Bohm Interpretation & Symplectic Flows / Elements of a Realistic Quantum Theory

0:00 So, this morning I'm going to talk about the craft, as I said, and I'm going to... So, this is pretty standard stuff, and this is really going back to the early Bible interpretation, and it's taking a look at it. But it's also, there was a question that was raised in 1953, but to answer that question, it took me on a journey through a subjective group, So that's how we make a connection with Basil's community that we're talking about. So, I'll take you through this. I think it kind of reinforces something about what Basil said. I also think we've put... Basil spoke quite counter critically. You'll see, what I'll do is give some representations of what he's talking about. You'll actually see some examples of this meta-peptic group in action. and also it's infected with the action as well. So it will allow us, allow me to answer some questions and propose way back. The motivation for this was that Boehm obviously produced this causal interpretation in 1952 and the idea was that not only do you have wave function but there was also a particle of definite position. Of course, we'll call this BI, bone interpretation, and in fact, it's expressed in coordinate space space-time domain, and it reproduces the predictions of all the Dox-Plan mechanics and it uses a formalism, and you'll see this from Jacobic formalism, and it's the connection without humanism and classical mechanics, which gives rise to the notion of their quantum potential. But in 1953, Epstein, in a letter, Fizz wrote a letter, suggested that references in this, of course, would have triggered people's minds off about canonical transformations and classical mechanics, and it wasn't long very long before someone wrote a letter to say, well, why can't we express this interpretation in under-representation? So how about it in under-representation? And it's a very short letter, and he answered it.

2:30 He said that he didn't think it was possible to express the court's interpretation in under-representation. And he didn't give very clear indication of what he meant by that, what were the constraints that had not been possible. he did however point out that he mentioned the canonical transformations and he implicitly referred to the canonical transformations of the operators but he said there wasn't a corresponding transformation of the wave function that was known at that time so such that when you canonically transform the operators you have a corresponding transformation of the wave function and it wasn't known what that group of transformations was. But the bottom line was that he had quite strong reservations that he could be done at all. And it's an interesting question because it's hardly been addressed since, although some people have actually cast the thing of momentum representation. There are one or two recent papers in the 90s that refer to the momentum representation as a possibility. and I'll come to it and some people are already thinking well that's just a Fourier transform I'll come to that in a minute but this question is quite an interesting one because trying to answer it has I think some ontological implications because you have the notion of a particle as well as a wave and a particle carries with it a whole bunch of baggage out of classical mechanics and the question is what do you mean by a particle particle to have for it to qualify as a particle. And when you do these sorts of transformations to other representations, is it possible to still retain the concept of this being a particle if you do these transformations? And so that's the question that I got interested in and tried to address. The second motivation was that that might have not been very interesting if there hadn't been a little, not so much work on the canonical transformations of the formulation. There'd been a lot of work on canonical transformations of being a distribution and phase-based representations of quantum mechanics. Plenty going on there. But very little going on in this area.

5:00 And basically because the mathematics have not come on the road to an end of the time. But it's related to what David hinted at, you know, there isn't a machine there to tackle this, what's happening to the wave function when you do these canonical transformations, what transformations are going on for example. So the basic thing was to try and understand and apply these transformations and then sort of attempt a reply to this question here, this actually I'm going to do. Along the way, we were talking yesterday about the impact of the baggage you carry with us with respect to interpretations and the so-called metaphysics that goes along with mathematics. And there is, classical mechanics is having a strong influence over the appearance of what is the quantum potential in this. and you can along the journey I've actually extended showed an extension, not making claims for it, but I've shown an extended Bohr interpretation where you extend it to include complex and momentum and complex energies it's not kind of related directly to this episode on question but it arose in going through to try and tackle the problem I'll show you an example of that and I'll show you parallels of play between classical mechanics and the entire conceptions. And then I'll just say something about the canonical transformation, the symplectic group, just to get the technology, and as it says on it, and then I'll anticipate part two. This is where, in the second part, this is where I sort of build up how to tackle this issue of casting this thing into different representations. And the first thing is that since the symplectic group is acting on the operators I'd better try to cast the whole thing into an algebra not get bogged down in representations so the first thing is we transform the whole Schrodinger equation into algebraic form and we also try to get out some current operators and the motivation for that is that

7:30 When David, when the wrong interpretation came out, there was this notion of trajectory, which had a guidance condition, which p grew at s, and immediately the people saw the class mechanics, right, p divided by m was velocity. Okay, let's integrate that. and that's quite, it's very seductive but it turns out that that won't hold if you go down the representations but what's con currently holding in the coordinate representation is also that Q dot or velocity is J divided by Rho so it's a code divided by Rho density so you have the motion of trajectories which are carried in the probability density flow What's remarkable about the Bohm interpretations in space-time is that both of these concepts have been carried simultaneously, namely that the two are equal. And that gives us the mechanistic idea of a particle being subjected to a force. And this is where the mechanical picture of a particle comes from. It's the concurrence of those two, which makes it, I think, the essential causal aspect. and so that's the motivation for trying to, well, what is the current in other representations and if I'm going to go algebraic I'm going to have some current operators so we actually create some current operators what I'm going to do is show what is happening in the Hamilton-Jacovi formulation in classical mechanics when you undergo a transformational And when you do that, we've got very, in all sorts of applications, it's in optics, we have a wavefront propagating through a bending system, also happening in classical mechanics where the action is propagating, and when it's propagating, and it propagates syntactically, what you have is the mapping of the transformation from the action of one end to the other end, in fact it can go straight there in a certain genre transformation. Now we're familiar with these, obviously, changing variables, but you can generalize this thing, you can change it to kind of a...

10:00 for a linear symphlectic group. What's remarkable is that that is the generator for a symphlectic group. And similarly, we can look at the transformation is the wave function. As the operators have been synthetically transformed, then the wave function is underground and generalized for a transform. And this, you'll see it, you'll see the parallel. It's good to do it here because you can see these parallels could get lost in subsequent swarms. You can see that the generator for this one, which is the exponential argument of the kernel, And this is this relationship, this meta-flectic covering. And so we're then set up to actually generalize this formulation to all the syntactic equidarian representations or representations which you cannot look for ever about transformations or according to representation. At least, I'm focusing on linear syntactic transformations here. I want to give you some examples that we sort of bring to life, I think, for the transform. I also bring to life the genre transform. I am just saying, well, what is the answer to this question? Can you rethink of this, this article? Or, well, there's special cases of it, where it's true or where it's not. So, let's take a look at the poem interpretation from scratch. are all familiar with it. I was showing an equation here, and the big move that Boehm made was to express the function in a proper form, and separating out the real imaginary parts with two equations. Actually, this is expressed in terms of a range of change of density, but actually what you really end up, before you scale it away, is an energy equation here too, there's a factor you take out which you divide on both sides and these are two energy equations from this graph here. So what you have here, you recognize this as a probability density conservation, I should say, rho here. Here we're working in a representation, working in coordinate representation, what we have is psi, psi star as the

12:30 and what you'll have here is a familiar form this is the Hamilton-Jacoby equation take this potential here take this P grad S as your momentum P squared over 2M very suggested form but what on earth is this and so David suggested that we could say we have a particle here with a momentum, has a momentum associated with it, with P grad S, and moreover, if you take a look at this bit here, that's the conservation law, of course, is the time-recognition density, time-recognition divergence for current, take that apart, rho V, P over M, so we have two things happening here, we have this relationship that's being used here, we see the kinetic relationship would appear from the use of halton mechanics on this expression here. So this is this concurrence of V equals J will appear to be nice. The key aspect of this was that if you had set these particles off on the initial wave function, you set the particle off the distribution of possible locations particle to be equal to the density of the wave function. I'll show you some pictures later. Then this tells you that those trajectories have always been portion of the density thereafter. So you can always go back and look at the density of the trajectories and say, well, So that really summarises the interpretation, a very brief summary, and the question, this This is an example of a not an eigenstate, a well here, as Brett was saying yesterday that it's not well by absolute zero results, you can get mixtures of things happening, see if this is a couple of eigenstates here mixed up together and it's a drop of the well and they interact and you get this oscillating state.

15:00 And you start off with some, and here is the density dropped in at the start, targets of time here, and this is the position. And you can see here, all these trajectories have been actually put broadly, have been laid down on the density, such that their closeness is dense in proportion to the density of weight proportion. it's a rather sort of interesting thing here that this I've been lazy here, this new this should have been done in the coordinate representation we're already anticipating later on it's the idea that I can do this in the representation, this is a 0.5 by 2 representation, by that I mean that the operators have been transformed sympatheticly transformed We'll come to that in a minute. And it's a bit like almost doing a Fourier transform, but we're not quite getting there. And so we have a notion of a fractional Fourier transform coming up here. This is just one part of the rotation subgroup of the symplectic group. We can do other types of transformation, not just rotation. what I wanted to do now is just to show you some parallels that occur when you look at classical mechanics and quantum mechanics and see how the Hamilton-Jakobian forms appear in both quantum mechanics and classical mechanics also to see you can get the trillion equation similar way to how the Hamilton-Jacobian equation is. It's quite a strong parallel to the two. So you've got the differential of the action, a fixed point along projection, determined by Hamilton's equations. So we've already extremised the action. And you have this Hamilton, and you have this complement. And with those circumstances you've seen from this, you've got p to the derivative of s by q and of h is minus ds by dt. And you put these two together, and this is effectively this equation, but of course you've now put the q in here. So you've got how much you go with the virtual quantum. Quantum mechanics, we've talked about this.

17:30 You define the action through a polar form. instead of putting the R in front of it now you actually push the R into here and pull this down and this imaginary part of the action and then you postulate this differential action and this is the problem and you say well okay I've got a derivative here and it's the same idea as a PQ and HQ so let's look at what DSQ here and you end up with a dsq like this and we see some interesting features appearing here you need to block nodes of the of the formalism then you can reformulate that and just multiply through by the yeah multiply this through by the away function and you collect up this this form here the key step is made here by defining what p and q is i'm kind of coming the other way if i in this way, whereas P is not necessarily an eigenfunction in the operator. So when I operate on this, I obtain some function times the wave function, or put the other way around, put P to Q as equal to the action of the operator divided by the wave function, similarly here. If you use that and you put this in here, you have these two equations, which arise from that, by slotting that into there, this into there, and you have this, and you have that. And these put together imply the Schrodinger equation. So you can actually see these sort of parallels, the suggested parallels between the quantum and the classical system. But what's really interesting here is that what David Bohm picked up was on the real part of the momentum, he picked up on the P grad S. But in this mechanism here, you can pick up only the imaginary part here because you've got a grad of this as well. And similarly for the energy here, the real part of the energy which of course is the differential perspective, the real part of the, you know, the real part of the imaginary trans. And these things appear here. So this is what you see in the traditional Bohm interpretation. But now we're talking about not a real action here, but we're actually talking about complex

20:00 actually gives rise to the idea that you can have complex mental energies and they're of that form and what the idea of sitting behind all this is that your momentum really is a measure of rate of change with respect to space or your energies which is a measure of change with respect to time and that's what we're bringing out here So let's just take us a little bit further because it kind of opens up with some interesting ideas. We can suggest that we can take an extremal action in the same sense as we do in classical mechanics. But since we have a P of Q which is because of the fact that your S is derived from a wave function in the first place, because Q dependence is already implicit in this, you do a variation, you can't do an independent variation of your Q and your P's as you do in classical mechanics. The helms and discretions come out because these are independent variations in action. You don't do this here, you can't do this here. And so then you take this variation and end up with an expression like this for this greater change of the momentum. Because with respect to Q. And what you find, if you put a Hamiltonian in it like this, you generate the real and the amount of parts of this energy. And the rates have changed, the momentum come out of these two. What's interesting is this term here, which is a sort of convective term, which has got qi here, and it's not the real part of the mass planet. And you think, well, what is this part here? Well, let's just look at this bit here. This is the gradient of the external energy, because this is the rate of change in real management. And so what you can get is when you examine it, this turns out to be the derivative of the quantum potential we're talking about. And the question is, well, what is this? And this turns out we can signify this to be the derivative of an imaginary part of the quantum potential.

22:30 So we have two parts here. We've got the Q and the U. This would be U, which is the derivative of this whole expression. This is the derivative of U. And this is the Q and this is the derivative of Q, which is the quantum potential. Now, if we were to do David's trick of setting up q equal to p over m, that would disappear. What I've just left with was two force equations, this one and this one. And you see there's a force equation. Imagine the part from the one. It's a forward-reserved rate of change in a force equation for the other part as well. and the real part has this external potential and the question is where is the complement of the imaginary external potential do you remember the idea of an optical potential and if you were to add that optical potential back into the equations you have this cause of interpretation as actually a source of density We've still got this term here, and this expression here, if you do some multiplication through, here's a rate equation for this. Okay, this one is related to the language. Would that extra term tell you how it was, the creation and violation of 20% of the directive? Did you ever think of that? I didn't, but I'm going to chase the pictures of these things. Yeah, I know, I'm just wondering if you're answering. No, I haven't thought so. I'm sorry, I know what's coming, so... I was talking to Simon Saunders when he came with Harvey Brown to a secret talk and I said I've got the Dirac equation with quite a potential in it now so you can create an annihilate particles and I said well I'm going to hold your horses but in fact this might be a way of actually putting it in I haven't been told about that I've been carrying the idea through though you've got nothing to do So this is interesting because obviously Dave is just on this but now we have this idea

25:00 because your wave function is not a complex thing and you're now talking about the space you're saying well look, let's regard the momentum as a measure of the spatial rate of change of the wave function. The other thing you're interested in is value. So, you're already seeing these notions of the gradient of something and the value of the point coming in here, this idea of complementary variables. So, the interesting thing about doing this is it's very suggestive. It's actually, I think it adds some metaphysics. It adds some difference. It fills down your storyline. Maybe you've asked some new interesting questions, as you said earlier. What's that? The new questions of what is the role of W is playing, can it be useful, et cetera? Yeah, well, I'll go on to that. Yeah, but it's interesting that here, for example, if I... I'm not going to go on this slide, I'm trying to show you this right here. Okay, so here it is, I've just added it again. Okay, so now these two equations are looking pretty symmetric. And, you know, what is Q dot? Well, select that. And here we are, these two of these equations. And it's interesting because of this minus sign here. And if you have a free particle, you can fire at a potential step. well you've got the gradient of this your real momentum is going to be zero so this thing is going to be switched off we're not talking about it this is zero this is zero as well and this is just a constant value And so what's happening is that as soon as you hit this step, this switch is on, this kicks in, this switches this on, this then kicks in, and this comes in. So what's happening is this term here is switching this equation. what actually happens is you go through a tunnel a tunneling situation in fact the real momentum what happens is that the momentum switches over

27:30 from the real stream of the imaginary and when you hit the step on going down the other side again it just flips back out again these two equations are operating coupling and the sign on this is switching switching you into the complex into the imaginary domain out again, as you come out the other side of the barrier. That's quite a nice storyline. The other thing it does is also says, well, in classical accounts, you don't have the opportunity of switching your momentum into the complex domain, so you just have to bounce back. Something else you can do, stay that side of it, so you can't go through it. You're not able to switch to it. This is a better way than I have to describe it. There's a better way than I have to describe it. I just say, of course, there's a barrier, but this is much better. Right. This is mine, because it says, well, this is the liberation of the way that the complex domain offers you this notion you can switch through, and it's triggered by it. But the other interesting thing is, if you went in the presence of a real potential, but rather not to a potential, a complex potential where you could reserve, the converse apply. So even in the presence and resolve of you being to famous patterns, and of course the nuclear physicists know this. Are you concerned in probability? Does this extra term concern probability? The size star, still give size star right all the way through? Yes, you can do that. Right, so you can still keep the probability, yes. I'd like to show you as an example. If you put this term on the Schrodinger equation, you can actually get a solution to this. It doesn't give you a non-unitory transformation, is what I think I'm asking. No, it still holds. As long as you're creating a non-unitory transformation somewhere else. I'll show you a picture of this. Do you come across anywhere in physics or whatever, this kind of form of those equations? Have you ever seen it anywhere before? I haven't seen it either. There's probably a lot of that, but I haven't seen it. I think there was some work done in nuclear physics, maybe you know,

30:00 where they had emerging potentials on it. Oh, emerging potentials are, yes. They've been around. Sort of like an imaginary potential. Well, it is. It is, in fact, because he's got a V plus IW. And if you think of V as an external potential, then you could think of IW as a complex potential. So you're in the optical model of the nucleus. Yes, you are. That's not a model at all. That's not a model at all. I think, in fact, that's where I suggested you should look. You came to me and said, what about this imaginary potential? I said, I've learned something in my school days. If you look it up there, yes it does. In fact, you do have this term, you can go and look up the cross sections and there it is at play. And what this does though, it just highlights you that these things, it doesn't go down a black hole, wherever the absorber is. The absorber has an effect on some way out, remotely in the same sense. a bit like you're collecting these connections come from different parts of the net and this is sort of doing it's not local it's not local it's making some sort of suggestion I'm not sure it's the exact analogy but it effectively says that you know even absorb potential with the scatter so yes this is this is a picture of it all in all happening this is there is a there is a I'm going to go into the details as I can give you some details and leave them all we have is here is a complex is the complex potential these colors the colors don't come out and so what I've density of the function. It looks like my printout. Remember I didn't switch the colour on my printout. I just got everything in black. Okay, let me just follow the thing. This is the density. And then you've got here, this is the eigenbound, this is the state. And then you've got the real part of the potential, which is in red, which is this one here, then you've got the imaginary part of the potential which is here because where the potential is imaginary it's acting as a source and where it's

32:30 negative it's acting as an absorbent, okay so here you see you have what you have here is these trajectories being motivated to move down this line and they've been spontaneously created here and as they move down this line and move down the potential, actually coming off the other side and if you generate the trajectories you see just this this is a space-time and you can see over here particles being generated in this area, being absorbed in this area. The interesting thing about this is that with David Bowman's, the big question for David Bowman to start was how do these trajectories end up being at the density of the wave function to start with? And you have to sort of lay them out on the wave function. He did give, he put together an argument, an equilibrium argument, which is not like Gibbs' argument, which I haven't followed through, which said that you would end up, trajectories particles would end up with that distribution when you set it up when you have to do the analysis you have to set it up but here these things are generated at the density of weight function at the start and then they disappear and the fact that they're being generated and then they disappear is actually creating the density so this is the kind of way of saying electron and violation of creation. Yeah, it's quite sort of suggested. It did suggest it well. It's probably going to show you a bit. Okay, so let's just move on there, I anticipate. It's about a little bit of a diversion. It was by me sort of looking at what it was meant by momentum, really. I don't understand. What do I mean by this momentum? Because I think I know what it is. I don't know what it is. I can believe I know what it is in classical mechanics. But what does it mean? and it's related to derivative derivative actually so that was the diversion let's just talk about symplectic group we're going to have to dive down there now because we're going to want to

35:00 transform these operations why do we talk about it well, when you're doing these transformations one of the things I faced was well, if I'm going to transform what am I going to hang on to when I'm going to transform invariant as I do these transformations and what's invariant is the relationship between the position and the momentum which is that maintaining this Poisson bracket and commutator bracket relations and this is just stringing out the position of the and then you end up with a form here, which is a matrix here, which has got a 1 here and a minus 1 here. That's what's preserved on the flag, actually. Sorry? That's the thing that's preserved on the flag system. Right. So, I've jumped ahead. I've jumped ahead. I've jumped ahead. I've jumped ahead. I've jumped ahead. formalities I it's a group is a group this is the group of majors is actually that of course it's just a representation of the group the basic key to this is that in relation to a vector space that you have this this symplectic form this is things preserve it has is as anti-symmetrics as you swap it around and it changes sign. It also has this linearity property here. It has this property here. It also has this common degeneracy. But what essentially that boils down to is its representation, its matrix representation is of this. And when you're doing transformations which actually preserve this form, then what you have is symplectic transformations. And the point about it is that as long as you're undergoing symplectic transformations, then you preserve the Poisson bracket or the commutative bracket. And these are the things that are key. These are the ones you hang on to. This means that you're canonical, and this really maintains a relationship between momentum and position. That's the thing. And one interesting thing about these symplectic transfers is one of the most liberal transformations you can get hands on. it's one of the least constraining of all the transformations and groups It has a very de-chargeable

37:30 dimensional structure on the line Yes, but it's obviously free and that's why it's perhaps remained invisible to us really to all the people, that it doesn't pop up sitting there behind the scenes and these are the things that are being observed Can I just say something, George where he was talking about my lens Do you see that one there? Do you recognise that? Yes, this is the one I use. So here I have the... That is the lens one. It's the beginning. That's one of the terms I use in the lens one. Yeah, I've said something about that now. I forget which. It's through a lens. It's actually through the curved lens. The gamma is the radius of curvature of the curved surface. doing classical mechanics and then suddenly for example I'm going to actually use a lens system to take a look at the performance look at as if it were taking an analog but looking through a lens system of the wave function just do the two in parallel so what you have let's take this one of these and then you say oh let's apply this edge which is a synthetic map and then what you're saying your omega here is defined as this, and what you want to do is preserve the, if you have a symplectic match, which preserves this omega, whether it's, you know, you transform this omega itself to magic transformation, and if it remains as the omega, if it remains as this matrix, then you know you've got your transformation to symplectic transformation, and you know, everything's fine, that we still have this lot intact. So those are the sorts of transformations we look at. We know that if we do a change of representation, if we transfer, transform the operators, according to that, then we're still going to have all this stuff in place, which is good. And it applies in classical mechanics, and it applies in random, yeah. It applies in logology. That's right. Yeah. That's why I think it's very important. I'm trying to do this not talk properly to the kids but I heard the other day that they don't come up with Chikovie theory anymore what level? degree level honestly degree level I'm doing the maths

40:00 don't talk to me about sorry I'm mature ok so here we are this is a major interpretation Then you say, well, okay, what are the conditions on A, B, and C, such that that's the case, and you end up with these conditions here. So it's going to turn on the wrong, it's special. And it has these properties, which you derive from just preserving. But there's an interesting, what Basel was referring to is this thing. You can actually decompose this thing into, uniquely into these factors. And that's really quite handy. is related to the rotation group, and these, I'll show you an example of that, and that term there, when you're operating with that on the operators, you're actually carrying out your wave functions on the going of fractional for a chance more. This one here just scales the distances and compresses the momentum. This is diagonal. This one here, if you think about the lens system, this is operating, if you imagine this operating on your Q and your P, what happens here, let's put a 1 here, for example, or something like that, so what happens here is that your new Q is just as it was, and your P has now got a gamma times the Q and a 1 times the original P. Okay, so what we're doing here is if you're in your lens system, you've hit the lens, and then during the lens, like it's thin lens, you can imagine that you're offset from the axis, which is your cube. This is unchanged. But actually in the centre of the lens, what we've just done is you've just changed the angle, and that angle is the momentum. So this thing is actually going through it, taking it, a parallel beam, just pushing it through the lens. Whichever way it comes on the side of this, of course, it could diverge or converge. So that's the optical analogue, and you can see each of these elements has an optimal. This was an interesting one because, you know, what on earth is that? It's actually a great refractive index of fibre, which has a great refractive index in the centre of the outside.

42:30 So this is called, it was our decomposition. and just read you actually get the lens thin lens formula from this 1 over u plus 1 over v is 1 over f which is full. No optics, just symplectic transformations. But only the theme? For the theme, yeah, of course. Then you would probably have to get a non-linear. Yes, non-linear symplectic. We're already talking about the linear symplectic. Right. So then you've got... Okay, so that's that. That's just the fact that you're able to get all the lens aberrations every week as well, which is truly amazing. Sorry about that. No, no, no, carry on. I'll ask a question. You also get the lens aberrations. Oh, yeah, that's right. The accelerator people use the same. Oh, right. Yeah, of course. Because the point of that is, instead of hiking your way all the way down the right to the green line, way over there. There's some collective map that takes you from one point to the next. Alex Drakt, he's an accelerator designer, he's done a lot of work on this. And he's gone off onto the non-linear ones as well. I was talking about the linear maps here. That's when you get these operations. So, you know, where the... So what I wanted to do here is just to anticipate what we're going to do. We've talked about this idea of mapping these operators and then what's happening to the wave function. So how can we formulate a transformation from interpretation of the representation of what is the technology? I'll talk about this in particular. This is the direction we're looking at. And how does that relate to the classical limits? And we've talked about it before some practice. But can we get back to the classical limits as well? how many see the remnants of classical behavior in the quantum world. And when is this crucial condition, and I think it's quite an important condition, but such that you can call David's interpretive cause or interpretation, some mechanical ontology, hang on to that. When is that true, for what sorts of events?

45:00 In Lonely language, that's which is the correct shadow metaphor. Yes, yes, that's right. condition that picks out the particular shadow manifold out of the more general commutative charge. So here, so are there any constraints on there, such that the bone projectors in phase space are simplected in the variant, and the idea there is that if I have trajectories in task mechanics, and I transform them, canonically, the frame I mean, then it's a straightforward point-to-point transformation and I'm talking about the same point it's just that I'm looking at it in a different way and it propagates in a point-to-point sense in that frame and that's the sort of notion that I've got in my head about what's a particle and I've taken that as perhaps one of the conditions that you want to allow the main causal interpretation That's just a viewpoint, and you may want to be free of them that also got this condition as well. You may not want to impose this constraint. So let's just put a look here. This diagram here tries to capture the parallels that are happening on classical scales. This is trying to measure the action scale. You know, to drop a wrapper, we're talking actions of about 4, 5, 6, and all the magnitude of that size. But if you're talking quantum mechanics, these actions are extremely small. 10 from minus 33 joules seconds are extremely small actions. And so the contrast between these two things is enormous. This is a beautiful way of not saying H-bar goes to zero. Oh, OK. You have a couple of problems. Right, so H-bar is always H-bar. H-bar, yes. You don't interfere with nature. You don't interfere with nature. You're just looking at small actions rather than large actions. And that's the way of characterizing it. When you set the equations up and put them into a union plus four, and then this disappears out of new agents and you left with the gamma. It's actually just a scale factor. I'm never in the deformation algebra. I haven't translated my deformations into gamma. I really should. And then you see the deformation algebra. Which I want to talk about.

47:30 It's the same thing that's going on. It's the same graph. Okay, so, yeah. So, very large actions here. We just talked about this. anticipating what it's about to come. We've talked about some ray optics. We've talked about this ray system. But I haven't said much about the Hamiltonian evolution itself. I've talked about these transformations of coordinate systems, but actually you also want to preserve... When you have a classical flow going along like this, you surely want to hang on to the bracket structure. You don't want to move outside that. So you also have that flow as a synthetic flow in itself. This is an effective mapping on the phase space, in time. But we're doing, that's why we call an active transformation, but you can also do passive things. And what we refer to these things is we have a flow happen, but we're just looking at it in different frames. So these are passive transformations. Let's make that point. But in ray optics, you've got some sort of flow down the axis. In taskimatic, you've got some flow in time. and in this frame in the phase space you have this synthetic operator happening on X and P and you have, I brought photo here of a single cover and I'll come to that in a minute what it means is that I need to illustrate, I'll come back what we have here is of course P is the this is a classical action, this is a real action you have the action itself there but the action front itself propagating and it propagates one point to the front propagates another point on the front so this is a point of transformation it's probably one that's not mixing maps like you were talking about so and you'll see this in the actual transformation the genre transformation itself so here have a certain for linear maps here in the space that you can have here in quantum mechanics you've got the this thing acting on the operators and what happens here is that we have a corresponding evolution of the wave function and this this is a quadratic for a transform this thing is an integral mapping this takes takes the wave front and it says okay Okay, what's the variable function over here, where I said, well, this weight is here, and this integral of transformation just maps the hole on the front, and it was weighted, of course, from this generator, and it just says, well, this is the Huygens construction, every part of one weight contributes to another weight.

50:00 Of course, this is the way Feynman got. Yeah, this is the way we've got. So, the way of doing Feynman up in that covering space. And that's why I was making all this solid notes about the menoplectic group, because that's essentially what Bill was talking about. This is the... That's what the menoplectic group did. It's a nice way of putting quadratics, because when I say menoplectic, everybody's sort of doing clams, but if you say quadratics furrier, people say, oh, I might have a chance. So I really probably want to put quadratics furrier. It'll really come to life, just down the line, just to get on to life. Sorry. But no, no, it's interesting. Just so many people go to sleep. Yeah, absolutely. this thing here this is an attempt to try and show wave functions sort of propagating this is time running up this axis here this is a classical system why isn't it coming out of colour no actually that wasn't that was actually so this is your wave function this is the classical limit and one of the things you'll see just down the line classical chain is that the flow is carried by classical trajectories and classical trajectories can actually cross over in the configuration space and this can lead to these sorts of What I was going to ask, I mean the language is very suggestive and it might be done with mathematics Can you actually give a fluid flow interpretation of this instead of thinking of it as you know the could you actually say that this is actually some sort of a fluid flow language for the Australian version and don't use the bone interpretation of there being a particle there but it's actually a fluid flow language for the Australian version I see you get into trouble there when you go to many particle systems it doesn't work as long as you stick to the single particle system you couldn't distinguish about that ontological interpretation you're actually finding a fluid flow

52:30 yeah but okay but when you've got more than one particle say you might not expect a simple fluid I mean maybe the fluid that's the reason why yeah so it doesn't yeah I'm not but I'm just I'm looking for it fluid mechanics, or any very... There is a word, there is a word, but people have done it, and you do it for one of the... The Horvacs in France do it. There's a chap, Muga, is it? Muga's are not young. Muga's done something. Yeah, okay. So that's from Tenerife. You go from the Tenerife people, and they... They said, they said, well, let's take a look at the Vigna function on phase phase, and look at the propagation of that density on phase phase, and let us then say, well, let's take moments, you know, hydrodynamic moments, just moments and plot those, and project those down to configuration specs. And it does turn out that if you take the first moment, if you project the first moment of momentum, with respect the density down, it's precisely the Bohm momentum. But that applies for one dimension, because when you get into vorticity, you really get trouble. But did you know the Bohm theory already appeared in Moyer's paper in 49, before Bohm? The quantum potential was already in there. Did you know that? I didn't know that. This thing happens all the time. It tells that it's impossible to be original, you realise that. The reason I say this is because a lot of people say, oh, yes, that's great, we can do that. Bohm, bha, bha, bha, bha. They're both exactly the same. different perspectives on exactly the same structure. I'd love to see that happen because it would reinforce my spin on things that space itself is some sort of complicated restructuring as I've talked about. Actually, Holland has just produced something on this 100.000 interpretation as well. Yes, I know you said me. Peter Holland. I should have chased that up. It's just very recent. So there's work going on now. Okay. But the conundrum comes. There is a conundrum where you can get terms which have got not just the grad, but the curl in them, and that causes difficulties maybe more to see this.

55:00 I think Peter does something about that as well. But here you have what the idea here is, is that you can just propagate these weights at a time. Yeah, I'll just continue. You don't necessarily expect that maybe a Stokes equation is something as simple as that. You would expect, you know, quite complicated structures. I know that. There was up for grabs. I mean, here's the mathematical structure and you're now looking at an interpretation of it. from a particular fluid point of view, so obviously this is a pretty basic level we're talking about here, I'm not talking about the other side so here the idea is that I'm propagating weight dash in the core representation, and here is another representation, somewhere between core and momentum, and I'm also propagating it back as well and I'm looking at how this propagates in classical mechanics the classical rejections just carry with them the density and they just go along with it and because in classical mechanics you can get fewer constraints on the phase-space propagation than in quantum mechanics it's possible that when you project down the phase-space onto the configuration space all the projectors line up and produce a focal point and you get this sort of crossing and this produces enormous peaks in the density and this is the sort of focal point issue Whereas in quantum mechanics, you don't get that. In the pure states, you get non-crossing of the trajectories. They don't cross. And that's something that Scully tried to show that the Bohm theory was surrealistic, unfortunately, on his physical point. So that's a nice point. That's to be correct. I've got a paper, I'm promising you. Sorry, Robert. I don't want to get out of the way of people. so this I'm just trying to say here here's again a visualisation of what this quadratic majority transform is doing it's mapping from this represent you also recall that this is a symmetric mapping as well and similarly here the quadratic transform being done here to get you from this coordinate representation to this representation some representation notionally between

57:30 momentum, it could be any of these rotations And you see this flow of these trajectories here being propagated with time in this representation. So this is what we're looking at, and we're asking, you know, can you say something about, can you preserve the relationship between the mechanical velocity relation and the current when you're doing this, when you move from, you know you've got it here in this representation, can you hang on to it when you go to these, can you hang on to that causal idea patients. That's what we're trying to do. And also, when you look at these bone trajectories here, are they a symplectic map of these trajectories here in the corner representation? In some cases they are, but no one knows. And so the idea that the old classical thing, that these in a classical sense, these trajectories here will definitely be a symplectic map of these. you can go any point here and you can do a syntactic transformation and tell you what the trajectories there's a point-to-point map that will take you from that trajectory and in certain cases will that happen here so not only are you losing potentially the relationship between current and momentum and velocity but you're also losing the mapping thing as well which for me is a bit of a preservation of the idea of the mechanical part this is anticipating what's coming this is pretty intense I have to skate over it so what I've explained before in order to do these representations I'm going to have to go algebraic do the transformations in the algebraic frame and then project down into representations so I have to just do matrix elements in each representation to actually get the evolutions of the wave functions. The first task is to get Gixart algebraic. And then the next thing is to talk about these two different transformations and show how they're related. Once we've done that, we've generalized the barren terms. We can write down Bohm's equations and we can write them down in an asynchronous sort of transformation. So here, Basel did this yesterday. He talked about ideals and he expressed this density here as this line potent and these two objects here the idea. I'm going to do too much detail about this, but what it ends up with...

1:00:00 I'll go through it again tomorrow, I think, when you've gone... Right, okay. Ah, I'll be going tomorrow. So what I, and I can go to, try to walk through it, but the idea is that you put this into here, put that in there, you end up with two equations, set each to each other, and these are independent, so you can set them to zero on both sides and you end up with these two points here and these are actually looking very likely and you can do a projection of these operators into a configurator space and that's you do just get the and it's complex sorry yes of course what you do is you set b equal to c dagger and that gets you to the state so this becomes a psi It's a kind of sign down there. It's a sign-style. Or the other way around. Whichever way. Yes, whichever way. You get two equations. But that takes you down to a representation. Let's stay off of the representation. Let's stay at the algebraic form. But then you say, what you then do is you just express these two operators in this form. Having done that, you then have this and this. and with a bit of... No, just so it's a duty. But here, what you can do is you can pre and post multiply by some B to E's and B to C's and you can get rid of the E and you can land up with two equations which are these algebraic forms. This one, you recognize it as the Vierbeer equation. This is the conservation equation. This is the second of Bowen's equation. and this other one here is actually an algebraic form of the Hamilton-Jacoby equation this is why I was twittering about at the dinner table yesterday just if you remember I was twittering about the two equations and I've forgotten the dinner table but that's the second equation was he taking notes? obviously my one was even awake I was rabbiting on I just thought I'd link up what I was rabbiting on about with the actual precise equations So these are the algebraic forms of the formulation, and what it means is that I can then just take matrix elements of these two, and I've got them in any representation I want.

1:02:30 But the key thing is, it's in here, because it's called h itself, it's not just h, it's a function of x and b operators. So I have to do, I obviously have to transport those to 0 over it, and then I should do the matrix element. I'm still challenging people, has anybody seen that second equation with the anti-commitator in any way of the physics of life? I gather it. Can we put up a prize for it, George? Whoever finds it will get a box of chocolates or something. It's got to be obscure. So, and obviously these, what's happened here, you've got mixed states because you've got from season B, but if you take it down to just the same, then you end up with this being the phase, algebraic form of the phase for the same way. So you're on these two complementary parts, you keep talking about these dual bits, you know, the real and the measuring bits, and you can put two equations in. And so, and this is the conservation, this is the quantitative equation. And there wasn't any... No quantum potential, yeah. No, no. But if you do take the matrix element of this, it does come out... You've obviously got to transform that to whichever representation you want to go in by using the canonical transformation. Take the Rx and see it come through. But the other thing is, one side line, you've been able to subtract those two equations, and how come these two? It totally suggests it. Sorry about when I was a bit too early, these are the equations. I mean, if I take VCs, and outcomes. So these are the representations. These are now in a representation. It's an I-value operator, or whatever it is, projecting a representation. But, you know, I still have a problem wanting to get trajectories out of it. So I still need to know how to get the currents. And so you need to, because, you know, in the vehicle question, remember we had the current coming out of here, but instead of having that grand dot J, you know, we did dot you've got this term how did you extract your algebraic counter well you can do it and there's a commentator relationship between that express like this provided you express your operators in these forms and this this is an interesting object it's um the born drawn symbolic derivative

1:05:00 Do you want to go back to the original paper? Yeah, it was 1926. 1926, the original paper, to find out how to do it. Jordan also worked quite deep with Albert Einstein. That's right, that's right. Jordan was the one who was the one who wanted to wear. And that anti-commutator there is really a Jordan product. If you're using the mathematical symbolism when talking about Jordan, then that should be called a Jordan product. But physicists like to call it a Jordan product. I could give you lock, stock, and barrel on the structure of this, but it's in too much detail. It's quite interesting. Well, now, given that now we've got, we've been able to ship the whole lot into a different presentation. We've also got to ship these all into a different presentation. Now, we're in business now. We can now generate some trajectories. We know how to transform the algebraic form. The question is now... Can I just get noticed the difference maniacs, I want to call this people, they derive the guidance condition from first principles for some reason and they refuse to recognise all this other rich structure that is lying there. And if you go into momentum representation, there is no guidance condition. There is a condition, but it's not a guidance condition by any means. It's x equals, if you, x equals minus D, it's grad P of S, and that's no guidance condition. X equals plus grad P Sorry, it's minus this. Minus grad, but it's grad P of the action. So yeah, that makes this possible, and what you end up with you do the transformations, you end up with products of P and Q, and they're symmetrized effectively. It effectively symmetrizes the operator, this thing. Okay, this is just one page on this generalized electron transfer, and recall that when you do a electron transformation of some function, you have a function over here, which is where you want to change variables. And the way you do that is you can take a stationary point.

1:07:30 You say, I want to find the stationary point of this. And at that stationary point, I am going to solve that relationship there. With respect to the old variable, I'm going to solve that. I'm going to solve it to the point Q. That finding that stationary solution actually defines the trajectory of this transformation. When you snap that solution back in to this object here, I have obtained that is the definition of the genre transfer you can see there's a point mapping and once you've got there so now we've got our function over here which is projected over here which was over here and the interesting thing about this if you follow this through if you take the gradient of this new function s prime, d by the s prime by the q prime, get the momentum out, compute what it is, it turns out to be that, and the initial one turns out to be that, and the final momentum turns out to be that. Please, ladies and gentlemen, make a note of this equation, because I'm going to use it in my transparency tomorrow. This is a very important relationship, so you don't know the entire time. where I'm mapped in a disk. There's two-point functions going in there. But these, so I'm now able to form a symplectic, I have a symplectic pair, I have a Q to P. And if you go through this in detail, you can show that that mapping from Q to P expressed where this is the Q here, where P is that, and Q prime, where P prime is that, subject to the fact you've already, you're now related to the transformed version here and you're doing a P over here on the wave front, as it were, on the action front thing, it turns out that for a quadratic, well, for any generator, you have a symplectic mapping. Symplectic mapping is the flow represented on the phase space generated by this symplectic flow. It doesn't matter how long it is. Applies in all cases. We focus on linear ones, and when it's linear, to have a quadratic, he shows the thing here, this generative quadratic, it just relates your q and your q prime and when it's a quadratic you have a linear synchronic map so I'll go to the next one

1:10:00 you'll see here this is a generalized Fourier transform I'm going to give all the results here, when you transform the wave functions it's not a mechanics manner, here's the wave function over here, you're now trying to obtain the wave function over here, you have an And this G here is this, these coefficients, don't worry too much about it, but here you see the very same generator appearing. Exactly the identical generator that was occurring in the classicals in the large gamma limit in the genre translation is appearing in here. And it's this generator, which is actually during this matter here. Now this relationship between, this coupling here is the coupling between the seplectic group and the metaplectic group, because these, there's various factors here, I won't go into some detail, but what this is saying here, these quadratics, some little detail about this thing. That was for the benefit of Morris, wasn't it? Maslow's index. Yeah. He's the king of Maslow's indexes. I'll say something about the Maslow's index, because that's something that relates to Basel's mirror experiment. So, therefore, what you have here is a quadratic Fourier transform. Here you are. There's this G here. There's a lot of factors in front of it. There's an interesting factor here, which is this determinant of B, where B is the top right-hand diagonal number of left-hand inductive matrix. And that goes to zero. And what happens is you pick up factors Here it's been taken out But when you go through these When you go through these focal points Or these turning points This generates an extra phase factor This index here clocks up one factor Each time you go through one of these Infinities, these singularities Of this V factor And that's what you're seeing And it collects this phase factor here Picks up an I each time Doesn't pick up one goes I, each time you go through the focal point, you go through again to get minus one. So there's your double covering coming through, whereas in classical mechanics, it just turns the, if the forward has been carried through a focal point, you just turn the element upside down here, minus side. But here, what's coming, happening, as the wave goes through, this picks up, picks up an additional value, an integer value, and then each time it goes through the monologues, it picks up a factor I, and that's why it takes a little bit longer

1:12:30 This is the non-algebraic way. I went to Georgia in the algebraic way. This is the analytic way of doing it. So here's the same generator, the identical one on the biggest page. So here's the relationship between this genre transform, which is point to point, and the polar transform, which is many to one. You can take a limit of this You can take a limit of it But what's happening is that This here If you take that limit Precisely, you take the stationary Phase limit of this You end up doing precisely the Transformation Which was done on the previous page And then that turns into a point-to-point Transformation That gives you the phase propagation That tells you how the classical limit is propagated And then this here The trajectories are determined by the solutions of the genre of transformation. And those trajectories carry the probability from the starting point to the end point. And that's what gives you a classical wave function. And there's a transformation point about this bit here is that the probabilities are actually not just the r's, but they're the r squared dq as the probability density. you have to carry the probability as these elements that's where the half form is sorry this is a private because nobody else needs to listen because it's in the literature where you see everything that's a half form and it's such a confusing notion it was something important I had no idea what it must have been yes I was just talking about that was it That's it. Where your B in your symplectic matrix is going to a zero, you're clocking up an m factor. It's much more complicated than that. When mathematicians get hold of something, they sort of really screw it up. But this is essentially from the physicist's point of view where it's coming from and what's clocking it up. if we have Horace here would you tell us exactly what I'm up there all I'm going to do there is that's what happens when you apply the transformations to the Bowen interpretation

1:15:00 you can just see that it just makes the equations a little more complicated you can just see these symplectic factors appearing here's the kinetic energy here's the potential, here's the imaginary potential term given some power factors here you talked about HQ what the quantum potential is what is the quantum potential in one of these other representations and that is where there is no dependence on gamma in other words I can take all factors out that haven't gone gamma the scale factor and take all those factors out and those we refer to to the other factors. And factors containing the gamma factor reveal the quantum potential term. Is this not a problem? Does this relate to what Moritz calls the boning? You're always thinking about it as HQ notation. It's different. You can just work through this. But basically, from the bit that hasn't got the gamma center, bit, the remnants of the classical bit, you can define this mechanical velocity. And here you've got your current. It looks pretty nasty because your currents have transformed. As soon as you go into another representation... But you work at all that. And you can get these out and you can integrate them. But you might also ask questions, when are these two equal? When do you have the mechanical notion and when do you have the flow notion? And when do they coincide? When is the causal So here's just two examples. You've seen this before. This is the quantum hallmark. The restricting states evolving at almost the momentum representation, not quite. So where I've got Q prime here, don't get locked onto position here. I'm now somewhat close up against the momentum axis in terms of representation. presentation. So you can see this evolution. This here is a projection of the real and imaginary momentum. So take each, take the, and these are the manifolds, this is the momentum manifold for the real and the imaginary momentum. And then you can project the velocities onto that. So every

1:17:30 velocity has associated with a real part and a matrix. And this is a two, this is an evolution of, this is the real momentum manifold that you can see, and this is the two different representations, one of them red, this is a quarter of right angle, and this was almost and you see in this particular case the only difference between these two is a phase shift and this is a plot here where you can see the Legendre transform and also the Metaplexing through the work where in the Legendre transformation you start off from the same densities and you start off with the same initial conditions but the mapping, the Legendre mapping into configuration space produces these crossovers as you were referring to this is the classical limit whereas in the quantum limit you never get these cross-interjectures in each case you're maintaining the density of the wave function this is the classical wave function this must be tied up with your structure like you keep twittering with my no no I'm talking about but I'm wondering in what sense well in fact you've got singularity there objectories go through the same point that's essentially a focal point actually it's not I've said that not to be precise before it's a course it's a course that's the answer that's the answer and course it gives it gives sense remember that sort of heart shaped thing where all the brains are flying not actually going to a focal point but through a through a distribution so what's the phase doing in the classical what's it doing where is the where is this where is the an expected group in this, well, this is just carrying the current, we're just talking about the densities here. But what was the blue, I thought the blue line was the eclectic line. Well, in the wave function, yes, yes, yes, in its general propagation, but where can you read, see that, you can't see the double covering in there, but you can see it in the

1:20:00 phase. And here, this is, a full period here is this distance, believe you me, if you look You can see it doesn't come back round until we come to here, okay? This is my 2pi. I should explain what this is. This is space and this is time. So this is a space-time line, and what a phase is. And so here, as I move along in time, a phase is evolving, and you can actually see it repeats itself up to 2pi. It isn't the high-action limit. But in the very low-action limit, around to 4 pi. Is it clear it's 4 pi? How do you know it's 4 pi? It's in the text, it's clear. It's not really on the diagram. It is actually clear. But what you're seeing is that the timescales are the same, but you can see there's a doubling up of the turnaround. It takes quantum mechanics twice as long to come round in the phase. And that's the sort of cover you think, which is related to the I as well. Yeah, you might as well do that. So, there is the harmonic. quantum linear potential you're just firing you're firing a pair of waves at this and of course and you're looking at how to represent this and when you transform this into the momentum representation you can swap around and you see you've just got very simple operator there. What happens in the momentum representation of trajectories actually this very line now, this Q is now the momentum rate. So if I look at this, and this is a function of time, so what I see is constant rates of momentum of momentum rates, should I say. So momentum is here sort of dropping down and down and down. The fixed rate of momentum rate of change of mental, fixed force and the linear potential was just walking into a fixed force

1:22:30 in the same draft. So you get a very classical picture in misrepresentation and in fact there's no potential in misrepresentation in the mental representation. And it's a good place to start to do the analysis because the solution is very easy. So the rosaries get a bit confused. Right. But then you need to make people happy. You need to transform in the coordinate presentation. So here's my analogy with the lens. Here I was. I've just shown you there. And here I am. I want to take myself around the corner. I want to go through minus 5.2. I'm going to come down now. And I'm going to then look at symplectic maps, which are equivalent to going through a lens system. I can imagine myself looking into this wavefront, to this system, through the end of the lens. and I can then at each of these planes I can look at different symplectic transformations of the system and the water function but the symplectic maps are represented by the wave and this is the gradient reflective index and this doesn't do a straightforward point to point transformation neither does going around the corner here this Fourier transform is not a point to point neither is this, this is a Fourier transform it sort of mixes things all up point-to-point. And what we're able to do is just here's the USR of the composition. This rotation here is what's happening when you're rotating and generating the fractional fluid transform. And these things we've talked about already is scaling the momentum and the space, and we've talked about the bending of the lens here, and you know what it does the opposite. And you... Yes? I have a screen for a particular question. That's it. Is it evident here now, as you've shown it, On your point of view, where in Bob's term, where's the explicate, explicate order in this particular? It's in, the explicate orders are in the various transformations that Wilfred's talking about. Okay, now, so you can either explicate in the position coordinate or you can explicate in the pi by two or in the... OK, so what's he doing here in this sense? We're not really talking about it, because we're not talking about the actual object itself, I mean, that's what I ask, is he? I could regard this as confusing

1:25:00 but what I wanted to do was to show the parallel between optics and the transformations that we're carrying out as you transform an image going through a multiple system here we are transforming away functions because you can't change the result of an expert in order because you're taking an image and then you're doing the folding transformation which then produces another expert in order and so on, so that these S's are the things which I have as M's in my folding, in the S's, if you translate them to M's then they're in your automorphisms, that's the, well they're automorphisms, they're similarity transformations and that's that's the unfolding process in David's language and so we're unfolding image and each image is an expected order and then we cross it through a lens system where we get another image upside down or inside out and so on so that each stage the image is the expected order. So I'm not sure that helps with a general question. No, I'll probably not answer that. This is the... Okay, so this is going up the slope. You go up the slope of this potential. And you start walking and they'll roll off. And because you've got two waves interfering with each other, what tends to happen is they have a tide running on the beach and coming back down. And this is going on all the time. Okay, so that's it. I've done the trajectory. It's a bit like having particles sitting on the sea. people just sitting there up to the head of the beach and back down again you're seeing this occurring and these this is looking at the this is looking at the trajectories it's one of those planes in the optical system it's looking at normal and if I just go back it's a plane to 2 and 5 you can see looking here and I set it up to try and get the thing at least classically upright and facing them. And not bad, they were pretty similar in these trajectories. There was a distinguishing that was the phase phases really got changed around. So that's sitting behind the scenes of the metacletic map of the interaction. And then

1:27:30 the last thing I wanted to do, I said, well, are these trajectories do they transform in a symplectic way? When I take the For example here, this is the coordinate representation, this is what the bone trajectory in phase space, this is the real momentum, this is the position, we're in the coordinate space now, and the trajectory is making a cycle around this thing. and when I go into other representations I also get a cycle and there's the red one you see this is the case of which plane is that that's plane 6 that's actually part of the way around the line that's only part of the transform of the coordinate representation before we start going into the lens but if I take the coordinate representation and I do a sympathetic map in the classical sense I get the black what it's telling you So the trajectories do not map symplectically. They don't pull together in the classical sense. And similarly here, this is plane 5, as far as far end. It's been through that gradient refractive index which is doing the Fourier transfer. You see it really messes up the symplectic map. So you don't have the same point-to-point mapping in the trajectories. So here we are, this is the end of the books. so what can I say about this Epsilon question, well the question was, are these identical and for which are they identical well, when any external potential, this is is that a symplectic transformation? what that means is it's for the first two brackets on the front of that decomposition, where you have simple lens type non-gradient refractive index types And it's certainly the case that this holds. You can talk about the cause. And also, for external potentials of quadratical degree less, and for quadratic potentials, you don't have that constraint. It'll apply to all of them. So this is true for all of them. Where you're talking about the quantum harmonic also, the QRP would be examples like that. This is certainly true. interpretation in that sense and finally for any external potential

1:30:00 and for any symplectic map in classical mechanics we'll always get this relationship holding so you've got a classical mechanics you always get the mechanical loss even to the current so that's the conclusion so that is our answer to the Epstein's question on whether you can get a causal interpretation in another representation and can you sum and finally my view on this is because of that you know you should be very careful putting too much faith on the mechanical ontology and really even when the ontology applies you know I think the current equal to mechanical they're not sympathetic and variant so things are looking a bit flaky in terms of transformation of the properties of a particle so I prefer to see these structures are the problems against him. I believe with that. I'm sorry, but you won't be invited to Rutgers to give it to him. No, I'm not. To Rutgers. That's all I have to say. It's a great discussion. Should we have a quick pause or some questions now? There's a great deal of work behind it. Well, if it's explicating it was a tricky bit, you know, actually just getting the answers out, that's the hard thing. Because these trajectories, because the bone potential goes infinite in certain places, you have to watch it with the numerics. I'm impressed by that can of work. Well, this is your thesis book. Yeah, yeah. Part of the thesis book. This is a sort of sketch of what happened. It's a tremendous insight into the structure that lies behind the walls. It's not a simple yes, no, it is. It's amazing that it's even there. but you see the sort of hangovers you get you get this carry through classical concepts it's a history of ideas maybe we ought to have an island where we breed people and never heard classical physics

1:32:30 the optical mechanical analogy is very deep very illuminating I didn't hear so the analogy between the structure and the objects is extraordinarily deep and illuminating obviously it was always right there since Hamilton since Hamilton well even since Euler but certainly since Hamilton Hamilton brought this out the trouble is you didn't know there was an electron you realised that even if it was an electron you would have discovered wave mechanics in which case is where we'd see the imprint of the quantum in the classical formalism much more clearly than we do of course it is there but yeah this business of how you see the current in the classical lecture just because it does reduce the classical velocity I think that was a very important contribution we got yes I got it there was a bad and you couldn't see how the currents are, and there was a lot of fitting around, trying to find out what kind of bridges were, and eventually we got there in the air. It's very satisfying actually to see something like this all fitting together so beautifully. but it's sort of there's been a journey of the storylines and what storylines can you tell each term and you can feel yourself seduced into different metaphysics, different ideas but you see the first resistance that businesses had, certainly resistance that I had was why are you messing about imaginary moment and imaginary talent but then nobody didn't take any notice of it which was a good thing In quantum field theory, you use imaginary time all the time. It actually has... Imaginary time. In fact, when I was writing my function articles up, they actually had... They were in Euclidean metric for E4. Right. And it's... There's actually good reason for it in quantum field theory. It actually selects out the ground state as your initial state. Right, okay. And it's quite a technical issue. Is that because it drops down? Is it sort of decay or something? you think in terms of source approach

1:35:00 you start with you create two particles out of the vacuum so to speak and not out of some other complicated state and at the end all you want is you ask the question what is the probability of you getting those two particles back again when the rest of the system is in at the ground state and to force the system to be in the ground state because that's what the experimentalists are working with they don't have an excited that lead to everything else. It turns out that mathematically you have to work with imaginary time. It's just because the exponential e to the i, e, t. If you take t imaginary, it's e to the minus e tau. And then if you wait long enough, it's only the ground state that survives because all the others decay even faster. Because the energy is higher. Because the energy is higher, so they decay. So it's actually a mathematical device to get out the physical questions. So if you're going to have a physical question, you have to use a nature to retire. It's kind of, it's bizarre. And that's why all the lattice gauge calculations are done in E4, you know, space, in equilibrium space and time. You're going to miss some interesting topological... If topological properties are important, you're going to miss them in that kind of calculation. Because a lot of Derham's theorems only work in E4 and once you go into a non-compact space. it's topological it won't go check first my computer I didn't realise how much you'd have to be can I take back my identity no that's very nice If you take it from me... Oh yeah, do you understand that? No, I'm just... Okay, so I guess if you like, the guts of what I want to talk about is based on a paper which appeared a couple of months ago which is an attempt to have a look at quantum theory and Schrodinger revolution and all this sort of stuff, but thinking perhaps this was my question I think about the question of exactly what the role of complex numbers is and related to aetonian fields

1:37:30 I have some ideas here about a kind of mathematical representation of these algebraic fields which in some sense doesn't work which is in some sense different to the conventional picture in the sense that it's known to no longer have a continuum structure. There's an intrinsic kind of mathematical process which prevents them being defined on, for example, the rational meridians of the Riemann sphere. And my claim is that this could lead to a possible resolution of some of these long-standing conceptual problems. Now, I could launch straight into this, but I thought, well, when George was going to invite me and ask me what I wanted to say, I thought, well, maybe since this is a fairly informal setting, I could sort of split this up into two parts and talk about the prevailing little baggage, if you like. Who am I to start with? And where on earth have I come to this idea from? And I mean, this has pros and cons. I mean, it has the pros, I think you'll probably understand, as I say, what my motivation is and why I've come to this point. The con is that if you don't like the metaphysics and the baggage that I'm going to present, it could be a kind of a barrier to, when I come to the more technical part. Now, all I can say is, if you don't find what I'm going to say today, if it doesn't resonate with you in any way, then don't please, just forget I ever gave this talk and come to the second talk with a blank slate. George has a tablet for that. No, no, no, no, no, no, no, no, no. Well, Anyway, so I thought I would start with actually so here's a little part of my world line I guess this is not in the frame that you would this is a pre-existing time time's going up so I have to start at the bottom I did a doctorate in classical general relativism

1:40:00 So, kind of pretty much, you know, from childhood, I was actually sold on the beauty of relativity thinking, so that's always been the ambition in life. And I worked under Dennis Sharma, and I don't know if you've known Dennis, but as a result of that, I have some very illustrious academic blood, I have to say, which is that Dennis' Dennis' supervisor was Paul Dirac, my academic grandfather, if you like, of Dirac, and as I'm sure everyone knows, Dennis' student, one of the students was Steve Hawking, so my academic brothers is Hawking, so I have in a vicarious sort of way, it's very illustrious. Dennis finished up in Trieste, did he? He did indeed, yes. He supervised Valentini? Yes, that's right, that's correct. Yes. That's right. In fact, the last time I met him, I was actually organising a conference in Trieste on Monsoon. Monsoon will come tonight before I do. And it's about six months before he died. I haven't seen him in that country for quite a while, so we've had lunch together and stuff like that. By various sort of slightly ill-defined and slightly rambling processes. I only love the general theme of my thesis. I don't really want to talk too much about this, but just say a few words. The theme of my thesis was actually on this kind of concept of energy momentum in space-time, the whole issue about gravitational energy momentum. I actually used some ideas in syntactic geometry to define sort of gravitational waves near black holes. This was in a kind of linearised regime, which brought in syntactic forms and Well, I think the interesting part of my thesis, and perhaps the more profound part, was actually just looking at the whole issue of energy momentum in space-times, which have no background disometries, so you don't do any linearisation or something like that. The problem is that, I'm sure you know, that if you write down a kind of theoretic energy momentum tensor based on the Lagrangian that you would use for electromagnetism, you end up with a Lagrangian which has these connection coefficients, or Christophelsen, if you like, in the Lagrangian.

1:42:30 So it's intrinsically non-covariant that can be reduced to zero at any level of suitable choice of time. This is often called the Einstein pseudotensor, but it's not a tensor because it's, you can always find coordinates where it's reduced to zero. So the thinking at the time was that really the concept of a gravitational energy was not really a well-defined concept unless you had the least sum asymptotic isometry. And so the standard thinking was that it's only defined, for example, when you had asymptotically flat spacepimes, and then Herman Bondy did some work in the 60s on defining gravitational energy in this sort of asymptotic regime. Well, I guess I would claim that I would sort of dispute that general claim, and the main, perhaps I see the most important part of the thesis was actually, well this is a, no I don't want to get into technicalities, but it's to do with looking at sort of tensor builds or differential geometry on the tangent bundle over space-time, where I was able to define perfectly covariance. expressions which had the right conservation properties for energy, for gravitational plus matter in curved spacetimes. The reason I mention this is it comes in slightly sort of tangentially towards the end of my talk, and it also slightly links actually with a couple of points that have been made today and yesterday, which is that the tangent bundle is at least locally diffeomorphic to the Cartesian product of space-time with itself. So it's located to m cross m. So you can actually map these quantities which I derived onto space-time, but they don't involve tensors, as you already saw, but tensors, these bitensors, so tensors which have say, differential arguments at two points, like your two-point functions. These would be two-point tensors. So you end up with quantities on space-time local, they're quasi-local, they cannot be localised, they're defined as a global concept. So this seemed to make sense to me actually, that energy, especially gravitational energy momentum, wasn't a localisable concept, but it didn't require going to the full asymptotic limit of a flat space time to have a world of quantum quantum. By the way, I'll talk

1:45:00 a bit later, Roger Penrose has, I guess sort of slightly obliquely, but made some references to possible connections between gravitational energy and non-locality in quantum physics. And I don't actually, I don't buy that at all, because I see this type of quantum non-locality is quite different to this quasi-locality, something that's not localised at a point, but definable, at least in the neighbourhood of the Diora region. Anyway, let's see. well I was I was offered a postdoc I mean I could have formed my life I suppose in doing relativity and de-quantum gravitation certainly one so I was offered a postdoc one problem at the time Steve was at the time his group were working on this super symmetric super gravity which really was I suppose a fairly use of conventional quantum theory quantum field theory to gravity and this is really where I kind of although I'd done all the standard quantum theory courses I found myself asking if I'm really going to do this do I really understand quantum theory well enough to get a launch into this big time this really was never stopped anybody else Anyway, I also I'd have to say you know, go to the park and people say what do you do? Actually, if you say, well, I work on black holes that's very impressive What exactly do you do? Well, if you start saying, well, I actually work on differential geometry or tangent black holes and you can see their eyes sort of glaze over within none of the seconds that you start to talk I don't know, I thought this stuff was too obscure to really spend a life on it so I actually ended up then in the late 70s making total for again reasons which we can talk about at the lunch a number of relatively random events occurred which made me get quite interested in climate and meteorology and stuff like that and I spent a few years in the States and I now actually work at a, well they know as well, a European organisation, a geological

1:47:30 organisation based in southern England, which has as its member states, for example Sweden, so I actually have a number of colleagues in Sweden, both at Stockholm University and and North Shopping and all those other places and this really has been my sort of life's work since 78 I guess in the early days I used to ask myself a lot whether I'd made the right decision here but I think over the years I've actually enjoyed myself and I've made a lot of good friends and colleagues around the world, I mean I've got a number of again in Australia I've got a number of really good colleagues and friends who there. And I guess if I made my infront of it, the only, the other thing which occurred to me last year was I was elected into the Royal Society as a result of some of this work. So actually that's made me slightly more relaxed about coming to meetings like this, because if you'd invited me, you know, a few years ago, I'd be thinking, shit, can I really afford three days to do something I should be finishing off paper X, Y, Z, and, you know, so on. But now I think it's not like paper X, Y, it's a few years ago. For at least 10 years I didn't do, I really put all this stuff out of my head, but I really had quite a major relapse in, I can almost date it to the day, which was, my wife, we don't live very far from Oxford, and my wife wanted to, for some reason, I can't remember, go to Oxford to buy some clothes or something. What else do they do? And in case, I knew I wasn't going to be much help with that, so I sort of made some excuses and ended up in Blackwells, which is the main book. And they just published a book on the 300th anniversary of Newton's Principia, which Roger Oh yes, I should say Roger was my internal examiner, so he got to know this stuff quite well, he wants to talk to him quite a lot. So you didn't manage to persuade him then on the non-local, quasi-local? Well in those days I didn't know anything about non-locality. I mean, this was the thing actually, I mean, I have to say, well you must know yourself, I mean, if you did quantum, studied quantum mechanics in the 70s, I mean,

1:50:00 the ulcerum and all that stuff was never talked, I'd never heard of it. where I was we were deeply in scotch 68 we started we were discussing all the experiments did I say some of the early experiments were pre-except of course when actually testing even Paul Davis who interviewed me for Science Today or whatever he said I don't believe in non-locality trouble with your theory or something like that but now he's changed his mind completely so when you're ahead of the field you just get ostracized I know that from those crazy people who move along their character I was tempted by Max Planck he said you never convert your enemies This is very true because I used to always I think we're only interrupting We'll talk about this one We're only interrupting the lunch My locality is going to be a big theme of this my talk No, it's just that I was pussy-putting around my locality because I'd been hammered from all directions A couple of youngsters in the audience said Why are you pussying around? We all know it's not that Get off with it the realisation that the whole thing had flipped I guess perhaps this as I say, so I spent a few hours in Blackwell so my wife was doing what she was going to do perhaps this was my first sort of exposure to non-locality and I guess I saw this article from Penrose as a kind of cry from the heart that here was this person who was absolutely wedded to GR and relativistic thinking reconcile non-locality. And I always remember this phrase in one of the sentences, that whilst it clearly doesn't violate relativistic invariance experimentally, it somehow violates the spirit of relativity. And I could see that he was a tortured soul. Well perhaps we were responsible for his torturing because he used to come to our seminars. And that was

1:52:30 the red hot topic of conversation in our seminars. and then I mean as you know he started to speculate about the role of gravity and it was very kind of well I think you probably still agree but it's still a kind of hand-waking and I started to that perhaps sowed the first seeds of thought about and he's talking about instability I was going to say I actually think it's true what he said about gravity I've actually made the claim actually think there's a mechanism in what I was talking about, where it's space itself wants to go classical, and if the matter is too non-local, the space will actually collapse the matter down. So you think you've actually got a mechanism for objective reduction of the wave? Absolutely, and I think it's the Penrose mechanism, it's an explanation for the, it's a mechanism for the Penrose conjecture, so I actually think that part of his thinking is right. Because Penrose, of course, just, I mean, paper, which is where I first read it, and it is an absolutely fascinating paper, at the end he just has this very speculative stuff about how partially formed, partially bifurcated in space-times. Oh yeah, I don't buy that, no, I don't buy that, that's a bit too, but... Well anyway, this was the starting point, as I say, a relapse, you know, which, but I have to say, I think my views actually are now diverged a bit from him in the sense that, well, I mean, this is really what I'll come down to, well, maybe I won't talk about that, but this is the base of my thought, what I now believe. But I'm not, I guess I don't believe in sort of separate collapse mechanism, and I have, well, let me give you my, I'll give you my spin on non-locality and see whether it appears or not. But I think it's probably slightly different to what you were saying. I mean, so, yeah, so basically something I just build around in my spare time. I mean this is what earns me my money so I have to keep on straight and narrow here during the day but the small hours of me. And John Bell told me exactly the same story when he used to see him on conferences and he said don't tell anybody from CERN I'm here. That's right. Don't tell anybody from CERN I'm here.

1:55:00 Now, Basil did kindly send me an email because I hadn't realised actually when you first George kindly invited me that you were actually somebody that had made friends and knew David Bohm quite well and I obviously began to sense there was Bohm was a common theme behind a lot of today's, this week's talk So I realise I'd better get to know some of this stuff in a pretty quick measure. Well, I was hoping you knew all this before. Yeah, well, yeah. Now, I have to say, I mean, one of the great things about the internet is all you have to do is type somebody's name into Google, and you've suddenly got everything that needs to be known into it in that one's seconds. and sure enough almost the leading on his own and had a mass of information and I clicked on something else by the way I realise now how these GCSE students kind of do their essays you don't even have to read the place and the university's which I don't understand why I think it's a perfectly respectable way of working with knowledge as long as you read it first fair enough I accept that as a footnote anyway I did so I have to confess that I just literally did this off the website two things Well, I was sort of quite, I have to say, pleased in a sense that two of the, I don't know what the word might be, motivations or sort of inspirations for Bowen's work, were actually taken from fluid mechanics. They're really metaphors. Metaphors, okay. And the first one, that's what you talked about, which is essentially a low Reynolds number flow, you know, dropping from glycerine and spirit around and so on and so forth. And then the other one here is actually about what I would call a high-levels number flow, where you have tokens, and this is talk about ever-changing patterns of all you see, and splashes and so on and so forth. Such transitory subsistence may be possessed by these abstract forms implies only a relative

1:57:30 independence, autonomy, rather than absolutely independent, so there's some underlying structure. So that was, actually I felt pretty good about that, because I still see myself basically in something that's spent the last, I don't know, 20 years, I don't know, 20 years, 20 years in an incredible, incredible mechanics, yeah. Now, I have to say, on the other hand, And this picture here of the ink actually reminded me much more, or reminded me of something which people play around with a lot in, you know, image of the climb and stuff, which is, if you like, ink in face space, as described by something like this. So here's Lorentz without the T. Ed is something I know very well, works at MIT, and these equations here are a kind of a simplification of fluids, what's called Rayleigh-Beynard convection, truncated just to three components. And this was published in our premier meteorological Journal of Antisperic Sciences in 1963 and it was the first demonstration of chaotic behaviour from a low order three component model and I'm sure you know the story that this explosion of interest in chaos actually occurred about at least a decade later and the mathematicians complained, why did he publish this stuff in such an obscure journal? Wait a minute, this is our premiere Anyway, now I have to say, just coming back to this Penrose thing, I did start to think about this issue about these types of non-linear systems and whether there was any sense in which they might be used to provide some insight, A, into this question of this apparent problem between non-locality and the spirit of relativity on the one hand, and this whole issue about kind of collapse models and trying to formulate ideas about collapse into a more quantitative

2:00:00 type of system. So what I want to take you through is actually my thinking pretty much in those days, because actually as it happens now, I haven't really, this is a line which in some sense was a bit of a dead end, but it sort of led on to this piece of work which was published recently. But I think, well, as I say, given that George had the time, I thought I would sort of just run through what I thought my thinking was, so you could perhaps see how things emerged. So let me just start with some basic, if you like, almost sort of statements of fact about these types of attractors. So that's a sort of a picture of the Lorentz attractor, it's the invariant set on which the system of the corrections evolves. So in particular, if you started the system, by a notion of invariance, if you start the system on this attractor, so this is what I'm calling the ink, the ink is the, if you like, the attractor, this fractal structure which sits in this bigger space containing all the glisterium and so on. But if it starts on the attractor, then by definition it will stay on the attractor for all time in the future. In other words, the states as evolved by those three equations on the right hand side of the last slide will ensure that it stays on that sort of geometric, fractal geometric structure. states we get inky stets. Conversely, a state not on the attractor cannot have evolved from any such state on the attractor. So if you pick a point which happens to be a glycerine state that it has no inky parent and will come from an inky parent, so to speak. Okay, now now, let's imagine at some time we impose a, the reason I'm going through this will become apparent in a few minutes, because I'm going to put these points in a much more metaphysical way in a few minutes, but this is basically a statement, or fact, if you like. If you take some perturbation, or imagine some perturbation, to that state at the time of t,

2:02:30 and in particular, let's say you displace it by some distance delta x, so delta x, x, y, z, these are the axes in phase space, so delta x is a perturbation in that x direction, and let's say you keep the other two components unchanged. so this is just a sort of a manual perturbation you're doing it, it's not implied by the equations, it's something you impose on the state the thing is that because this attractor has a if you like, if you go transverse to the trajectories, this attractor has this fractal structure, this sort of cantor set like structure then you're going to almost certainly in the measure of the You're almost certainly going to take your point which was on the attractor off the attractor. And you can say this actually even though the perturbation might be incredibly small. So that's in a sense a statement that, like when you said you wind the cylinder up a few times, the ink has disappeared. So if you sort of choose a point at random there's almost certainly going to be a glycerine state rather than an ink state. If you perturb in this sort of way, you'll take... It was like a continuous vector system. Yes, that's right. Yeah, these attractors are basically this fold, stretching and folding, stretching and folding. So, let's put this now slightly more metaphysically, not perhaps too much more metaphysically, but slightly more metaphysically, in the sense that I'm going to imagine when I first wrote this I said suppose the possible initial state of the universe and I thought that's a lot let's say I can take one of your S3s you've got plenty of S3s and I'm going to assume again we've had a little bit of discussion but I'm going to assume that the dynamics governed by a non-linear but deterministic dynamical system, which you know, so x this is a state vector and this is some non-linear functional and it's going to have this fractal type structure

2:05:00 notified by the rank it doesn't have to be the rank system and I'm going to assume that if you like the almighty when you chose the initial states, I mean you could have chosen, you know, one infinity, but I'm going to assume that he was constrained to choose the initial state on the attractor. So the system started from the point of the attractor. And so any state, a state that's evolved from a possible initial state on the attractor is clearly dynamically consistent with the equations. So again, if you started off on an inky state, you'd never get to, or a trend, this type of transition is dynamically inconsistent. And again, just coming down, restating these in a slightly more difficult way than the form, it's basically the same. If you were to consider a hypothetical, dynamically unconstrained perturbation, so now I'm, again, imagining imposing a perturbation onto the state T, and it's going to change of the state vector, leave the other components unchanged. When I say dynamically unconstrained, the perturbation takes no account of these dynamical equations, just as before when I just moved the system along the x direction, perturbed it along the x direction. I wasn't concerned about whether that perturbation is consistent with dynamics. Then, as before, my claim is that no matter how small in amplitude is this delta x, this perturbed off the ink, off the attractor, and so in some sense that I would be inconsistent. Perhaps, yeah, just, I mean, perhaps, sort of what I'm thinking here is that, and I don't know if I'm using the word, because I've come to this word explicate and implicate rather recently, so maybe I'm just using this phrase, but I would say in some sense the explicate measure of delta x might be indeed small but the implicate measure of delta x is actually very large because it's taken you in some sense it's taken you a large way from the attractor in fact it's taken you right off the attractor it's taken you from a state which is dynamically consistent

2:07:30 at the initial time, but something which is dynamically inconsistent. So, you know, I'm using the word small here, small in amplitude in this sort of perhaps, I don't know if I'm just using it, but in this sort of explicate way. But it actually has a big implication about whether the state is consistent or not. They're very general terms. I mean, this is one of the problems with them. They're too general. You have to make it specific, but that's okay. All right, so we're going to go through this once more again. Sorry to be tedious about this, but I'm going to introduce a little bit more, to lead you down the garden path of where I'm coming from. So again, let's consider a dynamically consistent state, so it's evolved from some initial state on the track. Now, this component, I've talked about the termine components of this state. And just to make things a bit more concrete, let's suppose I'm going to call this component X subscript A of X subscript U, the universe. I'm just going to call it the apparatus, in case we're part of XU. and I'm going to again consider a dynamically unconstrained perturbation delta x which maps xa to xa prime which changes some attribute of the apparatus and we'll call that change of attribute the orientation but leaving the rest of the universe unchanged in some sense again my claim is that with such a system no matter how small an amplitude of x to the extent x, this perturbed state, the resulting perturbed state of the universe, is almost certainly dynamically inconsistent in the sense that it has no conceivable parent in the set of possible initial states. Right, now we're almost there, I suppose. So I'm going to now imagine this kind of complement to xA in the universe includes something which I'll call the electron. and I want to consider a proposition which I call C which says that if at time this time T the apparatus this component of XU were to have been

2:10:00 aligned differently to its actual alignment then the apparatus would have interacted with the electron to return up. So that's a proposition. So the question is, is that proposition true or false? Obviously you'll recognize that in terms of standard quantum theoretic analysis. So my claim is that in thinking about what this proposition means, and thinking of it in this dynamical system sort of framework it's requiring us to consider a hypothetical it was actually aligned in some way but we're asking the question what would have happened were to it have been aligned differently so it's discussing a hypothetical dynamically unconstrained because when we make this proposition we're not worrying about whether it's dynamically consistent to have aligned the apparatus in a different way to its actual alignment, we don't consider that question. So it's dynamically unconstrained perturbation to the apparatus, which leaves the rest of the universe, including the electron, which we're trying to, in some sense, measure, unchanged. And then what actually occurs, in some sense, is then imagined to be associated with evolution of this perturbed state of the dynamics. So, this is my, just, this is kind of the crux end of my argument. This seems a rather, in some sense, innocuous statement from the point of view of sort of semantic English and so on. We use these types of statements the whole time. You know, if I hadn't given up relativity, what would I be doing now? I mean, this kind of seems a reasonable question in a sense. But my argument is that all questions of this sort imply, or yes, they directly imply some sort of perturbation to the universe, which is dynamically unconstrained. And my argument is that if the dynamics of the universe did have this type of fractal-type structure to it, then these types of perturbations will almost certainly result in dynamically inconsistent states, in the sense that one can find no possible initial state which would dynamically evolve into that hypothetically

2:12:30 the deterrent state. So, in that basis, this counterfactual really cannot be said to have any definite truth value. It just is, it makes no sense to say this has any truth value within the constraint that I'm talking about a universe evolved by dynamical, deterministic dynamical thinking and with this type of structure. this of course is now the key point here I mean when you go this is really what struck me when I read Penrose's article and he was talking about Bell's theorem and this sort of stuff and as I say I mean Bell's theorem wasn't taught to me at university so I had to go out and buy books so I bought I remember one of the first books I ever bought was has a nice sort of thing and you know consider the EPR, the bone version of EPR experiment and and then this is the argument for you imagine you have a hidden variable theory so you have some sort of spin function for your left-hand particles and one for your right-hand particles And in deriving the Bell inequality, you assume that if you take the right-hand particles, and you have a common alignment direction, then the common values of this hidden variable so these values take 1 or minus 1 for example, then if a is equal to 1 b must be equal to minus 1 so this is so if you were to have measured them with exactly the same direction, one is going to spin up they were emitted from this sort of singlet, zero, and the elemental state initially. Now this, I have to say, this is pretty much what I... I was trying to think about this,

2:15:00 you know, what this really meant in the context of these dynamical systems, and this is pretty much, you know, what I was thinking. Well, I mean, this does imply something... well, first of all, this does imply something that you don't actually... you can't actually do in your experiment, because you have particles, left-hand particles are measured with respect to some orientation, and right-hand particles are measured with respect to some other orientation in general, different to the left-hand ones. And so you can't actually take those right-hand particles and say what their spin would have been had they been measured in the same orientation as the left-hand particles. So this statement is a sort of counterfactual statement, it's that you can directly test on the particles that you measured with respect to the different orientations. So I was meant to think, well, if one had a kind of framework in which maybe this collapsed hypothesis or whatever somehow invoked these types of nonlinear systems, maybe one could bring to bear this type of argument that I've just been through. to say that actually this statement here is just not provably true. It's neither true nor false. It just has no sort of mathematical truth value within the system in which you set up. In other words, the truth of this seems to me to assume, as I say, a hypothetical dynamically unrestrained perturbation to the state vector of the universe in which one part, the apparatus measuring the spin of the left-hand part, is perturbed by some dynamically unconstrained perturbation while the rest of XU is left unchanged. And that was the sort of thinking. Using non-linear dynamical systems thinking, and I have to say I just added this but on the other hand it did seem to me it had some bearing on the fact because you know we're saying this is a tiny perturbation in the big scheme of things just changing the orientation of the apparatus seems to be a pretty harmless sort of thing and this is where this idea about a small perturbation in one measure is actually a very big perturbation

2:17:30 a differently defined measure. So it depends, you know, what your measure of thickness and performance is. But this is the question. Could we seek a deterministic non-linear theory in which such perturbations that were required to address the truth or falsehood of this statement here would actually result in some dynamically undefinable and hence in some says, computably undefinable states. I mention this phrase because, again, I was attracted by Penrose's thoughts about non-computability. That's always been, again, something which has some resonance in my thinking. Can I just interject something? Actually, David Bone and I tried to explain the non-locality through local non-minium dynamics. We had one paper on it, but it was before the non-computability ideas came And we got a hairy theory, which I didn't believe it, so we never followed it. But we actually did try it. I mean, you're an expert in it. We were sort of amateurishly trying something that way. Just for the record, we're not totally crazy. We do try all the way. All the approaches. All the local approaches. Now, I did actually I did actually initially try to think about this actually using these types of what I would call conventional chaotic systems I mean, because I'm going to talk about slightly more or less conventional I think you said it in this way I may have done, yes The idea was to actually link the idea was to link a chaotic oscillator like the Lorentz system to a potential well so you imagine a damped potential well, no, imagine a double potential and you have a sort of damp system, you've got some damping so clearly the system goes to the two minima while you have an unstable equilibrium at the top and clearly you can say exactly where the basins of attraction are so if you're anywhere to the left of the maximum potential then the thing will go down to this one so you can colour, if you like

2:20:00 the wells of attraction the attraction wells the basins of attraction the basins of attraction in colour and now what you do is you couple this sort of potential well this damp potential well dynamic to take oscillator. I'm not going to bother you with the equations, but the net effect of that is you can get what are called riddled basins of attraction, in the sense that what happens is that if you take a neighbourhood of anywhere in this potential world, there are actually points which go to one attractor, to one minimum of the potential weld, or to a different one. The closer you get to one of those minimum of the potential weld, more the number of points goes to that minimum. So if you coloured, let's say, the base of attraction at the left hand, my left hand well, red, and this one green, and if you get closer and closer to the red one, more of the points are red, but actually there'll still be a kind of a you know, there'll still be a few, let's say I mean, there's a multiplied measure where they can go to the green one and as you go near the kind of saddle point, then actually, yeah, it's 50-50 but they're actually completely riddled, they're interspersed in a kind of totally sort of way. And I thought at the time that this might be, A, a kind of a more concrete way of approaching Penrose's ideas about collapse models in a complicated way. And it might also link to these ideas about, well, about the Bell Theorem and a possible way of circumventing this in terms of Bell Theorem. I have to say at the end of the day it probably was unconvincing and it was unconvincing because of two fundamental reasons it's actually very hard to link all this sort of stuff to the Schrodinger equation I mean where does if you want to try and fit this idea into Schrodinger dynamics how do you do it I just didn't know the answer I saw that with very grave difficulty I saw that as a sort of technical issue at the time,

2:22:30 and a more sort of troubling one to me was actually this one, which I didn't really sort of talk about much. But the problem with these fractal, these little basins of attractors is that if you really want to get to this limit where you say a perturbation, no matter how small, will take you sometimes off the attractor or off to a point where you cannot say whether it's in the basin of attraction of one attractor or the other, actually requires you to integrate the equations, the chaotic equations, sort of forever, so they really have reached, they've asymptoted onto their attractor. So in some sense, it is a kind of a property of the system which is only truly realised after an infinite integration time. And I sort of thought that was at least a bit contrived. I have to sort of wait for eternity before you can apply this kind of idea. this is another common feature of people who try to do this kind of thing I'm thinking of Hepp's work and there he has to wait a long run infinite time before it actually collapses now in the framework in which I want to talk about tomorrow I believe I can overcome both of these problems so the system I will describe does not require infinite time in any sense and in fact it will be very, very, very strongly linked to the Schrodinger equation. It really gets in some sense to the heart of the Schrodinger equation. So I think I can convince you I've sort of made some progress to overcome these two problems. So, let me just what time should I finish? We've gone a quarter of an hour. He's gone. I went to Are we going out for lunch again? Yes. We will have some food. Same way now. I won't be able to do too much. I want to go, because actually, in meteorology, fluid mechanics, if you like, these models of chaos, low-order models, I mean, they're used quite a lot, but they tend to be, in some sense, for solving toy, what I call toy problems. Because in reality, the atmosphere, or a term in fluid, if you like, is not a low-dimensional system. And here's a picture of the atmosphere and it's got big eddies, it's got sort of medium-sized eddies, and it's got smaller ones, and here's a smaller one here, and in fact, and by the way, we were talking about hexagonal, so this is not a very good example, but you start to see the convection start to form hexagons, when it's very clear, you get very clear hexagonal structures.

2:25:00 anyway, even within one of those so each one of those, if it was really well formed hexagonal convection individual clouds would be sort of little thunderstorms if you like, and within those thunderstorms they're little heady and so on fascinating, it's a bit like Bernard's cells isn't this basically Bernard's cells but in a sense multi, I'm trying to stress a multi-scale So actually, when you, what I'm saying is relevant to quantum theory, it's a bit, it may not seem so quite, yeah. I've seen that picture before. Okay. So in reality, if we talk about how predictable is some, let's say, one of those large-scale headaches, how far forward in time can we forecast that in some sense of which there's a big symphonic circulation how far ahead can we predict that and more specifically what prevents us predicting it forever well the problem is that as I say we have this sort of coupled non-linear system which couples scales together and we might, we have these wonderful satellites which take measurements every maybe 10 kilometres or so. But sooner or later we're going to get down to scales below that 10 kilometre scale and then we just won't know how much information about if that's too good with the Atlantic. We won't have much information about those scales. So we'll get to a point where we can say at initial time we know this scale pretty well, we know this scale pretty well we know this one, but we don't really know that one. So there's uncertainty about that one. Or even if we could know that one there'll be an even smaller one that we don't know. So actually we can stick as many thermometers as we like in the Atlantic but there'll always be a scale that we don't know something about. So the question one can ask is well how long does it take the uncertainty in that small scale, whatever scale that happens to be to propagate up the spectrum and start to infect the ability to break this large scale.

2:27:30 there's some interesting sort of scaling analysis you can do on this type of problem because it's, I mean, as you know, of course, it's really quite difficult to integrate these equations exactly, although we have large computers to do it, but nevertheless one can apply scaling arguments and, for example, we can talk about a predictability there's a time scale, if you like eddy would have, so k is a wave number run over wavelength, so large k is a small eddy and we can associate a time scale with its length and its typical sort of overturning velocity and we can use a theory which actually originated with Kolm-Gorov about homogeneous isotropic turbulence, which in a three-dimensional system the argument I mean, I'm not giving you this argument, but the argument tells you that the energy I'm not talking about energy, I'm talking about basically energy per unit mass or something, a factor that the mass term but the energy scales as the wave number to the minus five over three, it's a very famous so-called five-thirds power of a three-dimensional homogeneous isotropic turbulence in the so-called inertial sub-range, that means between the scales in which the turbulence is being forced so the large scale forcing and small-scale dissipation. And that gives you a relationship between this time scale for an eddy, which will take us the kind of a predictability time as well for that eddy to its wave number. So it says as the wave number gets bigger, this predictability time gets smaller. So it's more difficult to predict. So the predictability time of small eddings is small, the predictability of big eddings is large, I mean that's fairly, I suppose, intuitive in some sense. So the weather system, charging across the Atlantic, you can think of its integrity for a couple of days, where we have a tiny little cumulus cloud that comes and goes in a matter of minutes. now the fascinating thing is actually if you sum all these, you imagine some octaves, what are called octaves but you imagine these wave numbers separated by

2:30:00 so you have a series of eggs which is wave numbers are rated by powers of 2 and you sum over all these predictability times as a measure of the total predictability time of this thing, so in other words summing the time it takes, uncertainty to propagate right through the spectrum. For three-dimensional turbulence, and this is only for three-dimensional turbulence, this sum converges to roughly two times your large 80 turn over time. For two-dimensional turbulence, this sum diverges, it's infinite. Now, what this says is for two-dimensional turbulence, what it says is providing you can make your uncertainty confined to small enough scales, you can predict your large scale as far ahead as you want. So you have a recipe for predicting large scales further. Just make your uncertainty on a small enough scale. But 3D turbulence you can't do that you'll always be hit by the uncertainty in a small scale, in a finite time, no matter how small scale that small scale is. Now, technically, what this really, this is, they're talking about going to infinitely small scales, they're actually really talking about some invisible limit of the Namius-Lokes equations, which are probably called the Euler equations. And in that limit, what this sort of scaling argument is saying is that uncertainty can propagate from arbitrarily small scales to some given large scale in a finite time. Now this is really what we use the butterfly effect and it's commonly used to describe low order chaos. Actually what the butterfly effect really is about is this thing here. It's about the fact that the butterfly flapping its wings. See the butterfly flapping its wings is not really a small perturbation. lie, it's actually quite a large preservation. I mean, it's like it's a large preservation. It's on the scale of that thing. But that uncertainty propagate through these non-linear interactions to affect this large scale in finite time, according to this paradigm. You realise, of course, that if we lived in a two-dimensional world, we'd have a job. Well, this

2:32:30 is the thing. We do actually live. This is what keeps me in a job. We do live in a two-dimensional world, and the of the Earth. The rotation of the Earth actually makes the flow in the atmosphere very quasi two-dimensional. It's not two-dimensional down on the scales of here. But on weather scales, it's actually, the rotation of the Earth actually makes, it's called geostrophic dynamics, it makes the dynamics of the atmosphere very quasi two-dimensional. So why do you get any difficulties relating to the world of that? Well, because we still, because we don't, we can't, well, because actually one, because we, because we can't, we don't measure, I mean, there is uncertainty in these little scales down here. We don't have instruments measuring every little eddy. And the upscale propagation still occurs with 2D turbulence. The upscale propagation is there in 2D or 3D turbulence. In 2D turbulence, in 3D turbulence, it's saying we wouldn't be able to ever predict more than about five or six days ahead. We can actually do a lot better than that. And the reason it can be better than that is because of the quasi-2D nature. but we are still hit by the fact that we don't have perfect knowledge down on the kilometre scale as well this might become the reason I've got this slide might become more apparent well it will become more apparent in the next talk another way of saying what I just said let's imagine we had let's say the Euler equation and let's take which was representing my state vector was something like vorticity then I could what I do in practice when we solve equations is we project and these are partial differential equations of course we on a computer we first have to have some basis so some people might use good points or some people have finite elements at my case we use spherical harmonics so we would represent the state vector in terms of vorticity of spherical harmonic coefficients at some particular time. I mean, at the moment, we have, like, 500 spherical harmonics to represent

2:35:00 all system and the planet. Now, what this 3D, this Colin Goroth 3D paradigm is saying is that there's some finite, there exists some finite time, I'm going to call it T sub P, P for predictability, horizon, if you like, such that that time, A1, the largest scale spherical harmonic coefficient, at this time T, is sensitive to, let's say, uncertainties in A subscript N at initial time, way down the sequence. No matter how large is N. Now the interesting thing is that Reg, you said, I caught the remark you said during Melvin's talk, he said, oh well talking about fluid flow for his system he said, well I'm not talking about something as simple as Navier-Stokes well, if you think Navier-Stokes is simple then you've actually made yourself a lot of money a million dollars because actually proving this rigorously proving this rigorously for Navier-Stokes is actually a problem that's up there with the Riemann it's killed people trying to prove it rigorously and you know made attempts to look at the literature on this stuff but I just get totally defeated by my complete lack of knowledge of functional analysis What's canon difficult is it? What's canon difficult is it? I can't really answer that I mean, they express, what they do is express this problem in a functional, analytic way, as I say, using these Sovolov space theories, and then it becomes a certain problem in functional analysis. But they can't prove or disprove that problem. I mean, maybe it's one of these unapprovable things, isn't it? But the point about this, and my claim is that although I think most physicists know about chaos And it's applied to everything. I don't think actually this kind of paradigm of what I call finite, this finite predictability horizon in what is otherwise a completely deterministic system is actually that well known.

2:37:30 And this kind of got me thinking, again, about this question, you know, I found it hard to link the chaos, but I could see that this sort of presented an opportunity to get over this waiting in infinite time type of problem. But then, again, how the hell do you link that to Schrodinger? You've got a real non-linear PDE, a complex linear PDE. Now, this is what I got, really thinking about this complex, what the hell are these complex numbers doing in the Schrodinger equation? yeah, of course this is the I'll come back to this equation this is obviously the simplest sort of solution to the Schrodinger equation this is mechanics 101 I suppose when you have a free particle and basically at any point in time and space I mean the wave function is just a complex number frequency and wave function by the energy but I started to think about numbers. And of course, you know, in euthrology we use them all the time to solve, particularly when you're doing perturbation and you're trying to look at some stability of a certain flow type, you know, it's a stable, unstable. Move into the complex plane. Move into the complex plane, you have all the usual contour integrals and stuff like that. So, you know, at the end of the day, of course, you actually only take a bit apart. So, you use complex to solve an equation, but I mean, if you didn't like complex analysis, you could always solve them without using them. Of course, when we did what we did, we weren't allowed to take the real part. It didn't mean anything, of course. Sorry. I always thought that you could only use that trick with linear systems. Then you could always separate the real and the major. Well, yes, and indeed, as I say, it tends to be mostly used for line phenomena. I mean, you know, in reading all the popular literature, people spend a lot of time

2:40:00 worrying about showing this cat in terms of superpositions, but I've never seen anyone further worrying about I mean, you know what? A live cat plus dead cat is problematic, but what about a live cat plus square root Academy, that seems to be people worse. So I just sort of had this thought in my mind about whether, by the way, I was always trying to, this is a lot of anecdotal things in which I should say, but I can always remember a friend of my wife's, I wasn't talking about quantum mechanics, I was just talking about science in general, and she said, she's quite a, I mean she's a very intelligent lady, but now does art and literature, she's a very maths at school and she got turned off when complex numbers were introduced. She said, what on God's name is that? You know, we've been hoodwinked or something. Now I understand, of course Reg, I mean I understand what you said. You can represent complex numbers as just pairs of real numbers in certain ways and of course 20th century, no, not 20th century, mathematics since, I don't know, Gauss's in that complex number, so you know, in some sense we can be very you know, I mean, well can't reject it, but on the other hand I do wonder whether, well this is the idea we're thinking, maybe are we being a bit complacent about the meaning of these terms? Now just the last point before I want to stop is that I went up to, oh yeah I got, I mean Ian Stewart who was writes a lot of popular, in fact he wrote a book on Does God Play Dice, but he's also somebody who's very interested in chaos theory. I got talking to him about, actually it was about weather stuff, and