Relativity, relational holisim and the bell inequalities

0:00 ...about 10 minutes to this afternoon. Well, apart from that, there's me, and then there is, of course, Teddy West on the show, and I shall be... Teddy, you're fine, okay? So, let's test this out. Testing, testing, testing, testing, testing. Cambridge, the 12th of June, 1987. Cambridge, the 12th of June, 1987. Testing, testing, testing, testing. In many respects, I think there's really nothing new in this paper. It's all an attempt to put things together in a different way, and I hope that you will find that I found that working this out, putting things together in this way, I'm feeling that, well, let me give you, to begin with, it's a very sort of snapshot of the kind of things I'm going to be doing in the paper. I'm going to be talking about the feeling that there is a conflict between relativity theory on the one hand and the fact that the other inequalities are violated on the other hand.

2:30 We want to see if we can understand more clearly why there isn't a conflict and what lessons we should learn. Two things. One is I'm going to be poking around at the status of this thing called locality. It's supposed to be known superluminally that very similar things trying to put them together in different ways and then I want that there are facts that the relevant facts about these relations deepen the relativity theory that the idea that with relativity theory that you're not supposed to be able to send superluminal messages or this is also referred to in the literature as the first signal principle and I'll be using both pieces of terminology.

5:00 We shouldn't be, we are convinced now after learning relativity theory that I can't write up a telegram and send it by some device and get you that telegram with some information on it by any system that travels faster than light. What locality means more than that though is very clear. Space-like related locations. In some sense, what goes on here has connections with what goes on there. Let me just take this. Now, here is the PR paper, I think. You can assume determinants. The Bellarmine qualities do actually have values, exact values, whether or not they influence by what you decide to measure over there.

7:30 Why is that a relatively clear application of locality? Well, if it were violated, you could send a message, okay? What, suppose I want to set one of my apparatus there with a code book, depending on what you observe over here, if I can affect these values by what measurement setting I make on the A apparatus to tell what I set, then I can use that to send a Morse code signal. And it turns out, and I can determine this in this kind of application of locality,

10:00 Here's the intuitive argument that values aren't determinants. In that case, of course, another feature of the setup used is that there is correlation. That is, if I make a measurement over here with a value, then I know maybe I decide not to actually make a measurement, but I know it's true that if I were to make a measurement, there is an exact value over here that should not depend on whether I actually carry out. The plan of making it over there. And of course, the other way around, so all the values have to, locality is right in this sense, all the things that locality itself is complex, rough, and what I'll call, which is what you need to get determinants.

12:30 It's also being used in what I always seem to theories, assume that all the values in question are exact point values. Relativity theory assumes that you've got space-back value at all spaces. Relativity is only going to apply a notion of locality which tells what causal, which puts restrictions on what causal connections are between values which are themselves exact point values. But this argument from locality to determinism has to envision at least the possibility that sometimes there aren't exact values, there aren't indeterminate values. The argument was, what I do here can't affect what happens over there, or what's the case over there, but what's the case over there might be a difference between being determinate and indeterminate. That's now a much broader sense of locality, which we have no reason to think is implied by relativity. So, the first approximation of the problem with this argument, let me start with what this that I have been glibly referring to as relativity,

15:00 And general relatives, like there's just one, or, and amplify with all sorts, in all sorts of ways, in all sorts of theories, and I'm going back up to four areas, and you could amplify this, this first, by, for example, talking about trajectories of massive particles. In particular, if you were counting on sending your telegram by writing it out on Western Union stationery and putting it in a little capsule and putting a rocket ship behind it, that way at least you're never going to get a message sent faster than the speed of light. You could send your signal by using light rays. Well, that won't get it there faster than the speed of light. So we have two particular realizations of relativistically invariant theories. Theories which, in those special applications, give the restriction of no way of getting messages faster than the speed of light. More generally, we can think of We can extend this theory to give a more general treatment of the electromagnetic field, where you have something similar in that the state at any one point is, if you fix the state of the electromagnetic field farther down or in a sandwich, farther back than the light comes, a yet more amplified way of we can give a faster than light signal.

17:30 However, well, let me introduce another piece. There are, again, a family of what we might call relativistic causal theories. In each of the applications that I've mentioned, we get the result. So far, using the kind of mechanism that you can't communicate messages, or more generally, you can't in any way talk about causal chains influencing each other faster than them. The trouble is, though, that consistent with this principle, we could think of all sorts of elaborations. It is estimated that it is consistent with this that we can have tachyons, or the light-speed barrier, and slower them. And again, it's perfectly consistent in ways that will become much clearer as the talk proceeds with relativistic invariance. And again, talk about counterfactual dependencies of all kinds have no obvious clear relation one way or another to the sorts of more specialized situations, Let's go back and look at this and see how we're doing. When we said that relativity implies locality, I was being obscure. What precisely is it that implies locality? If it's just the criterion that your theory be relativistically invariant, there's no reason at all to think that implies locality.

20:00 All of these theories seem to be the examples that we have of all the theories in which the quantities all have exact point values at all places at all times, and so in any of the applications that we know about, it won't give you the strong notion that we'll have to deal also with some of those theories. The kind of indeterminiveness of the question here is quantum indeterminism. What do we think we should ever get outside of this family? Well, we have a candidate, namely, quantum theory. But even in quantum theory, you do have exact values, all right, namely, you have the, always have an exact value of the state function amplitude. And so, why shouldn't we think that those amplitudes, I don't think that's right, and there's a straightforward reason, and there's a not quite so clear reason, only because there's a not quite so clear reason first, because... The amplitude has an exact value at each point, all right, is that to say that what you, what you, so looking at the amplitude, the square root of the amplitude, you know, that this really does characterize by giving the value of the amplitude.

22:30 How do you characterize what's going on? You have to say what information does this give you? A state which was a mixture corresponding to this coherent state would give you that same information. So if what we've got here really is a coherent superposition instead of a mixture, I have to be careful to distinguish between the coherent states. And of course I can do that only by looking at how the values of the state function at the different points interfere. What that really means is that what's going on at any one point in the collection of those facts doesn't give the whole story. More specifically, suppose you were to ask about the momentum of the particle in question. In order to determine the momentum, you have to perform an experiment which is in some way spread out, either you have to look at momentum exchanges at distinct points, or you have to look at that interference phenomena, but to look at interference phenomena, you have to give the wave some space, so you're going to have the wavelength and the frequency well defined, so there's a kind of operationalist flavor for thinking that distinguishing between the mixture and the... Now that kind of argument is a bit hand-waving and, as I mentioned, described in operation lessons later. I haven't done this, but I actually think that one ought to be able to make that argument precise. By referring to the fact that in giving the coherent superposition, what you require are not only the amplitudes, but the phases at each point. But the phase is always a relative phase. And to know what the relative phase is at a point, you have to know something about its relation to the phase at other points.

25:00 So the facts here really aren't facts about individual points, they're facts that connect the different points. I've been quick with that argument. I'm going to give a much simpler argument for the same conclusion, and that's when we have superpositions which involve more than one system, so-called entangled states, which, of course, is just what's in question for development qualities. So, if I have such thing as the amplitude at one point, All right, you look at an individual point and ask what the amplitude is, it does not give you, and you ask that question about each point, that does not give you what the state function is. To get the state function, you have to have the amplitude at all pairs of points. We don't hear, we see here at least one kind of clear-cut example where the kinds of facts represented by the statement aren't facts which can be covered by the sort of theory that these relativistic causal theories seem to be about. Okay, so this argument is not looking very good. They're a little bit clearer about what's wrong. We have exact correlations, and where your derivation of the Bell inequality isn't one of the so-called stochastic versions. Let me now look at what happens if we turn to the stochastic versions of the arguments for the Bell inequality. It's to talk about the joint probability of there being a probability. So we're going to look at the probability, given that the measurement apparatus on the A-wing here becomes a little bitty,

27:30 To make sure that we're looking just at the quantum facts and not hidden facts, we conditionalize this on all the exact value factors there might be, the hidden parameter lambda. Another way of saying the same thing, we're looking at so-called hidden variable theories. We want to find out the extent to which we can describe what's going on in terms of Definite facts that are true about one situation or the other situation. We can code those just with one variable to cover all of them. You look at the joint probability for outcome A given measurement setting A, outcome B, measurement setting B. And now looking at this probability you say the following. Since you believe in locality, we ought to have something called factorizability, and that's just a statement that the outcome here, A, well, first of all, it better not be dependent on the, on what happens, I'm sorry, on what measurement setting, you said over here at B, it also shouldn't be dependent on what the outcome is on B, alright, because if it was dependent, you would have... So what you would expect is that you have factorizability, which precisely is that the probability of A given capital A and lambda, I'm sorry, that this joint probability just factors into the two pieces, the probability for A and B given cap A and cap B and all those two parameters lambda is equal to the product probability that you have for little a given Again, the intuition here is, if this weren't true, if this were an inequality, there'd be some statistical connection to it, and it feels like that what happens on one wing is somehow affecting a causal effect on what's happening at the other wing.

30:00 From this assumption, Bell's inequality is followed quite easily. I'm not going to reverse that. None of that is controversial. So what we need to examine is whether there's some sense in which this condition follows from locality which follows from relativity, giving us Bell's inequality, or whether the parallel arguments also, in this now stochastic case, also go back. Initially, when I was working on this stuff, I said, no problem. I'm going to show that this argument equivocates in the same way as the more specialized arguments did by saying the following. This argument assumes that there are definite probability values at each point, probability values as factors. If there were definite, you can think of these probabilities as dispositional properties. This is a property that applies to the A-Wing system. It gives you the distribution for what you get if you were to set your measurements this way. This is a property that applies to this position. Dispositional properties represented by these two probability distributions, then there'd be some plausibility that they couldn't be independent, sorry, there'd be some plausibility to think that if they weren't independent, there'd have to be some breakdown on locality, and so relativity implies locality implies factorability, but then I'd want to say, yeah, but the problem is that in some sense these probabilities themselves are indeterminate. Well, people were quick to point out to me that this didn't seem very plausible because quantum mechanics always gives us these probabilities. They seem to be perfectly determined. So the difficulty that I found with the original formula of the argument doesn't seem to apply in this case. Let me give a very brief summary. I want to talk about something else, which I hope is going to throw luck on the whole situation. Here's where we've gotten to. We can derive Bell's inequalities either from the assumption of determinant values or from this factorizability assumption.

32:30 And there is some, although not very exact, there seems to be some plausibility to the idea that determinants or the factorizability conditions express some kind of locality which might be thought to come from relativity theory. So now with that summary for where we are, let me break and let me talk now some completely general wild and wooly metaphysics because I think that the way we are all accustomed to certain presuppositions is so deep that we're not aware that we And precisely the presupposition in question, which makes both of these arguments seem to be an unstated part of this classical heritage, is that if you start out with a picture of the world made out of physical things, and if you can say exactly where these things are and where they are going, then you could set the Laplacian superintelligence to work to calculate both the future and the past. It's part of this picture not only that the world is made up of individual objects, or perhaps individual space-time points, but that the objects have properties, in particular they have non-relational properties, properties which are just about the individual atoms.

35:00 Now, of course, there are also relations. There are facts like the fact that this pen is inches long. Sorry, this end of the pen is four inches away from that end of the pen. That's a relation that holds between that little silver bit. So it's not as if the classical world didn't have any relations in it, but I claim that once you lift up all the objects and you lift up all their non-relational properties, then in a sense everything has been set. Once all the objects, all the atoms, and their non-relational properties are set, then everything, including all the relations, are set. We can make this a little bit more precise with the notion of supervenience. So we have two objects here, which can be separated by one foot, say. And now I go over here to two other objects. Speaking metaphorically, and I don't take possible worlds for granted, We travel now over to another possible world, and I copy these two objects, and I give these two objects exactly all the same properties in this possible world, all the same in this possible world and in that possible world, and then I'll also automatically have reproduced all the relations. I call this world particularism and again my claim is that we tend to think of the world as particularistic and we aren't aware of it but it's the fact that we are unwittingly making that presupposition that makes all of this so. I wanted to argue that classical physics can... and there seem to be exactly two... there are two flies in that ointment.

37:30 One is that if the relationalists are right about space-time, then relational properties don't supervene on non-relational properties. A substance finalist would seem to be able to get out of that because he's got space-time points. But not really, because a substantivalist also has a different non-supervening relation he's got to worry about, namely the relation of being located at. And then it occurred to me, though, if you thought of spacetime points as properties, like math, then you really could see classical physics as being true and particularistic. And that's actually how I first got into that. Let me put that aside or I'm never going to get done here. Okay, let me introduce another piece of terminology that I'll be using. In application to the present discussion where we're talking about physical quantities that do or do not apply to space-time points, one can express the Particularist Doctrine as what I would call the Universal Point Value Assumption, or UPVA for short. And that, now, is the claim that there are only spacetime points thought of, if you like, substantively, or thought of as properties, and that all quantities always have exact values at each spacetime, at each spacetime. Now, could the universal point-value assumption be correct? Well, that's the denial of what I was munching about a moment ago, to deny that particularism or universal point-value assumption. And finally, we have a very plausible candidate for these non-supervening, or as I'm going to call them, incurrent relations, namely precisely the sort of superpositions or entangled states that give rise to the superposition.

40:00 There are all sorts of puzzles that you have in the development qualities, the ETR state, the double split experiment, and, well, that's not it, but at least the single state. You have, I mean, one property, and we have a state where the A system has one property and the B system has another property, that's all right, but they could have a superposition, which is somehow on the line of these two, right? There are no spin properties or, indeed, any others. The definition of property would be just properties of this system and just properties of that system on which this thing would, say, supervene. If there were, it would be to examine the way in which the Bell inequalities can be seen as an instance of inherent relations, or an instance of what I call relational holism. Now the way I propose to do this is by saying that there really are two kinds of localities that are in question here. The first kind of locality is just the universal point value assumption. It's a locality of values. It says that the values of quantities are themselves local. This is one way I think to understand Einstein's principle of separability. By the way, I can cite as some kind of evidence that indeed this is a very thorough going part, that he couldn't do physics if points weren't separable, by which I think he meant something like the universal point value assumption.

42:30 Now, there's then a second kind of locality, and that's the kind of locality that these relativistic causal theories are about. This is a space of possibilities. The world might be one in which the universal point-value assumptions fail, or the universe might be one in which the universal point-value assumptions hold. Now, in those universes in which the universal point-value assumptions hold, we might have a world in which In this case, we're looking at a world in which all quantities have exact values at all points. In that special situation, there is, perhaps, that we've dealt with in a pretty broad range of cases, relative to causal connections. It seems to tell us specific suggestions for what one means by causal connections. All of these terms can propagate faster than the speed of light. Relativity theory implies locality. What I claim is really going on is we've already got point value locality. Because it never crossed their minds that the world might be like this. We imagine that the only possibilities are these two, and of course, if we focus attention on these two, then, at least in specific examples of proposed causal mechanisms, relativistic causal theories seem to suggest we're in one of these worlds and not in one of those worlds. So, from now on, I'm going to accept this assumption of local causes, although it's still, I haven't done very much towards it. The only thing I've done is to try to

45:00 All of these terms can be used to give a plausible assumption on the assumption that there are point values, the kind of theories that we see in relativity theory that do suggest that. Now, the other thing I have to make this completely clear is getting them in there in some way. Indeed, it's not as if everything... I don't have any understanding of whatsoever. Clearly quantum mechanics is getting us in both worlds here, all right? And what the connection is, that's essentially the measurement problem. I don't have anything helpful to say about that. So let me proceed. How does this all bear on these two arguments? Well, with this, it's very straightforward, okay? The point that we're now seeing is that we have to admit this as a possibility, insofar as this is a possibility, relativistic causal theories don't give us any reason for thinking it's wrong, so we don't get the strong notion of locality out of any relativistic theories, which we need to argue that the world is really one of these worlds. Which, together with another application of locality, which at this stage would be clear, because it would be in one of these worlds, but they were seldom involved.