Quantum Reference Frames, Imprimitivity Theorems & Uncertainty

0:00 There's no way on earth that you may feel the relief, and if I did Section 5, it would be like I backed up and stopped here. Okay. First of all, the point of view that I'll adopt here can be described in terms of these conditions.

2:30 Don't worry too much about these names. I'm not going to be repeating these names throughout the talk. I just give them labels here. More precisely, it's the idea that spatio-temporal quantities can be measured only with respect to or relative to some physically accessible frame of reference. By physically accessible, I mean the phrase physically accessible to be fairly liberal, however. In particular, a frame of reference needs to be Defined in the obvious way by the four walls, floor, and ceiling of the laboratory to be physically accessible, I would also include, although I won't discuss this view here, I would also include, for example, the frame of reference obtained via an interesting but complicated algorithm in the Barber-Bertotti theory, for example. Why is it physically accessible? Well, because it's based on, the algorithm is based on data that you would measure, locations of stars and what have you relative to this. And then you apply this sort of best matching condition in the Barbara Bertani theory to obtain a frame of reference that's not the frame of reference that can be traced back to any single body. It's a frame of reference that arises out of the relative locations of many bodies. So that would be an example of a physically accessible frame of reference. It's not given by a lab, for example. Yes, I'm sorry, I didn't mean space-time there. Distinguishing measurement relationalism from a more general relational account of space-time. Oh yeah, that's coming. So measurement relationalism I take to be just about what we can and cannot do with measurements. And the claim is that when we actually perform measurements of spatio-temporal quantities, it's always with reference to, with respect to, some physically accessible reference frame.

5:00 It doesn't, but I haven't made any claims yet about whether there is an absolute space-time. I wouldn't say we have no access to absolute space-time. I mean, it sounds a bit odd in the context of general relativity that we seem to have access to the culture of space-time. What do you do with an exit? Well, I guess I should say this then. We don't have the sort of access you would need to be able to say, I am measuring the location of a given particle with respect to absolute space. We certainly have access to other features of it. Its curvature, whether it's expanding, things like that. Sure, sure. Yeah, so that would need to be made more precise. This is the more precise statement that I attempt right here. The principle of relationalism says something somewhat stronger, namely that whatever you might believe in, whatever you might believe, sorry, let me put it this way, even if you believe in something that you would be tempted to call absolute space or absolute space-time, that thing isn't the right sort of thing relative to which you can even define spatio-technical quality. So you might believe that There's such a thing as an absolute spacetime that's, let's say, Leibnizian in character, right? In which case, there may be this thing as absolute Leibnizian spacetime, but Leibnizian spacetime isn't the sort of thing relative to which you can define spatiotemporal quantities. You still, within a Leibnizian spacetime, have to pick a frame of reference and say, position is defined with respect to that, momentum is defined with respect to some inertial frame that I've chosen. That's principle relation. So the difference between principle and measurement is that in this case there could be such a thing, but we just don't have the right sort of access to it. In this case, there isn't anything that could even in principle be used to define. What one means by a given spatial temporal quantity. So you always have to specify some reference frame when you're defining spatial temporal quantities. So this clearly is a more relationalist statement in the traditional sense of that.

7:30 There's a comment on the right. I mean, that seems to me something you could say, without mentioning reference claims or anything else. Identify the locations based on top of the curvature. Okay. I mean, is that supposed to be ruled out by this? No, I certainly wouldn't want to rule out someone being able to say... Let's say I'm at a local maximum of curvature, something like that. I mean, I may be a little bit less inclined to allow you to say that I'm at the point of maximum curvature in all of spacetime, but then I'm at some local maximum. That I could accept. But doesn't that slightly qualify what you're saying that the culture and the necessity, based on what the necessity defines? Well, I guess in the case that you're considering, I could then go on to say, I'll take the origin of my coordinate system to be this local maximum of curvature, and then define, let's say, you know, some instantaneous Cartesian coordinate system, well, plus the right, some curved coordinate system that maps all the space at this time and make my position measurements with respect to that. And I would say, fine, then what you've done here is, I mean, I've assumed now that you've, all of this assumes that you, in fact, have physical access. You can, by means of measurements, experiments, determine that you are at a local point of maximum courage, at the point of local maximum courage. In which case, it's physically accessible. That's what's wrong with endorsing this viewpoint. Well, I do, I do, I would want to say that. You would? Yeah. I would. That's what I thought. It sort of gets you out of all of this.

10:00 It gets you into this point. Well, I think I can, I think I can accept what Michael was saying and still endorse this view here. It says, spatial number points are necessarily defined in terms of a physically accessible frame of reference. Yeah. And, uh... The term point of this is silly, but I'm going to get to it. You said, the point of my knowledge. And then you use that to construct a frame of reference. It's just going slightly the wrong way around the sequence. For example, there's a senator in class, I don't know if it's something that came off, but maybe you've identified a position with a maximum. You use that to construct a frame of reference, isn't it? Right. There's a senator in terms of the frame of reference. I mean, I guess... I guess all I mean to be saying in this statement here is that it is something like an endorsement of traditional relationalism that says that there may certainly be ways of picking out points in space-time, points of local maximum, whatever, right, but those things, those things, so suppose I picked out some point in this way. All right, even in this case, if I'm not claiming to have access, if I'm not claiming to be able to specify which point it is, but I just want to say, let there be, you know, let's consider the point of maximum curvature in all of space-time, you certainly then could go on to say, And that thing defines an origin of coordinate system or something like that. But the claim here is, and there's no, I don't think there's any issue, I don't think there's any problem or conflict between allowing myself to say that and this statement. Because, well, as long as I'm also going to say the following, there's nothing about being a point of maximum curvature that makes you the origin of the coordinate system relative to which you measure positions. I could just as well and just as legitimately have picked out some other point. That's all I'm saying. There's not some absolutely given origin with respect to which position measurements are made, for example. It doesn't mean that I can't pick out such origins in lots of different ways and the ways that make reference to something I think is independent, exists independently of me and stuff like that.

12:30 Well, this seems to bring in a little bit of conventionality in the discussion, that a reference frame involves conventional choices. Yes, that's what this is saying. And what you're saying there is that you have to make a choice in order to define the responses. Yes, that's exactly what I'm saying. That's exactly what I want to bring up in a moment about the inertiality of the reference. Now, this is a view, this is the second one that I'm actually willing to endorse, which is why I was willing to defend it for a moment. I'm not actually willing to endorse this last view, nor do I think it's, I think it's probably also not essential to my discussion, but I also don't know how to excise it from the discussion, so I'm going to endorse it for the sake of this talk, and we can discuss amongst ourselves if you wish how one might get rid of this assumption from the talk, but it's going to be there. And the assumption is that... Spatiotemporal quantities are necessarily defined in terms of a given physical body that is stipulated by convention. So this is the condition here that rules out the kind of procedure that you were mentioning a moment ago by defining my spatial temporal quantities in terms of, let's say, a point of maximum curvature in space-time, because that's not a physical body. And for reasons that maybe have already emerged from our discussion, I don't necessarily endorse this view, because I certainly do think you could... Say, let's consider the point of maximum curvature. Let that be the origin of position measures. But the question is based on this kind of, you know, one of the more existing ones, you know, relativity. We have one body, but the thing is sort of moving about with the same gravitation. Right, right, right. What would you say about that sort of thing? Well, in the context of this talk, I have to say that there's no sense in talking about position room. Position room, for example. Location of things. Am I willing to say that to court? Not necessarily. But for the sake of this talk, I'm going to endorse this view for reasons that have become clear. I do think that with some more work, one might be able to excise it, but you'll see that it would be a lot of work. I don't know. I'm completely confident that it could be excised.

15:00 So, what I'm saying intuitively is, and what I certainly do think is true for at least most, if not all, of the actual measurements that we make of the position is And I'm going to continue to use position as the example just because it's the simplest spatial temporal quantity to talk about. What I'm saying here is that position measurements are made with respect to what we normally call lab frames of reference. We pick some physical object, like our lab, with rods and clocks rigidly affixed to it, and we make our position measurements relative to that. Now, as a matter of fact, that is often what we do, not necessarily lab with rods and clocks, but some analogy of those things. But it's, I think, probably too strong, not probably, it is too strong a statement to say that we necessarily do things that way, but I am going to say for the sake of this discussion that we do necessarily do things that way. As soon as we adopt that view, we are in trouble in two separate ways. One is, in the sense that I will try to make clear when I get to the next section of the talk, to talk about the hydrogen atom as an example. Quantum theory, in a certain sense, takes for granted an absolute space. For example, the configuration space for n particles. I don't want to try to clarify that statement just now because I think it would be more clear when I just do the example and then I'll point out the stage at which, in the appropriate sense, quantum mechanics has forced us to grant ourselves an absolute space. So you're just gonna have to file this statement away for a moment and I'll try to clarify it in a moment. How are we gonna deal with this issue? Well, we're gonna deal with this issue by noticing that hint there. It is possible to describe relational facts in absolute terms. This is a very familiar point. Even if I'm Newton and I believe in absolute space, I can describe in terms of my absolute coordinates all the relational facts. And that's the strategy we're going to take here. So we're presented with this apparently, in a sense that I'll clarify, absolutist theory, but we can still describe relational facts in its language. And that's what I'll be doing. There's another problem to the point, obstacle to adopting the point of view that I'm trying to adopt here, and that is that in a certain sense,

17:30 It's impossible to have a quantum reference body. I'll try to say the word reference body when I mean a reference frame given by some physical object like a lab. Why is that? Well, let's suppose we've adopted this room without a reference body. So what is the location of the room relative to itself? Well, that's always got to be a well-defined... Question, right, because the origin, whatever point in the room I've chosen is at the origin. What's its location relative to itself? Well, it's at the origin. And similarly, with respect to its momentum, what is the momentum of the origin with respect to this lab frame that I've chosen? Well, it's got to be zero, by definition. And in classical physics, these facts don't raise any interesting or important issues, in quantum mechanics, and people have noticed this going back to Bohr, I said I wasn't going to say Bohr, people have noticed this fact for quite some time, that you can't have a reference body that behaves that way in quantum mechanics because then it has to have a definite position and momentum. And of course we don't know how to describe things quantum mechanically that have those. All of these have simultaneously a definite position and momentum, so we wouldn't be able to write down a state, if you like, for the lab itself. A hint about how we're going to deal with this issue, which is mostly we're not going to deal with it, we're going to sidestep it, and that is, in quantum mechanics, a reference body never has to play both roles at once. You can always ask yourself, is my reference body soon to play one role or the other? You don't have to ever ask yourself, is my reference body suited to play both roles at once, because you never are going to be measuring the position and momentum simultaneously of anything with respect to that given reference body. So it never has to serve those two purposes at once. Okay, now it's working. If you have any hidden values that postulate definite positions of different momentum... Then you have to tell the story a different way. Yeah, and there is, there are, I mean, with respect to bone or modal interpretation or something like that, there are ways of translating my own study into those languages, but that would take way too much time right now.

20:00 I bet you're absolutely right. Okay, let's think about, very briefly, about the hydrogen. Can I just make a clarification? Indeed. You're saying that this quantum reference body never plays both roles at once, which seems to make sense, but aren't you... You know, I've got in my mind that you want to use this reference body to essentially construct a familiar inertial frame, right? Well, it defines what I mean by an inertial frame, yeah. So inertial frames, though, need to know? I mean, they're foliations of space-time. Yeah, yeah, yeah. You can't sort of say, well, I'm just going to ask about position now, or I'm just going to ask about... Let me come back to inertia. That's a very, very important issue, yeah. And what I'm going to be assuming about my lab is that it's a miracle. But let me come back to say how that's going to be handled in a minute, because that's the crucial issue. But the point, so I do have to make some kind of assumption like that about the lab. The only point I was making in the little hint is... When I make measurements with respect to something that I'm going to call my reference frame, I'm only making a position measurement or a momentum measurement, I'm not doing both. So I don't have to work, I don't have, so even if I was going to measure the position of momentum of something that is in a sense defining the lab, I can only measure its position of momentum. This will certainly come back, this point will certainly come back in a moment. Okay, hydrogen atoms. Very familiar story, I take it. If you look in textbooks, good ones and bad ones, you see something like the following story. It is pointed out very quickly, sometimes in the first line, that the cooling potential depends only on the distance between the proton and the electron in the hydrogen atom. That's clear from this expression. And then the author says we can use this symmetry to reduce the problem of the hydrogen atom to a problem that looks like the problem of a particle of essential potential, mathematically it's the same problem, and then we perform separation of variables, followed by some messy but straightforward mathematics, and we get solutions for the radial and angular parts of the wave function that look like this. Clearly, separation of variables has been put in by assumption, essentially. We assume that the solution will take this form, and then we actually use that assumption to find out what the solution is.

22:30 That's a familiar story. A slightly more, I would say, accurate way of describing what's happening is the following. I'm going to justify my claim that this is more accurate in a moment. It's not hard to find if you're interested. He does it this way, and I think this is the way that, in some sense, it ought to be done. He first writes down the general Schrodinger equation for two particles, the proton and the electron. Here it is. And there's a sense in which, although I'm going to qualify this statement in a moment, but there's a sense in which we have granted ourselves an absolute space. We've told ourselves that the answer to our question is going to be expressed in these terms, and where did that come from? Well, we gave it to ourselves. Where did this absolute space come from? We gave it to ourselves. We just assumed its existence. I'm going to quantify that statement slightly more carefully in a moment. For now, I just want you to notice that there's some r to the sixth that we've given ourselves here. It's not clear where it came from or what its physical significance is yet. Well, what do we do then? As Schiff points out, we perform a change of variables, that is to say, we reorientize r to the sixth to a relational center of mass coordinates. And then we perform a change to spherical coordinates followed by the same separation of variables and we get an equation for the wave function with respect to These variables here, which are the relational variables and spherical coordinates, so the little x, y, and z with the relational variables, we just re-coordinatize thus, and we get a wave motion for the relational variables and spherical coordinates.

25:00 The point here is that we're working not just in spherical coordinates, which is frequently, in fact invariably pointed out in textbooks. But we're also working in relational coordinates. This is a wave function whose arguments are relational coordinates. Can I just ask a question? It seems to me you're doing yourself down a bit in the discussion about the x-eyes and y-eyes. It seems to me you've dealt with the question of the absolute space problem. It doesn't seem to me that it throws you, in a sense, towards an absolute space, because all you're going to do now is transform to these, as you call them, relational co-interests. But I can't see what the difference is. Yeah, okay, so let me say what the difference is now. This is the crucial point, actually. Look, you can throw this away right from the start, right? And you can start here, right? You can start here with R to the sixth, coordinate-ized, thus. Which is good. You usually want to do that. No, here's the issue, though. The issue is, it's this arena in which I know that the Schrodinger equation is valid. Why do I know that? Because I know, I assume, rather, that this is an inertial, that these things describe an inertial frame of reference, coming back to your point. I don't, right? I mean, I've just given, I've said, let there be an inertial frame in which the positions of the two particles are described. I haven't said anything about where that inertial frame comes from, but I must assume that it's inertial, otherwise the Schrodinger equation is not the evaluation of motion. So this is the arena in which quantum mechanics gets off the ground, if you like. I give myself a frame in which I claim that the Schrodinger equation is valid. And then I start my work.

27:30 And you can see that it's not just a matter of convenience. It's not, I didn't work in these relational coordinates just because it makes the mathematics easier to do. It does make the mathematics easier to do, and that's often stated, when it's explicitly noticed that I'm working in the center of mass and relational coordinates, it's often said, we do this for convenience. It's not just for convenience. It's the, I mean, if we try to transform back to absolute coordinates, we would find out that we are essentially asking a relational question when we solve the hydrogen atom. We're not just using the relational coordinates as a matter of convenience. So suppose we wanted to transform back. Once we get the answer, if you like, or once we get this expression here, which involves these relational coordinates. Suppose we wanted to transform back to absolute coordinates. What we would have to do is, first of all, we have to put back in the center of mass. Because one of the things we did when we separated the variables, we said, oh, the center of mass, that thing moves inertially. Get rid of it. The whole thing is going to factor into a part, the relational part, and the center of mass part. Put that back in. We're going to need to do that to do the transformation because the transformation, and this is the crucial point, the transformation back to absolute coordinates is going to mix up the relational and the center of mass coordinates. So we can't do the transformation without putting this back in. We put it back in, we perform this ugly transformation, and I didn't even bother to write down the result because it's a mess, but the crucial point about the mess is it's entangled. It entangles the two particles. In other words, we don't get, I mean, naively what you think you're doing is what I thought I was doing some 15 or 20 years ago when I wrote down the first time I thought about it. I was trying to solve the problem of the hydrogen atom. Naively, what I thought I was doing was I was finding out the energy of the hydrogen atom associated with the motion of the electron. That's not quite what you're finding out. What you're finding out is the energy associated with the relational degree of freedom. That is to say, the energy associated with the distance between the electron and the proton. That's what you're solving. When you solve the hydrogen atom, and when you try to transform back into some kind of absolute coordinates, to see if you can parlay that answer into an answer to a question like, what's the energy of the electron, you find out that the result is completely entangled. So you don't actually get an answer that factors into what's, let's say, apart from the electron and apart from the proton.

30:00 So that's what I'm saying in the box here. The standard so-called energy levels of the hydrogen atom are energies associated with the relative position in motion of the proton. Okay, I want you just to file that example away, and we'll come back to it, how much time I have to come back to it and really do it justice, we'll see. And I wanted to switch gears now, but I guess maybe just to say very quickly, in case I don't have too much time yet, the lessons that we should take from this example. There are really two lessons I suppose. The first lesson is that when we do quantum mechanics, let me back up, the first lesson is that quantum mechanics is absolutist in the following sense. We start off by writing down the Schrodinger equation in some coordinates which we assume to be inertial. Now that assumption doesn't have to be completely unjustified. We can say things like, well, you know, I have reason to believe that this lab is effectively inertial or things like that, but the point is that somehow or other we've convinced ourselves that we've got coordinates that are inertial or good enough, close enough to inertial so that the Schrodinger equation is indeed the valid equation of motion in those coordinates. So if the coordinates aren't inertial, then the Schrodinger equation isn't the equation of motion. That's lesson number one. And that then is supposed to be a more precise way of saying what I said very quickly early on, that quantum mechanics assumes a kind of absolute space. The second lesson is, nonetheless, even though we begin with this assumption, granting ourselves, justify it or not, something that we can call an absolute space in the sense that we know it to be inertial. And I would actually argue, and certainly if I'm adopting the point of view that I described in the first section, I would say always, necessarily, but whether it's necessary or not, very frequently, we find ourselves interested not in quantities described in terms of those absolute coordinates that we granted ourselves at the beginning, but relational quantities, and we see that in the hydrogen atom. What we're really interested in in the hydrogen atom is the energy associated with the relational coordinate, or set of coordinates.

32:30 Even though the question initially was necessarily framed in terms of what I'm calling absolute coordinates, I mean, the word absolutely is somewhat confusing here because I'm not supposing that it describes an absolute space in the sort of Aristotelian or Newtonian sense of those terms, but what it is is a space that I've granted myself, that I know is the inertial, and therefore in terms of which the Schrodinger equation is the propagation. Okay. So those are the lessons I want to take from that example, and whether we'll have time to put those lessons, bring the lesson back into the picture, I don't know. So I'm going to switch gears now, and talk about some mathematics. And here's the big picture, in case the mathematics becomes too much, or we don't have time to go through it in proper detail, I do want you at least to get this big picture. Observables can be defined, and I mean that, defined, in terms of their transformational properties. And I'll give you an example of how that's done in a moment. Second aspect of the big picture is, such observables, when I define observables in terms of their transformational properties, they form what's called a system of imprimitivity. And you can then apply certain very powerful mathematical results to derive facts about the observables in question, and one of those facts is the acerbity relations. And I'll say in a sketch how that's done. Are you saying that the acerbity relation is an imprimitivity theorem? Well, the acerbity relations follow from the imprimitivity theorem, yes. I don't know what the theorem is, but... Well, I'll sketch it for you, yeah. Third aspect of the big picture, the transformations that we use up here to define the observables also define, implicitly define, the direction of a reference frame once we've granted ourselves this absolute background frame that I've been talking about.

35:00 So once I've gotten one frame relative to which I can start making these transformations, I can tell you what all the other frames are by applications of, I mean in this case I'm thinking about the Galilean room. The absolute frame that I grant to myself, I can generate other frames, and translate them into other frames. And then the interpretation suggests, to my mind, the connection between the notion of a reference frame and the uncertainty of relations. And the hope that I want to generate is more than you, and maybe even generate more than me. I just don't want to turn into reality. We can use the notion of a reference frame to shed some light on the latter in certain relations. I have to move fairly quickly through mathematics. My strategy for the next few slides is going to be to make sure we all at least have the big picture, and those of you for whom this stuff is more familiar than others can need only a reminder. You can probably glance at the slides and remind yourself about various results. Just so you can read it, curly H is always a Hilbert space, L of H is always a bounded operator on it, S is some locally compact topological space, if that's gibberish to you, S is R3, mostly, and V of S is the sigma algebra, or else that's subsets of S, or nice subsets of R3. We can define then a notion of observable quantum mechanics that's equivalent to the usual notion. Well, it's an extension of the usual notion in the following way. We define a map. I'm just going to say it in words. What I'm going to say is what I really want you to get. We can define a map from this space S, which is intuitively the space of possible values of our observable. We define a map from that to operators on a Hilbert space. If we did this in a completely sort of standard textbook way, those operators would be projection operators.

37:30 And so what we would be doing is intuitively mapping eigenvalues to eigenprojections. It's not crucial for the type of discussion I'm giving here, but it is in fact crucial to carry out this program in detail, that we generalize that idea a bit and talk about maps not to eigenprojections, but to positive operators. I think for the sake of time I'm not going to describe that in detail. So the point is, just in the same way that we can take a self-adjoined operator and break it up into its eigenprojections, its spectral projections more generally, and the associated eigenvalues or sets of eigenvalues, we can work essentially in the opposite direction and start off with the eigenvalues, map them to eigenprojections, and define in this way an observable for ourselves. It's thinking about observables in that way that allows us to think more clearly about what it would mean for an observable to have certain covariance properties with respect to elements of some group. Locally compact topological group. Intuitively, it's the symmetries, the Galilean group, spatial transformations, rotations, that sort of thing. In my examples that I'll have time to give you anyway, G will always be the Galilean group. Another piece of notation, alpha, is the action of G on S. Remember, S is the set of possible values of the observable in question. And by action, I just mean that it maps elements of the group to some mappings of S to itself, so M of S here is the set of all mappings of S to itself, and alpha associates every member of our group, every symmetry, with some transformation on the space and possible results. And then U of G is some continuous unitary projective representation of the group G on our Hilbert space. I say projective here, but I'm actually going to ditch that very quickly and just say it's just a simple unitary representation, just for simplicity, so I don't have to carry phases around. Using these notions, then, I can tell you what I mean by an observable being invariant under elements of this group and covariant with respect to elements of this group.

40:00 Invariance is very easy. In fact, I don't need a lot of this machinery to talk about invariance. I just say that E of delta is going to be the projection operator corresponding to the set of possible results delta, where delta is a set of numbers or a set of elements of S, but numbers, real numbers or regions in R3 or what have you. So this is a projection operator, and this is just the standard way of transforming a projection operator according to some member of this group, and it just tells me that I get the same thing back again, and that's what I should mean by invariance. That can be stated without all that machinery above. This one can't, at least not easily. What I mean by covariance is just this. The transformation according to little g, which is just an element of this group, of delta. So let's think about positions. Suppose we're talking about the position operator. And let delta just be this region of space between my fingers, and let g be a spatial translation. What I'm saying is, take that region, translate it by g, and then find out what the projection operator is corresponding to that region of space. Okay, that's what the left hand side is. The right hand side is just this again. It's just take the projection operator corresponding to the region delta and then transform it in the Hilbert space according to G. And those two things should give you the same answer. It's, I mean, as long as this observable, the observable represented by the map E, it kind of respects transformations with respect to this group, right, then it shouldn't matter whether I first transform the values and then find the observable, or find the observable and then transform it. So good morning. Well, the only point is that in variance I don't have to worry about the action of the group on the space because I'm not doing any transforming of the underlying space of results. All the transformations happen in the Hilbert space. Whereas here, this is kind of a statement about the relationship between transformations in the results and transformations on the Hilbert space.

42:30 And it says that if E, as an observable, respects those transformations, then those two different ways of performing a transformation should end up with the same thing. I guess this whole slide is just a piece of it. I'm really going to be concerned now with the Galilean group. We can write as a product, a semi-direct product, a semi-direct product, a product, a semi-direct product, never mind. Velocity boosts and rotations. I'm really only going to be worried here about spatial translations and boosts. So how can we do position and momentum in this way? Well, in the case of position, as I've already said, the space of possible values of the observable is just R3, the single particle, and the Borel sets are just the propositions. The particle is located at some point whose coordinates are in the set delta, where delta is a set of points from R3. And we'll let Ex be the observable for position. If it happens to be a PV measure, that is to say a projection value measure, that's the only kind I'm considering, you can in fact recover the standard quantum mechanical position operator with this integral, but don't worry too much about how that's defined. You can do all the same thing, making the appropriate changes for momentum, and here's the crucial point. We haven't really said anything yet about which operator in the Hilbert space is the position operator. I've just said in a schematic form how you define it. But we're going to be able to find out which one it is by respecting this requirement. And the requirement is that position should be invariant under boosts and covariant under translations. Why do I say invariant under boosts? It's position at a moment. So if I take my space... And I give it an instantaneous velocity that doesn't affect positions at that time.

45:00 Yes, it's true that things will move in that boosted space, so the positions at later times will be different. But instantaneous position, which is all I'm thinking about right now, is not affected by a boost. And on the other hand, it is affected by spatial translations. So if I take my entire space and move it 10 feet to the left, then the position operator better change accordingly. So that, I mean, to give you the physical reason for that is, position operator insurance probability is finding particles in places, right? And the probability that the particle is here, right, better be exactly the same as the shifted probability that the particle is in the shifted area. So if I take the whole space and shift it five feet to the right, then I've got to do the right thing to my position operator so that it maintains the same probabilities in the shifted space. Well, if I had time, I would talk about angle and how you can define angle observables. I'll say later, quite briefly, why that's interesting and important to do angle as well as position and momentum. Okay, so what we have, if we have an observable that we've defined in this way, I said, give me an observable and I require that it respect members of the Galilean group in the right sorts of ways, so it's invariant under boost and covariant under spatial translation, let's say. So keep thinking about, if this is all new to you, keep thinking about the position of the observable because it's the only one that's going to matter in the end. Then I have defined what is called the system of imprimitivity. And the system just collects up all the objects that I've been talking about. The Hilbert space, the spectral values of the observable in question, the group whose invariances, with respect to which I want to have the right kinds of invariances and covariances, its action on S, its representation on H, and the observable in question. So, for our purposes, let that be... Let that be the standard Hilbert space of square integral functions on R3, let that be R3, let that be the Galilean group, let that be the omnidirection of the Galilean group on R3, let that be the representation of the Galilean group on this Hilbert space, and let that be the position observable.

47:30 Then what I have is a so-called system of imprimitivity. And why is that interesting? Well, there's a series of theorems, which we'll just call the imprimitivity theorem, because I'm not going to be precise enough to worry about that. Subtle differences between different versions of the theorem, but there's a theorem called the imprimitivity theorem which says the following, consider two systems of imprimitivity and so let's start off with, let's just consider a system of imprimitivity that involves a position operator and one that involves a momentum operator, right? The position one is going to have the position operator being invariant with respect to boosts, co-variant with respect to translations. The momentum operator is exactly the opposite. It's invariant with respect to translations and co-variant with respect to boosts. So we've got these two systems of impermissibility. The co-variants in particular, with respect to the members of the Galilean group, are in each case with respect to different subgroups of the Galilean group. So position is co-variant with respect to spatial translations, which I called. A, I think, in the, when you break the Galilean group up to this, and the momentum system of primitivity, the momentum observable, is covariant with respect to boosts, so it's, they're covariant with respect to different subgroups of the Galilean group. Each one of these, each one of the representations of the Galilean group, sorry let me back up. Now suppose I've got a representation of just, let's say, the spatial translations. I've got a representation of the spatial translations on Hilbert space. That's some unitary representation of spatial translations. That representation of a subgroup of the Galilean group induces a representation of the full Galilean group. If you're not familiar with this concept, you're just going to have to take my word for it, that you can induce a representation, that is to say, you can derive a representation of the entire Galilean group from a representation of one of its subgroups. This is the theory of induced representations. Well known, but I don't have time to delve into how this happens. Similarly, let's move over to the momentum side, and suppose now I'm considering a representation of the boosts.

50:00 That representation of a subgroup of the Galilean group also induces a representation of the whole Galilean group, so it sounds like I'm getting something for nothing, and I guess in a sense I am, but it's still true. Now I've got two different representations of the Galilean group. The imprimitivity theorem says that they're equivalent of unitary transformation. So the one I get from one system of imprimitivity is unitary equivalent to the one I get from another system. That's what all this mess is saying. This is only here for those for whom a reminder is sufficient. If a reminder is not sufficient, what you should know is just this. The position and the momentum systems of imprimitivity are equivalent up to unitary transformation. And you can prove that. In particular, what we learned then is that the representations of the Galilean group are essentially That all representations of the Gauss language are essentially equivalent to each other, that is to say, unitary equivalent to each other. That's the lesson we get. And from that lesson, we can derive the vile relations, which I'm taking here to be my way of talking about the assertive relations. I think for the sake of time, again, for those for whom a reminder is sufficient, you might glance at this, but the point I want to make here is the one in the box at the bottom. Simply by requiring the right covariance conditions on position and momentum, I can now, by way of the imperativity theorem and Stone's theorem with the other results, the uncertainty relation between position and momentum. What's often done, by the way, when you want to derive the vial relations... When you're deriving, I mean the usual way of deriving them, I suppose, is just to write down canonical-computational relations and then exponentiate, but if you really want to derive them and not just perform that formal trick, then very often what you do is you grant yourself a position operator, and then you say, you grant yourself the idea that momentum generates translations in position, and that's going to get you automatically to the law of relations very quickly.

52:30 But what you really only have to do is say position and momentum have the right covariances with respect to the Galilean group, and you can derive by the imprimitivity theorem that momentum generates translations of position, and vice versa. And so you only have to assume these covariance conditions to get an answer to the question. Well, remember what the covariance conditions are. They're conditions about how position and momentum behave with respect to elements of the Galilean group. What I get for free, if you like, is a recipe now for determining reference frames given a single reference frame. Because I know that position generates translations in momentum. I know that momentum generates translations in position. And I know all that just from the very minimal assumptions I put in from the start, namely that position and momentum have the right covariance with respect to that lane group. So knowing that now, if I give you a space that's four times in some way, and then you say to me, I want that space to be translated 10 feet to the left, I know how to do it. Because I know that momentum is generated by translations in position. And so I just write down the appropriate unit of your operator and shift the space. I also know the truth of the uncertainty relations. Now, these two things might seem unrelated to one another, but I think they're not. I think the fact that I get from these assumptions in the start, the fact that I get both the uncertainty relations and an account of how to generate reference frames, given a reference frame, are connected. And in the last two minutes, I'm going to try to say something about... What that connection is by way of a theory of quantum reference frames. This theory is, I mean, what I'm doing here is essentially taking stuff from a paper by a hard-knocking co-worker in the 80s. It's a fairly difficult paper to read because it's very badly written and almost incomprehensible in places. There's also some mistakes in it. Not important ones, but mistakes that can be very misleading.

55:00 So I'm giving you what I take to be what they were trying to say. You shouldn't necessarily compute what I say to them, and certainly not vice versa. And I'll tell you where I'm doing something that's me and not them when I get there, which will be very soon. So imagine the following situation. We have an external observer who's the kind of observer who wrote down that two-particle wave function for the proton and electron to start with. That observer simply grants himself some reference frame in which he takes it as given that the Schrodinger equation is valid. A kind of absolute view, or what I think of as a view from nowhere, and I won't try to justify that phrase right now for lack of time. So this guy out here is having a look at someone who's in a lab making a measurement, and remember that what I claim for the sake of this discussion is that all measurements are like this. All measurements are like people in labs measuring with respect to their physically specified reference range, in this case specified by a lab. She's got rods and clocks, and I'm only going to talk about position measurements right now, so rods are all we need. She's got rods that tell her, you know, where a particle is with respect to the four walls scaling the floor of the lab. But what we want to do is describe, we don't have a way, I claim, ab initio, of writing down a purely relational What we have to do, I claim, is write it down from his point of view, but then restrict our attention to the relational facts. So he's going to write down what she takes herself to be doing. But he's going to write it down in his coordinates. And that's what we're saying in the box here. So the question is, how do you transform? She's going to say, I just measured the position of an electron, for example. And he's got to transform that statement into his coordinates. And of course, what the transformation better give is, she just measured the position of that electron relative to the walls on the left. That better be the translation of what she says, I just measured the position.

57:30 And it's very easy now. We know exactly how you must write down that transformation. And I say must because that's one of the lessons that we got from the imperative theory. We don't have a choice about how to implement these transformations. Once we've made what I claim to be the right assumptions about position and momentum and their covariances with respect to Galangry. So this is what Ahrar, Neff, and Coffert remind us all the mess about in primitivity and so forth. They just, because they know quantum mechanics, know what to write down, and it's that. I'm not going to try to justify it at the moment, but it is essentially a Galilean transformation of the right sort. The point is that it generates the right answers. So if I have a position observable for the internal observer, and she's measuring the position of particle n, and I apply this transformation to it, then what I get is Position observable for the external observer, sorry, the position observable for the external observer for particle n minus, where zero is going to represent the lab itself, right, minus the position of the external observer, the external observer's position observable for the lab. In other words, I've taken her position observable and turned it into a relative position observable. That's just to kind of check to make sure this is the right answer. And you can do it very quickly and check that it is. So, that's the right answer. That tells me how to translate her statements about position into the external observer's language. And then I can start applying this transformation to other observables and see what happens. And I get interesting results like this. I don't have time to interpret it in full detail, but just the intuition is that when I translate The free Hamiltonian of the internal observer into the external observer's coordinates, what I get is free Hamiltonian for all the stuff inside the lab, that's the n greater than zero, free Hamiltonian for all the stuff inside the lab, plus a kind of drift term, which represents the drift of the lab with respect to the external observer, because it might be moving. Alright, this is the last slide. Given that we now have a way of transforming between what the internal observer says she's doing, which for the purpose of this talk is what I claim we all say we're doing when we make quantum mechanics, that's the sort of thing we say, into the external observer's language, we can now translate what happens in position measurement.

1:00:00 We can let the internal observer make a position measurement, and then let's transform that entire description to the external observer's point of view. So, in a very simple, and as I say here, over simple model of a position measurement, we have the internal observer's Hamiltonian, sorry that should be an equal sign, that first plus should be an equal sign, equals some free part plus some interaction part, interaction representing the interaction between some measurement device and some particle whose position we're measuring. And for this simple example, we can take the interaction part to look like this, where we're measuring the position of particle one, And our apparatus has a pointer observable. Our apparatus is labeled system number two, and it has a pointer observable that's canonically conjugate to this P. So intuitively, a pointer observable is another kind of position observable, and this is the canonically conjugate momentum observable. And then this represents the interaction between the two systems, and there's some scaling factors. So you could say, you know, g of t at minus infinity is zero and at plus infinity is zero, and it goes smoothly, spikes up at the moment of interaction, spikes back down. And this is a standard account of measurement in quantum mechanics. Apply a little bit of perturbation theory, and you get the standard result, namely that the observable canonically conjugated to this becomes perfectly correlated or nearly perfectly correlated with this observable here. That's the standard account of measurement. And the point of this discussion is I can now translate that internal description of what's going on, which is of course exactly what we always write down. I mean, plus or minus the fact that this is a highly idealized situation. But that's something like this is what we write down when we make quantum mechanical measurements, and let's transform it into what I'm calling, perhaps provocatively, a view from nowhere. And we can find certain results that are interesting, certain computational relations that are trying to be interesting. One of these... So H prime represents the transformation of this thing into the external coordinates, and we find out that it commutes with the total momentum. Well that's good, it better commute with the total momentum because nothing was entering nor leaving the lab, by assumption. It does not commute with the momentum of the lab. That's the first point. It also does not commute with the momentum of the measured particle.

1:02:30 This isn't terribly surprising because we interacted with the measured particle. We bumped it around a bit in this interaction, so you're not too shocked that its momentum can be changed by this interaction. This one may be a little bit more shocking, and I think there's a story you can tell about why it is non-zero, but for the moment I just want to notice that it is non-zero, and notice that therefore... Measuring the position of the particle disturbs both the reference frame and the particle in this sense. Their momentum, the time derivative of their momentum during this interaction is non-zero. So, what does that tell me about uncertainty? Does that give me any physical insight into the uncertainty relations? Well, it doesn't give me any insight into the numerics of the uncertainty relations. It doesn't tell me why there's upper bounds or product limits. But it does, I claim, give me some hope of understanding the qualitative uncertainty relations, the claim that when a position is well-defined, a metric isn't, in the following way. If I know that whenever I measure a position, while I'm measuring it, I'm disturbing the very thing that I take to define position in relation to my lab. If I know that I'm disturbing it while I'm doing that, then I know while I'm doing that, from the external point of view, that lab is inappropriate for defining momentum because it's not an inertial system. So it's not the sort of thing that could be used or ought to be used to define momentum. And so when I'm inside the lab, I should say to myself, remembering that when I'm inside the lab, I always... For the sake of this talk, I don't necessarily define position and momentum with respect to my lab, and I think to myself, well, is that an appropriate thing to do? In classical physics, I never have to ask that question, but in quantum mechanics, I do have to ask that question. Well, I do have to ask it in classical physics as well, but I'll set that aside. In quantum mechanics, I have to ask myself, is it appropriate for me to be taking this physical object to define momentum? Well, if I'm currently making a position measurement, the answer is no, because I know that there's some wider frame out there, represented by the external guide, relative to which this lab's momentum is not conserved during the interaction, and so that's why I claim, or that's one way of understanding why momentum, even from the point of view of the internal observer, should not be considered well-defined during position measurement. Okay, that was a long story, but there were some interruptions.

1:05:00 Some detail, a little bit of digestivity as it goes through it, but I suppose there's a sort of, if you stand back from this intuitivity stuff, I mean, what I want to look back at the whole thing is that quantum mechanics essentially operates with only half the phase space, and this is the first thing you notice when you do quantum mechanics, classical mechanics has Q and P and two-dimensional things, but quantum mechanics, you can either do it on the position that you're talking about. And of course you can pre-mark nice things like Q plus IP and Q or P, get what's called a space-dynamic function out of that. And you can polarize the phase-space various ways. You're always truncating and doing it in a half. And the thing is, once you've got the half, once you've chosen the half, that specifies the whole of the mechanics. There's quite a bit to worry about the big difference between classical mechanics and quantum mechanics. In classical mechanics, if you do half the base base, you've only got half the answer. If you do quantum mechanics, where you do half the base base, you've got the whole answer. And that is essentially what's going on with this. Yeah, I mean, I would, one qualification is what you said. You can start off asking, so we talked about observables as being maps from some space to operators on Hilbert space. You certainly can, and in some cases should, start off by taking that initial space. But then what you find out when you do that, you still end up having it, to use your language, but you have it in a different way. You don't split it into a position part and a momentum part. You end up having it in other ways. One way that I didn't get a chance to talk about is you can have it into an angular momentum part and an angle part.

1:07:30 So you can either talk about, and you get exactly the same kinds of file relations then between angular momentum and angle, and what that tells you is that if you pick a direction in space, I mean with respect to in the language of reference frames then you could ask the following question... I want a reference frame that determines for me a position in space, right, and you might want to know this, for example, if you're going to do the EPR experiment, I want to be able to share a direction in space with someone on the other wing so I can measure spin in the same direction as them, and you can do that, right, but that doesn't commute with, in the wild sense, that doesn't commute with angle, so that I can't both know... I can't have simultaneously good knowledge of the direction of space and some angle away from it. I mean, the reason I make this point is specifically that it's crucial if you want to talk about the Bohm version of the EPR experiment. I mean, I didn't even mention the EPR experiment, but if I want to apply this discussion to the EPR experiment, well, if I'm doing the original EPR experiment, I can just talk about position momentum. But you can come back, and I did this to myself, I came back and said, wait a minute, that's not enough, because we've got all these other versions that measure incompatible observables and other pairs of incompatible observables, in the most obvious case of spin, of course. And the question is, are reference frames involved there in a similar way? And I originally thought, no, that's not true. They're not involved in the same way. And then, after some more reflection, I convinced myself that they are involved in more or less the same way. In the case of Bohm's version of EPR, for example, if I share a direction in space with my space-like separated partner, we can certainly well do that, but then suddenly we become unable. That partner becomes unable then to measure, to know that he or she has measured the spin in some direction at some specified angle away from that shared direction of space, so that all we can do at that point is just measure spin in the same direction. The quantum mechanics doesn't let you do that. You have to choose. Once you've got the cue, you can answer any question you like about quantum mechanics.

1:10:00 I completely agree with that. And so there's this choice, you know, of half the space, which... It really allows room for complementality, right? That's what I tend to see. I just want to add that you're not always splitting it up into Q's and P's. Sometimes you split it up in other ways. There is another way, yeah. There are basic, what's called, geometric formulations. There are all kinds of other ways. Right, exactly. So, just coming back to the example of the hydrogen atom and so forth, Suppose you have the mass of the protons become infinite. Well, then you disentangle those two motions, yes, the integral and the central mass. In that sort of linear case, I mean, it seems to me there's no point to follow an example, because essentially this, I tell you, entangling everything. But if you take the two people, you can't disentangle them. Indeed. And the reason for that, and here you can see some of the analogy between, that I see at least, between that case and the case of measuring things with respect to a lab. Because the same is true of the lab. If you let the lab's mass become infinite, then it's not disturbed anymore by the measurement. And I think the reasons are the same in both cases. But you could say, in the case of the laboratory, it's a pretty good idealization to take the maths of the laboratories, not just the approximation in the case of the proton. Right, right. Not the band approximation. So that's, I'm just wondering, when you apply it now to the laboratory and those kind of things, Well, I think that's what we, we certainly do that in practice. I mean, in practice we certainly say to ourselves, well, I'm measuring some fact value, some tiny particle, and it's mass relative to the labs.

1:12:30 All of these terms are effectively nil, and therefore the interaction with the labs can be neglected, and that certainly will be due to practice, and I guess along similar lines, in practice, we could measure the position and momentum of middle-sized objects or even pretty darn small objects to any degree of accuracy that our instruments will allow, and the uncertainty relations are irrelevant to those things because we just don't have the ability to measure. Precisely enough to bring the uncertainty relations into play, but I guess the case, what I want to think about though, sorry, one more comment along these lines, and that is, it's exactly because of the points you're now making that I said, nothing I'm saying here has anything to do with the numerical content of the uncertainty relations. The only question I want to ask is, is there some way of understanding why it is that, in principle, we should say that momentum is undefined during a position measurement? And there, I think, if you're speaking in principle and so forth, you have to recognize that the lab is a finite object, and that it is, in fact, disturbed during this measurement. Well, that's, of course, true of the boards, and of course, I'm in some situations, but it could just take us. The title that you got, The View from Nowhere, is a kind of, you know, Oracle-Nagel, I suppose. So, of course, the way that Nagel sort of thought of The View from Nowhere was that if you want a particular perspective on something, that introduces it to some degree of subjectivity in the description. And when you get The View from Nowhere, you're trying to reach far within these more perspectival aspects. And this links with ocean values and the connection with relativity. Well, I just wondered how you sort of see that in the context of your own research. Right, so it comes in in the following place. When the internal observer, I'll be the internal observer for the moment here at my lab, and I ask myself the question, is this lab the sort of thing that could be used or ought to be used to define what I mean by momentum?

1:15:00 And that question is a question that cannot be answered by me sitting here inside this lab, because I don't have access, since the lab is the thing that, if I take anything to define positional momentum, it's going to be the lab, I don't have access to get to positional momentum, and so that question is a question that's sort of a question about, well, if I could look at it from nowhere, from this absolute point of view, what would my answer be? And take that to determine my answer. And see, I can do that, even while I'm in the lab. Even though I don't have access to the position of momentum in the lab, I can, if I know quantum mechanics, I can say to myself, well what, if I did have this absolute point of view, what would I say about the lab? And what I would have to say about the lab from that point of view is that it's disturbed by momentum. And so it's a way of, it is an attempt to say how someone who's necessarily looking at things from a certain perspective Could escape, if you like, an answer, escape from that perspective in a certain sense and answer a completely objective question, namely, is this lab the right sort of thing to determine the momentum? You said you wouldn't mention Bohr at any price, but you were, in fact, close to mentioning him once. How close do you think your own point of view is to Bohr's? After I reread the words by the EBR, where the notion, and what became clear to me is, it's a classic case of rereading it in preparation for teaching it to undergraduates and realizing that what I thought it said is not what it said. And then looking at the, and then I wrote to various people and said, gee, you know, did I just not understand all these years what's going on? Am I the only person who missed the boat here? And they said, well, you know, you should go look at various papers that have come out in the 90s about Borg's reply to PPR, which I did, and felt that they were also wrong.

1:17:30 And what struck me about Borg's reply is that he does make heavy use of the notion of a reference frame in the reply. And that doesn't make it into the official story, if not the official stories that I've seen about Borg's reply. And so then I started reading earlier boards and tried to figure out, well, how are you thinking about these reference frames and try to work out what's going on. And so that's the history behind the project, and then I started thinking more carefully about what should you say about reference frames, never mind what the board thought about them. Now, to answer your question, I think it's very close to what the board thought. Minus, of course, a lot of mathematical justification. There's one word in Bohr's reply that still causes me mild panic, and that is the word uncontrollable. So he often says this interaction between the particle and the lab frame is uncontrollable. And I have speculations about what he means or ought to mean by that, but I don't find in Bohr a completely consistent account of what is meant by uncontrollable. I can plug that word into my story in a certain spot, and it has to do with the fact that I don't have access to the lab itself, but I wouldn't want to claim that the core had anything like what I have in mind when I plug that word in. So the short answer to your question is I think it's 97.5% what the core would say. Yes, for my clarification, at the end when you were speaking about the transformations, I suppose, just because you mentioned the different systems and properties of the spherical coordinates earlier, at the end of this, it can't really matter which kind of coordinates you are referring to. Absolutely not, that's right, yeah. I mean, whatever way you do it, it must be the same. Absolutely. Just to be clear. So, the points you made earlier on, did I miss a connection to the Amazon? The points of that, which points are there? The points you made about being transformed in different coordinate systems, and you said there were relational things. If we come back to molecular architecture, I missed something there. Right. There were two types of transformation one could consider. One, I claim, along with you, makes no difference whatsoever.

1:20:00 And the other, I claim, does make a difference. The kind of transformation that makes no difference is the kind of transformation where I have space and I transform between Cartesian and spherical coordinates. That surely is a matter of convenience, always. However, suppose I have a space, and for a single particle, any kind of transformation you make whatsoever is just a re-coorganization of the space for that single particle. But when I move into a two-particle case, then transformations that mix up the coordinates for the two particles, those transformations I claim are of a different sort. I mean, mathematically they're not of a different sort. They're all just re-coordinations of the space. But from a physical point of view, they should be thought of quite differently. When I mix up the coordinates for the two particles, as I do when I transform from x1, y1, z1, x2, y2, z2 to center of mass and relational coordinates, that mixes up the coordinates of the two particles. And that should be thought of as physically significant, because now I'm answering questions about different systems, if you like. Having the systems oriented to be physically different is the point here. Well, the point is that if I have the case of the hydrogen atom, right, both kinds of transformations are in play. One kind of transformation is just a transformation of the spherical coordinates, and that is clearly for convenience. Another kind of transformation is a transformation to the center of mass and relational coordinates, and that is not just for convenience. That is a necessity of defining the question in a way that I can... The question that I can answer, or the question at least that I want to answer, apparently, since it's the question that is answered in every textbook is, what is the energy associated with the relational degree of freedom? And to answer that question, I moved to those relational coordinates. And the point of saying when you transform back you get entanglement is just to point out that it was the relational question I was asking all along. But I had to do it starting from this absolute point of view and then transforming to the relational coordinates. I don't have, if I just write down the relational stuff from the beginning, right, then I don't have a reason, at least not one that pops into my mind immediately, for thinking that the Schrodinger equation in those coordinates is valid for this physical system.

1:22:30 My justification for thinking the Schrodinger equation is valid in that system of coordinates, the Schrodinger equation as written down in that case, which it's transformed as well because of the transformation of coordinates, the reason I think that that equation is valid is because I think that the pair, the electron and the proton, are in this absolute frame that I've given myself, moving inertially. That's the justification for writing down an equation. That's the argument I hear for saying that really, from a physical point of view, you should think of yourself as having first written down the r to the sixth, coordinate-ized in the simple x, y, z, x, y, z way for each particle, write down the Schrodinger equation in that space where you do think it's legitimate to say proton and electron are free with respect to... And then start doing the transformations. And you learn from those transformations how to write down the Schroeder equation for the relational degree of freedom, or degrees of freedom. These things which don't depend on the reference frame, yes, and those two things seem to be a little bit of intention in your thought. I mean, the usual thought is, oh, of course you have to make measurements with respect to the reference frame and set up some sort of system for ordering. But the remarkable thing is, with the distance between two particles, this doesn't depend on whether we've calculated the reference frame or not. So, you know, I'm not quite sure. I want to take your message, yeah. You seem like you're going to say, oh, well, I think, yeah, I do. The sorts of questions, well, there are, let's consider three different sorts of questions. One is the sort of question that has the same answer in every reference frame, the distance between two.

1:25:00 Well, you know, if the Galilean group is the one that's defining the reference frame. I certainly think there are such questions and they're physically important and interesting. I don't think the existence of reference frame invariant facts has any impact on the claims that I've been making. I think you can certainly perfectly well acknowledge the existence of reference frame independent facts. In fact, from the point of view that I've been describing, one discovers which ones those are by applying Galilean transformations. And see whether you get the same answer in all the frames. All right, so there we do have a disagreement. I just want to know what your reasons are for doing it in your way, Rob. Well, the reason would be that I don't see where we get the information that certain things are referenced very clearly. The fact that, from where do I get the fact that the distance between two particles is a reference to a invariant thing? I can certainly say, well let's take that as a starting assumption and build a physical theory on that assumption. But I don't see that I know that to be a true fact about the world starting off. But it seems to me that's... There are reference frames involved there, you've got a couple of clocks, things going up, and then you've got a spatial reference frame, and then you notice if you want to analyse the philosophy of light, the components, then you've got a spatial reference frame coming in, and then you notice the results of each of those, but it seems to me the starting point is that what are the physical facts? Which of the facts would you depend on some sort of convention about?

1:27:30 I guess I don't see why you'd want to give yourself that much stuff from the start, because you still have to know what it means to be a reference frame. If you're going to say the distance between two particles is a reference frame in independent quantity, you already know what you mean by a reference frame. And so, if you have to give yourself both of those things, why wouldn't you just give yourself the notion of a reference frame and then notice that the distance is invariant under transformation from one to another? So you sort of get that for free in this case instead of having to give yourself more strings. And then introducing the transformation groups under which these programs are adapted from that project. But how do you, what do you mean by objective facts if you don't already have a transformation group? Because I thought objective facts are very much under the transformation group. The whole idea of objective facts is that you start by just generating the world. And then you notice how much you can do in mathematics. But you started off by saying it must have meant something by invariant. No, no, no, I didn't say that. I didn't use the word in fact. I said these are the physical facts. These are the way I'm able to further the job. So give me an example of something that's objective.

1:30:00 And what do you mean by objective? Objective is that it belongs to the world, not to my conventional choice of something that's being subjectively developed into the description of the world. And what do you mean by subjective versus how do I know which description of the world to choose for the last of the convention? And how do I know whether a choice is a convention or not? So there's a variety of choices within what they want to choice, and the choice between the variety of choices. You see where I'm leading. I don't think that you're going to be able to tell me in a precise way what you mean by objective versus subjective without specifying the group of transformations relative to which an objective fact is apparent. No, it's a very specific definition of objectivity here. The philosophical understanding of objectivity, where objectivity is meant to be opposed to subjectivity in some general sense, not with this notion that we all take for granted having dealt with physics a lot as if we're dealing with some invariant feature, and I think that's... Surely there's something, the objectivity claim is more of a, well I certainly think in order of discovery, if you like, it certainly could happen, right, that we just have a kind of intuitive, imprecise notion of the objective versus the subjective, and the subjective is something that depends on the way I look at the world, so something sort of... And so on and so forth. Let's rationally reconstruct what happened here. And rational reconstruction is, well, my loose and intuitive notion of all of us agreeing about things actually is more precisely captured by the notion of these transformations.

1:32:30 So what I really ought to have meant from the start about being objective and being very respectful of these transformations is that I just got lucky. That I didn't have to be too precise about that in order to solve it. So, I mean, it's a principle of the law this time, but you say, oh, we know all the right transformations on a capture subject. I mean, not in the sense of mental, but how do you know which there might be? I mean, you said together then, you mentioned together then, well... It's certainly true that we could be wrong about our assumptions about which group this is, but what I'm claiming is that in order to tell your story, you also have to or ought to start with some group or other, and that you're committed whether implicitly or explicitly. You're committed to having chosen some group or other as the group that defines what I mean by objective or invariant. No, I don't think, all you have to do is say objectivity for me means that it really exists. Or maybe there's some other. Sometimes it means that, right? So we have to have the same story again. What do you mean by it really exists? What does that mean? I think I can push it back into having the same. Well, it's the same under some set of transformations. That's just the structural feature of how we represent things. So what do you mean by it really exists? I don't know. But that's not a very good starting point. No, I'm just saying, I'm certainly... I don't understand. These are all conjectures of how the world is, and really our future. You see, I think those conjectures implicitly involve these transformations. When you say things like, it's objective, or it really exists, and things like that, it's the same for all people who look at things from a point of view, all of which are related by some transformation.

1:35:00 This pen and that pen. That's all right. I've never said anything about boots, people, and... Exactly. That's a fact about the world. I've got to start my business with that. Now, if I want to represent that system, I have to have different choices. Some systems are preserved numerically. Well, I don't think that you had a lot of... Well, it's not a matter to start with unless... I mean, to say it's a fact doesn't matter. I mean, again, I don't know just how it happened. What do you mean by it's a factor? But didn't you, you helped yourself to, you said that in your examples, I don't know if it's new from nowhere, it comes in when you write down the truth real quick. And so you're helping yourself to this structure of this reference framework. Yeah, I think it's a feature. Then you come down to these, so this is, why should, why is that more fundamental than just, I mean, taking a piece of, ...setting a grid on the seafloor. We were going to do some surveying of a shipwreck. And then you throw away the grid. Why is it a view from nowhere and not just... Because you do throw away the grid. ...being bad at recapturing. ...being bad at the falsification. Well, I would put it slightly differently. I would say that this is a good example of the claim I made earlier

1:37:30 that the order of discovery doesn't have to be the order of justification. And what Klein did was he realized that what people meant, or should have meant, or implicitly were assuming when they said such and such is a fact of the matter about the world is, it does in fact have these appearances. And the fact that they proceeded as if that's what they meant is what got them to the right answer. Well, I'm not saying what Klein said, I'm saying what I said. Can I press on the issue of the invariances? I mean, with respect to which group? I mean, doesn't the, if that's really going to help you define objectivity, then what do you do about the fact that you want to have, that you have different groups that are, I mean, you may want to say something's objectively the case because it's invariant under the Galilean group. And then you have other contexts in which you want to talk about something that could be the case because it's invariant under the Poincaré group. So part of the whole, I mean, so my argument at this stage has simply been that in order to tell the story that Michael wanted to tell, he has to assume some group just as much as I do. I'm assuming it's explicitly, and I'm claiming it's implicitly assumed, but it doesn't mean that we've made the right assumption. That's a matter for further, you know, discovery, but we both had to make some assumption, that's what I'm saying. I'm claiming that Michael makes it implicitly, and I make it explicitly. Let's go back to Kahn's program, and then we'll stop. I think I dealt with the sub-superior geometry and the projective geometry. I look at subgroups of the projective. So you stop by saying, the thing that this geometry is about, there's nothing about objectivity there, but the thing that this geometry is about are just the incidence between lines, and what we have, or predicted, and those which preserve that feature. If you introduce parallel lines, then you change your view, because you still haven't got the notion of angle or distance, you want to go to that, you know.

1:40:00 Probably a lot of it, the common point is to say, which is this geometry about? Is it about incidence of lines, or sort of quantities? I don't deny that there's another important reading of the story. That is what I'm talking about. When I talk about distances, when I talk about momentum, what are the objects that I've described, or one of the physical quantities, if you like, that I wish to describe, my only claim is, when I say that the distance between, when you say that the distance between two particles is a fact of the map of the world, It's the fact of the matter, it's objective, whatever language one uses, then implicitly you're saying that it's invariant under some group. And I would rather say that, yes, it is true that we want to describe distances or position measurements or something like that, and then I'm going to start off by saying which invariance it's about to have, rather than discovering it. You said invariant under which group? And I would say, oh, well, it's not invariant under the objective group. But that doesn't make this state of the decision. It's not invalid under that fine rule. So you see, again, it's the equivalent of this thing of me specifying the rules. But haven't you implicitly specified it to yourself? And that's what allows you to say, when I point out that it's not a very kind of objective group state of the matter, that's not what I meant by objective. It seems to me that one has to make maybe what you are pointing out is that if you want knowledge to be possible of objective facts then you need something like environment but I mean that doesn't mean that the notion of fact of the matter.

1:42:30 I think that when someone says it's a fact of the matter, it's objective. That when pushed, they will find themselves having to say, what I mean by objective, or it's a fact of the matter, is a very uncertain transformation. But I don't think that that's a generic feature of objectivity, that you always mean something like that when you say something is a fact of the matter or it's objective. I have no theory to offer whatsoever about generically what one means. But I mean, if someone offers a theory which somehow releases a gap between objectivity and its bias... Then it wouldn't be conceptually inconsistent. It just means that it would be a theory which would be very difficult to know. Some of the quantities it postulates would be very difficult to know, maybe impossible. Are you saying that conceptually, or as a matter of fact, when we talk about position or momentum, we need them? That's what we mean, but we can talk about other notions of position or momentum. And then the problem would be that maybe we won't be able to know them if there is no connection to the environment. And I thought at the beginning he was saying that there is no meaning to the notion of fact of the matter if it's not related to or based on the notion of the environment. But that seems to be hardly too strong. Well, here's what I would endorse. I think two issues may be getting put together as one now. With respect to the issue that's been discussed in the last 10 minutes or so, I would say that if you're talking about position and momentum in a way that makes those notions susceptible to physical theory, and you claim that certain quantities involving position and momentum, maybe something as simple as the distance between two particles, maybe something more complicated, if you claim that

1:45:00 If you think that those quantities are physically objective, then my claim is that you have implicitly either do or ought to have in mind the idea that they're invariant under some group of transformations. That's one statement. It was my point of disagreement. You say under some transformation. It might challenge your research transformation. You say, oh, it's a transformation. I'm going to do this, you do that. How do you choose the Euclidean group? I mean, you're really not thinking about the natural path of the Euclidean group. I bet you say, in the back of my mind, I thought the path was the Euclidean group. But no, no, no, that's not quite right. What I mean is that, if I'm going to go out into the world and make... Sorry, if I'm going to then offer you... So suppose I say the distance between two particles is invariant. I just stopped right then, right? And I don't specify explicitly. So let's say it's a fact of the matter. Okay, it's a fact of the matter. Sorry. Forced to use something more. Yeah, it's a fact of the matter. I'm making the question. I'm making you the matter. That's a fact of the matter. It's a fact of the matter. I don't want to use that one again. I don't know why you're going to make the question. I don't want to give you that new question. Understood. Or the specific type of question. Understood. So if I'm going to say it's a fact of the matter, and then I'm going to... Use that claim to build a physical theory of some sort. My claim is that in doing so, you will have to make an assumption about how fact of the matter is implemented in terms of the variances. Now I realize you disagree with that, but that's my position, that you will have to make an assumption about it. You might do that explicitly, you might do it implicitly, but you have to make some sort of assumption about how fact of the matter is implemented in terms of the variances. Now, we could be wrong about that, right? I mean, you could proceed in an incorrect manner, and similarly, if I start with the idea that certain variances must hold, for example, well, I'm wrong about that, right? And I find this out by trying to construct a physical theory that respects those variances, and I mean, I think that we actually don't proceed in very different ways, but, I mean...

1:47:30 In terms of trying to discover which of the right invariances and covariances, but I just start off by saying, here, I think that, you know, this is what I mean by invariant, and I point to the Galilean group, and then I try to write down a physical theory that works, and I find out it doesn't work, because it predicts, you know, something wrong about, you know, the length of the moving rod or whatever, and so I then try to find some other group. That's my starting point. You two are exactly the same, as I can see, in the implementation. There's no difference, because all he's doing is, Michael's making, he's making a generalized claim of the spot that you'll find yourself. It's only a generalized claim. He's quite prepared to be wrong. ...about the groups under which the ignorance takes place. He's quite happy about that, but he can make a mistake in that case. I could be wrong about the facts of the matter. And I guess what I'm claiming is that they were wrong about that. Precisely. He will find out possibly later than you, you might be starting positions, and he will find out later, you could say. But it was only a generalized claim. But essentially, I mean, just a story of the law. I mean, I'm sure I'm not on that group. But when Einstein started, he put this whole thing off quite often. Deep conservation groups produce deep physics. Yeah. Invalid, unparalleled. But then later on, he said it himself that relativity theory should be called invariant theory. Remember this quick thing that Einstein said. It's a theory about the invariant. It's not a theory about the groups or the... Well, I agree with that. I think that's important to keep. I think it's important to, for other reasons, I think it's important to say that about relativity theory, just because people get the wrong impression of it. Everybody recognizes there's a close connection between symmetries and things like groups. Some people take symmetries to merely be heuristics, but others take them to be something more physically real, and groups are a representation of some piece of physical reality or some constraint that physical reality presents.

1:50:00 But I was just curious, when you say that you explicitly have to say something about some notion of invariance under a group, How do you relate that to symmetries? Is there something real about the symmetries and the groups are then just a representation of that, or that symmetries are kind of heuristic? I guess this sounds dangerously close to a question about realism versus anti-realism, about which I haven't... It's more triggered by von Prossen's discussion of these sorts of issues and wondering about you guys going back and forth about what seems to be sort of the anchor for each one of you. I guess I don't really understand, and I'm not being facetious here, I don't think I really understand the distinction between apparently distinct positions that you outlined about the nature of symmetries. I mean, I guess I would say if symmetry is part of a true physical theory, then there's some very important sense in which the world must respect that symmetry. Isn't that true? Right, and the group, let's take it within the, just the mathematical way we represent that in the theory, or implement the symmetry in the theory, or attempt to. We might get part of the symmetry wrong, for example. We could use the Galilean group, and we could use the Poincare group, which expresses... Maybe the following statement will clarify my position with respect to it. I would certainly agree to the following. Supposedly, try to write down a theory that's Galilean invariant, that is to say, the influence of the Galilean group and its invariants, the appropriate sorts of objects in the theory are invariant under the appropriate sorts of transformations from the Galilean group. And I look at the world and I find out, by golly, the world isn't like that. I speed up a rod. Then at that point I certainly would be willing to say that the world does not respect the symmetry that boosts invariance of position.

1:52:30 The world does not respect that symmetry. So I'd be inclined to try again. I don't know if that answers your question or not, because I'm not really clear quite on what... It's basically when you say something about the world of symmetry... It's very important. Yes, yes. And then, well, if we had more time, I would have also liked the other Michael's sense for how symmetry plays out in your story of saying, well, it's a fact of the matter that there's an objective distance between these two things. Well, I mean, to talk a bit briefly, I mean, since Einstein wrote this idea of general principles of characterizing, I mean, general principles, and that's what dominates, because since Einstein, that's kind of the dangerous part. I'm just raising a sort of possible point that I wish I may have been slightly mistaken about that. I think he was mistaken about something. I mean, another way of illustrating the importance of symmetries in my way of thinking about things is clearly, symmetries play a crucial role in defining what you mean by various physical quantities or positions. They play a crucial role in defining what you mean by... Well, that's how I defined them in the end, isn't it? I said the position operator is the thing that has the right symmetries with respect to, and it turns out that not only, in my view, must you say that, but that is all that you have to say in quantum mechanics. Thank you very much. Thank you all very much.

1:55:00 This is the mathematical... Thank you for your attention.

1:57:30 Thank you for watching. But we had the cake, so we had the cake before our talk.

2:22:30 And he's very happy Carter-Wayne is gone, but in his case quite literally still doing some writing. He may be doing some writing, he doesn't seem to be at all interested in giving seminars or anything. If he is doing anything, his line was that he just is not.

2:25:00 That was exactly what he said to us, and so I made an exception to this one, this is the only one I'm ever going to give, but I suspect you will, now that he's broken the ice, you might be able to persuade him. You could promise him that nobody else will invite him. Double-edged, a bit double-edged, I think, you know, John might do. I was just saying, the reason John came into the conversation... Thank you for your attention. Thank you for your attention. Yes, I'm sure it's exactly how you would, and I think that's wonderful. I mean, I don't have any pretense or subconsciousness at all. Anyway, it's been nice to see you again. I do think your back is up and down a bit, a bit better than it has been. I'll get on with you as soon as I've got the date that I'm moving, actually, that we've been in France, and then obviously the address. Well, no, you've got to have given it, because I've got emails, so I just have to weigh in on that. Yes, all right. Thank you very much.

2:27:30 If I give you a line, which is my line, then you can just terribly pop it, so it's got the email address. Okay? And then I can forward it to my phone. Yes, that's wonderful. Okay, and then you can connect to my phone. Has my phone already gone off? I think so. Oh, I wanted to. No, don't do that. I'm sorry. No, don't do that. I'm sorry. No, don't do that. I'm sorry. This is one of the... I mean, the sound is a bit of a... Have a safe journey back home. Take care. No, it's really kind of you. I'm fine. I actually have a couple of things around here.