Double Covering — Metaplectic Group & Covering Spaces in Hamiltonian Quantum Formalism

0:00 Well, good afternoon, all of you, to the last session of today. This is another string of local heroes, I see, as they are called people from Oxford. I assure you it's prettiest to believe that you're a hero if you were born in Oxford. Her speaker is from Theoretical Physics in Oxford, Carl Dalby, Simultiating the Concept of Parton. Thanks. And let me start thanking Peter and Harvey for half an hour of whiteboard time. I would feel we're all aware that to some extent of a certain conceptual tension that exists between the role of time as it appears in quantum mechanics, which is effectively of the absolute time, and the model of time, as it appears in something like general activity, has just one coordinate in a coherent theory. Well, I'm not going to spend half an hour addressing that. Indeed, I'm going to address what seems to be a completely unrelated topic, the topic of particle creation and interaction backgrounds, and particle creation as seen by non-inertial servants. But we'll see that addressing those problems really brings us to the heart of this conceptual tension regarding the role of time in the two theories. And so perhaps by tackling particle creation we can bring a little bit of light on this conceptual tension that exists between a different role of time and time in the two theories. Of course I'm also doing, I will try to explain what I think is quite a deep connection between our concept of particle and concept of vacuum and our concept of time and simultaneity. So I'll start with a brief description of particle creation and gravitational backgrounds and a brief historical of it. I'll just lay down the base of the task of particle creation and gravitational backgrounds and the problem of polliation dependence that emerges in those backgrounds. I'll explain how radar time can be used to relate this problem of polliation dependence with the known phenomena but not well explained phenomena of observer dependence and then I'll start talking about examples of particle creation and compare it with particle creation and electromagnetic backgrounds and things like that.

2:30 The first prediction of particle creation and gravitational backgrounds was in a rather remarkable paper by Schrodinger in 1939. This was well before we knew whether or not the universe was expanding, and he wrote in a rather impressive paper, if it's expanding it will mean the creation of matter merely due to its expansion, and the background is here somewhere. That wasn't picked up on really until the late 60s, when Parker and a series of papers addressed again particle creation and expanding universes. Even then, the subject really didn't hit the headlines until 1975, with the prediction of Hawking radiation from black holes. That was certainly an exciting discovery, but at least for me, a much more intriguing King discovery was the one made later that year by Andrew and independently by Bulling and by Davies, that an observer who accelerates uniformly in flat, empty space will observe a thermal bath of particles, even in the state that all inertial observers claim to make. So it forces you to recognise that particle creation, that this phenomena is not just a peculiar property of black holes, but it has to have an element of observer dependence. This is a statement, all inertia observers agree contains no particles, and an accelerating observer merely due to their acceleration sees particles in it. So at that point, the race was on to try to characterise the observer dependence of this notion of particle and this notion of vacuum, and I think it's fair to say that even 25 years after the discovery, no systematic consistent observer dependent definition of particle that could be applied to any given observer. Anyway, to explain how this observer dependence comes about and the problems that I've been describing it, I've been first of all to the basic task of describing particle creation in a gravitational background. This is the only piece of hard core maths, if you want to call it that. equations describing some sort of field operator and any time I get a concrete example I will talk about direct fermions

5:00 that's just my habit because it's in a background by definition that means we're ignoring electron-electron interactions and we're just looking at what this background does to this sea of otherwise free particles so we can still expand the field operator in terms of modes and creation and annihilation operators but the only problem is now our modes are not simple So there's no obvious prescription as to which modes we just put next to the A's and think of as representing the particles and which modes we just put next to the B data and think of as representing the other things of antiparticles. So the general trick is to, by whatever means possible, assign two such choices. One choice of which seems natural at early times and defines what you mean by N particles in the N vacuum which seems most natural at late times, and defines what you mean by out particles in the out vacuum. And if your two choices differ, then the overlap between negative energy in states and positive energy out states defines particle creation. But that, of course, brings us to the question of how do we choose these in-and-out modes? Well, there are various approaches to choosing in and out modes and it's perhaps a bit simplistic to try to categorise them in these terms but very roughly speaking they can be categorised into two main types those that rely on symmetries that are present in the space-time or those that rely on an arbitrary choice of foliation of the space-time into space and time the methods that rely on the symmetries that are present in the space-time are obviously limited by the fact that you need enough symmetries in your space-time, otherwise the method doesn't work. The methods that rely on a choice of foliation, well, once you've made your choice of foliation, particularly Hamiltonian diagonalisation, allows you to say, effectively, whatever you would like to say. You can say how many particles are created, you can say where they are created to a certain extent, sorry, you can say when they were created, to a certain extent you can say where they are created, but everything you say depends on the foliation you chose. So as long as that foliation was chosen arbitrarily, you have to conclude that everything you

7:30 said was meaningless. Meanwhile, although the practitioners of each of the approaches will often mention a certain observer when justifying a certain choice of coordinates or a certain choice of foliation, there is no systematic way in which the choice of observer is incorporated into the maths of any of these approaches. A more systematic approach to understanding the observer dependence of the notion of particles is to use particle protectors which provide a more operational particle concept. However, you can't use them to define particle and antiparticle modes firstly because all it can actually say is whether or not it detected one it can't say what it would mean for none to be there. But also for the rather important reason that it would be served in. So long as a particle detector is anything that successfully detects particles, you can't also say that a particle is anything detected by a particle detector. You run for the problem, for instance, that I could say that my figure packing on this table is a particle detector, and every time it packs it's detecting a particle, and I could call it a friend on, and we could discuss the problem with that. And you would all agree that that's a stupid thing to do. but if a particle is by definition you can detect by a particle of detection you have no way of explaining why So, a more complete treatment of the observer dependence of the particle interpretation would be to try to take something like this, this foliation dependent approach, but try to find some unique prescription to assign a choice of foliation to any given observer So if you could define a unique affiliation for any given observer, then you could use Hamiltonian Diagnisation with it and you'd have a satisfactory concept of possible. That brings us to the concept of radar time. Although not particularly well known, there has been, for about 50 years, a perfectly consistent way of uniquely assigning a choice of affiliation observant. And it's really rather straightforward. It's due to Sir Herman Blondie's in the 50s. He introduced it in his work on K-calculus. However, he himself warned against applying it to anyone other than an inertial observer in flat space. And he warned against it effectively because at the time he didn't trust proper time. We've since done the experiments to

10:00 check that clocks agree even when they're accelerated. It's actually no longer have any reason to distrust proper time like he did, so we come to in fact take his definition wholesale. Let me say exactly what it is. We want to assign some time to some event that is distant from us. So we simply send a ray of light from ourselves to the event, making note of the proper time at which we sent the ray of light, we receive a ray of light back from the event, making note of the proper time at which we received the ray of light back. We average the two and that's the time we assign to that event. so obviously if you're an inertial observer in flat empty space that becomes Einstein's synchronicity condition and it simply reproduces your inertial frame the important thing to note is that all you need to do that is a well defined proper time once you've got a well defined proper time you can do that of course it won't cover all of space time in general if there are any particle horizons in the quite the surfaces of simultaneity so defined will converge as a particle horizon Of course, that's exactly what we need to describe such effects as the other effect in Horton radiation, so that's not an undesirable feature. We would also consider it fairly unnatural if a person's concept of simultaneity was able to be applied to places where they clearly couldn't see. So it's independent of the choice of coordinates, because I didn't have to mention a choice of coordinates to introduce it. It's single-valued in the causal on the life of the observer, that is in the history of the future of this class. It's covariant under affine reframatisations and time reversal, that is if you use a class that tends at a different rate but still a steady rate, or even if it tends backwards, then you'll still use the same hypostens as a similarity, you'll just label them differently. It agrees with proper time on the trajectory. The naturally incorporates particle horizons, and the large-scale structure of the coordinate is not sensitive in a way that I will explain in a second. So having defined a foliation that relies only on the observer, it's extremely straightforward to write down the particle definition that relies only on the observer. Basically we've just taken this Hamiltonian diagonalisation prescription, but now we can say, given our observer, the Hamiltonian at time tau is just the diagonalisation of this t mu nu, where k nu is now the time

12:30 translation vector field, which I didn't the time translation vector field is defined such that it's everywhere orthogonal to the hyperservice in question and it's length which reports as to how far you are from the next hyperservice. So it takes you from one moment in time to the next if you like. So now we just do Hamiltonian diagonalisation where this is our k vector, this is our my preserve of simultanity and we've now got a well defined particle prescription that applies to any observer and any gravitational background at any time. So, it's independent of the choice of coordinates or gauge, it generalizes Gibbons' definition to non-stationary space times. Gibbons has effectively said if your space time has got a time like Killing-Vector field then that's really useful because that gives you modes that look like plane-weight modes that from e to the i omega t and e to the minus i omega t. So Gibbons says if you've got a time like Killing-Vector field, use that for your k and then it doesn't matter which hyperservice you use. it leads to a concept that particle is non-local on small scales but is effectively local on scales larger than the constant length of the particle concerned. The fact that it's not particularly well known and should be more well known is that that is true even in ordinary non-interacting field theory in flat Minkowski's face for an inertial observer. Anyone who's done quantum field theory will have seen the derivation of the Hamiltonian in terms of creation and annihilation operators, starting from the Hamiltonian written in terms of the field operator, for instance, and it's a local function. After they've shown that, the next step in the textbook is normally to show how you get the charge operator in terms of creation and annihilation operators from the charge operator written in terms of the field operator. But this one's always just written down. The derivation is unique, so you can do it backwards just as easy as I can assure you that that is the operator, the number operator written in terms of the field operator, and this is a non-local kernel, these are vessel functions, and so this is non-local in small scales, effectively local on scale, which is about what you would expect for the everything in quantum field here, it's consistent with the leak that particles are either about a constant wavelength across or shielded by stuff that spreads out across it also places no asymptotic requirement on the behaviour of the background so it allows

15:00 finite time to be considered it allows you to say when the particle creation occurs, also we can write down a finite volume number operator which has controllable fluctuations by controllable I mean take a bigger volume you can get smaller fluctuations so if you try to be too fussy you'll get fluctuations to draw your results but at least you can say to a certain extent are the particles I've created. Now, before going into genuine particle creation, let's just have a quick look at some examples of radar time. First one, of course, the uniformly accelerating observer in flat space. Radar time is the time that it reproduces the Killing vector that's used in all derivations of the under-effect and effectively reproduces the under-effect wholesale. So we can breathe a sigh of relief on that one. Another couple of similar examples Now, what I presented with radar time is effectively a concept of simultaneity that applies to an observer In flat space, there is another possibility that we could use as a concept of simultaneity namely the instantaneous rest frame So I decided to do a quick comparison in this overhead to show some of the problems with the instantaneous rest frame which would make it quite unsatisfactory for constructing a number operator and in line with me quite unsatisfactory generally. This is a diagram that appears in a lot of textbooks on the twin paradox, for instance. There are your hypersurfaces of simultaneity of the traveling twin. And they have this obvious problem that they're not single-valued here and they don't apply at all over here. So even though Barbara can see these events perfectly well, when she tries to write a new logbook when they haven't, her prescription simply doesn't give her an answer. whereas over here, even though she only saw it once, you have to write it in your log book twice. Whereas if you simply step to the definition, send your right of life, receive it back, no risk for two, then if you're this observer when you see it and you're that observer when you receive it back, then you end up with hyperservice. There's no idea to just do that, and the separation there is a doctorship relative to the separation there. Similarly, you can look at the gradual turnaround basis often said in the textbooks that, you know, this problem by taking note of the fact that you actually have to turn around during some period of time with finite acceleration. Of course that gets a bit of a gap, but now you're assigning three times where we've been over here. I can also explain briefly what I meant by the fact that the large scale structure is

17:30 insensitive to small scale details by imagining what would happen if Barbara, during her turn-around, was to go to the bathroom, for instance, in the room next door, and then quickly return to her position. Obviously, if you were using instantaneous rest frame, then she would now have another whole set of hypersurfaces that are going into something like that, and, well, they're messy. So now she's assigned three times to every event over here, five times to every event over here, and it's all kind of a fair shame. Whereas if she goes to the loo and is using radar time, Like, this leads to a very small change in the nature of her hyperservices, a simultanity just here and here, and similarly down here and out here, and you get the picture. Small changes in her trajectory, small changes in her radar time coordinate, which is desirable. A couple of quick cosmological examples. Decider space is another one where we feel we know what we're doing in the particle creation, and so it's another one where we'd better check that we agree. Decider space, sorry I didn't actually write the metric down, but you know what an FIW metric looks like, and this is just an exponentially increasing scale factor. your time translation vector field is simply the standard killing vector that's used in Gibbons' definition of derivation of the Gibbons' temperature for instance your radar time and radar distance lead you into static coordinates and so again the standard derivation of the distance space temperature comes through effectively unchanged the Milne universe is rather an interesting one the 1 plus 1 dimensional Milne universe is an FRW universe with a scale factor proportional to T now if you're a cosmologist and you pick one of your particle creation formalisms from your textbook that's designed for cosmology so you'll generally use unless you're into city space curiously you'll generally use cosmic time or conformal time or something like that to decompose your mom of creation in this kind of a universe, which is qualitatively much the same as you'd get in any other kind of universe. However, all the time, knowing that all you wrote down

20:00 was flat space and a strange set of coordinates. Indeed, unlike the Unruh case, where you wrote down flat space and a strange set of coordinates that corresponded to a non-inertial observer, this time, the first observer you're talking about is an inertial observer in flat empty space. So if your coordinate system is going to trick you into describing particle creation According to that observer, I think it's better say that's a slight embarrassment amongst cosmological models, and it's convenient to notice that if you transform back into radar distance and radar time, you'd find yourself back in the underlying flat space. So the definition's not so easily tricked. How much time do I have? Five minutes. Okay, well, very quickly, we've said a little bit about, well, quite a lot about radar time, but not really very much about particle creation. In order to get some sort of intuition for quite how much you can say about when particles are created and where they are created, it's convenient for the moment to have a quick look at particle creation and electromagnetic backgrounds. Even in this case there has been some subtleties and mistakes made in trying to get the appropriate definition of particle. It's common in the almost entirely everyone puts the Ea0 on the right-hand side. Hamiltonian diagonalisation quite naturally requires that the Ea0 go on the left-hand side with the DLT, and the Hamiltonian defines particles and antiparticles as the rest. Effectively, an antiparticle or a particle with a negative kinetic energy, not a negative total energy. If you put this on the other side, for instance, you could add a large constant a-naught that steps up to a dimension. So having written it in that form, you can now see, remember the particle creation has caused any time a negative energy density evolves to pick up a positive energy component. And there's two ways in which that could happen here. Either a-naught could be non-zero, so that it acts as a potential in which particles could tunnel, or the spatial part of a could be time-dependent, mean that something that started as a negative energy eigenstate won't stay a negative energy eigenstate. If neither of those are true, if you can find a gauge in which this is zero and this is time independent, then obviously this equation takes the form of a time independent

22:30 Schrodinger equation. Eigenstates remain eigenstates and no particle creation occurs. That is, time independent magnetic fields don't create particles. Actually, another Another thing I could briefly mention, approaches that put this to zero and this not to zero rely on something called the tunneling approach, which works perfectly well in those cases but gets the wrong answers in any other case. Approaches that assume that A0 is going to be zero and this is going to be time-dependent rely on something called the gobble-you-goff approach and they get the right answers in that approach but the wrong answers in any other time. But if you just put the annual on the appropriate side, then the two approaches are kind of combined into this, you've got tunneling and you've got that kind of evidence. So how much does that allow us to say? Just one very quick over it. Particle creation in an electric field that looks like that. It's spatially uniform, both slowly turned on and slowly turned off. The reason for using one that's slowly turned on and slowly turned off is so that I can compare what this particle definition would say to what other definitions have already said, the ones that require asymptotically well-behaved backgrounds give the asymptotic answer, which is that a time-dependent concept of particle without you to say not just how many were created, but when they were created. And you can see that at early times all you've got is this peak around the origin saying that particles that are not moving are easier to create than particles that are moving. Once the particles are created, they get accelerated by the fact that they're an electric field. you know, entirely classically they behave like a classical particle so the height of this peak is proportional to the current height of the electric field while this bulge is slowly being filled out due to all the stuff that has been created up from that point until you eventually end up with the static result so it all seems to be a bit simple another very brief example you can do consider a potential barrier and for starters let's just consider a potential barrier a potential barrier would be less than 2 n the time paradox would certainly predict there's no particle creation here and in the conventional sense there certainly isn't particle creation here but what there certainly should be is vacuum polarisation around this barrier here and vacuum polarisation around this barrier

25:00 this point here. We would expect that this sudden, strong spike-delegate field that's on this slide would cause the vacuum fluctuations to be distorted in some way, given vacuum polarization, as is described by the Great Goodson-Fond and Field theory. if you've got finite-point number operator with controllable fluctuations, then you can actually say, we're the there's an R in this case and you get a little bit of antiparticles concentrated just inside the line a little bit of particles concentrated just outside of the line and I should mention that if you measure the fluctuations in those measurements the fluctuations are always bigger than the measurements to be best into it so it's you can understand why they don't call it particle creation and why they do call it vacuum polarisation it is just a slight distortion in the vacuum fluctuations and if you turn the potential up above 2m then as you might expect what happened antiparticles start appearing inside the well while the particles start getting created and being spat out and there's an alphabet I would be sorry you can really enter taking the intuition back to the uniform I already told you there was a thermal spectrum created. We can say where those particles are created and roughly speak, but that's the total particle distribution there and within a certain frequency. You can say that's the spatial distribution of particles with a certain frequency. The variable on the bottom is radar distance. It's not this distance which goes to zero here. Radar distance goes to negative infinity here. And you can see this particle creation is effectively rather than the mouse pack being higher or lower, which is why particles do the same as the antiparticles rather than the opposite. Okay, Radar time provides an observant in a collation of spacetime. It is independent of the choice of coordinates, agreed to prop time on the trajectory, a single value in the quadriple, etc. You can use it to define an observant definition of particle-antiparticle modes, which it applies

27:30 to an arbitrary observer in any of its mechanical gravitational backgrounds. It's consistent with detector models, but allows you to say what it is that the test is detect. It's independent of the choice of coordinates or gauge. It generalises given this definition to non-stationary spacetimes. It's not local on small scales, but effectively local on scales like in the complement. It allows finite times to be considered and allows you to say with definable precision are created. And a slightly bold final claim, which really just relies on the fact that we have right our time at our disposal, is that there needn't be any inconsistency between the foliation dependence of quantum mechanics and the coordinate covariance of general relativity, provided we correctly take into account the role, the important role played in both of those theories by the observed. Thank you. Antony. Can I ask about the definition of radar time? What happens, say the Friesen model, in the very early universe, near the singularity? You've only got a tiny vision where you can have this signal going backwards. If there is an initial singularity. I've got the overview here, but I told you that. because I think I was going to run over time anyway. Again, you've got a hyperservice of simultaneity that doesn't go all the way across, but you do have a slightly different boundary condition at the edge. In the case of, I don't know what I said, where your radar time doesn't cover the whole space-time, your k vector goes to zero at the edge, which applies a certain boundary condition so that your eigenvalue problem is still well-defined. But it's going to cover just as the causal horizon. Yeah, it's going to tell you that according to that observer, But that's all that an observer who was there could possibly discuss. And that doesn't cause any problem to this, using it to define particle mode? No, obviously there's an interesting question which I have not addressed at all, and which cosmologists have to address in whichever fashion they can, which is what state you would expect the universe to be in. And that question has nothing to do with which observer is trying to look at it. In the case of dissatisfied space, they generally use quite a state's vacuum, or they use the conformal vacuum, or whatever. I certainly have not said anything about what state you should have expected the universe

30:00 to be in. But given a state, and given an observer, I can tell you what the observer sees in the state. And obviously, given... Simon. Simon. Simon. Simon. I mean, I think the obvious question is, given the observer dependence, or world-land What, if any, conditions of consistency would you require? Is it okay for two entirely different worldlines to disagree systematically? Obviously in the flat space we had the rather convenient fact that all inertial observers agree with all other inertial observers. And if we want to... Well, first of all, let me point out the number operator doesn't appear in the governing equation. If it did, this would be a complete disaster. the energy momentum tensor if you were going to try to find something to put on the right hand side of some sort of semi-classical back reaction problem then you'd better make sure that the things that appear in the equation don't become which observer put them in but that's not the case with the number algorithms, there's nothing wrong with that when you observe an image but if you wanted to make some systematic definition of what you thought the energy momentum tensor was in an independent fashion and to link that to what particles are there that, then obviously you need some sort of consistency to it. And it would be nice if we could show that to at least within the uncertainty that is inherent in measures of fluctuations and things like that, free-pauling service through the same event would hopefully agree on the particle content after that event. If that were true, you could to a certain extent say this is the piece of the creation that's due to the background and this is the It's just due to the observer. It's one where that remains a conjecture and one that I'm certainly interested in trying to show you, but it's not a straightforward line. Okay. Next speaker from the philosophy faculty in Oxford, Oliver Pooley, The Philosophical Significance of a Heritage Violation. The floor is yours. okay well first I'd like there's another local person who had nothing to do with the organization of this week to add my thanks to everyone else's thank Peter and Harvey for organizing it

32:30 Okay, while preparing this talk, I was slightly embarrassed about the title, because it started to seem a little bit grandiose. I mean, it would be much better to call it The Significance of Parity Violation for Some Questions in Philosophy, or something like that. So you're definitely about to experience a change of gear down this. paper is sort of talking about some similar things okay so and these these four papers that I mentioned here are ones which sort of got me interested in this topic but references to those here and there okay so there are two questions in particular that I want to raise and say something are the philosophical questions that you might ask concerning parity violation. First of all, is it an example of this sort of thing, that the failure of a space-time symmetry implies the existence of some sort of absolute symbol? And the obvious analogy I have in mind there is Newtonian mechanics, so I'll just say something about that in a second. The second question is, does parity violation show that being of a particular hand for this intrinsic property of the handed object. Now that might seem a peculiar question to ask. The point is that the account according to which handedness is not an intrinsic property is by far the most attractive one philosophically, and it may seem the obvious one to all of you, in which case you'll be able to close your eyes for a bit. But some people, well, you can set things up in such a way that the fact that nature distinguishes between left and right suggests that this perhaps is untenable. I want to answer that it isn't, so that's a question I'll be looking at. And then, finally, in answering these two, I want to just make some very vague remarks about the link between explanation, ontology, and things like that. Excuse me, do you mean you will say no to question two? I would say yes but I would say parity violation and now I will say no

35:00 right so that's the very general thing just to set the scene make sure we're all talking about the same thing Okay, so the parent transformation is just spatial reflection through the origin and in spaces with a non-number of spatial dimensions this maps objects at one hand in a somnative air encumbering counterparts so for example left hands get mapped to right hands and the next question, well what does it mean physically for this transformation not to be a symmetry of physics One claim, in fact, while I was preparing this, I realised that this was falsified by the example that Nick Hubbard discusses. But if you're considering realistic theories, this is going to have to mean, if it's not a symmetry, that for some sort of handed processes, the probability of a process of one handedness differs from the probability of the process of the other handedness. If the probabilities were the same, in the deterministic case, if they were either always possible, always happened or always didn't happen, then parity would just be mapping one physical state of affairs to a physical state of affairs. So, for it to map physical state of affairs to something which is unphysical, this has to be the case. Now, a different way of putting things is that the laws do not take the same form in coordinate systems related by parity. And this is meant to make you think of Einstein's statement as a relativity principle, where the fact that you do have the symmetry means that the laws do take the same form in coordinate systems moving with uniform velocity relative to each other. But it's not a symmetry means that the laws written with respect to a left-handed coordinate system are just going to be different in forms than the ones when written with respect to a right-handed coordinate system. And we'll see more precisely that the differences are rather trivial, but we'll see what those are later. Perhaps we should forget three, because I need to speed up at four. Well, this is couched in terms of the form of the laws with respect to coordinate systems. In terms of quantum mechanics, that might seem a bit unsatisfactory.

37:30 Perhaps what I should be saying is that the representation of parity on the Hilbert space sends physical states into unphysical states or something like that, but I'm not really sure about that, so if anyone has any comments of that sort, Okay, so now turning to the first question I mentioned, the analogy, or the example that makes us ask the question is Newtonian mechanics, and what we say about absolute space, the reality of inertial structure and that sort of thing. And a sort of standard line is that the fact that the laws of classical mechanics are invariant under boosts, in addition to being invariant under time-independent translations and rotations, means that there is no absolute standard of rest, and therefore Newton's concept of absolute space in a preferred frame is undermined. However, that the laws are not invariant under time-dependent rotations and arbitrary time-dependent translations indicates that there are dynamically preferred frames of matching up the points of space at different times that are independent of matter and its relative motion. In particular, we have preferred coordinate systems to inertial frames and if we do write the physics in a coordinate general way, the inertial structure represented by any fine connection becomes explicit and this is naturally given a realistic interpretation. I mean, obviously that's a contentious claim, and I personally would disagree with it. But I think that the reason where you have room to disagree with it is the fact that the structures in question are non-dynamical. And so if you consider the case of general relativity, it is generally covariant, but that's because all the structured fields are in there already, and they're also dynamical, And it's the fact that they're dynamical, which means that you don't have any preferred coordinates. And that's just, you know, my soundbite view of the general covariance of GR, that it in no way supports the sort of anti-realism about space-time structure. Okay, so do we, can we say similar things in the case of parity?

40:00 Well, are there preferred coordinate systems? The answer, I mean, I want to say yes and no. It's true that the two otherwise equivalent coordinate systems, the laws, will look different when written in the right-handed, with respect to a right-handed coordinate system than when written with respect to a left-handed coordinate system, or there is the form of the laws for which that will be true. But there's no sense in which one or the other is preferred. And that's different from the case of inertial frames, where obviously the inertial frames are preferred, the laws take a simple form in those. So, you know, there's a disanalogy of the Newtonian case. And is there an associated space-time structure here? Well, yes. I mean, if you want to write the physics coordinate independent way then you're going to have to introduce some orientation fields such that the physics does take the same form in both right-hand and left-hand of the coordinate systems and Hagrid for example suggests that we should take that structured field seriously as representing something real in the world so this is his paraphrasing Erwin. Erwin was talking about de-afine connection and inertial structure Huggitt says I mean Huggitt's actually worrying about the implications of this for the reality of space time an intermediate step is to take the orientation field seriously as a real physical field so I want to come back to the question of whether doing these things taking that line is a good thing to do when there are these differences with the inertial case. Okay, so I'll be coming back to those questions, but now it's the question of handedness. Right. Okay, so consider an object lacking any plane of mirror symmetry and its mirror image. two such objects are incongruent in the following sense you can't, through a continuous transformation that doesn't alter the metrical relations internal to the object map one of them onto the other and the sort of nice standard example that gets trotted out to illustrate that is that you can't fit a left-handed club on a right hand or a left hand in a right club even

42:30 but they are counterparts which is just to say that they're identical in terms of their geometrical relations. Something that I realise is blindingly obvious, but as I have a picture, I should put it up. So, left and right hand, but you can put the points upon one correspondence so that the distances and angles between, you know, the corresponding points of the stone. Anyway, so why is this interesting? Well, it may be a sort of historical accident, But Kant certainly thought it was interesting because he believed that it refuted Leibniz's reductive account of space. The point being that if all spatial facts are supposed to supervise on facts about the relative distances between bodies, and if these facts are the same for left and right hands, how is the relationist supposed to be able to, one, explain the incongruence of left and right hands, and two, in fact Kant eventually said that that's something that you can't understand or explain, in perception. And secondly, how can the relationist explain what it is to be the one-handedness rather than the other? That was the question that I raised at the beginning. And Kant said, well, consider the possibility where the first creative thing is a hand. Surely it's either a left or a right hand, but according to relationism it can't be either. And he thought that was a patent of absurdity. Of course it's not either a left or a right hand, but we should to reject the premise, but that was Kant's line. Okay, so with that sort of historical origins of philosophers' interest in these questions, I mean, I would like to put it like this, that more generally there are two basic options. You could either have some story according to which incongruent counterparts do different intrinsically, being left-handed to become intrinsic property of left hands. And, I mean, okay, that already explains what it is to be of a particular handedness once you've said what such intrinsic differences are, or intrinsic properties are. But then you go on to exploit such properties in explaining the incongruence. So either you adopt that position, or you say that left and right hands are intrinsically identical, and then you've got to explain not only the incongruence, but then you have to explain or have some story about what it is to be of one handedness rather than the other, which doesn't really arise there. Now, why this is the obviously preferred option, I would say, is that there are these various problems with the first.

45:00 One and two are related, and the basic point is, for example, that if you, I mean, take Fs, which is the standard example, these are incumbent counterparts in two dimensions, but if you can rotate the F and the plane, you can map the left-handed version onto the right-handed version and similarly for all handed objects and also if you're in a non-orientable space then you can sort of continuously move one round into the other now John Ehrman sort of responded to this by saying well actually this isn't so so someone who holds this account at least has to say something like well, whether or not these intrinsic properties can be instantiated depends on the dimensionality of space or whether these intrinsic properties are always instantiated but whether or not they ground in congruence depends on the dimensionality of space I mean, that's clearly the logical room to say that but of course, what that amounts to is entirely mystical now Irwin says, well, it's not at all mystical that certain properties can depend on the dimensionality of space the fact that a set of vectors is maximal in one space it won't be in a high dimensional space but of course the disanalogy is we can completely understand why it is that whether or not a set of vectors is maximal and depends on the dimensionality of space but we haven't even been told what these intrinsic properties are to be able to evaluate the claim that their possession or non-possession depends on the dimensionality of space and also I mean there is no such account but you would have to ask well how does that fit in with our ability to recognise that as that, and that sort of thing. So, clearly, this is, you know, this is beginning to seem the obvious account of what it is to be handed in this. So, a brief summary of the account. You say incongruent counterparts are intrinsically identical, and then you say, contra-count, that the incompetence can be understood in terms of the spatial relations that hold between them. So, for example, the obvious point to make is that, let's just consider Euclidean distances. If the distances between the hands are constrained to remain Euclidean, you can understand that it's a sort of mathematical theorem that for given sets of distances

47:30 that are instantiated between the left hand and the right hand, it's impossible to alter them continuously, keep them Euclidean, keep the internal distances of the hand unaltered, and yet send all the distances between the parts of each hand to zero. So that's the relation's basic explanation of incongruence. Of course, he must have a different story about, well, what constrains the distances to be Euclidean or whatever, but that's just a different question. And then, given incongruence comes first, and then you explain what it is to be left and right in terms of incongruence, the relation of congruent for a particular sort of object partitions a set of a particular sort into two and then left and right are just like arbitrary labels and we recognise a particular hand is left by recognising that it's congruent to one that we've learned by abstention to call left so there's nothing intrinsic to being left handed that's all to do with being congruent to the hands that other people call left and this is very consonant without There's a sort of thought experiment, how would you explain to an alien galaxy what we meant by left, if you couldn't point to some asymmetrical object in common, putting a side parity violation, which would enable you to do it. And so hands, lone hands, it would be either left or right. And this collection of people have discussed such an account, and basically these are on the side of people who speak the truth. Okay, so then two questions and now I'm going to review rather quickly just some facts about how we actually some details of our parity violating theory, the electroweak theory you've got this four component object Dirac spinner which, I mean considering master's fields, the Dirac equation and the gradient is invariant under parity, which turns out to send psi to this. So you're not only, x doesn't only get sent to minus x, but you pick up this gamma nought factor. Now, sometimes you see this justified in terms of keeping this invariant, but it seems to be much more preferable to actually work the other way around. And actually moving on to the next slide, if you start with your expansion

50:00 of the free field in flat space, and interpret these as annihilation and creation operators for different sorts of particle, and then you say parity should send a particle of spin lambda momentum P to the particle of spin lambda momentum minus P and work out how these have to transform, that's where you get the gamma naut in front of there. And then it turns out that the equation is going to be a grand-handed probability. That seems to be a conceptually preferable route to go. But OK, so that's the parity-symmetric general situation, but now you need to find these. You split the field into these two bits, the left-handed and right-handed bit where you've got this 1 minus gamma 5 or 1 plus gamma 5 projector, where gamma 5 is that. And these, I mean, to just motivate the left and right, parity now sends the left one into the right one, roughly, and if you consider this, it turns out that in the master's case, these are states of, the left corresponds to a to Felicity and right to positive Felicity. So, I'm drawing a picture that I know some of you will object to, because these are not classical objects. You've got a spin like that, and the direction of motion like that. And the thought is that these are intrinsically identical, right? There's no intrinsic description that picks out a right-handed one rather than a left-handed one. It's only a convention particular vector with a certain direction which we have to get a hold of by saying what we mean by clockwise or something like that, that allows us to say, allows us to pick one of these out in terms of its relations to us. So the basic claim I'm making is that these chiral components, although more heterical hands, are pretty much the same. They're intrinsically the identical, certainly in the non-interacting case, and this is just choosing a particular representation you've got. I mean, this is just what we had before, but now choosing a certain representation of gamma matrices. So you can see now gamma-naught is sending

52:30 the file now corresponds to the top two components, the file, the bottom two components in the whole spinner, and gamma-naught is just interchanging them. And in this representation, you can see explicitly that these two spinners transform under incompatible, sorry, under inequivalent representations of the Lorentz group. And that, you might think, well doesn't that mean they're sort of somehow differing intrinsically. But of course, this is relating the spinner to the coordinate system. And in particular, if we switch coordinate system by parity and choose a left-handed coordinate system, then the sign of beta will change and phi r turns out to transform, as you can see, just as phi l does under the old coordinate system. So my claim is that this is all consistent with there being no intrinsic difference between the left-handed right-handed spinners. And the mathematical difference between r relates them to our coordinate systems and conventions and that sort of thing. Okay, but of course we do introduce a difference when we introduce interactions. And we take this parity-symmetric Lagrangian and now just reorder it by grouping the left-handed electron and neutrino fields together. I mean, this obviously... In fact, I have no left-handed neutrino, I think, or right-handed neutrino. If I put it in, I think it would make my story better. You'll notice it's absent at the moment. But you now, you first of all, to take this sort of freely grunge in and turn it into a gauge theory where left and right behave differently, you choose a global transformation where the left-handed field transforms differently to the right, in particular under SU2, you've got this non-trivial transformation on this SU2 doublet of the left-handed component of the neutrino and electron but the electron right-handed field is a scalar and then for the U1 group you've got different transformations of left and right and of course when you turn that into a gauge theory and introduce counteracting interaction fields to make the whole thing invariant interaction between certain fields and the left-handed fields, but no corresponding interaction

55:00 between the same interaction field and the right-handed fields. Just the same thing again but in the physical, where I've chosen the physical, the mass eigenstates of the gauge field so that you've got, you can see that the left-handed fields are coupling to the the W bosons, but no right-handed field does, and it's actually only the coupling to the photon that's parity-symmetric. Okay, so that's the basic theory, and now, at the end, we just have to ask the questions we were asking to start with. Does this theory suggest that left and right-handed particles differ intrinsically? Well, I mean, what we had before we introduced interactions was clearly the spin of the particle is an intrinsic property. I mean, it's intrinsic to the fact that we're representing it by this spinner, that it's got this property. But in terms of what's differentiating the left-handed and right-handed particles, it's just their interactions. And in the theory, those interactions aren't explained or grounded in any particular intrinsic properties. Unless you're suggesting something like, you know, the hypercharge, the value of hypercharge in a particular particle has its categorical ground or is interacting with the field in the strength that it does. So that could be one thing you would say, that left and right-handed fields do differ intrinsically, but it's not, it doesn't mean that being left-handed or in particular being a left-handed field is an intrinsic matter. It's just a brute law-like fact that all left-handed fields have this property, and it's because they have this property that they interact in the way that they do. So the claim is that the law gives us no reason to suppose that handiness is an intrinsic matter. Although we don't actually have to explain it in terms of attributing intrinsic properties to left and right-handed fields, we can introduce this orientation field. So, okay, the fact that the way we distinguish left and right really related the spinner, the different components of the spinner to our coordinate system means that if we wanted to coordinate pre-expression, we would have to introduce an orientation field explicitly. basically, you don't, I mean, you could actually make sense of the theory without taking that field seriously or suggesting that there are these intrinsic properties. You could just

57:30 say that the basic fact is that particles that interact in a certain way are handled in the same way. It's not grounded in anything. It's just a law-like fact. The problem with taking that line is that this is a highly non-local fact, right? You've got particles here and over the other side of the universe coupling to W bosons, say an electron and it so turns out that they're handed in the same way and no matter where you go in the universe you won't find a right handed electron coupling to a W boson so at least if you take the orientation field seriously you've got some sort of level explanation of this correlation I mean it's not just a monocality you've been having determinism at the very first occasion but there are equally problems regarding this field as physical I mean is it really giving an explanation of why only left handed rather than right handed fields are interacting in this way or is it just encoding the fact that they do and I think this is related to the fact well you might think that this is related to the fact that the field is non-dynamical but actually I'm not sure it is because in GR you've got a dynamical metric which we want to regard as a real field clocks and behaviors they do in any course of growth. Okay, so maybe I'll stop there. Come on, please. And I just wonder whether, at perhaps a lot more level, it seems to me that there are two problems. Maybe this is what you're saying. There are two things that happen. There is the appearance of objects of different parities, such as our hands, which, unless I'm not naive, theoretical business point of view, I describe by saying that you get equal numbers of rights and lefts because they're degenerate states in some kind of Freudian, which does not itself, but it's in it. And then there is the problem, or then there is the question as to whether some kind of Freudians create symmetry

1:00:00 and create particles which are preferentially not for what they are. Do you have problems with, do you have, I was going to say problems with both these problems, or do you address both questions, on just the second one. Well, I certainly, I'm not sure what the, I think, I think the, I think, I think what my position is, is that the, at first glance, the first problem is an interesting problem. There's a very attractive philosophical story to be told about it. And it then comes a question of whether that story can be upheld in whether that's compatible with explaining the second state. The question is, could a classical, at least one question, a classical universe, where the laws were not paradigm-breaking, reduce paradigm-breaking objects, a function of the universe can. Right. Could a classical universe ever reduce paradigm-breaking objects if it wasn't intriguing to the physical laws? I mean, most solutions involve asymmetric Does that mean the difference would be another universe might have particles oppositely oriented with respect to the orientation? Exactly, yeah. And it seems that if you're taking the film seriously or, you know, you think there's some sort of asymmetric structure of space-time itself, which, you know, it's and it's the particle field's relation to that which explains the ability to interact one way or the other. It seems that you're always going to have the option of a world where things with exactly the opposite relation behave in the way that the opposite orientation behave in the way that particles do in the actual world.

1:02:30 So if you had, say, an empty space-time with this orientation field in it and then meaning for the distinct worlds, but they could, they have to be distinct, do we start the distinct from where we start saying, well, in one world, there are particles in this way in perspective, from that world that may, even though we can't protect that. Yeah, I mean, at the observable level, the matter fields in the two models are going to be identical, but the only difference is the sort of congruence of the matter fields to the orientation field, which we're treating as a real field, and it seems that you can't just sort of treat the field as real, and at the same time, these two are different that was the last question thank you all our next speaker is from Alexander Atria altering the remote past we're all hoping you're going to do something radical A couple of words of a misleading title to begin with, by remote I mean either temporally It's patiently remote. Anyway, I'm... Basically, it's a little more than a semantic trick. My approach here is to... Is to have... I have a very abstract approach to the... both to the Bell formalism and to the relevant bits of quantum mechanics. So I begin without any, I don't have any reference to a particular ontology at first. So we're used to ontology, sort of Bell inequality ontology, referring to proton pairs, angles, directions, and so on, polarizers, or spin meters, or something along those lines.

1:05:00 So I begin with this abstract scheme of, as I say, developing quantum mechanics and the Bell formalism. And in that abstract context, I define a, you might want to call it a locality condition, but I'm not sure that's exactly the best term. I use the term parameter independence, which is slightly misleading, because the term, of course, already exists, and it has a very well-defined meaning, which isn't exactly the one that I have in mind, which is slightly different, the same idea. And anyway, depending on the particular semantics one adopts, the violation of that parameter independence can take on various different meanings. And, of course, But, in the case we're familiar with, in which the semantics is both angles and so on, the violation of the locality condition means, well, the locality condition itself, this parameter independence, means that says that the rotation of an apparatus, on, say, the right side, can't change the value on the other side, on the left side. So the violation suggests that it can. That's in one semantics. In another semantics, which I would consider, in which the parameters in question will be times over angles, well, it's no longer a matter of rotating an apparatus on the right. So it's no longer a matter of choosing between two times on the right, between two directions on the right. It's a matter of choosing between making a measurement at one time rather than another time. And so if that kind of parameter independence is violated, it means that by weighting and making a given measurement, exactly the same measurement, at one time rather than another, the value on the other side is changed.

1:07:30 And as far as I know, well, put it this way, in the framework I consider, there don't seem to be any particularly significant restrictions on the time ordering. So the two times at issue on what I call the right could lie in the absolute past of the time in which the value is considered on the left, or in the absolute future, or they could be space-like separated, or... So that's why I called it, I called the talk of the remote past. By the way, what time did you get exactly? You have 20 minutes from now. Including... Well, usual framework source produces these pairs. This is a slight difference. So I'm assuming that each object, I guess this thing is in my phone, has these dichotomous properties, which depends on two parameters which are called M and N. So the S is just the subsystem, s is 1 or 2, k is the parent, k is 1 to n. And so these are the, what I'm calling the parameters, and the independence will refer to the second one. Anyway, so until I've defined that independence, this quantity would in principle depend, aside from what It will depend on both the subscripts. Anyway, having defined that, for the kth pair... By the way, quantum mechanics has yet to end of the picture. These are objects of any kind. They could be just about anything. So I defined this quantity for the kth pair.

1:10:00 And until I've said anything about the dependence of the second index, It could be as large as 4 or as small as 4 or 4 minus 4. And when I've written it, the second index of the first factor of every term is the same as the first index of the second factor. Anyway, now I introduce the assumption. I say that there's no dependence. I don't justify it physically. I say that there's no dependence on the second index. So once I've done that, I can rewrite the same expression in this form. This is something one finds in, well, without the second index, but Mermin and Redhead's book, Celery has something similar, there's nothing terrible for you about this. The idea of doing Belgian parties with times rather than angles, let me see, Tony Leggett gave a talk in Utrecht about a year ago, I suppose, in which he did Belgian parties with times, but that just referred to a single system. Girardi, Grassi, and Debra worked on it six or seven years ago maybe. There was another talent, Tommaso Calarco, he was in Intervalle at the time, I think about seven years ago, who did something similar to Tony Leggett. and of course the salary as well anyway so once this expression is rewritten in this form we can have the bound of the modulus from 4 to Well, then I just sum over k, y by n, and that average, the modulus of that average can exceed 2. And that's why my bell inequality, which I've written in this form, where these correlation

1:12:30 functions have that form. So, again, so far, nothing to do with quantum mechanics. Now, this is the relevant part of quantum mechanical formalism, which I will use. There must be a better term for these objects. I call them unitary self-adjoined zero-trace operators acting on C squared. There's probably a word for them, but... Well, as I say, I call them generalized Pauli operators, because, of course, the Pauli operators of matrices refer to three particular directions, and these are just generalizations. Maybe sigma dot n, maybe that's the thing called. Anyway, they're characterized by a pair of angles, well, we just need one. So, this is the thing that I'm referring to, where this and this are orthogonal and normal. Anyway, in C squared times C squared, I define this average value, where the B is this linear combination of products of things of this form, where the superscript, of course, represents the subsystem, and this vector, sigma, is the vector in the average value. And, well, for instance, with this choice of angles, this will reach a maximum of 2 root 2. Anyway, so how am I going to replace angles by times? Well, take the angular version of the parameter independence. It says, as I said before, by rotating an apparatus on the right, you can't change the value on the left. Well, the rotation of the apparatus involves an initial angle and a final angle. To both angles will correspond one of these generalized powering things. And so physically that's what's happening. Mathematically, corresponding rotation is carried out by a unitary operator.

1:15:00 So, u times sigma sub alpha, and then the conjugate of u, usually if you're not written. So, in fact, for any angular difference, there will exist a unitary operator which will get you from the initial angle to the final angle. So that rotation, which figures in the primary defense condition, is represented by a unitary operator, and there will be a cross-bonding group of unitary operators. Well, suppose that group happens to be the time evolution group. So, well, I've written spectral representation, spectral expansion of the unitary operator, which rotates this angle difference. Suppose these happen to be the eigenvectors of a time-dependent Hamiltonian, where the important thing to get the right kind of time evolution, or to get a time evolution, because usually one considers a triple time evolution. The important thing is that these e's be different. So basically I'm, well anyway, I end up taking an operator like this on the left, another one on the right, so it's a product evolution, and the time evolution generates beats, oscillations like two tuning forks. You get beats whose frequency is equal to the difference of the frequencies of the two tuning forks. And the beats, of course, in this case, are in configuration space. And so the effect in question is a measure of those beats. The violation of belting ecology will be an indication of, will be an expression of those beats. So this is basically the same operator I had on a previous page, just with times as a subscript. and well I've rewritten it in terms of this is the average value which has a maximum 2 and 2. This is all the same as I had on the previous slide. So the correlation functions, these average values, assume that form. That's again the

1:17:30 the difference between the energy eigenvalues, and this is the time difference, of course. So those are times that will give you a maximum violation. Now, again, first I didn't make any reference for quantum mechanics. This now is the relevant part of the quantum mechanical formalism. And here, I compress a number of very delicate issues into a paragraph. Well, these are not really the issues that I want to talk about. I know I'm sort of vulnerable, all kinds of objections, but now I try to draw a connection between the two things I've been talking about. And I assume that the pairs that I was talking about in the first part are accurately described by this vector, which has been at issue in the second part. Furthermore, that these generalized boundary operators faithfully reveal the corresponding properties that I was talking about, plus or minus bondings, that I was talking about in the first part. So if I make all those assumptions, then the observables represented by B or B tilde, so the tilde just means the time version of B, would violate the parameter independence that I mentioned. Well, anyway, so for the past while we just had one substitute rather than two, but now since primary independence is again at issue, it'll be best to reintroduce, well, at least the possibility of having a second subscript. So occasionally the second subscript will reappear. Where there is, well, notation becomes a bit of a problem at this point, but where I do write just one subscript, that doesn't necessarily mean that there's no dependence, because again, the second subscript, remember, is always, the second subscript of a first factor is always the same as a plus-substant for the second factor.

1:20:00 Now then. So for the modulus of this to exceed 2, the modulus of... there will be at least one term the expression, so one pair, possibly more, which does not, which is not equal to 2 or minus 2. So, suppose the k0 pair is to blame, there must be at least a time h such that there's a dependence on the second parameter. So, where j is not equal to j prime, where these are simply times. Suppose that, in fact, we're talking about, so these are what, the last two terms. So, I'm supposing that this will depend on whether we've got u or u prime. Well, if you think about the meanings that we've assigned as various symbols, it looks very strange. I mean, how can that possibly be, given the semantics in question? And I agree that there's something extremely strange going on, and I'm not really sure how to interpret it. But just for the sake of discussion, I'm just exploring the implications of this temporal semantics. Again, it's not entirely telling me what they are exactly, but let's try to explore the implications. And suppose that, for the time being, that quantum mechanics is not at fault, and that the assumption of realism remains valid, and sort of play tricks with time. So we have to wonder how it is that, so this is the problem if I understand, how it is this is equal to say minus 1 and this is equal to plus 1. So, well, one possibility, a pretty absurd one, is, but I think the alternative is just

1:22:30 as absurd, so don't be too surprised that I'm mentioning this possibility, is that it's just a kind of conceptual or notational fact, that it's just by considering this symbol beside the first one somehow that influences the first value so it's just by association that this is equal to whatever it was, say plus one and this is equal to minus one but how can that be? I mean notation doesn't doesn't have that effect there must be something more physical there surely just on a page doesn't change, or just the fact that one expresses more of an interest in one angle, one time rather than another, in one term of expression, surely that can't change the value. So it must be something more concrete, it must be like measurement. But measurement, of course, was assumed to do no more than faithfully reveal the underlying value. Of course, that's an assumption of the question, but here I'm not. So, measurement refers to, so maybe those two terms in which the second factor is different. So they refer to two different experimental situations, in which, on the one hand, so So on the left, suppose on the left the measurement is made at time t, and on the right, in both cases, and on the right it's made at t' on one hand and t' on the other hand. So whether the measurement on the right is made at t' or t' if the measurement does actually have an effect, it seems to affect the outcome on the left. But first of all, there are a number of problems with that possibility. For one thing, in the case in which the parameter was an angle, we did something far more spectacular and physically obvious. We actually turned it up right. In this case, we just wait. We do exactly the same thing at one time rather than at another. So why is it that that should have an effect on the value on the other side?

1:25:00 And furthermore, what value is the other side supposed to have before the measurement is made? So if there is this dependence on measurement itself, and not on something a bit more obscure, which I don't really know how to characterize, well then what kind of value is it to be assigned before the measurement? Anyway, there are a number of problems, I think, almost as many, associated with seconds or measurement interpretation I've discussed. Anyway, to make the effect more spectacular, the values can, of course, be linked to larger circumstances. So, they can be linked, for instance, to railway tracks or something. And so, in fact, I'm saying that, well, at 12 o'clock, January 1st, 2000, two trains pass each other without incident at the central station of Tokyo, if we make a measurement today at 5 o'clock in Oxford. Suppose we wait an hour and do exactly the same thing we would have done at 5 o'clock, but at 6 o'clock instead. Well, in that case, same time, so 12 o'clock, January 1st, 2000, in Tokyo, the trains collided instead. I think I've... Well, that sounds like a spectacular ending of your book. Do you want to end here? If I run out of time. You have a few minutes. Ah. Well, should I give a few more minutes for questions, or should I go on? I'll give a few more minutes for questions. at the very beginning of Bell analysis one considers anti-correlated or correlated pairs the one word you didn't use in the whole story was correlation you didn't tell us that these two D's whatever they are, are correlated or uncorrelated and how they're correlated did I have to? if you're doing Bell analysis I would think so I don't think I have to. You don't think correlation plays any role? That's not what I said. I said that I don't think I have to derive that inequality.

1:27:30 All I need is the fact that they have values of plus and minus one, so that kth term in the average, well, in what becomes an average later, will be equal, once I've made the assumption, so before I make the assumption, is equal to, it is bounded by plus or minus four. Once I've made the assumption of parameter independence, it's equal to plus or minus two. Regardless of correlations. Perhaps, in that case, you're just playing with a fact that dichotomic arithmetic trivially and tautologically satisfies bell inequalities and has no significance. Oh, no, no, no. But then I connect polymechanics to the arithmetic than a fest clock. But then you're doing something that's entirely irrelevant to Bell analysis because Bell's point was by assuming, by trying to reinstate locality and reality into a greater theory that he could then do two things. He could average it down back to quantum mechanics by halve reality and locality in the figure theory. his whole point was that locality and reality impose certain limits on this bigger theory. And his question was, what limits do they impose? And those limits are supposed to be a Bell inequality. So everything from the start of the derivation to the end of the derivation of a Bell inequality is supposed to be in a classical realm. And then you take the results and go to the lab and see whether nature satisfies that or satisfies quantum mechanics, which runs entirely independent of any Bell analysis. Ah yes, but I haven't really talked about nature itself. I've talked about the relationship between the quantum mechanical formalism on one hand, and my arithmetic for the first part. I think of this question. Thomas. My remark goes a little bit in the same sense that if you take this experiment with trains and collisions... Sorry, if I... If you take this experiment with trains and collisions in Tokyo, to establish a correlation that will violate causality, then you need quantum mechanics, you need quantum correlations, and if you do this experiment every day during one month that you make the statistics, in average there will be no correlations. So there is weak locality for microscopic observables. Sorry, there will be no correlations? There will be no correlations. If you make an average, the correlation disappears.

1:30:00 Between the collision in Tokyo and the fact that every day you choose to make an experiment at 5 o'clock or 6 o'clock, If you do it many times, then there is no correlation anymore. Because the train is macroscopic? Yes, and because it is weak locality that will prevail. Well, it could be, but... No, I agree that it's very difficult to take advantage of this sort of thing in any controlled way. But I think in a single instance, there does seem to be this dependence on the... Well, I don't mean that it does seem to be. If one makes a number of assumptions, one seems to be able to conclude that the assumption of parameter independence is violated. And I think that directly leads to the train scenario. Anthony, and then Jeremy, and then... I'd just like to ask, in the usual EPR-type thing, where you have space-length separated events, you conclude that there has to be some sort of nonlocality, but it can be just one way. It can be just one in one direction. If I have this non-local influence, it can just be in one direction. So presumably in an analogous situation where the events are time-like separated, this influence can be in just one direction. It could just be forward Yeah. Now, I singled out the time choice as being the cause rather than the effect, by an appeal, for instance, to free will, or to a random number generator, which uses something that can be shielded in an appropriate way from the effect of the other measurement, which could lie in the past. Now, of course, collapse represents a problem, and there are a number of ways of dealing with that, but I can introduce an arrow by infield to things like that, maybe if you will. Does that make any sense?

1:32:30 Well, I think I was very confused, and now I want to make a comment. And I congratulate you on the imaginative construction, but I'm going to make a comment that sounds a bit deflationary. The only way I can avoid being confused about, my goodness, could the passage of time be a cause of a distant event, and so on, or marks on a paper or consideration, is to hold on tight in my mind to your equation that for a single wing, a single sigma matrix, said that sigma T dashed is a kind of Heisenberg evolute of sigma T. So when I fix on that, I think, ah, so we are now concerned with the four observables, two in each wing, in which they, within a wing, they're related by this Heisenberg evolution. Sigma T dash is a certain Heisenberg evolution of Sigma T. Now when I do that, I think, well that's fine, one could consider what the quantum correlations are for a pair of observables in each wing related by this unitary transformation. And you may or may not get a Bell inequality violating quantum prediction. Depending on the choice. Depending on the choice. And you could label, helpfully, sigma with T and T dash. But that's got nothing to do with the passage of time or waiting through time being a cause, the discussion of such an experiment and its quantum predictions and its local realistic predictions would proceed with the usual requirements that there has to be a free choice of whether you measure sigma t or sigma t dashed and you don't on any single particle measure both. And instruction sets in the local realistic whatever the 3D choosing experimenters choose to measure on the particular pair. You know, the instruction set a pair carries is independent of what the two parts of it are later destined to be measured for. So, in a sense, I don't see

1:35:00 how you get the verbal argument for the train scenario, because I think you've just labelled what we normally call kind of sigma A and sigma B as sigma T and sigma T dash. And if they're just non-commuting spin components, you've got to proceed in the ordinary discussion of the experiment. You can't measure both, et cetera, et cetera, on a single component. Yes, that sounds very reasonable, but it still happens when, how does one come to dependence? well I mean one thing is you were saying we're going to get an argument for parameter dependence well a quantum person would say no there's only output of dependence well actually no but the kind of parameter dependence that I use is very different again I'm using a pretty misleading terminology that's not the standard I'm just saying that there's no dependence the assumption of parameter independence just says that there's on the first subscript and not on the second one. So I'm trying to account in some way for the dependence on the second parameter, which seems to emerge from the violation. Okay, thank you. No, thank you. Before we thank the speaker, I want to point out on the program there is sort of a brainstorm session, six o'clock, about the conference next year. Okay. Well, thank you all the time. Thank you.