What cannot go faster than light? / subsequent discussion

0:00 Perhaps violating some speed regulations, and suddenly, spontaneously, a bomb detonates, an immense tachyon, or non-tachyon, which goes off into space, and I do the public service of catching the tachyon in my handgun, so that it is not wandering about doing damage. And most people, and certainly I, feel that this sort of thing should be included both by the laws of the state and also by the laws of nature. But somehow we have to ask, is that sort of thing going to be incorporated into physical theory? See if we can introduce a notion of a cause and effect and limit the cause-effect relationship systematically by the philosophy of life. So I tried to formulate that idea. Here is something called the principle of local causality, which first of all I formulate in a rough way, and later I try to make it more precise.

2:30 The rough idea is that the direct causes and effects of events are nearby, not along the odds. And even the indirect causes and effects are no further away and permitted by the limit philosophy C, the philosophy of Y. That's to say that in this space-time diagram, in which time goes this way and space goes that way, if I am interested in events in such a space-time region, I'm going to look for the effects of those events in the forward life form, for the causes of those events in the backward life form, and over here in an entirely spatially displaced space-time region, I'm not going to look for rather the causes or the effects. Now, that of course doesn't mean that what happens here is independent of what happens there. There may be correlations, but these correlations will not be cause-effect correlations. They will be correlations caused by the effects of common causes. In the overlap of the backward-right point, something might happen here which has consequences here and consequences there, and these consequences are mathematically correlated. But that's not cause-effect, and I'm not going to worry about correlations. To turn that into something we check in the news for mathematics, I say the following. I will say that theory is locally causal, respects local causality. The probabilities of events in space-time region 1 are unaltered by specification of events in space-like region 2. I omit the qualification for the moment, and you would see that this would be a stupid formulation of the idea of local causality, because events here are correlated, and specifying something about 2 certainly changes the conditional probability for events in 1. So I have to qualify what I say in the first sentence, and I say, this is so when what happens in the causal cone of one is already sufficiently specified. For example, by fully specifying events in space-time region three.

5:00 So the idea is that everything that I can use to account for events in this region is contained in such a region. And anything that I might say about that is then superfluous, should not change the conditional probability, because what that should tell me is about something that's happened here, which also had an influence there, but which is already fully represented probably from here to here, when I tell you what all happened in this space at that time. So this is the idea of local results that I'm going to discuss. And it's that idea which you should attack if you don't like my conclusion, at least in my opinion. Of course, like the conjurer, I may be diverting your attention, but it is indeed my impression that the art of the matter is here. And if you can find a better formulation of what we feel we mean by local fatality, I would be interested. Anyway, that's all I'm going to introduce for the next few slides. The first thing to assert is that ordinary quantum mechanics is not locally causal, and that's an obvious observation. The kind of situation in which you see that that must be so is the kind of situation which Einstein, Podolsky, and Rawson mentioned in 1935 and which Bohm made more easy to illustrate in 1951. They arise, for example, in experiments involving spin correlations. You can form in quantum mechanics, for example, something called the singlet spin state of two spin-half particles, which is the up state of one, the up state of one, and the down state of the other, coherently superposed with the down state of one and the up state of the other. And this state is such... But if I do an experiment on the right-hand side and find that it is up, then I know that this term is irrelevant, and according to quantum mechanical rules, I strike it out, I collapse the root bracket, and then I see that the result on the left will certainly be down. That's to say, specifying up on the right changes the probability for down on the left.

7:30 Before I told you what happened on the right, there was a fifty-fifth chance of up or down on the left. But when I specify what happens on the right, the probability in the left changes, so this does not meet the requirements that I give for a locally-caught theory. Moreover, in quantum mechanics, there just is nothing other than this real function by which I could attempt to specify events more fully in the relevant space-time region. As far as predictions of measurements on spin are concerned, this, for quantum mechanics, is the whole story. Most physicists actually supplement that story in their mind by so-called hidden variables. Even if they deny that when it gives them very rigorous discussion, most physicists do really believe that there is more in nature than there is in a country. And that that more, in some way, accounts for these correlations, so that what happens to both regions is really just a consequence of what has already happened in their relevant factory-like columns. But to postulate such hidden worlds, strictly speaking, should go beyond quantum formalism. Most physicists do it half-consciously, half-systematically, and badly. There are people who have tried to do it consciously and well, and usually they have been speared at for their feelings in so-called invariable periods when they are explicitly formulated or regarded as in some way not absolutely kosher. But in my opinion, it is a thoroughly responsible reaction to this. So quantum mechanics, strictly speaking, are not supplemented either consciously or unconsciously by invariables. It's not locally causal. And the question arises whether you can imagine supplementing it by extra so-called hidden variables which would make it locally causal. And you can't. That's the development, simply said, of Einstein's adjustment model. And I will remind you a little bit of the demonstration with that. Quantum mechanics is not only locally causal, but it cannot be turned into a locally causal theory by completing it with extra variables.

10:00 This is a space-time diagram and I consider something like this experiment in which you do experiments on true spin half particles which are going in opposite directions and which have been prepared in a critical way. So I have, these are space-time regions and in these regions there is a counter whose result is a All of these terms can be used to define an academic approach to mathematics and physics, and can be used to define an academic approach to mathematics, and can be used to define an academic approach to mathematics, These are all variables which are under the control of the experimenter. They are, in this particular example, they are the angles at which certain analyzing equipment is fed. It could be the angles of certain polarizers or photons or of stern girl-like magnets. All we need to know is that there are two variables under the control of the experimenter. And then, in addition, I made the hypothesis that there are some variables lambda, whose nature I do not know. They might be a set of numbers or a set of functions, what you like. By means of which I can form some hypothetically complete description of this region, and indeed of that region, which fully identifies the backward life forms of both the regions of primary interest. There will also be certain variables C, which are under the control of the experimenter, and are things like whether he switches on the current. And things like that, which are perfectly normal variables which even quantum mechanics recognize. These numbers are the hypothetical variables that perhaps quantum mechanics does not know about.

12:30 Then I define a joint probability distribution for the results of experiments A and B, given the various things A, B, C, under the control of the experimenter, and the hypothetical extra variables which are not under the control of the experimenter. And I then will use my idea of local causality to simplify this expression. Just by ordinary probability theory, such a joint probability distribution can be written as a conditional probability distribution, in which P is made one of the conditions, all the other variables, and then the probability of B given all the other variables. That's probability theory. And now I apply to that locality. This A cannot depend on capital B or little b. According to my idea of control, little b and big B are in a region which is spacelike separated from the Earth, and anything that they could be relevant to is already represented in the numbers, which I have assumed to be fully specified. So that this first expression simplifies in that the only arguments which are relevant are A, C, and lambda, not capital B and lambda. And in the same way, the second practice simplifies. Only B, C, and lambda survive because capital B cannot depend on beta A when lambda is given. Now, this formulation is already interesting in itself and often is taken as a starting point. What it says is that the joint probability distribution factorizes the probability for A given certain conditions and the probability for B given certain conditions. That is to say, when the lambdas are given, the A and the B are uncorrelated. And in our experiments, we average over some distribution of lambdas that the outputs from the experiments seem to be correlated. So what one is saying here is that correlations in distant regions must be reduced to common causes.

15:00 And when those common causes have been identified and health-fixed, any residual fluctuation is independent in the two spatial regions. So that is also a perfectly reasonable formulation. I would say that correlations which satisfy this requirement are locally explicable. And it is a fact that quantum mechanics presents us with correlations which are locally inexplicable. To show that these ideas have consequences, well there are many that are doing that and here's one, I will define a correlation function. Which is a function of the experimentally controllable variables a, b, and c and it is got by multiplying together the results of the two experiments a and b and weighting that with the probability of that particular result a and b which is a function of a, b, c and lambda and then of course I must take into account the probability distribution over the lambdas Which, for themselves, in some hypothetical theory, which I have not constructed before, will also be given by the experimentally controllable variable. Now, a very important assumption is made. I assume that A and B are three random variables. And one has to insist on this assumption. It is so easy to take it for granted. It was taken for granted in the Einstein's Adelphi Wall of Paper. One imagines that the experimenter is quite free to choose to put the polarizer where he pleases. But this choice is not dictated by anything that has happened before, and in particular, it is not dictated by these variables lambda. Does not inform us about the variables lambda. So the total distribution of lambda given c, a, and b cannot depend on a and b. They can be a function only of c. So I'm putting in here the idea either that experimenters are free or that I could find something that is free which I would use to dictate which doing one experiment rather than another, the Swiss national lottery machine, radioactive decay, coin tossing, pseudo random numbers generated on a computer, whatever you are prepared to treat as free, I will put in there and couple it up to some mechanics which will orient the polarizer. Wrap down.

17:30 The expression simplifies, and also using locality to simplify that expression into the factorizable form, I come to such expression. And now I have mathematics. That expression is in fact an extremely restrictive one, and one of the properties that it has is the Klaser-Holm-Chan-Chamoni inequality. If you consider how that expression can depend on its arguments A and B, you find that it is limited by this inequality. Here is A of B minus A of B prime, where I have considered two different values, B and B prime, of the setting of one piece of equipment, and here I have considered A prime, sorry, here I have considered B and B prime, here I have considered also A prime, a second possible setting alternative to A for the other piece of equipment, and this combination must be less than two. Now, one can determine these things experimentally, and one can also calculate them theoretically. Just by construction, that expression is less than 4, because the a and the b from which I formed it are each plus or minus 1, and I weighted it by probabilities, which are all less than 1, and I took a sum which has four terms, and then I added, well, I took an average, sorry, so that this remained less than 1, and then I have four terms here to show here. No, that's right. There are four terms and it is less than four by definition. It is less than two by local causality. So the local causality is not saying something trivial. It's saying that it must be less than half of what it might have been. In quantum mechanics, it is less than 2 times the square root of 2. So quantum mechanics also restricts this thing. This is proved in this paper of Sealson and in a much nicer way in this paper of Lundi. The locality, however, is the most restrictive. It says that it's less than 2, and when you find, as you do, situations in quantum mechanics where 2 root of 2 is realized or nearly solved, you have a complication, which is local causality in quantum mechanics.

20:00 But that's not a question of experiment. That's just a fact about quantum mechanics. Quantum mechanics could not respect local causality and cannot be turned into a theory which respects local causality by adding hypothetical variables. Experiments have been done to see whether nature really respects quantum mechanics in this strange situation. And it is found that quantum mechanics is predicated and therefore in a certain sense local causality is excluded. One must think some proviso about that. The experiments that are done are not the Gedanken experiments that the peers would like to have done. The experiments that are done are the ones that can be done. And they have difficulties. If you look at this expression, if you look at this correlation function and give a longer definition of it, you find that it is the probability of two yeses plus the probability of two noes minus the probability of yes-no and minus the probability of no-yes. Well, without the signs, this thing must just add up into one, the total probability of all the possibilities. So they can be rewritten as one minus twice the probability of yes or no, minus twice the probability of no or yes. In practice, the most likely thing for a counter to say is no, because in real experiments, usually particles go in their own direction. They don't have a chance to trigger the counter. And your counters are inefficient. In the best experiments that have been done, I think they are 10% efficient. So nine out of ten times you have a no, which is because the counter is not working properly. So overwhelmingly often the counter says no, for perfectly trivial reasons, and in no circumstances the dependence on the orientation of the other polarizers is rather small. So this expression tends to be a little less than one and weakly dependent on A and B, and if you substitute that sort of thing in there, you find it is trivially satisfied. So it is, strictly speaking, incorrect to say that experiments have been done which violate the locality inequality.

22:30 What has been done is that experiments which have approached All the elements that you would need have been done and then have been extrapolated to better experiments by using reasonable ways to correct for the inefficiency of the counters and for the fact that many of your particles were going in the wrong direction and so on. So it is only in that sense of an extrapolation from existing experiments that you can say experiments have decided. There are people who insist very strongly on this point, I don't. I find the experiments that have been done, I don't regard them as proofs, but I do regard them as a good guide to where I at least should make my own efforts in the future, and I will make them on the hypothesis that nature does not respect both of us either. Having said that, we nevertheless come back to the fact that we do seem to be limited in what we can do. For example, we can't use these funny correlations to build superluminal telegraphs. I used to have a sneaking hope that that would turn out to be possible, and that people would call it the Bale telegraphs. The reason that that can't be done is because, in this diagram, in order to make the difficulty, I have to consider a correlation involving four quantities, capital A, little a, capital B, and little b. But of these, only little a and little b are under experimental control. If I had control of b and wanted to use it to somehow send a signal to people in this community, I would have to admit that capital B is out of my control.

25:00 That's the experimental result which I just have to accept which I can do nothing about. I can't use that to send messages and therefore the fact that there is some correlation between this and these two things is not useful to me. I have to ask whether there is some correlation between capital A and little b, not counting big B, and there is no such correlation. And yet, nevertheless, there are correlations between capital A and capital B which are inexplicable unless I assume that what happens here does depend on what I do with little b. So it's as if you were tossing a coin and I had the power to give it an extra round while it's falling. You wouldn't know that I had that power because you didn't know whether it was going to fall heads or tails. And yet here it seems that in order to account for the correlations when we see both things long afterwards, we have to introduce such possibilities. That's a very frustrating thing. Now, as I just pointed out in that particular example, there are certainly general discussions in quantum mechanics which exclude the possibility of such signals. Since that is the best form of local causality that we have, I'll discuss that a little bit. So, a general assertion. In quantum mechanics, the probabilities of measurement results in spacetime region 1 are unaltered by specification of three variables in spacetime region 2 when the result on region 2 is not taken into account. My three variables, I mean the ones that I can control or imagine being under my control or which I can imagine being random processes. I have to then talk a little bit about how three variables occur in theoretical physics. And since I'm interested in the question especially of signaling, I have to discuss in what way such variables can be introduced.

27:30 Now, the very notion of a signal, for me, is essentially human. I don't know how to define the concept of signal in inanimate nature. If somebody of you can define the concept of signal without referring to people or animals or some creatures to which we normally attribute. I would be very interested to hear of such a definition of signal, but I don't know of any definition which does not involve animals or people, and I will restrict myself to people. The question is, how are people represented in quantum mechanics? And there are two ways of doing it. Can you explain why you don't accept mechanical signals if I may call it that naively. You've got two pieces of elevators and I some switch in one. So, if apparatus 2 is switch, why does that not constitute a signal? Is it a self-operating switch involved? You are not involved in switch. So, what controls the switch? What tells it to, why does it go off or not go off? I'm sorry? Radioactive atom. Anything. Anything. It may even be a determinate cause. Well, I mean, I do not say why you switch it then to the signal. The burden is not on the signal but on the origin of the signal. So what we want is to identify a cause, a free cause. So somewhere in nature, you want something that can be identified as a free cause. Now, a radioactive vacuum is certainly something which I would consider. It could be that there are, in nature, random events, and it may be that the behavior of a radioactive vacuum is one of them. And then I would use that. And then you could say that you could call the effects of that spontaneous signal, and I would buy that. But now you're making a hypothesis. But the radioactive decay is such a thing, whereas just now we were contemplating the possibility that quantum mechanics could be completed by adding extra variables.

30:00 It might be that by adding extra variables we could make what looks like a random event into a determined event. That might be. But nevertheless, I'm willing to admit that there may be spontaneous events in nature, and then it will be possible to define signals. So this is a retreat from what I just said. If, however, you have only a deterministic mechanical system, I have no idea how to define the concept signal. Because for me, such a thing is just one big machine. And I do not know how to separate one activity of the machine. I call it a signal rather than the general activities of this machine. It is just following its equations, whatever they may be, and it would be hard for me to point to one activity and say, aha, that's a signal. Well, it's true that the bell telegraph system would be most useful for transmitting meaningful signals, but I would, I would, I think at this level of discussion I would tolerate almost anything as a signal, which could be identified as the effect of the signal. Now, there are two ways in which human beings can be inserted into a quantum mechanical calculation. You can put into the calculation external fields, and you can allow measurements to be made. When you put in an external field, you say that, in addition, if you work, say, with a Hamiltonian to define your mechanical system, you say, let's imagine adding to the Hamiltonian a term involving a field b, I'm sorry, a term involving a field phi, which multiplies some operator b belonging to the system. And this field I, we will say, is a free field. I don't give you any equations for the propagation of that field. I don't relate it in any way to anything else that has happened.

32:30 That's something at the disposal of the theoretical physicist, and you can use such a thing to represent the free interventions of an experimenter. Somehow that allows you to put in things from outside which are in no way dictated or even conditioned by what has happened before. So you can imagine, and physicists do that, you can imagine adding such a thing and you can ask, what are the effects of small changes, and of course you can integrate that into big changes, what are the effects of small changes in such an external field? You can't show. And it's easier to work with the Heisenberg operators, that the change in a Heisenberg operator due to such a small change is given by the commutator of the operators involved, that A is the operator that's going to be observed using a language that I hope means something to you, although it's not very good language, that A is what is going to be observed, the way in which it changes with respect to phi is given by the commutator of A and B. The operator which is involved in the interaction multiplied by a step function which says that the time of X, the time of the observation must be later than the time which the signal is the same. Now in quantum mechanics we have a thing called local commutativity which says that operators belonging to different places which are separated by a space-life distance commute. And so you see that the X and Y are space types of the spectral knowledge. This is zero. Changing an external field does not change the statistical results of measurements in a distant future. So, no signalling there, even if I have external fields at my disposal. The other way in which human beings are allowed into quantum mechanics is that you can allow them to make measurements. When I measure b and observe a particular eigenvalue, the width function collapses. I must strike out all those terms in an expansion of the width function which has eigenvalues different from the one that has been seen.

35:00 You can show that this again has no statistical consequences for what goes on in the distance region. The demonstration goes something like this. I can imagine the total wave function of the system, that's the whole experimental setup, being expanded into joint eigenfunctions of the two observables in question, the one A, which somebody might measure on one side, and the one B, which I'm going to measure on the other side. Calculated in such an expression is given by the sum over a of a b squared. In this expression, there are no interference terms between different values on b. That means that if I measure b and collapse the wave function and measure it again and get a different result, when I have averaged over everything, I find that I will get just the same result as if I had not measured b squared. So the fact that the result of the measurement is not under my control means that I cannot use my ability to do one measurement rather than another as a source of signals in the sense that I cannot influence the statistical outcome of the report experiment. So this is the sort of local causality that does exist in quantum mechanics, and one might as well ask, why should we not just contend with that? Why do I want more? Well, one reason for wanting more is that I do not like inexplicable correlations, even if I count 50 times. The fact that these happen in different regions of space and time and are correlated in an inexplicable way, inexplicable in the sense that I have explained, that is disturbing and really obliges me to go on trying to treat them. It gives me the impression that I have not understood the situation and that I should continue to try to understand it.

37:30 The no-signaling concept in itself is unsatisfactory. It involves concepts which are not very nice. For example, it talks about measurements. The concept of measurement is terribly obscure as soon as you start to think about it. It's given very glibly in the first few pages of the textbooks of quantum mechanics that measurements have such and such results and that after a measurement the wave function changes and so on. But when you look around at nature all going on and you try to identify here a measurement, there a measurement and so on, you get into trouble. It's very difficult to identify anything in inanimate nature which can be thought of as a measurement, and then you say, well, so it's people who make measurements, and then you ask, how well qualified must the person be? Does the person already have a primary degree? Can a cat make a measurement? Can a computer make a measurement? A calculator? Can the concept evaporate? It's one of these concepts which you can use very glibly provided you don't think about it. And then there was the other way of getting museums into the game by using free variables or external fields. Now the external field is a very poor model of the freedom of an experimental physicist. For example, an experimental physicist is operational only in a very limited temperature range. You have got to heat the lab to something not too far below zero and to keep it cooled to something not too near to boiling point. An external field, on the other hand, works absolutely regardless of the conditions. Whatever the state of the world may be, that external field tells what you have described. So somehow it's too free. If you really wanted to represent the freedom of an experimenter, you would have to have something which was a bit more constrained by circumstances. As soon as you decided to do experiments on the philosophy of signals between human beings, and I'm now working on a hypothesis that it's only human beings that can originate signals, you would be faced with the difficulty that the human brain is an extended object, not very well bounded in fact, and you would want to know whether the signal originated in this bit or that bit, or some other bit, at which...

40:00 Precise point of time before you could very carefully calculate the velocity with which the signal is propagated. So here again, this is the problem. So, we are left with this conclusion. That the one reasonably precise definition of local causality that I could give doesn't work is in contradiction with the theory of quantum mechanics that we all believe in. We call these very big concepts, and so it may be that the limitation to philosophy is less than life is not a very exact law of nature, but it's something that I analogous to the laws of thermodynamics. Now you know that the laws of thermodynamics are things which you do not find in the exact laws of nature. They are large plane approximations. And the closer you look at the fundamental equations, the less you see of the laws of thermodynamics, and it may be in the same way that the closer that we look at the microscopic laws of physics, the less we can extract the idea of the limitation of velocities to less than the velocity of light, even though, like the laws of thermodynamics, it is something that we are absolutely bound by. Can we make the travel arrangements to come and give talks like this? Thank you. Well, thank you very much for the very clear position of the very mysterious subject. We have said a good bit. Now, just to try to see whether we've understood who had to apply in this subject to the case of Wayne Starwood. Down to earth, by the way, Wayne Starwood. Do you think that the hysteria's correlation has any relevance to the description of the fact that locally distant situations, but presumably do not predict the outcome of the experiment?

42:30 I think I would put it differently. I would say that... What we see is that we cannot divide the world up so neatly as Einstein taught into space-time regions, one is there and we can dismiss this one, but that thing can no longer be done. On the other hand, we do have the attack itself, the variables which are free in this region. If somebody might be deliberately manipulated, do not statistically affect things over here. But to the extent that I am concerned with an experiment here, I can ignore the activities of someone in a remote place. So we still have this amount of separation of all of ours here from the rest of the world. And in this way, we do not have to deal with the whole world and with the activities of all of our natural species every time we discuss our experiment. If I ask, could you put up this slide with the LC theorem on it again. In this slide you have a theorem of classical probability, or rather you have the definition of classical. Conditional probability, which is what you call, well, it's just the first line which you don't call anything else. Now, a quantum theorist, somebody who really believed in quantum theory wouldn't use classical probability at this point of the analysis. He would describe the states not by probability distributions but by density matrices. I have not at all inquired into how you do your calculations of probabilities. Even quantum physicists, even though he uses wave functions or Feynman graphs or path integrals or whatever in his calculations, is busy calculating ordinary probabilities.

45:00 And his ordinary probabilities have to respect the laws of probability theory, so that his conditional probabilities have to satisfy this theorem. This is a theorem about how the results of experiments, whether calculated by squaring amplitudes or however, can depend on variables a, b, and c, which simply define the experimental setup, and hypothetical variables lambda, which are of the same nature as a, b, and c, but unknown to us. So it is just a question of extracting from that a certain subset which meets certain extra requirements. And this standard theory of probabilities is just the question to draw thin diagrams for what you like. You see that that's just the way it is. And it has to be like that even if the theory that you are subscribed to calculates each of these things by spurring an amplitude. This is true, I would say, by definition of probabilities. For me, the word classical in front of probability is absolutely redundant. Probabilities are positive and they satisfy the commoder of axioms and so on. No, I have joined probabilities conditional on possibly non-commuting. On the contrary, all of these things to use, these describe an experimental setup and these are hypothetical variables of the same nature which we do not know. There is nothing non-commuting here. These are results of experiments with non-commuting arguments. Could be you should use non-commuting objects. If you use non-commuting, you would not be investigating what I have defined as local causality. Maybe you define another one which has non-commuting arguments. But to call that local causality, my feeling would be that you were abusing language. You could say, for example... Indeterminism is just a new form of determinism involving non-commuting objects, and I would love to regard that as a bouncing theorem.

47:30 Sir, there is a hypothesis in this diagram. If we slide down a little, you've got two regions, the left block in the left circle and the right circle. Now, this involves measuring their position. What I was going to go on to say... Would you like to make it a cootsie, please? Go on. I've only got a very naive view of quantum mechanics. It's a decade or more since I've taught any kind of talk. It's still the same. Good. What I was meant to believe in my naive way years ago was that you cannot measure positions. Now, the usual way of expressing that is to say that if you want to know the position of anything within a given degree of error, then you have to allow a certain arbitrage in the momentum. But, in fact, I think probably a safer way to put it would be to say that any measurement position has a certain probability distribution. And the tail of that problem goes off to infinity. It is never zero. So, there is always a certain probability in any given experiment that those two regions work near the last. I think that is probably a very good one. But it brings me back to the very first thing that I said about the necessity in theoretical physics for local variables. And for the fact that these are only roughly approximated by the variables that we do have in the theory, namely experimental results and the disposition of experimental equipment, you are quite right to face the possibility that perhaps experimental equipment more or less extends throughout space and that an experimental result more or less occurs throughout space. It could be like that. And to the extent that that is so, the idea that philosophies are limited to be less than the philosophy of life simply becomes silly. We would have no point to have such a concept. So I insisted that if you did want to investigate this concept, you had to envisage a theory in which the rough concepts of contemporary physics are replaced by exact concepts called local variables.

50:00 Strictly speaking, it is in a theory where that refinement has been made that I could prove a theorem, and so long as that refinement is not made, the doubts that you have in mind, and therefore the possibility of formulating local causality, persists. This one, yes, we have E of A B minus the rest of them, and these three define all the probabilities, and if you create it by definition, they have to be less than three, whereas the maximum there, those are okay, so it should be less than three, because the maximum value for that E of A B minus should be zero. I'm sorry, I only heard one word in three, because the electronics are helping me with my English. Speak much more loudly. The kind of probability, the maximum probability of that D of A and B prime, for that question to be a maximum, that has to be zero, which would make that less than three, not four. You're saying that you're challenging the statement that this is less than four by definition. Yeah, okay, it's not one of my most essential statements, but nevertheless, I believe it can equal four. You can imagine, oh, by the way, that's the point. This E is not necessarily positive. It's the expectation value of the product of two things, each of which can be plus or minus one. So it ranges from minus one to plus one. And you can imagine that this particular one is minus one, and the others are plus one. That could happen. What I wanted to remark was not a question, but to make a historical remark, if I might. Now, I was extremely interested in when you spoke about local causality being, as it were, quasi-thermodynamical, because Einstein, when he was formulating, or getting towards the stage of formulating special relativity, did it himself, and he was particularly guided by the example of thermodynamics. Which he regarded as very technical, but ethical physics. But rather interesting, isn't it? It is.

52:30 Sir, I have a few more questions. I have a few more questions. To what extent does Rogers defend the assumption of the existence of data? What I have said is an examination of the consequences of assuming different variables. No, no, talking about free variables. Free variables. Oh, if I have no free variables, let's suppose that the world is a completely deterministic system. That even radioactive decay is determined in some way that we don't know, and that even experimental physicists are determined in some way that we don't know, then it might be that the theory could be locally causal. Certainly my theorem does not hold in this respect. So what we should say is that three wheels and local causality are incompatible. Right. Sir, I think there's a short remark on history of thermodynamics, not on the history, but I'd like to add earlier to what Professor Whitraer has said, towards the end of your talk, you did make a remark about thermodynamics disappearing if you take too fine a view of it. And that's absolutely true. But quantum dynamics must not disappear. And I think implicit throughout your discussion, you've taken that time is going on in some respect, unidirectionally, it doesn't do a trick and come back on itself. You don't have in classical or quantum mechanics any means of defining the direction of time, do you? You have to go to thermodynamics and algorithms in order to provide that direction of time, which, in my view, is implicit in everything you said. Is that so? Well, I make definitions as little as possible, and one of the things that I would not define is the direction of time. Now, when you make definitions, you have to define in terms of other things which are not defined. And I would take time as one of these undefinable. Now, it is a puzzle that the manifest asymmetry with respect to direction and time is not contained in the laws of microscopic physics. But somehow, the laws of microscopic physics have to be supplemented by binary conditions. And we have not, or at least most of us, have not yet got to the stage of spinning boundary features.

55:00 So that's as far away as I can push this problem. We're dealing with the velocity of light. Do you think that the fact that the limiting, or the limiting velocity phenomena is velocity of light is connected to the fact that all the apparatus we have, or everything at this scale, is electromagnetic? All the phenomena of everyday life are electromagnetic in our opinion. Well, it could be like that, that this restriction was for electromagnetic phenomena and not for other phenomena. I would find it difficult to imagine embedding that idea in a coherent theory, because at some level or other, everything utilizes the original. There is, for example, in quantum electric dynamics, there is the phenomenon of vacuum polarization, according to which even the propagation of a photon involves all the other particles that could possibly exist, which can come in virtually. And so, even the propagation of photons depends on everything else that might be, and it would be very difficult if those other things were not bounded by some such very general restriction. If you could not find, also for photons, some peculiar case in which some aberration would show up. But that's my guess. If you have a theory in which you can put such a limit on electromagnetic signals and not on others, I would certainly be willing to do. Yes, well, you said a moment ago, you said a moment ago that the quantum mechanics are not the standard for mathematics, which sounds very reasonable, but that theory has a number of different interpretations, degree of belief, frequency, proper suggested propensity. Do you have any views or interests in the question of which is the most appropriate interpretation of the problem? It's intended as a philosophical question. No, it's a question which I hope you're not going to ask. Go ahead. What is my conception of probability, whether it's a frequency or something else?

57:30 I have no answer to that. I know how physicists choose probability. They have formulas which allegedly calculate probabilities, and during the calculations, you do not have to worry what probability is, and then the rule is that to test such an assertion, you do many experiments, as many as you can afford, and see whether the frequency that you observe is equal to the so-called probability that you calculate. So if you wish to invent a fair probability to justify what this is doing, that is what you have to keep in mind. Well, I'd like to make a few quick points. The first one is the, uh, the Searleson inequality that you have there, 2 squared and 2. I wanted to just quickly point out, uh, I just wanted to quickly point out in connection with that Searleson inequality that it holds not only in quantum theory, but in any distributive algebra. So it holds the Jordan algebras and distributive Siegel algebras, and can be violated in non-distributive algebra. I just wanted to mention that. But another point is your diagram claiming that quantum theory is not local, where you do the space-time diagram and the time slice and the circle above it. Well, actually, I mean, quantum fields are local differential equations, and you can express the field in there as a function of the field of time zero. Quantum theory, from that point of view, is strictly local because the observables, they are actually expressed as observables in time zero. So really that is a consequence of... Let me interrupt him. Everywhere that you say observable, would you say operator? Yes, yes. But that is what quantum theory is. Would you continue with that? Alright, okay. So I'm saying that when you make a measurement of a field, that is associated with an operating quantum theory. That operator can be expressed in terms of operators at time zero, so therefore the non-locality, the non-causality that you're pointing to could be more carefully expressed in terms of properties of the time zero algebra already itself, and not in the propagation in time, because those operators can actually be all measured at time zero.

1:00:00 And you can actually express them to your left-hand piece with a simple interval, like when is the same time zero. So it is actually a causal expression in that sense of the piece. Yes, I think that what you have said is related to this topological equation that the Heisenbergs, especially in the relativistic theory theory, have certain structures. And one of them is that the differential equation is the satisfier in how we fit factors in the life form. As an operator assertion, that is offset to me.