12th UK Foundations of Physics — Bell Inequalities for Beables & Collapse Theories

0:00 Thank you. Alright, in spite of the spatial transformation, So our first speaker will be Peter Morgan from the University of Oxford. On the derivation of value to appeals, we know the intent is to think about the assumptions needed to derive value to appeals. Now, my intention, and I hope you'll join me in it, is to think of the local Behables theories as more about a classical statistical field theory, rather than as a classical product of Behables. I take the well-known causes for Behables to be more suitable for field theory, precisely Mostly because the picture, so to speak, that goes with the, I don't know, the variables, has hidden variables A and lambda, C and nu, B and mu, associated with space-of-time regions, rather than with particles or anything that sort. If you, you can, of course, describe a particle sort of system in these terms by saying that any particle that happens to be within this region would contribute its hidden variables to the hidden variable description associated with this region. I take that to be a level of indirection that is inappropriate for a particle theory, but

2:30 it's very much appropriate for a field, but on the other hand, this description is very much appropriate for a field theory. In any case, I suppose I'd say that the history of the late 19th century through all of the 20th century has been precisely in which the laws feel there is, and it seems a little late for us to be, for us still to be thinking in terms of classical particle model if you want to think in terms of hidden variable models of any kind. So, and all this is part of a program, as far as I have a program, in which I try to see more clearly what the distinction is between a classical statistical field theory and a quantum field theory point B. Now Bell derived from inequalities for vehicles in 1936, and it's two papers, this one and this one, both in his book, of course, and Morris, straight after he published it, he published a paper, finding a flaw in it of sorts, and Bell admitted that, but said, oh, but in order for that to work, hidden vehicles associated with this region here, the intersection of the past of this region, the past of this region, have to correlate with instrument settings in here and in here. And he felt, and I think it's fair to say that everyone seems down as well, but that actually is a reasonable assumption to make about classical field theory, about classical theory in general, and really go to say that very few people are actually sitting in this situation for a classical field theory. Now, whereas there's presumably thousands of papers on As far as I know, there are only 20 or 30 papers Precisely because it is perceived that he more or less broke a line underneath the situation with this paper. There are some interesting contributions, but they don't really make much difference to the situation. Now, the distinction that's made between A, B, and C, and lambda, mu, and U, it really is very, very minimal. From a mathematical point of view, the only mathematical difference is that we integrate over lambda, mu, and U. And the difference from a physical point of view is taken that

5:00 we know what A, B, and C are because we've measured them or we've controlled them in some sort of way, and so they're not hidden. And on the other hand, lambda mu and u, precisely because they are hidden, you don't know what the values are, you have to integrate it with all the possible values you might have. Now, in order for the derivation of the analysis to be particularly meaningful, it must be robust under different choices of where you split between a and lambda. Let's say, someone who wants to deny that they can be given variable theories can't say, well, for this particular choice of A, the lambda, you can't construct a model. And clearly, they didn't very well just come back and say, well, no, that's not the choice I want to make. I'm going to make this choice. So whatever our derivation does, it must be robust for different separations. Okay, now Bell's fundamental definition is that for a locally causal theory, where variables associated with this region here, X cap is associated with this region here, then the philosophical way of saying this is that X, X cap, and XP screen off X from Y, so there's no correlation between X and Y conditional on X cap and XP. Now, I point out this definition is very, very vague in this way, just exactly what some might mean. Does that mean all except one, or does it mean just one? or does it mean you can choose whichever variable I like that you'd like? So I'd say this is a delicate definition in some sort of way. Of course, he uses it in a particular way, and from that, he uses what he actually means. On the other hand, Simone, Hall, and Clauser say that for locally causal theory, where XP now is all of the hidden variables associated with this, which is a relatively well defined in classific terms definition, Xp screens off X from Y, and that essentially moves the not only common cause assumption in that if you know this, then that screens off any correlation between here and here. And in terms of a Reichenbachian sort of way of thinking, that's essentially a common cause assumption.

7:30 Now, this is much more natural from a classical physics point of view. For a hyperbolic equation, a general hyperbolic equation, we would say, well, that we would accept as an assumption. This, on the other hand, is rather a strange piece from a classical point of view. As I say, in some sense, you're introducing a common cause as a criterion that has to be satisfied by a classical field theory, and that's not something that's a history of force. But if you make this assumption, then you have to make several assumptions. There's no conspiracy assumption, which is the only one of these, which something previously, as far as I know, has been identified as particularly, well, indeed has been discarded or people are not concerned by it. Which is to say there are no correlations between instrument settings and different variables. You also have to make a no correlation assumption, as I call it. There should be no correlations between hidden variables, between hidden variables. And also, you must make a known contextuality assumption, as I say, in terms of known correlations between the instrument's epics. And once I've gone through the way that it is derived by Bell, at least I should say by the Spaniard, since I was used to derivation of equals, and if you take it subsequently, I'll discuss the way the point of field theory can be looked at in terms of describing a Bellin equivalence in the experiment. So, suppose that we actually observe a and b in those regions of r a and r b. We can apply equation one or equation two either way because So here we have C, mu, here we have lambda, mu, lambda, a, excuse me, here we have mu, Here we have B, here we have A. A and C and lambda and mu are together, all the hidden variables are here, and so we can say that we will screen off mu and B by virtue of the fact that A, C, lambda, mu are there. So that's fine, either equation 1 or 2, because we've

10:00 of the variables involved actually in this conditional probability equation. To calculate the gene, this is all very standard, albeit it's slightly different from the derivation of all three derivations of low inequalities not for variables. Straight conditional probability gets you to here. The application of these two equations gets you to here. Then what we can do is rewrite this. is a matter of conditional probabilities separated into these three separate conditional probabilities. Then we can apply equation one, or we can apply equation two, together with additional assumptions, to derive this equality and these two equalities. As I say, what we have here is that we have lambda, so that's here, we have mu, mu, a, b, and c, and we want to get rid of mu and b, essentially. However, what we have here is mu and a and c. Mu and a and c, we do not have the lambda in variables, which would allow us to apply equation one, the dulls version of what a lowly causal theory should be. Now, we don't have what we need for equation two. We do have enough to be able to apply equation one. Now, this I'm going to call no correlation. Strictly speaking, it's no correlation in a sense, but it's conditional on the boundaries of mu and a and c, So there's no correlation between lambda and mu, conditional on mu and the MNC. And, of course, that doesn't mean there's no correlation between lambda and mu, only conditional on. Nonetheless, I've called it no correlation, and remember that the mass is actually not exactly the same as this much mutation. You also have no non-local, since you're not going to law on that, but I didn't feel on the first and the second foil, that lambda and b are not going to be dependent on each other and since those are in space-wide separation and so on, it's not something that particular to go on. It would not concern me if they work our relations, but nonetheless, it's not something that has to be.

12:30 Then once we have these three things, we feed that back into here, we feed that into here, we can carry out the lambda and new integrations The mean of A given AC to the new, the mean of B given AC to the new. Right. Other than that, we suppose, without any loss of generality, obviously this is, we're not dealing with the probability version of the construction of bone inequalities, some CSH sort of derivation, then the average is also going to be less than equal to 1. Now, we're going to assume, together with Bell, essentially, this, which can't be derived on his assumptions about both the causal theories or Schoenberg and Krausers. Then, we also have to assume that we can change A, B, and C, because what we're going to do is we're going to add the mean when we take the measurement directions, A, B, and C, we're going to answer that A, B dash, and C. We're going to claim that we can change B to B dash without changing either A or C. Now, in sort of immediate sort of way of looking at that, well, I've got some instrument setting over here. I've got an instrument setting a mile over there. Surely there's no reason why I can't do that. And I agree completely, except that in fine, fine detail, the class purpose is really doesn't want to agree with that. It depends just exactly how you do this. So we're going to add those together, add these together, add these together, and lo and behold, we get our less than equal to two. Now, in contrast, if you have two spin-and-half particles, just for a typical example, you find that this only has to be less than or equal to 2 squared root 2. However, if any of these conditions, of which only this one, as people have talked about, are not satisfied, then you can only prove it's only less than or equal to 4. So there's really not a great deal of difficulty reaching 2 root 2 in some sort of loose sense. It's about as loose as the way that Bell talks about it. So I don't know if I can say, because it's not a particular loose set. Now, first of all, I'd point out that no contextuality is closely related to no conspiracy, and it's really on the basis of this point where I say that the no contextuality issue really should be taken a little more seriously than what we might assume.

15:00 assume. Now, if we suppose that the vehicle C that we're measuring is actually complete information, in other words, we're going to measure everything about here, or we're going to claim that we know everything about what's happening in here. Now, we have that the variables new and null, so we're just not going to do an integration overview, and if we assume no correlation and no non-local conspiracy, then we can just reduce this. Now, to write on inequalities requires only that we have no contextuality without no conspiracies. So, in some sense, we can trade off no conspiracy and no contextuality by simply changing our opinions as to what we've measured as opposed to what we haven't measured and what we're going to integrate over and what we're not going to integrate over. Now, in quantum field theory, the Rich-Leyer theorem is usually thought quite awkward, precisely because if we measure every single observable associated with this region here, that, in fact, determines all the observables everywhere else in space-time. At least it does up to the sort of changes that one would be requiring here, instead of setting A1, you'd have to have setting A2. So there's a finite change between the settings here for one measurement, for one experiment, or for another. So, yeah, we don't take that contextuality to be a problem in common field theory. We take it to be a problem, but it's a matter of how are we going to deal with this rather than saying, well, that rules out common field theory. And I'd say, I'd suggest that that's the same sort of attitude we ought to take to, you know, contextuality that we need to project morality in a classical field theory. We should allow this sort of arrangement. On the other hand, suppose, on the contrary, that we take new to reconcreated information and integrate over it completely, and then we're going to assume that there's no correlation, then the mean comes out more or less the same as we had before, we are at C because we just don't have C. Then, for us to be able to derive them in the policies we simply have, we have to assume the no-conspiracy and no-conspiracy in a similar sort of way, except that now we don't have C involved.

17:30 Now, in favor of no-conspiracy, Bell argues that the dynamics can be made very chaotic, and hence, the settings may be taken to be at least effectively free to the deficit at hand. Now, this should be a thing from a classical point of view, that either are correlations or there aren't. If I have a particular configuration in space-time of my field, that I'm taking to be describing the experiment, then if there are correlations, there are correlations. I simply measure them between here and here. It may be that five seconds later, it's very difficult to measure those correlations, but nonetheless, classical physicists would simply say, Well, the correlations are there. It just will take a very, very fancy experimental setup in order to determine what the correlations five seconds later will be. And in principle, it matters not at all for classical physics, whether the dynamics are chaotic or not, as to whether there is a correlation between these two space-like separated regions. Now this is clear enough to my mind that just because A and B are free for the purposes at hand, that just does not imply that U and A and B are mutually independent. It's clear enough that if you have a correlation between U and AB, that doesn't determine A or B, or indeed U, but it just describes a relationship between AB and U. Now, if in fact there is a correlation, and I managed to set AMD successfully, then it just must be the case that I did manage to set new, even though I didn't apply any effort to it happening, nonetheless, it must have been the case that you set successfully, it wasn't promised precisely because I didn't succeed at it because I didn't try to do it. Okay, so moving on to the no-correlation, which in some ways I think is the place where are rather more vulnerable, I think, and firstly there's just no way to justify the no-correlation assumption. You cannot measure the variables you set you're not going to measure, or you can't measure, And I take it that you would agree that we can't measure the infinite number of variables

20:00 that we're taking mu to b. On the other hand, of course, it's a metaphysical claim to say that there are correlations between mu, founder, and mu, which are not screened out by A, B, and C. So the classical physicist, I think, is free to take either choice, depending on which is suitable for him. to determine that they can't construct a hidden variable theory is actually, it's reasonable for them to insist that they can't use this as a way of constructing a hidden variable theory for bell inequalities. Very much in a field sort of sense. Now moving on, now Shimoni and Paul and Clauser argue this way, they say that even though the space-time region in which lambda is defined, I should have said this earlier. Lambda and A is not associated with the space-time hyperplane. It's associated with the whole of this past region. When you talk about lambda, and that's why this talks in this way, even though the space-time region in which lambda is located extends to negative infinity in time, so lambda is associated with this whole region. Nu, A, and C are all the vehicles other than lambda itself, so you have A, Nu, and C. are in the backward light cone, and indeed it's all, that is all the hidden variables apart from lambda itself going back here. And mu and b do refer to vehicles with space-like separation from the lambda region, so mu and b are at space-like separation from this region here. And so therefore we ought to be able, it should be okay for us to say, just do this. But now this is, that's fine, I mean, you can argue like this, you might even say it's a strong this argument is exactly the same as to say, well, we're going to use Bell's definition of a locally causal theory. So if you're going to make, if you're going to actually use a properly classical definition of this, of what a locally causal theory is, that's, in other words, you're not going to have any reference to a common cause of this. Then, there seems to be a point making this argument in a supplementary way to get yourself back to the situation that Bell allows, so that you So this is the situation that Bell allows, where the fact that landerism there doesn't pretend to be able to deduce this, or to form his principle of the causality theory definition.

22:30 Okay, so moving to quantum field theory, description of the way that Bell inequality quality-violating experiment might work. We're very used to using a four-dimensional field space to describe an experiment like this. However, I suppose physics would doubtless, and should, have a problem with this statement, since obviously the theory reduction is always problematic. This is less problematic, however, which are many theory reductions one might introduce. And a physicist would certainly say, oh, yes, well, quantum field theory is what underlies this four-dimensional field of space description, and we'll go with that. Now, however, the means is that when you see it, when someone in some sense says, ah, I've got this beautiful experiment, you say, oh, well, what's your quantum field state? And you say, gosh, that's Jolly Carter. You've constructed a system in which you get inequalities violated is really remarkable experimental results. And we say, gosh, that's very clever of you, to build a quantum state that does that. And indeed, it's just the fact. It is a remarkable thing because it's not easy to actually do these experiments. You actually have to work very hard at making a quantum field state actually violate the values of any kind. And it is precisely the fact that there are non-trivial correlations in that state interest and marks those states out as very special. In quasi-probability terms, because after all we can use a Wigner sort of description of this, what we have is that at this time here, we have a particular quasi-probability describing the quantum field state, we can use that way, and what we would say is that at some previous time, at some earlier time, so we have some Wigner function here, And at an earlier time here, we'd have we dashed Q and P and T dashed, yeah, that we would say, ah, well, we've arranged that the Vigna function at this time will indeed evolve to this Vigna function here, and whatever the dynamics are determines what this has to be in order for this Vigna function to be the case.

25:00 Now, in the classical case, we have exactly the same situation. We would set it up so that we would have a probability distribution here, describing what's happening here, and that would require, given a particular demo, that we would set up the probability distribution in the past to be what would be required to make it because of this. So there's no greater conspiracy in this kind of description than there is in this kind evolving quasi-probabilities, deterministically evolving quasi-probabilities, as opposed to deterministically evolving probabilities. So, there's no violation of binary-probability we can look at. Let's just add two minutes. Let's get back. We will just pick them. Concluding the correlations we discussed here, I think probabilistic very little, no more the quantum field theory sort of correlations are committed to. If we take an equally impressive quantum field theory to make quantum field theory, they're just all correlations. And again, classical physics always is set. We have initial conditions. These initial conditions, we've given you a place where it's by P or whatever, instead of a Ligner function. So whatever initial conditions are now, we just have to make sure that in the past is not now. And at only times, one of the ways Bill argues is that, indeed, there is these violations have to be violated, these assumptions have to be violated in order to construct yourself in the model. But it's not necessarily to be violated by much. Once you've got back ten years or whatever, you have this enormous great space, you have ten light years across and whatever the description is back here just has to be something that gives you what you observe here. And you can have infinitely small, sorry, arbitrarily small modifications back here which result in very peculiar things here because that's the way that the field series work. Okay, so this is the

27:30 and so many more than houses description. The no-contextuality assumption, although that doesn't... I don't want to put too much weight on that, but it does extend the no-conspiracy assumption, makes it a no-conspiracy assumption that requires just a little bit more from the person who wants to prevent a classical model from being constructible. And then we also have the no-correlation assumption, which, as I say, the quantum theorist has no right to, essentially, to prevent someone from making a decision like that there are correlations. Okay, now Bell accepts that the mayor theorists and such conspiracies inevitably occur, and these conspiracies may then seem more digestible than not because of the theories, and then says, well, I'm not going to make such a theory, I'll wait for someone else to produce one, and then I'll be in trouble with interest to see how they manage to do it. Well, I now have a story that suggests that once you've taken apart this derivation a little, there's actually more space there for a classical field theorist to construct the model than previously as we thought. But I think that the space that's available for a field theorist is not a space that's easily available for a particle theorist. And that's because for a particle theorist, essentially you're talking in terms of things coming out of here, two particles coming out of here and landing here and here, you don't have these hidden variables in the same way. These hidden variables, the variables, are only the A part, the measured part, are actually relevant to the particle for the particle theorist, whereas lambda and u are non-trivial for a field theorist. Okay, now I emphasize that although I put this in terms of the local description, There are reasons for thinking that a vehicle's model must be non-local and a cable belt. Well, I won't put this on. I'm sorry. I will take a minute out of my time. This is essentially an advert for 1PHO 304171. What I do is, in that paper, is I say, if I've got a quantum field that is commutative at space-like separation, and particularly if I take a time or quantize finding the field, then I can calculate a probability distribution over pi at a particular time. And the result

30:00 you get out is this object here. Let's say it's an exponential of a Gaussian. You have this kernel in here which is root k squared to sem squared. The root k squared to sem squared is precisely a non-local, it corresponds to a non-local classical dynamics. If you have this non-local classical dynamics description described by this particular kernel, you get precisely this probability of distribution. On the other hand, there's no other classical dynamics which will give you this thing. Now, on the other hand, when you go into real space description of this, this is a modified Vessel function. This modified Vessel function has exponential tail long to one side. So you have to, I think, if you're going to go and construct a classical field theory model, you do have to have a non-local theory, but it's in a very specific way. And it's essentially, this is an extension of something like the heat equation, which obviously has exponential tails also, which is deemed acceptable by classical physics. But nonetheless, this is a relatively invariant, a relatively distinct invariant type of non-locality. And I'd point out, too, that H bar in this is playing exactly the same role as KT plays in a classical Gaussian or a Klein-Gordon rule, where you'd have K squared to 10 squared there and 1 over KT there. So what H bar is doing with this is it's acting exactly the same way as KT amplitude of fluctuations in the field. And as such, you can then adopt something like a Heisenberg fluctuation approach to understanding precisely what what quantum field theory like this is doing. Okay, excuse me, I'm four minutes over there. Three minutes, I'll start. I think I'm about 30 minutes. All right. Next is Guido Bacchigaluti from Berkeley. Thank you very much.

32:30 OK, so, my title is Glass Theories as Cheable Theories, and let me quickly explain the motivation behind his work. If you remember Bill's paper on G.L.W., he has this remark saying that the G.L.W. hits, the multiplication with the Gaussians, ought to be understood as the variables of the theory. And I never understood quite how that was supposed to work, and so I'm working out how it can work. There are also other motivations about the time asymmetry of the collapse theory and the 80-0 rule and so on, but these motivations will not be apparent in the rest of the talk. Okay, so, to set up things first, a very small summary of collapse series, or rather the original GRW model. So we've got a stochastic evolution of the wave function in position representation, and what we have is that at random times, with a certain distinctive frequency, the Schrodinger equation is supplemented by the multiplication with Gaussian, these Gaussians all have a fixed width, sintered at some point x, so that the wave function goes over into that one, with that normalization factor, and there's a certain

35:00 probability density for the center of the multiplying Gaussian, which is in fact mutual to that normalization. And indeed this thing integrates to 1 over, if you That's the same as an elliptical thing, so that's what integrates do well. So, that's the original VW model. Now, of the B-A-Ball theory, I'll take a term to refer basically to Bill's 1990, 1984 construction, which Tumas has already reminded us of in this paper, Wiebel's quantum field theory. And the idea is, let's take some quantum mechanical observable, specifically basically, now it is a sharp observable, a spectral measure, as I think of the eigenprojections of a separate joint operator, and in his example that was supposed to be fermion number density on a lattice, but let's just take it abstractly, we assume that this always has some definite value, irrespective of what the quantum state is. And they construct a dynamics which is generally stochastic, such that the usual quantum mechanical probability distribution is the asymptotic distribution for the process. That means, in particular, this distribution is preserved by the dynamics in some kind of time-dependent equilibrium. And Bill has a canonical construction of this dynamic, you know, among all the probability currents that satisfy that continuity equation, chooses one in particular,

37:30 And then he defines the infinitesimal parameters of the process, which by standard theorems then determine the dystochastic process as a positive part of current divided by the density. Actually, it's not quite through standard theorem that restricts the process because this density can have zero, but you have to avoid some care to show that this has global solutions and unique ones, but it can be that way. So, that's the vehicle theory in the sense of the Bill 1984 paper. And the intuition one should have is that it's some kind of analogue of generalization of Debray-Bone theory. We've got a wave function that equates a Schrodinger equation, which guides the evolution of the vehicle according to a certain equation. And this equation is in fact quite analogous to the guidance equation in the standard degree bone theory, one can show that in the sense that with an appropriate definition of continuum limit, if you start with the vehicle theory that uses a discretized position as the vehicle, you will actually get the bone theory out again, and that's being shown by Tony Semperi here, Edmund Bink, and, yeah, as I've heard, similar to that. And, yeah, I mean, that particular dynamic is sort of canonical, maybe because

40:00 of that, and another nice feature of that specific dynamic is that if the viewable, in fact, is a decoherent variable, on the scale of decoherence, the transition probabilities of the viewable theory are the same as the standard transition probabilities of the bond. Okay. So, if we want to put things a bit forcefully, if we compare the collapse picture and the viewable picture, they look diametrically opposed. In the Veerle case, there's no collapse, and, okay, I'm sort of forcing the two things apart. The ontology is provided by the Veerles, whatever that means, and the wave provides evolution, while in the GW case, we do have collapse. perhaps, the wave provides the ontology of the theory, whatever that means, and the hits, the supposed vehicles, according to Bill's remark, provide the evolution. So, how could it's ever be understood as vehicles. Yeah, I say, I put it forcefully because, of course, one can also say, oh, we've got a dual ontology, both vehicles and waves exist, but like this I'm just making the contrast stuff. Small remark, wave ontology, whatever it means, viable ontology, whatever that means, here are just a couple of suggestions for the intuitions we might have, say for the wave ontology behind G.I.W., that a very natural thing to say is just eigenstate, eigenvalue, the wave that we consider just as a one-dimensional ray in the oval space,

42:30 a quantum logical proposition, a portive operator that happens to be an eigenstate or that has a value, end of story. Bell later remarked about thinking of the wave as density of stuff, and Gabby and co-workers took that suggestion up, and I'll say later, I think, something about density of stuff, but a prima facie looks a bit like a variant on streaming at the original intuition of what the wave function was. And, well, Beobel ontology, what would that mean? Well, again, that might mean Beobel has a value, in the same spirit of ontemologic subspace of Gilbert space that's connected with what's art there. But maybe we don't need to do that, because if we just have that, the vehicle having values, everything is commuted to. There's no need for thinking of it in global space terms, and it sort of breaks down in a continuum, that we just give the configuration space. So maybe another possibility is just to think of the values of the vehicle as values in some state space which is defined by the vehicle. Okay. So, how could we understand it as being? Let's start first with a sort of simplified toy collapse theory, where it might be easier to make that connection. So, just consider the usual von Neumann measurement collapse.

45:00 The state goes over to some pI psi normalized with a certain probability where the pIs are the eigenprojections of some type of joint operator. Now, we can define a toy collapse theory by taking this process and declaring that it occurs spontaneously at random times. That's our toy collapse theory. We're not worrying about solving a measurement problem now or anything like that. And so, the advantage of this, for our purpose, is now that the same mathematical object, the eigenprojections of a certain observable, which we had in the vehicle theory, are now are associated with the hits in this collapse theory. And, okay, this gives us room to play. So, notice, first of all, that at least at the times of the collapses, given the standard eigenstate-eigenvalue rule, in this toy-collapse theory, this observable has values. And if we want to talk somehow about the evolution of these values at the collapse times, which in terms of some kind of stochastic trajectory, well, we can write down the probabilities for it, and they're given by the standard Born rule for successive measurements. And there it is. But, notice also that the collapse theory, or the collapse evolution, includes more than just this evolution of values of this observable. In fact, the quantum state evolves also between the collapses, and even at the collapse times,

47:30 if the projections are not one-dimensional, the quantum state is not uniquely determined by the value i that will lie somewhere inside this, the substance. So, we've got elements like in a viable theory, but we've got extras. Now, the question is, do we need these extras? And the idea, now, if we want to reinterpret this toy collapse theory in terms of variables, is just keep the evolution of the values of this observable and just drop the extras. and say, if we want to interpolate what happens between collapse times, just take pi as constant between collapses, and what we get, then, is the following picture. We've got a certain vehicle that always has values. Our theory now describes the evolution of this vehicle, how it jumps around, and the wave function side is now reinterpreted merely as the object that enters the probability formula that defines the probabilities for the various predictions. And the difference between the collapse theory, thus reinterpreted, and the standards-beable theory which we can set up for that beable becomes quantitative, it's just that we get different probabilities for the possible projections. So, for instance, we can compare

50:00 the single time distributions in the standard Bebel theory. It is constructed in such a way that the single time distributions are always given by the quantum mechanical distribution, while for this collapsed Bebel theory, to take the probability for that whole trajectory and sum over all preceding stages to get the probability distribution at the time t. So, we've got this quantitative difference, except in one special case, by the way, when the vehicle is, in fact, also a decohering variable for that evolution, in which case the transition probabilities are the same and these single-time probabilities reduce to these. Okay. So, thus far, the toy will answer you. You have . That's fine, thank you. The G-A-W-his themselves look rather different. I may not have convinced you get that it's possible to institute the original GW as a viable theory. So, let's do it in two steps. First step, we generalize the notion of a vehicle, and what we do is, instead of taking an arbitrary self-adjoint operator, we take an arbitrary POV measure as our So these things are self-adjoint and positive operators, but not necessarily related projections. And now we assume that this be-able has values, whatever that means. But, you know, we can just assume that it's values in some states. We don't need to talk about fuzzy quantum propositions.

52:30 And then we construct the dynamics for these values such that the quantum account for distribution is the time-dependent equilibrium distribution, and that can be done exactly in the same way as Bill did with the projection value missions. And, okay, I actually haven't seen this generalization described in these terms, but I have seen an example of such a theory, and that's the phase-space-bone theory by Gonzalo García de Colabieja, which, in fact, takes the distribution defined by coherent states in phase space, the quantum distribution, and then has a dynamics defining trajectories in phase space, such that the distribution is preserved. but the coherence states just define precisely a POV measure in this sense, okay, it's a continuous version, here I have just written it down discreetly, and I should check whether this continuous version is the appropriate continuum limiter of some discrete theory with exactly this dynamics, but I think it's already clear enough that this is an example of this kind of generalized viable theory. Stage two. GRW can be construed as a spontaneous process POV collapse, spontaneous measurements of a POV measure happening at random times, and that can be seen very easily, such a measurement of a POV observable, in the simplest case of Q operations, goes like that. matter, instead of having a projection operator here and here, we have some operators, A i,

55:00 such as A i star A i, uses back the effect E i, and GRW has exactly that form if we take as AX, it's a continuous function, and that we take as AX the multiplication with the Gaussian, multiply with the Gaussian and renormalize, and the probability, and AI star AI gives is indeed an EX which integrated then gives one. So that this GW has this form. And now we can play the same game as in the toy case. Let's interpret the hits as values for this beable, which we can just interpret as position because the value space is just a position space, or if you want to interpret it as some kind of fuzzy position if you are so inclined. The collapses are reinterpreted as jumps in the stochastic evolution of the beable, and And we don't need to think of psi as a collapsing object, and so we've reinterpreted GW as we want. How much? More? Just a few more minutes. Okay, that's fine. I've got two transparencies. So, I could have stopped here. I wanted to show we can construe GW-adreviewable theory, but they are sort of follow-up questions. And, you know, to be honest, I should also consider whether this reinterpreted theory is just really a reinterpretation of the matter of taste and how we choose to understand the

57:30 theory, or whether it's actually a different theory with different predictions. And it seems to me that, in fact, it gives a different account of macroscopic localization. Let me just sketch how that goes. You know, in the GW collapse, when we have N particles in the original version, the hits of the different particles are independent of each other. But, under certain circumstances, because of the form of the wave function, one single hit will suffice to localize the whole of the wave function. And if it is the wave function that matters, then in one instant we have our macroscopic localization. In the vehicle picture, the single hits are reinterpreted as jumps in the trajectories of each single particle. So, once under the same conditions, after the first particle has jumped to a design localized place, all the others will also cluster around there, talk about macroscopic localization we have to wait until a macroscopic number of particles have all clustered one by one around there and that seems to me rather rather problematic i mean i mean then we can start figuring if we say that if all particles move together then that empirical adequacy is restored, but we don't have exactly the sort of one-to-one correspondence, it, and, and, and, and, and, and, and, and be able. Um, now come sort of the open-ended remarks. Um, on the other hand, even in the further development of the GRW theory, one went over to talking about the collapse triggered rather by mass density, so maybe it is kind of legitimate to go along this way also in the indivisible reinterpretation. And this is something I still

1:00:00 have to do, perhaps having a look at it. CSA, formalism is again a tighter formal collapse and evolution of mass density as a vehicle, but I don't know that yet, I'll have to look at it. And I'll just go back to the density old stuff. Maybe thinking of mass density as a vehicle for GRW gives us an intuition about how one understand the idea of density of stuff, because, in fact, you can't really interpret, say that the wave function is density of stuff, because the value of density of stuff in configuration space does not fix the wave function. There's no phase information, so you can't really identify the wave function with density of stuff, but there's no problem in thinking as a vehicle in this sense, which is guided by the wave function. And it could be, in fact, that if mass density is what we are talking about, that that's, in fact, a deep-cohering variable for the appropriate true quantum theory of gravity, so that we would be in the case in which the reinterpretive class theory and the Beobel theory are not only about the same thing, but are also quantitatively indistinguishable, at least on that time scale. what I'm suggesting is some kind of perverse blurring of distinction between different interpretations of quantum mechanics. Thank you. Okay, we've been dodged with one quick question. Well, I'm just tempted to an even more perverse blurring, which would be to collect some of these vehicles together on trajectories and forming mines, and then you have a mini-mining interpretation which would solve, locally at least, the problem with GRW, which is the problem for the rules.

1:02:30 I mean, any good interpretation of quantum theory, to some extent, should be interpreted as any of the others in some sense. Yes, I mean, at the final stage here, when we start saying, okay, they're quantitatively indistinguishable, the form of the laws is different, but in practice they're indistinguishable, You can add further ones which become indistinguishable from different ones or many minds, and then if the form of some of the laws is relativistic and the form of other laws is not, then maybe you put the criteria to decide. Okay. Sure. Okay. Shall I allow some time? Okay. Let me see. I think Yass, did you introduce me that first? Both of them. Okay. And then start. Raise your hands again. Now, do you have the review of the Schrodinger version? What actually is the version of the version of the version? Sure. This is the version of the version. Yes, sir. Yes. Yeah. Well, with the original GIW, I mean, if you perversely reinterpret it, wave function just does what it always did. It just comes in a non-standard way in I mean, there's a non-standard guiding equation for the vehicle, which is just a generalized boardroom. It's slightly perverse, but that's the game I'm playing. I sort of lost track of the toy, as it was, as you are, that individually you had, after

1:05:00 perhaps the state, the vehicles just stay constant between jumps. And you're imagining that's the prime to the prime of GRW. The original particle state. But the jumps for each individual particles are actually rather there in GRW. Yes, precisely. That's a problem. ...mets up with lots of particles, but some of them get over there and we'll be able to catch up to them. Yes, that was the trouble, but when we talk about macroscopic localisation, it will take an enormous time until all the particles have all bunched up to make our macroscopic object localised. Yes, so in this form it is problematic. then you have to start sort of fiddling with it, or you have to start saying, okay, we've just explained the idea of collapse theories as vehicles, let's see, with other collapse theories, maybe CSL, or something that has more to do with mass density anyway, as it seems, a way that we can play a similar game, and once we've got mass density as a B of all, that problem disappears, but I'm not sure that CSA all leans itself exactly to this kind of formal analysis. Okay, I guess I'll have to carry it into the break. Okay, thank you. And our next speaker is Marcus Adelby from the University of London. Yeah. Well, I'm going to talk about Bell-Coach and Specker theorem. It's often referred to just as the Koch and Specker theorem, though in fact it was first proved by Bell in a paper

1:07:30 that appeared a year earlier. And it's quite a complicated story, in fact, and I'm not going to say much about it. I think it's not exactly an historical injustice to Bell that it's tended to be called the Koch and Specker theorem. The way Bell presents it, he really um, being ignored, I'd say. But anyway, in fact, he did, um, prove it first, and that's very relevant to what I want to talk about today, because, first of all, Itamar Petowski, about 20 years ago, he didn't actually claim to have nullified the theorem, which is the claim that these people made, but let's say he cast serious doubt on the theorem. These people much more recently made the claim that the theorem is nullified. I'd like to point out that the word nullified does not mean the same as the word refuted. It's still a perfectly valid mathematical theorem, however you look at it. So what they're claiming is have the implications that it's usually being thought to have. And that, of course, raises the question as to what exactly those implications are. At that point, the story gets quite complicated, because what Bell says about the implications of the theorem, and what Cochin and Speckham say about the implications, I would say have zero intersection. It's not just that they're different, they're totally different. And so there is a question as to what exactly they have nullified. My own feeling about it, by the way, is that although they haven't nullified the theorem, what they've done is they've certainly made it look quite different. I think I would want to say that they've revivified the theorem. Before they got going, I didn't think the theorem was very interesting. I mean, I thought, well, okay, here's this theorem. I didn't get excited by it. I only started to get interested in it when they claimed that it wasn't true. Well, no, that they'd nullified it. So I think they've made it more interesting than it was before. The opposite, in fact, of what they say. But that isn't

1:10:00 actually saying that what they've done isn't valuable. It's just valuable for a different reason from what they've said. Okay, well, I'll start off with what Bell says about the implications, because, after all, he was the one who came first. I'm actually, I could give a whole talk, actually, on what Bell says about this, and I can't, there isn't the time. It's, I'm not going to call it, it's almost schizophrenic, actually, in the sense that he's pursuing two, he says different things about it, almost. Anyway, I'd certainly time to mention that. There's a lot I'm skirting over here. What I'm calling Bell's view, you could consider to be an oversimplification. Anyway, the point I want to stress is this one, that the ordinary concept of measurement is that it's what you use in science to find things out. You measure some quantity like a length or a max or whatever, and after doing you know what the value was. And that's absolutely central, really, to the classical conception of physics. It's where our empirical knowledge comes from. And the point about the Bell-KS theorem is that it shows that although you can construct a theory like Bones, in which particles move along trajectories and behave in a superficially classical manner, what you can't recover is that basic classical assumption that when you measure the properties of these things, you find out what the property was before you measured it. And that's pretty fatal, really, to the classical worldview. So that's quite an important fact about quantum mechanics, really quite central to it. I would say Bell himself never does make this point I would say that really you can say that the Bell-KS theorem and contextuality are the explanation of why a hidden variables theory is a hidden variables theory. It's what makes the variables hidden. That's perhaps a slide over simplification. And Bell doesn't say that, and there are reasons why he doesn't say it, and I'm not going to talk about it.

1:12:30 Okay, let me just put up on the screen what Cochin and Specker say about it. Now, they're very interested in the logical and mathematical structure of these quantum mechanics. And they've got an idea which, I must say, actually appeals to me. I tend to feel that a hidden variables theory breaks some kind of symmetry on a sort of loose, intuitive level. I couldn't make that very precise. And as I understand it, Cochin and Specker were trying to make that exact, they say that in a quantum theory, in quantum mechanics, the observables are operated on Hilbert space, in classical physics, their functions on base space, and so the hidden variables theory ought to map the quantum observables onto the classical observables in a way that preserves that's their basic logical structure, and they have this criteria. I don't want to say anything more about that, because I actually do think that Witowski, Mayer, and so on have, in fact, invalidated that argument, and it's not what I want to concentrate on. Okay, the theorem, which I imagine you all know, so I'm just going to flash this up on the screen very, very quickly, one wants reminding. You consider a spin one particle with a spin vector s hat. You consider three orthonormal unit vectors, three directions at right angles. You consider the squared spin components. They add up to two. And if you measure one of them, you've got to get zero or one, which means that if the measurement is revealing values that were already there that values assigned to a set of three orthogonal directions must be one of these three combinations. And the theorem then shows that, in fact, you can't do that. Now, that seems very impressive at first, until you ask, well, okay, how many vectors can you assign values to in that way?

1:15:00 and it doesn't make any statement about that at all. In fact, that statement that you can't assign vectors to the whole unit two-sphere consistent with this requirement, I mean, it could be that you could assign it to everything but one. That isn't true, actually. But, I mean, it isn't excluding that possibility. And that's really the point that... I'm going to call them PNKC now, for short. The Towski, Mayer, and Clifton. That's the weakness they work on. Schatowski was the guy who came first, and he had a very intricate set-theoretic construction which involved some very sophisticated arguments from axiomatic set theory to show that you could assign values subject to that requirement to nearly every triad, in some sense of the word nearly, which in fact is quite problematic the set of triads which are wrongly coloured are also not measurable with respect to the standard probability measure. And so that is a serious weakness in this argument, and I think it's the reason why it didn't attract a lot of attention at the time. Though I think actually, if you look at it, it's quite impressive. It's got reasons for that, what he says. Anyway, 20 years later, Mayer, Kent, and Clifton came along, and they took quite a different approach. They consider countable dense subsets of the unit 2 sphere. Mayer originally considered all the points on the unit sphere which have rational coordinates, of which that's obviously countably many, And they said that this is empirically indistinguishable from a sphere where you're assigning colors to every point, which I think is true. I mean, we can't actually prove that any... We might be living in a world where every length is rational. There are no irrational lengths. And they showed that if you restrict yourself to the rational two-sphere, then, in fact, you can assign colors according to that rule. And that actually is quite an impressive result, I think. And it certainly makes the theorem look very different.

1:17:30 Yes. Now, okay, I think that does invalidate Coach and Specker's argument that the killer is that they actually anticipate this, because in their paper they say it's not physically meaningful to assume there are a continual number of quantum mechanical propositions. So I think they accept in advance that the mayor, Ken Clifton Constructions, will kill much. I don't think it affects Bell's argument, because if the measurement did ascertain some pre-existing property of something, any instrument involved playing a purely passive role, it seems to me that interpreting words in ordinary English, that means the variables wouldn't be hidden. And MKC would actually have constructed models which are not hidden variables theories, which sounds a bit surprising and I think is not true. So let's look at it a bit more closely. If you look at the, what I would, what they say is, their actual title of Mayer's paper was Finite Precision Nullifies the Fenton-Specker Theorem. I would say it's exactly the opposite. Finite precision saves the Koch and Specker theorem, but infinite precision would half nullify it, only half. The point about Mayer's model is, if you look at it, it's very, very violently discontinuous. In fact, it's maximally discontinuous. The function is discontinuous at every point of its domain of definition. So in a continuous colouring, if you have a red point, then it's got a neighbourhood which entirely consists of red points. And so if you try and align your apparatus to try and find out the colour of that point, the fact that you can't be quite sure how your apparatus is aligned won't matter. It's still going to pick up red, which is the colour you're looking for. But what you find in the mayor colouring is something like this. I didn't have enough time to fill in all the points. I mean, there are infinitely many blue and infinitely many red mixed together like that.

1:20:00 And under those circumstances, it's obvious that unless... If you could actually align your apparatus with infinite precision, then of course you could discover the value of any particular point. Given you can't, then I'd say you obviously can't, still a hidden variable at the moment. And I think that Bell's point, therefore, remains valid, at least for the Meyer covering. It's interesting to ask, I mean, I'm not suggesting that anyone could perform infinite precision measurements, but it's interesting to ask what would happen if you could. And the answer is, I think that you could tell the values. Also, interestingly enough, however, it would violate signal locality. There are some deep connections between the two theorems proved by Bell. His non-locality theorem and his contextuality theorem are very, very closely linked technically. I think they're very, very closely linked conceptually as well. and in fact certainly in the case of the MKC models if you could perform infinite precision measurements you could actually send messages faster than light as well the two go together of course there's no question of actually doing that because we can't perform such measurements I think it's got some sort of connection with some ideas of two but I'm not going to have to say more about that Now, the theorem, what I wanted to do was to show that this isn't just an accidental feature of Mayo's colouring, it's actually something absolutely necessitated by quantum mechanics. So I wanted to show that the same thing is going to happen in any similar model. So to show that, I started off with the concept of a regular colouring, as I call it, which I'm assuming it's a KS colouring, so it's not a KS colouring of the whole of S2, but of some open subset, and it's almost everywhere continuous. The motivation for insisting that it be almost everywhere continuous should be obvious, I think, in view of what I said about Hensley. So it's kind of what you think of as a colouring. I like a political map of the Earth, nice patches of colour with some white bits

1:22:30 which you can't manage to colour, of course. And the question is, how big are the white patches? How small can you make the white patches? So this is just kind of strengthening the Kochel Specker theorem, really. It's saying something about how big a part of the sphere you can cover. And you can prove two things. The first is that the areas of white must have non-empty interior. least one circular patch which is pure white, no color in it at all. And the other thing you can prove is that there's a fixed model-independent number lower bound on the total area of white. It's interesting to know the exact value of this. There are indications it's around 0.01, something like one You can color, I suspect, something like 99% of the sphere, leaving out an area, if it was the Earth, about the size of Australia. I'm not quite sure about that. That's kind of a little bit iffy. Okay. Now, I will then want to compare that with what I try to define the most general possible colouring of the kinds, this includes both Petowski's colourings and the very different Meyer-Kent Clifton colourings. They're both examples of what I call a pseudo-KS colouring. And I've characterised that, I'm not going to bother to go over that because there isn't really time. And then, having got the concept of such a colouring, then define the concept phenomenological color, which you can think of intuitively as what you'd see if you looked at the sphere through finite resolution eyes. So if you've got a patch which is all intrinsic red, counts as phenomenologically red, ditto if it's intrinsically blue, it counts as phenomenologically blue. If it's a patch where, if you've got a point, each neighborhood of which contains at least one red and at least one blue, then that counts as black. Actually, if you mix red and blue, you get purple, I think, but I prefer black.

1:25:00 The idea is that black comes from mixing countries of color. So black corresponds to the discontinuity variance. And the theorem, now, the Petowski colorings, so a pseudopausque coloring does not have to be Borel measurable. the Petowski colorings are really very badly behaved indeed. The nice thing is that if you replace the original intrinsic coloring, which is possibly not a Borel measurable function, with what I call the phenomenological coloring, you get something that is guaranteed Borel measurable. So you can use standard measure theory on it. And you can then prove the patches which are phenomenologically blue or red, I count as the continuity region, the patches which are phenomenologically black is the discontinuity region, and you can prove that the phenomenological coloring is a regular KS coloring of the continuity region, which means that the discontinuity region, the black area, has got to be the same size as the white patch that's excluded by a regular coloring. So that means that the situation you're left with is this. If you've got a regular colouring, you have some nice patches of red and blue with some white, which isn't coloured at all. And what Petowski, Mayer and Clifton allow you to do is to fill in those white patches and make them black. But from the point of view of actually knowing what the values are, there's really nothing to choose between these two cases. You've got the same kind of really pathological violent discontinuity here that you have in the Meyer coloring, and it means you can't know what the values actually are. So in terms of actually learning anything, it's no different. I've got about seven minutes, is that right? Okay. Now, I'm just going to go back to what Bell says about it. I put that little bit up from four, because this, I think, is the bit where Bell explains it best. He makes two points. He makes this point that I quoted earlier, the word measurement strongly suggests

1:27:30 the ascertaining of some pre-existing property of something, which I'm calling his epistemological point. It's a point about what we can know. And then he also relates this to Bohr, this passage from Bohr, about the apparatus and the system forming an indivisible whole. And so the results have to be regarded as a joint product of system and apparatus to complete experimental setup, which I'm calling his Bohrian point. It's kind of contextuality, basically. Bell's idea is that the epistemological point follows from the Borean point, and I think the Petowski mayor, Kenton Clifton, make you see that Bell doesn't get it quite right, because the epistemological point doesn't really follow from the Borean point. It's It's following from the discontinuity. In fact, that pause point about the indivisibility of the system and apparatus still applies to their models, and it's actually quite important. It's what makes them non-local, in fact. But it's quite separate from, well, it's not the same thing that makes the colours empirically unobservable. So these two different points become disconnected. make the whole thing appear different. Yeah, I'll just point out the way in which the Borean point works in their models. If you look at a GXZ type setup, so you have three particles coming out to three detectors and you measure these three observables at these three points, what happens models, you can't just give N1, N2, and N3 any value you like. They have to be restricted to a certain dense subset of the Cartesian product of the unit sphere with itself three times. And the set of allowed values cannot themselves take the form of a Cartesian product, which means that the properties which particle 1 has depend on what's being measured at 2 and 3, and that's a form of contextuality. It's showing the properties depend on the setup.

1:30:00 And it also means that if you change the alignment of one of these detectors, that forces a change in the alignment of the other two, which gives you a way that's a form of non-locality. And it would be signal non-locality if you can measure to infinite precision. Okay. Thank you. We have time for questions. of course. I'm not sure of that. I was just it's at least at least up to 13% and I've got a bound. It's rather complicated. I think it might be. I'm not sure. It's going to be somewhere between 90% and 99%. Do you have a question? Just a small comment, one has a little room to play with what constitutes the results of a measurement, I mean, one could choose a model in which the results is the phenomenological color, or one could use these models, supplementing them with something like, if you measure in in a black region and you just get random results of red and blue. I think your points about non-locality and so on still hold, but in slightly different variants.

1:32:30 Sorry. I mean, I think you were making the point about signal locality and so on, assuming some specific filling in of the details of, okay, what does it mean to, well, no, it was just with the infinite precision, But maybe one could say, okay, when we make a measurement, it's maybe we'll be using precision, but we get a random result out of the black redermon, and then one would have to run arguments slightly differently. Now, with finite precision, you might still get a result, say, a random result from that creature. And I'm just saying there is some room for defining what the Clifton in particular modern would be, this somehow ambiguity, how one could understand the details, and correspondingly, then, your analysis of the signal and so on can have to be adjusted, but I think it still stands. The signal and locality is specific to their particular models, and that argument I gave on the last slide, is to their particular models. The other argument I suppose was much more general. I mean, I think actually their actual models are discontinuous everywhere, the whole thing, their act, shape, circumcision, and that kind of thing. I mean, I'm not going to contact with this. And the criterion says that the Ben-Rouin is a radical discontinuous, and in that point.