From the Feynman Chessboard to a Deterministic Construction of the Propagator

0:00 As an outsider, let me say, it's been great. I'm very lucky to be here. I'd like to thank Lou for telling me about this, and also the organizers for letting me to invite myself to this meeting. it's pretty clear to me that Ampa, among other things, really is an explorer's club. I hope this talk will at least be in that flavor that it is an exploration. I've changed the title somewhat. To, you know, differential calculus hide this space-time algeverse, I figure this is as good a place as any of any to sort of gently complain about Newton's dominance of our language in physics. So, what I'm actually going to talk about is a toy model related to the Feynman chessboard model. The chessboard model is a very simple model. It's a toy model because it's one plus one dimension, and it's exactly solvable, and it gives us a bit of a picture of the Dirac propagator. It's just in one dimension. It's just in one dimension. Okay? So it's very, very simple. Now, the route I'm going to take, because quantum mechanics, in a sense, is there's all sorts of different mysteries associated with quantum mechanics, so I'm going to stick within classical statistical mechanics, because, you know, that's my supervisor used to say that the system mechanics is very simple. All you ever do is count. And, in a sense, that's all we're ever going to do. we're going to stick with kinetic theory, and we're going to look at binary random walks, sort of in the discrete picture, and look at how we go over to the continuum, the diffusion version. We're going to note from that that there are problems in going over to the continuum, that the continuum prescription is actually a convenient fiction. So we're going to go back and we're going to correct the binary random walk, and find a continual amendment that goes to something called the telegraph equations. Telegraph equations are just that diffusive, except they have a finite signal velocity,

2:30 and they have mean-free paths and other things that you associate with real diffusion. Once we're at the telegraph equations, we'll see that we're just a formal analytic continuation away from the Dirac equation. We just replace 1 by i at some point. of the number system gets us to the Dirac equation. But of course, this is formal, and we don't want to do that. As soon as we do that, we've lost classical statistical mechanics in the sense that we no longer have a realistic picture of what's going on. So at that point, we've got to ask the question, well, can we bring this Dirac equation back into classical stat and meth, including this formal analytic continuation? Okay, and my claim is I don't really have to worry about time in this talk a pretty new idea in this entire course, and that's contained in a single slide. This is the aha slide, and if I've prepared you well enough at this point, you'll all look at this slide and go, oh, of course, and we'll see how we actually bring in this formal analytical continuation right into classical statin met. So then I'll talk about the latest thing, which is a deterministic version of this entwined paths, and just a couple of words, the words on Boehm versus covenate, because I know, although I don't really know very much about the Boehm picture, but a lot of people here do. So that's the plan. Before we actually get into that, we need to ask the question, well, why are we doing this? So here's sort of the big picture. Now, so this is sort of standard, what we do in quantum mechanics. It's possible to do this, so think of mutant's laws. And the paradigm that we're operating under there is that single particles move on smooth trajectories, right? The fact that we've got this second derivative here means we're in the domain of smooth trajectories. And, you know, what we do is we canonically quantize, replace these dynamical variables by operators. And the thing is, there are two things associated with this. there's this formal analytic continuation, we're bringing in the square root of minus one, but also there are smoothness assumptions built into these differential operators. And the thing about all this is it's fantastically elegant. You just sort of slip by from the Hamiltonian formulation of classical mechanics

5:00 to quantum mechanics, which is a completely different domain. When we're here, the paradigm has changed completely. We're into wave propagation here. Everything goes well, of course, until we try to get back to classical physics. And that's where it's very, very tricky. This is where the measurement process come in. And that part is really tricky. So we're going to avoid that and say, well, let's have a closer look at this, because this is where we go to classical physics. May I just ask one sort of quick question of clarification? These things that take part in the random walk, can they overtake? Can they overtake? particles overtake each other we're talking about one dimension yes so this is all happening like a long straight street yes can they overtake can they pass each other can they pass each other okay well actually um in the end we'll see that there's only one path and so when you say they pass each other they actually pass each other in different segments of the trajectory but yes they I think that's the short answer to the question. OK, so let's have a closer look at what we really want to do. What we really want to do is we want to look at this canonical quantization. Here's the canonical quantization. And it really has these two parts that we really want to inspect. One is this formal analytic continuation, which brings in the algebra, It takes us from the real number system in the Schrodinger equation, real numbers to complex numbers. But there's also this geometric thing, the differential operators. So we're going to look at that fairly closely. And one easy way to do that is to say, well, okay, let's look at canonical quantization, but take out the square root of minus 1. So then we're just asking about geometry. So here's the canonical quantization without the formula of the continuation. We're just going to replace E by this differential operator, E by this. What do we get? Oh, well, we get the diffusion equation. Well, that's nice because we know everything about the diffusion equation. We know there's an underlying realistic model for the diffusion equation, namely Brownian terms, random walks.

7:30 So what we're going to do is we're going to have a look at the geometry random walk, so I'm going to lie the fusion equation. And we can do this just by considering a discrete random walk, OK? So here, we're looking at a binary symmetric random walk. So if we just look upstream, this is t, this is x. We've got lattice delta n to delta t. If you're sitting here, you ask, OK, where did I come from? Are you going to come from here or here? Probability 1 half of each. So here's the consummation of probability. Now we make a slight deeper phase and say, well, okay, supposing we refine the lattice, and so this, in a sense, the street function is really an approximation to a function defined on the real line, and that it's smooth. If we do that, we can expand U in a Taylor series and just collect a lowest order of terms. Well, because of this direction minus x and plus x, the first order of terms and space cancel out, and you're left with this. Okay? So you're left with this, and so you can say, okay, well, if I refine the lattice in just the right way, so that this ratio is a constant, you go over to a partial differential equation, right, which you recognize as being the diffusion approach. Now, the first question I ask is, well, does this make sense? It doesn't make sense to take the limit as these two lattice basings go to zero in such a way that d is constant. And here, you have to take part of this on faith, but in actual Brownian motion, that is sensible. Okay. You look at Brownian motion, you find that this ratio does, in fact, approach a constant. the result that gives that to you from a random walk is this statement you may recall from your statistical physics the mean square n-to-n distance of an n-step random walk increases n and the thing here is the square of the n-to-n distance that increases with n another related statement is the fractal dimension of a Brownian path is 2 And if that doesn't ring about, there's another way of looking at this, and that is that if you look at the speed of the particle, at lattice space is delta x, delta t, the speed is, of course, delta x over delta t.

10:00 And so since there are only the two speeds, either plus this v or minus this v, v, over using the symbol delta here, the variation of the velocity is proportional to delta x and delta t. So this statement here, being a constant, is the same as this thing. The product of delta x, which is a measure of your accuracy in x, times delta v, which is measure of the variation of the velocity, is asymptotically a constant, okay? So that probably does look familiar, but from a different context, namely quantum points, an uncertainty principle looks like this. And this, in a sense, is the classical uncertainty principle. It actually calls for Brownian particles if you measure it, and you measure Brownian particles. And you measure this delta x, delta x is the distance over which you measure in time of is the variation in the velocity, some measure of that, and that will be asymptotically constant over whatever range of delta x you choose, from the size of the box down to the order of the mean for your power. Below that, this will fail. So this is the class of uncertainty principle, and the thing to notice is that in this context, it's completely geometry, It's not algebra. It has nothing to do with algebra. It's just the geometry of the powers, these very crinkly fractals. I mean, you must be very, very close to the fractals, I think. I mean, the thing about the way it's dangerous to throw away high, is it's not clear that you can almost do the analysis. Absolutely. But we're going to come back to high. But the reason we haven't got I here is I in the sense muddies the water. So we don't really know what we're dealing with since we put the I in. Here we know exactly what we're dealing with. I understand that. As soon as you put I in, you know you're dealing with a completely closed. And we know exactly what we're dealing with. Well, maybe you have a quantum mechanical brain and I have a classical brain. But, you know. It seems to me, you know, we know what the density represents when we're looking at the solution of the fusion equation, but no one here, I think, can really tell me what a wave function is. None of it really knows. It's part of an algorithm.

12:30 So, this is a summary of looking at this canonical quantization without the formality of continuation. This requirement, this canonical quantization, it converted these smooth classical paths to fractal ones which obeyed this thing. So it took those smooth paths, wrote them into these non-differentiable curves, And the final result was that the description at the end, according to densities, is that's what's made them smooth. So, two sort of opposing things. It also shows that the implied continuum limit that we looked at is a convenient fiction. And it's a convenient fiction because it leads to silly things, like there's no inner scale for diffusion. And the signal velocity is infinite. this thing, if that thing is asymptotically a constant, then as delta x goes to zero, delta b goes to infinity. So the velocity goes up to infinity. Which is just saying that Wiener paths, they're only an approximation to Brownian paths. Real Brownian paths flatten out at some inner scale and are no longer frank. So what we want to do is we want to We want to remove these silly things that the limit is saying. And how do we do that? Well, Mark Katz, the model of the telegram relations, does this. And this model is not terribly well known, but all it does is the following. What we're going to do is we're going to fix the speed of the particle and then just have it be buffeted back and forth in space with its speed fixed with its velocity being, you know, plus or minus its speed. So what this looks like is if you zoom out on a trajectory, it just looks brownie. It looks like a random wall, right? So on scales from, say, meters to microns, it still looks just like diffusive paths.

15:00 But if we get down to the scale of the mean-free path, it now just looks like these broken line segments, same slope of each segment, a single speed. So that's the analogy of having, say, a particle being bucketed about in air. Because it has a mean-free path, eventually this crack of behavior smooths out. Yeah? It's just a slight historical question. Feynman, who came up with this basic idea first? I don't know. I wish I did. Because Feynman just says, actually it's a problem example. He says, you know, right, these functionals could have various many forms. blah, blah, bing, bing. Of course, this one page is made history. I wonder if it's conceptually fond of this place on the hundreds. No, it looks just like, no. That's not true. If you read it, it looks just like that. I just fond of it. I don't know the one that's not pleasant. I like it. Just because when you blow it up, you get to fine detail, of course, there is self-similarity. It doesn't, no. But you actually find that there is statistical self-similarity when you do this. If you govern this by basically a Poisson process, then once you're back in here, it is self-similarity. So you don't notice the difference between the math and the math. How do you know? Well, basically, one thing you can do is you can do a computer simulation. But there are analytic methods that you can look at these things and see. In actual fact, at this scale, you really can discriminate between this and Wiener-Pounds. because only when you get close to seeing this that you see, oh yes, there's a difference we have a fine scale here what about the second source what happened first? well let's so let's just write down the difference equation that corresponds to this picture okay, it's the same sort of thing as we did before, we're going to break it up into two two directions, a plus and a minus

17:30 And we can write a conservation equation, right? So plus means going in the plus direction, minus means going in the minus direction. And if we're going along one of these paths that's, you know, almost everywhere straight, you just run in these occasional corners, it's going to go straight with some large probability, like 1 minus m delta t. Delta t is a ladder spacing. We're going to think of shooting that as 0. So it's going to persist in the same direction with a large probability and change directions with a small probability. So this is just conservation equations for this little model, written in a discrete way. And we can take the continuum limit of that in such a way that the velocity, delta x over delta t, remains 1, or c, or whatever you want to call it. When you do that, and just write the equations in two-component form, this is what you get. Now, you may not recognize this as being how you usually view the telegram equations. It's just one way of writing it that Marquette's found. So, sigma z is, these are the two power matrices. It's a two-component system. The sigma z, the plus and the minus, this is a remnant of the two different directions. The sigma x refers to the scattering term. And if you iterate this twice, you get a second-order equation that will look more like the telegraph equation you're used to seeing. It's just like the Dirac case, right? You iterate the Dirac equation twice, you get the Klein-Gordon equation. You iterate this thing twice, you get the Klein-Gordon equation with the change in sign of the mass, or the mass squared. So here, we know all about this in classical Citroën mechanics. Notice that we just add an i here. We get the Dirac equation in one dimension. So this thing here, which is all we've done is add a knife going, so just add the end. This is a representation of the Eran equation in one dimension. Now, that's pretty standard, right? You can go from classical equations to quantum equations just by a formal analog continuation. You'll be all familiar with this, right? Here's a fusion equation. Just stick in an eye at the right place, and you have Schrodinger's equation. So, what we want to do is we want to have a closer look to see whether we can get rid of this formulation, or at least understand how it works. So, now we've finally come, run it on route, to the Feynman chessboard model.

20:00 So, here's Feynman's prescription for the pattern integral version of his model. So here, it's a sum over r where r is the number of corners in the top Here are these, you know, cast paths but here we colored them with two colors The reason we colored them with two colors is simply that if we put an i in here then, you know, the periodicity of i is 4 So, if we want to see what the i does to this let's just expand out this propagator Well, what it does is it, if we write this in terms of real and imaginary parts, here's the real part, so we just skip every two corners, and we get this minus 1 to the power k here, and minus 1 to the power k here. So the effect of the formal analytic olytic continuation is to take what would have been a partition function, right, and made it into an alternating And so, in this expression, this i is, in a sense, unimportant. Because this i, all it does is distinguishes between the i to the left. The actual algebra induced by i comes from the alternating portion up here. So, hence the two colors, right? If we look at this trajectory, we can say, okay, the weighting of this thing is 1, the weighting of this thing is i, the weighting of that thing is minus 1, minus i. OK? So to distinguish the plus and minuses, it's just by color. And we can distinguish the direction just by the direction. So let's just have a look at what this is doing to base. And this may be of interest to sort of this whole program of looking at discrete physics. Notice what is really happening here, right? The actual directories, they operate on a discrete clock, right, blue here going to the right is 1, plus i, minus 1, minus i. The only thing that actually occurs in these trajectories are the four points of this circle. And it's the actual geometry of the trajectory that actually smears this out to give you this. defocusing a bit and saying,

22:30 okay, what about the regular fine and pattern rule? Fine and pattern rule And the non-real-grist one starts here. It starts in the continuum. So you don't see any of the geometry which really gets this from the discrete. But the discrete in this chessboard model, clearly each trajectory only knows this discrete alternation of colors. So at this point we ask the question, is there a discrete characteristic function that will allow us to parameterize the entire propagator with a single power now there's a good reason to ask this question when you think of the path integral formulation of quantum mechanics it falls short of being a realistic model of quantum mechanics in the way that linear powers are a realistic model for the diffusion equation in the sense that you can't think just moving along a finite path, right? A diffusive path, you can think of moving along the analog of a Wiener path. But if you try the same thing with a finite path, it all falls apart, right? So you imagine that, and you say, first of all, I don't know what phase is. And second of all, wave functions have nodes, right? And the nodes are places where the particle cannot be seen, whereas paths go through nodes. every physical particle has to be associated with a whole ensemble of Feynman paths. So, if we could find this parameterization of the entire ensemble of Feynman paths, covered by a single path, we'd have a link between particle and wave in the actual construction of the problem. Okay, so that's what we want to do. And all you have to do to do that is have a slightly closer look at these final patterns. Okay, so here's one of them. And here, remember, the colouring is important. It's actually the colouring that extracts the pattern that we want. In fact, we've got two colours here, and the colours change at every two corners. Well, there's one. Here's another one, same sort of thing. And the distances here are just governed by a Poisson process. Now, what I want you to notice is what happens when

25:00 we superimpose these two special, ah, they're special in their relationship. Okay, so of course this is the special slide. And here I just want you to focus on color. Now, our initial problem was to say, well, how do we get this peculiar alternating pattern of colors on the trajectories? Feynman has to do it by inventing this rule which doesn't seem to pertain to nature. In other words, when we extract the Durant propagator using one of these, we're putting in the coloring by hand, and we don't know how nature could possibly work. However, if we pair these guys up like this, then we can say, oh, well, all major has to do is go forward in this direction on the blue, and then come back on the red. If this axis is T, then the coloring is automatic. You associate a plus one with the blue, a minus one with the red. the other nice thing you get for free is the fact that once you've traversed this whole thing and come back you're set up to just do the next member of the ensemble you're back at the same point so you can write the entire ensemble of pairs with a colouring with a single pair alright, so that was the aha slide everything else I have to say really sort of follows from this, you can test this. There's a potential fly in the ointment in the sense that quantum mechanics itself, in all this form, when you start trying to find these propagates, you realize it's actually ill-conditional. The patterns you get from the Schrodinger equation and the Dirac equation are very delicate. And if it's embedded in a diffusive background, any slop over from the fusion will wipe out the pattern. So, with this prescription, you're not guaranteed that the thing will converge. If any of the fluctuations that come in and slop over into the eigenspace that corresponds to these patterns, you're out of luck. Then you'd have to wipe the entire final ensemble completely without any fluctuations.

27:30 So here's a rather bad drawing of the propagator, and this isn't done with weight. This is a single part. This is 10 to the 3 passes, 10 to the 5, 10 to the 7. But the sort of janginess is part of the fact that it's a single matrix, and so it's out of that. At this level, at 10 to 7 parts, it's visually indistinguishable than just solving the discrete direct equation and writing this deep down. So you can do things like put this thing in a box and see if you get the energy levels. Let me just summarize this point. How am I doing on time? Oh, good. Well, 20 minutes. 20 minutes, all right. Okay. So, there are, in a sense, three ways of looking at the climate propagator as well. The first way is the traditional way and say, okay, the climate propagator, what it really does is it counts these chessboard paths, and of course it counts them with the rule that it's a plus one for the blue and a minus one for the red. So it's a sum of little powers or a sum of little histories, just like the regular one. Now, having looked at these entwined powers, we can also say, well, another way of interpreting this, to give us a better handle on what's being counted, is to join these guys out and say, what's really being counted are these virtual pairs. We look at these two things and say, well, all right, what is this thing? say an electron, in which case this guy here is a positron. And we reverse the signs on this and say, oh, well, okay, this is a birth, death, birth, death. And all we're doing is we're counting those. What we're doing is we're noticing there are populations shuffling back and forth. This is, we can call this sort of an adolescent pair. It's just been born. And this is a senescent pair because it's just approaching the knowledge. those two components of the propagator, all they're doing is they're watching, they're counting the flow back and forth between the adolescent and the senescent. So that's sort of the picture in terms of these virtual pairs. But the picture that I'm sort of promoting is to say, well,

30:00 look, we get a little bit extra by seeing that we can And that gives us a little bit extra to play with. Now, so you can sort of work with this, and I wasted vast quantities of time on a parallel processor, a parallel machine at NASA, looking at a particle in a box. but let's just ask a question that is unfair in the Copenhagen interpretation so let's notice in the entwined path picture the dynamical process builds in the uncertainty principle with each trajectory each one of these trajectories when you stand back from it is distraught it has fractal dimensions as building in the uncertainty principle geometrically with each path. Just like in the non-relativistic path interval, the paths themselves are these fractal curves, and it builds in the uncertainty principle with each path. So, you can ask the question, well, does the uncertainty principle actually come from the dynamics, or not? really that strictly speaking Come and Hager interpretation is mute on that question because it doesn't believe that the such thing is a particle and a different theory Bohm on the other hand is quite explicit about this Bohm says no you don't need that because in Bohm there really is a particle and a path and that path is smooth so it can't principle geometrically. So, this suggests the question, what about entwined paths? Can we remove the fractal geometry from the dynamic process? So, can we get rid of the stochastic part in the propagation? And the answer is yes. I'd sort of like to show you a demonstration. Unfortunately, we don't have an overhead projector. I don't know whether everyone would be able to see it if I just turn the screen around. Maybe we can, let's just try and see.

32:30 It doesn't work on the top of that. So, okay, the idea is simply to learn from this chessboard model. what really makes this thing work is this alternating blue and red let me just see I don't know if anybody can be able to see this I think it is can anyone see that it takes you 10 If you sort of move in a bit, you'd be able to see it. Now, okay, so what we're going to do is we're going to construct one of these trajectories on the left here. So on the left would be the path. And on the right, we're going to construct the densities that we get just by counting the path as it goes by with a plus or minus sign or a minus sign, depending on whether it's going forward in T or backwards in T. So, this is the problem. Okay, so here's a directory that's going forward in T and it's registering these two components. And we just go there and back. We're going to do it again, just slightly shifted. And you can see that what you're building is, in actual fact, something that looks like a square wave. We're going to keep on going, shifting a bit more, just to see what we get. I mean, the way you put these pires together, you can put them together any way you want. I'm just putting them together in a way that's, I think, fairly obvious as to what it's going to do. So here we have this complicated-looking trajectory over here, okay? And what it's produced is this series of alternating delta functions, shifted by pi by 2 from the two different directions, okay? And it's very simple. Here's the trajectory. Here are the densities that it creates. And now I'll just see, show you the next one, and I'll copy this out the thing upside down.

35:00 Okay, so we're going to go a little bit farther with this. We're going to do what we did before, but then we're going to use those double functions to create something that's very familiar. Same thing with trajectories on the left. We're building densities on the right. So these two guys, they're just, think of little gnomes sitting on space-time. They're just counting. counting plus one when it goes forward and minus one when it goes backwards. So we've now speeded this up by a factor of ten. So here's the complicated-looking directory. Here's what we've built. And they're the two components of E to the IMT, one on each side, separated by direction. and really the thing to know is our method of generation was quite arbitrary but you could do it in a realistic way so you could write this on space the thing to know the argument with Newton over language is simply that when we start with quantum mechanics here is where we start we don't start here we start here so we miss all this Here we went from the discrete to the continuous, and if we start here, we miss this. And the point is, in a sense, we created the algebra by going from here to here. So that's sort of it. Well, let me just show you the sort of things we can do with this, or what we have done. We haven't got very far. Basically, we're looking at a particle in a ring. Let me just show you what these things look like. And, you know, all we've done is that the B is equal zero case, we can get the, we can get the rest of the propagator simply by doing more S-transformations. So here's, here's the sort of, the B is equal zero case, we just shift this to either side of the cone.

37:30 When we do this, this of course is cheating because all we're doing is relying on the printer, interfering with these things, right? But this is the sort of pattern that we get. If we do it properly, you can do it properly. Here's an example. This, which of course is trivial to generate using waves, this is just a manifestation of a single trajectory going back and forth in that same way, generating this thing. And that is the, you know, that's one of the components of the propagator. That turns out to be a very, very big statement. I can mention this, please. I can prove it. You will. I can show you a paper. We're Jessel. It's a very, very great expansion. It proves my statement that you're using higens, higens, formalization, higens, higens. I think it works in freedom actually anyway because you can get a formalism for this. I'll talk to you about it. I'd love to hear it because the next thing that needs to be done is, if there is a formalism for this, you can't do it. While we're interrupting you, we don't have your email address on our list. well I can easily fix that up can you just write it on the board just to tell it now it's easy So, the latest thing we've done is simply to put one of these things in a box. So it's just georundie at ryerson.ca So this is sort of fun in the fact that it's incredibly ordinary, but the only thing that's not quite so ordinary about it is the same thing, a single particle going back and forth on X and T. And then this, so this is the standing wave, the first excited state of the particle on the ring, single particle trajectory, and you can see this nice pretty

40:00 picture that's easy to generate from waves, but it does take a bit more effort to generate it from this single path. So you want to check that you get the eigen, the energies right, so here's, just from doing that, you know, plotting the energies versus n squared. So here these dots are the actual things. It's a straight line that just gives you the behavior. So it doesn't do well. This error bar is assuming the worst, that you are out by what we call a Planck length, which is just the smallest length of the discretization. So here, when you build those, in a sense, the trig functions, you build them from these histograms and the We let them go long enough that the errors are pretty small. So this just says, in a sense, this is roughly at the stage of all quantum theory. In that it says, put these things in a box, they understand what the stationary states are. I don't want to be confused with the other stationery states. Okay, so having said that, I've skirted around absolutely the most difficult part of the problem. We all know that you can create all this stuff quite easily in waves. The problem comes in when you get to the measurement process. And here, all the rest of this is speculation. But the good thing is it's testable speculation. So, you know, we've got this single-pound propagator which gives us the U-process. What about the out-process? So this is the reduction. So the Copenhagen version of this, imagine you have some source of electrons, and, you know, the particle spends all its time drawing this propagator on space-time, and you imagine the detectors here, and the particle can sort of go through and come back. When there's no detection, all it does is spend a lot of time drawing this propagator on space-time. But then imagine the detector goes click. Well, in this system, to make it look like Copenhagen, what happens is when it goes click, it says, alright, you're no longer allowed to rewrite history. It can no longer go back within time to add to this propagator.

42:30 So from then on, it's pushed into the future, and so you have this analogue of the wave function collapse. Now, in order for this to work, of course, for it to be meaningful, is when you test it, the Born Poster must come out of this. There's no point in putting the Born Poster into this, otherwise you've lost the advantage of having this being a realistic model. So if this thing means anything in terms of nature, we should be able to do this and have it so the detectors do indeed detect according to the square of the amplitude of this thing. But of course that has to be done, but it's a doable problem, we just haven't got there yet to program it. We're pretty close, I think. So that's the Copenhagen picture But of course we have this other picture Which in some ways is I think better than the Copenhagen picture Not quite so popular That's the Bohm picture And remember that in the Bohm picture There's this quantum potential Which you get from Stein But then you have this other feature You have this smooth particle trajectory now just notice what happens with these entwined paths because it's worthwhile seeing that with the entwined paths even in the Copenhagen point of view when the detector goes click we're left with one unpaired entwined path right, these things go back and forth like this, once you prevent this from rewriting history you've got a blue line going all the way from here to here and that's one path plus the drawing on space time this entwined picture has within it really these two things. So this seems to me to suggest, well, Bohm is probably a pretty good picture of this. So the idea would be if you wanted to do this in the final picture, you'd say, okay, let the particle draw a sign on spacetime and then use the accumulated pattern to determine where the source is to on the last departure. So it just works its way along the pattern that it's drawn, which is, you know, in a sense, exactly what the bone picture is saying.

45:00 So, you know, that's where it is, and, you know, it's still very much in the exploration phase. But just to conclude, let me read you a concluding paragraph of a very lovely paper that was written in 96. You probably recognize the authors in the language. But I think, you know, this sums that point quite well. This is talking about the chessboard model. And it says, one may wonder, why does this simple combinatorics occur at a level so close to making one distinction, the solutions to the Dirac equation in continuum 1 plus 1 physics. We cannot begin to answer the question except with another question. If you believe that simple combinatorial principles underlie not only physics and physical law, but the generation of space-time itself, then these principles remain to be discovered. What are they? What are these principles? It is no surprise to the mathematician that I ends up as central to the quest. For I is a strange amphibian, not only either 1 or minus 1, I is neither discrete nor continuous, not algebra, not geometry, but a communicator of both. In this essay, we have seen the beginnings of a true connection of discrete and continuous physics. And the last statement is, very nice, Here we have a glimpse of the possibilities inherent in the complete story of discrete physics and its continual limit. the continuum limit will be seen as a summary of real physics. It is a way to view, with glass darkly, crystalline reality of simple quantum choice. Of course, that's Peter. That's Lou and Pierre, and I certainly couldn't say it in my name as well. Thank you for listening. now a question may I write something on the board because I think there is a formalism which can do this which can this is what you've done is incredibly useful to me personally because now I have a model of what it looks like what something looks like propagators. This is conventional propagator, something like this. And then there's a lot

47:30 of stuff, it doesn't matter about that. And then there's two terms, minus theta, t minus t prime. And then summation over two spins. R equals 1, 2. This is x, of course, is four vector. This is rosavistic. And this is just a conventional one. Right, that's conventional propagator. And there's two terms. I'm sorry, I've only written one of the terms. The other one is cross i theta, and notice the i there, t dash minus t, that's the reverse time one, as opposed to the two different colours, which is presumably what that's all about, you know, the two side directions, and then there's 4 r equals 3, and then side x, something like that. That's conventional propagator. Summations over four spin states, but two particles It's an anti-particle, basically, verse time, forward time. Now, in the nil-potent formalism, there's only one propagator. There's only one term in the propagator. It looks very similar, but there's a big difference there. Let me write the thing out. Sf, x minus x dash equals, and then there's d cubed p, and then all this stuff. And then the operator, t minus t dash, and then psi x, psi back. So there's only one time direction. Well, there's two time directions, but the time direction matches up with this. And the reason is because when you write down the Dirac equation, what you write down is for, you match up your energy terms with your time terms. There's two energy terms plus, so that should be d by dt. d by dt minus, there's four terms in the row or column matrix. d by dt, sorry, that should be. And then there's the p terms and the d del terms and all that. Not interested in those. But there's four energy terms. Well, actually, I'm, because that's spin up, spin down, spin up, spin down.

50:00 And they match exactly the energy terms in the wave function bit. So the plus e will always go with a plus t, and the plus e again will go with a plus t, and the minus e will go with a minus t. there's a matching up, exact matching up of terms in that and that and that's why you can write it as one propagator, so you can actually get one formalism for the whole thing which is precisely what you want it's also four dimensional and so it shows that your model will fit four dimensions as well as two dimensions that's great to hear I mean I'm very, your model is extremely useful to me because it gives me a representation of what I'm trying to do So, you know, can you see what I'm getting at, though? This propagator's only got one term, and this is where your eye comes in, because you split the terms and get a pole between them. In conventional theory, there's a pole between two terms, and that's where your eye comes in. Here there's no pole, because there's only one set of terms. There's a package of four terms which all go together. Right, and I think unless the speaker wants to respond to this, we'll go on to another question. Yeah. but I think that's very important to the speaker this particular point I think there's something to talk about there just have a look at that, that's the page on which that's written no, does anybody else want to say it? I think that's true for me instead of splitting it up into two it's all packaged into one this is another sort of historical thing There's a funny story, you may have heard it too, about Wheeler calling up finally in the middle of the night and telling him that, I figured it out, there is only one. Why all the electrons look identical? There is only the one. And Feynman said, remarks to this, I never understood what that crazy guy was telling me. Have you got a light chair of that? No. Wheeler. Yeah, he actually was Wheeler calling out a climate. And in a matter of fact, that's all this is. Instead of having a one-electron universe, you've got a one-electron Dirac-C. So all this thing is doing is building the Dirac-C in a single time. You have to track this.

52:30 Have you sent a copy of all this? No, I haven't. I love talking about it. How long were you there? I'm actually going back on Monday. I'd love to chat to anyone about it. I don't want to make a background noise. Did you wonder at all about my question about whether these things in the random walk can pass each other? Can pass each other? Yes, in actual fact, they have to, in the sense that... Well, I thought they'd have to, yes, so I couldn't quite see how to be one of them. can see that. Here's this slide. In a sense that there's a bit of a cheat here, in the sense that here we're thinking about one dimension, and these guys, if this is one path, it has to hop over itself, right at these intersections. And we're sort of forgiving that in the sense that we're thinking, well, okay, this is embedded in a three-dimensional space, and there's plenty of room to hop. Actually, there is another way of looking at this in which you don't have to expand nearly as much work to build those trig functions. If you think of this as a projection from three dimensions of a double helix, so the guys are doing this, then the square wave that we looked at is actually what you get from that when you just project it. So this could be a representation of a double spiral that comes back down. The thing is, I don't know whether the physical interpretation of that works for the graph equation.

55:00 Well, yeah, I mean, if it works great, then I... Supposing a vacuum. Not just a model. That's where they actually get anything. Can you give me a simple explanation? What is the difficulty when I'm thinking about that's more than two people? I'm just hearing that there is a difference. Okay. There is a considerable difficulty in the sense that we built this thing on a lattice, right? And so we now say, okay, suppose you want to do this in three-dimensional space. Well, the reason these things move with constant velocity is simply that c is this characteristic velocity that you've got to pay attention to, right? It's there. It's part of the spectral relativity. But the idea behind the sum of all paths is that when you build in all these half paths, if you're sitting in a three-dimensional space, you expect the path to have extent, not just in one dimension, but in three dimensions. When you try and put that on the lattice, You can't have the constant speed and the sum of all the paths and all the directions and maintain both of those at the same time. You know, you keep an average speed as being c, but you can't keep a fixed speed as being c. So you have to try and drop through hoops to go around these things. And I haven't seen one that really works. there's anecdotal evidence that a friend of mine and I found, which was a peculiar thing about the Dirac equation, that if you ask for the propagator between two points here, you try and write that as a sum of the powers, you're out of luck unless you do the following thing, you rotate the z axis to that direction when you do that, you see that you have two 2 by 2 systems instead of a 4 by 4 system and then you can write the propagator as two chessboard models. And those two chessboards actually correspond to these two guys there. That's what I mean, those sets of equations are in the back. Two by two. Yeah, I'd love to see how this thing works. But, I mean, that is sort of the barrier. If you embed this thing in three dimensions and you want to bring in the paths in the other two dimensions, you've got a big problem finding a lattice and doing that.

57:30 And, you know, right now, at this point, the evidence really seems to be more anecdotal that you should be able to do it. Somehow the geometry of the Dirac equation knows how to do it, but we don't. Well, that's a drink of coffee, and thank you very much. Thank you. But I don't need it here. This is just composition. So, it is... Lots of things are not associated with it, so... But let's go on, because there's certainly more to cover here, unless you think there's something more important. taking the permutator two things is not slow but this operation of taking three derivatives on something but this is so you never check anything and you have to do it at all you're never checked no i never thought it i never checked it through i i find that i often So I'm trying to set up this thing with as few assumptions as I can. So here we are. Commentator equations say that we're operating in a more or less black context. but there is some, but that, about my description of that is, um, um, actually I should have said we're operating in a curve context, I mean, right, because the whole point of that is that the field is coming from the curvature, so, um, sorry about that, but there's no space in it, and let's let go of the rest of the assumptions, so there will be no commutator equations.

1:00:00 so then what assumptions will we have we'll have three concurrent processes and we'll have the process derivative x dot and we'll have definitions of derivatives d sub i will be the commutator of f with xi and dt is this where that means sum over i And that's it. Those are definitions of derivatives. So this derivative here is, even as in the old derivation, it's really a covarian derivative. It's got curvature. But on top of that, it doesn't delta i, j, b, xi. It's just giving you sort of some answer. Well, when you say three concurrent processes, you're getting towards space now, you mean? Yes, that's correct. Would you, is it obvious it's the same J as five species? No, not obvious, but it would make matters quite complicated to have different Js happening for different processes, so I didn't try to figure out how that would look on any map. and then what this definition is is telling you what space and time are and no commutator so we're going to go through the Feynman-Dyson derivation again but I won't have any commutator equations we just assume that we self-calculus in this way and find out what happens It's interesting to hear what it means to see this homogized people. Excuse me? It might be interesting to hear what it means to see this homogized people. Oh, yes, indeed. I agree. It means that it's like a program universe where the tick of the clock causes these three sequences to approve a new message. It's not the same thing, but I knew it was, yeah. On the previous slide, you said we're operating at this curvature. Yeah. Just below that, because we have no space here. Right. Ah, no, about curvature. I better, I use the word curvature without using the word space. And I better clarify that a little.

1:02:30 Here's a derivative, bk, defined by the communicator with xk dot. And then xk dot, formally, could be written in terms of something else, like pk and something else. I just want to show you a curvature combination. If I figure out how these, so the pk is supposed to be the one that looks more like an ordinary derivative. They commute with one another, and they delta ij with respect to the xk. define dfdxi to be equal to the commutator of f with pi, and I assume this, then those two really would really look like d by the xi. So I can try referring to a flat coordinate system like that. Curvature has to do with the relationship between one way of differentiating, one way of making distinctions, and another way which you've decided is flat. And the other point about abstract curvature, I'm only talking about in the most abstract way, is that by definition, curvature is happening if some derivatives are not commuting with one another. So this matches in with what turns into curvature when you do actually have space. But I use the same term. I hope that's all right. You look like you want to say something. But is everybody clear about what I'm saying? I'm saying that, for me, curvature is the fact that something doesn't commute in the sense of taking derivatives. So I want to find a difference, and then I find another difference, and I end up in a different place. So, for example, if you were actually talking about curvature, and you watch the difference that you get by going around a closed loop, then you will find that something changes, and that indicates curvature. So, for example, I do my best to keep the tangent vector to the sphere going in the same direction,

1:05:00 and I go up here and down here and come back to where I started. But in fact, it's turned by 90 degrees and that's measuring the curvature. That change going around the world is measuring the curvature. What happens here is that if you refer to a flat background in that way and then measure the curvature for this guy, you find out that it's equal to a very familiar formula for the curvature of the case. But that's referring it to as flatness. Let's not worry about that. Let's continue with these assumptions. The assumption that's most interesting to think about, I agree, is the one of Simon Panini. Why should I be taking an automaton, which is ticking everybody at the same time? I think it would be interesting to have to go with that. So now we have this die, this covarian derivative. And there's no t variable, but on the other hand, if you did want one, I could give you one. It would be an discrete t variable. t dot should be one, just as a sign. So jt prime minus t is one. So that says that the tick is at the operator level. The values of time are J inverse, J inverse, plus J inverse, J inverse, plus J inverse, plus J inverse, and so on. Perfect. And an ET value is given by F dot minus that sum of derivatives. So that defines the fundamental relationship of space and time in the model. So now let's find out what happens. now I have another bit of machinery of course because this is going to look this is going to be electromagnetism and that bit of machinery is the vector cross product there are curls and so on in the calculation I won't do all the calculations of course but I'll do some of them and it's fun the basic identity that I need about epsilon is written just beneath that diagram so diagrammatically this is epsilon epsilon 1, 2, 3 is 1, epsilon 1, 3, 2 is minus 1, epsilon 1, 1 is 3, it's equal to 0. If there's a repetition, it's 0, right? It's the sign of the permutation otherwise.

1:07:30 In the diagram, this is the 1, 2, 3 direction clockwise around the vertex. And then this says that epsilon A, B, I times epsilon C, C, D, I equals delta A, C, delta B, D, minus delta A, B, delta B, C. Which you may have seen before, but if you haven't, I'll slow it down for just a moment. I use this all the time, and I'm just going to use it without worrying about the indices. It's easy to see what's going on here. Suppose you put a 1 and a 2 in here, and a 1 and a 2 in there, right? there would be a 3 in the middle, and this would be 1, 2, 3 plus, and this would be 1, 2, 3 minus. So the total evaluation to the left would be minus. On the other hand, if it's 1 and 1 here, as I said, sorry, let me write it down. So you can see it 1, 2, and a 3 in the middle, a 1 on the left and a 2 on the right, there. so then that has value minus 1 because this is 1, 2, 3 in this direction and this is 1, 2, 3 in the other direction and over on the right hand side one matches 1 and two matches 2 so it's coming from this one which has got the minus and if you flip one of them it'll go over to the other one so the two deltas conspire together to do what the two epsilons did All right, it's the basis of everything. For example, here's the vector cross product. The vector cross product of two vectors, a vector has an index, that's an index, is epsilon ijk, aibj. All right, and here's the dot product. And here's the triple cross product. And then if we use the formula that lets us expand two interacting epsilons, when they go parallel they get a minus sign

1:10:00 and when they cross over they don't and you read off here things were commutative minus a times b dot c plus a dot c times b and you find that if you're in a non-commutative situation you've got a slight generalization of the formula because b is sitting in the middle it can't be uncommuted for fair ok, so we'll use that in fact, we'll start with this here's h, x dot cross x dot and I certainly have already checked because I have a previous slide which I didn't show you but I'm checking it again on this slide the divergence of h is here well, the divergence of h is d dotted with the cross product and that d is taking the commutator with x dot and the commutator with x dot means you take this product in this order this product in the other order. But now you see these are exactly the same. So the diagrammatic proofs let you do it without worrying about how the indices went, and that's why the divergence of H is zero. So we get an explanation of why the divergence of H is zero that way. Then what about E? I hope I've said enough to you so you can kind of have some fun following this algebra. Because some of it's cute. We're going to define E by that, Okay? So that's the same as saying what's x dot cross h. So we have to compute x dot cross h. Oh, so here's x dot cross h. Why do we not just say that? Because we must be x-spatic as well. No, no, no, no, no. No, no, no, no. That's scale. That's scale. That's actually because the scale's a different number. That's better. The microscopic scale is not a scale. Because it's based on a different set of interactions. It's based on atoms and molecules, which are electromagnetic. You mean there's some clear forces. Well, what sort of force is driving the edge function? Some sort of unknown force, is that the other day? The expansion is gravitational. I mean, we're talking standard. I don't want to give too much time to talk about that. The gravitational scales are different from microscopic scales. Well, my question is not changed. Both big people didn't have ratios.

1:12:30 Since we brought that up, and I don't mind it being brought up a question for you, James. The Hubble Laws was first understood as linear. What comment did anyone offer about how to think about that in the extreme where you're looking so far out that the speed would be greater than light speed? That's a big problem. That's very important. And in fact, you can throw an even worse electron, which is like it's accelerating if you look far enough away. And let me, just so that you see why people think that, I'll try to explain it instead of a little bit. When stars blow up, they have certain, what's happening is there's nuclear They're fusion, they're supernova because eventually they run out of fuel. That happens when they hit iron, when they fuse enough so that there's iron. So there's a certain isotope of iron that occurs. It's like you have this hot fire and you're throwing ice on it because you can no longer get fuel out. So you cool it and then there's a unique way that the star blows up. And then you look at that isotope of iron, and then you know the speck, that it admits, that it's a speck, that it's a speck, that it's a speck, that it's a speck, that it's a speck, and so when this thing's going away from you, if you know how rapidly it blew up, you kind of have an idea how much ice you threw on the fire as it's blowing up. So you have an idea of this particular amount of iron in the ground. So these curves of the supernova truck, as they're going away, you can see this Doppler thing. So the experiment is sort of verifiable in lots of different ways. So it looks like it's expanding. Now, let me respond to that because one is estimating the speed of the receding star on the basis of the Doppler effect. The equations one is using to make that estimation are the ones from special relativity theory. They cannot yield an answer from velocity that is in excessive speed.

1:15:00 And that means that the further away the star is and the faster it's really going, the more in error the estimated velocity may be if this other model actually is one at a time. And that means it looks like it's slow if it's really far away, and if it looks like it's slow and it's really far away, that means it looks like it's fast when it's closed in, and that means it looks like it's accelerating. So you're saying that you're able to explain doctorships, not when. Yes, yes, exactly. So you're saying it is receding. Well, it's old, isn't it? Yes, it is. In fact, it's a thousand million years old. Yes, that's absolutely true. Let's take into account that. Let's take into account. Looking into the distance is considered being able to look into the past. Yeah, exactly. And so they say going at what they believe to be a certain speed in the past. Let me slightly correct this. Is that what the universe was doing then? It's not really doctorship, it's the actual expansion of the way. But still, you can think of that when you get out there. Yeah. Just as a social experiment, how many people here, by show of hands, and for reasons that Cynthia has stated, and many, many other common sense reasons, find this, well, there's two camps, I'm sure. There are people who embrace this Big Bang cosmology type of stuff, and people like myself who think it just smells bad. It's crap. How many people here would be in each camp? Hold on. You have to ask one question to get answered. Yeah, okay. How many people here support the concept of Big Bang cosmology? No, no, I'm not putting my hand up. I didn't find Cynthia's object. So the next one, see the next one. The notion that it's going to happen again. No, no, that's all. How many people don't support it? How many people are against? I think that was a fall.

1:17:30 Can I call it back when I report it? Sure. The way that I ask the question when I raise my hand is that is the most, that is the easiest way for me to remember all the consistency, including the reason that the Big Bang is as cool as it is now and I can actually see the Big Bang. which is a big thing. And also the doctorships, right? All these things... Well, this is sort of repressive. Well, it represents one of the... That's all I'm talking about. How do I challenge him? What you were saying is wrong. He's saying that I'm saying the way that I think is because that's the easiest way for me to remember it. That's all I'm saying. You come up to the way for me to remember it easily, Yes. This is exactly the right attitude to take. This is the attitude of tolerate the best thing we've got and keep an open mind for what may come down the pipe. That isn't what happens, though, is it? No, it isn't. No, particularly with regard to that theory, because there is complete absurdity to not believe in it. I know. And that's what we do. It has taken on this day and become a religion. It has become a religion. In this matter, I'm in a God's day. Now, not very many people say what you say. Most say it's obviously true. It's not the same thing. I'll buy your argument every time, but I won't buy that. It's obviously true. It's not obvious. It's true. It might be true. It might not. My feeling is against, but I'll buy that argument for yours. And remember, I went into exploring this. I went into exploring the Hayden as though it is. Not believing the bad thing of claiming this, uh, is uniform. I haven't returned it for my life. The easiest way for me to understand that now is to have some global That's fine That was after I'm committed to this because I play around with different ideas, but there is a view which we can take right. The real time is our time.

1:20:00 And the relatives, despite making new signals and stuff around the services and the way we've been trying quite a long way beyond within our own galaxy. Now, well, the real time is our time. And the nature of university knowledge is that. Thank you.