Wigner-Moyal Transformation, Quantum Potential & Quantum Algebraic Structures

0:00 Thank you. Seven hours. Oh, wow. That makes it big. It can make it bigger and smaller. Okay. Well, that's pretty. That looks like it's pretty in focus, doesn't it? Yeah. So now you can... Don't know. That's just a remote control. Oh, don't worry. I'll... Okay. Well, I think it's not far away. I'll just... Put it on. So... Okay, well thanks everyone for coming to the last possible seminar of the year, it's a great place to talk to Basil Heide from Birkbeck who's going to talk about how to break quantum mechanics to form algebra as a focus of the mutation. Thank you, James. I was a little bit torn when I was doing this, because I've been invited to a philosophy-orientated group, and yet some of the things that I've been doing are more in a mathematical line. Now, and yet I feel I want to try and get out what it is I've been doing here, so that philosophical then can begin to try to look at it and see if there's anything in it from that point of view. I'm not going to be very serious with the mathematics, but I just want to paint the picture. And one of the things that I've been exploring just recently, and I know it's been around for a long time, is the idea of deformed algembres. Because I'm particularly interested in long-committed geometry. and of course the quantum phase space as it were is a typical example of something which would be amenable

2:30 to the ideas of quantum geometry and what I'll be concentrating on in this talk is looking at the algebraic structure in phase space, now why algebraic and what do I mean by algebraic, I really mean the matrix mechanics or the work that Howard does in particular structure of the theory is fully characterised by the algebraic relations in a net of observable algebras. In other words, the basic object is the net of algebras as opposed to their representative algebras on a Hilbert space. So I'm rather saying let's not worry about the Hilbert space side of things. Let's look at the algebraic structure and see if we can say something more general about these problems. So the sort of mathematics that these people like Hart do are von Neumann algebras and C-star algebras. You won't need to know any of that just to be aware that in the background these things lurk. My interests have been much more in the orthogonal Clifford algebra. That's essentially Dirac's equation and relativity and so on. But the thing I've been absolutely fascinated with in recent years is the symplectic Clifford algebra. Now, it was a long time, I thought there was this structure lying around somewhere, and I remember talking to a fellow physicist and they'd never heard of it, and I said, well, it's got to be there. Unfortunately, some mathematicians, I think it was Grunov, either a Belgian or a French mathematician, who actually had a brilliant book on symplectic algebras, and from that, I've been able to gain a lot of very interesting insights. But the symplectic side of things often says that if we've got an orthogonal clitoral algebra, we know about spin, because that describes all the spin properties, and one of the things we know about the spin is the covering group. The clipper group is actually a covering group and that's the one we're all very familiar with. What many physicists don't realise is there is a double cover of the symplectic group and this is the metaplectic group. And it's in this metaplectic group that some of these things that I'm going to talk to you about are actually taking place.

5:00 okay, so first of all let's have a quick look at what the advantages, disadvantages of the algebra well the advantage is that it's a very rich structure and you can actually encompass thermodynamics, phase transitions and so on within the algebra and it gives you more degrees of freedom rather than limiting it to Hilbert spaces you've also got the fact that since you've got the field theory you've got these ideas of inequivalent, Hilbert spaces switch an interaction on you and we should go into the Hilbert space. So you don't really want the base things on Hilbert spaces and what I'm saying here is the dynamical symmetries are manifest, that means that the symmetries in the quantum algebras, such as the symplectic symmetry in particular, is much more apparent in the operator form than in the eigen. In fact you don't see it the eigenvalues. So that seems to me to suggest that we should be looking at the algebra seriously from an interpretational point of view. Why? Because there doesn't seem to be any simple physical interpretation of the algebraic approach. Now what I've been developing, and I should perhaps address more on that topic here, but I felt I didn't have enough mathematical structure to actually go into that. But this is an idea that rather than having particles or fields in space-time, what the algebra suggests is that we're dealing with process. And this is more like the Whiteheadian idea of process. And therefore one should be looking philosophically in those terms to see whether you can make a sense of the mathematics in terms of that. I was very struck by somebody who pointed out to me that Hamilton, way back in the 1850s They were very interested in algebra. Hamilton contributed quite a lot to the foundations of algebra. And he says here, in algebra, the relations which we first consider and compare are relations between successive states of some changing thing or thought. and it was rather interesting that he not talking about material substances as such but rather trying to be a little bit more general, that there doesn't seem to be any distinction between a thing and a thought. The numbers are the names or nouns of the algebra

7:30 marks or signs by which one of these successive states may be remembered and distinguished from another and the relations between successive thoughts thus viewed as successive states are more thought. And these are the primary relations of the algebra. In other words, there's the beginnings of the process type of thinking, way, way back there. Okay, so that's the background. And what I particularly want to do here is I want to talk about the algebra as a phase space, and they turn out to be all non-commutative. I want to discuss the Vigna distribution, and I want to see how you can look at that theory as a deformed prosomal algebra as you introduce a new type of product in there which then enables you to look at it algebraically. Next thing I want to show, I don't know how many people are aware of this, but the Bohr model is already in the Vigna-Moyal approach. It is already in Vigna's 1949 paper in the appendix and I'll show exactly how it gets there. And then when we're dealing with the Vigna distribution, we're always dealing with density of states, density etc. And I want to show that the Louvian equation has got to be supplemented. I mean I was under the impression about five years ago that all you needed to do statistical quantum mechanics was the Louvian equation. And it seems that this is wrong and there's another equation that you need. And in fact I found it when I was looking a way of trying to understand where the quantum potential comes from in the bones theory from the point of view. It drops out of a simple Mickey Mouse calculation the way Boehm did it and you know we all know Einstein said he's got it too cheap, but the question is where is it in the general structure of quantum mechanics? Because I've always been claiming the Boehm theory will always give you the same predictions as quantum theory, experimental predictions, and people have always tried to prove that it's not going to work and it's going through something wrong. the reason why I say that is because the mathematics we use is just simply the mathematics of quantum theory he had nothing new it's interesting then to see that if we go to the algebraic approach, you'll see that the quantum potential doesn't appear in the two defining equations, but if you project them from the algebra down onto a representation, suddenly the

10:00 quantum potential appears so it's a projection from a higher structure and we should bear that in mind metaplectic group is a covering space of the underlying symplectic structure. The symplectic structure is where the classical domain lies, but what we'll find is that we've actually, when we've got a non-commutative structure, we don't have a unique underlying phase space, we have shadow manifolds as they're called in commutative geometry. I want to show that the Bohm method and the Viva method are just examples of these shadow manifolds. Okay, so that's the aim Now, just to put it a little bit firmly When we've got classical mechanics, of course We've always got trajectories in phase space And we've got the possible bracket relation there When we've got quantum mechanics We now have to go to the algebraic structure And that algebraic structure, as we know If you want to find the position you have to diabolize the position matrix and then you just find the position of the space like this which you can't say anything about the momentum space and then you can have a P operator diabolize that you cannot diabolize these simultaneously so you cannot have a phase-based structure and yet the linear distribution claims to be a phase-based structure and it's to do with functions of all operators So let's see if we can see where the phase space arises in the Vigman distribution. And all the time in the work I do in this talk I'm only dealing with pure states. Don't get confused and think that if I've got the density matrix there I'm dealing with a mixed state. Leave mixed states alone. Don't confuse the idea. Just think of this as just an ordinary pure state with quantum characters. No mixtures. what we do is a density matrix actually is a two point function in configuration space but what we can do is we can actually make a density matrix in a phase space the way we do that is very simple we simply take the Fourier transform of the wave function write the density matrix with these two terms Fourier transformed and then what we do is we change coordinates we have a mean a difference of position, a mean momentum and a difference of momentum.

12:30 In other words, we're as if we're looking at cells in our phase space. When we do that, we can then, putting that in this equation, we then find that there is a relationship between the density matrix in position and the difference in position space and the density matrix in terms of x and p. So there is a one-to-one correspondence between these two. This is quantum mechanics with no approximations in it at all, and what Wiener and Moyer did was to regard these two, X and P, as the position of the momentum of the particle. Notice they're not from the way we derived it. So when you're doing that, you're making an assumption, and then you shouldn't be surprised if these probability distributions turn out to be negative. because you're doing something which is not much of a matter. Anyway, the idea was because we can now integrate over these districts, treat it now as a classical distribution, and then we can find the integration over dp which is the probability of psi star psi in x-space and integrating over dx gives you the probability in p-space. and then of course we know that you can get mean values if you've got an operator function of x and p you can show that the mean value of that operator can then be calculated in a way that looks as if you're doing classical probability theory but you have to know what a x p is and here's the relationship between x and p So what you have here is a mapping between the operators and functions and the problem is that this can be negative and you can very quickly show it's negative. Zakos has worked out the Wiener function for harmonic oscillator n equals 3 and you can see that in the dip there the distribution actually goes negative. Okay, so now let's have a look and let's see what exactly are X and P. As I said, mathematically we've simply taken a density matrix in configuration space for X space

15:00 and made it into a density matrix in XP space. And so far there are no approximations as long as you don't interpret it in that way. So why can we have x and p simultaneously? Well because if we treat x as the eigenvalue of this operator and p is the eigenvalue of that operator, you can show that cap x cap p is actually zero. the commutator between them is actually zero. And therefore these are legitimate eigenvalues in that structure. And so what we're doing when we're dealing with this, we're actually averaging over cells in phase space. Notice already we're getting a kind of non-locality built into quantum mechanics right from the beginning. It's a kind of two-point function configuration space. OK, now let me formalise this so that we can see what this has got to do with deformed algebras. But we've seen we've got this mapping and therefore what we're doing is we're having a mapping from a vector in a Hilbert space to a function in function space. And we can formally identify A equals sym, where this sym is shorthand for that integral that I had in the previous slide. Now, let's introduce a product, and then we want the product of, the same product of AB to be equal to same A products in B. So we introduce a new product into the algebra, such that the product is non-commutative, but it's associative. Now, of course, hiding behind the Wigner approach, there's always this order ambiguity, and you won't worry about that, Don't worry about that, just leave that aside. Having mentioned it, people will now start worrying about it. People who know this will immediately say I'm worried about it. Okay, now what Moyard did was actually introduce the star product defined in this way, where this was an exponential, where this operator with the arrow that way operated on there, this arrow operated on here, this one then on there, and this one then on there. So that was a kind of two-way operation going on there.

17:30 Okay, then what you can see is at Qualic-Ax we have the commutator whereas in the deformed fossil algebra we have the Maillard bracket and the Maillard bracket is like a commutator only it's this expression, you just take the real part or the imaginary part of this explanation. Now the interesting thing about that deformed algebra let h-bar, here it is, this is the deformation parameter, if you let h-bar be zero, your Moyer bracket goes straight into the Poisson bracket. So the way to get the classical correspondence is from the deformed Wigner algebra rather than trying the way Dirac suggested that there's some kind of analogy between the commutative bracket and the Poisson bracket. Here it is rigorously it becomes this. Tell you about an equation that I was worried about, and this is only half of that, if you like. So there is another term, a Baker Bracket, which, look, it's 1958, which I read it when it came out, sorry. And I didn't know why it was even bothered to introducing. But it's a bigger bracket and it's equivalent to an anticomputator or if you're a mathematician to a children product. And it's this, the equation for this, that introduces some new insights I think into these approximations. Now just to very quickly give you an idea of how can use these star products, you can actually solve a Stardon value, sorry I didn't invent that name, equation which is essentially take a Hamiltonian, use your density function here, and find out what the energy is. It's like an energy item value of a star product. And then what you can find is, go through the formality of this, find the solutions, and in fact these solutions are the solutions that slide earlier on with the well, showing you that you get negative contributions. So that's the way you can use it. Now then, my problem, remember, is that I hadn't seen, nobody has seen what we do with all sorts of things, what to do with this bigger bracket. Okay.

20:00 is it necessary? As I say, I thought this was everything. Once you got this, you didn't need anything else. But John Barker pointed out to me a couple of years ago, it's well known in the trade, that if you look for an eigenvalue of f using this equation here and taking a stationary value of f, and what f is for that, you find that the solutions are not unique. and you can multiply it by an arbitrary constant. But you have to therefore supplement the solutions with the anti-commitation equation. So it seems as if you need both of these to get a completely defined equation for the stationary states. And it looks as if we've got somehow two equations here, a difference of energy equation and a sum of energy equation, which is something maybe one needs two Schrödinger's equations, I'm just putting this up in the air for a moment. Okay, so what we've got there is here's the commutator equation and here's the anti-commutator equation and that's a very nice review article where this is all reviewed if anybody's interested. Okay, now let's say, okay, you've been giving us some mathematics now. Let's just have a look at what's involved physically and come back to the physical interpretations. Now I always get flat because I've published a book to do with the Bowen interpretation as a group of people who think it's just a waste of time, etc, etc, etc. Now, in other words when we don't believe in quantum mechanics and so on, but we believe in quantum mechanics. I believe in quantum mechanics. What I don't understand is what is it telling us about the way things are going on in the real world and not becoming subjective, as we always have to do as bear information about our knowledge, nothing to do with our knowledge. So let me make it absolutely clear, I believe in the uncertainty principle. Make it a confessional. The uncertainty principle comes about it simply because we cannot measure X and P simultaneously. I emphasise, measure. if we cannot measure that

22:30 simultaneously what does it imply? well the first thing it could imply is that the particle has simultaneous values of B and Q but we just can't measure them simultaneously if they do have simultaneous X and P then we get trajectories and when you sort of think about electrons and so on your interference, you know, maybe a lecture, God knows what it is, so you get away with not giving it an extra, but then when you find buckyballs giving you interference, you begin to say, well, maybe, but I'd also, you know, just to notice, there was a paper just not so long ago by Dieter Zay, who absolutely slayed, he told people not to encourage youngsters to do these calculations because the trajectories are near factors. It's very interesting because the other approach is the one that certainly in many many treaties that I've seen is a party who does not have some time position. and this is John Wheeler the image of a quantum phenomenon this is like a two mirror interferometer and you argue you have a sharp tail where the electron actually comes in and your detector goes so the dragon actually bites a bit but god knows what goes on if that's not a phantom I don't know what is So, you know, here you, I'm sorry, I'm just getting a little bit carried away. Here you've got Zay saying these trajectories are phantoms. At the same time, you've got Wheeler, and this is quite acceptable. Remember, Wheeler worked with Bohr, and Wheeler actually said he was describing what was in Bohr's mind. What was Bohr's mind? I don't think he got it right, but I mean, Bohr was much more subtle than that. Okay, now brothers, the Bohm approach, very naive the Bohm approach, very naive, but it says let's take the first possibility that particles do have a position on it. How can you say something about the trajectory?

25:00 And Moe found that if he very trivially wrote the wave function as R into the IS I know people say there are other solutions which can't be written like that, forget them just look at most of the solutions that I know can be written like that Put it into Schrodinger's equation, just separate out real and imaginative from it taught in the undergraduate school. And then what we find is that Schrodinger's equation gives two equations. The real part of the equation is this, and anybody who's done classical physics, they don't teach Hamilton-Jacobey theory anymore, apparently, will recognise that this is just a Hamilton-Jacobey except it's got an extra term in there. what Denny Lifford called the quantum potential. It's something that I've studied quite a lot myself. Merely out of curiosity of what it looks like, what it doesn't, and so on. That's the term which I was looking for, which I will be looking for in a purely algebraic approach to quantum mechanics. Okay, now what Bowen simply said was, look, I know in my classical mechanics, I have these canonical relations the action, in the classical case, let me simply assume that when S is the phase, I have these two equations. And from those two equations you can then calculate all the trajectories. But notice that when you're doing this the bone P is not the eigenvalue of the operator P. That's why he gets away with it. he has another P and the P is this it's the real part of what I think Peter Holland calls a local expectation value so you've thrown away some of the kinetic energy as it were and therefore it's not surprising we have quite a potential to correct it because this equation is simply this equation is simply this is a conservation of energy equation and that's a conservation of energy equation but now we've got a new quality of energy coming in look at it that way and the imaginary part is just the concentration probability ok so if you start with the quantum probability, do the bottom thing you'll end up with the quantum probability

27:30 and then there are these famous trajectories that Christy is very good at producing here's the, well everybody's seen this here's the two set experiment, they'll always say you can't explain the interference by the particle trajectories, well you can because these are potential particle trajectories so a particle will come in somewhere around on one of these and will follow it and end up on the screen. So you produce the interference particle. But then you say well between that slits and that screen there's no force, why the hell are these things? Bending? Calculate the quantum potential when you see that there is a dip in the quantum potential force. And so you see where the quantum potential is playing the role of organising these trajectories to form the switch. You do the same with the barrier. Here's the barrier, here are the particles going through. The only reason why it's dense here is because we were looking at what happens at the bifurcation points. And there you find that some particles don't even reach the barrier and the reason why they don't reach the barrier is because the quantum potential actually Some of them stay in the barrier for some time, and some of them go through. This is for, incident energy is half the potential energy, half the energy barrier. So it shouldn't be going through. Anyway, you couldn't play around with that. Okay, now the question. What do we make of these trajectories? Do quantum particles actually follow these trajectories? I'm sorry, I don't know. And no one can know. Because we can't measure them. But because we can't measure them, my claim is you should not therefore dismiss them as meaningless. So what use of it? Well, certainly we have no a priori predictive power because we can never produce experimentally a particle with a known X and a known B. However, it does have an apostolary interpretive value. Take our classical example of a cat with a half-silver mirror here. If you put the particle in the front of the wave packet, the cat dies.

30:00 If you put it at the back of the wave packet, the cat lives. So you've got an explanation saying, when I see the cat's dead, I know the particle was in the front of the wave packet. but you can't put the party on it in the front of the way you've had it. And that's the difference. Whether you like that kind of interpretation or not, it's open to discussion. Now, I always felt that this is a good way of exploring these narrow devices. Never did it myself because I'm not particularly interested in doing anything useful. I've never done anything useful in my life. But I was very happy to see that John Barker, who's in Glasgow, has actually taken the bone picture seriously. And this is a quantum dot here, and he's looking at what could possibly be going on in terms of the trajectories inside the quantum dot. And enough, and I won't go into the details of this, we said the advantage of the Bohm picture is that it permits the extraction of unambiguous philosophy fields and spatial trajectories for electrons injected into a quantum dot without taking classical theory. So here's a man who actually has spent some time in the real nanotechnological field saying that the Bohm theory is useful, that this is an impurity, you've got three impurities in here, he's looking at what the position of those, how the position of those impurities affect the conductivity, if you like, or the resistance of the electrons. that's my that's the I don't want to play about the Boehm thing I claimed earlier that Boehm was actually sitting in the Moyer approach and this is Moyer's paper Boehm's paper was 1952 so this was actually What Moyal does Remember Moyal is treating that As a classical probability distribution With x and p as a position He then says Let me form a mean value Of the nth power of p

32:30 Defined by the standard governs That he's got That I've shown you very briefly How it arises it's just playing with those density matrices and then what you find is that if you put them for a wave function you've got the two wave functions here if you put a wave function RE to the IS I've written R as root to the density you find that you get P double one is equal to del S and that's just the bone condition for his momentum so this momentum is the same as bone's momentum Now, he's got the transport probability equation, which is not very interesting, but are you sure, BJ, that that is the bone momentum? Okay, well then what he does is he looks at the transport equation for P double bar. And here it is. These are equation numbers in his paper. And then if you take that and just manipulate the symbols a bit, there it is with the quantum potential in the real display. In other words, in equation A45, there is wrapped up in that term the quantum potential. So that shows that the P double bar of Moyle's paper is exactly the same as the P that Bohm uses. Okay. Now then, I want to get on to this business about something that's missing. I've always shown you that there is a mapping between those two objects. I've also shown you for the time independence of these equations, we have those two equations. when you put in time dependence we saw that that was the Wigner-Moyal with the Moyal bracket that classical limit was just the Poisson bracket, I don't want to stress that point, that when you take this deformation algebra and let H become small, you get this, reduce it to the Poisson bracket, and the quantum equation is that which we're all familiar with the question is, what is the time dependence which involves the beta bracket. Farley, David Farley has spent quite a bit of his life actually worrying about this. He calls it the third equation.

35:00 And what he does is actually, I think it's in this paper, suggests that you really need two time parameters in order to get a time-dependent equation. And that in itself is rather interesting, in white, white, white time. I'll show you how to get it without two time parameters but just with one time parameter. Then you've got some classical limit which if you think about it it's mainly that's the classical and then the question is what is that time dependence for the anti-commitator? Never seen an equation like that written down. So that set me thinking. And the thought ran like this. What is the operator equivalent of the Schrodinger equation? Remember the Schrodinger equation is almost done in terms of representation, in terms of Hilbert's face. But what really is the operator equivalent of that? Most people would immediately say, I did, Heisenberg. It's obviously Heisenberg. But all Heisenberg does is take the time into the operators and leaves the wave function, T0, just sticking there, not doing anything. Now, from what I've been doing on slides previously, we must have the density operator in there but the question that I kept bashing my head against the problem where is the phasing, the algebra and then I was reading it's funny how these things suddenly pitch in them I was reading Dirac's classic book yet again fortunately I have the third edition because this is not in the first or the second edition And Dirac introduces an idea, an object which he calls the standard ket And I don't mean that which we all use and are very familiar with Dirac writes it, operator, and a pick with no bar in front of it And he also has a standard bra, which is turning this way.

37:30 What on earth is he trying to do? If you read him carefully, you'll see what he's trying to do is do a completely algebraic quantum mechanics. A completely algebraic quantum mechanics. and if you don't put the bars down here and still treat these as operators, this thing here is just an impotent. So the fact thing, that symbol is actually just an impotent in the algebra. And what stopped me for a long time was that the Heisenberg algebra is nilpotent and it doesn't have impotents because it's nilpotent. But if you extend the which is essentially the same as introducing a delta function you find that there is an impotent in there and therefore you can play this game. And then what one's looking at is that these objects here are essentially symplectic spinners I don't know how many people have heard of symplectic spinners, they've heard of ordinary spinners which are orthogonal spinners But these are symplectic spinners which are the objects that double cover the metropathic group. What I did was to say, alright, let's look at these spinners as algebraic elements and then they became elements of the left ideal in the algebra and the standard problem would be the operator conjugate to the wave function which is just the right idea. And then you have two operator equations, one for the left ideal and one for the right ideal. And that then is similar to the two equations that we have coming through from early Glock. It's very simple what you do. Just add the two trillion equations, as everybody does, this is standard in the text books. And you get this particular Luvian equation here, and that gives you a conservation of probability. Then what you can also do is you can subtract the two equations, and you get this horrible looking thing. But if you polar decompose the left ideal, you in fact find that you've got an equation, which is that equation, these are all the hop origins here, and you've got a time dependence for the anti-commutator. Now has anybody seen that equation? Is it right or is it? I think it's right. So what I've done is I filled in one of these question marks. In other words that question mark is actually that object there.

40:00 so now what I should do is I should have a look here and see if I can find the same thing in fact what I do is to say alright let me take the Vinkum IR let me take two non-stationary state pure equations if I add these no if I subtract these two I get the Newton equation with the IR bracket in it if I add these two I get what looks like a horrible mess I kept going around and uttering anti-derivatives. But in fact, if you clear that mess up you find that you get, when you go to the powers of order of h, you get the anti-commutator as just ds by dt, f, and if you fold the h, it's just a classical Hamilton-Jacobie equation. So in other words, this other equation, the anti-commutator equation, is just in the limit, it's just an algebra equation. And here it is without two times in it. The trouble is you can't find any simple way of expressing that, whereas you can when you're in the algebra. Now then, the way I've dealt with this equation here is to use these things called left ideals and right ideals, and I haven't used Hilbert's space, and now I must come into Hilbert's space, and there is a theorem which you can find in this, I don't know, I don't know, mechanics by Trudy and Robson there's a theorem which says that if you've got an ideal you can actually find a projection to send it into Hilbert's space so let's use a particular projection that form is very simple you take equation 1 and you project it and this is the probability of finding the particle with the probability A which is the mean value of that and here is the phase of the wave function coming in here and now when you say we've got a phase coming And if you choose the X representation, then what you find is that if you work out this, it just gives you the conservation.

42:30 I've chosen the oscillator Hamiltonian here for reasons which will become very apparent in a minute. If you then work out that expectation value or that projection, you find the quantum potential appears. In other words, it's only when you're projecting into the underlying spaces from the algebraic structure that the quant potential appears. Now, one thing people might worry about is that you've got to have an equation with a phase in it and you've got to be very careful. that's that equation there we've got to ask questions about this engaging invariant and so on and to my amazement I found that within a couple of lines you can show that you get the Bohm-Arm effect out of this equation very very simple the important other thing for me was if you take the projection now into P space then you find, this is why the quadratic Hamiltonian is important you get a quantum potential in the P space as well And one of the criticisms of the Bible theory was it loses the symmetry, the x-p symmetry, that's in quite an account. It doesn't. It's there. The reason why you don't see it is because of the naive way Bohm said about doing it. If you'd said about doing it, the p would have got the same reason. But the point is that this comes out of that algebraic structure. So it comes out of essentially a non-cognitive structure. Here is, instead of a guidance condition, P equals S, you get this object here which former old Shelley Goldstein, when he saw it, just spat blood at me and told me I didn't know what I was talking about because I completely destroyed his theory. Very angry with me about this. But what you can do is you can actually find streamlines, trajectories in P-space so that you've got complete symmetry between the two. Okay? Now you might say, well, why have you got face spaces here? One is defined x is the eigenvalue and p is the real value of that. The other one is p is the eigenvalue and x is the real value of that. These face spaces are not eigenvalues of the operators.

45:00 In fact, Melvin Brown, who's done a very nice thesis for me, has shown that there's a whole series of shadow phase spaces that you can actually create here. And the reason that I'm proposing for it is because you're essentially quantum mechanics working in a non-committed geometry and not in a phase trace. And all our attempts to put quantum mechanics in the phase space is going to fail. that all these theorems and arguments that we've been having we will to a moment at the beginning but the emphasis was not on this geometry that has now become so popular ok now just to begin to round up I think using this action up shadow-faced faces. If you go to, and I've been using symplectic spinners, and where else do they appear in the discussion of quantum mechanics? Well, they actually appear in geometric quantization, and for many years there was this thing here, which is called a half form, which people were calling a symplectic spinner, and I just couldn't understand why the two were connected, compared with the way I was coming to this field. But in fact, what you find is that if you look at this structure, if you look at the symplectic cliff of algebra, you have the symplectic group, and this is where Hamilton's equations live down here. But if you project, if you lift the symplectic transformation up into the metapocalyptic group, then you can actually prove Schrodinger's equation rigorously. Schrodinger's equation lives in here. Remember Schrodinger, where he got his equation from? If you read his original paper, he does something fanciful, and he's got some kind of very confusing integral over actions. And he's got a little star against it. At the footnote of the paper, he says, I am aware that this step is not unambiguous is not

47:30 unambiguous in other words he's done something which he can't justify at all but if you take the menoplectic structure then you can show rigorously that you can lift Hamilton's equations of motion and you get this one problem it only works for quadratic Hamiltonians, linear and quadratic, it doesn't work for higher Hamiltonians, and it's this metaphlectic group that works for quadratic Hamiltonians, and I think people who were in optics know this stuff. Now there is a more general structure which you'll find in Goleman and Sternberg, and the metapleptic group is actually a subgroup of a thing called ham, which is the general ham-tony group. Unfortunately, the mathematical properties of this general ham are not well defined. Goleman says we've got lots of work to do with this thing, so it's not, as far as I know, it's not been solved, and I know a man who knows a lot about this, Maurice de Grusin, who has helped me a lot with the work I've been doing here. He claims, I'm still not 100% clear that his proof is correct, but he claims that with a proper lifting technique, you can show that the Schrodinger equation actually lives in hand. So that you can actually deduce Schrodinger's equation from the Schrodinger as he calls them. Where here now, instead of using the Hamiltonian to define movements, you actually use what he calls the Bohmian, which is the Hamiltonian plus the quantum potential. And the whole thing fits mathematically very neatly into each other. If anybody's interested in a good book, I'd recommend that one by Bruce. It's a fairly recent book. But But you have your shadow matter fault there, okay. So, let's, this is all I'm gonna say as far as the mathematics is concerned. I'll just a few further remarks to work on. I'm essentially saying that I don't believe quantum mechanics lives in space or in phase space. It lives in some deeper structure than it is.

50:00 And its essence is it's non-communativity. And all our attempts to understand quantum mechanics Has always been to force it Into our classical notions of manifolds Face spaces, space time If you take the non-conjunctive algebra as basic Then the philosophical suggestion here is That we're really talking about processes And how do we actually discuss this This could be a whole other hour possibility now that may be vital, etc. This is what we should be thinking in terms of the process. And look at this beautiful quote I found from Einstein. Perhaps the success of Heisenberg's method points to a purely algebraic description of nature, that is, to the elimination of continuous functions of physics, then, however, we must give up, in principle, the space-time continuum. you read that card and then you've got this beautiful poet John Wheeler who was criticising for a smoky dragon and now I'm going to praise him for a very deep perception here and that is he says this particularly comes about when you're trying to quantise Robert you say not day one geometry that is you're giving the manifold quantum physics downside, which is what we've done so far and why we can't, I believe, quantum algebra. Rather it should be day one the quantum principle, day two geometry. In other words the geometry should emerge from quantum principle, whatever the quantum principle is. I'm trying to suggest that that quantum principle has got something to do with non-cognitivity, to do with not with particles moving in space-time or fields interacting in space-time. Yes, you can get quite away with that, but fundamentally we really should be thinking in terms of processes, and processes from which space and time, phase-space, are actually abstracted. Now, have we got any examples where we can do this kind there's some work that was done in the 40s

52:30 by a great Russian mathematician Gil Planck and he showed that there are two ways of looking at commutative algebras the traditional way was you start with a topological space a metric space and then you construct a commutative algebra of functions on that space but what he also showed was that if you take a given commutative the topological features and the metric properties from the algebra itself. In other words the algebra contains the geometry of your space. At point is essentially the maximal ideals. We'll define a point for you. Notice I was using left ideals because I've seen a more structure but it's a similar ideal. The problem is that this program cannot be carried through for a non-commutative algebra. When you look at a non-commutative algebra, the best you can do, I find a nice discussion of this in this paper, is you have shallow manifolds. So you're forced by the non-commutivity into this notion of shallow manifolds. If you like, you're forced into complementarity in a way. I mean, this is ontological complementarity and not epistemological complementarity. Okay, so let me now completely round that. So what are the conclusions? What I've tried to show you here is that you can construct face spaces, which actually account for quantum processes, but you have to be careful and be clear on what you're actually doing. Not just saying you're giving a particle position momentum and then doing classical position, not doing that anymore. The Wigner method uses cells in phase space. The X and P there are the main values of these cells. It gives the correct expectation values, but you cannot use this as a probability distribution because it becomes negative this can all be captured very beautifully in these deformed cross on altruism which naturally have the classical limit in them or you can take Bolt's method

55:00 in which the X and P either the mean value either a mean value or an actual value remember that was the business around PR and X, XR and P in that slide I had previously. Once again, it gives the correct. Both of these are exact. They produce the exact value. Although someone told me the other day that he'd seen a paper which someone had proved that the Wigner doesn't give the exact value in certain cases. Unfortunately, I haven't been able to follow that. From the way it's done, it seems to me that you always get exact values. You cannot get disagreement with quantum mechanics. unless you introduce some new assumption. So you get the correct expectation values, you've got the quantum potential, do you give it a physical mission? But then we've seen what the quantum potential is, it's not a real potential if you come to it from the projection. Because it only emerges when you project it onto the phase space marathon, in a shadow marathon. So it's a bit like in the gravitational field. curved geometry there is no force as it were, but when you project it onto Euclidean space then you see the thing curved and you see there's a gravitational force present. So it's like that, only different. And I claim it emerges from the four algebraic quantum mechanics and in fact you need to extend Heisenberg algebra to do that and this now suggests, We should look at different ways of trying to interpret the algebra and maybe going along these lines will get us closer to quantum gravity. Maybe. Thank you very much. thank you very much I'm sorry it was rushed but I hope to have a bit of general pictures yeah there was a lot in there any questions what is the definition technical definition of a shadow manifold

57:30 I'm using it there is a technical definition, sorry Mike I'm going to referred you to this framework here. Okay, okay. It's technically defined here. But it's connected with the Gale-Fan construction? Yes, it's a generalisation of the Gale-Fan construction. And I think Michael might also have something on this. Remember we were talking about Mike the other one. But you'll find it, I'm not prepared for the technical talk here, but I think you'll find it in that It's a very nice paper, that, if you want to get an insight into the mathematical structure. I learnt a lot from it. I think Heather's in that paper as well. Does it have anything to do with the role of automorphism groups? That I don't know. I mean, I'm getting it, you know, I have this view of algebraic quantum mechanics where you get these equations which are also coming up in knot theory in Luke Atherman's. And if you use those projection operators, as I suggested, here, this, the x projection operator, then what I haven't shown here is that there actually is the idempotent in exactly the same way as the idempotent in Luke Atherman's work. Yes, in knot theory. In knot theory. statistical average. It's that is the projection. I mean, here is the actual projection operator on Newton. So, in a sense, that is defined. I mean, I don't know what more you want. theorem is essentially the theorem. So, that's another way if you really want a technical discussion, then you ought to go with Battaglia and Robinson and see the proof of that theorem. Because that's the theorem that we're using. I'm sorry, what I've done is something very different, obviously. Is it true I'm doing a very different approach to these questions? To normal... To normal interpretations of chronomechanics. I guess, yeah. But, I mean, recognisably in the same arena. Yes, yeah. So, at the end, you're suggesting that non-communativity of algebras is really,

1:00:00 non-communativity of algebras, the non-communativity of operators that represent, and what are the operators representing? I'm talking wrong with them to represent measurements. They're usually represented. Yeah, so they're processes. These represent processes. now um so now we have the the instead of saying that you can't simultaneously i mean instead of saying the two operators don't commute what we say is that the two processes can't coexist or something that's what the order of the unfolding of the processes is vital yeah so it's more activity it's more the process is more like activity we're puzzled by non-comitivity in classical things because Because objects don't, there's no such thing as non-commutivity with objects, really, in P&Q anyway. But non-commutivity is all around us in activity, because I cannot walk through that door before opening it. So non-commutivity is natural in an activity-based. One idea that was inspired by Grassman actually was, and this is Grassman in his Grassman algebra that we normally, he actually said when he was developing his Grassman algebra was to say, mathematics is not about material process, it's about thought. that's why I was in when I were to remark that thing and talk while I was laboring usually thick ones on the book the thick ones of the overhead projector should be some thick ones oh there's some thick ones that one and what Graston said was the following he said let's think about thought, thought is really about process I mean I've never read anyone Let's think about two thoughts. I want to use the idea of thoughts because I want something abstract rather than particles. Do we actually say that these two thoughts are independent of each other? Or do we say that the old thought contains the potentiality for the new thought and the new thought contains the trace of the old thought?

1:02:30 and he said then what I'm interested in is actually putting a bracket round there and considering a process if you like, which actually is some kind of action with two endpoints and what's in between is important because if you subdivide that you get another process and what he was able to do by putting strings of these together is to produce a grasp of algebra But we have lost, from those graphs, you know, the DX and DX hub, DY, etc. But we have lost that activity that he put in at the beginning and just think of them as rigid structures. So, what do you think of this, what of the original bone motivational for interpretation is left? The ontology. the aim you see when you look at it when Dave was around in the 50s Bohr was thought to provide an epistemological argument for quantum mechanics and Heisenberg with information knowledge and so on there's a great deal of people using epistemological accounts of quantum mechanics I've never been happy because when I'm not there it's still going on and therefore we have always looked for an ontology of some the 52 was a very primitive ontology you've got a close relation with classical mechanics and therefore you could call it Bohmian mechanics trying to keep that structure without becoming too outrageous as I've become here but Bohm was well aware he was never happy with his work with that Bohmian mechanics in fact when I said Locke Shelley Goldstein's calling it Bohmian mechanics he said why on earth is he doing it that hasn't he read anything I've written because he wasn't thinking about it in terms of mechanics he was going much into a much deeper philosophy which was to do with the implicate explicate order and these things fit very nicely with or you think of the algebra as being a description of the implicate order

1:05:00 you can't make it manifest, it's implicit the structure is implicit in the non-community structure but the way you actually make orders which are classical like is through these projections and therefore you project out the explicit orders so when you said early on in your talk you said you know we've got the uncertainty relation And then you emphasised that you can't measure position and momentum at the same time. But on the original Bohm theory, the thought was, look, the particles do have position and momentum at the same time. But have you... In the early days, Bohm hoped that he would find some way that they would not be hidden, if you like, as I've seen the Spaniard talk about. as time went on he dropped that position and he said no this is not going to work that way you know we've explored too much of the structure and it's not working and he had reasons for it not working when you look at what the quantum potential does and so forth and you soon realise you're not going to be able to do it and then he said well ok well this is just a provisional ontology but I've shown can exist therefore let's try more refined ontologies, more sophisticated ontologies and my hope is that the algebra is going to provide an ontology where these things are actual physical processes and you can well I've played a about with that kind of thing and I've actually shown that you can get orthogonal flippant algebras out of it and I'll let it over to my research students thesis coming out, showing how you can do that. So that you've got, and Clifford was very much in favour of activity, all these things, rotors and motors, they were not objects moving in space and time, but some sort of activity. So is there, so, in the position that you ended up with, Is there a kind of approach more between you and a more orthodox approach more, kind of coming together? Well, someone said to me in Germany, you're just doing ordinary quantum mechanics.

1:07:30 Yeah. But I claim we were never doing anything else. Now, when the David was, you remember I only came to work with David Rowe in 62. Yeah. Now, certainly there we were never trying to maintain, for the first ten years I worked with David Bowen, we didn't discuss his invariable paper. We were much more onto this process, onto the, we were with Penrose, the spinners, the twisters, looking at them algebraically as some kind of philosophical activity, you know, philosophical terms, some kind of process, some kind of activity going on. now what I think what I've done here I think unfortunately it's not here to pass an opinion on it but I think I've actually moved that program a lot further forward I don't know whether it's any good or not it's not a matter but I think it's moving in a direction which will enable us to discuss these things in that way that's why I tend to be a little bit ambivalent about other activities that are still trying to explain it as a mechanics or you know like that valetina we were talking about earlier I think we've got to be much more radical. Now you say repulsion, I've never felt we were anything else for the repulsion in any sense. For me the staggering thing was why does this work? Because I was bored up by saying look you can't talk about Traderone. I was a physicist started working with David Bohm, didn't talk about this 52 paper because I thought it was a load of crap because I picked up what everybody was telling me to say and then one of my students came up to me and said look why don't you talk about this old crap and he said have you read the paper and then started asking some rather penetrating questions and it was quite clear that I haven't read the paper even. And it was when I went away and read the paper and I suddenly thought, my God, you can do these things. And then with Chris Dudley and Chris Lippides, we started doing the things and showing they existed. And I thought, when I first got the trajectories out, this was 70, 70-something, I thought, now people are going to have to, you know, debate it.

1:10:00 but when I talked about it there were people, very distinguished people in the world, they sort of fidgeted a little bit and they said you can't do that I said you can't do that suddenly and I said well let me explain it's very simple, what we've done and he said oh that's trivial so I don't quite know how you can't do something suddenly becomes quite trivial I mean I think it is trivial I agree with John, it's telling you something that may be useful in trying to deal with these nanotechnological devices when you've got one or two particles present. It might give you new insights into how to design things. I don't know, John certainly says it helps him. So to dismiss it I think is wrong. To say this is the way it works is wrong. because, you know, I think the way it works is a lot more subtle than something I've carried on. It's a lot more subtle than... I was struck when you were talking about this this idea of process and so on. Do Lovere's ideas come into any of this? I mean, he's obsessed by this, you know, by variation and process. papers. I had a research student, Gerard Keturian, a very remarkable man. Gerard Keturian, yes. You know him. I've met him. Okay, you've met him. And it was in the days of plenty. I'm sorry, students, there were days of plenty. I'm sorry you don't have them there. And he came to me and said, look, I've got some money to do a PhD for three years. I want you to teach me quadrumechanics. And he wanted me to do apply category theory. That's right. He wanted me to use category theory in quantum mechanics. And I said, hey, look, man, I've got no idea what category theory is. And he said, well, I've got no idea what quantum mechanics is. So I said, all right, let's do a thing. I'll teach you quantum mechanics, you teach me category theory. Unfortunately, we failed completely. No, there were some very interesting, I mean, one of the reasons why I was fascinated by what he was doing was this Bill's, at least I thought it was Bill's idea, until I had

1:12:30 dinner with him a month or two ago, and that was that you do take the morphism as the basic thing which is the process, not the sex. You abstract the sex from the morphism. But when I talked to him, maybe I had too much to drink or something, but it seemed to me he was saying the opposite, that there is a space-time manifold out there, and we have to understand its properties. Whereas the Calgary theory idea was that you construct the space-time out of the metaphors. And I thought we'd get on board together, you see, but somehow we... All you would have to do to get Bill to buy you drinks is to say that you thought Grassman was a great man. Really? Well, I don't think he's a great man. I mean, when I read his ideas on the algebras, I was absolutely staggered. But in some ways, those ideas actually appear from the 20th century standpoint extraordinarily naive, and as do Hamilton's remarks, they really seem to reflect this kind of cantilever. But I feel we need a bit of novelty re-injected into the discussion. But how do you make talk of structure in the setting of algebras rigorous in the first place? I'm not by waving remarks about thoughts and processes. As you know, I'm not going down that philosophical direction because I'm trying to look at the mathematics and get the mathematics to tell me what's going on, which is why I did the lecture this way rather than the philosophical, I mean, I can do that, but I'm not convinced by it. whereas this seems to be showing one reason why you should be looking at the algebraic structures and relating them to the process I could also be motivated just to look at the algebraic structures just as it were a pure exercise in a deeper level of structural understanding of the structure that one has in the phase space which is itself an interesting exercise in fact that one would cover so much from a Hamiltonian approach that ties up deeply with these ideas in classical optics That's right, that's right, that's right. There's a really more system. The more purely structural approach to the theory is interesting enough in itself without perhaps having to invest a lot of speculative...

1:15:00 I like speculation. Well, at any rate, without having to buy into any particular ontology. Yes, I know, but I hope that what we've left for people to realise is that we need ontologies explored I think that's the position that David and I had Sure, but the task of the explorer of ontological alternatives the task of systematic ontologists and the task of algebraists are two, I'm going to say two very different tasks I always like to keep them going side by side as you know for various because I'll have a nice petition. Has anyone else got any questions? Okay, well, thanks very much, Basil, for a really rich, interesting talk. That's a really nice way for us to end up. I mean, it's so nice. With a nice headache. That's really great. Well, thanks to all visitors from far away, George and Michael, and I look forward to you all hearing about the new programme of Boston Physics Seminar's next academic year, coming along to some of those.