Free Will Theorem

0:00 ...the equation of geodesic deviation, which is to say, where you have two 3D balling particles and it's the distance. between them that you look at, and its second derivative is discerning by the Riemann tensor. And I'm just saying, maybe that's the next step for you to get that much just out of the mathematics. We can do that. There is no doubt about that. No, it's clear now that the next step is to really look at all the basic mathematics and general relativity and see how physics is. Yeah, just one observation, of course, if you take the Newtonian to those new field equations, that is the weak field approximation where you only have one potential covering everything, you find that when you get the spherical solution of that thing, it's not just 1 over r as it is in Newtonian gravity, but 1 over r plus some extras. And as long as the extras have very small coefficients, you're all right. But you will get some small new effects. Yes, and one of the new effects seems to be a kind of hobo-puddle expansion of particles moving apart at a speed proportional to their distance. Is that acceleration? Oh, yes. So, do you think there's any satisfactory way to reconcile, on the one hand, a vision where you have discrete space for continuous time with, on the other hand, a mixing of What was the second on the other hand? The mixing of space and time that you have the Warren's transformation. Oh, oh, the Warren's transformation. I don't know how to deal with the groups at the discrete level in any satisfactory way.

2:30 There are hints about, you know, there are algebraic moves that people have done over the last years to take a V algebra and form it to form what's called a quantum verb. And Bonnberg formalism actually is about discrete derivatives in a lot of ways. So it might be that what you're asking for is something like a symmetry under the action of the algebra generalization of the Bonnberg. But that's too fancy remark to answer your question. There should be some simple sequence of ideas that shows what you really want to do. Yeah, I personally have suspicions about this side of the argument, about this relativity theory, and I'm not too hopeful about reconciling these things and just wanted to see if you were at all hopeful about it. Of course, it might be that symmetry is something that really is best understood at the continuum level. On the other hand, I wouldn't take that as any kind of assumption. I really don't want to. I'd like to see a different interpretation of this. For example, when people talk about particles and identify them with irreducible representations of some group, like S3, And then, what they really use, in a lot of cases, is just some of the common rhetorics of the situation, like the fact that the representations correspond to Young diagrams, and there's a rule for combining two Young diagrams to get a bunch of others, each one of which corresponds to one of the representations. So then, you could forget everything else and say, well, there's this peculiar combinatorial rule which takes two young diagrams and gives me these other outputs which are the particles that could happen when these two collide well maybe there's another interpretation of all of that at the level in the discrete world which happens to go over to the symmetries in the continuous world and then that would be more satisfying yeah what's so appealing about all of this is you you really get so much, at least there's a small, common origin, something like that. I mean, I think it is a big problem that you don't get the symmetry.

5:00 And this problem is not really there in the logistical level. It's very all that it's out of there in case. I mean, there's no way, I've ever seen, of doing sort of discrete calculations which are still there in some terms. Well, we really are going to start again at 11.30, so I guess we probably really ought to... That's the end of Lou Kaufman's delivery. John Conway now talks of the free will theorem. ...producing something out of nothing, but free will? Wow. Well, actually the theorem I'm talking about is really very weak because it doesn't produce free will or from nothing. It produces almost nothing but from free will. So, but it's at the level of mathematical proof, which is really quite something to say anything about, if we will, at the level of mathematical proof. But you know, I haven't come equipped with all these other people. I'd love to rise on this thing. Oh, maybe I can do this. Okay. I'll use another person's let's go. Yeah, but then I have to rub up that. Oh, let me rub it off. I'm good at that. Okay. There's a rather function like one. I like the old function like one. You're being recorded here by the way. Behind the... Through that curtain, would that be a problem? If we drove through those curtains? I might be switching my lines. It's accurate, actually. There's some more chalk down here. Yes, there is a piece of chalk here. There's more chalk on that. They've got a blackboard rubber as well. No, no. I think it's behind the blackboard. That's good. Okay. So, there's an if. I suppose it is. I really like to say, if then, on this blackboard, I don't think there's really room, so I shall say, so suppose experimenters have a free will, if there exist experimenters have a free will.

7:30 perhaps I should say by the way physicists will think all of this is really very ancient and a good deal of it is but the last 5% or something isn't and the degree of simplicity and provability if there exist experiments with free will elementary particles and however on the way I have to make some physical assumptions of axioms And I call those axioms spin, twin, and thin. And perhaps I should say that, perhaps I should say there's really quite a lot about my collaboration with Simon and Cauchin. I mean, this theorem is a joint theorem with Simon and Cauchin, who some of the speculative physicists will know as half an author of the Cauchin-Specker paradox, or whatever it's called, which is 40 years old. And this is just a revamped version of the Cauchin-Specker paradox. So let me just say that straight off. So if you think it's all 40 years old, then you're sort of partly right. Say 5% that. Okay. Okay. But, the difference, well the differences are quite considerable, but anyway, let me tell you what these are. These are the physical axioms. And fin, you will know. It's basically the assertion coming from relativity theory, that information can't be transmitted faster than a certain speed.

10:00 Now, that speed we know is the speed of light, but we're not worried whether it's the speed of light or not. If there's any finite bound on the speed of information transmission, that's good enough for us. So, say, ten times the speed of light, if you like. However, I'll suppose it's the speed of light, for simplicity. And what do the various axioms mean? well let me concentrate on spin first there are things called spin one of the particles perhaps I should say something this is really about quantum mechanics it's part of quantum logic I'm therefore not an expert in it I mean my acquaintance with quantum mechanics started a long time ago at that time I was rather more expert than I am now when I was an undergraduate at Cambridge and I took lectures from Dirac and succeeded in passing examinations for those lectures, which I never thought I'd do. Then I didn't think about it for twenty or thirty more years until I got talking with Simon and Koshin. Then we were talking about questions inside quantum logic, which in a sense he's the author of, Ian Specker. There's an earlier kind of quantum logic, definition of quantum logic really due to von Neumann, but it's sort of wrong. And the equation is better kind of is right. But anyway, spin is the following section. I'm rigorously going to obey any theory, so to speak, any mathematics of quantum mechanics or anything because I want to make it very clear that only certain very simple things are required. You see, it's reasonable to be a bit suspicious of quantum mechanics. It's a funny theory. So, on the other hand, one shouldn't be suspicious of some of the consequences of quantum mechanics because it's one of the most extensively verified theories in the history of science. But anyway, there are things called spin-1 particles, and what happens there is this. You can do something called measuring the spin of a spin-1 particle, measuring the component of spin in a certain direction.

12:30 Well, I don't quite want to do that. I want to measure... Oh, by the way, if you did that, you'd get the answer 1, minus 1, or 0. However, I don't want to do that. I want to measure only the absolute value, or if you like the square, because the 0, 1, minus 1, the square is the absolute value, and so you might call it the absolute spin. So if you measure the absolute spin of a spin 1 particle in some direction, you get the answer 1 or 0, okay? And if you measure in two orthogonal directions, well actually I wanted to go straight to three orthogonal directions. Then first of all, those three measurements commute, it doesn't actually matter which order we do them in. That's not true for the spin itself, but for the absolute spin it's true. And you always get two ones and a zero. In some order. Can't say which of the three things is going to be the zero. Hi. John, is there a similar rule to spin a half one? No. No. So your whole thesis is going to depend on spin 1? Not really. The simplicity of the argument depends on spin 1. But I don't know if you know this GHZ experiment. They use a certain analogous sort of property of spin half particles. But you need one more. And it's more complicated to describe, so I want to talk about that. Okay, so that's a very simple thing to understand. I mean, you know, you take this thing and prod it in three perpendicular directions, and you always get two ones and a zero. And I think that's enough for me to start discussing the old Cauchy-Specker paradox, the 40 year old version. You see, it's natural to think, naively, that the thing called measuring been one particle in a certain direction means what it says. That is, that there is something there, so to speak, the particle has decided already what answer it will give you. What the particle knows somehow, because it can feel itself spinning or something, what answer it will give you in any particular direction. But, you know, such naive ideas were killed

15:00 a long time ago. This is one of the best and simplest ways of killing them. So let me say what actually happens. What we do, I mean, you see, if this were true, then one could sort of, well let me put this particle in the cubical box for simplicity. And then I'm going to mark on the face of the cube some directions that I'm going to think about asking about. Okay? And here's the particular directions I'm going to choose. a circle in each face of the cube and then, well, those circles will touch the edges of the point, so I'm going to mark those. But then I'm also going to inscribe a square in each circle. Like that, and then I'm going to divide that square, sort of go with the that is in that space. And that gives me knowing more points. Now, let me say it follows from this spin axis, the 1-1-0 axis or 1-0-1 axis together with the assumption that the thing's decided what it's going to tell you anyway, look, if I let's ask about the exact reverse of one of these directions what would the answer be? well, we see these two things a one and a zero but we know there's got to be two ones and a zero so this has got to be a one And if you think about it for a moment, you'll see that the answers given for exactly opposite directions are always the same. It's just squaring. Yes. But you see, this operation is called measuring the squared spin, but it might not mean that. So I don't want to assume that that's what it means. It's just, as far as I'm concerned, it's just something I poke with the appropriate kind of equipment, and I get an answer, one or zero.

17:30 So, you see, but anyway, I just want to remark that we might as well identify opposite directions that consider them as the same, because they're going to give the same answers. So, how many directions are there here? Well, there are nine in each face, but we only need to count half the faces, so that's nine trees, making 27. And then there are 12 edges, but again, we only need to count that half, so we get six, 27, and six is 33. So there are 33, so to speak, bi-directions. And now what I'm going to tell you is, it's impossible to label these consistently with this rule. Now that's, it doesn't terribly much matter about the proof of that statement. The proof is a little bit complicated, but it's totally elementary. I think the best place for me to do it is over here. Is there another colour of this? Would you like a black felt tip? Yeah, that's fine. Let me get my head down here. Got it. I don't know where they are. Oh, that's an angel. Three more. Oh, lovely. There's a whole door that's closed. Oh, okay. Oh, look at these. Oh, wow. Well, I think I've got a bunch of houses and trees and... Anyway, so let me just sort of discuss what happens here. So, you see, what I'm going to do is disprove the idea that these answers exist before you measure them. If they do, then, well, let's consider the idea of measuring the three coordinate directions. One of those has got to be a zero, and two have got to be ones, so I'll make, with Atlas General, I'll make that one be a zero, and then the view of the fact that opposites are the same. Now, now everything orthogonal for that, everything

20:00 with right angles to it, has to be a one. I'm only marking the particular directions of this system. And I'm sorry, I'm taking a rather Picasso-like view of the cube. My apart that I can see from the sides. Right. Now what am I going to do? Well now I want you to notice that this distance here from this lower left spot to the upper right spot on this face is exactly the same as the distance from here to here. Or from here to here. Because they're diamonds at the same circle. And therefore, these two subtend a right angle at the center of the cube, and therefore these two do. Ok? That's an important remark. And so, this dot, this dot, and this dot are at right angles. I mean, they are a cool frame. So, let me just say what I've just said. This dot, this dot, this dot are at right angles. So, one of these two dots has to be a zero and the other one a one. In other words, there's a sort of arrowhead pointing to the zero. It either goes up or it goes down. And there's a similar arrowhead in that direction. So, what I'm saying is, one of these has to be a zero and one a one, one of these has to be a zero and one a one. Now, without loss of generality, that gives me two arrowheads. But two arrowheads on the square face of the cube, it always looks the same, if you think about it. So I might as well suppose that those two arrows are both going up. In other words, if those numbers are zero, and those numbers are one. And then translating that into the fact that opposites are alike, I get that. And now everything goes wonderfully. What I want you to do now is to look at this line here. and the way it sort of goes to the top that's a plane, do you understand me? okay, and at right angles to that is the corresponding line here

22:30 so, look, because this is a zero this thing here must be a one okay, that's got to be a one one, all these things here, I know something about it, what do I know about it? Oh yes, I see a one, and a one, and therefore that must be a zero. Those are three directions of right angles. All I'm using, if you don't follow the argument, it doesn't matter a dam, all that I'm using is the fact that whenever three things are at the right angles, two of them have got to be ones, and 1 or 0, which is what the particles managed to do. Now because this is a 0, it's orthogonal to all these three, and therefore they've got to be 1s. And now exactly the same thing happens on the other side. Because this is a 0, this thing has to be a 1, and then because this is a 1 and this is a 1, this has to be a 0. Then because this is a 0 and it's orthogonal to these three, to be 1's. Okay. So I've got the entire front face now labelled with 1's. Aha! But look, here's a 1 and here's a 1. They're at right angles, I told you before. And the third number of that is this place here. So let me stick in green. That place has to be the zero because of that one and that one. Let me emphasize that one there. And put the same zero down there. And then in view of this one, and this one, that has to be a zero. Oh, the theory here, I've now got two zeros to write down. And that's impossible. So anyway, the summary is, it's just not possible to label these things with 1s and 0s in such a way that whenever three of them are at right angles, they're two 1s and 0s. Nonetheless, that's what the particles do, so to speak. They give answers. Now, this is really rather like the game of 20 questions, the way I used to play it with my sisters, which is, you know the game, you're supposed to think of an object, and then they'll ask you whether it's animal or whether it's a world animal, you can tell them and then they can ask questions about this.

25:00 And, but of course, you know, when my sisters got anywhere near the object I was thinking of, I changed the object I was thinking of. That involved being clever, because, you know, you have to make the new object consistent with all the answers you've given them, and that was often quite hard. particles are doing with. You know, they haven't got answers in mind to all the possible questions you might ask them, but they think on the fly. Okay? Um, now, so, the, uh, the free will theorem is going to talk about, sort of proves they think on the fly. That is to say that the answers don't exist ahead of time. You see, this doesn't prove it. And this, how can I say it? You see, the strategy that the particle might have might be a bit more complicated than we're given credit for. The answer to a given question might depend not just on the direction of that question, but on what previous questions you asked. And And if so, then the particle can behave in the way that I did. You know, it can have a strategy, and it sort of relies on the fact that if you ask certain questions, you see, when you ask questions that aren't all part of them, then all bets are off. I mean, in technical language, those questions don't permute, and therefore, you know, that means you're not bound by any promises, so to speak, to give the same answer as you would have done in other circumstances. It's covered all over with counterfactual conditionals, which is, I mean, in normal sort of treatments, you know, in fact quantum mechanics is largely a way of sort of studying counterfactual conditionals, really. You get the answers, the concept of the state gives you the answers, to some extent, to questions, whether you're going to ask them or not. And if you don't ask them, on the other hand, they needn't have answers. And in the proper way of formulating things, which is really very difficult to live up to, one doesn't have any of these counterfactual conditionals.

27:30 And so we're going to avoid them anyway. So let's introduce you now. Well wait a moment, what do I mean by free will? Well, Well, what I mean is very simple. I mean the opposite of determinism. So what I mean basically is, am I going to drop this piece of chalk? Well, I decided not to. But, to mention a counterfactual conditional, I have a strong feeling that I could have dropped it if I wanted to. Look, there, it's done. And so what do I mean by free will? Well, I think that whether I've got this chalk or not is not a function of all of previous history. It's not determined. Now, of course, both the idea that we have free will and the idea that we don't have free will, that the world is deterministic, have long, respectable philosophical histories. You can believe whichever one you like, basically, and you'll find plenty of ancient names on your side. However, I think most of it, I don't know, is it true? Who are the determinants here? Can I raise a corollary question? Do you just rule out the possibility there is a determinant model possible for which not everything is observable? I don't need to consider that possibility, so to speak. I mean, you hear all the assumptions I make, they're quite precise and limited. It's difficult for me to answer your question, though. Yes, but you see, look, I don't want to rule out something when there's no reason to rule it out. I mean, I want to make minimal assumptions. And so, no, I'm not going to make any assumptions I don't need. I'll tell you exactly the assumptions I do want. It just means I couldn't answer your question at all. Oh, well, you're a determinist. Because I'm in terms of that sort, as it happens. I'm not quite sure what that sort means. The sort is explaining it. There's an underlying model, but not all of it. If you try and make observations on it. Oh, you mean just incomplete? The observational is incomplete, but the model is complete itself. Yeah, all observations must be incomplete because they interact with it. Yeah, yeah. I see what you mean. As a slightly soft deposition, yes. Well, let me say something. I mean, what I'm going to rule out, I mean, what I'm going to suppose is,

30:00 well, basically the free will I'm going to suppose that the experimenters have is that they can set their apparatus to measure in one of these directions independently. They can just decide on this. Where are they? Oh, I'm going to measure this direction. And that, you see, you shouldn't really have asked me what I rule out when you were trying to explain your philosophical position because, you know, what my philosophical position is is irrelevant and I don't actually want to reveal it, necessarily. I mean, because for this theorem, It's a mathematical theorem. It doesn't depend on any philosophical presumptions, unless you count this free will assumption as something. But I'm not saying I believe in this necessarily. I mean, I'm just saying if experimenters have free will, then so are the possibilities. Anyway. Coming from Birkbeck, I should perhaps make a comment that's even in a deterministic world. You're talking about compatibilism? Yes, but then if it was, so to speak, decided at the beginning of time... But we could still have the sense of making that decision. Yeah. Well, there's a philosophical position, which I think was started by Hume, called compatibilism, which is the idea that one can have free will in a perfectly deterministic universe, I find it hard to... I find it hard to leave that statement by itself, but if you put the initial conditions as the point where it's occurring, I can relate to it better. Don't say where it's occurring, you have to say where it occurred. You don't need to put the initial conditions at the beginning of the universe. If you say, I am watching systems and people and the way they behave, and even though the system I'm watching is deterministic, my initial observations will be in error and if it's the right kind of determinism, it could be chaotic, in which case I very soon can't relate people's decisions and actions to the beginning. some of these observations are really about theories of the world

32:30 in other words what we introduce from various things now this free will thing I'm talking about is not about theories it's about the world which is rather different and that's one of the main ways in which its strength differs from the old paradoxes of this type let me continue however, I mean this is an interesting discussion about whether we have a decision as to what experiments you will base your argument on for setting them as happening in the world as distinct from other things isn't that an element of arbitrariness? yes, maybe but I mean, this is what I need to prove my theorem as I'm proving my theorem, I make certain assumptions I mean, it's an evidence they depend on your assumptions yeah, of course, I mean, look I'm writing down all the assumptions and, you know, these are the only assumptions are I'm also well as I'm speaking trying to make them more clear that anyway let me let me carry on and I'm sorry I'm trying to think so well maybe I should just sort of try to prove the thing where's something to rub off this one these they say it's behind there oh it is it is it down on the no no you can't So anyway, I will make one sort of thing now. When you were talking about initial conditions as being here all the time, then they probably shouldn't be called initial. But anyway, let me come back and say what's going on. So the basic idea is, well, you know, the twin axiom here is the fact that one can put two particles into the so-called singleton state we're asking, they promise to give the same answers. So, in other words, here we are. I have two spin-1 particles, and some magic has happened to them, called preparation, and if I ask this particle the same question, parallel, or if we're in a current universe, directions related by parallel transport, then they promise to

35:00 give the same answers. That's the twin axiom. That's something that's been verified at small distances, and we're going to suppose it holds interplanetary distances roughly. So I've got a particle here, A, and a particle B, and I'm actually not going to use the full strength of the twin axiom. On A, I'm going to measure it in three orthogonal directions, X, Y, and Z. Okay? And on B, I'm going to measure it in a direction I'll call W. Because I don't need the full strength of the axiom, and it's important that I thin it down. And the assertion is, well, let's suppose the contrary. that there is a function, phi to a, of various information that tells you what the answer is. Now, let me just for a moment write i, equals answers to x, y, and z. And what I'm going to suppose is that, which I'm going to prove is not the case. So, you know, if you think this is a big supposition, then fine. I guess suppose that the answer that particle A will give is a function called theta A of all the information available to A. And where is that information? Well, roughly speaking, it's in the backwards line tone of the time when the experiment takes place. That time may not be terribly precisely defined. It may take a few seconds or something for the experiment to go on. this is perhaps a slightly fuzzy light gun. I don't quite know where its boundaries are. So this contains answers to any frame that you... Well, you see, if this thing is a function of everything that's ever happened to us, or anything that it can access, any information that it can access, then it only depends on what's in this past light gun. But you're allowed to ask it about any frame whatsoever? Oh, I'm allowed to ask these XYZ, any free or formal directions I like, But in fact I'll promise to do it just for directions that are visible in that little scheme over there, because that's enough.

37:30 So there's a sufficient axiom. Well, there's a slight problem here, which is every now and then two of these are orthogonal and the third one isn't there. But let me supplement that by adding the third one of those. And then I'm going to find that in many particular triples I'm going to ask. So you could make a twin axiom for arbitrary directions, but yours is 40. Yeah, I only want this much. And then again, W is going to be one of those directions. Okay? And so we're supposing, in order to get a contradiction, that this is equal to P to B of certain information. Well, just let me for the moment call this information accessible A and information accessible C. And that information will be a function of everything I'd like. Okay. But now let's suppose that these two experiments take place, say, one on Earth and one on Mars, which is always at least five light minutes away from Earth. So anything that's accessible to both is five minutes out. Okay. Right. Well now, the first thing I'm going to do is remark that X, Y, and Z, this particular triple, I ask, are actually accessible to A, at least by the end of the experiment, and I'm not terribly fuzzily worried about these boundaries. So rather than doing this, I'll put X, Y, and Z here. And then I'll call I A prime the rest of the information in the history of the universe that's inside this light. And a similar thing will happen over here. This will be W, and that will be . And perhaps I should mention a little convention, which is if I put a question mark against one of these vectors, It means, the answer you'll give to that vector, to that direction, if it's part of this triple, in that order, if it's the third member of the triple, etc. So, what that would mean is, I mean, to say that that is equal to one, say, means that in this situation, z will give in, so on.

40:00 It's just a way of concentrating, just a notation device. Okay, and the axiom says, the twin axiom says, that if w is one of x, one of z, and now, you see, because I'm only going to measure in when you find that in many directions, there's a certain positive probability that will happen, so to speak. If w is one of x, y, and z, then, and then this is, well, let me just say e, g, z. Then, this happens. That the answer b will give to question w is the same as the answer a will give to question z. is W in this case that I'm talking about. So that's the Trimaxe. It says if W should happen to be one of X, Y, and Z, then the answer B will give to W is the same as the answer A would give to X. So the action says that you can't prepare No, the action doesn't say you can't prepare. If this happens if you, you see, okay, so let's set up the situation. My colleague went to Mars here and completely independently of me, is going to decide at 12 noon universal time or something that's a reasonably precise, not terribly precise concept, it's going to decide which vector to ask and ask it. I'm going to decide which triple to have to ask and we ask it. Okay? And if it should just happen, which it might, that w is one of x, y, and z, then we have to have this equality. that's the twin axis which is a consequence of quantum mechanics and that's the last thing I'm going to suppose from quantum mechanics. I suppose one thing, the spin axis, is the second thing, the twin axis, as follows from properties of spin on particles. I wish you'd start scratching your beard like that. I mean, is there something that... Have you said anything about correlation or anything in this department? No, I mean, but this is the correlation. I mean, so yes. But this is all I'm saying. You see, I'm not supposing any theoretical apparatus.

42:30 And it's really important that I'm not supposing any theoretical apparatus. There's no wave function or anything. wave functions of a funny non-relativistically invariant concept. I can't say anything about that because I'm supposing, you know, everything I say is going to be relativistically invariant. But this fact seems to be relativistically invariant. It's true. But let me quickly say what happens. Because what we do now is we separate this information, the rest of the information available to A into the part that's available to A only and the part that's available to A and B. And I do a similar thing over here. And let me go back these dashes in because they were some time ago. The names, the symbols have changed their meaning a little bit. Now, let's see. Let's try the dependencies here. Well, you see, this thing here, there cannot in fact be any dependence on information available only to b. Because, take two different values for that information, and suppose that these two things are different, well they can't be. Oh, just let me make a little remark. The experiment has made up their mind only at the last possible moment. So, you know, we're free to change our money until a few seconds before we do the experiment. And so, we can wait, what am I trying to say? So, the fact that this depends on Ia and doesn't depend on Ib tells you that this cannot depend. You know, because this doesn't depend on Ia, this can't, and because this doesn't depend on Ia, this can't. So, they can only depend on the common information and on the particular questions we ask. Okay? Yes? Can I say, um, so where you're thinking of the single system and the ambient observer, so that they're both closely seeing the same thing, two different, and, um... No, they're not seeing the same thing. They're not. BC is W. I mean, BC is this particle, AC is this particle. So, there are two separate particles to start with? Yeah. Well, to finish with, anyway.

45:00 There are ones on Earth and ones on Mars now. They've been separated. So, what's the intention of they did interact at some time? They did interact at some time, yes. So, we brought them together, measured their total spin. That was zero. It's a standard entanglement thing. I'm not saying anything terribly new. it's the simplicity and the depth of precision of what I'm saying that's new not the facts, the facts are old okay, now, here's a new thing, I mean we have no idea what this information available to A and B is I mean it might have been something to do with the start of the universe or something and normal arguments can't cope with this so they have to suppose that the information is or isn't contextual or something which is a way of ridding it with this dependence on this common information. But it doesn't matter. What we do is, look, these experiments can be performed. That's part of the free will assumption. I can set X, Y, and Z to any three-orthodox reactions out of those 30 degrees, and my colleague can set W to be any one of those reactions totally independently. So there are circumstances under which these experiments can be performed, and therefore there is a value for IAB for which these functions are defined and that value I'll stick with the I0 some particular value of that information that involves entire suppositions about what was going on in the whole universe and allowing it to depend for instance on the fact that these particles may have heard the experimenter's conversation when they were talking maybe looking at cosmic rays which contain information from the early universe But now, I'll call that function, I'll call that function, I'm sorry, theta nought of w. It now only depends on w. Okay? Of course, I would get a different function, say theta 1, if I put a different value of information in it. But I'm saying there is one function here, and it depends only on direction. Okay? And this must be theta nought of this. And similarly, theta naught of x, say, must be theta a of x-query, y, c, and i naught. Okay? I mean, do you understand this?

47:30 I mean, because this function depends only on w now, we've cleared it with the dependence on everything else. The three functions here that I get by putting the question mark at any one of these three variables must, in fact, depend only on that variable by the axiom, the twin axiom, because they're equal for this one, which obviously depends only on W. And you see, I can choose, just before the experiment, I was free to make XBW, I was free to make YBW, I was free to make ZBW. I was free to do any of those things. But now you see it follows that we have a function of a single vector here that has this 101 property. Because if x, y, and z are any orthogonal triple, two of these have to be one and one is zero. But there isn't such a function as what I've proved before. So there's the proof of it we will see. However, I mean the proof is not of the consequences what it means is that the those functions don't exist if you make the three-wheel assumption and the three physical assumptions I'm making those functions don't exist and therefore what the particle gives to your experiment is not a function of the entire part of the history of the universe that's successful today. That's the consequence. We're making of course precisely the same assumption about ourselves. The questions we ask are not dependent, are not functions of the entire previous history of universe. If they are, then we're deterministic. But if we're in a deterministic world, then, so to speak, if you believe you're in a deterministic world, then I don't see much point in talking to you. Because, you know, Of course, I can't stop it according to your theory, because, well, you know, whatever I was going to say was laid down at the beginning of time, and I'm just obediently saying this. But, anyway, it's rather peculiar. If you don't believe in the free will of science, then you really can't do science. I mean, science is based on the idea that we could do an experiment different from the one we did. If what buttons we're going to press are totally determined, then, so to speak, the physical theory only has to exist for the particular history of the universe we're in.

50:00 It's still not exactly clear to me in what way this is different from saying we cannot have free will in a determinate universe or stop without all this mechanism. I'm sorry, I think it's quite unstopped. Just say the last bit again. The free will theorem says if we have free will, what are the particles from free will? Yes. Now, let me just stop there for a moment and say this seems very, very good to me. We've got 20 minutes. Okay. It seems to me that the converse, which I can't prove, is more interesting. and that suggests that we obtain our free will from the free will of the particles of which we're composed but let me hear your question again I'm assuming that you can do that purely by saying how else would we get it? yes, but that doesn't sound like a mathematical proof to me and this does one of the ways in which this differs from previous versions is the degree of precision and acceptability, so to speak, of the arguments. I mean, we have to make this free will assumption. You can't prove that anything... Let me give you a simple proof that we cannot prove the existence of free will. It's very simple. You go to a film, the first time you go, you don't know what's going to happen. But then, was so great that you'll take your significant brother to it and so you go together to the second shell and now you know exactly what's going to happen of course you don't tell your friend um well the proof that uh we can't prove that we have free will is essentially we might be in the second shelly universe movie okay suppose it already happened and now for some being run again, and we're being given exactly the same sense of impressions as we were the first time, then, you know, that's entirely compatible with everything we observe, but the second time, everything's determined. So we cannot possibly prove we have free will.

52:30 And you'll notice, therefore, that there were two mentions of free will in this theorem. There was one of the tail of the error. If we have free will, something I can't prove, but really very strongly, then they have free will, too. Okay? But the way I think of the universe now has really changed. All these little particles... Oh, by the way, there is a version of the theorem which you can take from the GHZ type experiment, which will work for split-half particles, too. Well, never mind, I don't want to Yeah, okay, but it's a bit more complicated and it doesn't say anything different than the act but you require three particles Well, you can do it with spin-off You can do something with spin-off You can do something which establishes this result with spin-off, too It's a little bit trickier But you can do it. And of course, you could sort of try and say, well, only if particles have been entangled with previous particles in their past do you get anything. And maybe if they're not, then they're totally free. You know, I mean, totally unfree, determined. But then that's not a very physical sort of assumption. So the most reasonable thing to believe is that these answers that particles give are really, to some extent, free decisions. They're rather funny because of the phenomenon of entanglement. They're very strange. You see, you know, what one particle gives... The way I like to say it is that these two particles each have half a mind. But they do have half a mind of their own. they can decide what to do and I think of this totally differently now I mean I think of the world totally differently to the way I've been thinking of it most for the rest of my life here are these little things taking little decisions fluttering all the time in a manner that's independent of their previous history it cannot be determined from their previous history I should say, you know, okay yes I do believe in free will You see, let me say something. The idea that we have free will is not outlandish. The idea that the universe is not deterministic. I mean, if you actually go out in the real world or read history books, you'll see some momentous decision that Churchill took in his study in 1943, which vitally affected history. You'll see discussions of what would have happened had Churchill not taken that, and so on.

55:00 if you really think the universe is deterministic you think, you know Churchill is some kind of automaton but I mean in the rest of the world, in the sort of humanistic world, we're thoroughly accustomed to the idea that the universe is deterministic what happened to physicists and scientists generally is a rather curious historical accident Newtonian mechanics is a deterministic theory And the tremendous success of that made us rather fond of determinism, and sort of built determinism into our thinking. And we find it rather hard to think more deterministically now. But there's no evidence in favour of determinism. And what about if you... Can you rule out the possibility that there will be a deterministic model for which operations of agents as ourselves being observers could That is all the dualism I think you're getting up to No, you'll find I'm not sure of that the words may sound bad, the words always are dualistic but the thing I'm alluding to is not if you have a theory to deterministic which gives rise to agents that as part of the mechanism which would be unable to resolve this question. Then you'll have a determinism by model, which is not resolvable when we regard ourselves as part of the model. And that won't let you get to an answer of this. Well, you see, look, if you say determinism right at the beginning, okay, then I think it becomes lackless. I mean, because it has an assumption of free will in it. So, you know, if you want to believe that the world is deterministic, with any extra provisos or... It means it's more like a paradox that you can take one side or the other and get your answers in... Well, gee, many of the difficult questions of this world are of such a kind that we don't know what the answers are. Well, I just proved it for you. I proved that you, by the second showing argument, that you cannot disprove determinism.

57:30 so I mean if you want to have determinism you can have it with any kind of mix of philosophical preconceptions you like but then my theory is obviously irrelevant but true of course, that's usually true because it says the paradox is left with you well the fact that we cannot tell whether the world is deterministic on us is left with us that's absolutely I mean What you cannot do, really, is consistently maintain a kind of dualism according to which we are left out. We survive as sort of free agents just because we happen to be human. We survive as free agents, and the world, the elementary particle world is all deterministic. That won't work. That's what they show. What about the reverse? Could the elementary particles have free will and we don't? yes and to a certain extent that's true because you know you can't just fly out of the window even if you want it I mean you're bound in various ways our free will is not total and indeed it's very much hand in you know we're bound by the laws of the universe and to a large extent there's this averaging goes on I mean if the averaging didn't go on the averaging washes out the freedom but what I believe is the case is that somewhere up here Now, the washing out is not as strong as it usually is. So do you believe that our free will actually is tied to the free will of our silent life? Yes, I believe we derive it from that. I believe that somehow evolution has equipped us with something up here that does not wash out this free will of the particles. So this verges on believing that our consciousness is related to monochrome? Yeah, I'm sure, yes. I'm absolutely believe in that. Well, perhaps I shouldn't say absolute. I feel that very strongly. Peter? Can I ask something completely different on a much more fundamental level from right at the beginning of the talk? I don't think it affects your argument in any way, but it's just something that, when Keith was talking about this with me a bit last night, I thought, well, is spin really a property of three-dimensional space? in three-dimensional space?

1:00:00 Well, I mean, you want an answer to that. I don't know. I mean, because I haven't discussed questions of sort of robustness and scale very much yet. And I think I should just sort of briefly summarize what happens when you do. First of all, this theory is robust. Oh, no, I don't think it is. No, no, no. I'm sorry, I'm not directly answering your question. I'm setting up some things first. In practice, directions are never totally at right angles and so on. Well, it doesn't matter. I mean, if they're nearly at right angles, then nearly always you'll get to once and a zero. And very similarly, the dimension of space-time does not really enter into this. I don't mean n dimensions. I mean that spin is a property. Let me sort of continue. So what actually happens is, if we, I mean, for instance, Because I was supposing special relativity in there, you can change it to general or anything, provided that you're not in a black hole sort of thing, you know, as space-time is sufficiently close to the Lorentzian model for, you know, it ought to be roughly true. The robustness question is tied in with your question. Now your question is a sort of rather more subtle one, it's how should we view spin? Is it part of the ordinary dimensionality you know three dimensions but it doesn't matter because it would solve this 33 business in a way because it might solve the paradox well you know you see the point is the you see i didn't refer to spin i mean i referred to certain experiments i actually happen to mention the word spin but what i'm really referring to is something where you get a lot of equipment and do some measurements and a little light goes off. I mean, and, you know, the prevailing theory of quantum mechanics and science is phrased in terms of subtle things like spin. Okay. But I didn't really mention it. I didn't mention the results of... I understand that your argument would still hold. I don't disagree with that at all. I'm just thinking that the reason why it behaves oddly is because spin isn't any true three-dimensional properties. but I don't think it affects your own I don't know what the reason why it behaves ugly means, I mean it does behave ugly, period I mean, you see, if you start talking

1:02:30 about the reasons why mathematical theorems are true the reason why the physics is truly interested not the mathematics, the physics well ok, whatever it is, I mean the reasons why the physical world are the way it is that's a meschatological question incidentally, I don't believe Newton himself and I don't believe you can call his theory deterministic. His theory, Newtonian dynamics as following him, is deterministic. Well, his theory isn't, because he has infinite numbers of interactions instantaneously. So he can't be deterministic. But that kind of determinism is, so to speak, allowed in what we're talking about. And it still can't happen. For instance, our theory really disproves... There's this theory what is it called GRW-Garrady-Lumenium-Babonism, a sort of theory of a sort of mechanism for reduction. And they frankly admit that their theory is not relatively low, but they say they have great hopes of making it so. The free-will theorem, one of its positive consequences, perhaps I should call it the negative consequences, is that they won't be able to do it. there can be no believe us in free will and we just addressed believe us in free will believe us in free will are forced to believe that there is no mechanism for redemption it just happens we have five minutes for questions or more questions believe us in free will are forced to believe there is no mechanism for redemption let me say can happen. There are certain kinds of reactions, usually called decoherent reactions, I prefer to call them coercive, that force the world to give answers. Okay? That's a separate thing. I mean, one of them is measuring the squared spin of a spin-one particle in a certain direction. When you do that, it's like the 20 questions game. If my sister asks me a question, I must Okay, but those reactions, although they coerce the existence of an answer, cannot tell you what the answer's going to be. I mean, there can be no mechanism that says what the answer is,

1:05:00 because the answer doesn't exist ahead of time. It's not, you see, previous versions of this say that quantum mechanics, or some extra theory with the properties of quantum mechanics or something, does not determine the results of certain experiments. Arthur doesn't say that. Well, it does say that, as a trivial consequence. What it says is that the results of those experiments are not determined ahead of time. They could go either way. And obviously, I mean, you know, we watch Churchill going into his study. We don't know what he's going to decide. Period. You know, there's no amount of consulting of his colleagues which will decide what the great man is going to decide. Similarly, there's no amount of theory that will decide what answer this particle will give, because it hasn't decided yet. Doesn't this destroy the version of the bone theory that says everything can be projected back to the beginning of time? Well, you see, the bone theory is deterministic. I have no quarrel with that. If it's a deterministic theory, it's deterministic. However, let me say something. It's not really deterministic. There is a subtle way you can open up a theory. If you say, look, you've got various hidden variables, and the standard way of thinking of these things is that we don't know the values of these hidden variables. They exist, but we don't know the values. And so let's sort of think of the set of values that this variable might have. It's a little sort of open set or something, a little region. now if instead you take Bowen's theory or any other one of these theories and make the hidden variables in it be set valued instead of point valued ok, so at the moment here's this set of hidden variables, now we make an observation, and what this set was shrinks, because some values are incompatible with this observation, and that's what always happens now that I call opening up a theory, ok We'd be talking about the set of all possible trajectories. Well, talk about the set of hidden variables. Think of it in terms of my 20 questions again. I'm going to freely confess to you now that I'm not thinking of an object. Okay? Now you can ask me 20 questions. And now the set of objects that are compatible with the answers gradually decreases. Okay? Now, if you take Bohm's theory and do that to it, then it's a non-deterministic theory you see the way it shrinks

1:07:30 wasn't determined there isn't a particular value inside this set the way the set shrinks is limited by the questions we ask so what am I trying to say and the answers were not determined ok, that's all so any theory with hidden variables that's deterministic has an equivalent theory that predicts at Amnesty, and that I believe is righter. You're actually describing a slightly different version of the bone theory than the one I was thinking of. Yeah. Varieties of it. The strong one that I'm thinking of is the one that projects all the projectors right back from the beginning of the universe. Yeah. And sets all the randomness there. Yeah, well, I'm thinking, I mean, my words apply to that. I mean, you see, the point is, you think there's a particular point here which is derived from the initial conditions of the universe, we don't know exactly where it was now in the open version of the theory exactly where it was isn't defined it's like I mean it's like my sister saying we don't know what what object little Johnny is thinking of but that's precisely equivalent in its predictive consequences to the assertion you know Johnny is thinking of one of the following set of objects and that's all not one of them Johnny is thinking of the following set of objects and every question, that question goes down, that that set goes down. I don't have to be thinking of a particular object. All my answers will be compatible with the idea that I'm thinking of a set of objects, which is initially the entire universe. As soon as you say it's got to be animal, vegetable, or mineral, and you say vegetable, then my set shrinks to the vegetable portion of the universe. What you seem to be saying there, you used the word writer just now is a key point in there if you it's all to do with whether you could have the underlying deterministic model and you're saying it's writer, to use your word, not to now, if you consider we have all been brought up, not necessarily in a Newtonian world, but at least in some sort of causal relationship world then the problem I have as an engineer explicitly in work is that there's a disjunction as you go over to the probabilistically righter model, and that makes it a wronger model. Now, if those two are alike and give the same results,

1:10:00 which I'm going with you, yeah, very probably you can do. We don't need the disjunction. Therefore, to carry on with the causality to the bottom, even though you can't observe it, is a better method because it creates a continuity of mentality. Well, look, I have no objection to carrying on that way. I mean, as I say, as soon as somebody wants to believe in a deterministic world, they'll have to shake their hands. It's only for that reason. Only for this reason. It doesn't matter what you're reaching out to me. Can we continue this later? It's now the lunch hour. We will convene again at 3.30, and I suggest we express our appreciation of both speakers. that was the end of John Connolly's talk well the Dirac equation for accountants now my father was a chartered accountant I wonder what he'd make of this Well, actually, Carnot Horde talks about the Dirac equation for a canton. Here's the title, so as not to, um, um, bother any feathers in the audience. If I was an accountant, I may fail at that. Uh-oh. Well, the real reason, so, um, So, my background is actually classical statistical mechanics. A bit closer would be good, I'd just say. This was drawn by my daughter, so he may not be properly dressed. So, you know, the life of a classical system is really pretty easy, because all you ever do is you count objects, but I can't usually say that in a physics audience, because some

1:12:30 will get annoyed that you're trivializing what you do. So I do want to actually focus on counting. So this talk is really directed towards a very simple operation of a count, of keeping track of distinguishable objects. So that's why it's going to be the direct equation for the countants. You may well ask, well, why we want to do this, you know, put a direct equation in accounting perspective. I apologize, I'd actually put for the quality of these slides. They did not come through well. They may be a bit hard to see. To start with, I think it's better for you. Can you read that at the back? I don't count. Maybe if we turn off the lights. Then if you don't go to sleep, it won't be. So, you know, just thinking about the physics of the Dirac equation, there's a lot of real physics in there, right? There's special relativity, there's aspects of weight positivity, and certainly principles, spin, these are all things that we think about in terms of physics. I added point E, that's my perspective, I think there really is a pragmire of interpretations in that. I mean, it's really, it's really, well, some might think it's really awful, but on the other hand, maybe it's just very interesting. Now, the mathematics associated with the graduation involves, well, a number of things, but here I've listed real numbers, complex numbers, throughout matrices. You can think of that in a way as a progression of counting systems, of number systems, the differential operators. And at the end, again, could be slightly facetious. You've got a wave function. But the physical significance of the wave function is simply unknown. And if you don't believe that, we could have a, you know, one of the proofs of that is I don't think two people in this room would agree exactly what the wave function is. To me, is just the basis of the probability calculus. That's it. So, the idea is that our mathematics, what it does, it allows us to describe precisely the evolution of this wave function, but we haven't a clue what the wave function is. So, one thing we can do is say, well, can we come up with a context in which we know exactly what the wave function is?

1:15:00 Okay. Now, in order to know what the wave function really is, I'm going to ask you all to forget of your physics tools, right? All I'm never going to ask you to do is count. So the first thing we're going to throw away, actually, is real numbers. Because real numbers, you can't add, subtract, multiply, divide, and finite time, so they're not useful for us. Of course, we lose the differential calculus, you know, we lose everything but basically our fingers and toes for counting. And, you know, say, why bother to do that? Again, it And then it comes back to quantum mechanics that, you know, one of the positives for quantum mechanics is the wave function contains all the information we can know about the system. And although we don't really know what a wave function is, we possibate that it contains all of the information about the system. And I really want to weaken that positive. To me, the wave function, if it really has a reflection in nature, is sort of an equilibrium. So in this so-called derivation, what's going to happen is that the waveform itself is going to appear as an equilibrium pattern formed by a simple dynamical process. And we'll have a look at that dynamical process and see that it really is simple. and all you ever do is count and then when you want to get to the Dirac equation fill it just again and just take a continual minute okay so that's how would you weaken the statement? we can this statement says that the wave function is it it's got all the information about the system well we're going to see the Dirac equation appear in which the wave function itself is an equilibrium object-like probability density for the fusion equation. Well, it doesn't have all the information about the system because it's partly determined by boundary conditions. Yes. And then, okay, you might say, well, of course it's determined by boundary conditions and therefore it does contain some information about the boundary conditions. everything on its head. I think what people will say is, well, yes, the wave function feels the boundary conditions, and so contains... And then becomes discreet, you know, or rather

1:17:30 its content becomes discreet. Right, and in any point, the discreetness does come from the boundary conditions. We'll actually sort of comment a little bit on that later. So, is simply to put this, put the Dirac equation into a very simple context where all we ever do is count. So, what we're going to do is we're going to look at things that are called stochastic entwined paths. So again, we're going to count that the, I'll describe what the stochastic process is, and in words it looks like a complicated, I'll then show you a figure where it's obvious what's going on. We're on a lattice in the x-y plane, it's got some lattice facing epsilon, and there are four different directions that a particle can pop, you know, given by these two, or these four arrows, and we'll call the blue ones, we'll code this going forward, it's blue going backwards, it's red. and the rules for this war are the following we use a binomial distribution for going forward in y, we maintain direction with probability 1 minus epsilon m epsilon is this lattice spacing m is some positive rational some positive rational constant and this epsilon m is very much less than 1 so with probability 1 minus epsilon m we just keep, when we maintain our direction, we're going forward. And with probability epsilon m, we change state, and changing state means either we change our direction, or on every other state change, we drop a marker. When we go in the backwards y direction, we just follow the markers that we drop. It's a little bit like Hanson's record. So let's to see what one of these things look like. So here's one of these entwined paths, okay? And here the lattice is pretty small, we can't, you know, it's too small to plot on the scale. And so we imagine the walker starts out here, and it starts out going in the plus-x direction, and, you know, the most that goes straight, but then it turns up one of these things that

1:20:00 changes the state, so it then goes left. At this point, it has another one of those things that says change state, so it drops a marker. It keeps on going, another change of state, changes direction, drops marker, changes direction, drops marker, changes direction, drops marker, and so on. So at some point, determined by something else, it then reverses and goes back along the lattice through the marker points. Right? So, this is sort of a Hanson-Gradle warp where you have this thing that gives you these oriented rectangles. Okay? So, the warp looks like this, right? Now, what does that do? Well, if we look in the plane, we can look at these rectangles and say, oh, well, these things are oriented, right? If we use the right-hand rule, the first rectangle, and we use the right-hand rule, these things are now pointing out in this direction. The second rectangle, the orientation is opposite. Okay? So you can imagine sitting on this lattice, you know, if you sit your chair somewhere in here, and you let this process just run, then on average, you're going to say, well, okay, the orientation is positive. If you sit your chair anywhere near this boundary where there's some sort of average length involved change, then this boundary will shift, and you will watch the boundaries shift, and sometimes it'll be positive, sometimes it'll be negative. If you're sort of sitting in the middle here, it'll be mostly negative. So there's this alternating pattern that you get from this. Okay, so the question is, how are we going to average this? How are we going to see this on average if you let this process keep on going? Well, so the question is, how do you count these things? Well, the easiest way to count these things is to notice that the orientation is actually defined by, say, the right-hand set of arrows. So let's forget about the left-hand set of arrows, which you can reconstruct from the right-hand anyway, and then you get this path, which I'll call the enumerative path. Okay, and the thing to notice here is that there's this funny rule that the color of the path changes place with every two corners, right? So it goes blue, blue, red, red, blue, blue, red, red, blue, blue. So there's this rule for the counting which says, okay, stick in a minus sign for every two corners.

1:22:30 Because when we have a left-hand corner, it's a plus, we're not changing our orientation. orientation. When we get to the right-hand corner, there is a minus sign, because we are changing orientation. Remember, that's the thing that we want to figure out in the end. We want to get an ensemble average of the orientation of these rectangles. Okay, so in a way, sort of looking for an average for toxicity. So that's what we're going to count. So how do we do it? Well, we just do it just as I suggested. We use these enumerative paths, and the blue segments will say contribute to plus one, the red segments will contribute to minus one. And we can either do that directly, if we do that directly, we can set up what turns out to be a path integral. But if you want to do this quickly and easily, all we have to do is just assume that an equilibrium is reached. So assume this process has gone on for a long time and sort of covered the plane with these oriented rectangles, and then look at the conservation equations for this, for a density that counts the number of paths by orientation. And there are, you know, if we sit at a ladder site, supposing we're, we're, we're, we're, uh, this is at x, at y, um, sorry, this is x, y plus epsilon, and we ask, where does this density come from? Well, with probably 1 minus epsilon m, it comes from the downstream blue, right? So there it is. There's, uh, there's the downstream blue. Now, the only other possibility is it can come from a right, from a right-hand term, but it's from a red. If it comes from a red, that means there's a sign change. So The only other one is down here, so the other direction, we have partitionings by direction. The probability is it comes from a downstream blue with probably one minus f on n. Or the other possibility is there's a left turn involved here. But this doesn't change color, so you get this plus. these two equations just count contributions from all of the piers in equilibrium

1:25:00 now because of this 1 minus epsilon m as we step through the lattice we're going to get an exponential decay we can get rid of that by keeping track of this w which just takes out this exponential decay and if you do that what you get is You mean it multiplies up so the thing doesn't get smaller and smaller? Yes, that's right. So you filter out the exponential to k, and then what you get is this equation here. And let's just pause and make sure we know what we're doing at this point. We're assuming that the left one is rational, so the prime initial position is rational, then this w plus and minus rational, right? deal with with rational numbers all we're doing is just counting these oriented areas there's no abstractions in here at all other than elementary probability right we haven't had to use anything terribly fancy the only thing we have assumed is that there's this whole walking process reaches an equilibrium i mean it could be that you sit yourself down on the plane and watch this thing go about you and an equilibrium is never breached right but So now, let's be physicists again and say, okay, we're, you know, discrete stuff is just too tedious for us to deal with, and we've got all these tools to get rid of the discreteness. So we're going to take a continuum limit. So what we do is we go back to this equation and say, oh, well, if we subtract W from each side of this and then divide by epsilon and take the limit as epsilon goes to zero, this is what we get. We get a partial, two, a couple of partial differential equations. That was the first thing I thought of when I saw your four-point lattice because four-point lattice is needed to evaluate by differences second all the partial differential so that's really what you're doing now of course the dynamics makes it move yes the dynamics makes it move and and what the stochastic process is doing is it's building up with a dynamical process a pattern that that gives you this partial differential equation in equilibrium, okay? So, this is what we get if we just make

1:27:30 it a bit more compact, we can write two components, an object, here's sigma z, here's this real matrix with an anti-symmetric real matrix, which is minus i sigma y, and writing that in that compact form, you have this, which is just, it's just a form of the Dirac equation, the Dirac equation in one dimension. So, if we, you know, to check that, if we iterate that equation, as you, you know, as you pointed out, you get the second order version of this, and because, you know, sigma z and sigma y is new, you lose the intermediate terms, and you get the Klein-Gordon equation. Okay, and it's just a Klein-Gordon equation in which A equals C equals 1, we've set it up that way, and, you know, so that we understand it in terms of a dynamical process, Y here is T. Okay, so, the thing to notice about all of this is to say what it isn't. We haven't said anything about the uncertainty principle. We haven't quantized anything. We haven't invoked Schrodinger's equation. analytic simulation, you know, there are no complex numbers around. We haven't said anything about special relativity, and there's no interpretation of quantum mechanics around. We know exactly what the wave function is. So is it a real variable you've got? Yes, W here is real. Nevertheless, if you do what Louis was talking about this morning, you can set up a kind of homomorphism between the differences here, the finite differences, and some complicators, or rather. I don't know whether that would be good or not. I think that would be very good, and a little bit later on, if we just come back to that, there is sort of an interesting relationship between, I think, the discrete physics that Lou was talking about and this. Okay, so we have, on the one hand, we have this how to form the solutions of the Dirac equation from a completely transparent process, a physical process, and it's the solution of the Dirac equation

1:30:00 represent the options for the equilibrium. On the other hand, we have the context of quantum mechanics in which the Dirac equation is really quite complicated. It's got all this physics associated with it. So, it's sort of worthwhile just going back to the model and saying, okay, what are all these complicated physical things that are associated with it in the other context? Now, some of this will be obvious, some may not be so obvious. First of all, the uncertainty principle. Okay, where does that appear in the countenance model? Well, the enumerative paths really are just random walks on scales where, you know, delta x is very much greater than m. So if we defocus, you know, these enumerative paths, they're just random walks. Random walks have a practical dimension of two, and it's that geometry that actually gives you the uncertainty principle. That may or may not be well known, but I think it is true, so it's the random aspect that does it. It's like the difference between the Schrodinger equation and the fusion equation, the path integral versus the final integral. And the paths in both cases are these deep of two fractals. And that's the scaling that makes it all work. the uncertainty principle just comes in from the geometry, the macroscopic geometry the quantization in the complex numbers it comes from the following, unfortunately the color isn't coming through very well but we have poor directions, right the directions in plus y were blue in minus y were red and that was the input to the accounting model, right, that's all there is and we're counting orientation what happens with the stochastic process is these four directions get smeared into basically the unit of a circle. So instead of, if we neglected the color here, we'd get a characteristic exponential k that applied e to the minus mt. But what happens when you put these directions in that? It preserves this length, and the e to the minus mt is converted, or a convenient way of describing it, is e to the i mt. So that's really what happened. Here, the stochastics, instead of taking you over to an exponential decay, they build in this aspect of phase. Okay, so that's

1:32:30 where the, in a sense, that's where the i comes from. Yes, that's right. I think we'll come back to that in a minute. That's right. It's the fact that every two corners change the sign. So that looks like two operations. You square minus one which is the analog of the square root of minus one so that's where this moment came from that's right that's where this came from and and and just taking care of that using the square root minus one is is um is convenient okay so special relativity okay how does this where does that Well, when you set this thing up, the powers all have links only in the four directions, you know, plus or minus one, plus or minus one, and the xy. Now, for us, y goes over to t, so the velocity is plus or minus one. So those are special. And so the velocity speed of the particle is always c. And it's very interesting to know that the model with orientation, as long as you take orientation into account, You get the oscillatory solutions, and you also get the branch transformation. The geometry is intimately linked to this, in a sense, this analytic continuation. If you take out the orientation and say, OK, I'm not on the channel.