11th UK Foundations of Physics Conference — Memorial Session for Robert Clifton

0:00 So you get Hamilton's equations, which are, however, also attainable, and in a way this is more fundamental, at least to the Hamiltonian way of thinking, these are also attainable by a so-called modified Hamilton's principle, where the q's and p's are varied independently in a principle that requires the stationarity of this integral where the Lagrangian has now been rewritten in terms of the Q's and P's. Now in the mid-19th century, I don't know this history but the books tell me that it was Ostrogrodsky and others who brought out The fundamental mathematical principle underlying the use of variational principles in dynamics is that, obviously, because the variation is an enormous subject, there are various things you can consider, but one thing you can consider is a variational principle in allowing higher derivatives of the ques than just the first derivatives, up to, say, the nth derivative and to impose the condition that not just the q's but all their higher derivatives up to the n minus 1 should be fixed at the endpoints of the path in q comma t space. And the variational principle, a variational principle of that kind corresponds, when you do the action integral, the modified Hamilton principle and so on, the genre transformation that I had on the previous slide, it corresponds to the system of first order differential equations having the same function, which we call H, on the right hand side. So turning that around, if you have a dynamical system whose time evolution is controlled by a system of first order equations in which, wonderful piece of good luck, there is a single function h whose derivatives specify all the first order rates of change of your cubes.

2:30 If there is such a thing, then you can Legendre transform, and you can get all the perspective and insight from the calculus of variations. Now, what about Hamilton-Jacobey theory? Again, I'm going to spend longer on this, because I think, first of all, it's less familiar, but it's also the one about which I'll come up with the question. So in Hamilton Jacobi's theory, I just want to say in terms of the calculus of variations what I think would take the fundamental idea of the theory to be, and then I want to ask you a technical question. So there are really some notions not yet stated and definitions that I'd just like to remind you of. We're going to be concerned in what is called extended configuration space or event space, which is the space of the q's comma the t time. So that if there are little n little q's, there's going to be r to the n plus 1. And I'm going to be concerned with a simply connected local region of r to the n plus And I'm interested in a family of hypersurfaces that cover that region simply, they foliate that region, they don't intersect, and every point in G has a single hypersurface going through it. I'm also interested in congruences of curves that cover region G simply, in the sense that through every point there goes a unique curve in the congruence. and there'll be an n-parameter congruence where my parameters are written with u's. So alpha also runs from 1 to n. And if I have a congruence of curves like that, then at every point I in fact have a definition of tangent vectors. dt by dt is just 1. And this will also give me therefore values throughout g of my Lagrangian L. I'm imagining L thus defined on my g. it the q dots and this will give momentum and a set of two n functions that assign qi pi pi equal to dl by b q dot throughout the region g that's

5:00 called the field and it's called canonical if they if the q's and p's satisfy Hamilton's equations which in other languages that the curves of this congruence are extreme ones, and we say that a curve, and indeed also its congruence, belongs to a family of height of surfaces, if and only if the PIs are ds by dq, okay? Well, this, of course, is the familiar from previous slides. This is famous to anyone who's looked at Hamilton-Jackaby theory in a textbook, but this is what's going on a lot, that P is ds by dq. Now, to me, with these preliminary definitions, the content of elementary Hamilton-Jacques theory is well-caught by the following statement, that the following three conditions are equivalent on a function s defined on g and taking real numbers as values. The first is that S is a solution of the Hamilton-Jafferby equation, the second is that the field belonging to S is canonical, and the third, it's a bit of a mouthful as I've written it down, is intuitively that the hyperservices of constant s are as it's written geodesically equidistant I should warn you there isn't a metric that behind this formalism that merit that makes us say geodesic and some books say geodetic which is a good warning with the team that there isn't a metric but The idea is that if you, sorry, there's a metric on the three space, you're saying the three plus one, oh sorry, I mean n space. Yeah, there is a, there is a, there's this Amityan background metric in R n plus one, but we're not using it. This equidistance is defined in terms of the Lagrange, that's what I'm saying. So in any case, I just want to say that the value of the action integral, the integral of L, along curves of the congruence from one element of the hypersurface family to another is independent of the curve you choose.

7:30 So any pair of curves between any two hypersurfaces, you'll get the same integral of L along them. So this is the kind of thing that Hamilton Jacobi theory is concerned with, and in effect the flavor of the subject mathematically is given by this equivalence. The initial value problem is solved approximately as follows, that under some mild conditions about the non-vanishingness of a Jacobian, any n-dimensional surface, a co-dimension one surface in G, together with a prescription of S's values on it, and the values of S's derivatives at a single point of it, defines a unique solution of the Hamilton-Jacobey equation. Well, it is... So, this abstract formalism of pure mathematics is connected to mechanics by, as the books indicate, having S represent an ensemble of systems which have at all times a strict correlation between their Q and P, which is given by this map that P is dS by dQ. And going back to the earlier session, I'd like to stress that this application is, of course, mostly discussed in the mechanics books as a way of solving problems by the introduction of a suitable S, typically by so-called complete integral of the Hamilton-Jacobi equation, typically found by the separation of variables. great contribution and it's a remarkable fact that many problems in mathematical physics can be solved by the separation of variables and that paradoxically faced with solving Hamilton's equations an array of first-order ordinary differential equations it can be ironically best to adopt the strategy of considering an associated partial differential equation which is typically it's ironic because they're normally harder but this was Yacobi's great it's useful to reverse this, go from the ordinary to the partial.

10:00 But going back to the pilot wave session we had before T, I would want to stress that, as the mechanics books do, that, of course, the solution to a mechanical problem is considered, if you like, as just computing a trajectory in phase space, but given initial Q and P, this solution is independent of the choice of s that you made the s represents a fictitious possible population of systems which you introduce for the purposes of problem-solving but of course Q and P at the initial point in time determine the later trajectory independent of what s anybody might wish to write down that of course is not true about the s that is the phase of the quantum wave. But my question to you is, what is the structure of the set of solutions? I mean, I have indicated here how to specify a solution uniquely, but there's great redundancy in this. Many different S functions will be, sorry, many different Ns with specifications of values 1 and 2 on them will, of course, yield the same solution know more about the structure of that set now having said that I'm going to skip section 5 which is a bit about the structure of that set about complete integrals and the geometry of completed and go to the to the to the philosophical punchlines. And the first comment is to say that as a philosopher, I'd rather like to distinguish three kinds of modal involvement that a physical theory that postulates the state space could have. Philosophers will recognise a joke here about Quine, the famous logician who hated modality, and talked about three grades of modal involvement, each of than the previous. I love modality, but it's true that it gets richer from one to three. So in the first grade of modality, you simply consider the problem, you fix the problem on the actual problem, a specific system with a specific Hamiltonian or Lagrangian, and

12:30 And you, however, consider counterfactual, contrary to fact, merely possible, non-actual, initial conditions or final conditions, or perhaps boundary conditions. Obviously, the postulation of a state space is precisely doing that. In the second grade, you perhaps fix the laws, but vary the problem. And we've implicitly done that ever since slide one, because I generalised over all possible finite values of little n, the number of configurational degrees of freedom of my system. And we do that all the time when we make generalisations across problems, like proving the conservation of energy. I would make the philosophical remark that these two grades, though straightforward, do show a richness of local involvement in analytical mechanics which is not well caught by the philosophy class from standard example that if all A's are B's as a law, then that has counterfactual force that if anything were an A, it would be a B, in terms of which the philosophical debate is normally conducted. And, of course, in the classroom, I'm all for focusing on simple core examples, but it is a richer and more varied situation in science that that classroom style suggests. The third grade, however, going back to my very first slide, is to notice that variational principles do consider contralegal histories of the system, of course. And there is a venerable philosophical principle, which is that if something is actually true, there should be something in the actual cosmos that makes it true. And I'll call this, for short, actualism, though philosophers will recognize that's usually used for a different doctrine. But the idea that actual truths, things that are plain true, are made true by something in the actual cosmos, is plausible. law. And it is, when you first meet it, worrying to see somebody like Hamilton, for all those geniuses, state what is purportedly an actual law, the laws of classical mechanics, state this while mentioning merely possible histories. Indeed, they're not just merely possible as in modality grades one and two.

15:00 They actually violate the law that he's trying to state. So it's a little bit weird. Well, I don't think it's really paradoxical, but it's worth pausing for thought. I think that there is a reconciliation that can be made, and that actually Lewis's, David Lewis's doctrines about possible worlds and counterfactual conditionals come to our aid in this regard. that in that discussion, we speak of counterfactual conditionals, called a counterfactual for short, as being true at a possible world, and indeed made true by the nature of a possible world, the one at which they are true. But nevertheless, it is useful to state the truth conditions of these conditionals by talking about other possible worlds. But it's still true that the truth-makers, as the philosophers call them, are in the world concerned. And in the same way, I would say that you can think of Lagrange's or Hamilton's equations as made true by the nature of the actual world. But to see them as only Lagrange's equations from a variation principle is useful because of getting the perspective of the calculus of variations. And thank you to Ostrogrodsky and the great 19th century mathematicians is at least in this simple formalism in which that kind of perspective is available. It's nature's kindness to give us a single function h that controls the dynamics of all the cubes through first order equations. Okay, well never mind about final kind. And I have six minutes. Well, I have two slides and I'll short, therefore quickly finish off. Now, my final two slides are about Hamilton Jacobi theory, and I said that my main message about them was to ask you a question, but let us try to be a philosophical magician looking at Hamilton Jacobi theory and ask what are its modal involvements. Well, we've already said, in a way, the punchline, that S represents an ensemble of systems chosen by us, or actually by a good textbook or by the divine Yannity, for solving a hard problem. And so the obvious philosophical comment is in italics

17:30 that this is what I call the first grade of modality. But in spades, I mean, that is to say, it is the fix the problem, both considered as the number of degrees of freedom, as the actual one, fix the Hamiltonian or Lagrangian as the actual one, but consider different initial or final or boundary conditions. Well, in postulating an S function, you are obviously doing that in space. That is English jargon for doing it a lot. I think the Greek word would be the apotheosis of the first grade, most of my people. The vulgar island people in the northwest corner of Europe, we call it space. apotheosis of the first growing. And the attempt to go beyond that to talk about modality would go something like this. I've done it in two ways, and I'll only share with you the first. I'll never mind my last line. You could think of Q, T, what I call an event or an extended configuration, as like a philosopher's possible world, surface of constant X is a set of words. And if we go back to Hamilton's original version of the S function, in which it was just the line integral of L, along a trajectory that is in fact taken by the system, a so-called natural motion of the system, from initial Q star T star to later QT, then the surface of constant S is in fact in effect a sphere, sometimes called the geodesic or geodetic sphere of radius R with center the initial point. Well of course as soon as a mobile metaphysician talks about spheres, they think about the famous Lewis in 1973 analyzing counterfactual conditionals in terms of spheres of worlds ordered by similarity. And indeed, it's true that one can just write down a Lewis Stalmaker-style counterfactual conditional, which is the analogue for extended configuration space of the Lewis conditional. It is, however, not something, apart from the length of it, I admit to you that it is not something that we often need to say in mechanics.

20:00 But it is available. And I will then stop for the question project. Thank you. This is cited from a good book by Root, but I have forgotten. actually I don't think his discussion does the sensible thing of just saying why it should be so it's in his but that's that I think is the in the classical early 19th century theory from Cauchy and Fapp I think that's the that is the answer but thank you for that I don't know question so the question was um to get a unique solution to the hamilton jack of the equation so it's a first-order partial differential equation for this function s and the function position of time so one would expect that if you're given the function s as a function of x at the time zero everywhere that should give you a unique solution but it seemed that you need to know the value of the derivative of s at a point as well with respect to time and i think that i suppose the short one would be the analogy with the ordinary case it's very it's on the it's an arbitrary point on the surface but but the surface oh so you're not

22:30 I think you need the initial data on the state space, which is P and Q. I think my last one was about Hamilton-Jackobey theory in face space, but I think the question should be posable and answerable only in the event space that I wasn't stating it. If you're just able to see one particle in a three-dimensional space in time, your surface is not three-dimensional? No, it isn't. Ah, right. The particle surface across this four-dimensional green G. Two surfaces. No, it's a three-dimensional model. So if it's a single-point particle, my event space has a dimension of all, and there's a three-dimensional surface. Ah, so it is a three-dimensional surface on which you give the values of S everywhere. Okay, so I'm still puzzled. Yes, I think there is a puzzle. I've got some other questions later, but did you have something to say specifically about it? I wonder if you could comment on this whole tentative mark, which has two aspects. The first is that, I think the first two kinds of modality that you read are in a sense long vicious, in the sense that if a philosopher counterfactually happened to dislike modality, he could sort of live with all these kinds of modalities and translate them and put it in another language, it's just about generality, something like that. And the second aspect is that the third kind of modality is much more strict in that sense. And it also has to do with the question whether you would regard a variational principle itself as a law of nature. or whether you, as you at one point said, it's just regarded as a method of finding the lost nature themselves, such as the equation of motion,

25:00 and which in itself doesn't have any... Yes, well that's a good question. I wish I could answer it properly. I think it is true that those suspicious of modality would regard the third as worse. And I suppose I think though that those who say modality first and second aren't really modality, they're just a bad way of talking about innocent things like generality, which have no mobile involvement. I'm afraid I'm sufficiently disagreeing with that, that I'm not sure I see, well I don't see such a clear line between one and two on the one hand and three on the other but about the second part of your question I think that obviously it's been a real gift from God to receive the least action principle in all its various forms in classical mechanics and then field theory and then transmogrified in quantum theory And I was only saying that an actualist can, in this context, say quite specifically what it is about their preferred actualist language that licenses the variational principle. But there ought to be a mathematical physics community, a clear understanding of the limits also in the field theory case, and of course quantum is enormously what's going on, but I suppose a similar thing should be said about field theory, and if somebody knows this I'd like to hear it, but of course some, going back to variation principles of stating the laws of nature, I should just say out loud the obvious thing that many textbooks say, which is they're only really useful for understanding symmetry and conservation laws, and so on,

27:30 which is a wonderful subject, but because they lead to partial differential equations, the Euler-Nagrange equations, they are not much used in the practice of this. They're used for guessing, but they're not daily work. Okay, returning to the issue with Anthony, in the very first line of that part, you mentioned some mild conditions or mild requirements? yes yes which was it's about the structure of this certain determinant so it doesn't address the issue of of of why a single point is why it is necessary at some point or other to specify the great i think i mean if and the textbooks are right then the answer has to be that in some way the ordinary the analogy between ordinary function in this regard breaks down, but it's just not true for first order that you need no gradients. Anyway. But the Hamiltonian function, it was the gradient of s, so you have to keep that when you solve the Hamiltonian equation. But if you know s everywhere as a function of x, then you automatically know the gradient of s. Right. So you should know the S by the T at every point times the error. So you should be able to reach forward in time. I think we're going to have to leave that point where it is. But anyway, please join me in thanking. Next speaker is Michael Dixon, who is in the University of Indiana at Bloomington, and now a Spanish student here in Austin. He's from Boys Who Play to EPR. Right. Most recently I'm from the Isle of Wight, where they don't sell overhead projection sheets. So, what little pictures I need, I'll draw. First of all, just a very, very brief note. Jeremy said what needed to be said about Rod Flippen, since this session is in fact in his memory.

30:00 Rob and I worked on several projects, and then in the last year or so, 18 months, we went our separate ways in terms of our projects, and discovered about six months ago that we were in fact working on the same thing, from very different points of view, but coming to very similar conclusions, and I take some comfort in that fact. What I'm going to talk about today is in some ways extraordinarily simple, and sometimes when I talk about it, I'm embarrassed, because I think at the end, everyone's going to just say, why did he spend 20 minutes saying that? But inevitably, at the end, instead, people all disagree with me, so I guess there's something at least worth saying to provoke questions. And it's about Laura's reply to EPR, which I thought I understood very well, and then And as so often happens, trying to teach it to 18-year-olds, I got confused and went back and read Borg's reply to EPR, got a little bit more confused, and after reading it several times and then reading early Borg, it seemed to me to start making some sense. Not in the sense of being the incontrovertibly correct reply to EPR, but in the sense of being a reply to EPR that is internally consistent. And that's what I want to try to share with you very briefly here. First of all, a remark about the EPR experiment. This is what the EPR experiment is not. Well, as far as saying, the EPR never proposed an experiment in the entire paper. But the experiment they did not propose, but wrote down some equations, is not the following. and I did an impromptu survey which only included one Brit so I can't implicate you in this but it included many Americans and over half of them said this is what the EPR experiment was and I wasn't asking just anyone I was asking people who walked to and have so this is what some people teach their students actually what I ask them is what do you tell your students the experiment was so that's a slightly unfair question because we all lie to our students so here's the non-EPR experiment there's a source which spits out a pair of particles with equal magnitude but opposite velocities we assume the particles have the same mass for convenience

32:30 and so we have this particle here which has minus P for velocity this one has plus P and at any given time of course they will be equidistant from the source so we have perfect correlation and position of momentum measure this guy's momentum you automatically know this one's momentum measure this guy's position relative to the source you always know this guy's position relative to the source why isn't that the EPR experiment well first of all it's not quantum mechanical nothing I said relied on quantum mechanics at any stage purely classical description the correlations are established purely on the basis of classical conservation laws and nothing else. There's nothing quantum mechanical whatsoever about that. Second of all, if that were the EPR experiment, Bohr has a very quick reply. And the quick reply is, apply the uncertainty principle in its epistemic form. Whenever I say the words uncertainty principle in this discussion, I mean in the simple epistemic form in which by 1935 it was not in dispute, at least not by Einstein and company. Apply the uncertainty principle to the source. solved the problem well in order for me to infer this guy's momentum from this guy's momentum I have to know the momentum of the source because what's true is they're meant to add up to whenever this guy whatever the source of momentum is well similarly in order to infer this guy's position from this guy's position relative to the source because that's the thing they're equidistant from I have to know the position of the source it's not enough just to measure this guy's position and say aha it's here so then we must be unless you know what the source is. But you can't know the momentum and position of the source, so in fact, from a quantum mechanical point of view, if we accept the epistemic uncertain principles, you can't even know that you've prepared this state. Although, possibly, of course, you can know that you've prepared this state. So that is the non-EPR argument and the non-Borian reply to it, although it would be a very quick reply if that had been their argument. Okay. What is the argument? Well, the EPR argument, in fact, as you all know, and of course, as soon as you point this out to someone, they say, oh, of course, that's not the EPR argument. Here's the EPR argument. They write down a generic, well, they mention the generic fact that if I measure one of two incompatible observables on a system whose initial state is whatever, then as a result

35:00 of collapse, they don't use the term collapse, but they perform the corresponding mathematics, with a system that's in one of two incompatible states. And then they go on to actually give an example in terms of position and momentum. If I had time, I would tell you in a little bit of detail why the example is particularly unfortunate. It would have been much cleaner if they had chosen other examples. And Bohr's reply, to some extent, at least its persuasiveness, if it has any, actually relies on their choice of position and momentum as opposed to the generic fact that I could have chosen any pair of non-competing observables and generated the same argument. But I don't have time to go into the details of why that choice is particularly unfortunate, so I won't for the moment. But I'm going to restrict my discussion to that choice, because that isn't the choice that they're making. Okay, the logical structure of the argument, then, goes something like this. And it's, again, slightly unfortunate, but here's how it goes. I'll make up some notation, which I hope will be clear, but I'll read it out to you as I go. This little symbol here stands for, it is possible that. So it is possible that we measured the position. M is measured. This is position. M is the position of particle 1. and of course and in that case I'll find it later in that case the position of particle 2 is definite so this is a possibility this is a possibility that everyone involved in the debate acknowledges we apply the criterion for physical reality and so on and so forth perfect correlations, quantum correlations the situation where we particle one and low and low particles two, particle two's position was definite. We can say the same thing, that's an and by the way, right? We can say the same thing for momentum. And if you like, you can make, if you want to imagine one of these things is actual instead of merely a possibility, it makes no difference to the logical structure of the argument. And then we conclude either possibly or actually, it's not exactly clear because they don't have their modality sorted out nice and cleanly that's just either possible or actual, that in fact

37:30 both position and momentum of particle 2 are definite. Now there's a very simple logical modal fallacy in this argument from the possibility of two things does not follow their joint possibility and this is recognized immediately simply by the fact when I put these two possibilities together because everyone acknowledges EPR included that I can't co-measure these things so how do they get to this conclusion well I mean they recognize it this by itself and not explicitly but how could they not recognize this by itself would be a logical fallacy they get there with the help of some kind of principle of non-disturbance and that's The gap between these premises and this conclusion. What kinds of principles of non-disturbance do they need? Well, here's the one that they actually need. They need a principle, and it is, I'm very glad, dear Jeremy, that you talked about counterfactual, so I'm not going to go back over this. The principle they need is a counterfactual principle, and it goes like this. Suppose, I'm going to imagine for the sake of this discussion, at least for the moment, that we're sitting in a world where no measurements are actually made, going to consider what we could have done okay so we asked me so we say what would have happened had we measured the position of particle one well of course one of the things we know what would have been true is that particle two's position would have been definite in that case let us EPR's principle criteria to physical reality and so on and so forth allows us then to infer the defineness of particle two's position in fact it's Now, put yourself in that world, he is Jeremy's terminology, Jeremy by way of other people, other people is by way of Jeremy, put yourself in that world where we have in fact now measured the position of particle one and particle two's position is definite, and ask what would have happened having instead measured particle one's momentum. So now we're sitting in this world where we actually measured particle one's position Well, what would have happened if we instead had measured Particle 1's momentum. And what EPR need is that Particle 2's position would still have been definite.

40:00 That is the counterfactual principle that they need. So our first jump is to a world where we measured Particle 1's position. Our second jump is from there to a world where we measured Particle 1's momentum. And Particle 2's position remains definite even so. And they consider this a kind of principle of non-disturbance because they are conceiving of its violation as requiring the following idea. Measuring something over here on particle one, may make momentum, can't have made any difference to whether particle two's position was definite. That's why they're thinking of this as a principle, not a principle. Bohr does not accept the principle. He accepts a weaker principle. By the way, I should say that I'm not going to give you any evidence whatsoever that anything I say has any connection with historical Bohr. But I don't have nearly enough time to make the case, so I won't. Bohr does not accept that principle. That's clear enough. But he does accept the weaker principle. He accepts a principle that says, suppose I measured Particle 1's position. Then, everyone acknowledges Particle 2 would certainly have been at a definite position. Now jump to a world where I didn't measure Particle 1's position. But not a world where I measured Particle 1's momentum. I'm just a world where I didn't measure Particle 1's position. Would Particle 2's position still have been definite? Bohr's answer is yes. In other words, the measurement of Particle 1's position did not bring Particle 2's definiteness of position into being. And so the question then is, how can Bohr resist the move to the strong principle of non-disturbance that EPR need? need, module the other principles that are not in question, how can Bohr resist the further move to that strong principle of non-disturbance? When do we start? How much time do I have? I don't care about it. Ten more minutes, that's fine. Okay, that's one way of asking the question, what is Bohr's reply to EPR? How does he resist the move from the weak principle of non-disturbance which he accepts, to a strong principle of non-discrimination which he rejects and which would, in fact, be sufficient for EPR's argument to go through. Another way of thinking about Bohr's reply, and I think it's historically a more accurate way, is to ask the question, or to think of Bohr's reply as an explanation, does this

42:30 thing rotate so much? To think of Bohr's reply as an explanation of the non-commutativity of these two observables. One of Bohr's great concerns, in fact, was to explain, in what he considered to be physically intuitive terms, or at least physically intuitable terms with some word, the non-communativity of various pairs of quantum mechanical observables. And Bohr's claim that his reply to EPR is nothing more than, wait, I've already been saying all along, did I say it wrong? It is zero. It is zero. It is zero. Oh, did I just, did I just? Oh, minus sign. Sorry. It's a big sign. I'm not saying sign. Sorry, sorry, sorry. Yeah, probably minus two. What does that mean? Signs. It doesn't matter. I'm talking about the fact that the observables that are correlated, sorry, the correlation of observables do not commute. Let me continue and see if it makes any sense. Now I've got to compute where I am. Right. One of Bohr's claims that he makes when he responds to EPR is that... This is just a racist, so you don't worry about it. All he's doing is saying again what he had said all along. And that claim is often taken to be disingenuous. But I don't think, in fact, it is disingenuous. I think it is, in fact, from one point of view, exactly correct. Okay. So, what was Bohr saying all along? What is Bohr's explanation of uncertainty? The central, I think, the central notion for Bohr

45:00 is the role of reference frames in quantum mechanics. And specifically, the role of reference frames in the definition of position and momentum. It's a common place to say that the notion of position doesn't make any sense outside of the specification of a reference frame in which that position is being measured or being specified, if you don't want to talk about measurement. And same for momentum. If I say the position of the particle is 5 meters, 5 meters, 5 meters, you're going to ask me where's the origin. And similarly for momentum. One of the points that Bohr makes very early on, in the mid-teens, is that the specification of a reference frame in quantum mechanics must be done by dictum. You must simply stipulate that this lab, let's call this our lab, is in fact our reference frame, come what may. And the point here is not the point that is true even in classical physics. post-Aristotle, namely that there's not a given reference frame and a given absolute frame. The point is rather that because of although it wasn't called uncertainty at that time, because of what would turn out to be the uncertainty relation, because of the fact that I cannot know both position and momentum of something simultaneously, I cannot even from an external point of view keep track of this lab frame and know that its position and its momentum are, even from some external The consequence is that if I, within this lab frame, exchange momentum with the frame itself, then as long as this frame continues to be my frame of reference, I cannot take that exchange into account. Because this thing is my lab frame. It is my reference frame. And come what may, it is at rest. That is our choice. And of course, it is always possible to move to a bigger frame and take into account the exchange of momentum. That's self-evident. But the point is that as long as this thing is my reference frame, then come what may, I must declare it to be at rest. Even if I know in my heart of hearts that relative to some bigger frame, it exchanged momentum or something.

47:30 Okay, that fact leads to a consideration of what he often calls conditions required or conditions necessary for the measurement of fill-in-the-blank. His favorite two blanks were position or momentum. Although it's interesting that he did do the same thing for other things, notably spin. But we have to set that aside. But the claim that I will make, and not have any time to defend, at least during my talk proper, is that a measurement of position requires the possibility of an exchange of momentum with the lab frame. By requiring the possibility of an exchange of momentum with the lab frame, What I mean is, after I'm done measuring position, I have to admit that I might have exchanged some momentum between the lab frame and the measured system. And I don't know whether I did or not. Well, as long as I'm restricting my consideration to this frame here. I can always go outside of the frame and see whether the lab frame jumped a little bit, and even how much it jumped. But as long as I'm standing here inside this frame, taking this to be, come what may, my frame of reference, after a measurement of a particle's position, that it might have exchanged momentum with the lab frame, and I don't know whether it did or not, and if it was so, I don't know much. Conversely, that's a claim that I haven't defended, by the way. I'm just stating it. Conversely, a measurement of momentum requires the absence of exchange of momentum with the lab frame. That's a little bit easier claim to defend without much argument, because if, in fact, it exchanged momentum with the lab frame, And if I continue to take the lab frame as my frame of reference, come what may, well, I didn't do a very good job of measuring the momentum, did I? Because some momentum was exchanged, some momentum was exchanged with the lab frame, and I have to just ignore that for reasons that I've already stated. So, in fact, I haven't measured the momentum to think at all. So what is Bohr's account of uncertainty? In the case where I measure position, some momentum may have been lost or gained. glossed into or gained from the lab frame, I have no way of determining whether this happened and if so, how much momentum was exchanged. And therefore, measurement of position renders me uncertain about the particle's momentum.

50:00 And a similar story can be told in reverse about momentum, position uncertainty, and the other direction. That is Bohr's account of uncertainty. And the point I want to make here is that he insisted that that account was required even after Heisenberg derived the uncertainty relations because the mathematical derivation of the relations was insufficient in Bohr's mind to understand why they were true. He wanted to give this kind of physical intuitive account of why they were true. Okay, now, very quickly, this is all very brief, but I hope that I'll provoke you to sing something. What about Bohr's reply to EPR, then? well first of all Bohr proposes a thought experiment that realizes EPR the equations of EPR right now it is in recent years been in some dispute whether or not it does in fact realize those equations I will stick my neck out and say that it does but it does so precisely for some of the unfortunate facts that follow from EPR's choice of position nonetheless I don't think that Bohr's reply actually relies on itself, on the details of the experiment itself, so I'm not going to mention in detail what the experiment is. Instead, I'm simply going to ask a question. Suppose we measured the momentum of particle 1. Then, of course, everyone will admit that the momentum of particle 2 is definite. And now we ask ourselves, what would, I'm going back to those two counterfactual principles Now we ask ourselves, what would have happened if we had measured the position of particle 1 instead? And remember, Bohr wants to resist the conclusion that momentum of particle 2 is still that way. Well, there are two things that are different in this world where we are now sitting, namely one in which we have measured position of particle 1. There's two ways in which that world differs from the world where we just were, where we measured Particle 1's momentum. The first difference is the difference in what we measured. But Bohr's acceptance of the weak principle by itself tells me that that's not going to be sufficient to make a difference in the definiteness of Particle 2's momentum. What's the other thing that's different? The other thing that's different is the presence of the correlations themselves.

52:30 Because when I'm in a world where I measure particle 1's position, I have to admit the possibility of an exchange of momentum between particle 1 and the lab frame of reference. As soon as I do that, the particle 1 plus particle 2, as soon as I make that admission, what I'm really admitting is that the compound system composed of particle 1 and particle 2 exchanged possibly some undeterminable amount of momentum with the lab frame. In that case, they are no longer perfectly correlated in momentum. To put it differently, I cannot any longer predict from particle 1's momentum, particle 2's momentum, or vice versa, because the very act of measuring particle 1's position destroyed the correlation in their momentum. You can write this all down upon mechanically, even for spin, and it just falls right out of the equations. But Bohr wants to actually explain why it's true in the case of position of momentum. because of the fact that the measurement of the particle one's position involves this indeterminable possibility of exchange in momentum. So the correlations in momentum are no longer in place. So I've jumped to a world where I've measured particle one's position. And more things, I'm running out of time so I'll just stay from the words, more things, that that is enough to resist the conclusion that in the world where I am now, Particle 2's momentum is definite. Because the very basis I have for concluding that Particle 2's momentum is definite. One of the ingredients that I have for concluding is definite is missing in this world. Namely, it's perfect correlation with one's momentum. I think I'll just stop there. I mean, I've already said enough, I think, so I'll just stop here. You seem to be suggesting the four gigs to make mystic, but maybe the most vocal recognition. Well, I didn't, yeah, what I was about to, when I bit off the end of my tongue here, right at the end. I was about to talk about morality. What Bohr says, as you know, is that there's not a mechanical disturbance from one

55:00 to the other. And I think that I mean, it's clear that that's right. He's not proposing some direct mechanical influence on the other guy. He says instead that what happens is an influence on the very condition which allowed definability, blah, blah, blah, you know, this phrase. And what I think he's about is the condition he's talking about is in fact the presence of this perfect correlation. Now, what kind of, whether or not there's non-locality involved, I think is in fact an extremely subtle question depending a lot on what you mean by non-locality. What is clearly not involved is a non-local action of some sort, for our physical sense of the term. I haven't, I mean, I'm being a little bit hesitant because I haven't really sorted out in my own mind exactly what say is about locality here. First of all, I don't know what the interesting notion of locality is. I'm not denying that there is one. I just don't know exactly what the relevant notions of locality are, or how to express them in a way that makes them amenable to evaluation in the face of what we're the same. So that's not a very helpful answer. You seem to draw a distinction between and doing measurements on particle 1 may be introducing disturbances and normally one thinks while disturbances are on particle 1 I understand that you've said that white things are not just one is somehow disturbing the reference frame in respect to which article one and article two uh no it's not that you're just let me stop here let me that's not that's not this is that right that's not quite the claim uh that i think is being made i think the claim is being made is that if we were to go that route then we would have to admit that in fact the reference frame somehow, by this interaction. But that's precisely what we don't do there. In taking it to be a reference frame, come what may, we are saying that any exchange of momentum with that thing is just lost into the world. Until or unless I wish to consider some enclosing

57:30 reference frame, and I didn't go down that route to consider what the logic of the argument would be in that case, so let's not for the moment. Rather, what happens is if I have this compound system, pair of particles that are correlated in momentum, which is just to say that their total momentum adds up to something that I know. Then, if I interact with one of those particles in a way that alters its momentum in a manner that I cannot account for, I can no longer predict the second particle's momentum given the first. Because my interaction with the first has broken that correlation. This is not true, this is not true of merely a position of momentum, it's true if you take any pair of spin observables you'd like, right down the singlet state, measure one of the spin observables on one of the system, and it breaks the correlations that are in the singlet state. Just to follow that up, and it's all I'm saying you can put momentum into the frame, but of course you can put it into the frame instantaneous. Now I know that Boris's timing probably wasn't considering whether you could get color-based space-like separated objects, but a physical frame can only, a you can only get momentum to its center backs after some delay in time but yeah I know there are many many reasons why the discussion in directly there are many ways in which the discussion has to be made more precise one of those in fact is what Rob and his student Hans Halberson have worked on quite a bit which is the discussion of the position and the relevant notions of position and momentum in the first place, because we were taking it as unproblematic that, for example, the EPR state exists, which is not, at least not in its traditional sense of being a state of global space. And we've taken the notions of position and momentum to be unproblematic, and you're pointing out yet another sense in which this whole discussion is very much idealized. And what I found remarkable when Rob and Hans and I put our heads together, I guess we're eight months ago, and realized that we were coming to similar conclusions, is that once one does make various notions precise, and there's a choice about how to do that, many of the points, many of the main philosophical points still hold. Now, the particular point

1:00:00 that you're making, I don't know how to account for that properly in this context. In other words, I mean, I understand the basic point, which is that the exchange of momentum doesn't immediately and instantaneously, it doesn't happen in some instant. But how to properly take that into account, I don't know. I'm afraid I'm going to have to bring that to the session to a close. Thank you very much. The last speaker of today's session is Thomas Burr, H. Wollard Burr. He's speaking to us on a Koscius-Specker theorem for inaccurate measurements. I'm going to speak about the topic which was raised by Rob in a paper with Dave and Kent not long ago. This may perhaps serve as a clarification why some of you may have heard part of what I'm going to say already. So it's about the Cochins-Becker theorem for finite precision spin-on measurements. I'll be briefly talking about the traditional Cochins-Becker theorem, about the circumvention of that theorem proposed by Indifferent-Verenberg, Mayer, Kent, and Robb. And finally, about this result, this Cochin-Specker theorem for finite precision measurements, minimum measurements. Just to set the scene, the traditional Cochin-Specker theorem is dealing with the question whether the measurement outcomes predicted by quantum mechanics can be regarded as resulting from detecting hypothetically predetermined values of the observators. The answer which Co-transpector gives to that question is no, that the outcome of sharp

1:02:30 spin one measurements and in general of sharp quantum measurements cannot be predetermined in a non-contextual way. They establish this by constructing a set of observables which cannot all be assigned non-contextual truth values. Just let me remark that their construction is based on an unsolvable coloring, on a coloring problem on the sphere, which they show to be unsolvable. And they, this theorem applies to infinite precision measurements in two ways or in two senses. The first sense is that if we make a measurement in a certain direction, spin one measurement, say, then the eigenvector of that observer we measure, all three eigenvectors, we get values. And so they assume that the components of these triangles are exactly orthogonal as our eigenvalues of some observable. And they also assume that two measurements, which are intended to pick out one eigenvector as a member of two non-maximal observables, pick out exactly the same vector. And what this argument, this circumvention of the Cochin-Specker theorem is about is exactly a relaxation of this second assumption of infinite precision. Now, let me briefly relate to you these arguments circumventing the Couch's Becker theorem, which in an early primordial form were due to Ketowski in 85, then for the spin-1 case Mayer in 1999, then Kent in 1999 was generalizing that to Hilbert's basis of arbitrary dimensions and also to unsharp observers. And Rob and Adwin were in fact constructing the imperative theory in that paper in 2000. Now, the notion of inaccuracy which they are dealing with

1:05:00 picking up is the following. If we are making a measurement, for example, of spin in a certain direction, we want to measure spin in some direction, we are in fact not certain that we're actually measuring spin in that direction. I mean, we don't have ultimate control about the measurement device, and therefore it might well be that we are in fact measuring spin in a direction which is slightly deviating from the direction which we want to measure. of the second infinite accuracy requirement. They use a mathematical result, which was established earlier, or they actually prove it also in some cases, so that there is a dense set of quantum mechanics observators for which all measurement results can be predetermined in a non-protectual way. it's a dense set. And that's a mathematical result which is firmly established. So there can't be any discussion about that. They then go on to give an interpretation of that. They say that in fact there is a theory predetermining the results of all quantum mechanical measurements in a non-contextual way. Well, not precisely, but with an accuracy. That is, so the, that's a very ambiguous idea in fact. I mean they say that we can redetermine the results of spin measurements in a dense set of directions. Now if we want to measure spin in some direction and we're not, not absolutely sure about the fact that we measure exactly in this direction but we only can for example specify a small neighborhood in which we are certain to measure, then in that small neighborhood, there will already be observables whose value is predetermined in a non-contextual way, and which we, and the result of that predetermined, or that predetermined result we take to be the result of the measurement in the direction which we wanted to measure.

1:07:30 Okay, so this was established for sharp measurements and also for unsharp measurements, and there have been, in the literature, quite some criticisms of that interpretation by people like M.D. Cabello, Mermen, and so on. I am not going to add anything to that criticism. In a way, pointing at the price which one has to pay for these invariable models. Okay, now we're coming to the third section where we see this Coche-Specker theorem for finite precision speed one measurements. So first of all, sharp spin measurement, that's a little repetition of the formula. S, X, Y, and Z are the spin-1-poly matrices, so these three-dimensional matrices. For example, spin in center direction is represented by this poly matrix here. Let us denote by psi Z1, Z0, Z-1, the eigenvectors of spin in Z direction of this poly matrix. observable here, and p set i to project this onto these eigenvectors. So for example, the project onto the eigenvector of spin set with value eigenvalue plus one is this one here with one here and zero anywhere else. If we're doing that in an arbitrary direction, call it n, then the corresponding spin operator is given by this one here. And so we get a projection value measure, which is representing a sharp spin observable. It associates to every positive outcome, one, zero, or minus one, a projection operator. Okay, P n i is a projection operator onto the eigenvector of spin in direction n corresponding to eigenstate i. And the probability of getting an outcome in a sharp measurement of spin in some direction and on a system in size given by this trace formula. That's the usual thing. So this is the sharp case. Now let us come to the unsharp case,

1:10:00 this inaccuracy. So we said the kind of inaccuracy we are talking about is that we are not certain about which direction we are actually measuring, we are actually measuring. So there's some uncontrollable degree of freedom in the apparatus. So let's suppose I want to measure spin in a certain direction, n. This is the intended direction. If we have this kind of inaccuracy, there will be a kind of error distribution, which is and that's this picture here where WN of M is the probability density that we in fact do measure in direction M although we want it to measure in direction M. Okay? So this is this probability density. And the probability of getting an outcome I in such spin one in an intended direction n on a system in state psi is the following. You just integrate this trace here with this density of the error distribution and you can pull that out because everything is weakly continuous. And defining f in this way here, you come back to the original a trace formula, now not with a sharp projection operator for the observable, but with an unsharp operator, with an effect operator, positive operator. And this positive operator is this one here, defined in this way. So it's a sharp spin operator integrated with the density of the error distribution. And these are, this is indeed an unsharp spin property. So one can show that they are positive operators with spectrum in the interval between that F and I, if we sum those up for I equal 1, 0, and minus 1, that they add up to the identity and that we've got a positive operator-valued measure, which are the standard tool for describing inaccurate measurements. So, nothing more. Now, let us introduce an additional requirement on this error distribution.

1:12:30 The requirement is that this error distribution is covariant under rotations. So that means that if we are measuring spin in a direction Rn, then the probability density of actually measuring in the direction Rn is the same as the density of measuring in direction M when we're actually intending to measure in direction n. I have got a little picture about this. This is a very natural requirement, right? So this is the original spin measurement where we want to measure in direction n, but in fact, by a client misalignment, we do measure in direction n, described by this error distribution. Now, if we rotate the whole device to measure in a direction n, rn, but in fact we do measure in a direction rn, then this will be described by this error distribution and the claim is that these two are the same. Right? So if... That's covariance. And I think it is quite a natural condition on this error distribution. Now, if we, from this assumption of covariance, we will get some results. First result, these unsharmed spin operators, they transform like spin operators, in the sense that this formula here holds, d1 of r is the spin 1 representation of the rotation group. And this is the traditional transformation equation for angular momentum. That means that if we have this covariance condition, then these unsharp spin operators do transform like angular momentum operators, and they can, with full classification, be called spin operators. Second lemma, if we have this covariance of the error distribution, then the unsharp spin properties F have the same eigenstates as the sharp spin operators, as the sharp spin properties. I will very briefly show you a proof of that. Well, it's messy, but

1:15:00 first of all, you can, for example, first of all, in the case where n is equal z, you can write down, actually, the spin property in direction z. Oh, sorry, in some direction an element this way, and then you make the integration and you get something which is diagonal. Okay? So F set of plus one is a diagonal matrix, alpha one, alpha two, alpha three, and these diagonal elements are given by some integrals. If we compare this to the, and you can do the same thing for the other eyes. If you compare this to the shark spin which have this formula 1, 0, 0 0, 0, 0, 1 and we have a look at these eigenvalues and we evaluate these integrals in the case of where the error distribution approaches a delta function so that we have sharp measurements, then we indeed see that alpha 3 goes to 0, alpha 2 goes to 0 alpha 1 goes to 1 Right? So as this error distribution approaches the delta distribution. So indeed, in the limit of infinitely precise measurements, we get back our sharp spin operator. And then we can do the same thing for an arbitrary direction and by just doing the rotation and exploiting the covariance property. Okay, but here you see, of course, in the case where n equals z, you already see that these here have the same eigenvalues and the same eigenvectors as these here, right? It's the x, y, and z vector. They are the eigenvectors. and then we can one can show a third lemma which says that the iron values is alpha 1, alpha 2, alpha 3 and there was also an alpha 4 of these unsharp spin properties they are these four numbers here

1:17:30 all these numbers are in the interval between 0 and 1 the I-ing values of any of these unsharp spin profits add up to 1 and as the inaccuracy goes to 0 the unsharp spin operators converge to the sharp spin operators and that is the thing which we need in the sequel Now, let us remember quickly the old coach and speck of group, or the traditional one. Predetermining the results of the spin 1 measurement, so whether the result will be now 1, 0, or minus 1, is equivalent to picking one of the projectors, the projector of 1, 0, or minus 1, right? zero p and one of p and minus one. And when you pick a projector, a one-dimensional projector, you can equally well pick a vector, right? And this leads to the coloring rule. So predetermining the results of the sharp measurement is equivalent to assigning one color to one of the vectors in color to the two other vectors in a tripod, in a tripod of eigenvectors of this spin observable. And what Cochin and Specker then show in the argument that there is a finite set of observables such that not all eigenvectors can be colored according to this coloring rule here. Now for us the situation is not so easy because these unsharved spin operators are not projection operators and therefore picking one of these unsharp spin operators to be the realized or actualized one cannot be translated into picking an eigenvector, right, because they're not projection operators onto a vector. Therefore we must introduce a slightly different coloring group and it's a following coloring We choose an unsharpness tolerance, call it delta, which is between 0 and 0.5. And we say that, well, at which level you exactly choose your delta is irrelevant,

1:20:00 but it will be somewhere in this range, right? That's a matter of taste, and the results don't depend on that. And we say that, well, an eigenvector, an eigenvalue, which is closer, which is above 1 minus delta, shall count as 1. and what an eigenvalue is below delta shall count as zero. So we say that an inaccurate measurement in direction n, if it has a result i, then the eigenstate of this of the unsharp spin property with an eigenvalue bigger than one minus delta gets one color and the eigenstate of the property with an eigenvalue smaller than delta gets another color. That's the new coloring group. see from this special form that these unsharp spin properties have, that if the measurement error is small enough, these alphas go either to 1 or to 0. So at some time, they will be above 1 minus delta or below delta. And therefore, according to this coloring rule, one eigen vector will get one color, the two others will get the other color. So we're back to the traditional coloring problem of the Cotius Mecca theorem. And we know already from that, from there that this coloring is impossible. So we arrive at the following theorem. If the measurement errors transform covarity, and the measurement errors are sufficiently small in order for these eigenvectors to be above one, eigenvalue to be above one minus delta respectively below delta, so if these two conditions are met, then there exists a finite set of observables for which the results of inaccurate measurements cannot be predetermined in a non-contextual way. Good. Now I can sum up. To sum up, let me put this result a little bit in a historical perspective. So coach and specker showed that there is no non-contextual predetermination of results of accurate measurements. Meyer, Clinton and Kent showed that there is no contextual or sorry that there is a non-contextual predetermination of results of inaccurate measurements, okay?

1:22:30 And what this result here shows that there is no non-contextual predetermination of results of inaccurate measurements with a covariant error distribution. So this, the price which my attempt at Clifton have to pay is that the covariance condition fails in their cases. And therefore, the observable describing their actual measurement statistics in these models don't transform coherently, and they don't have the right commutation relations, and one would not actually in the literal sense to be able to talk about spin observers. measurement statistics don't in the literal sense describe spin observers and one could not say that they're really talking about the spin one particle because um that the square of the angular momentum won't be conserved in their case good that is the result uh let me point to two to some open questions where i would be grateful to get some remarks or suggestions first open question do without this coloring rule which basically treats eigenvalues above 1 minus delta as 1 and below delta as 0. Something like that would be possible if one had a Gleason type theorem for inaccurate measurements and so the The question is, can one in, is there anything like that, like decent type theory for, for Ein Schar effect operators, not for, not for the projection letters of Schar's big properties. Another question is, can results like these here be generalized to, for example, to position of spin. That's not entirely trivial because it's an infinite dimensional Hilbert space and when you smear it with some error distribution you get a continuous number of eigenvalues and that's not the reason to treat it. These are the open questions.

1:25:00 I wonder a little bit whether your conception of inaccurate measurement involving un-sharp would coincide with their conception in which you just have an orthodox sharp spin measurement that would be placed by a spin measurement in a slightly different direction? Well, in this very crude sense, I think it would coincide, because in both cases we would say that we want to measure spin in some direction, but in fact we do measure it in another direction. But in your case there is probability distribution. Yes, yes. And although I haven't checked the details, I'm pretty sure that if you take this positive operator value to measure the Gaussians, all these positive operators would commute with a sharp spin operator in the original direction. yes yes in the Maya if you change the original you get something which is not community so yes the mathematical properties of your two functions seem to differ already in a very elementary and fundamental way. Yes, yes. I would guess that in their case, you would have to assume not a Gaussian distribution, but a delta function, which is just non-central, which is of the origin. Yes, yes. So that's a different error distribution, right? if the error distribution has this covariance property, then this result holds. And obviously, in their case, if there at all exists an error distribution, then it cannot have this covariance property, reading my result as a positive. That even includes that the k-law is supposed to be improper error distribution.

1:27:30 Well, yes, yes, that's no problem. Okay, so it's independent? Yes, okay. Marcus? Yes, I mean, I'd accept what you just said now. You said something that you're at the end of your talk. You said, you kind of implied that it didn't really count as a spin measurement at all. It wasn't a cadaric Arabist view. Well, I mean, that's... I thought that was much too strong a statement for me. Yes. Yes, I mean, yes, I agree with you. Don't take me literally. I mean, what I wanted to say, if, from a background, orthodox point of view, you are defining a spin observable to be an observable which has an angular momentum transformation property, in that very orthodox sense the statistics of their spin measurement is not described by a spin operator. But that's all I wanted to say. Yeah, so there are some people like also who define observance via their transformation properties under the kinematic group also. And if you define a spin observance to be one which has the angular momentum transformation properties, then what they treat as spin is It's not anymore a spin product in that sense, but I didn't want to claim anything substantial. I think that's true. They don't claim to be treated in a more continuous space. Yes, yes, yes. I mean, that's not my point. I just wanted to say this operator describing their measurement statistics do not have the transformation properties of angular momentum, that's a technical point, however you want to weigh that and interpret that. That's your point. I mean, I would think that what you're really talking about is the significance of the theory at the end is the thing that describes the probability distribution to the results. I wouldn't think one could really talk about it as being observable in the context of the Merck-Kenton typical results. You're talking about the probability distribution describing an apparatus that measures it less than perfectly, for whatever reason. Which an apparatus does. But I think these are related. I mean, in fact, the measurement results can in this

1:30:00 formalism of POEM be tracked back to operators producing by, to positive operators producing by the trace formula, the statistics of the measurement results. And these positive operators do not, if you do that for their measurement statistics, will not have the right transformation I'll say something later. Yeah, so if you assume POVM, then you actually get a stronger result that you can find the born rule as to specify probabilities even in a two-dimensional type of thing. But I'm not sure that result necessarily leads to any claims about non-protectual variable theories because what you get is a new kind of situation where, for instance, one POVM would be identity identity over two, like two elements, some identity that were positive, and they have the same probability. And so if you make a naive statement that any, you know, two effects are the same, then they should have the same outcome, you know, in a measurement, you should have the same outcome, and then you get this contradiction immediately, which is that I can't get outcome one for both the outcomes in this measurement that the trivial will Yeah, well, if I'm not mistaken, I mean, I don't remember Paul Busch's result in every detail, but I think he's having a G-type theorem for the set of all effect operators. Is that true? What would be needed here would be a G-type theorem for a subset of effect operators, namely for these unsharp operators, for these Fn1, 0, minus 1. And they have got a very special algebraic structure. So, but thank you very much for that effort. I think I'm going to bring this session to close. Just before I rethink the speaker,

1:32:30 let me just make one or two more announcements. The library closes officially at 5.30, but the librarians have allowed for the pew room in the corridor, which is outside the lecture room, to remain open. If people want to use after us talk it's possible to do that. The request is simply that the last person in that room close the sliding door. We will make an attempt tomorrow as promised today to do tea and coffee both in the entrance lobby and in the common room so there isn't so much congestion. Tomorrow we start the parallel sessions in the afternoon. I would just make an appeal in order to avoid what could be a lot of congestion beside the sessions I would ask you to try to choose your sessions in block because I try not to move around within the hour and a half session because I think otherwise we may have a major problem. Well, it only remains for me to thank Thomas Boyd-France. Thank you. Thank you.

1:35:00 Thank you. Thank you. Thank you. Thank you. In that case, it will be, it will actually be at the ball. As I say, I don't think we've had time to go over the ball yet, that's why I sort of thought. They'll presumably be at that, so that's what's happening. Or maybe it's a sort of, so it's a nice cafeteria time.

1:37:30 Thank you. Thank you. and arguing that this actually in everything that the sound is really going on with the Berm approach and potential that it's actually all being called a Bermia which functions just like Alan Henry in the classical theory I think it's very nice work it's very suggestive the only thing is it's not clear, they're explicit it's passed on Bermia at Bermia I mean, Basel is, I think, a very, very authoritative answer. He's, everything sort of works kind of, brackets him as, oh no, David Byrne's collaborator, where the idea of discussing more than coming forward, kind of glorified, and more than coming out of the system. In fact, I think he's actually a little, I mean, well, put it this way for a physicist, he's a very good mathematician. Very good, especially a very, very good scientist. And I think he has a strongly good algebraic insight. And whether you actually buy into the various physics or not, I think it's taught us that it's very, very interesting for the unification of the structure and the forms, especially when they're not very starting to explain that, which I think is very interesting.

1:40:00 and where there is also where it comes from. But we don't know anything about philosophy, but this is still the start of that. We actually understand what it's about, you know. That's really cool about that. I don't know, I don't know. Of course, I was going to tell you, that she was measuring a dying trial instead of philosophy of physics. That's why we're looking at that loss. I don't know if you're going to have to say that, but I'm not going to be stable. I'll be able to run, yeah, that's the one that I'm going to talk about, that's how great people. You know, that's because it is, and it's very young at that time, because I was a person who was a musician, but I actually want to fall like that. And he said, for people's sake, can you really affect the minister's sake? Thank you. It's basically my transparency in my computer. Oh, of course. How do you get inside out? It might be something to do. It's something to do. It's something to do. It's a box. What, back in DC? Yeah. I'm trying to get to sleep or something. If you go, it's why you don't have to write a key. Oh, you mean an app? Sorry, what do you remember in the department? Oh, shit. I know, there's nobody else who's got a key. What are you talking? You understand? I've still got that talent. I love that talent at the top. Well, I'll have to go all the way back to Leeds tomorrow, it'll be a bit of a pay than the others to go in my way. Or you'll have to get somebody in your school with the kids. I hate to say, I don't think you've got me. I can't think of a third or third. Well, I can sort of read with those. I know. Well, it's all about reading my whole thing, which is probably more stressful.

1:42:30 Is there anything you can expect here? No, it's fine. Well, you've got the key, yeah? Yeah. No, I've got the key until you're in my house. Well, do you know when I saw that? No, I went from Leeds, and it's sort of a Bentley on the way. Yeah. But who's got the key to the room where your transparencies are? Oh, my account. They ain't gonna go down to Leeds. They don't know how to use computers or anything. and there's only two people can go out and get a key on them it's not a close one there are only false sides it's not that far is it there are only false sides it's still very frustrating well I think I am going to go over I'm going to talk is it the computer that's just Well, I mean, it's just, I got no sleep at all last night, because immediately this meeting is over, I've got to go. Thank you. Thank you.

1:45:00 What is the biggest way to get to Teddy Hallman? you can't you've got to walk up to um I don't think they might have seen that they're probably going to stand up to be raised to the welfare system. I've got to reflect very much of the reclining or talking, um, the only one, of course, I think they've recorded it's an amazing Mr. Memorick and let him emulate the human being the human being the human being hello, welcome, that's my point I'm a passport and we'll get to the bit that's it on the side we'll see that I've been talking about I'm not going to have a thing that was great I'm not going to have a thing but why don't you still have this I'm not going to have a problem it's backing back in the air print your talk is actually crowded It was another piece, it wasn't an American, but it must have been there. No, no, that was the one thing, this was something that started, because I think, Oh, I'm sorry, I'm getting terribly confused.

1:47:30 Oh, back, back, back, that's absolutely great. I'm a bit of a feeling. Yeah, so it's not going to be a... That's a strong kind of a thing. It's all! It's a big deal. It's a big deal. It's a big deal. It's a big deal. Well, it's a big deal. It's a big deal. It's a big deal. Was that you? Yeah, that was it. Thank you. Thank you for that one so far, it's time for you to do something else. So, it's just up here, everyone. Okay, well, see you tomorrow. Thanks for being in the line. Oh, my God.

1:50:00 All right, I hope you've got me on your list. Michael Wright, I was almost able to check in earlier, but I think the illiteracy need to start. It's a long time. Thank you very much indeed. Thank you. Alright, I have to go to the... Yeah. Yeah. Yeah, I have to go to the break, so... Yeah, 18 lines, so... 18 lines are on the... I just said... Yeah, I have to come back in... Yeah, I have to come back in... Thank you.

1:52:30 I was going to start with the screen, but that is going to be finished. But then, what's the 15 years ago? One year, one of the things, and the USB, it's just one, one, two, one day, it's two, here, oh, I see, I'm sorry, I'm sorry, one, one, there's a, it's going to be a, I've been there, so many, you want to make a lot of this. And then, it's . Thank you. I don't know. I'm terribly sorry, I've got a run-up, and they told me, well I thought they said that's what they told me, but I want to misunderstand, I do apologise for disturbing, no he doesn't, well I think number one and five, I thought they said two and five, actually I thought they said it was Bessie's fifteen,

1:55:00 That's what I did not say. I thought that S15 would be S15. I think S15 is the IT group, so that's not the... That doesn't sound right, does it? So Betsy 15 would be, would that be on here, or would there be a little bit of a conversation? Would there be a conversation? Betsy 15 is the implicit, the Betsy one. I'm so sorry to pay more attention, I'm afraid I'm just so tired, and I've just assumed just a little bit of the standards and quantum outputs. Okay, I'm not sure what's going on really. I'm going to try to do that like this. Keep looking. Oh, my God.

1:57:30 I don't know. Oh, my God.