondi-K calculus, which actually goes back to Page way back in 1930-something, you can get relativity out of it. I don't want to go through all this because it's going to take too much time to talk. And what you can do is, from this calculus, you can actually generate the Lorentz transformation in a standard way. My point here is not to go through the details, You can see it in Luke Halpern's book of physics and knots. The point is that this structure of quantification rules that I put up earlier on enables you to get the full ion transformation out. Are you OK? If you want me to go through it, I will go through it. But I'd rather go on and just say, it can be done. If you want to do it, go and look at it in Luke's book. Because it'll probably be correct in this book, but it's okay, it's going to be correct on me. Okay, so now what have we got there? I pointed this out. These are essentially like coordinates, because I've got C equals 1. And I've written them as T times 1 plus X times sigma. And we can think of sigma as a diagonal Poley matrix. In other words, the sigma is equal to 1 minus 1, 0, 0. And then the transformation, the Renn's transformation, was like this. And in fact, I can get all the power spin matrices out of this. What I want to bring out in this slide was the fact that my idempotents are essentially the light rays. They describe my light rays. And the origin of the idempotents is the epsilon. so my light ray as it were grows out of the point so this is really the directional calculus as far as we go here we have not enabled us to relate one light cone to another again it's in the algebra it's not outside the algebra so that what we've got here is we've got the interpotent of epsilon 1 there, the interpotent of epsilon 2 there and I can relate the structure of my space

30:00 sorry, I can abstract the structure of my space from the algebra itself so I don't start with the space-time coordinates and build it up, rather it's already contained in the algebra ok, that's the point I've tried to get across in this part of it but, there's no quantum mechanics yet What I've been talking about is really the orthonormal clifford algebra, if you want to go into it in all its gory detail. But quantum mechanics doesn't come out with the clifford algebra, this is why I think that Penrose stopping with twisters is not correct, it's got to be more general. We have the commutation relation there, we have this picture of either or. and yet, just let me remind you what I'm trying to get to is we've got bone trajectories and we've also got the Wigner distribution that constructs a phase space with a probability distribution in your phase space ok, there's a problem it's only a quasi-probability distribution because the distribution goes negative and one doesn't understand what a negative distribution is I just want you to look at this formula here which is the something that Wigner wrote down in 1932 and I still to this day don't know where he got it from because he doesn't explain it, he just says thanks to a discussion with Szilard we can write down distributions being that, plucked out of thin air so we've got a phase space, how do we get the phase space how does it fit in with what I've been saying the reason why I put Lou Kaufman's analysis through the k-calculus in there was because rather remarkably this is something that Walter Schemp pointed out to me the other point I had was stationary suppose this is an aeroplane moving I'm looking at something moving and I want to find its coordinates with light rays So what I do, well let me send off, rather than just a light ray, let me put a frequency on it. So then it's, I'm doing radar, I'm doing radar.

32:30 Outgoing signal, incoming signal. Time delay, the incoming signal, and that's measured the distance away, that's the way we see where the aircraft are, and the Y in there is just the frequency difference which object, Doppler shift. Then let me write down the autocorrelation function, in other words just make this quantity and integrate over all t prime. Slight change of variables and what I find is that this autocorrelation function is identical to the Wigner distribution and all we're doing here is we're just making this point move so we're looking more at a kinematical geometry rather than a static geometry and I'm getting this Wigner distribution and this Wigner distribution is just the main value of the Heisenberg group so I really should be looking at the Heisenberg group that's what's missing, that's the second part of the story. Now the Heisenberg group emerges from the Heisenberg algebra and it's defined in this way. The point I want just to make as a reference is that the enveloping algebra of the Heisenberg algebra is in fact an object called the symplectic Clifford group. This is very poorly treated in the literature. And the only, the first reference, serious reference I saw of it, this is Cumberon in 1990, where he actually puts the orthogonal structure down and then shows that you also get a symplectic Clifford algebra. And it's now, I'm just concentrating now on the symplectic Clifford algebra, and it comes out of the Heisenberg group. There's also a very the Clifford group, which is a metaplectic structure, which is the double cover of the symplectic group. Just as your spin is the double cover of the orthogonal group, here you have a double cover of the symplectic group. And that's where the analogy between the orthogonal Clifford algebra and symplectic Clifford algebra comes in. It's about double covers. Okay, now, this held me up for a long time. The Heisenberg algebra is new potent.

35:00 And if you've got a nil potent algebra, it does not contain idipotence. And that looks as if my whole scheme is just wrong. And if you look at the way people deal with representations of nil potent algebras, it's a nightmare. I don't know if anybody's familiar with idipotence. It's quite a nightmare. There's a much simpler way of dealing with it, non-ions. What is a non-ion? It's just a generalization of a quaternion. I can write my quaternions in this way with an I in there, sorry with an omega in there, where omega is exponential pi I which is equal to minus one and my quaternions follow from that. The non-ion comes about instead of taking two pi over two, you take two pi over three and you close it cyclically like this and then you get instead of the four quaternions you get nine, I'm sorry there's only eight up there and the one that's losing is the identity. nine objects, that's where the non-europe comes from. And this was discovered by Sylvester in 1884. We know it as a vial finite algebra, CN2, where in general it's the nth root of unity that you're interested in. The interesting thing about this is that when n goes to infinity this object goes to the call it will explain why it's extended in the middle. But it's not just the Heisenberg algebra. The reason why it's extended is because it actually contains idempotents. So I've got a bigger algebra than the Heisenberg algebra and that bigger algebra contains an idempotent. So once I've got that idempotent I can now do what I did with the Sovano group, I can now do with this group, then I can generate finite phase spaces. And here we go.

37:30 Impotents exist in this algebra now, because it's finite. It's not the impotent. And here is a typical impotent which I've pulled out, which is in terms of the basic operators u and v satisfying this with u n equals 1 so it's really a I'm really doing a doughnut tutorial. I'm just looking at a bit of it because I don't want to simplify things. These idempotents are orthogonal and there's a complete set of them and these primitive hidden potents are just the points of our space and we can generate the points in our space by means of this inner automorphism. So I can generate the points of the space. They're in the algebra, they're not outside, they're in the algebra. And this operator T, I mean I know where I'm going, it's just the displacement operator and here's the momentum operator, discrete delta x. even label these and the labeling this is just the left ideal and what we can what we show is that in the continuum limit it just becomes the k. So the algebra contains a matrix. But then equally I can find another set of importants which will generate a new set of points with the generator t prime now related to exponential x. So I've got either the p space which is this, p's can be related like that, you can show that in the limit this does become the momentum and in x the limit that does become the position. But because of this inner automorphism in the algebra you can never display both the position space and the momentum space together you can either do one or the other and this is typical of a non-commutative geometry where we're trying to find the properties of manifolds manifolds don't exist but shallow manifolds do exist now the continuum limit this is taken straight from vial's group theory in quantum mechanics. He actually shows that

40:00 here is your impotence and your US enables you to go from one impotence to the other. The V gives you the eigenvalues if you like. And then if you take the limiting process you get the left ideal becomes the Y function times this factor square root of essentially and then you get trillion representation. So you have in the continual limit, but the important now is here in the algebra and the question is where is it, oh and that's just saying that if you do the Fourier transform, this is like Fourier transform, something interesting there. OK, the question is, how has Bohm got away with it? Well if you look at it in a very simplified way, this P is not the ideal value of the operator B. So all the no-go theorems do not apply to this P. That's the first reason why you can do it. But the interesting thing now that remains is why the quantum potential, where the devil is the quantum potential? And it has to be there, some people have tried to, it has to be there because you need it to conserve energy. This is the total energy and this does not equal that. So it's got to be there. The big question I've asked myself was where is the quantum potential that required rethinking asking what is the operator equivalent of the Schrodinger equation most people would just say Heisenberg that was my first reaction just go to the Heisenberg picture and there we are we've done it but the Heisenberg picture puts the time into the operators but it leaves psi of t of nought sitting there doing nothing it's not taken into the algebra it's still left as an element in Hilbert space notice I haven't said anything about Hilbert space in what I've been doing at all well I've occasionally hinted at it but it's not based on Hilbert space this well one way of looking at it is

42:30 this will carry information about the state of my system using a density operator. But the problem is, where is the phase? Because it's the phase which gives you all the interference properties, therefore the algebra ought to contain the phase in some sense. Now in order to solve that problem, I had to see where this idempotent was in the formalism as we know it already. Remember from the discrete violin algebra, I've got an idempotent sitting there. where is the impotent in that more closer to the normal way we do quantum killings? And it's sitting in Dirac's book. Not his first edition, not his second edition, but his third edition. And he introduces something which he calls the standard kept. And by standard kept, I don't mean that. I mean this without the bar in front. I think probably most people if people have read it would not know why he introduced the standard ket I mean I certainly didn't know most people I've asked say didn't know he introduced it corresponding to the standard ket there is also a standard bra and that putting the standard ket and standard for bar together is just the impotent that we've been looking for in other words Dirac had the impotent in there dealing with essentially an algebraic structure and therefore your density operator can now be written like this and what this means is that we actually have symplectic spinners in there. We have a left ideal and this is the operator equivalent to the wave function and we have a right ideal which is the conjugate to the wave function in the algebra, not in Hilbert's place, in the algebra and therefore I have two algebraic equations in place of the Schrodinger equation and I distinguish the two and normally we say the complex conjugate wave equation is not important but when you split it up into left and right ideals you've got a more general situation so you have to take both of them Could you make it more explicitly clear what is the difference between a sterile scale and a illusion scale?

45:00 It's got operators on this side. In other words, you see, the way Dirac did it was to say, you write it as something like Simon, you've got operators on this side, and then you put the operators inside, and you put the eigenvalues in. He says, why put it in in the first place? Keep it all out in operator form. And then you can see you're making a left ideal. Yeah? Because you've got something you don't want to operate from this side. So you've got a singularity. Whether he knew it was idempotent I don't know. Obviously not because it's much easier if he was to say what he meant. And turning it the other way stops you multiplying this side and you've just got the operators coming that side. So two constructing left and right appeals. Right now what I want to do is just to take those two equations and I want to sum the two Schrodinger equations and I get all this is not operators please this is all in operator space not Hilbert space let me emphasize that time and time again first equation I get which looks very which is just the Livio's equation, the operator form. But there's another equation. Take the difference between the two. Polar decompose the operator, not the ideal, the operator with the idempotent in there, it's singular. So I can't polar decompose. If I take the idempotent out and what's left is not singular, I can then polar decompose it. If I polar decompose it, I get the following equation which is a phase equation where this is a phase operator now and I've got an anti-commutator here as opposed to a commutator there now I have not seen that equation in the literature anywhere I'm claiming it's a new one until someone contradicts me And it's essentially a conservation energy equation, actually, in terms of that. Would you mind if I ask you, could you explain how you got the time dependent Schrodinger equation in the first place? I was merely saying, I've got these two ideals and I want to know how these ideals unfold in time,

47:30 and I'm going to assume they satisfy the Schrödinger equation. So I'm really making an assumption there. In the same way as you make an assumption using the Schrödinger equation in Hilverspace forward. I'm just saying now, when I'm in the algebra, I'm still looking for the algebraic equivalent of the Schrödinger equation. And I'm taking the simplest look at it. And then I'm just arguing that by doing that, I get the movie of the equation, so at least I know I'm not wrong. or not very long, because otherwise I wouldn't have to move in the equation. In fact I know I'm right. But I have this other equation. And the reason why I'm doing this is because remember where is the quantum potential in the Bohm interpretation or in this algebraic approach? Where is the quantum potential in the algebraic approach? We still haven't seen it yet. now because this is an equation which I've never seen before and quantum mechanics has been around 100 years give or take surely they're either something wrong in what I've done or else it exists somewhere else couldn't find it anywhere else but David Farley emailed me as soon as I put on the net. He said, I think I've seen something like that in the Vigno-Moyal approach. Farley. Farley. David Farley from Durham. So that immediately sent me scurrying to Vigno-Moyal. And Vigno-Moyal is based on a deformed Poisson algebra from a mathematical point of view. And there is a thing called a Moyal product where the commutator is so all of this is done in phase space functions and faceless. I've got a, the commutator is replaced by this Moyer bracket which is the sine bracket and the beauty of this is that as you go to the order of the Moyer bracket to order of h bar it's just Poisson, classical Poisson bracket. In other words this is the way to get classical physics out of quantum mechanics through this star algebra.

50:00 there is also the Baker bracket and that's what David Farley said to me what about the Baker bracket now the Baker bracket is equivalent to the anti-computator and this is really a Jordan product it's not an anti-computator mathematicians would call it a Jordan product and this product when you go to order H is just the straightforward product so now you do the same thing other equations with in this game and you get the Poisson, the Alluvian equation and you also get this vastly looking thing. The amazing thing about that, see it's got the Baker bracket in it. This one's the equation with the Moyle bracket in it but you've also got an equation for the Baker bracket. And you look at that, and when you go to the limit, the first order of an H bar, it's just the classical Hamilton-Gokowi equation. Which is why I say when we go to this equation, this equation here, this is just the generalization of the Hamilton-Jacobian equation, the quantum equivalent of the Hamilton-Jacobian equation. There should be a quantum potential in there, so where is it? But in order to get it out, what you have to do is you have to go to a representation of the Hilbert space, and then there's a theorem, I put this just to impress my mathematical colleagues, that there is a projection which takes you from the left ideal back to a Hilbert space. and so what you're doing is if you take the projection operator to be XX in the standard sense what you get is this equation here where this is the sandwich between X and X take the harmonic oscillator just to illustrate what happens this equation becomes the conservation of probability this equation contains the quantum potential so the quantum potential only arises when we project from the algebra into an underlying space. So, the space can be anything. The space is in. Yeah, you can go from momentum, you can go from momentum.

52:30 So you can have any projection you like, and you can go anywhere between, you can have combinations of p and h, some fractional Fourier transform, perhaps. it would work. I had a guy who's got a thesis on, that's beautiful. Okay so you also, this is what Shelley Goldstein got very cross about, because he didn't think it was possible to do it in p-space, because the way he calls this the guiding equation in the x representation, and I don't know what it is, nothing like a guiding equation there. It just tells you what the x is that you use in the representation of your phase space. So what do we do? What we've got is that the non-commutative geometry, which is the algebraic structure of quantum mechanics, which contains these essentially, when you project that, what you see is the Bohm interpretation. So the quantum potential here appear because you're projecting into a space which is not the right space for it, if I could put it that way, It's a bit like general relativity where you take a curved space and you project onto Euclidean plate and the things become curved. So it's in the power force of your life. So I mean there's a family of these shadow spaces? Yes, yes. What do they sort of parameterise that? Well, whichever representation, it's equivalent to taking a representation. I haven't solved any problems here, I'm just showing where this fits in with this idea of countering representation or something like that. Can you say anything about what sort of variety of such things there are? That is, sorry, that's something, work under progress. That's something I'm working on at the moment. I said it was only a couple of years ago when I actually got all this stitch, stitched up and feel confident about it. Obviously we have to go on from here that's what I've been trying to do I wonder if I should oh just to to sort of back up the same sort of structure if you think about geometric quantization which is another that's the first place where you introduce symplectic spins and they're introduced as half forms okay I've introduced them algebraically

55:00 And those spinners live in the covering space, in this metaplectic space that I talked about earlier on when I was talking about a symplectic cliff analogy group. And they are just projections from this metaplectic group down into space-time. And you can see the Hamilton equations live down here, whereas Schrödinger's equation lives up here. And Maurice de Gosson, Gosson, got to be careful. French speaking, do you speak French or French? I speak French. Gosson, is that Gosson? Gosson, de Gosson. De Gosson, he's French. It's aristocratic language actually. Okay, now there is a difficulty with that upper structure. and Gunther Stenberg actually called it Ham 2N and my claim is, although I wouldn't put any of my money on it is that the non-computative structure that I've been talking about actually lives in Ham and then underlying that are these various shallow manifolds so that each quantum potential is actually specified by the particular representation we're using OK, well that's just about it, I'll come to the end now summarize. My aim was not to start with a given space type structure or a given space of any kind. Now in order to start I had to go away from polyclips and fields and look at a purely algebraic point of view. There are many people known led by Haag and others particularly in Germany who feel that the proper approach to quantum mechanics is purely algebraic. Forget Hilbert's space. It's there, but it's the algebra that is more important, it's more general, it enables you to not only do quantum mechanics, but you can also bring thermal properties in, you can get phase transitions and so on. It really is the place where you should be teaching, actually, rather than the standard approaches. Then I explored the properties. Most of are in orthogonal and symplectic cliffid algebra. So that's the mathematical overarching one. Then my claim is that we can regard the idempotence in those structures as the points of our geometry

57:30 and I'm hoping that we can actually now, talking with Lou Kaupin about this, that we can actually talk about curved spaces as well as flat spaces. I haven't said anything about them. That's really where the non-community of geometry comes in. But that's work under progress. Okay, then I've shown that the Bohm interpretation fits very nicely, I mean it's not something arbitrary that's pulled out of a hat or anything, it's a very natural result of this projection from this non-connective structure. And then I'm speculating at the end here, and again this is ongoing work, the generalisation, you see what I'm doing is, if I'm looking at these spaces I'm actually looking at how are related to each other and Jones towers are actually in impotence related to each other that's where the Brady comes in and therefore you're looking at say more and more particles being involved in your space till you go to the infinite field and then with the discussions again sorry it's it's it's something in progress that maybe global topological properties such as the which is what Lou has some ideas about. What he talked about last week to me was that you can think of a knot as some kind of non-local feature because no matter where you cut it, it'll always fall to pieces. So you can't say a knot is located in one position of space to make. And in fact wave function, when you look at it, I haven't done it in the algebra yet, something I wanted to do before I came here when I got sidetracked. So the algebra is actually containing global topological features and that is where there might be a connection with a lot of hearing about it. Okay, thanks, Josh. Yes, are there any questions? Samson? So, I mean, how many choices are there for the Clifford algebras that one could begin with? I mean, what constraints are there on the Clifford algebras in order to, so to speak, have solutions of the equations for the dynamics? Well you've got, there is, if you look at the Clifford Algebras per se, then the only ones that appear in physics, let me start that way, are the ones I had on there.

1:00:00 But there is a cyclicity of order 8 in Clifford Algebra, so as you go to bigger dimensions, you just essentially repeat what you've got before. Okay, so there's this, I forget what the name of the guy is, who has done this box symmetry or something. I'm sorry, I just can't remember at the moment. I think so, yeah, that's it. That sounds as if it's, but I'm... So in fact, there's a rather limited... It's a very limited amount, and the largest group is the conformal group, and that's all equations of physics with zero S mass are invariant of the conformal group. And then that's it. Now the dynamics comes in when you start wedding the orthogonal with the symplectic. Now at the moment the Dirac equation just brings them in as a direct product because you have the d by the x and then you have the gamma mu. d by the x mu and the gamma mu. So it's a direct product so you're looking at the dynamics in that sense. So the Dirac equation is pinning together the symplectic structure with the orthogonal structure. But there's some work that has been done by Walter Schemp and Ernst Bintz and Sonja Polis which shows that the symplectic structure is actually carried in the flag of the spinner and that is revealing some very interesting structure which I have been looking at just recently. So you've got a very different way of combining the symplectic and the orthogonal to produce something different from the Dirac theory. Issues of convergence and so on don't really, infinities and all the rest of it, don't really appear here, is that right? Because everything is done algebraically. done algebraically. Now whether you've hidden them somewhere of these problems or not, because you've still got to project on the Hilbert space to get physics out of it, but at least you've got another way of looking at it, and so hopefully it might throw some light on that problem. I'm not saying it will, but I just feel. And also I've essentially come from a discrete structure here, because my discrete vial algebra, the limiting process was rather

1:02:30 I'm using, I'm sort of defending myself by saying it's taken from vile, in other words. Although I can't see anything wrong with it, vile did it before, maybe there's nothing wrong with it. There's always a problem when you go to infinity, as you know. And I think that leaves looking out a little bit more carefully. But you see, the sort of thing that Lou and I have been talking about is discrete structures. maybe the continuum is the thing that's at fault this has been a theme that has been around infinities in quantum mechanics come from the continuum and I mean is there a relationship between your discrete structures and the Planck scale or something like that that I don't know, that's something that we've got that's something we've got a lot of there's a possibility there because I had a delta x in my u or my v I forget which one, one of those t's And that's just putting a fundamental length in there. But then with the fundamental length, you've got the problem with relativity. There's still a lot of problems that can be sorted. Harvi? Basil, can I just come back to this business of the origin of the quantum potential theory? Yeah, yeah. I thought you'd be interested in this, Harvi. Well, you have the strange function for the quantum potential that's del squared r over r and so on. And you want to know, why is this sort of a natural component in some theory involving forces? Why does the potential take that form? Of course, just within standard quantum mechanics, you say, well, it's just a consequence of the form of the Schrodinger equation because you just expressed the Schrodinger equation in terms of the r and the s fields, and that form just pops out. Okay. Now, when you say that you're giving a kind of a natural account of the quantum potential in the case of the algebraic approach, the fact that you put the Schrodinger equation or the analog of the time-dependent Schrodinger equation by hand, aren't you really putting in what you want to get out? Because again, that is going to be a very, very non-trivial constraint on the nature of the potential. Yeah, that's very true. But there's something that I haven't told you, which I'm extremely interested in, and that is that Maurice de Gausson claims, and has proved in his book, that you can start with Hamiltonian dynamics on your symplectic space.

1:05:00 and if you lift Hamilton's equations up into this covering space you actually recapture Schrodinger's equation normally we just say Schrodinger's equations pulled out of his hat and that's the way I got around my answer to you because I didn't know whether you wanted me to go into this deeper theory that Morris has been playing around but how do you know for example I mean when you say you impose Hamilton's equations How do you get out the fact, for example, just the simple fact that it's a first-order differential equation in time? I mean, normally, Hamilton's principle leads to second-order equations of motion. What is it? There's something very fundamental being put in here that gives you a first-order equation of motion, it seems to me. I've never thought about that. so I mean I am very happy to find that if I have Hamilton's equations of motion which comes from my classical theory, are you questioning Hamilton's equations from the classical theory? You see basically a classical theory classical mechanics is all about a symplectic group if you want to look at it in group theoretical terms and you're looking at symplectomorphisms in there now I'm assuming that is a valid way describing classical physics. Are you questioning me as to whether that should be where we start? I'm starting there. Okay. Okay to make it clear whether we should have something else down there or not I don't know. But I'm sorry I feel very strongly about the global properties, the group properties rather than the nitty-gritty details. If you've got a symplectic structure and you've got the whole classical mechanics as we know it fitting, well Hamilton's equations of motion classical mechanics in that sense. I mean there are other processes going on to generalize that but just staying there it's a symplectic structure, symplectomorphisms. Now what you can do is mathematically you can take the symplectomorphisms on your phase space and you can lift them up into this come this way. If you just take quadratic Hamiltonians up to quadratic

1:07:30 you can show trivially that Schrödinger's equation lives up here. You can deduce Schrödinger's equation from those simplectomorphisms. What Morris has done is actually say not only for quadratic Hamiltonian with quadratics in it, but every Hamiltonian you can generate in the same way. I would like to see his ideas tested. and in fact I'm meeting him at Mike's next weekend and we're going to have a go at this. Whether there's something radical here that he's put his finger on. So you're deducing the Schrodinger equation. You're not assuming it, like I said, when it was down there. Now does that help you? Or does that answer your question? Well, I mean, if I may be permitted to make one more remark in this connection. this is supposed to be a theory that's fundamental I mean it seems to me that you should be deriving the Hamiltonian nature of classical particle mechanics as a lemma in the classical limit rather than making that a constraint on your fundamental dynamics. I've also assumed the dimensionality of my space around that. There are a number of assumptions I put in there and I will stand up and say yes, that's an assumption I put in. And the reason why I have to put those assumptions in is because I don't know any better. So when you, for example, when you say you can generalize the derivation of the Lorentz group from two dimensions on up, how do you know how many dimensions space has? So again, you're just putting that... I don't know. Thanks. I don't know. That's something that we have to work at. There is an argument based on division algebras, that if you have a structure and you don't want structures to disappear on the transformations, you want a division algebra over the reals. And the Quaternion algebra is the biggest algebra that does that. This is why people thought the Quaternions were something very special, and I've got the Quaternion, you know, this gives me the Quaternion straight away. But it doesn't tell me why it's not any dimensional. And I would love a theory which told me about the dimensionality. This is the first time I've seen the algebraic sort of stuff,

1:10:00 so can I press for a more intuitive understanding of a few points? To follow Holly a little bit, because you lift the Schrodinger equation comes out from the Hamilton-Jacoby, classical Hamilton-Jacoby equation. I look at, without the algebra, just from the classical Hamilton-Jacoby equation, and you've got the extra potential that comes from the Schrodinger equation, the quantum potential. Is there any intuitive reason, any intuitive that gets you from that classical Hamilton-Jacoby equation to that extra force of nature? No, because my feeling is that this is the wrong question to ask. I don't think you can do it that way. That's what all these no-go theorems were about. Kirchner, Specker, von Neumann, and so on. You've got to start somewhere else. And the somewhere else is the thing which depends upon process. Now, I know process physics is not well developed. I mean, I'm making this assumption. and therefore it's within that context that I see the quantum potential emerging Can I make my question specific now to you? What is it that makes it come out to the form of the quantum potential and nothing else? I think possibly Harlewine may have asked the same thing but what is it? Look, you have just had to look all I can tell you is what happens there's the slide comes out of. It's just simply a consequence of that anti-commutator sandwiching it and putting it in very simple terms between the x ket and the x bra. I don't know any more, any intuition more than that. It's just a simple step you do, and out it pops in that form. Now if you can see why it pops out in that form, great. I mean, you can see why it's coming out because you've got this P squared here which when you put it as D by Dx, you're bound to get a second order differential of some kind here. But if you want some sort of cosmic wisdom

1:12:30 be like that. I can't give you the answer. The difficulty I find is I can't see the connection between quantum mechanics and classical mechanics. Something new was put in. Now, how the clever the algebra gets, how this new principle enters in from classical physics, quantum physics follows. This is the part that I have a lot of difficulty with. It would be great if there is a reason and you can turn it back up. I agree with you. What I would like to concentrate on is the defamation algebras. To try and understand how they work because they give you a much clearer transition from the quantum world to the classical world. The trouble with the defamation algebras is that it's not unique. You can have all sorts of defamation algebras. And therefore you've now got the problem define what deformation algebra you start and why should you start with deformation at least with the way I'm coming through the non-computative structural ideas is that you don't have that problem because it seems to me now you might not like the philosophy of process but for me it's the only way I can make sense of this and again it's not crystal clear there are still problems with this but it seems to me that this is the way we should be going and this is the way we should be looking at things rather than do what we've always been trying to do come from classical physics and see how quantum physics arises. If you see this the non-committed one of the most what we should really be as physicists I know don't think this way but what we should really be worrying about it, why is classical physics commutative? If I'm right about that, actions is where non-commutatively hits us in the face every day, through action, because we know we can't walk through the door without opening it first. Now for some reason, and if you look at momentum and position, look at the derivative, and there's all sorts of hairy problems with that limiting you're really questioning that whole process and I'm just trying to suggest another way, I'm not saying that this is going to this is the answer, I'm saying this is another

1:15:00 way to think about things and for me it has something appealing to it I have a question, have I corrected there are actually two different steps you're making first of all you're like taking you're going away from the space time theater in which things take place, but it's just one possible projection. Especially space, let's forget about time, let's say space. Secondly, instead of thinking of another space from which you project on this one, you look at the things acting on that, which is happening in there. So you make two steps. One, you sort of go out of the... This is like something Poincaré already suggested, that we go out of... I don't know which of these... We go out to take physical space as a theater, but it's just a projection like which you see through the curtain of the theater and something like that and then secondly you actually care about the dynamics where things happen and because there are specific processes there adding potums like projectors and quantum mechanics which allow you to focus on the points of space you sort of through this double step you can get back to your original project that's the just a very basic question well not a question just actually how would you shortly then define or say what the process is if you're already living space time and want to define space time through the concept of process well we actually do it this is what we do when we're learning was Bondi who once said that by the time you're seven you have educated yourself into classical physics. If you look at Piaget's analysis I don't know if this will answer your question but David Bowen and I spent some time looking at Piaget's work on the way kids form concepts. And it's a great struggle for the youngster to actually understand what we've taken self-evident when we're in our teams. there it's much more primitive it's the action you actually you know you move about space experience that you fall over and so on until you gradually build up a picture of a geometry which allows you to avoid all these objects that are in your life so you're creating your space out of activity before you realize it before you really become conscious that's like you hit the wall and just by pain and this movement means wall means that's the point then you don't do that

1:17:30 again. So it's a learning process through action. And what are we doing when we find our properties of space? We are actually, we explore our space around us by light rays. And that's the whole point of the K-Calculus. We don't go out there and put metre rules down and measure things like the Euclid, Euclidean used to do. Rather we probe the space. And from our probings we construct what is out there. Notice I'm saying what is out there questions, is it out there or inside? I don't want to go down that road, but we construct the properties. So we're really sort of faced with a pre-geometric structure, as Wheeler said, and we're putting geometry on it. And part of the geometry of quantum mechanics is that it gives rise to the kind of thing that Pan is worrying about, why the quantum potential in that particular form. What I'm suggesting is this is a way we could perhaps explore further why it arises and But it means changing to process dominated thinking and not through things moving in space in time. Otherwise I have another question. Somebody else? I have some trivial questions, but I don't know if you want to extend it. There's some general questions, but I don't know if you want to. I can ask privately afterwards. Well, maybe you can just go upstairs to the lounge and everyone's people. You've got some tea out there, have you? Yeah, yeah, tea or coffee. Sorry, I don't want to be pushy, but I'd like a cup of tea. No, no, but it's actually a better idea to do that. There's sunlight there. Yeah, good, yeah. Was that...?

1:20:00 Um... Yes, thanks very much. Yes, yes. That's why I got the... I think I got the general... Whatever, I won't say to... No, I... I think so. Let's speak. Thank you. Yeah. Thank you. Well, in a word, Harvey... Um... What do you think? There's some less mathematics. and I would put that in a bottom-down that I said the way to do physics. That's very interesting. But, you know... The nice mathematics does certainly have some very interesting conceptualization attached to it, which might not directly pair on how to do physics, but... But I think by putting in a standard formula, you're putting in so much... And then it's not at all surprising to get out that, that board for the, um...